Tue Aug 21 11:24:27 PDT 2012

Author: brad
Date: 2012-08-21 11:24:25 -0700 (Tue, 21 Aug 2012)
New Revision: 20616

Added:
   short/3D/PyLith/branches/v1.7-trunk/modulesrc/mpi/mpi_reduce.i
   short/3D/PyLith/branches/v1.7-trunk/unittests/pytests/mpi/TestReduce.py
Modified:
   short/3D/PyLith/branches/v1.7-trunk/doc/userguide/benchmarks/benchmarks.lyx
   short/3D/PyLith/branches/v1.7-trunk/doc/userguide/benchmarks/savageprescott/savageprescott.lyx
   short/3D/PyLith/branches/v1.7-trunk/doc/userguide/benchmarks/strikeslip/strikeslip.lyx
   short/3D/PyLith/branches/v1.7-trunk/doc/userguide/boundaryconditions/boundaryconditions.lyx
   short/3D/PyLith/branches/v1.7-trunk/doc/userguide/components.lyx
   short/3D/PyLith/branches/v1.7-trunk/doc/userguide/extending/extending.lyx
   short/3D/PyLith/branches/v1.7-trunk/doc/userguide/fileformats/fileformats.lyx
   short/3D/PyLith/branches/v1.7-trunk/doc/userguide/governingeqns/governingeqns.lyx
   short/3D/PyLith/branches/v1.7-trunk/doc/userguide/intro/intro.lyx
   short/3D/PyLith/branches/v1.7-trunk/doc/userguide/materials/altformulations.lyx
   short/3D/PyLith/branches/v1.7-trunk/doc/userguide/materials/materials.lyx
   short/3D/PyLith/branches/v1.7-trunk/doc/userguide/preface.lyx
   short/3D/PyLith/branches/v1.7-trunk/doc/userguide/runpylith/runpylith.lyx
   short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/3dhex8/3dhex8.lyx
   short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/3dhex8/dike/dike.lyx
   short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/3dhex8/friction/friction.lyx
   short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/3dhex8/gravity/gravity.lyx
   short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/3dhex8/quasistatic/quasistatic.lyx
   short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/3dhex8/static/static.lyx
   short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/3dhex8/surfload/surfload.lyx
   short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/3dtet4/3dtet4.lyx
   short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/greensfns2d/greensfns2d.lyx
   short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/shearwave/hex8.lyx
   short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/shearwave/quad4.lyx
   short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/subduction/subduction.lyx
   short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/twohex8/twohex8.lyx
   short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/twoquad4/twoquad4.lyx
   short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/twotet4-geoproj/twotet4-geoproj.lyx
   short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/twotet4/twotet4.lyx
   short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/twotri3/twotri3.lyx
   short/3D/PyLith/branches/v1.7-trunk/doc/userguide/userguide.lyx
   short/3D/PyLith/branches/v1.7-trunk/examples/3d/hex8/step14.cfg
   short/3D/PyLith/branches/v1.7-trunk/libsrc/pylith/feassemble/ElasticityExplicit.cc
   short/3D/PyLith/branches/v1.7-trunk/libsrc/pylith/feassemble/ElasticityExplicitLgDeform.cc
   short/3D/PyLith/branches/v1.7-trunk/libsrc/pylith/friction/RateStateAgeing.cc
   short/3D/PyLith/branches/v1.7-trunk/libsrc/pylith/friction/RateStateAgeing.hh
   short/3D/PyLith/branches/v1.7-trunk/modulesrc/friction/RateStateAgeing.i
   short/3D/PyLith/branches/v1.7-trunk/modulesrc/mpi/Makefile.am
   short/3D/PyLith/branches/v1.7-trunk/modulesrc/mpi/mpi.i
   short/3D/PyLith/branches/v1.7-trunk/pylith/apps/PyLithApp.py
   short/3D/PyLith/branches/v1.7-trunk/pylith/friction/RateStateAgeing.py
   short/3D/PyLith/branches/v1.7-trunk/pylith/problems/TimeStep.py
   short/3D/PyLith/branches/v1.7-trunk/tests/2d/frictionslide/ratestate_stable.cfg
   short/3D/PyLith/branches/v1.7-trunk/tests/2d/frictionslide/ratestate_weak.cfg
   short/3D/PyLith/branches/v1.7-trunk/unittests/libtests/friction/TestRateStateAgeing.cc
   short/3D/PyLith/branches/v1.7-trunk/unittests/libtests/friction/TestRateStateAgeing.hh
   short/3D/PyLith/branches/v1.7-trunk/unittests/pytests/mpi/Makefile.am
   short/3D/PyLith/branches/v1.7-trunk/unittests/pytests/mpi/testmpi.py
   short/3D/PyLith/branches/v1.7-trunk/unittests/pytests/problems/TestTimeStepAdapt.py
   short/3D/PyLith/branches/v1.7-trunk/unittests/pytests/problems/TestTimeStepUniform.py
   short/3D/PyLith/branches/v1.7-trunk/unittests/pytests/problems/TestTimeStepUser.py
Log:
Merge from stable.

Modified: short/3D/PyLith/branches/v1.7-trunk/doc/userguide/benchmarks/benchmarks.lyx
===================================================================

--- short/3D/PyLith/branches/v1.7-trunk/doc/userguide/benchmarks/benchmarks.lyx	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/doc/userguide/benchmarks/benchmarks.lyx	2012-08-21 18:24:25 UTC (rev 20616)
@@ -1,197 +1,197 @@
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-\textclass book
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-\html_be_strict false
-\end_header
-
-\begin_body
-
-\begin_layout Chapter
-\begin_inset CommandInset label
-LatexCommand label
-name "cha:Benchmarks"
-
-\end_inset
-
-Benchmarks
-\end_layout
-
-\begin_layout Section
-Overview
-\end_layout
-
-\begin_layout Standard
-The Crustal Deformation Modeling and Earthquake Source Physics Focus Groups
- within the Southern California Earthquake Center and the Short-Term Tectonics
- Working Group within CIG have developed a suite of benchmarks to test the
- accuracy and performance of 3D numerical codes for quasi-static crustal
- deformation and earthquake rupture dynamics.
- The benchmark definitions for the quasi-static crustal deformation benchmarks
- are posted on the CIG website at Short-Term Tectonics Benchmarks 
-\begin_inset Flex URL
-status open
-
-\begin_layout Plain Layout
-
-geodynamics.org/cig/workinggroups/short/workarea/benchmarks/
-\end_layout
-
-\end_inset
-
- and the definitions for the earthquake rupture benchmarks are posted on
- the SCEC website 
-\begin_inset Flex URL
-status open
-
-\begin_layout Plain Layout
-
-scecdata.usc.edu/cvws/cgi-bin/cvws.cgi
-\end_layout
-
-\end_inset
-
-.
- This suite of benchmarks permits evaluating the relative performance of
- different types of basis functions, quadrature schemes, and discretizations
- for geophysical applications.
- The files needed to run the 3D benchmarks are in the CIG SVN Repository
- 
-\begin_inset Flex URL
-status open
-
-\begin_layout Plain Layout
-
-geodynamics.org/svn/cig/short/3D/PyLith/benchmarks
-\end_layout
-
-\end_inset
-
-.
- In addition to evaluating the efficiency and accuracy of numerical codes,
- the benchmarks also make good test problems, where users can perform simulation
-s based on actual geophysical problems.
- The benchmarks are performed at various resolutions and using different
- element types.
- By comparing the runtime and accuracy for different resolutions and element
- types, users can evaluate which combination will be best for their problems
- of interest.
-\end_layout
-
-\begin_layout Standard
-\begin_inset CommandInset include
-LatexCommand input
-filename "strikeslip/strikeslip.lyx"
-
-\end_inset
-
-
-\begin_inset CommandInset include
-LatexCommand input
-filename "savageprescott/savageprescott.lyx"
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-SCEC Dynamic Rupture Benchmarks
-\end_layout
-
-\begin_layout Standard
-The SCEC website 
-\begin_inset Flex URL
-status open
-
-\begin_layout Plain Layout
-
-scecdata.usc.edu/cvws/cgi-bin/cvws.cgi
-\end_layout
-
-\end_inset
-
- includes a graphical user interface for examining the benchmark results.
- Benchmark results for PyLith are available for TPV205-2D (horizontal slice
- through a vertical strike-slip fault), TPV205 (vertical strike-slip fault
- with high and low stress asperities), TPV210-2D (vertical slice through
- a 60 degree dipping normal fault), TPV210 (60 degree dipping normal fault),
- TPV11, TPV12, TPV13, TPV14-2D and TPV15-2D (horizontal slice through a
- verticel strike-slip fault with a branch), TPV14 and TPV15 (vertical strike-sli
-p fault with a branch), and TPV 16 and 17.
-\end_layout
-
-\begin_layout Standard
-The benchmark results indicate that triangular and tetrahedral cells generate
- less numerical noise than quadrilateral or hexahedral cells.
- Results for all of the benchmarks listed above except TPV14-2D, TPV15-2D,
- TPV14, and TPV15 have been updated using PyLith 1.7, which provides much
- closer agreement with the other codes compared to previous versions of
- PyLith.
- We attribute the closer agreement to small adjustments of the fault implementat
-ion and some small bugfixes to the slip-weakening friction formulation.
- 
-\end_layout
-
-\end_body
-\end_document
+#LyX 2.0 created this file. For more info see http://www.lyx.org/
+\lyxformat 413
+\begin_document
+\begin_header
+\textclass book
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+
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+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Chapter
+\begin_inset CommandInset label
+LatexCommand label
+name "cha:Benchmarks"
+
+\end_inset
+
+Benchmarks
+\end_layout
+
+\begin_layout Section
+Overview
+\end_layout
+
+\begin_layout Standard
+The Crustal Deformation Modeling and Earthquake Source Physics Focus Groups
+ within the Southern California Earthquake Center and the Short-Term Tectonics
+ Working Group within CIG have developed a suite of benchmarks to test the
+ accuracy and performance of 3D numerical codes for quasi-static crustal
+ deformation and earthquake rupture dynamics.
+ The benchmark definitions for the quasi-static crustal deformation benchmarks
+ are posted on the CIG website at Short-Term Tectonics Benchmarks 
+\begin_inset Flex URL
+status open
+
+\begin_layout Plain Layout
+
+geodynamics.org/cig/workinggroups/short/workarea/benchmarks/
+\end_layout
+
+\end_inset
+
+ and the definitions for the earthquake rupture benchmarks are posted on
+ the SCEC website 
+\begin_inset Flex URL
+status open
+
+\begin_layout Plain Layout
+
+scecdata.usc.edu/cvws/cgi-bin/cvws.cgi
+\end_layout
+
+\end_inset
+
+.
+ This suite of benchmarks permits evaluating the relative performance of
+ different types of basis functions, quadrature schemes, and discretizations
+ for geophysical applications.
+ The files needed to run the 3D benchmarks are in the CIG SVN Repository
+ 
+\begin_inset Flex URL
+status open
+
+\begin_layout Plain Layout
+
+geodynamics.org/svn/cig/short/3D/PyLith/benchmarks
+\end_layout
+
+\end_inset
+
+.
+ In addition to evaluating the efficiency and accuracy of numerical codes,
+ the benchmarks also make good test problems, where users can perform simulation
+s based on actual geophysical problems.
+ The benchmarks are performed at various resolutions and using different
+ element types.
+ By comparing the runtime and accuracy for different resolutions and element
+ types, users can evaluate which combination will be best for their problems
+ of interest.
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "strikeslip/strikeslip.lyx"
+
+\end_inset
+
+
+\begin_inset CommandInset include
+LatexCommand input
+filename "savageprescott/savageprescott.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+SCEC Dynamic Rupture Benchmarks
+\end_layout
+
+\begin_layout Standard
+The SCEC website 
+\begin_inset Flex URL
+status open
+
+\begin_layout Plain Layout
+
+scecdata.usc.edu/cvws/cgi-bin/cvws.cgi
+\end_layout
+
+\end_inset
+
+ includes a graphical user interface for examining the benchmark results.
+ Benchmark results for PyLith are available for TPV205-2D (horizontal slice
+ through a vertical strike-slip fault), TPV205 (vertical strike-slip fault
+ with high and low stress asperities), TPV210-2D (vertical slice through
+ a 60-degree dipping normal fault), TPV210 (60-degree dipping normal fault),
+ TPV11, TPV12, TPV13, TPV14-2D and TPV15-2D (horizontal slice through a
+ verticel strike-slip fault with a branch), TPV14 and TPV15 (vertical strike-sli
+p fault with a branch), and TPV 16 and 17.
+\end_layout
+
+\begin_layout Standard
+The benchmark results indicate that triangular and tetrahedral cells generate
+ less numerical noise than quadrilateral or hexahedral cells.
+ Results for all of the benchmarks listed above except TPV14-2D, TPV15-2D,
+ TPV14, and TPV15 have been updated using PyLith 1.7, which provides much
+ closer agreement with the other codes compared to previous versions of
+ PyLith.
+ We attribute the closer agreement to small adjustments of the fault implementat
+ion and some small bugfixes to the slip-weakening friction formulation.
+ 
+\end_layout
+
+\end_body
+\end_document

Modified: short/3D/PyLith/branches/v1.7-trunk/doc/userguide/benchmarks/savageprescott/savageprescott.lyx
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/doc/userguide/benchmarks/savageprescott/savageprescott.lyx	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/doc/userguide/benchmarks/savageprescott/savageprescott.lyx	2012-08-21 18:24:25 UTC (rev 20616)
@@ -1,523 +1,530 @@
-#LyX 2.0 created this file. For more info see http://www.lyx.org/
-\lyxformat 413
-\begin_document
-\begin_header
-\textclass book
-\begin_preamble
-
-\end_preamble
-\use_default_options false
-\maintain_unincluded_children false
-\language english
-\language_package default
-\inputencoding latin1
-\fontencoding global
-\font_roman default
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-\end_header
-
-\begin_body
-
-\begin_layout Section
-\begin_inset CommandInset label
-LatexCommand label
-name "sec:benchmarks:savageprescott"
-
-\end_inset
-
-Savage and Prescott Benchmark
-\end_layout
-
-\begin_layout Standard
-This benchmark problem computes the viscoelastic (Maxwell) relaxation of
- stresses from repeated infinite, strike-slip earthquakes in 3D without
- gravity.
- The files needed to run the benchmark may be found at 
-\begin_inset Flex URL
-status collapsed
-
-\begin_layout Plain Layout
-
-geodynamics.org/svn/cig/short/2.5D/benchmarks/savageprescott
-\end_layout
-
-\end_inset
-
-.
- An analytical solution to this problem is described by Savage and Prescott
- 
-\begin_inset CommandInset citation
-LatexCommand cite
-key "Savage:Prescott:1978"
-
-\end_inset
-
-, which provides a simple way to check our numerical solution.
- A python utility code is provided in the utils directory to compute the
- analytical solution.
- Although this problem is actually 2.5D (infinite along-strike), we solve
- it using a 3D finite element model.
-\end_layout
-
-\begin_layout Subsection
-Problem Description
-\end_layout
-
-\begin_layout Standard
-Figure 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "fig:benchmark:savageprescott:geometry"
-
-\end_inset
-
- shows the geometry of the problem, as described by 
-\begin_inset CommandInset citation
-LatexCommand cite
-key "Savage:Prescott:1978"
-
-\end_inset
-
-.
- The analytical solution describes the surface deformation due to repeated
- earthquakes on an infinite strike-slip fault embedded in an elastic layer
- overlying a Maxwell viscoelastic half-space.
- The upper portion of the fault (red in the figure) is locked between earthquake
-s, while the lower portion (blue in the figure) creeps at plate velocity.
- At regular recurrence intervals, the upper portion of the fault abruptly
- slips by an amount equal to the plate velocity multiplied by the recurrence
- interval, thus 'catching up' with the lower part of the fault.
-\end_layout
-
-\begin_layout Standard
-There are some differences between the analytical solution and our numerical
- representation.
- First, the analytical solution represents the earthquake cycle as the superposi
-tion of uniform fault creep and an elementary earthquake cycle.
- Uniform fault creep is simply the uniform movement of the two plates past
- each other at plate velocity.
- For the elementary earthquake cycle, no slip occurs below the locked portion
- of the fault (blue portion in the figure).
- On the locked (red) portion of the fault, backslip equal to plate velocity
- occurs until the earthquake recurrence interval, at which point abrupt
- forward slip occurs.
- In the finite element solution, we perform the simulation as described
- in the figure.
- Velocity boundary conditions are applied at the extreme edges of the model
- to simulate block motion, steady creep is applied along the blue portion
- of the fault, and regular earthquakes are applied along the upper portion
- of the fault.
- It takes several earthquake cycles for the velocity boundary conditions
- to approximate the steady flow due to steady block motion, so we would
- not expect the analytical and numerical solutions to match until several
- earthquakes have occurred.
- Another difference lies in the dimensions of the domain.
- The analytical solution assumes an infinite strike-slip fault in an elastic
- layer overlying a Maxwell viscoelastic half-space.
- In our finite element model we are restricted to finite dimensions.
- We therefore extend the outer boundaries far enough from the region of
- interest to approximate boundaries at infinity.
-\end_layout
-
-\begin_layout Standard
-Due to the difficulties in representing solutions in an infinite domain,
- there are several meshes that have been tested for this problem.
- The simplest meshes have uniform resolution (all cells have equal dimensions);
- however, such meshes typically do not provide accurate solutions since
- the resolution is too coarse in the region of interest.
- For that reason, we also tested meshes where the mesh resolution decreases
- away from the center.
- In the problem description that follows, we will focus on the hexahedral
- mesh with finer discretization neear the fault (meshes/hex8_6.7km.exo.gz),
- which provides a good match with the analytical solution.
- It will first be necessary to gunzip this mesh so that it may be used by
- PyLith.
-\end_layout
-
-\begin_layout Description
-Domain The domain for this mesh spans the region
-\begin_inset Formula 
-\begin{gather*}
--1000\leq x\leq1000\ km,\\
--500\leq y\leq500\ km,\\
--400\ km\leq z\leq0.
-\end{gather*}
-
-\end_inset
-
-The top (elastic) layer occupies the region 
-\begin_inset Formula $-40\ km\ \leq z\leq0$
-\end_inset
-
- and the bottom (viscoelastic) layer occupies the region 
-\begin_inset Formula $-400\ km\ \leq z\leq-40\ km$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Description
-Material
-\begin_inset space ~
-\end_inset
-
-properties The material is a Poisson solid with a shear modulus (
-\begin_inset Formula $\mu$
-\end_inset
-
-) of 30 GPa.
- The domain is modeled using an elastic isotropic material for the top layer
- and a Maxwell viscoelastic material for the bottom layer.
- The bottom layer has a viscosity (
-\begin_inset Formula $\eta$
-\end_inset
-
-) of 2.36682e+19 Pa-s, yielding a relaxation time (
-\begin_inset Formula $2\eta/\mu$
-\end_inset
-
-) of 50 years.
-\end_layout
-
-\begin_layout Description
-Fault The fault is a vertical, left-lateral strike-slip fault.
- The strike is parallel to the y-direction at the center of the model:
-\begin_inset Formula 
-\begin{gather*}
-x=0\ km,\\
--500\leq y\leq500\ km,\\
--40\ km\leq z\leq0.
-\end{gather*}
-
-\end_inset
-
-The locked portion of the fault (red section in Figure 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "fig:benchmark:savageprescott:geometry"
-
-\end_inset
-
-) extends from 
-\begin_inset Formula $-20\: km\leq z\leq0$
-\end_inset
-
-, while the creeping section (blue) extends from 
-\begin_inset Formula $-40\: km\leq z\leq0$
-\end_inset
-
-.
- Along the line where the two sections coincide (
-\begin_inset Formula $z=-20\: km$
-\end_inset
-
-), half of the coseismic displacement and half of the steady creep is applied
- (see 
-\family typewriter
-finalslip.spatialdb
-\family default
- and 
-\family typewriter
-creeprate.spatialdb
-\family default
-).
-\end_layout
-
-\begin_layout Description
-Boundary
-\begin_inset space ~
-\end_inset
-
-conditions On the bottom boundary, vertical displacements are set to zero,
- while on the y-boundaries the x-displacements are set to zero.
- On the x-boundaries, the x-displacements are set to zero, while constant
- velocities of +/- 1 cm/yr are applied in the y-direction, giving a relative
- plate motion of 2 cm/year.
-\end_layout
-
-\begin_layout Description
-Discretization For the nonuniform hexahedral mesh, the resolution at the
- outer boundaries is 20 km.
- An inner region is then put through one level of refinement, so that near
- the center of the mesh the resolution is 6.7 km.
- All meshes were generated with CUBIT.
-\end_layout
-
-\begin_layout Description
-Basis
-\begin_inset space ~
-\end_inset
-
-functions We use trilinear hexahedral cells.
-\end_layout
-
-\begin_layout Description
-Solution We compute the surface displacements and compare these to the analytica
-l solution 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "fig:benchmark:savageprescott:solution"
-
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float figure
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\align center
-\begin_inset Graphics
-	filename figs/model_descript.eps
-	scale 33
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Plain Layout
-\begin_inset Caption
-
-\begin_layout Plain Layout
-Problem description for the Savage and Prescott strike-slip benchmark problem.
-\begin_inset CommandInset label
-LatexCommand label
-name "fig:benchmark:savageprescott:geometry"
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsection
-Running the Benchmark
-\end_layout
-
-\begin_layout Standard
-After checking out the benchmark files from the CIG SVN repository, change
- to the 
-\family typewriter
-meshes
-\family default
- directory.
- Decompress the gzipped files in the 
-\family typewriter
-mesh
-\family default
- directory,
-\end_layout
-
-\begin_layout LyX-Code
-gunzip *.gz
-\end_layout
-
-\begin_layout Standard
-Alternatively, simply gunzip the mesh you want to use.
- There are a number of 
-\family typewriter
-.cfg
-\family default
- files corresponding to the different meshes, as well as a 
-\family typewriter
-pylithapp.cfg
-\family default
- file defining parameters common to all problems.
- Each problem uses four 
-\family typewriter
-.cfg
-\family default
- files: 
-\family typewriter
-pylithapp.cfg
-\family default
-, 
-\family typewriter
-fieldsplit.cfg
-\family default
- (algrebraic multigrid preconditioner), a cell-specific file (e.g., 
-\family typewriter
-hex8.cfg
-\family default
-), and a resolution specific file (e.g., hex8_6.7km.cfg).
- You can then run the problem by typing
-\end_layout
-
-\begin_layout LyX-Code
-pylith hex8.cfg hex8_6.7km.cfg fieldsplit.cfg
-\end_layout
-
-\begin_layout Standard
-This will run the problem for 10 earthquake cycles of 200 years each, using
- a time-step size of 10 years, for a total simulation time of 2000 years.
- Ground surface output occurs every 10 years, while all other outputs occur
- every 50 years.
-\end_layout
-
-\begin_layout Standard
-Once the problem has run, results will be placed in the 
-\family typewriter
-output
-\family default
- directory.
- These results may be viewed directly with a package such as ParaView; however,
- to compare results to the analytical solution, some postprocessing is required.
- First, generate the analytical results by running the 
-\family typewriter
-calc_analytic.py
-\family default
- script.
- This will produce files with displacements and velocities (
-\family typewriter
-analytic_disp.txt
-\family default
- and 
-\family typewriter
-analytic_vel.txt
-\family default
-) in the 
-\family typewriter
-output
-\family default
- directory that are easy to use with a plotting package, such as matplotlib
- or Matlab.
-\end_layout
-
-\begin_layout Subsection
-Benchmark Results
-\end_layout
-
-\begin_layout Standard
-Figure 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "fig:benchmark:savageprescott:solution"
-
-\end_inset
-
- shows the computed surface displacements for the 10th earthquake cycle
- compared with the analytical solution.
- The profile results were obtained as described above, and then all results
- (analytical and numerical) were referenced to the displacements immediately
- following the last earthquake.
- We find very good agreement between the analytical and numerical solutions,
- even for meshes with uniform refinement.
- We have not yet explored quantitative fits as a function of mesh resolution.
- For this benchmark, it is also important to consider the distance of the
- boundary from the region of interest.
- Also note that the agreement between analytical and numerical solutions
- is poor for early earthquake cycles, due to the differences in simulating
- the problem, as noted above.
-\end_layout
-
-\begin_layout Standard
-\begin_inset Float figure
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\align center
-\begin_inset Graphics
-	filename figs/soln_profiles.eps
-	scale 66
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Plain Layout
-\begin_inset Caption
-
-\begin_layout Plain Layout
-Displacement profiles perpendicular to the fault for a PyLith simulation
- with hex8 cells and the analytical solution for earthquake cycle 10.
- 
-\begin_inset CommandInset label
-LatexCommand label
-name "fig:benchmark:savageprescott:solution"
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_body
-\end_document
+#LyX 2.0 created this file. For more info see http://www.lyx.org/
+\lyxformat 413
+\begin_document
+\begin_header
+\textclass book
+\begin_preamble
+
+\end_preamble
+\use_default_options false
+\maintain_unincluded_children false
+\language english
+\language_package default
+\inputencoding latin1
+\fontencoding global
+\font_roman default
+\font_sans default
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+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100
+\font_tt_scale 100
+
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
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+\spacing single
+\use_hyperref false
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+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\use_refstyle 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\leftmargin 1in
+\topmargin 1in
+\rightmargin 1in
+\bottommargin 1in
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\quotes_language english
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Section
+\begin_inset CommandInset label
+LatexCommand label
+name "sec:benchmarks:savageprescott"
+
+\end_inset
+
+Savage and Prescott Benchmark
+\end_layout
+
+\begin_layout Standard
+This benchmark problem computes the viscoelastic (Maxwell) relaxation of
+ stresses from repeated infinite, strike-slip earthquakes in 3D without
+ gravity.
+ The files needed to run the benchmark may be found at 
+\begin_inset Flex URL
+status collapsed
+
+\begin_layout Plain Layout
+
+geodynamics.org/svn/cig/short/2.5D/benchmarks/savageprescott
+\end_layout
+
+\end_inset
+
+.
+ An analytical solution to this problem is described by Savage and Prescott
+ 
+\begin_inset CommandInset citation
+LatexCommand cite
+key "Savage:Prescott:1978"
+
+\end_inset
+
+, which provides a simple way to check our numerical solution.
+ A python utility code is provided in the utils directory to compute the
+ analytical solution.
+ Although this problem is actually 2.5D (infinite along-strike), we solve
+ it using a 3D finite element model.
+\end_layout
+
+\begin_layout Subsection
+Problem Description
+\end_layout
+
+\begin_layout Standard
+Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:benchmark:savageprescott:geometry"
+
+\end_inset
+
+ shows the geometry of the problem, as described by 
+\begin_inset CommandInset citation
+LatexCommand cite
+key "Savage:Prescott:1978"
+
+\end_inset
+
+.
+ The analytical solution describes the surface deformation due to repeated
+ earthquakes on an infinite strike-slip fault embedded in an elastic layer
+ overlying a Maxwell viscoelastic half-space.
+ The upper portion of the fault (red in the figure) is locked between earthquake
+s, while the lower portion (blue in the figure) creeps at plate velocity.
+ At regular recurrence intervals, the upper portion of the fault abruptly
+ slips by an amount equal to the plate velocity multiplied by the recurrence
+ interval, thus 'catching up' with the lower part of the fault.
+\end_layout
+
+\begin_layout Standard
+There are some differences between the analytical solution and our numerical
+ representation.
+ First, the analytical solution represents the earthquake cycle as the superposi
+tion of uniform fault creep and an elementary earthquake cycle.
+ Uniform fault creep is simply the uniform movement of the two plates past
+ each other at plate velocity.
+ For the elementary earthquake cycle, no slip occurs below the locked portion
+ of the fault (blue portion in the figure).
+ On the locked (red) portion of the fault, backslip equal to plate velocity
+ occurs until the earthquake recurrence interval, at which point abrupt
+ forward slip occurs.
+ In the finite element solution, we perform the simulation as described
+ in the figure.
+ Velocity boundary conditions are applied at the extreme edges of the model
+ to simulate block motion, steady creep is applied along the blue portion
+ of the fault, and regular earthquakes are applied along the upper portion
+ of the fault.
+ It takes several earthquake cycles for the velocity boundary conditions
+ to approximate the steady flow due to steady block motion, so we would
+ not expect the analytical and numerical solutions to match until several
+ earthquakes have occurred.
+ Another difference lies in the dimensions of the domain.
+ The analytical solution assumes an infinite strike-slip fault in an elastic
+ layer overlying a Maxwell viscoelastic half-space.
+ In our finite element model we are restricted to finite dimensions.
+ We therefore extend the outer boundaries far enough from the region of
+ interest to approximate boundaries at infinity.
+\end_layout
+
+\begin_layout Standard
+Due to the difficulties in representing solutions in an infinite domain,
+ there are several meshes that have been tested for this problem.
+ The simplest meshes have uniform resolution (all cells have equal dimensions);
+ however, such meshes typically do not provide accurate solutions since
+ the resolution is too coarse in the region of interest.
+ For that reason, we also tested meshes where the mesh resolution decreases
+ away from the center.
+ In the problem description that follows, we will focus on the hexahedral
+ mesh with finer discretization near the fault 
+\begin_inset Newline linebreak
+\end_inset
+
+(
+\family typewriter
+meshes/hex8_6.7km.exo.gz
+\family default
+), which provides a good match with the analytical solution.
+ It will first be necessary to gunzip this mesh so that it may be used by
+ PyLith.
+\end_layout
+
+\begin_layout Description
+Domain The domain for this mesh spans the region
+\begin_inset Formula 
+\begin{gather*}
+-1000\leq x\leq1000\ km,\\
+-500\leq y\leq500\ km,\\
+-400\ km\leq z\leq0.
+\end{gather*}
+
+\end_inset
+
+The top (elastic) layer occupies the region 
+\begin_inset Formula $-40\ km\ \leq z\leq0$
+\end_inset
+
+ and the bottom (viscoelastic) layer occupies the region 
+\begin_inset Formula $-400\ km\ \leq z\leq-40\ km$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Description
+Material
+\begin_inset space ~
+\end_inset
+
+properties The material is a Poisson solid with a shear modulus (
+\begin_inset Formula $\mu$
+\end_inset
+
+) of 30 GPa.
+ The domain is modeled using an elastic isotropic material for the top layer
+ and a Maxwell viscoelastic material for the bottom layer.
+ The bottom layer has a viscosity (
+\begin_inset Formula $\eta$
+\end_inset
+
+) of 2.36682e+19 Pa-s, yielding a relaxation time (
+\begin_inset Formula $2\eta/\mu$
+\end_inset
+
+) of 50 years.
+\end_layout
+
+\begin_layout Description
+Fault The fault is a vertical, left-lateral strike-slip fault.
+ The strike is parallel to the y-direction at the center of the model:
+\begin_inset Formula 
+\begin{gather*}
+x=0\ km,\\
+-500\leq y\leq500\ km,\\
+-40\ km\leq z\leq0.
+\end{gather*}
+
+\end_inset
+
+The locked portion of the fault (red section in Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:benchmark:savageprescott:geometry"
+
+\end_inset
+
+) extends from 
+\begin_inset Formula $-20\: km\leq z\leq0$
+\end_inset
+
+, while the creeping section (blue) extends from 
+\begin_inset Formula $-40\: km\leq z\leq0$
+\end_inset
+
+.
+ Along the line where the two sections coincide (
+\begin_inset Formula $z=-20\: km$
+\end_inset
+
+), half of the coseismic displacement and half of the steady creep is applied
+ (see 
+\family typewriter
+finalslip.spatialdb
+\family default
+ and 
+\family typewriter
+creeprate.spatialdb
+\family default
+).
+\end_layout
+
+\begin_layout Description
+Boundary
+\begin_inset space ~
+\end_inset
+
+conditions On the bottom boundary, vertical displacements are set to zero,
+ while on the y-boundaries the x-displacements are set to zero.
+ On the x-boundaries, the x-displacements are set to zero, while constant
+ velocities of +/- 1 cm/yr are applied in the y-direction, giving a relative
+ plate motion of 2 cm/year.
+\end_layout
+
+\begin_layout Description
+Discretization For the nonuniform hexahedral mesh, the resolution at the
+ outer boundaries is 20 km.
+ An inner region is then put through one level of refinement, so that near
+ the center of the mesh the resolution is 6.7 km.
+ All meshes were generated with CUBIT.
+\end_layout
+
+\begin_layout Description
+Basis
+\begin_inset space ~
+\end_inset
+
+functions We use trilinear hexahedral cells.
+\end_layout
+
+\begin_layout Description
+Solution We compute the surface displacements and compare these to the analytica
+l solution in Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:benchmark:savageprescott:solution"
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float figure
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\align center
+\begin_inset Graphics
+	filename figs/model_descript.eps
+	scale 33
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Problem description for the Savage and Prescott strike-slip benchmark problem.
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:benchmark:savageprescott:geometry"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Running the Benchmark
+\end_layout
+
+\begin_layout Standard
+After checking out the benchmark files from the CIG SVN repository, change
+ to the 
+\family typewriter
+meshes
+\family default
+ directory.
+ Decompress the gzipped files in the 
+\family typewriter
+mesh
+\family default
+ directory,
+\end_layout
+
+\begin_layout LyX-Code
+gunzip *.gz
+\end_layout
+
+\begin_layout Standard
+Alternatively, simply gunzip the mesh you want to use.
+ There are a number of 
+\family typewriter
+.cfg
+\family default
+ files corresponding to the different meshes, as well as a 
+\family typewriter
+pylithapp.cfg
+\family default
+ file defining parameters common to all problems.
+ Each problem uses four 
+\family typewriter
+.cfg
+\family default
+ files: 
+\family typewriter
+pylithapp.cfg
+\family default
+, 
+\family typewriter
+fieldsplit.cfg
+\family default
+ (algrebraic multigrid preconditioner), a cell-specific file (e.g., 
+\family typewriter
+hex8.cfg
+\family default
+), and a resolution specific file (e.g., hex8_6.7km.cfg).
+ You can then run the problem by typing
+\end_layout
+
+\begin_layout LyX-Code
+pylith hex8.cfg hex8_6.7km.cfg fieldsplit.cfg
+\end_layout
+
+\begin_layout Standard
+This will run the problem for 10 earthquake cycles of 200 years each, using
+ a time-step size of 10 years, for a total simulation time of 2000 years.
+ Ground surface output occurs every 10 years, while all other outputs occur
+ every 50 years.
+\end_layout
+
+\begin_layout Standard
+Once the problem has run, results will be placed in the 
+\family typewriter
+output
+\family default
+ directory.
+ These results may be viewed directly with a package such as ParaView; however,
+ to compare results to the analytical solution, some postprocessing is required.
+ First, generate the analytical results by running the 
+\family typewriter
+calc_analytic.py
+\family default
+ script.
+ This will produce files with displacements and velocities (
+\family typewriter
+analytic_disp.txt
+\family default
+ and 
+\family typewriter
+analytic_vel.txt
+\family default
+) in the 
+\family typewriter
+output
+\family default
+ directory that are easy to use with a plotting package, such as matplotlib
+ or Matlab.
+\end_layout
+
+\begin_layout Subsection
+Benchmark Results
+\end_layout
+
+\begin_layout Standard
+Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:benchmark:savageprescott:solution"
+
+\end_inset
+
+ shows the computed surface displacements for the 10th earthquake cycle
+ compared with the analytical solution.
+ The profile results were obtained as described above, and then all results
+ (analytical and numerical) were referenced to the displacements immediately
+ following the last earthquake.
+ We find very good agreement between the analytical and numerical solutions,
+ even for meshes with uniform refinement.
+ We have not yet explored quantitative fits as a function of mesh resolution.
+ For this benchmark, it is also important to consider the distance of the
+ boundary from the region of interest.
+ Also note that the agreement between analytical and numerical solutions
+ is poor for early earthquake cycles, due to the differences in simulating
+ the problem, as noted above.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\align center
+\begin_inset Graphics
+	filename figs/soln_profiles.eps
+	scale 66
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Displacement profiles perpendicular to the fault for a PyLith simulation
+ with hex8 cells and the analytical solution for earthquake cycle 10.
+ 
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:benchmark:savageprescott:solution"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document

Modified: short/3D/PyLith/branches/v1.7-trunk/doc/userguide/benchmarks/strikeslip/strikeslip.lyx
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/doc/userguide/benchmarks/strikeslip/strikeslip.lyx	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/doc/userguide/benchmarks/strikeslip/strikeslip.lyx	2012-08-21 18:24:25 UTC (rev 20616)
@@ -1,918 +1,982 @@
-#LyX 1.5.6 created this file. For more info see http://www.lyx.org/
-\lyxformat 276
-\begin_document
-\begin_header
-\textclass book
-\begin_preamble
-
-\end_preamble
-\language english
-\inputencoding latin1
-\font_roman default
-\font_sans default
-\font_typewriter default
-\font_default_family default
-\font_sc false
-\font_osf false
-\font_sf_scale 100
-\font_tt_scale 100
-\graphics default
-\paperfontsize default
-\spacing single
-\papersize default
-\use_geometry true
-\use_amsmath 0
-\use_esint 0
-\cite_engine basic
-\use_bibtopic false
-\paperorientation portrait
-\leftmargin 1in
-\topmargin 1in
-\rightmargin 1in
-\bottommargin 1in
-\secnumdepth 3
-\tocdepth 3
-\paragraph_separation indent
-\defskip medskip
-\quotes_language english
-\papercolumns 1
-\papersides 1
-\paperpagestyle default
-\tracking_changes false
-\output_changes false
-\author "" 
-\author "" 
-\end_header
-
-\begin_body
-
-\begin_layout Section
-\begin_inset LatexCommand label
-name "sec:benchmarks:strikeslip"
-
-\end_inset
-
-Strike-Slip Benchmark
-\end_layout
-
-\begin_layout Standard
-This benchmark problem computes the viscoelastic (Maxwell) relaxation of
- stresses from a single, finite, strike-slip earthquake in 3D without gravity.
-  Dirichlet boundary conditions equal to the analytical elastic solution
- are imposed on the sides of a cube with sides of length 24 km.
- Anti-plane strain boundary conditions are imposed at y = 0, so the solution
- is equivalent to that for a domain with a 48 km length in the y direction.
- We can use the analytical solution of 
-\begin_inset LatexCommand cite
-key "Okada:1992"
-
-\end_inset
-
- both to apply the boundary conditions and to compare against the numerically-co
-mputed elastic solution.
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Subsection
-Problem Description
-\end_layout
-
-\begin_layout Standard
-Figure 
-\begin_inset LatexCommand ref
-reference "fig:benchmark:strikeslip:geometry"
-
-\end_inset
-
- shows the geometry of the strike-slip fault (red surface) embedded in the
- cube consisting of an elastic material (yellow block) over a Maxwell viscoelast
-ic material (blue block).
- 
-\end_layout
-
-\begin_layout Description
-Domain The domain spans the region
-\begin_inset Formula \begin{gather*}
-0\leq x\leq24\ km,\\
-0\leq y\leq24\ km,\\
--24\ km\leq z\leq0.\end{gather*}
-
-\end_inset
-
-The top (elastic) layer occupies the region 
-\begin_inset Formula $-12\ km\ \leq z\leq0$
-\end_inset
-
- and the bottom (viscoelastic) layer occupies the region 
-\begin_inset Formula $-24\ km\ \leq z\leq-12\ km$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Description
-Material\InsetSpace ~
-properties The material is a Poisson solid with a shear modulus
- of 30 GPa.
- The domain is modeled using an elastic isotropic material for the top layer
- and a Maxwell viscoelastic material for the bottom layer.
- The bottom layer has a viscosity of 1.0e+18 Pa-s.
-\end_layout
-
-\begin_layout Description
-Fault The fault is a vertical, right-lateral strike-slip fault.
- The strike is parallel to the y-direction at the center of the model:
-\begin_inset Formula \begin{gather*}
-x=12\ km,\\
-0\leq y\leq16\ km,\\
--16\ km\leq z\leq0.\end{gather*}
-
-\end_inset
-
-Uniform slip of 1 m is applied over the region 
-\begin_inset Formula $0\leq y\leq12\ km$
-\end_inset
-
- and 
-\begin_inset Formula $-12\ km\leq z\leq0$
-\end_inset
-
- with a linear taper to 0 at y = 16 km and z = -16 km.
- The tapered region is the light red portion of the fault surface in Figure
- 
-\begin_inset LatexCommand ref
-reference "fig:benchmark:strikeslip:geometry"
-
-\end_inset
-
-.
- In the region where the two tapers overlap, each slip value is the minimum
- of the two tapers (so that the taper remains linear).
-\end_layout
-
-\begin_layout Description
-Boundary\InsetSpace ~
-conditions Bottom and side displacements are set to the elastic
- analytical solution, and the top of the model is a free surface.
- There are two exceptions to these applied boundary conditions.
- The first is on the y=0 plane, where y-displacements are left free to preserve
- symmetry, and the x- and z-displacements are set to zero.
- The second is along the line segment between (12, 0, -24) and (12, 24,
- -24), where the analytical solution blows up in some cases.
- Along this line segment, all three displacement components are left free.
-\end_layout
-
-\begin_layout Description
-Discretization The model is discretized with nominal spatial resolutions
- of 1000 m, 500 m, and 250 m.
-\end_layout
-
-\begin_layout Description
-Basis\InsetSpace ~
-functions We use trilinear hexahedral cells and linear tetrahedral
- cells.
-\end_layout
-
-\begin_layout Description
-Solution We compute the error in the elastic solution and compare the solution
- over the domain after 0, 1, 5, and 10 years.
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float figure
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Standard
-\align center
-\begin_inset Graphics
-	filename figs/geometry.png
-	scale 33
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Caption
-
-\begin_layout Standard
-Geometry of strike-slip benchmark problem.
-\begin_inset LatexCommand label
-name "fig:benchmark:strikeslip:geometry"
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsection
-Running the Benchmark
-\end_layout
-
-\begin_layout Standard
-After checking out the benchmark files from the CIG SVN repository, change
- to the 
-\family typewriter
-quasistatic/strikeslip
-\family default
- directory.
- Decompress the gzipped files in the meshes and parameters directories,
-\end_layout
-
-\begin_layout LyX-Code
-gunzip meshes/*.gz parameters/*.gz
-\end_layout
-
-\begin_layout Standard
-Change to the parameters directory.
- Each benchmark uses three 
-\family typewriter
-.cfg
-\family default
- files: 
-\family typewriter
-pylithapp.cfg
-\family default
-, a mesher related file (
-\family typewriter
-strikeslip_cubit.cfg
-\family default
- or 
-\family typewriter
-strikeslip_lagrit.cfg
-\family default
-), and a resolution and cell related file (e.g., 
-\family typewriter
-strikeslip_hex8_1000m.cfg
-\family default
-).
- A few examples of running the benchmarks (elastic solution only) are
-\end_layout
-
-\begin_layout LyX-Code
-pylith strikeslip_cubit.cfg strikeslip_hex8_1000m.cfg
-\end_layout
-
-\begin_layout LyX-Code
-pylith strikeslip_cubit.cfg strikeslip_hex8_0500m.cfg
-\end_layout
-
-\begin_layout LyX-Code
-pylith strikeslip_lagrit.cfg strikeslip_tet4_1000m.cfg
-\end_layout
-
-\begin_layout Standard
-To run the time-dependent (viscoelastic) problem, it is necessary to append
- 
-\family typewriter
-timedep.cfg
-\family default
- to the above commands, for example
-\end_layout
-
-\begin_layout LyX-Code
-pylith strikeslip_cubit.cfg strikeslip_hex8_1000m.cfg timedep.cfg
-\end_layout
-
-\begin_layout Standard
-This will run the problem for 10 years, using a time-step size of 0.1 years,
- and results will be output for each year.
- The benchmarks at resolutions of 1000 m, 500 m, and 250 m require  approximatel
-y 150 MB, 960 MB, and 8 GB, respectively.
-\end_layout
-
-\begin_layout Subsection
-Benchmark Results
-\end_layout
-
-\begin_layout Standard
-Figure 
-\begin_inset LatexCommand ref
-reference "fig:benchmark:strikeslip:solution"
-
-\end_inset
-
- shows the displacement field from the simulation with hexahedral cells
- using trilinear basis functions at a resolution of 1000 m.
- For each resolution and set of basis functions, we measure the accuracy
- by comparing the numerical solution against the semi-analytical Okada solution
-\begin_inset LatexCommand cite
-key "Okada:1992"
-
-\end_inset
-
-.
- We also compare the accuracy and runtime across resolutions and different
- cell types.
- This provides practical information about what cell types and resolutions
- are required to achieve a given level of accuracy with the shortest runtime.
-\end_layout
-
-\begin_layout Standard
-\begin_inset Float figure
-wide false
-sideways false
-status open
-
-\begin_layout Standard
-\align center
-\begin_inset Graphics
-	filename figs/soln.png
-	scale 33
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Caption
-
-\begin_layout Standard
-Displacement field for strike-slip benchmark problem.
-\begin_inset LatexCommand label
-name "fig:benchmark:strikeslip:solution"
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsubsection
-Solution Accuracy
-\end_layout
-
-\begin_layout Standard
-We quantify the error in the finite-element solution by integrating the
- L2 norm of the difference between the finite-element solution  and the
- semi-analytical solution evaluated at the quadrature points.
- We define the local error (error for each finite-element cell) to be
-\begin_inset Formula \begin{equation}
-\epsilon_{local}=\frac{1}{V_{cell}}\sqrt{\intop_{cell}\left(u_{i}^{t}-u_{i}^{fem}\right)^{2}\: dV},\end{equation}
-
-\end_inset
-
-where 
-\begin_inset Formula $u_{i}^{t}$
-\end_inset
-
- is the ith component of the displacement field for the semi-analytical
-  solution, and 
-\begin_inset Formula $u_{i}^{fem}$
-\end_inset
-
- is the ith component of the displacement field for the finite-element 
- solution.
-  Taking the square root of the L2 norm and normalizing by  the volume of
- the cell results in an error metric with dimensions of length.
-  This roughly corresponds to the error in the magnitude of the displacement
- field in the finite element solution.
- We define the global error in a similar fashion,
-\begin_inset Formula \begin{equation}
-\epsilon_{global}=\frac{1}{V_{domain}}\sqrt{\intop_{domain}\left(u_{i}^{t}-u_{i}^{fem}\right)^{2}\: dV},\end{equation}
-
-\end_inset
-
- where we sum the L2 norm computed for the local error over all of the 
- cells before taking the square root and dividing by the volume of the domain.
- CIG has developed a package called 
-\begin_inset LatexCommand htmlurl
-name "Cigma"
-target "geodynamics.org/cig/software/packages/cs/cigma"
-
-\end_inset
-
- that computes these local and global error metrics.
-\end_layout
-
-\begin_layout Standard
-Figures 
-\begin_inset LatexCommand ref
-reference "fig:benchmark:strikeslip:tet4:1000m"
-
-\end_inset
-
- through 
-\begin_inset LatexCommand ref
-reference "fig:benchmark:strikeslip:hex8:250m"
-
-\end_inset
-
- show the local error for each of the three resolutions and two cell types.
- The error decreases with decreasing cell size as expected for a converging
- solution.
- The largest errors, which approach 1 mm for 1 m of slip for a discretization
- size of 250 m, occur where the gradient in slip is discontinuous at the
- boundary between the region of uniform slip and linear taper in slip.
- The linear basis functions cannot match this higher order variation.
- The trilinear basis functions in the hexahedral element provide more terms
- in the polynomial defining the variation in the displacement field within
- each cell compared to the linear basis functions for the tetrahedral cell.
- Consequently, for this problem the error for the hexahedral cells at a
- given resolution is smaller than that for the tetrahedral cells.
- Both sets of cell types and basis functions provide the same rate of convergenc
-e as shown in Figure 
-\begin_inset LatexCommand vref
-reference "fig:benchmark:strikeslip:convergence"
-
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float figure
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Standard
-\align center
-\begin_inset Graphics
-	filename figs/error_tet4_1000m.png
-	scale 33
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Caption
-
-\begin_layout Standard
-Local error for strike-slip benchmark problem with tetrahedral cells and
- linear basis functions with a uniform discretization size of 1000 m.
-\begin_inset LatexCommand label
-name "fig:benchmark:strikeslip:tet4:1000m"
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float figure
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Standard
-\align center
-\begin_inset Graphics
-	filename figs/error_hex8_1000m.png
-	scale 33
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Caption
-
-\begin_layout Standard
-Local error for strike-slip benchmark problem with hexahedral cells and
- trilinear basis functions with a uniform discretization size of 1000 m.
-\begin_inset LatexCommand label
-name "fig:benchmark:strikeslip:hex8:1000m"
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float figure
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Standard
-\align center
-\begin_inset Graphics
-	filename figs/error_tet4_0500m.png
-	scale 33
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Caption
-
-\begin_layout Standard
-Local error for strike-slip benchmark problem with tetrahedral cells and
- linear basis functions with a uniform discretization size of 500 m.
-\begin_inset LatexCommand label
-name "fig:benchmark:strikeslip:tet4:500m"
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float figure
-wide false
-sideways false
-status open
-
-\begin_layout Standard
-\align center
-\begin_inset Graphics
-	filename figs/error_hex8_0500m.png
-	scale 33
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Caption
-
-\begin_layout Standard
-Local error for strike-slip benchmark problem with hexahedral cells and
- trilinear basis functions with a uniform discretization size of 500 m.
-\begin_inset LatexCommand label
-name "fig:benchmark:strikeslip:hex8:500m"
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float figure
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Standard
-\align center
-\begin_inset Graphics
-	filename figs/error_tet4_0250m.png
-	scale 33
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Caption
-
-\begin_layout Standard
-Local error for strike-slip benchmark problem with tetrahedral cells and
- linear basis functions with a uniform discretization size of 250 m.
-\begin_inset LatexCommand label
-name "fig:benchmark:strikeslip:tet4:250m"
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float figure
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Standard
-\align center
-\begin_inset Graphics
-	filename figs/error_hex8_0250m.png
-	scale 33
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Caption
-
-\begin_layout Standard
-Local error for strike-slip benchmark problem with hexahedral cells and
- trilinear basis functions with a uniform discretization size of 250 m.
-\begin_inset LatexCommand label
-name "fig:benchmark:strikeslip:hex8:250m"
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float figure
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Standard
-\align center
-\begin_inset Graphics
-	filename figs/convergence.png
-	scale 33
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Caption
-
-\begin_layout Standard
-Convergence rate for the strike-slip benchmark problem with tetrahedral
- cells and linear basis functions and with hexahedral cells with trilinear
- basis functions.
-\begin_inset LatexCommand label
-name "fig:benchmark:strikeslip:convergence"
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsubsection
-Performance
-\end_layout
-
-\begin_layout Standard
-Figure 
-\begin_inset LatexCommand ref
-reference "fig:benchmark:strikeslip:summary"
-
-\end_inset
-
- summarizes the overall performance of each of the six simulations.
- Although at a given resolution, the number of degrees of freedom in the
- hexahedral and tetrahedral meshes are the the same, the number of cells
- in the tetrahedral mesh is about six times greater.
- However, we use only one integration point per tetrahedral cell compared
- to eight for the hexahedral cell.
- This leads to approximately the same number of integration points for the
- two meshes, but the time required to unpack/pack information for each cell
- from the Sieve data structure is greater than the time required to do the
- calculation for each quadrature point (which can take advantage of the
- very fast, small memory cache in the processor).
- As a result, the runtime for the simulations with hexahedral cells is significa
-ntly less than that for the tetrahedral cells at the same resolution.
- 
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float figure
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Standard
-\align center
-\begin_inset Graphics
-	filename figs/summary.pdf
-	scale 50
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Caption
-
-\begin_layout Standard
-Summary of performance of PyLith for the six simulations of the strike-slip
- benchmark.
- For a given discretization size, hexahedral cells with trilinear basis
- functions provide greater accuracy with a shorter runtime compared with
- tetrahedral cells and linear basis functions.
-\begin_inset LatexCommand label
-name "fig:benchmark:strikeslip:summary"
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Figure 
-\begin_inset LatexCommand vref
-reference "fig:benchmark:strikeslip:scaling"
-
-\end_inset
-
- compares the runtime for the benchmark (elastic solution only) at 500 m
- resolution for 1 to 16 processors.
- The total runtime is the time required for the entire simulation, including
- initialization, distributing the mesh over the processors, solving the
- problem in parallel, and writing the output to VTK files.
- Some initialization steps, writing the output to VTK files, and distributing
- the mesh are essentially serial processes.
- For simulations with many time steps these steps will generally occupy
- only a fraction of the runtime, and the runtime will be dominated by the
- solution of the equations.
- Figure 
-\begin_inset LatexCommand ref
-reference "fig:benchmark:strikeslip:scaling"
-
-\end_inset
-
- also shows the total time required to form the Jacobian of the system,
- form the residual, and solve the system.
- These steps provide a more accurate representation of the parallel-performance
- of the computational portion of the code and show excellent performance
- as evident in the approximately linear slope of 0.7.
- S linear decrease with a slope of 1 would indicate strong scaling, which
- is rarely achieved in real applications.
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float figure
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Standard
-\align center
-\begin_inset Graphics
-	filename figs/scaling.pdf
-	scale 75
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Caption
-
-\begin_layout Standard
-Parallel performance of PyLith for the strike-slip benchmark problem with
- tetrahedral cells and linear basis functions with a uniform discretization
- size of 500 m.
- The total runtime (total) and the runtime to compute the Jacobian and residual
- and solve the system (compute) are shown.
- The compute runtime decreases with a slope of about 0.7; a linear decrease
- with a slope of 1 would indicate strong scaling,  which is rarely achieved
- in any real application.
- 
-\begin_inset LatexCommand label
-name "fig:benchmark:strikeslip:scaling"
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_body
-\end_document
+#LyX 2.0 created this file. For more info see http://www.lyx.org/
+\lyxformat 413
+\begin_document
+\begin_header
+\textclass book
+\begin_preamble
+
+\end_preamble
+\use_default_options false
+\maintain_unincluded_children false
+\language english
+\language_package default
+\inputencoding latin1
+\fontencoding global
+\font_roman default
+\font_sans default
+\font_typewriter default
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100
+\font_tt_scale 100
+
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry true
+\use_amsmath 0
+\use_esint 0
+\use_mhchem 0
+\use_mathdots 1
+\cite_engine basic
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\use_refstyle 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\leftmargin 1in
+\topmargin 1in
+\rightmargin 1in
+\bottommargin 1in
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\quotes_language english
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Section
+\begin_inset CommandInset label
+LatexCommand label
+name "sec:benchmarks:strikeslip"
+
+\end_inset
+
+Strike-Slip Benchmark
+\end_layout
+
+\begin_layout Standard
+This benchmark problem computes the viscoelastic (Maxwell) relaxation of
+ stresses from a single, finite, strike-slip earthquake in 3D without gravity.
+  Dirichlet boundary conditions equal to the analytical elastic solution
+ are imposed on the sides of a cube with sides of length 24 km.
+ Anti-plane strain boundary conditions are imposed at y = 0, so the solution
+ is equivalent to that for a domain with a 48 km length in the y direction.
+ We can use the analytical solution of 
+\begin_inset CommandInset citation
+LatexCommand cite
+key "Okada:1992"
+
+\end_inset
+
+ both to apply the boundary conditions and to compare against the numerically-co
+mputed elastic solution.
+\end_layout
+
+\begin_layout Standard
+
+\end_layout
+
+\begin_layout Subsection
+Problem Description
+\end_layout
+
+\begin_layout Standard
+Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:benchmark:strikeslip:geometry"
+
+\end_inset
+
+ shows the geometry of the strike-slip fault (red surface) embedded in the
+ cube consisting of an elastic material (yellow block) over a Maxwell viscoelast
+ic material (blue block).
+ 
+\end_layout
+
+\begin_layout Description
+Domain The domain spans the region
+\begin_inset Formula 
+\begin{gather*}
+0\leq x\leq24\ km,\\
+0\leq y\leq24\ km,\\
+-24\ km\leq z\leq0.
+\end{gather*}
+
+\end_inset
+
+The top (elastic) layer occupies the region 
+\begin_inset Formula $-12\ km\ \leq z\leq0$
+\end_inset
+
+ and the bottom (viscoelastic) layer occupies the region 
+\begin_inset Formula $-24\ km\ \leq z\leq-12\ km$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Description
+Material
+\begin_inset space ~
+\end_inset
+
+properties The material is a Poisson solid with a shear modulus of 30 GPa.
+ The domain is modeled using an elastic isotropic material for the top layer
+ and a Maxwell viscoelastic material for the bottom layer.
+ The bottom layer has a viscosity of 1.0e+18 Pa-s.
+\end_layout
+
+\begin_layout Description
+Fault The fault is a vertical, right-lateral strike-slip fault.
+ The strike is parallel to the y-direction at the center of the model:
+\begin_inset Formula 
+\begin{gather*}
+x=12\ km,\\
+0\leq y\leq16\ km,\\
+-16\ km\leq z\leq0.
+\end{gather*}
+
+\end_inset
+
+Uniform slip of 1 m is applied over the region 
+\begin_inset Formula $0\leq y\leq12\ km$
+\end_inset
+
+ and 
+\begin_inset Formula $-12\ km\leq z\leq0$
+\end_inset
+
+ with a linear taper to 0 at y = 16 km and z = -16 km.
+ The tapered region is the light red portion of the fault surface in Figure
+ 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:benchmark:strikeslip:geometry"
+
+\end_inset
+
+.
+ In the region where the two tapers overlap, each slip value is the minimum
+ of the two tapers (so that the taper remains linear).
+\end_layout
+
+\begin_layout Description
+Boundary
+\begin_inset space ~
+\end_inset
+
+conditions Bottom and side displacements are set to the elastic analytical
+ solution, and the top of the model is a free surface.
+ There are two exceptions to these applied boundary conditions.
+ The first is on the y=0 plane, where y-displacements are left free to preserve
+ symmetry, and the x- and z-displacements are set to zero.
+ The second is along the line segment between (12, 0, -24) and (12, 24,
+ -24), where the analytical solution blows up in some cases.
+ Along this line segment, all three displacement components are left free.
+\end_layout
+
+\begin_layout Description
+Discretization The model is discretized with nominal spatial resolutions
+ of 1000 m, 500 m, and 250 m.
+\end_layout
+
+\begin_layout Description
+Basis
+\begin_inset space ~
+\end_inset
+
+functions We use trilinear hexahedral cells and linear tetrahedral cells.
+\end_layout
+
+\begin_layout Description
+Solution We compute the error in the elastic solution and compare the solution
+ over the domain after 0, 1, 5, and 10 years.
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float figure
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\align center
+\begin_inset Graphics
+	filename figs/geometry.png
+	scale 33
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Geometry of strike-slip benchmark problem.
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:benchmark:strikeslip:geometry"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Running the Benchmark
+\end_layout
+
+\begin_layout Standard
+After checking out the benchmark files from the CIG SVN repository, change
+ to the 
+\family typewriter
+quasistatic/strikeslip
+\family default
+ directory.
+ Decompress the gzipped files in the meshes and parameters directories,
+\end_layout
+
+\begin_layout LyX-Code
+gunzip meshes/*.gz parameters/*.gz
+\end_layout
+
+\begin_layout Standard
+Change to the parameters directory.
+ Each benchmark uses three 
+\family typewriter
+.cfg
+\family default
+ files: 
+\family typewriter
+pylithapp.cfg
+\family default
+, a mesher related file (
+\family typewriter
+strikeslip_cubit.cfg
+\family default
+ or 
+\family typewriter
+strikeslip_lagrit.cfg
+\family default
+), and a resolution and cell related file (e.g., 
+\family typewriter
+strikeslip_hex8_1000m.cfg
+\family default
+).
+ A few examples of running the benchmarks (elastic solution only) are
+\end_layout
+
+\begin_layout LyX-Code
+pylith strikeslip_cubit.cfg strikeslip_hex8_1000m.cfg
+\end_layout
+
+\begin_layout LyX-Code
+pylith strikeslip_cubit.cfg strikeslip_hex8_0500m.cfg
+\end_layout
+
+\begin_layout LyX-Code
+pylith strikeslip_lagrit.cfg strikeslip_tet4_1000m.cfg
+\end_layout
+
+\begin_layout Standard
+To run the time-dependent (viscoelastic) problem, it is necessary to append
+ 
+\family typewriter
+timedep.cfg
+\family default
+ to the above commands, for example
+\end_layout
+
+\begin_layout LyX-Code
+pylith strikeslip_cubit.cfg strikeslip_hex8_1000m.cfg timedep.cfg
+\end_layout
+
+\begin_layout Standard
+This will run the problem for 10 years, using a time-step size of 0.1 years,
+ and results will be output for each year.
+ The benchmarks at resolutions of 1000 m, 500 m, and 250 m require  approximatel
+y 150 MB, 960 MB, and 8 GB, respectively.
+\end_layout
+
+\begin_layout Subsection
+Benchmark Results
+\end_layout
+
+\begin_layout Standard
+Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:benchmark:strikeslip:solution"
+
+\end_inset
+
+ shows the displacement field from the simulation with hexahedral cells
+ using trilinear basis functions at a resolution of 1000 m.
+ For each resolution and set of basis functions, we measure the accuracy
+ by comparing the numerical solution against the semi-analytical Okada solution
+ 
+\begin_inset CommandInset citation
+LatexCommand cite
+key "Okada:1992"
+
+\end_inset
+
+.
+ We also compare the accuracy and runtime across resolutions and different
+ cell types.
+ This provides practical information about what cell types and resolutions
+ are required to achieve a given level of accuracy with the shortest runtime.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\align center
+\begin_inset Graphics
+	filename figs/soln.png
+	scale 33
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Displacement field for strike-slip benchmark problem.
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:benchmark:strikeslip:solution"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsubsection
+Solution Accuracy
+\end_layout
+
+\begin_layout Standard
+We quantify the error in the finite-element solution by integrating the
+ L2 norm of the difference between the finite-element solution  and the
+ semi-analytical solution evaluated at the quadrature points.
+ We define the local error (error for each finite-element cell) to be
+\begin_inset Formula 
+\begin{equation}
+\epsilon_{local}=\frac{1}{V_{cell}}\sqrt{\intop_{cell}\left(u_{i}^{t}-u_{i}^{fem}\right)^{2}\: dV},
+\end{equation}
+
+\end_inset
+
+where 
+\begin_inset Formula $u_{i}^{t}$
+\end_inset
+
+ is the ith component of the displacement field for the semi-analytical
+  solution, and 
+\begin_inset Formula $u_{i}^{fem}$
+\end_inset
+
+ is the ith component of the displacement field for the finite-element 
+ solution.
+  Taking the square root of the L2 norm and normalizing by  the volume of
+ the cell results in an error metric with dimensions of length.
+  This roughly corresponds to the error in the magnitude of the displacement
+ field in the finite element solution.
+ We define the global error in a similar fashion,
+\begin_inset Formula 
+\begin{equation}
+\epsilon_{global}=\frac{1}{V_{domain}}\sqrt{\intop_{domain}\left(u_{i}^{t}-u_{i}^{fem}\right)^{2}\: dV},
+\end{equation}
+
+\end_inset
+
+ where we sum the L2 norm computed for the local error over all of the 
+ cells before taking the square root and dividing by the volume of the domain.
+ CIG has developed a package called Cigma 
+\begin_inset Flex URL
+status collapsed
+
+\begin_layout Plain Layout
+
+geodynamics.org/cig/software/packages/cs/cigma
+\end_layout
+
+\end_inset
+
+ that computes these local and global error metrics.
+\end_layout
+
+\begin_layout Standard
+Figures 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:benchmark:strikeslip:tet4:1000m"
+
+\end_inset
+
+ through 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:benchmark:strikeslip:hex8:250m"
+
+\end_inset
+
+ show the local error for each of the three resolutions and two cell types.
+ The error decreases with decreasing cell size as expected for a converging
+ solution.
+ The largest errors, which approach 1 mm for 1 m of slip for a discretization
+ size of 250 m, occur where the gradient in slip is discontinuous at the
+ boundary between the region of uniform slip and linear taper in slip.
+ The linear basis functions cannot match this higher order variation.
+ The trilinear basis functions in the hexahedral element provide more terms
+ in the polynomial defining the variation in the displacement field within
+ each cell compared to the linear basis functions for the tetrahedral cell.
+ Consequently, for this problem the error for the hexahedral cells at a
+ given resolution is smaller than that for the tetrahedral cells.
+ Both sets of cell types and basis functions provide the same rate of convergenc
+e as shown in Figure 
+\begin_inset CommandInset ref
+LatexCommand vref
+reference "fig:benchmark:strikeslip:convergence"
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float figure
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\align center
+\begin_inset Graphics
+	filename figs/error_tet4_1000m.png
+	scale 33
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Local error for strike-slip benchmark problem with tetrahedral cells and
+ linear basis functions with a uniform discretization size of 1000 m.
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:benchmark:strikeslip:tet4:1000m"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float figure
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\align center
+\begin_inset Graphics
+	filename figs/error_hex8_1000m.png
+	scale 33
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Local error for strike-slip benchmark problem with hexahedral cells and
+ trilinear basis functions with a uniform discretization size of 1000 m.
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:benchmark:strikeslip:hex8:1000m"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float figure
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\align center
+\begin_inset Graphics
+	filename figs/error_tet4_0500m.png
+	scale 33
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Local error for strike-slip benchmark problem with tetrahedral cells and
+ linear basis functions with a uniform discretization size of 500 m.
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:benchmark:strikeslip:tet4:500m"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float figure
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\align center
+\begin_inset Graphics
+	filename figs/error_hex8_0500m.png
+	scale 33
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Local error for strike-slip benchmark problem with hexahedral cells and
+ trilinear basis functions with a uniform discretization size of 500 m.
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:benchmark:strikeslip:hex8:500m"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float figure
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\align center
+\begin_inset Graphics
+	filename figs/error_tet4_0250m.png
+	scale 33
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Local error for strike-slip benchmark problem with tetrahedral cells and
+ linear basis functions with a uniform discretization size of 250 m.
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:benchmark:strikeslip:tet4:250m"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float figure
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\align center
+\begin_inset Graphics
+	filename figs/error_hex8_0250m.png
+	scale 33
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Local error for strike-slip benchmark problem with hexahedral cells and
+ trilinear basis functions with a uniform discretization size of 250 m.
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:benchmark:strikeslip:hex8:250m"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float figure
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\align center
+\begin_inset Graphics
+	filename figs/convergence.png
+	scale 33
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Convergence rate for the strike-slip benchmark problem with tetrahedral
+ cells and linear basis functions and with hexahedral cells with trilinear
+ basis functions.
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:benchmark:strikeslip:convergence"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsubsection
+Performance
+\end_layout
+
+\begin_layout Standard
+Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:benchmark:strikeslip:summary"
+
+\end_inset
+
+ summarizes the overall performance of each of the six simulations.
+ Although at a given resolution, the number of degrees of freedom in the
+ hexahedral and tetrahedral meshes are the same, the number of cells in
+ the tetrahedral mesh is about six times greater.
+ However, we use only one integration point per tetrahedral cell compared
+ to eight for the hexahedral cell.
+ This leads to approximately the same number of integration points for the
+ two meshes, but the time required to unpack/pack information for each cell
+ from the Sieve data structure is greater than the time required to do the
+ calculation for each quadrature point (which can take advantage of the
+ very fast, small memory cache in the processor).
+ As a result, the runtime for the simulations with hexahedral cells is significa
+ntly less than that for the tetrahedral cells at the same resolution.
+ 
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float figure
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\align center
+\begin_inset Graphics
+	filename figs/summary.pdf
+	scale 50
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Summary of performance of PyLith for the six simulations of the strike-slip
+ benchmark.
+ For a given discretization size, hexahedral cells with trilinear basis
+ functions provide greater accuracy with a shorter runtime compared with
+ tetrahedral cells and linear basis functions.
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:benchmark:strikeslip:summary"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Figure 
+\begin_inset CommandInset ref
+LatexCommand vref
+reference "fig:benchmark:strikeslip:scaling"
+
+\end_inset
+
+ compares the runtime for the benchmark (elastic solution only) at 500 m
+ resolution for 1 to 16 processors.
+ The total runtime is the time required for the entire simulation, including
+ initialization, distributing the mesh over the processors, solving the
+ problem in parallel, and writing the output to VTK files.
+ Some initialization steps, writing the output to VTK files, and distributing
+ the mesh are essentially serial processes.
+ For simulations with many time steps these steps will generally occupy
+ only a fraction of the runtime, and the runtime will be dominated by the
+ solution of the equations.
+ Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:benchmark:strikeslip:scaling"
+
+\end_inset
+
+ also shows the total time required to form the Jacobian of the system,
+ form the residual, and solve the system.
+ These steps provide a more accurate representation of the parallel-performance
+ of the computational portion of the code and show excellent performance
+ as evident in the approximately linear slope of 0.7.
+ S linear decrease with a slope of 1 would indicate strong scaling, which
+ is rarely achieved in real applications.
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float figure
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\align center
+\begin_inset Graphics
+	filename figs/scaling.pdf
+	scale 75
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Parallel performance of PyLith for the strike-slip benchmark problem with
+ tetrahedral cells and linear basis functions with a uniform discretization
+ size of 500 m.
+ The total runtime (total) and the runtime to compute the Jacobian and residual
+ and solve the system (compute) are shown.
+ The compute runtime decreases with a slope of about 0.7; a linear decrease
+ with a slope of 1 would indicate strong scaling,  which is rarely achieved
+ in any real application.
+ 
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:benchmark:strikeslip:scaling"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document

Modified: short/3D/PyLith/branches/v1.7-trunk/doc/userguide/boundaryconditions/boundaryconditions.lyx
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/doc/userguide/boundaryconditions/boundaryconditions.lyx	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/doc/userguide/boundaryconditions/boundaryconditions.lyx	2012-08-21 18:24:25 UTC (rev 20616)
@@ -1,7860 +1,7937 @@
-#LyX 2.0 created this file. For more info see http://www.lyx.org/
-\lyxformat 413
-\begin_document
-\begin_header
-\textclass book
-\use_default_options false
-\maintain_unincluded_children false
-\language english
-\language_package default
-\inputencoding auto
-\fontencoding global
-\font_roman default
-\font_sans default
-\font_typewriter default
-\font_default_family default
-\use_non_tex_fonts false
-\font_sc false
-\font_osf false
-\font_sf_scale 100
-\font_tt_scale 100
-
-\graphics default
-\default_output_format default
-\output_sync 0
-\bibtex_command default
-\index_command default
-\paperfontsize default
-\spacing single
-\use_hyperref false
-\papersize default
-\use_geometry true
-\use_amsmath 1
-\use_esint 0
-\use_mhchem 1
-\use_mathdots 1
-\cite_engine basic
-\use_bibtopic false
-\use_indices false
-\paperorientation portrait
-\suppress_date false
-\use_refstyle 0
-\index Index
-\shortcut idx
-\color #008000
-\end_index
-\leftmargin 1in
-\topmargin 1in
-\rightmargin 1in
-\bottommargin 1in
-\secnumdepth 3
-\tocdepth 3
-\paragraph_separation indent
-\paragraph_indentation default
-\quotes_language english
-\papercolumns 1
-\papersides 1
-\paperpagestyle default
-\tracking_changes false
-\output_changes false
-\html_math_output 0
-\html_css_as_file 0
-\html_be_strict false
-\end_header
-
-\begin_body
-
-\begin_layout Chapter
-\begin_inset CommandInset label
-LatexCommand label
-name "cha:boundary:interface:conditions"
-
-\end_inset
-
-Boundary and Interface Conditions
-\end_layout
-
-\begin_layout Section
-Assigning Boundary Conditions
-\end_layout
-
-\begin_layout Standard
-There are four basic steps in assigning a specific boundary condition to
- a portion of the domain.
-\end_layout
-
-\begin_layout Enumerate
-Create sets of vertices in the mesh generation process for each boundary
- condition.
-\end_layout
-
-\begin_layout Enumerate
-Define boundary condition groups corresponding to the vertex sets.
-\end_layout
-
-\begin_layout Enumerate
-Set the parameters for each boundary condition group using 
-\family typewriter
-.cfg
-\family default
- or 
-\family typewriter
-.pml
-\family default
- files and/or command line arguments.
-\end_layout
-
-\begin_layout Enumerate
-Specify the spatial variation in parameters for the boundary condition using
- a spatial database file.
-\end_layout
-
-\begin_layout Subsection
-Creating Sets of Vertices
-\end_layout
-
-\begin_layout Standard
-The procedure for creating sets of vertices differs depending on the mesh
- generator.
- For meshes specified using the PyLith mesh ASCII format, the sets of vertices
- are specified using groups (see Appendix 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "sec:MeshIOAscii"
-
-\end_inset
-
-).
- In CUBIT the groups of vertices are created using nodesets.
- Similarly, in LaGriT, psets are used.
- Note that we chose to associated boundary conditions with groups of vertices
- because nearly every mesh generation package supports associating a string
- or integer with groups of vertices.
- Note also that we currently associate boundary conditions with string identifie
-rs, so even if the mesh generator uses integers, the name is specified as
- the digits of the integer value.
- Finally, note that every vertex set that ultimately is associated with
- a boundary condition on a cell face (e.g., Neumann boundary conditions and
- fault interface conditions) must correspond to a simply-connected surface.
-\end_layout
-
-\begin_layout Subsection
-Arrays of Boundary Condition Components
-\end_layout
-
-\begin_layout Standard
-A dynamic array of boundary condition components associates a name (string)
- with each boundary condition.
- This dynamic array of boundary conditions replaces the boundary condition
- container in PyLith 1.0.
- User-defined containers are no longer necessary, and the predefined containers
- are no longer available (or necessary).
- The default boundary condition for each component in the array is the Dirichlet
-Points object.
- Other boundary conditions can be bound to the named items in the array
- via a 
-\family typewriter
-.cfg
-\family default
- file, 
-\family typewriter
-.pml
-\family default
- file, or the command line.
- The parameters for the boundary condition are set using the name of the
- boundary condition.
- An example of setting the array of boundary condition components and changing
- the types of boundary conditions in a 
-\family typewriter
-.cfg
-\family default
- file:
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.problem]
-\end_layout
-
-\begin_layout LyX-Code
-bc = [x_neg,x_pos,y_pos,z_neg] ; Array of boundary conditions
-\end_layout
-
-\begin_layout LyX-Code
-# Default boundary condition is DirichletPoints
-\end_layout
-
-\begin_layout LyX-Code
-# Keep default value for bc.x_neg
-\end_layout
-
-\begin_layout LyX-Code
-bc.x_pos = pylith.bc.DirichletBoundary ; change BC type to DirichletBoundary
-\end_layout
-
-\begin_layout LyX-Code
-bc.y_pos = pylith.bc.AbsorbingDampers ; change BC type to AbsorbingDampers
-\end_layout
-
-\begin_layout LyX-Code
-bc.z_neg = pylith.bc.Neumann ; change BC type to Neumann (traction)
-\end_layout
-
-\begin_layout Section
-\begin_inset CommandInset label
-LatexCommand label
-name "sec:Time:Dependent:BC"
-
-\end_inset
-
-Time-Dependent Boundary Conditions
-\end_layout
-
-\begin_layout Standard
-Several boundary conditions use a common formulation for the spatial and
- temporal variation of the boundary condition parameters,
-\begin_inset Formula 
-\begin{equation}
-f(\vec{x})=f_{0}(\vec{x})+\dot{f}_{0}(\vec{x})(t-t_{0}(\vec{x}))+f_{1}(\vec{x})a(t-t_{1}(\vec{x})),
-\end{equation}
-
-\end_inset
-
-where 
-\begin_inset Formula $f(\vec{x})$
-\end_inset
-
- may be a scalar or vector parameter, 
-\begin_inset Formula $f_{0}(\vec{x})$
-\end_inset
-
- is a constant value, 
-\begin_inset Formula $\dot{f}_{0}(\vec{x})$
-\end_inset
-
- is a constant rate of change in the value, 
-\begin_inset Formula $t_{0}(\vec{x})$
-\end_inset
-
- is the onset time for the constant rate of change, 
-\begin_inset Formula $f_{1}(\vec{x})$
-\end_inset
-
- is the amplitude for the temporal modulation, 
-\begin_inset Formula $a(t)$
-\end_inset
-
- is the variation in amplitude with time, 
-\begin_inset Formula $t_{1}(\vec{x})$
-\end_inset
-
- is the onset time for the temporal modulation, and 
-\begin_inset Formula $\vec{x}$
-\end_inset
-
- is the position of a location in space.
- This common formulation permits easy specification of a scalar or vector
- with a constant value, constant rate of change of a value, and/or modulation
- of a value in time.
- One can specify just the initial value, just the rate of change of the
- value (along with the corresponding onset time), or just the modulation
- in amplitude (along with the corresponding temporal variation and onset
- time), or any combination of the three.
- The facilities associated with this formulation are:
-\end_layout
-
-\begin_layout Description
-db_initial Spatial database specifying the spatial variation in the initial
- value (default is none).
-\end_layout
-
-\begin_layout Description
-db_rate Spatial database specifying rate of change in the value (default
- is none).
-\end_layout
-
-\begin_layout Description
-db_change Spatial database specifying the amplitude of the temporal modulation
- (default is none).
-\end_layout
-
-\begin_layout Description
-th_change Time history database specifying the temporal change in amplitude
- (default is none).
-\end_layout
-
-\begin_layout Subsection
-Dirichlet Boundary Conditions
-\end_layout
-
-\begin_layout Standard
-Dirichlet boundary conditions in PyLith prescribe the displacement of a
- subset of the vertices of the finite-element mesh.
- While Dirichlet boundary conditions can be applied to any vertex, usually
- they are applied to vertices on the lateral and bottom boundaries of the
- domain.
- There are two types of Dirichlet boundary conditions, DirichletBC and Dirichlet
-Boundary.
- Both provide identical constraints on the solution, but DirichletBoundary
- is limited to vertices of a simply-connected surface, which allows diagnostic
- output of the prescribed displacements.
- DirichletBC can be applied to a set of unconnected vertices.
-\end_layout
-
-\begin_layout Subsubsection
-Dirichlet Boundary Condition Parameters
-\end_layout
-
-\begin_layout Standard
-The properties and components common to both the DirichletPoints and DirichletBo
-undary boundary conditions are:
-\end_layout
-
-\begin_layout Description
-label Label of the group of vertices associated with the boundary condition.
-\end_layout
-
-\begin_layout Description
-bc_dof Array of degrees of freedom to be fixed (first degree of freedom
- is 0).
-\end_layout
-
-\begin_layout Standard
-DirichletBoundary contains an additional component:
-\end_layout
-
-\begin_layout Description
-output Manager for output of displacements on boundary with specified displaceme
-nts.
-\end_layout
-
-\begin_layout Standard
-By default the output manager does not output any information.
- The specified displacements and velocities can be output by including ``displac
-ements'' and ``velocities'' in the output manager's 
-\family typewriter
-vertex_info_fields
-\family default
- array parameter.
- An example of setting the Dirichlet boundary condition parameters in a
- 
-\family typewriter
-.cfg
-\family default
- file is:
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.problem]
-\end_layout
-
-\begin_layout LyX-Code
-bc = [mybc]
-\end_layout
-
-\begin_layout LyX-Code
-
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.problem.bc.mybc]
-\end_layout
-
-\begin_layout LyX-Code
-label = group A 
-\end_layout
-
-\begin_layout LyX-Code
-bc_dof = [2] ; fixed displacement in z direction
-\end_layout
-
-\begin_layout LyX-Code
-db_initial = spatialdata.spatialdb.SimpleDB
-\end_layout
-
-\begin_layout LyX-Code
-db_initial.iohandler.filename = disp_A.spatialdb
-\end_layout
-
-\begin_layout LyX-Code
-db_initial.query_type = nearest ; change query type to nearest point algorithm
-\end_layout
-
-\begin_layout LyX-Code
-db_rate = spatialdata.spatialdb.UniformDB
-\end_layout
-
-\begin_layout LyX-Code
-db_rate.values = [displacement-rate-z]
-\end_layout
-
-\begin_layout LyX-Code
-db_rate.data = [1.0e-06*m/s] ; velocity is 1.0e-06 m/s
-\end_layout
-
-\begin_layout Standard
-We have created an array with one boundary condition, mybc.
- The group of vertices associated with the boundary condition is group A.
- For the database associated with the constant displacement, we use a SimpleDB.
- We set the filename and query type for the database.
- For the rate of change of values, we use a UniformDB and specify the velocity
- in the z-direction to be 1.0e-06 m/s.
- See Section 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "sec:spatial:databases"
-
-\end_inset
-
- for a discussion of the different types of spatial databases available.
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float table
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\noindent
-\align center
-\begin_inset Caption
-
-\begin_layout Plain Layout
-\begin_inset CommandInset label
-LatexCommand label
-name "tab:dirichlet:output"
-
-\end_inset
-
-Fields available in output of DirichletBoundary boundary condition information.
-\end_layout
-
-\end_inset
-
-
-\begin_inset VSpace defskip
-\end_inset
-
-
-\begin_inset Tabular
-<lyxtabular version="3" rows="6" columns="3">
-<features tabularvalignment="middle">
-<column alignment="center" valignment="top" width="0">
-<column alignment="center" valignment="top" width="0">
-<column alignment="center" valignment="top" width="3in">
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Field Type
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Field
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Description
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-vertex_info_fields
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-displacement_initial
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Initial displacement field in global coordinate system
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-velocity
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Rate of change of displacement field in global coordinate system
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-velocity_start_time
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Onset time in seconds for rate of change in displacement field
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-displacement_change
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Amplitude of change in displacement field in global coordinate system
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-change_start_time
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Onset time in seconds for the amplitude change in the displacement field
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsubsection
-Dirichlet Boundary Condition Spatial Database Files
-\end_layout
-
-\begin_layout Standard
-The spatial database files for the Dirichlet boundary condition specify
- the fixed displacements.
- The spatial database file may contain displacements at more degrees of
- freedom than those specified in the Dirichlet boundary condition settings
- using the 
-\family typewriter
-bc_dof
-\family default
- setting.
- Only those listed in 
-\family typewriter
-bc_dof
-\family default
- will be used.
- This permits using the same spatial database file for multiple Dirichlet
- boundary conditions with the same displacement field.
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float table
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\noindent
-\align center
-\begin_inset Caption
-
-\begin_layout Plain Layout
-Values in the spatial databases used for Dirichlet boundary conditions.
-\end_layout
-
-\end_inset
-
-
-\begin_inset VSpace defskip
-\end_inset
-
-
-\begin_inset Tabular
-<lyxtabular version="3" rows="4" columns="2">
-<features tabularvalignment="middle">
-<column alignment="center" valignment="top" width="0">
-<column alignment="center" valignment="top" width="4in">
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Spatial database
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Name in Spatial Database
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-db_initial
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-displacement-x, displacement-y, displacement-z
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-db_rate
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-displacement-rate-x, displacement-rate-y, displacement-rate-z, rate-start-time
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-db_change
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-displacement-x, displacement-y, displacement-z, change-start-time
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsection
-Neumann Boundary Conditions
-\end_layout
-
-\begin_layout Standard
-Neumann boundary conditions are surface tractions applied over a subset
- of the mesh.
- As with the DirichletBoundary condition, each Neumann boundary condition
- can only be applied to a simply-connected surface.
- The surface over which the tractions are applied always has a spatial dimension
- that is one less than the dimension of the finite-element mesh.
- Traction values are computed at the integration points of each cell on
- the surface, using values from a spatial database.
- The tractions are integrated over each cell and assembled to obtain the
- forces applied at the vertices.
- See Section 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "sub:Tutorial-twoquad4-traction"
-
-\end_inset
-
- for a tutorial that uses Neumann boundary conditions.
-\end_layout
-
-\begin_layout Subsubsection
-Neumann Boundary Condition Parameters
-\end_layout
-
-\begin_layout Standard
-The Neumann boundary condition properties and components are:
-\end_layout
-
-\begin_layout Description
-label Name of the group of vertices defining the mesh boundary for the Neumann
- boundary condition.
-\end_layout
-
-\begin_layout Description
-up_dir This is a 3-vector that provides a hint for the direction perpendicular
- to the horizontal tangent direction that is not collinear with the direction
- normal to the surface.
- The default value is (0,0,1), which assumes that the z-axis is positive
- upward.
- This vector is only needed for three-dimensional problems where the positive
- upward direction differs from the default.
-\end_layout
-
-\begin_layout Description
-output The output manager associated with diagnostic output (traction vector).
-\end_layout
-
-\begin_layout Description
-quadrature The quadrature object to be used for numerical integration.
- Since we are integrating over a surface that is one dimension lower than
- the problem domain, this would typically be set to something like 
-\family typewriter
-Quadrature2Din3D
-\family default
- (for a three-dimensional problem).
-\end_layout
-
-\begin_layout Standard
-By default the output manager does not output any information.
- The specified tractions can be output in global coordinates by including
- ``tractions'' in the output manager's 
-\family typewriter
-cell_info_fields
-\family default
- array parameter.
- An example of setting these parameters in a 
-\family typewriter
-.cfg
-\family default
- file is:
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.timedependent]
-\end_layout
-
-\begin_layout LyX-Code
-bc = [x_neg,x_pos,y_neg]
-\end_layout
-
-\begin_layout LyX-Code
-bc.x_pos = pylith.bc.Neumann ; Change BC type to Neumann
-\end_layout
-
-\begin_layout LyX-Code
- 
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.timedependent.bc.x_pos]
-\end_layout
-
-\begin_layout LyX-Code
-label = x_pos ; Name of group of vertices for +x boundary
-\end_layout
-
-\begin_layout LyX-Code
-db_initial = spatialdata.spatialdb.SimpleDB
-\end_layout
-
-\begin_layout LyX-Code
-db_initial.label = Neumann BC +x edge
-\end_layout
-
-\begin_layout LyX-Code
-db_initial.iohandler.filename = axialtract.spatialdb
-\end_layout
-
-\begin_layout LyX-Code
-db_initial.query_type = nearest
-\end_layout
-
-\begin_layout LyX-Code
-quadrature.cell = pylith.feassemble.FIATLagrange
-\end_layout
-
-\begin_layout LyX-Code
-quadrature.cell.dimension = 1
-\end_layout
-
-\begin_layout LyX-Code
-quadrature.cell.quad_order = 2
-\end_layout
-
-\begin_layout Standard
-These settings correspond to the example problem described in Section 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "sub:Tutorial-twoquad4-traction"
-
-\end_inset
-
-.
- It is necessary to set the boundary condition type to 
-\family typewriter
-pylith.bc.Neumann
-\family default
-, since the default value is 
-\family typewriter
-Dirichlet
-\family default
-BC.
- Constant tractions are used for this particular problem, so a quadrature
- order of one would have been sufficient; however, for problems involving
- more complex variations (e.g., a linear variation), a quadrature order of
- two will provide more accurate results.
- Note that there is no advantage to specifying an integration order higher
- than two, since linear elements are being used for this problem.
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float table
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\noindent
-\align center
-\begin_inset Caption
-
-\begin_layout Plain Layout
-\begin_inset CommandInset label
-LatexCommand label
-name "tab:neumann:output"
-
-\end_inset
-
-Fields available in output of Neumann boundary condition information.
-\end_layout
-
-\end_inset
-
-
-\begin_inset VSpace defskip
-\end_inset
-
-
-\begin_inset Tabular
-<lyxtabular version="3" rows="6" columns="3">
-<features tabularvalignment="middle">
-<column alignment="center" valignment="top" width="0">
-<column alignment="center" valignment="top" width="0">
-<column alignment="center" valignment="top" width="3in">
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Field Type
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Field
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Description
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-cell_info_fields
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-tracton_initial
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Initial traction field in global coordinate system
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-traction_rate
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Rate of change of traction field in global coordinate system
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-rate_start_time
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Onset time in seconds for rate of change in traction field
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-traction_change
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Amplitude of change in traction field in global coordinate system
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-change_start_time
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Onset time in seconds for the amplitude change in the traction field
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsubsection
-Neumann Boundary Condition Spatial Database Files
-\end_layout
-
-\begin_layout Standard
-The spatial database files for the Neumann boundary condition specify the
- applied tractions.
- The number of traction components is equal to the spatial dimension for
- the problem.
- The tractions are specified in a local coordinate system for the boundary.
- The names of the components of the traction vector are:
-\end_layout
-
-\begin_layout Description
-one-dimensional 
-\family typewriter
-normal
-\end_layout
-
-\begin_layout Description
-two-dimensional 
-\family typewriter
-shear
-\family default
-, 
-\family typewriter
-normal
-\end_layout
-
-\begin_layout Description
-three-dimensional 
-\family typewriter
-horiz-shear
-\family default
-, 
-\family typewriter
-vert-shear
-\family default
-, 
-\family typewriter
-normal
-\end_layout
-
-\begin_layout Standard
-Ambiguities in specifying the shear tractions in 3D problems are resolved
- using the 
-\family typewriter
-up_dir
-\family default
- parameter.
- In the case of a horizontal surface, users will need to pick an alternative
- vector, as the default 
-\family typewriter
-up_dir
-\family default
- would coincide with the normal direction.
- In this case, the orientation for the 
-\family typewriter
-vert-shear-traction
-\family default
- component will correspond to whatever the user specifies for 
-\family typewriter
-up_dir
-\family default
-, rather than the actual vertical direction.
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float table
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\noindent
-\align center
-\begin_inset Caption
-
-\begin_layout Plain Layout
-Values in the spatial databases used for Dirichlet boundary conditions in
- three dimensions.
- In one- and two-dimensional problems, the names of the components are slightly
- different as described earlier in this section.
-\end_layout
-
-\end_inset
-
-
-\begin_inset VSpace defskip
-\end_inset
-
-
-\begin_inset Tabular
-<lyxtabular version="3" rows="4" columns="2">
-<features tabularvalignment="middle">
-<column alignment="center" valignment="top" width="0">
-<column alignment="center" valignment="top" width="4in">
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Spatial database
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Name in Spatial Database
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-db_initial
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-traction-shear-horiz, traction-shear-vert, traction-normal
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-db_rate
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-traction-rate-horiz-shear, traction-rate-vert-shear, traction-rate-normal,
- rate-start-time
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-db_change
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-traction-horiz-shear, traction-vert-shear, traction-normal, change-start-time
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsection
-Point Force Boundary Conditions
-\end_layout
-
-\begin_layout Standard
-Point force boundary conditions in PyLith prescribe the application of point
- forces to a subset of the vertices of the finite-element mesh.
- While point force boundary conditions can be applied to any vertex, usually
- they are applied to vertices on the lateral, top, and bottom boundaries
- of the domain.
-\end_layout
-
-\begin_layout Subsubsection
-Point Force Parameters
-\end_layout
-
-\begin_layout Standard
-The properties and components common to both the DirichletBC and DirichletBounda
-ry boundary conditions are:
-\end_layout
-
-\begin_layout Description
-label Label of the group of vertices associated with the boundary condition.
-\end_layout
-
-\begin_layout Description
-bc_dof Array of degrees of freedom to which forces are applied (first degree
- of freedom is 0).
-\end_layout
-
-\begin_layout Standard
-An example of setting the point force boundary condition parameters in a
- 
-\family typewriter
-.cfg
-\family default
- file is:
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.problem]
-\end_layout
-
-\begin_layout LyX-Code
-bc = [mybc]
-\end_layout
-
-\begin_layout LyX-Code
-bc.mybc = pylith.bc.PointForce
-\end_layout
-
-\begin_layout LyX-Code
-
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.problem.bc.mybc]
-\end_layout
-
-\begin_layout LyX-Code
-label = group A 
-\end_layout
-
-\begin_layout LyX-Code
-bc_dof = [2] ; fixed displacement in z direction
-\end_layout
-
-\begin_layout LyX-Code
-db_initial = spatialdata.spatialdb.SimpleDB
-\end_layout
-
-\begin_layout LyX-Code
-db_initial.iohandler.filename = force_A.spatialdb
-\end_layout
-
-\begin_layout LyX-Code
-db_initial.query_type = nearest ; change query type to nearest point algorithm
-\end_layout
-
-\begin_layout LyX-Code
-db_rate = spatialdata.spatialdb.UniformDB
-\end_layout
-
-\begin_layout LyX-Code
-db_rate.values = [force-rate-z]
-\end_layout
-
-\begin_layout LyX-Code
-db_rate.data = [1.0e+5*newton/s]
-\end_layout
-
-\begin_layout Standard
-We have created an array with one boundary condition, mybc.
- The group of vertices associated with the boundary condition is group A.
- For the database associated with the constant force, we use a SimpleDB.
- We set the filename and query type for the database.
- For the rate of change of values, we use a UniformDB and specify the rate
- of change in the force to be 1.0e+5 Newton/s.
- See Section 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "sec:spatial:databases"
-
-\end_inset
-
- for a discussion of the different types of spatial databases available.
-\end_layout
-
-\begin_layout Subsubsection
-Point Force Spatial Database Files
-\end_layout
-
-\begin_layout Standard
-The spatial database files for the Dirichlet boundary condition specify
- the fixed displacements.
- The spatial database file may contain displacements at more degrees of
- freedom than those specified in the Dirichlet boundary condition settings
- using the 
-\family typewriter
-bc_dof
-\family default
- setting.
- Only those listed in 
-\family typewriter
-bc_dof
-\family default
- will be used.
- This permits using the same spatial database file for multiple Dirichlet
- boundary conditions with the same displacement field.
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float table
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\noindent
-\align center
-\begin_inset Caption
-
-\begin_layout Plain Layout
-Values in the spatial databases used for point force boundary conditions.
-\end_layout
-
-\end_inset
-
-
-\begin_inset VSpace defskip
-\end_inset
-
-
-\begin_inset Tabular
-<lyxtabular version="3" rows="4" columns="2">
-<features tabularvalignment="middle">
-<column alignment="center" valignment="top" width="0">
-<column alignment="center" valignment="top" width="4in">
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Spatial database
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Name in Spatial Database
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-db_initial
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-force-x, force-y, force-z
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-db_rate
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-force-rate-x, force-rate-y, force-rate-z, rate-start-time
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-db_change
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-force-x, force-y, force-z, change-start-time
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-\begin_inset CommandInset label
-LatexCommand label
-name "sec:absorbing:boundaries"
-
-\end_inset
-
-Absorbing Boundary Conditions
-\end_layout
-
-\begin_layout Standard
-This boundary condition attempts to prevent seismic waves reflecting off
- of a boundary by placing simple dashpots on the boundary.
- Normally incident dilatational and shear waves are perfectly absorbed.
- Waves incident at other angles are only partially absorbed.
- This boundary condition is simpler than a perfectly matched layer (PML)
- boundary condition but does not perform quite as well, especially for surface
- waves.
- If the waves arriving at the absorbing boundary are relatively small in
- amplitude compared to the amplitudes of primary interest, this boundary
- condition gives reasonable results.
-\end_layout
-
-\begin_layout Standard
-The Absorbing boundary condition properties and components are:
-\end_layout
-
-\begin_layout Description
-label Name of the group of vertices defining the mesh boundary for the absorbing
- boundary condition.
-\end_layout
-
-\begin_layout Description
-up_dir This is a 3-vector that provides a hint for the direction perpendicular
- to the horizontal tangent direction that is not collinear with the direction
- normal to the surface.
- The default value is (0,0,1), which assumes that the z-axis is positive
- upward.
- This vector is only needed for three-dimensional problems where the positive
- upward direction differs from the default.
-\end_layout
-
-\begin_layout Description
-db The spatial database specifying the material properties for the seismic
- velocities.
-\end_layout
-
-\begin_layout Description
-quadrature The quadrature object to be used for numerical integration.
- Since we are integrating over a surface that is one dimension lower than
- the problem domain, this would typically be set to something like 
-\family typewriter
-Quadrature2Din3D
-\family default
- (for a three-dimensional problem).
-\end_layout
-
-\begin_layout Subsection
-Finite-Element Implementation of Absorbing Boundary
-\end_layout
-
-\begin_layout Standard
-Consider a plane wave propagating at a velocity 
-\begin_inset Formula $c$
-\end_inset
-
-.
- We can write the displacement field as
-\begin_inset Formula 
-\begin{equation}
-\vec{u}(\vec{x},t)=\vec{u^{t}}(t-\frac{\vec{x}}{c}),
-\end{equation}
-
-\end_inset
-
-where 
-\begin_inset Formula $\vec{x}$
-\end_inset
-
- is position, 
-\begin_inset Formula $t$
-\end_inset
-
- is time, and 
-\begin_inset Formula $\vec{u^{t}}$
-\end_inset
-
- is the shape of the propagating wave.
- For an absorbing boundary we want the traction on the boundary to be equal
- to the traction associated with wave propagating out of the domain.
- Starting with the expression for the traction on a boundary, 
-\begin_inset Formula $T_{i}=\sigma_{ij}n_{j},$
-\end_inset
-
- and using the local coordinate system for the boundary 
-\begin_inset Formula $s_{h}s_{v}n,$
-\end_inset
-
- where 
-\begin_inset Formula $\vec{n}$
-\end_inset
-
- is the direction normal to the boundary, 
-\begin_inset Formula $\overrightarrow{s}_{h}$
-\end_inset
-
- is the horizontal direction tangent to the boundary, and 
-\begin_inset Formula $\overrightarrow{s}_{v}$
-\end_inset
-
- is the vertical direction tangent to the boundary, the tractions on the
- boundary are
-\begin_inset Formula 
-\begin{gather}
-T_{s_{h}}=\sigma_{s_{h}n}\\
-T_{s_{v}}=\sigma_{s_{v}n}\\
-T_{n}=\sigma_{nn}.
-\end{gather}
-
-\end_inset
-
-In the case of a horizontal boundary, we can define an auxiliary direction
- in order to assign unique tangential directions.
- For a linear elastic isotropic material, 
-\begin_inset Formula $\sigma_{ij}=\lambda\epsilon_{kk}\delta_{ij}+2\mu\epsilon_{ij},$
-\end_inset
-
- and we can write the tractions as 
-\begin_inset Formula 
-\begin{gather}
-T_{s_{h}}=2\mu\epsilon_{s_{h}n}\\
-T_{s_{v}}=2\epsilon_{s_{v}n}\\
-T_{n}=(\lambda+2\mu)\epsilon_{nn}+\lambda(\epsilon_{s_{h}s_{h}}+\epsilon_{s_{v}s_{v}}).
-\end{gather}
-
-\end_inset
-
-For infinitesimal strains, 
-\begin_inset Formula $\epsilon_{ij}=\frac{1}{2}(u_{i,j}+u_{j,i})$
-\end_inset
-
- and we have
-\begin_inset Formula 
-\begin{gather}
-\epsilon_{s_{h}n}=\frac{1}{2}(u_{s_{h},n}+u_{n,s_{h}})\\
-\epsilon_{s_{v}n}=\frac{1}{2}(u_{s_{v},n}+u_{n,s_{v}})\\
-\epsilon_{nn}=u_{n,n}.
-\end{gather}
-
-\end_inset
-
-For our propagating plane wave, we recognize that
-\begin_inset Formula 
-\begin{equation}
-\frac{\partial\vec{u^{t}}(t-\frac{\vec{x}}{c})}{\partial x_{i}}=-\frac{1}{c}\frac{\partial\vec{u^{t}}(t-\frac{\vec{x}}{c})}{\partial t},
-\end{equation}
-
-\end_inset
-
-so that our expressions for the tractions become
-\begin_inset Formula 
-\begin{gather}
-T_{s_{h}}=-\frac{\mu}{c}\left(\frac{\partial u_{s_{h}}^{t}(t-\frac{\vec{x}}{c})}{\partial t}+\frac{\partial u_{n}^{t}(t-\frac{\vec{x}}{c})}{\partial t}\right),\\
-T_{s_{v}}=-\frac{\mu}{c}\left(\frac{\partial u_{s_{v}}^{t}(t-\frac{\vec{x}}{c})}{\partial t}+\frac{\partial u_{n}^{t}(t-\frac{\vec{x}}{c})}{\partial t}\right).
-\end{gather}
-
-\end_inset
-
-For the normal traction, consider a dilatational wave propagating normal
- to the boundary at speed 
-\begin_inset Formula $v_{p}$
-\end_inset
-
-; in this case 
-\begin_inset Formula $u_{s_{h}}=u_{s_{v}}=0$
-\end_inset
-
- and 
-\begin_inset Formula $c=v_{p}$
-\end_inset
-
-.
- For the shear tractions, consider a shear wave propagating normal to the
- boundary at speed 
-\begin_inset Formula $v_{s}$
-\end_inset
-
-; we can decompose this into one case where 
-\begin_inset Formula $u_{n}=u_{s_{v}}=0$
-\end_inset
-
- and another case where 
-\begin_inset Formula $u_{n}=u_{s_{h}}=0$
-\end_inset
-
-, with 
-\begin_inset Formula $c=v_{s}$
-\end_inset
-
- in both cases.
- We also recognize that 
-\begin_inset Formula $\mu=\rho v_{s}^{2}$
-\end_inset
-
- and 
-\begin_inset Formula $\lambda+2\mu=\rho v_{p}^{2}$
-\end_inset
-
-.
- This leads to the following expressions for the tractions:
-\begin_inset Formula 
-\begin{gather}
-T_{s_{h}}=-\rho v_{s}\frac{\partial u_{s_{h}}^{t}(t-\frac{\vec{x}}{c})}{\partial t}\\
-T_{s_{v}}=-\rho v_{s}\frac{\partial u_{v}^{t}(t-\frac{\vec{x}}{c})}{\partial t}\\
-T_{n}=-\rho v_{p}\frac{\partial u_{n}^{t}(t-\frac{\vec{x}}{c})}{\partial t}
-\end{gather}
-
-\end_inset
-
-We write the weak form of the boundary condition as
-\begin_inset Formula 
-\[
-\int_{S_{T}}T_{i}\phi_{i}\, dS=\int_{S_{T}}-\rho c_{i}\frac{\partial u_{i}}{\partial t}\phi_{i}\, dS,
-\]
-
-\end_inset
-
-where 
-\begin_inset Formula $c_{i}$
-\end_inset
-
- equals 
-\begin_inset Formula $v_{p}$
-\end_inset
-
- for the normal traction and 
-\begin_inset Formula $v_{s}$
-\end_inset
-
- for the shear tractions, and 
-\begin_inset Formula $\phi_{i}$
-\end_inset
-
- is our weighting function.
- We express the trial solution and weighting function as linear combinations
- of basis functions,
-\begin_inset Formula 
-\begin{gather}
-u_{i}=\sum_{m}a_{i}^{m}N^{m},\\
-\phi_{i}=\sum_{n}c_{i}^{n}N^{n}.
-\end{gather}
-
-\end_inset
-
-Substituting into our integral over the absorbing boundaries yields
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula 
-\begin{equation}
-\int_{S_{T}}T_{i}\phi_{i}\, dS=\int_{S_{T}}-\rho c_{i}\sum_{m}\dot{a}_{i}^{m}N^{m}\sum_{n}c_{i}^{n}N^{n}\, dS.
-\end{equation}
-
-\end_inset
-
-In the derivation of the governing equations, we recognized that the weighting
- function is arbitrary, so we form the residual by setting the terms associated
- with the coefficients 
-\begin_inset Formula $c_{i}^{n}$
-\end_inset
-
- to zero,
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula 
-\begin{equation}
-r_{i}^{n}=\sum_{\text{tract cells}}\sum_{\text{quad pts}}-\rho(x_{q})c_{i}(x_{q})\sum_{m}\dot{a}_{i}^{m}N^{m}(x_{q})N^{n}(x_{q})w_{q}|J_{cell}(x_{q})|,
-\end{equation}
-
-\end_inset
-
- where 
-\begin_inset Formula $x_{q}$
-\end_inset
-
- are the coordinates of the quadrature points, 
-\begin_inset Formula $w_{q}$
-\end_inset
-
- are the weights of the quadrature points, and 
-\begin_inset Formula $|J_{cell}(x_{q})|$
-\end_inset
-
- is the determinant of the Jacobian matrix evaluated at the quadrature points
- associated with mapping the reference cell to the actual cell.
-\end_layout
-
-\begin_layout Standard
-The appearance of velocity in the expression for the residual means that
- the absorbing dampers also contribute to the system Jacobian matrix.
- Use the central difference method, the velocity is written in terms of
- the displacements,
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula 
-\begin{equation}
-\dot{u}_{i}(t)=\frac{1}{2\Delta t}(u_{i}(t+\Delta t)-u_{i}(t-\Delta t)).
-\end{equation}
-
-\end_inset
-
-Expressing the displacement at time 
-\begin_inset Formula $t+\Delta t$
-\end_inset
-
- in terms of the displacement at time 
-\begin_inset Formula $t$
-\end_inset
-
- (
-\begin_inset Formula $u_{i}(t)$
-\end_inset
-
-) and the increment in the displacement at time 
-\begin_inset Formula $t$
-\end_inset
-
- (
-\begin_inset Formula $du_{i}(t)$
-\end_inset
-
-) leads to
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula 
-\begin{equation}
-\dot{u}_{i}(t)=\frac{1}{2\Delta t}(du_{i}(t)+u_{i}(t)-u_{i}(t-\Delta t))
-\end{equation}
-
-\end_inset
-
-The terms contributing to the system Jacobian are associated with the increment
- in the displacement at time time.
- Substituting into the governing equations and isolating the term associated
- with the increment in the displacement at time t yields
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula 
-\begin{equation}
-A_{ij}^{nm}=\sum_{\text{tract cells}}\sum_{\text{quad pts}}\delta_{ij}\frac{1}{2\Delta t}\rho(x_{q})v_{i}(x_{q})N^{m}(x_{q})N^{n}(x_{q})w_{q}|J_{cells}(x_{q})|,
-\end{equation}
-
-\end_inset
-
-where 
-\begin_inset Formula $A_{ij}^{mn}$
-\end_inset
-
- is a 
-\begin_inset Formula $nd$
-\end_inset
-
- by 
-\begin_inset Formula $md$
-\end_inset
-
- matrix (
-\begin_inset Formula $d$
-\end_inset
-
- is the dimension of the vector space), 
-\begin_inset Formula $m$
-\end_inset
-
- and 
-\begin_inset Formula $n$
-\end_inset
-
- refer to the basis functions and 
-\begin_inset Formula $i$
-\end_inset
-
- and 
-\begin_inset Formula $j$
-\end_inset
-
- are vector space components.
-\end_layout
-
-\begin_layout Section
-\begin_inset CommandInset label
-LatexCommand label
-name "sec:fault"
-
-\end_inset
-
-Fault Interface Conditions
-\end_layout
-
-\begin_layout Standard
-Fault interfaces are used to create dislocations (jumps in the displacement
- field) in the model.
- The dislocations arise from slip across a fault surface.
- Both shear and tensile dislocations are supported.
- For fault interfaces, dislocations in 1D correspond to fault-opening (and
- closing), in 2D lateral-slip and fault opening, and in 3D lateral-slip,
- reverse-slip, and fault opening.
- PyLith supports kinematic (prescribed) slip and dynamic (spontaneous) rupture
- simulations.
-\end_layout
-
-\begin_layout Subsection
-Conventions
-\end_layout
-
-\begin_layout Standard
-Slip corresponds to relative motion across a fault surface.
- Figure 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "fig:fault:orientation"
-
-\end_inset
-
- shows the orientation of the slip vector in 3D with respect to the fault
- surface and coordinate axes.
- PyLith automatically determines the orientation of the fault surface.
- This alleviates the user from having to compute the strike, dip, and rake
- angles over potentially complex, nonplanar fault surfaces.
- Instead, the user specifies fault parameters in terms of lateral motion,
- reverse motion, and fault opening as shown in Figure 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "fig:fault:slip:motions"
-
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float figure
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\align center
-\begin_inset Graphics
-	filename figs/faultOrientation.eps
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Plain Layout
-\begin_inset Caption
-
-\begin_layout Plain Layout
-Orientation of a fault surface in 3D, where 
-\begin_inset Formula $\phi$
-\end_inset
-
- denotes the angle of the fault strike, 
-\begin_inset Formula $\delta$
-\end_inset
-
- denotes the angle of the fault dip, and 
-\begin_inset Formula $\lambda$
-\end_inset
-
- the rake angle.
- 
-\begin_inset CommandInset label
-LatexCommand label
-name "fig:fault:orientation"
-
-\end_inset
-
- 
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float figure
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\align center
-\begin_inset Graphics
-	filename figs/slipmotions.eps
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Plain Layout
-\begin_inset Caption
-
-\begin_layout Plain Layout
-Sign conventions associated with fault slip.
- Positive values are associated with left-lateral, reverse, and fault opening
- motions.
-\begin_inset CommandInset label
-LatexCommand label
-name "fig:fault:slip:motions"
-
-\end_inset
-
- 
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsection
-Fault Implementation
-\end_layout
-
-\begin_layout Standard
-In order to create relative motion across the fault surface in the finite-elemen
-t mesh, additional degrees of freedom are added along with adjustment of
- the topology of the mesh.
- These additional degrees of freedom are associated with cohesive cells.
- These zero-volume cells allow control of the relative motion between vertices
- on the two sides of the fault.
- PyLith automatically adds cohesive cells for each fault surface.
- Figure 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "fig:fault:cohesive:quad4"
-
-\end_inset
-
- illustrates the results of inserting a cohesive cell in a mesh consisting
- of two quadrilateral cells.
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float figure
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\align center
-\begin_inset Graphics
-	filename figs/quad4cohesivekin.eps
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Plain Layout
-\begin_inset Caption
-
-\begin_layout Plain Layout
-Example of cohesive cell inserted into a mesh of two quadrilateral cells.
- The zero thickness cohesive cell (shown with dashed lines) controls slip
- on the fault via the relative motion between vertices 3 and 7 and 2 and
- 6.
-\begin_inset CommandInset label
-LatexCommand label
-name "fig:fault:cohesive:quad4"
-
-\end_inset
-
- 
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsection
-Fault Parameters
-\end_layout
-
-\begin_layout Standard
-The principal parameters for fault interface conditions are:
-\end_layout
-
-\begin_layout Description
-id This is an integer identifier for the fault surface.
- It is used to specify the 
-\family typewriter
-material-id
-\family default
- of the cohesive cells in the mesh.
- Material identifiers must be unique so this value cannot be the same as
- any of the material models or any other fault.
-\end_layout
-
-\begin_layout Description
-name Name or label for the fault.
- This associates sets of vertices with the fault surface in addition to
- being used in error and diagnostic reports.
-\end_layout
-
-\begin_layout Description
-up_dir Up-dir or up direction (used in 2D and 3D simulations).
- In 2D the default in-plane slip is left-lateral, so we use the up-direction
- to resolve the ambiguity in specifying reverse slip.
- In 3D the up-direction is used to resolve the ambiguity in the along-strike
- and dip-dir directions.
- If the fault plane is horizontal, then the up-dir corresponds to the reverse-mo
-tion on the +z side of the fault.
- The only requirement for this direction is that it not be collinear with
- the fault normal direction.
- The default value of [0, 0, 1] is appropriate for most 3D problems.
-\end_layout
-
-\begin_layout Description
-quadrature Quadrature object used in integrating fault quantities.
-\end_layout
-
-\begin_layout Description
-output Manager for output of diagnostic and data fields for the fault.
-\end_layout
-
-\begin_layout Standard
-By default the output manager outputs both diagnostic information (e.g., fault
- normal direction) and the slip at each time step.
- Tables 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "tab:fault:kin:output"
-
-\end_inset
-
- and 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "tab:fault:dyn:output"
-
-\end_inset
-
- list the fields available for output for a fault with kinematic (prescribed)
- earthquake rupture and a fault with dynamic rupture, respectively.
- The fault coordinate system is shown in Figure 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "fig:fault:slip:motions"
-
-\end_inset
-
-.
- The vectors in the fault coordinate system can be transformed to the global
- coordinate system using the direction vectors in the diagnostic output.
- An example of setting these parameters in a 
-\family typewriter
-.cfg
-\family default
- file is:
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.problem]
-\end_layout
-
-\begin_layout LyX-Code
-interfaces = [fault]
-\end_layout
-
-\begin_layout LyX-Code
-
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.problem.interfaces]
-\end_layout
-
-\begin_layout LyX-Code
-fault = pylith.faults.FaultCohesiveKin ; default
-\end_layout
-
-\begin_layout LyX-Code
-
-\end_layout
-
-\begin_layout LyX-Code
-label = fault A ; Group of vertices defining the fault surface
-\end_layout
-
-\begin_layout LyX-Code
-id = 100 ; Value for material identifier associated with fault's cohesive
- cells
-\end_layout
-
-\begin_layout LyX-Code
-up_dir = [0, 0, 1] ; default
-\end_layout
-
-\begin_layout LyX-Code
-quadrature.cell = pylith.feassemble.FIATLagrange
-\end_layout
-
-\begin_layout LyX-Code
-quadrature.cell.dimension = 2
-\end_layout
-
-\begin_layout Standard
-The group of vertices has the label ``fault A.'' We replicate the default
- values for the fault ``up'' direction.
- These settings apply to a 2D fault surface embedded within a 3D mesh, so
- we use 2D Lagrange reference cells.
- The spatial database for elastic properties is used to determine the approximat
-e shear modulus and condition the equations for faster convergence rates.
-\end_layout
-
-\begin_layout Subsection
-Kinematic Earthquake Rupture
-\end_layout
-
-\begin_layout Standard
-Kinematic earthquake ruptures use the FaultCohesiveKin object to specify
- the slip as a function of time on the fault surface.
- Slip may evolve simultaneously over the fault surface instantaneously in
- a single time step (as is usually done in quasi-static simulations) or
- propagate over the fault surface over hundreds and up to thousands of time
- steps (as is usually done in a dynamic simulation).
-\end_layout
-
-\begin_layout Subsubsection
-Governing Equations
-\end_layout
-
-\begin_layout Standard
-The insertion of cohesive cells into the finite-element mesh has the effect
- of decoupling the motion of the two sides of the fault surface.
- In order to impose the desired relative motion, we must adjust the governing
- equations.
- PyLith employs Lagrange multiplier constraints to enforce the constraint
- of the relative motion in the strong sense.
- That is, we enforce the slip across the fault at each degree of freedom.
-\end_layout
-
-\begin_layout Standard
-In conventional implementations the additional degrees of freedom associated
- with the Lagrange multipliers results in a complex implementation.
- However, the use of Lagrange multiplier constraints with cohesive cells
- provides for a simple formulation; we simply add the additional degrees
- of freedom associated with the Lagrange multipliers to the cohesive cells
- as shown in Figure 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "fig:fault:cohesive:quad4"
-
-\end_inset
-
-.
- As a result, the fault implementation is completely confined to the cohesive
- cell.
- Furthermore, the Lagrange multiplier constraints correspond to forces required
- to impose the relative motions, so they are related to the change in stress
- on the fault surface associated with fault slip.
- If we write the algebraic system of equations we are solving in the form
-\begin_inset Formula 
-\begin{equation}
-\underline{A}\overrightarrow{u}=\overrightarrow{b}\,,
-\end{equation}
-
-\end_inset
-
-then including the Lagrange multiplier constraints results in
-\begin_inset Formula 
-\begin{equation}
-\left[\begin{array}{cc}
-\underline{A} & \underline{C}^{T}\\
-\underline{C} & 0
-\end{array}\right]\left[\begin{array}{c}
-\overrightarrow{u}\\
-\overrightarrow{l}
-\end{array}\right]=\left[\begin{array}{c}
-\overrightarrow{b}\\
-\overrightarrow{d}
-\end{array}\right]\,,\label{eq:fault:cohesive:lagrange}
-\end{equation}
-
-\end_inset
-
-where 
-\begin_inset Formula $\overrightarrow{l}$
-\end_inset
-
- is the vector of Lagrange multipliers and 
-\begin_inset Formula $\underline{C}$
-\end_inset
-
- is composed of rotation submatrices, 
-\begin_inset Formula $\underline{R}$
-\end_inset
-
-, associated with the direction cosines relating the relative displacements
- across the fault to the vector of fault slip, 
-\begin_inset Formula $\overrightarrow{d}$
-\end_inset
-
-.
- Note that by using the direction cosines to relate the relative motion
- across the fault, the slip vector and Lagrange multipliers (forces required
- to impose the slip) are in the local fault coordinate system (lateral motion,
- reverse motion, and fault opening).
- 
-\end_layout
-
-\begin_layout Paragraph
-Non-diagonal A
-\end_layout
-
-\begin_layout Standard
-The Lagrange multipliers contribute to both the system Jacobian matrix and
- the residual.
- Because we enforce the constraints in a strong sense, the terms do not
- involve integrals over the fault surface.
- The additional terms in the residual are
-\begin_inset Formula 
-\begin{gather}
-r_{i}^{n}=-C_{ji}^{pn}l_{j}^{p},\\
-r_{i}^{p}=d_{i}^{p}-C_{ij}^{pn}u_{j}^{n},
-\end{gather}
-
-\end_inset
-
-where 
-\begin_inset Formula $n$
-\end_inset
-
- denotes a conventional degree of freedom and 
-\begin_inset Formula $p$
-\end_inset
-
- denotes a degree of freedom associated with a Lagrange multiplier.
- The additional terms in the system Jacobian matrix are simply the direction
- cosines,
-\begin_inset Formula 
-\begin{gather}
-J_{ij}^{np}=C_{ji}^{pn},\\
-J_{ij}^{pn}=C_{ij}^{pn}.
-\end{gather}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Paragraph
-Diagonal A
-\end_layout
-
-\begin_layout Standard
-When we use a lumped system Jacobian matrix, we cannot lump the terms associated
- with the Lagrange multipliers.
- Instead, we formulate the Jacobian ignoring the contributions from the
- Lagrange multipliers, and then adjust the solution after the solve to account
- for their presence.
- Including the Lagrange multipliers in the general expression for the residual
- at time 
-\begin_inset Formula $t+\Delta t$
-\end_inset
-
-, we have
-\begin_inset Formula 
-\begin{equation}
-r_{i}^{n}(t+\Delta t)=A_{ij}^{nm}(u_{j}^{m}(t)+du_{j}^{m}(t))+C_{ki}^{pn}(l_{k}^{p}(t)+dl_{k}^{p}(t)),
-\end{equation}
-
-\end_inset
-
-where we have written the displacements and Lagrange multipliers at time
- 
-\begin_inset Formula $t+\Delta t$
-\end_inset
-
- in terms of the values at time 
-\begin_inset Formula $t$
-\end_inset
-
- and the increment from time 
-\begin_inset Formula $t$
-\end_inset
-
- to 
-\begin_inset Formula $t+\Delta t$
-\end_inset
-
-.
- When we solve the lumped system ignoring the Lagrange multipliers contributions
- to the Jacobian, we formulate the residual assuming the values 
-\begin_inset Formula $du_{i}^{n}$
-\end_inset
-
-(t) and 
-\begin_inset Formula $dl_{k}^{p}(t)$
-\end_inset
-
- are zero.
- So our task is to determine the increment in the Lagrange multiplier, 
-\begin_inset Formula $dl_{k}^{p}$
-\end_inset
-
-, and the correction to the displacement increment, 
-\begin_inset Formula $du_{i}^{n}$
-\end_inset
-
-, and by setting the residual with all terms included to zero; thus, we
- have
-\begin_inset Formula 
-\begin{gather}
-A_{ij}^{nm}(u_{j}^{m}(t)+du_{j}^{m}(t))+C_{ki}^{pn}(l_{k}^{p}(t)+dl_{k}^{p}(t))=0\text{ subject to}\\
-C_{ij}^{pn}(u_{j}^{n}(t)+du_{j}^{n}(t))=d_{i}^{p}.
-\end{gather}
-
-\end_inset
-
-Making use of the residual computed with 
-\begin_inset Formula $du_{i}^{n}(t)=0$
-\end_inset
-
- and 
-\begin_inset Formula $dl_{k}^{p}(t)=0$
-\end_inset
-
-,
-\begin_inset Formula 
-\begin{gather}
-r_{i}^{n}+A_{ij}^{nm}du_{j}^{m}+C_{ki}^{pn}dl_{k}^{p}=0\text{ subject to}\\
-C_{ij}^{pn}(u_{j}^{n}(t)+du_{j}^{n}(t))=d_{i}^{p}.
-\end{gather}
-
-\end_inset
-
-Explicitly writing the equations for the vertices on the negative and positive
- sides of the fault yields
-\begin_inset Formula 
-\begin{gather}
-r_{i}^{n-}+A_{ij}^{nm-}du_{j}^{m-}+R_{ki}^{pn}dl_{k}^{p}=0,\\
-r_{i}^{n+}+A_{ij}^{nm+}du_{j}^{m+}+R_{ki}^{pn}dl_{k}^{p}=0,\\
-R_{ij}^{pn}(u_{j}^{n+}+du_{j}^{n+}-u_{j}^{n-}-du_{j}^{n-})=d_{i}^{p}.
-\end{gather}
-
-\end_inset
-
-Solving the first two equations for 
-\begin_inset Formula $du_{j}^{m-}$
-\end_inset
-
- and 
-\begin_inset Formula $du_{j}^{m+}$
-\end_inset
-
- and combining them using the third equation leads to
-\begin_inset Formula 
-\begin{multline}
-R_{ij}^{pn}\left((A_{ij}^{nm+})^{-1}+(A_{ij}^{nm+})^{-1}\right)R_{ki}^{pn}dl_{k}^{p}=d_{i}^{p}-R_{ij}^{pn}(u_{j}^{n+}-u_{j}^{n-})\\
-+R_{ij}^{pn}\left((A_{ij}^{nm+})^{-1}r_{i}^{n+}-(A_{ij}^{nm-})^{-1}r_{i}^{n-}\right).
-\end{multline}
-
-\end_inset
-
-We do not allow overlap between the fault interface and the absorbing boundary,
- so 
-\begin_inset Formula $A_{ij}^{nm}$
-\end_inset
-
- is the same for all components at a vertex.
- As a result the matrix on the left hand side simplifies to
-\begin_inset Formula 
-\begin{equation}
-S_{ik}^{pn}=\delta_{ik}\left(\frac{1}{A^{nm+}}+\frac{1}{A^{nm-}}\right),
-\end{equation}
-
-\end_inset
-
-and
-\begin_inset Formula 
-\begin{equation}
-dl_{k}^{p}=(S_{ik}^{pn})^{-1}\left(d_{i}^{p}-R_{ij}^{pn}(u_{j}^{n+}-u_{j}^{n-})+R_{ij}^{pn}\left((A_{ij}^{nm+})^{-1}r_{i}^{n+}-(A_{ij}^{nm-})^{-1}r_{i}^{n-}\right)\right).
-\end{equation}
-
-\end_inset
-
-Now that we know the value of the increment in the Lagrange multiplier from
- time 
-\begin_inset Formula $t$
-\end_inset
-
- to time 
-\begin_inset Formula $t+\Delta t$
-\end_inset
-
-, we can correct the value for the displacement increment from time 
-\begin_inset Formula $t$
-\end_inset
-
- to 
-\begin_inset Formula $t+\Delta t$
-\end_inset
-
- using
-\begin_inset Formula 
-\begin{gather}
-\Delta du_{j}^{n-}=(A_{ij}^{nm-})^{-1}C_{ki}^{pn}dl_{k}^{p}\text{ and}\\
-\Delta du_{j}^{n+}=-(A_{ij}^{nm+})^{-1}C_{ki}^{pn}dl_{k}^{p}.
-\end{gather}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsubsection
-Arrays of Kinematic Rupture Components
-\end_layout
-
-\begin_layout Standard
-Multiple earthquake ruptures can be specified on a single fault surface.
- This permits repeatedly rupturing the same portion of a fault or combining
- earthquake rupture on one subset of the fault surface with steady aseismic
- slip on another subset (the two subsets may overlap in both time and space).
- A dynamic array of kinematic earthquake rupture components associates a
- name (string) with each kinematic rupture.
- The default dynamic array contains a single earthquake rupture, ``rupture''.
- The 
-\family typewriter
-eq_srcs
-\family default
- is the 
-\family typewriter
-FaultCohesiveKin
-\family default
- facility for this dynamic array.
- An example of setting the array of kinematic rupture components in a 
-\family typewriter
-.cfg
-\family default
- file:
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.problem.interfaces.fault]
-\end_layout
-
-\begin_layout LyX-Code
-eq_srcs = [earthquake,creep]
-\end_layout
-
-\begin_layout Standard
-The output manager includes generic fault information (orientation) as well
- as the final slip or slip rate (as in the case of the constant slip rate
- slip time function) and slip initiation time for each kinematic rupture.
- The name of the slip and slip initiation time vertex fields are of the
- form 
-\family typewriter
-final_slip_NAME
-\family default
- and 
-\family typewriter
-slip_time_NAME
-\family default
-, respectively, where 
-\family typewriter
-NAME
-\family default
- refers to the name used in the dynamic array of kinematic ruptures, 
-\family typewriter
-eq_srcs
-\family default
-.
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float table
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\noindent
-\align center
-\begin_inset Caption
-
-\begin_layout Plain Layout
-\begin_inset CommandInset label
-LatexCommand label
-name "tab:fault:kin:output"
-
-\end_inset
-
-Fields available in output of fault information.
-\end_layout
-
-\end_inset
-
-
-\begin_inset VSpace defskip
-\end_inset
-
-
-\begin_inset Tabular
-<lyxtabular version="3" rows="8" columns="3">
-<features tabularvalignment="middle">
-<column alignment="center" valignment="top" width="0">
-<column alignment="center" valignment="top" width="0">
-<column alignment="center" valignment="top" width="3.5in">
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Field Type
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Field
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Description
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-vertex_info_fields
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-normal_dir
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Direction of fault normal in global coordinate system
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-strike_dir
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Direction of fault strike in global coordinate system
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-dip_dir
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Up-dip direction on hanging wall in global coordinate system
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-final_slip_NAME
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Vector of final slip (in fault coordinate system) in meters
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-slip_time
-\family default
-_
-\family typewriter
-\noun on
-NAME
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" bottomline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Time at which slip begins in seconds
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-vertex_data_fields
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-slip
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Slip vector at time step (in fault coordinate system) in meters
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-traction_change
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" bottomline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Change in fault tractions (in fault coordinate system) in Pa
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsubsection
-Kinematic Rupture Parameters
-\end_layout
-
-\begin_layout Standard
-The kinematic rupture parameters include the origin time and slip time function.
- The slip initiation time in the slip time function is relative to the origin
- time (default is 0).
- This means that slip initiates at a point at a time corresponding to the
- sum of the kinematic rupture's origin time and the slip initiation time
- for that point.
- An example of specifying the kinematic earthquake rupture properties and
- components in a 
-\family typewriter
-.cfg
-\family default
- file:
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.problem.interfaces.fault]
-\end_layout
-
-\begin_layout LyX-Code
-eq_srcs = [earthquake,creep]
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.problem.interfaces.fault.eq_srcs.earthquake]
-\end_layout
-
-\begin_layout LyX-Code
-origin_time = 0.0*s ; default origin time
-\end_layout
-
-\begin_layout LyX-Code
-slip_function = pylith.faults.StepSlipFn ; default slip time function
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.problem.interfaces.fault.eq_srcs.creep]
-\end_layout
-
-\begin_layout LyX-Code
-origin_time = 10.0*year ; start creep at 10.0 years
-\end_layout
-
-\begin_layout LyX-Code
-slip_function = pylith.faults.ConstRateSlipFn ; switch to constant slip rate
- slip function
-\end_layout
-
-\begin_layout Subsubsection
-Slip Time Function
-\end_layout
-
-\begin_layout Standard
-The current release of PyLith supports specification of the evolution of
- fault slip using analytical expressions for the slip time history at each
- point, where the parameters for the slip time function may vary over the
- fault surface.
- Currently, three slip time functions are available: (1) a step-function
- for quasi-static modeling of earthquake rupture, (2) a constant slip rate
- time function for modeling steady aseismic slip, and (3) the integral of
- Brune's far-field time function 
-\begin_inset CommandInset citation
-LatexCommand cite
-key "Brune:1970"
-
-\end_inset
-
- for modeling the dynamics of earthquake rupture.
- Additional slip time functions will likely be available in future releases.
- The default slip time function is the step-function slip function.
-\end_layout
-
-\begin_layout Paragraph
-Step-Function Slip Time Function
-\end_layout
-
-\begin_layout Standard
-This slip function prescribes a step in slip at a given time at a point:
- 
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula 
-\begin{gather}
-D(t)=\left\{ \begin{array}{cc}
-0 & 0\leq t<t_{r}\\
-D_{final} & t\ge t_{r}
-\end{array}\right.\,,
-\end{gather}
-
-\end_inset
-
-where 
-\begin_inset Formula $D(t)$
-\end_inset
-
- is slip at time 
-\begin_inset Formula $t$
-\end_inset
-
-, 
-\begin_inset Formula $D_{final}$
-\end_inset
-
- is the final slip, and 
-\begin_inset Formula $t_{r}$
-\end_inset
-
- is the slip initiation time (time when rupture reaches the location).
- The slip is specified independently for each of the components of slip,
- and the slip and slip starting time may vary over the fault surface.
-\end_layout
-
-\begin_layout Description
-final_slip Spatial database of slip (
-\begin_inset Formula $D_{final})$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Description
-slip_time Spatial database of slip initiation times (
-\begin_inset Formula $t_{r}$
-\end_inset
-
-).
-\end_layout
-
-\begin_layout Standard
-An example of setting these parameters in a 
-\family typewriter
-.cfg
-\family default
- file is:
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.problem.interfaces.fault.eq_srcs.rupture]
-\end_layout
-
-\begin_layout LyX-Code
-slip_function = pylith.faults.StepSlipFn
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.problem.interfaces.fault.eq_srcs.rupture.slip_function]
-\end_layout
-
-\begin_layout LyX-Code
-final_slip.iohandler.filename = final_slip.spatialdb
-\end_layout
-
-\begin_layout LyX-Code
-slip_time.iohandler.filename = sliptime.spatialdb
-\end_layout
-
-\begin_layout Standard
-The spatial database files for the slip time function specify the spatial
- variation in the parameters for the slip time function, as shown in Table
- 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "tab:step-function-db-params"
-
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float table
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\noindent
-\align center
-\begin_inset Caption
-
-\begin_layout Plain Layout
-\begin_inset CommandInset label
-LatexCommand label
-name "tab:step-function-db-params"
-
-\end_inset
-
-Values in spatial database used as parameters in the step function slip
- time function.
-\end_layout
-
-\end_inset
-
-
-\begin_inset VSpace defskip
-\end_inset
-
-
-\begin_inset Tabular
-<lyxtabular version="3" rows="5" columns="3">
-<features tabularvalignment="middle">
-<column alignment="left" valignment="top" width="0">
-<column alignment="left" valignment="top" width="0">
-<column alignment="left" valignment="top" width="2.5in">
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Spatial database
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Value
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Description
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-final_slip
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-left-lateral-slip
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Amount of left-lateral final slip in meters.
- Use negative values for right-lateral slip.
- Applies to faults in 2D and 3D only.
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-reverse-slip
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Amount of reverse slip in meters.
- Use negative values for normal slip.
- Applies to faults in 3D only.
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-fault-opening
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Amount of fault opening in meters.
- Negative values imply penetration.
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-slip_time
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-slip-time
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Slip initiation time (
-\begin_inset Formula $t_{t})$
-\end_inset
-
- in seconds.
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Paragraph
-Constant Slip Rate Slip Time Function
-\end_layout
-
-\begin_layout Standard
-This slip function prescribes a constant slip rate for the evolution of
- slip at a point: 
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula 
-\begin{gather}
-D(t)=\left\{ \begin{array}{cc}
-0 & 0\leq t<t_{r}\\
-V(t-t_{r}) & t\ge t_{r}
-\end{array}\right.\,,
-\end{gather}
-
-\end_inset
-
-where 
-\begin_inset Formula $D(t)$
-\end_inset
-
- is slip at time 
-\begin_inset Formula $t$
-\end_inset
-
-, 
-\begin_inset Formula $V$
-\end_inset
-
- is the slip rate, and 
-\begin_inset Formula $t_{r}$
-\end_inset
-
- is the slip initiation time (time when rupture reaches the location).
- The slip rate is specified independently for each of the components of
- slip, and the slip rate and slip starting time may vary over the fault
- surface.
-\end_layout
-
-\begin_layout Description
-slip_rate Spatial database of slip rate (
-\begin_inset Formula $V)$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Description
-slip_time Spatial database of slip initiation times (
-\begin_inset Formula $t_{r}$
-\end_inset
-
-).
-\end_layout
-
-\begin_layout Standard
-An example of setting these parameters in a 
-\family typewriter
-.cfg
-\family default
- file is:
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.problem.interfaces.fault.eq_srcs.ruptures]
-\end_layout
-
-\begin_layout LyX-Code
-slip_function = pylith.faults.ConstRateSlipFn
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.problem.interfaces.fault.eq_srcs.ruptures.slip_function]
-\end_layout
-
-\begin_layout LyX-Code
-slip_rate.iohandler.filename = slip_rate.spatialdb
-\end_layout
-
-\begin_layout LyX-Code
-slip_time.iohandler.filename = sliptime.spatialdb
-\end_layout
-
-\begin_layout Standard
-The spatial database files for the slip time function specify the spatial
- variation in the parameters for the slip time function, as shown in Table
- 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "tab:const-slip-rate-db-params"
-
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float table
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\noindent
-\align center
-\begin_inset Caption
-
-\begin_layout Plain Layout
-\begin_inset CommandInset label
-LatexCommand label
-name "tab:const-slip-rate-db-params"
-
-\end_inset
-
-Values in spatial database used as parameters in the constant slip rate
- slip time function.
-\end_layout
-
-\end_inset
-
-
-\begin_inset VSpace defskip
-\end_inset
-
-
-\begin_inset Tabular
-<lyxtabular version="3" rows="5" columns="3">
-<features tabularvalignment="middle">
-<column alignment="left" valignment="top" width="0">
-<column alignment="left" valignment="top" width="0">
-<column alignment="left" valignment="top" width="2.5in">
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Spatial database
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Value
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Description
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-slip_rate
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-left-lateral-slip
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Slip rate for left-lateral final slip in meters per second.
- Use negative values for right-lateral slip.
- Applies to faults in 2D and 3D only.
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-reverse-slip
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Slip rate for reverse slip in meters per second.
- Use negative values for normal slip.
- Applies to faults in 3D only.
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-fault-opening
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Slip rate for fault opening in meters per second.
- Negative values imply penetration.
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-slip_time
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-slip-time
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Slip initiation time (
-\begin_inset Formula $t_{t})$
-\end_inset
-
- in seconds.
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Paragraph
-Brune Slip Time Function
-\end_layout
-
-\begin_layout Standard
-We use an integral of Brune's far-field time function 
-\begin_inset CommandInset citation
-LatexCommand cite
-key "Brune:1970"
-
-\end_inset
-
- to describe the evolution in time of slip at a point: 
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula 
-\begin{gather}
-D(t)=\left\{ \begin{array}{cc}
-0 & 0\leq t<t_{r}\\
-D_{final}\left(1-exp\left(-\frac{t-t_{r}}{t_{0}}\right)\left(1+\frac{t-t_{r}}{t_{0}}\right)\right) & t\ge t_{r}
-\end{array}\right.\,,\\
-t_{0}=0.6195t_{\mathit{rise}}\,,
-\end{gather}
-
-\end_inset
-
-where 
-\begin_inset Formula $D(t)$
-\end_inset
-
- is slip at time 
-\begin_inset Formula $t$
-\end_inset
-
-, 
-\begin_inset Formula $D_{final}$
-\end_inset
-
- is the final slip at the location, 
-\begin_inset Formula $t_{r}$
-\end_inset
-
- is the slip initiation time (time when rupture reaches the location), and
- 
-\begin_inset Formula $t_{\mathit{rise}}$
-\end_inset
-
- is the rise time.
-\end_layout
-
-\begin_layout Description
-slip Spatial database of final slip distribution (
-\begin_inset Formula $D_{final})$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Description
-slip_time Spatial database of slip initiation times (
-\begin_inset Formula $t_{r}$
-\end_inset
-
-).
-\end_layout
-
-\begin_layout Description
-rise_time Spatial database for rise time (
-\begin_inset Formula $t_{\mathit{rise}}$
-\end_inset
-
-).
-\end_layout
-
-\begin_layout Standard
-An example of setting these parameters in a 
-\family typewriter
-.cfg
-\family default
- file is:
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.problem.interfaces.fault.eq_srcs.ruptures]
-\end_layout
-
-\begin_layout LyX-Code
-slip_function = pylith.faults.BruneSlipFn
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.problem.interfaces.fault.eq_srcs.rupture.slip_function]
-\end_layout
-
-\begin_layout LyX-Code
-slip.iohandler.filename = finalslip.spatialdb
-\end_layout
-
-\begin_layout LyX-Code
-rise_time.iohandler.filename = risetime.spatialdb
-\end_layout
-
-\begin_layout LyX-Code
-slip_time.iohandler.filename = sliptime.spatialdb
-\end_layout
-
-\begin_layout Standard
-The spatial database files for the slip time function specify the spatial
- variation in the parameters for the slip time function, as shown in Table
- 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "tab:Brune-slip-db-params"
-
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float table
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\noindent
-\align center
-\begin_inset Caption
-
-\begin_layout Plain Layout
-\begin_inset CommandInset label
-LatexCommand label
-name "tab:Brune-slip-db-params"
-
-\end_inset
-
-Values in spatial database used as parameters in the Brune slip time function.
-\end_layout
-
-\end_inset
-
-
-\begin_inset VSpace defskip
-\end_inset
-
-
-\begin_inset Tabular
-<lyxtabular version="3" rows="6" columns="3">
-<features tabularvalignment="middle">
-<column alignment="left" valignment="top" width="0">
-<column alignment="left" valignment="top" width="0">
-<column alignment="left" valignment="top" width="2.5in">
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Spatial database
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Value
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Description
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-slip
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-left-lateral-slip
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Amount of left-lateral final slip in meters.
- Use negative values for right-lateral slip.
- Applies to faults in 2D and 3D only.
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-reverse-slip
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Amount of reverse slip in meters.
- Use negative values for normal slip.
- Applies to faults in 3D only.
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-fault-opening
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Amount of fault opening in meters.
- Negative values imply penetration.
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-rise_time
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-rise-time
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Rise time (
-\begin_inset Formula $t_{r})$
-\end_inset
-
- in seconds.
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-slip_time
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-slip-time
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Slip initiation time (
-\begin_inset Formula $t_{t})$
-\end_inset
-
- in meters.
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Paragraph
-Liu-Cosine Slip Time Function
-\end_layout
-
-\begin_layout Standard
-This slip time function, proposed by Liu, Archuleta, and Hartzell for use
- in ground-motion modeling
-\begin_inset CommandInset citation
-LatexCommand cite
-key "Liu:etal:2006"
-
-\end_inset
-
-, combines several cosine and sine functions together to create a slip time
- history with a sharp rise and gradual termination with a finite duration
- of slip.
- The evolution of slip at a point follows: 
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula 
-\begin{gather}
-D(t)=\left\{ \begin{array}{cc}
-D_{\mathit{final}}C_{n}\left(0.7t-0.7\frac{t_{1}}{\pi}\sin\frac{\pi t}{t_{1}}-1.2\frac{t_{1}}{\pi}\left(\cos\frac{\pi t}{2t_{1}}-1\right)\right) & 0\leq t<t_{1}\\
-D_{\mathit{final}}C_{n}\left(1.0t-0.7\frac{t1}{\pi}\sin\frac{\pi t}{t_{1}}+0.3\frac{t2}{\pi}\sin\frac{\pi(t-t1)}{t_{2}}+\frac{1.2}{\pi}t_{1}-0.3t_{1}\right) & t_{1}\leq t<2t_{1}\\
-D_{\mathit{final}}C_{n}\left(0.7-0.7\cos\frac{\pi t}{t_{1}}+0.6\sin\frac{\pi t}{2t_{1}}\right) & 2t_{1}\leq t\leq t_{0}
-\end{array}\right.\,,\\
-C_{n}=\frac{\pi}{1.4\pi t_{1}+1.2t_{1}+0.3\pi t_{2}},\\
-t_{0}=1.525t_{\mathit{rise}},\\
-t_{1}=0.13t_{0},\\
-t_{2}=t_{0}-t_{1},
-\end{gather}
-
-\end_inset
-
-where 
-\begin_inset Formula $D(t)$
-\end_inset
-
- is slip at time 
-\begin_inset Formula $t$
-\end_inset
-
-, 
-\begin_inset Formula $D_{final}$
-\end_inset
-
- is the final slip at the location, 
-\begin_inset Formula $t_{r}$
-\end_inset
-
- is the slip initiation time (time when rupture reaches the location), and
- 
-\begin_inset Formula $t_{\mathit{rise}}$
-\end_inset
-
- is the rise time.
-\end_layout
-
-\begin_layout Description
-slip Spatial database of final slip distribution (
-\begin_inset Formula $D_{final})$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Description
-slip_time Spatial database of slip initiation times (
-\begin_inset Formula $t_{r}$
-\end_inset
-
-).
-\end_layout
-
-\begin_layout Description
-rise_time Spatial database for rise time (
-\begin_inset Formula $t_{\mathit{rise}}$
-\end_inset
-
-).
-\end_layout
-
-\begin_layout Standard
-The spatial database files for the slip time function use the same parameters
- for the slip time function as the Brune slip time function shown in Table
- 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "tab:Brune-slip-db-params"
-
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Paragraph
-Time-History Slip Time Function
-\end_layout
-
-\begin_layout Standard
-This slip time function reads the slip time function from a data file, so
- it can have an arbitrary shape.
- The slip and slip initiation times are specified using spatial databases,
- so the slip time function, in general, will use a normalized amplitude.
-\end_layout
-
-\begin_layout Description
-slip Spatial database of final slip distribution (
-\begin_inset Formula $D_{final})$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Description
-slip_time Spatial database of slip initiation times (
-\begin_inset Formula $t_{r}$
-\end_inset
-
-).
-\end_layout
-
-\begin_layout Description
-time_history Temporal database for slip evolution.
-\end_layout
-
-\begin_layout Standard
-An example of setting these parameters in a 
-\family typewriter
-.cfg
-\family default
- file is:
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.problem.interfaces.fault.eq_srcs.ruptures]
-\end_layout
-
-\begin_layout LyX-Code
-slip_function = pylith.faults.TimeHistorySlipFn
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.problem.interfaces.fault.eq_srcs.rupture.slip_function]
-\end_layout
-
-\begin_layout LyX-Code
-slip.iohandler.filename = finalslip.spatialdb
-\end_layout
-
-\begin_layout LyX-Code
-slip_time.iohandler.filename = sliptime.spatialdb
-\end_layout
-
-\begin_layout LyX-Code
-time_history.iohandler.filename = myfunction.timedb
-\end_layout
-
-\begin_layout Standard
-The spatial database files for the slip time function specify the spatial
- variation in the parameters for the slip time function, as shown in Table
- 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "tab:Brune-slip-db-params-2"
-
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float table
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\noindent
-\align center
-\begin_inset Caption
-
-\begin_layout Plain Layout
-\begin_inset CommandInset label
-LatexCommand label
-name "tab:Brune-slip-db-params-2"
-
-\end_inset
-
-Values in spatial database used as parameters in the time history slip time
- function.
-\end_layout
-
-\end_inset
-
-
-\begin_inset VSpace defskip
-\end_inset
-
-
-\begin_inset Tabular
-<lyxtabular version="3" rows="6" columns="3">
-<features tabularvalignment="middle">
-<column alignment="left" valignment="top" width="0">
-<column alignment="left" valignment="top" width="0">
-<column alignment="left" valignment="top" width="2.5in">
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Spatial database
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Value
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Description
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-slip
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-left-lateral-slip
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Amount of left-lateral final slip in meters.
- Use negative values for right-lateral slip.
- Applies to faults in 2D and 3D only.
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-reverse-slip
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Amount of reverse slip in meters.
- Use negative values for normal slip.
- Applies to faults in 3D only.
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-fault-opening
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Amount of fault opening in meters.
- Negative values imply penetration.
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-rise_time
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-rise-time
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Rise time (
-\begin_inset Formula $t_{r})$
-\end_inset
-
- in seconds.
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-slip_time
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-slip-time
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Slip initiation time (
-\begin_inset Formula $t_{t})$
-\end_inset
-
- in meters.
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsection
-Dynamic Earthquake Rupture
-\end_layout
-
-\begin_layout Standard
-Dynamic fault interfaces use the FaultCohesiveDyn object to specify a fault
- constitutive model to govern the fault tractions (friction) and the resulting
- slip.
- When friction is large enough such that there is no sliding on the fault,
- the fault is locked (slip is zero) and the Lagrange multipliers assume
- their values just as they do in kinematic ruptures.
- In this case, the Lagrange multipliers correspond to the forces necessary
- to keep the slip zero.
- When the driving forces exceed those allowed by friction, we reduce the
- values of the Lagrange multipliers to those consistent with friction from
- the fault constitutive model.
- When we reduce the Lagrange multipliers, we must increment the slip accordingly
- to maintain consistency in the algebraic system of equations.
-\end_layout
-
-\begin_layout Subsubsection
-Governing Equations
-\end_layout
-
-\begin_layout Standard
-The algebraic systems of equations for dynamic earthquake rupture are the
- same as those for kinematic rupture
-\begin_inset Formula 
-\begin{equation}
-\left[\begin{array}{cc}
-\underline{A} & \underline{C}^{T}\\
-\underline{C} & 0
-\end{array}\right]\left[\begin{array}{c}
-\overrightarrow{u}\\
-\overrightarrow{l}
-\end{array}\right]=\left[\begin{array}{c}
-\overrightarrow{b}\\
-\overrightarrow{d}
-\end{array}\right].
-\end{equation}
-
-\end_inset
-
-Enforcing the limits imposed on the Lagrange multipliers by the fault constituti
-ve model requires determining the increment in slip for an increment in
- the Lagrange multipliers.
- The increment in the Lagrange multipliers is the difference between the
- value computed for the current slip (either zero or the slip at the previous
- time step) and the value computed from the fault constitutive model.
- Starting from our system of algebraic equations,
-\begin_inset Formula 
-\begin{equation}
-A_{ij}^{nm}u_{j}^{m}+C_{ji}^{pn}l_{j}^{p}=b_{i}^{n},
-\end{equation}
-
-\end_inset
-
-we compute the sensitivity for the given loading and boundary conditions,
-\begin_inset Formula 
-\begin{equation}
-A_{ij}^{nm}\partial u_{j}^{m}=-C_{ji}^{pn}\partial l_{j}^{p}.
-\end{equation}
-
-\end_inset
-
-Computing the increment in the slip requires computing the increment in
- the displacements.
- Solving this equation rigorously would require inverting the system Jacobian,
- which we do not want to do unless it is diagonal (as it is in the case
- of the lumped formulations).
- 
-\end_layout
-
-\begin_layout Paragraph
-Non-diagonal A
-\end_layout
-
-\begin_layout Standard
-In general A is a sparse matrix with off-diagonal terms of the form
-\begin_inset Formula 
-\begin{equation}
-A=\left(\begin{array}{ccc}
-A_{0} & A_{1} & A_{2}\\
-A_{3} & A_{n-} & 0\\
-A_{4} & 0 & A_{n+}
-\end{array}\right),
-\end{equation}
-
-\end_inset
-
-where the degrees of freedom on either side of the fault are uncoupled.
- We formulate two small linear systems involving just the degrees of freedom
- associated with vertices on either the positive or negative sides of the
- fault,
-\begin_inset Formula 
-\begin{gather}
-A_{ij}^{nm-}\partial u_{j}^{m-}=-R_{ij}^{pn}\partial l_{j}^{p},\\
-A_{ij}^{nm+}\partial u_{j}^{m+}=R_{ij}^{pn}\partial l_{j}^{p},
-\end{gather}
-
-\end_inset
-
-where we have replaced 
-\begin_inset Formula $\underline{C}$
-\end_inset
-
- with 
-\begin_inset Formula $\underline{R}$
-\end_inset
-
- to denote the explicit inclusion of the signs for the terms in 
-\begin_inset Formula $\underline{C}$
-\end_inset
-
- associated with the positive (
-\begin_inset Formula $n^{+}$
-\end_inset
-
-) and negative (
-\begin_inset Formula $n^{-}$
-\end_inset
-
-) sides of the fault.
- After solving these two linear systems of equations, we compute the increment
- in slip using
-\begin_inset Formula 
-\begin{equation}
-\partial d_{i}^{p}=R_{ij}^{pn}(\partial u_{j}^{n+}-\partial u_{j}^{n-}).
-\end{equation}
-
-\end_inset
-
-The solution of these two linear systems gives the increment in slip assuming
- all the degrees of freedom except those immediately adjacent to the fault
- remain fixed.
- In real applications where the deformation associated with fault slip is
- localized around the fault, this provides good enough approximations so
- that the nonlinear solver converges quickly.
- In problems where deformation associated with slip on the fault is not
- localized (as in the case in some of the example problems), the increment
- in slip computed by solving these two linear systems is not a good approximatio
-n and the nonlinear solve requires a large number of iterations.
-\end_layout
-
-\begin_layout Standard
-We use the PETSc Krylov subspace solver (KSP) to solve these two linear
- systems.
- The PETSc settings for the KSP object are set in the same manner as the
- main solver, except we use the prefix friction_ in all of the settings
- related to the KSP solver for these two linear systems.
- For example, to use the recommended additive Schwarz preconditioner in
- the friction sensitivity solves, the settings in a .cfg file are:
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.petsc]
-\end_layout
-
-\begin_layout LyX-Code
-friction_pc_type = asm
-\end_layout
-
-\begin_layout Standard
-See the examples in Sections 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "sec:Tutorial-3d-hex8-friction"
-
-\end_inset
-
- and 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "sec:tutorial:shearwave:quad4"
-
-\end_inset
-
- for details.
-\end_layout
-
-\begin_layout Paragraph
-Diagonal A
-\end_layout
-
-\begin_layout Standard
-With a lumped Jacobian matrix, we can solve for the increment in slip directly,
-\begin_inset Formula 
-\begin{equation}
-\partial d_{i}^{p}=-C_{ij}^{pn}(A_{jk}^{nm})^{-1}C_{lk}^{pm}\partial l_{l}^{p}.
-\end{equation}
-
-\end_inset
-
-By not allowing the fault interface to overlap with the absorbing boundary,
- the terms in 
-\begin_inset Formula $A$
-\end_inset
-
- for a given vertex are identical and the expression on the right-hand side
- reduces to
-\begin_inset Formula 
-\begin{equation}
-\partial d_{i}^{p}=-\left(\frac{1}{A^{n+}}+\frac{1}{A^{n-}}\right)\partial l_{i}^{p}.
-\end{equation}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsubsection
-Dynamic Rupture Parameters
-\end_layout
-
-\begin_layout Standard
-The components of the FaultCohesiveDyn object include
-\end_layout
-
-\begin_layout Description
-open_free_surface If true, enforce traction free surface when the fault
- opens, otherwise apply prescribed tractions even when the fault opens (default
- is true); to mimic a dike opening use false.
-\end_layout
-
-\begin_layout Description
-zero_tolerance Tolerance for detecting zero values (default is 1.0e-10);
- should be larger than absolute tolerance in KSP solves.
-\end_layout
-
-\begin_layout Description
-traction_perturbation Prescribed tractions on fault surface (generally used
- for nucleating earthquake ruptures, default is none).
-\end_layout
-
-\begin_layout Description
-friction Fault constitutive model.
-\end_layout
-
-\begin_layout Standard
-An example of specifying the dynamic earthquake rupture properties and component
-s in a 
-\family typewriter
-.cfg
-\family default
- file:
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.problem.interfaces.fault]
-\end_layout
-
-\begin_layout LyX-Code
-open_free_surface = True ; default
-\end_layout
-
-\begin_layout LyX-Code
-traction_perturbation = pylith.faults.TractPerturbation ; not default
-\end_layout
-
-\begin_layout LyX-Code
-traction_perturbation.db_initial = spatialdata.spatialdb.SimpleDB
-\end_layout
-
-\begin_layout LyX-Code
-traction_perturbation.db_initial.iohandler.filename = tractions.spatialdb
-\end_layout
-
-\begin_layout LyX-Code
-friction = pylith.friction.StaticFriction
-\end_layout
-
-\begin_layout LyX-Code
-friction.db_properties = spatialdata.spatialdb.SimpleDB
-\end_layout
-
-\begin_layout LyX-Code
-friction.db_properties.iohandler.filename = friction.spatialdb
-\end_layout
-
-\begin_layout Standard
-The prescribed traction perturbation is specified using the same fault coordinat
-e system as the slip directions in the kinematic ruptures.
- The perurbation has the same functional form as the time-dependent boundary
- conditions (and same spatial databases).
- Table 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "tab:fault:cohesive:dyn:prescribed:tractions"
-
-\end_inset
-
- gives the values in the spatial database for the prescribed tractions.
- Table 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "tab:fault:dyn:output"
-
-\end_inset
-
- shows the fields available for output.
- Additional fields are available depending on the fault constitutive model.
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float table
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\noindent
-\align center
-\begin_inset Caption
-
-\begin_layout Plain Layout
-\begin_inset CommandInset label
-LatexCommand label
-name "tab:fault:cohesive:dyn:prescribed:tractions"
-
-\end_inset
-
-Values in spatial databases for prescribed tractions.
-\end_layout
-
-\end_inset
-
-
-\begin_inset VSpace defskip
-\end_inset
-
-
-\begin_inset Tabular
-<lyxtabular version="3" rows="22" columns="4">
-<features tabularvalignment="middle">
-<column alignment="left" valignment="top" width="0">
-<column alignment="left" valignment="top" width="0">
-<column alignment="left" valignment="top" width="0">
-<column alignment="left" valignment="top" width="2.5in">
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Spatial database
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Dimension
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Value
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Description
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-db_initial
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-1-D
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-traction-normal
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Normal traction (tension is positive)
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-2-D
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-traction-shear
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Left-lateral shear traction (reverse shear for dipping faults)
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-traction-normal
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Normal traction (tension is positive)
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-3-D
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-traction-shear-leftlateral
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Left-lateral shear traction
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-traction-shear-updip
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Reverse shear traction
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-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsubsection
-Fault Constitutive Models
-\begin_inset CommandInset label
-LatexCommand label
-name "sec:fault:constitutive:models"
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-PyLith provides four fault constitutive models.
- Future releases may contain additional models, and a template is provided
- for you to construct your own (see Section 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "sec:Extending:FaultConstitutiveModels"
-
-\end_inset
-
-).
- The fault constitutive model implementations are independent of dimension
- and work in both 2-D and 3-D.
- In solving the governing equations, PyLith will use a scalar representation
- of the shear traction in 2-D and a vector representation of the shear traction
- in 3-D, with the shear traction resolved in the direction of current slip.
- The fault constitutive models contain a common set of properties and components
-:
-\end_layout
-
-\begin_layout Description
-label Name of the friction model.
-\end_layout
-
-\begin_layout Description
-db_properties Spatial database of the friction model parameters (default
- is SimpleDB).
-\end_layout
-
-\begin_layout Description
-db_initial_state Spatial database for initial state variables (default is
- none).
-\end_layout
-
-\begin_layout Paragraph
-Static Friction
-\end_layout
-
-\begin_layout Standard
-The static friction model produces shear tractions proportional to the fault
- normal traction plus a cohesive stress,
-\begin_inset Formula 
-\begin{equation}
-T_{f}=\begin{cases}
-T_{c}-\mu_{f}T_{n} & T_{n}\leq0\\
-0 & T_{n}>0
-\end{cases}.
-\end{equation}
-
-\end_inset
-
-The spatial database file for the static friction model properties specifies
- the spatial variation of the parameters given in Table 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "tab:static:friction:properties"
-
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float table
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\noindent
-\align center
-\begin_inset Caption
-
-\begin_layout Plain Layout
-\begin_inset CommandInset label
-LatexCommand label
-name "tab:static:friction:properties"
-
-\end_inset
-
-Values in the spatial database for constant friction parameters.
-\end_layout
-
-\end_inset
-
-
-\begin_inset VSpace defskip
-\end_inset
-
-
-\begin_inset Tabular
-<lyxtabular version="3" rows="3" columns="2">
-<features tabularvalignment="middle">
-<column alignment="left" valignment="top" width="0">
-<column alignment="left" valignment="top" width="2.5in">
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Value
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Description
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-friction-coefficient
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Coefficient of friction, 
-\begin_inset Formula $\mu_{f}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-cohesion
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Cohesive stress, 
-\begin_inset Formula $T_{c}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Paragraph
-\begin_inset CommandInset label
-LatexCommand label
-name "sec:friction:slip:weakening"
-
-\end_inset
-
-Slip-Weakening Friction
-\end_layout
-
-\begin_layout Standard
-The linear slip-weakening friction model produces shear tractions equal
- to the cohesive stress plus a contribution proportional to the fault normal
- traction that decreases from a static value to a dynamic value as slip
- progresses,
-\begin_inset Formula 
-\begin{equation}
-T_{f}=\begin{cases}
-T_{c}-(\mu_{s}-(\mu_{s}-\mu_{d})\frac{d}{d_{0}})T_{n} & d\leq d_{0}\text{ and }T_{n}\leq0\\
-T_{c}-\mu_{d}T_{n} & d>d_{0}\text{ and }T_{n}\leq0\\
-0 & T_{n}>0
-\end{cases}
-\end{equation}
-
-\end_inset
-
-The spatial database files for the slip-weakening friction model properties
- and state variables specify the spatial variation of the fault constitutive
- model parameters given in Table 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "tab:slip:weakening:properties:statevars"
-
-\end_inset
-
-.
- As long as the fault is locked, the initial state variables are zero, so
- specifying the initial state variables for slip-weakening friction is rare.
- The slip-weakening friction also includes a parameter, 
-\family typewriter
-force_healing
-\family default
-, to control healing.
- In quasi-static simulations, one usually wants slip confined to a single
- time step (
-\family typewriter
-force_healing
-\family default
- = True), whereas in a dynamic simulation slip occurs over many time steps
- (
-\family typewriter
-force_healing
-\family default
- = False; default behavior) and fault healing is often neglected.
- The properties include:
-\end_layout
-
-\begin_layout Description
-force_healing Flag indicating whether healing (cumalative slip state variable
- reset to zero) is forced after every time step.
-\end_layout
-
-\begin_layout Standard
-An example of setting the properties for the rate and state friction component
- in a 
-\family typewriter
-.cfg
-\family default
- file is:
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.problem.interfaces.fault]
-\end_layout
-
-\begin_layout LyX-Code
-friction = pylith.friction.SlipWeakening ; Change friction model from the
- default
-\end_layout
-
-\begin_layout LyX-Code
-friction.force_healing = False ; default value
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float table
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\noindent
-\align center
-\begin_inset Caption
-
-\begin_layout Plain Layout
-\begin_inset CommandInset label
-LatexCommand label
-name "tab:slip:weakening:properties:statevars"
-
-\end_inset
-
-Values in spatial databases for slip-weakening friction.
-\end_layout
-
-\end_inset
-
-
-\begin_inset VSpace defskip
-\end_inset
-
-
-\begin_inset Tabular
-<lyxtabular version="3" rows="7" columns="3">
-<features tabularvalignment="middle">
-<column alignment="left" valignment="top" width="0">
-<column alignment="left" valignment="top" width="0">
-<column alignment="left" valignment="top" width="2.5in">
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Spatial database
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Value
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Description
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-db_properties
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-static-coefficient
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Static coefficient of friction, 
-\begin_inset Formula $\mu_{s}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-dynamic-coefficient
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Dynamic coefficient of friction, 
-\begin_inset Formula $\mu_{d}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-slip-weakening-parameter
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Slip-weakening parameter, 
-\begin_inset Formula $d_{0}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-cohesion
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Cohesive stress, 
-\begin_inset Formula $T_{c}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-db_initial_state
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-cumulative-slip
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Cumulative slip, 
-\begin_inset Formula $d$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-previous-slip
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Slip at previous time step, 
-\begin_inset Formula $d(t-\Delta t)$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Paragraph
-Time-Weakening Friction
-\end_layout
-
-\begin_layout Standard
-The linear time-weakening friction model is analogous to the linear slip-weakeni
-ng friction model with time replacing slip.
- It produces shear tractions equal to the cohesive stress plus a contribution
- proportional to the fault normal traction that decreases from a static
- value to a dynamic value as time progresses,
-\begin_inset Formula 
-\begin{equation}
-T_{f}=\begin{cases}
-T_{c}-(\mu_{s}-(\mu_{s}-\mu_{d})\frac{t}{t_{0}})T_{n} & t\leq t_{0}\text{ and }T_{n}\leq0\\
-T_{c}-\mu_{d}T_{n} & t>t_{0}\text{ and }T_{n}\leq0\\
-0 & T_{n}>0
-\end{cases}
-\end{equation}
-
-\end_inset
-
-The spatial database files for the time-weakening friction model properties
- and state variables specify the spatial variation of the fault constitutive
- model parameters given in Table 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "tab:time:weakening:properties:statevars"
-
-\end_inset
-
-.
- As long as the fault is locked, the initial state variable is zero, so
- specifying the initial state variable for time-weakening friction is rare.
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float table
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\noindent
-\align center
-\begin_inset Caption
-
-\begin_layout Plain Layout
-\begin_inset CommandInset label
-LatexCommand label
-name "tab:time:weakening:properties:statevars"
-
-\end_inset
-
-Values in spatial databases for time-weakening friction.
-\end_layout
-
-\end_inset
-
-
-\begin_inset VSpace defskip
-\end_inset
-
-
-\begin_inset Tabular
-<lyxtabular version="3" rows="6" columns="3">
-<features tabularvalignment="middle">
-<column alignment="left" valignment="top" width="0">
-<column alignment="left" valignment="top" width="0">
-<column alignment="left" valignment="top" width="2.5in">
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Database
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Value
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Description
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-db_properties
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-static-coefficient
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Static coefficient of friction, 
-\begin_inset Formula $\mu_{s}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-dynamic-coefficient
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Dynamic coefficient of friction, 
-\begin_inset Formula $\mu_{d}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-time-weakening-parameter
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Time-weakening parameter, 
-\begin_inset Formula $t_{0}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-cohesion
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Cohesive stress, 
-\begin_inset Formula $T_{c}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-db_initial_state
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-elapsed-time
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Elasped time of slip, 
-\begin_inset Formula $t$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Paragraph
-\begin_inset CommandInset label
-LatexCommand label
-name "sec:friction:rate:state:ageing"
-
-\end_inset
-
-Rate- and State-Friction with Ageing Law
-\end_layout
-
-\begin_layout Standard
-The Dieterich-Ruina rate and state friction model produces shear tractions
- equal to the cohesive stress plus a contribution proportional to the fault
- normal traction that depends on a state variable,
-\begin_inset Formula 
-\begin{gather}
-T_{f}=\begin{cases}
-T_{c}-\mu_{f}T_{n} & T_{n}\leq0\\
-0 & T_{n}>0
-\end{cases}\\
-\mu_{f}=\begin{cases}
-\mu_{0}+a\ln\left(\frac{V}{V_{0}}\right)+b\ln\left(\frac{V_{0}\theta}{L}\right) & V\ge V_{\mathit{min}}\\
-\mu_{0}+a\ln\left(\frac{V_{min}}{V}\right)+b\ln\left(\frac{V_{0}\theta}{L}\right)-a\left(1-\frac{V}{V_{min}}\right) & V<V_{min}
-\end{cases}\\
-\frac{d\theta}{dt}=1-\frac{V\theta}{L}
-\end{gather}
-
-\end_inset
-
-where 
-\begin_inset Formula $V$
-\end_inset
-
- is slip rate, 
-\begin_inset Formula $V_{min}$
-\end_inset
-
- is a minimum slip rate, 
-\begin_inset Formula $a$
-\end_inset
-
- and 
-\begin_inset Formula $b$
-\end_inset
-
- are coefficients, 
-\begin_inset Formula $L$
-\end_inset
-
- is the characteristic slip distance, 
-\begin_inset Formula $\theta$
-\end_inset
-
- is a state variable.
- With an interative solver in quasi-static simulations with a small, but
- nonzero residual tolerance we never encounter zero slip rates in quasi-static
- simulations.
- Instead we want to avoid significant variations in the coefficient of friction
- for slip rates on the same order as our residual tolerance.
- We regularize the rate and state friction model by iposing a linearization
- of the variation of the coefficient of friction with slip rate when the
- slip rate drops below a minimum slip rate, 
-\begin_inset Formula $V_{min}$
-\end_inset
-
- (
-\family typewriter
-min_slip_rate
-\family default
- property with a default value of 1.0e-12).
- Note that this is different than the popular inverse hypoerbolic sine regulariz
-ation proposed by Ben-Zion and Rice 
-\begin_inset CommandInset citation
-LatexCommand cite
-key "BenZion:Rice:1997"
-
-\end_inset
-
- to permit zero slip rates.
- Following Kaneko 
-\shape italic
-et al.
-
-\shape default
- 
-\begin_inset CommandInset citation
-LatexCommand cite
-key "Kaneko:etal:2008"
-
-\end_inset
-
-, we integrate the evolution equation for the state variable keeping slip
- rate constant to get
-\begin_inset Formula 
-\begin{equation}
-\theta(t+\Delta t)=\theta(t)\exp\left(\frac{-V(t)\Delta t}{L}\right)+\frac{L}{V(t)}\left(1-\exp\left(-\frac{V(t)\Delta t}{L}\right)\right).
-\end{equation}
-
-\end_inset
-
-As the slip rate approaches zero, the first exponential term approaches
- 1.
- Using the first three terms of the Taylor series expansion of the second
- exponential yields
-\begin_inset Formula 
-\begin{equation}
-\theta(t+\Delta t)=\begin{cases}
-\theta(t)\exp\left(-\frac{V(t)\Delta t}{L}\right)+\Delta t-\frac{1}{2}\frac{V(t)\Delta t^{2}}{L} & \frac{V(t)\Delta t}{L}<0.00001\\
-\theta(t)\exp\left(-\frac{V(t)\Delta t}{L}\right)+\frac{L}{V(t)}\left(1-\exp\left(-\frac{V(t)\Delta t}{L}\right)\right) & \frac{V(t)\Delta t}{L}\ge0.00001
-\end{cases}.
-\end{equation}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-The properties include:
-\end_layout
-
-\begin_layout Description
-min_slip_rate Slip rate at which linearization occurs, 
-\begin_inset Formula $V_{min}$
-\end_inset
-
-.
- Should be about one order of magnitude larger than absolute tolerance in
- solve.
-\end_layout
-
-\begin_layout Standard
-An example of setting the properties for the rate and state friction component
- in a 
-\family typewriter
-.cfg
-\family default
- file is:
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.problem.interfaces.fault]
-\end_layout
-
-\begin_layout LyX-Code
-friction = pylith.friction.RateStateAgeing ; Change friction model from the
- default
-\end_layout
-
-\begin_layout LyX-Code
-friction.min_slip_rate = 1.0e-12 ; default value
-\end_layout
-
-\begin_layout Standard
-The spatial database files for the rate and state friction model properties
- and state variables specify the spatial variation of the fault constitutive
- model parameters given in Table 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "tab:rate:state:ageing:properties:statevars"
-
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float table
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\noindent
-\align center
-\begin_inset Caption
-
-\begin_layout Plain Layout
-\begin_inset CommandInset label
-LatexCommand label
-name "tab:rate:state:ageing:properties:statevars"
-
-\end_inset
-
-Values in spatial databases for Dieterich-Ruina rate-state friction.
-\end_layout
-
-\end_inset
-
-
-\begin_inset VSpace defskip
-\end_inset
-
-
-\begin_inset Tabular
-<lyxtabular version="3" rows="8" columns="3">
-<features tabularvalignment="middle">
-<column alignment="left" valignment="top" width="0">
-<column alignment="left" valignment="top" width="0">
-<column alignment="left" valignment="top" width="2.5in">
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Database
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Value
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Description
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-db_properties
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-reference-friction-coefficient
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Steady-state coefficient of friction at slip rate 
-\begin_inset Formula $V_{0}$
-\end_inset
-
-, 
-\begin_inset Formula $\mu_{s}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-reference-slip-rate
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Reference slip rate, 
-\begin_inset Formula $V_{0}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-characteristic-slip-distance
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Slip-weakening parameter, 
-\begin_inset Formula $L$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-constitutive-parameter-a
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Coefficient for the 
-\begin_inset Formula $\ln$
-\end_inset
-
- slip rate term, 
-\begin_inset Formula $a$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-constitutive-parameter-b
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Coefficient for the 
-\begin_inset Formula $\ln$
-\end_inset
-
- state variable term, 
-\begin_inset Formula $b$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-cohesion
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Cohesive stress, 
-\begin_inset Formula $T_{c}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-db_initial_state
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-state-variable
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-State variable, 
-\begin_inset Formula $\theta$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsection
-\begin_inset CommandInset label
-LatexCommand label
-name "sec:fault:cohesive:impulses"
-
-\end_inset
-
-Slip Impulses for Green's Functions
-\end_layout
-
-\begin_layout Standard
-Computing static Green's functions using the GreensFns problem requires
- a specialized fault implementation, FaultCohesiveImpulses, to setup the
- slip impulses.
- The parameters controlling the slip impulses include the components involved
- (lateral, reverse, and/or fault opening) and the amplitude of the pulses
- (e.g., selecting a subset of a fault or including a spatial variation).
- The GreensFns properties and facilities include:
-\end_layout
-
-\begin_layout Description
-threshold Threshold for non-zero amplitude; impulses will only be generated
- at locations on the fault where the amplitude excees this threshold.
-\end_layout
-
-\begin_layout Description
-impulse_dof Array of components associated with impulses, e.g., [0, 1] for
- slip involving the lateral and up-dip components.
-\end_layout
-
-\begin_layout Description
-db_impulse_amplitude Spatial database for amplitude of slip impulse (scalar
- field).
- Default is SimpleDB.
-\end_layout
-
-\begin_layout Standard
-An example of setting the properties and facilities for FaultCohesiveImpulses
- in a 
-\family typewriter
-.cfg
-\family default
- file is:
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.problem.interfaces]
-\end_layout
-
-\begin_layout LyX-Code
-fault = pylith.faults.FaultCohesiveImpulses ; Change from the default
-\end_layout
-
-\begin_layout LyX-Code
-
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.problem.interfaces.fault]
-\end_layout
-
-\begin_layout LyX-Code
-threshold = 1.0e-6*m ; default
-\end_layout
-
-\begin_layout LyX-Code
-impulse_dof = [0] ; lateral slip-only
-\end_layout
-
-\begin_layout LyX-Code
-db_impulse_amplitude.iohandler.filename = myimpulse.spatialdb
-\end_layout
-
-\begin_layout LyX-Code
-db_impulse_amplitude.label = Impulse amplitude
-\end_layout
-
-\begin_layout Section
-Gravitational Body Forces
-\end_layout
-
-\begin_layout Standard
-Many problems in geophysics require the consideration of gravitational body
- forces.
- For example, it is often important to include the effects of the lithostatic
- (overburden) pressure.
- In future releases of PyLith that permit nonlinear bulk rheologies, body
- forces will affect plastic yield criteria and the deformation field for
- large deformation/finite strain problems.
- As described in Chapter 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "cha:Governing-Equations"
-
-\end_inset
-
-, the body forces contribute to the residual,
-\begin_inset Formula 
-\begin{equation}
-r_{i}^{n}=\int_{V}f_{i}N^{n}\: dV.
-\end{equation}
-
-\end_inset
-
-For gravitational body forces the body force per unit volume, 
-\begin_inset Formula $f_{i}$
-\end_inset
-
-, is given as the product of the mass density, 
-\begin_inset Formula $\rho$
-\end_inset
-
-, the scalar gravitational acceleration value, 
-\begin_inset Formula $g$
-\end_inset
-
-, and the gravitational acceleration orientation vector, 
-\begin_inset Formula $a_{i}$
-\end_inset
-
-:
-\begin_inset Formula 
-\begin{equation}
-f_{i}=\rho ga_{i}.
-\end{equation}
-
-\end_inset
-
-The mass density is a property of every material model, and is thus included
- in the spatial database with the physical properties for each material.
- The gravitational acceleration is assumed to be uniform and constant for
- a given problem, with a default value of 9.80665 m/s
-\begin_inset Formula $^{\text{2}}$
-\end_inset
-
-.
- The orientation vector will depend on the dimension of the problem as well
- as the coordinate system being used.
- The default orientation vector has components (0, 0, -1).
- This is appropriate for three-dimensional problems where the gravity vector
- is aligned with the negative z-axis, as would be the case in a geographic-proje
-cted coordinate system or a generic Cartesian coordinate system.
- For cases in which the curvature of the earth should be considered, the
- spatialdata package provides an earth-centered, earth-fixed (ECEF) coordinate
- system and a local georeferenced Cartesian system; in each of these cases
- the orientation vector is computed automatically, although this feature
- has not been tested.
- For problems in one or two dimensions where the orientation vector is constant,
- the vector will need to be explicitly specified.
- For example, in a two-dimensional problem, the vector might be specified
- as (0, -1, 0).
- The vector still has three components, although the extra component is
- not used.
-\end_layout
-
-\begin_layout Standard
-Gravity is turned off by default.
- To include gravitational effects in a simulation, you can turn it on as
- follows:
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.timedependent]
-\end_layout
-
-\begin_layout LyX-Code
-use_gravity = true
-\end_layout
-
-\begin_layout Standard
-By simply adding this flag, the default gravity field values will be used
- and a 
-\family typewriter
-gravity_field
-\family default
- component will be assigned for the problem.
- The default values may be changed by altering the properties of 
-\family typewriter
-gravity_field
-\family default
-:
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.timedependent.gravity_field]
-\end_layout
-
-\begin_layout LyX-Code
-acceleration = 100.0*m*s**-2
-\end_layout
-
-\begin_layout LyX-Code
-gravity_dir = [0, -1, 0]
-\end_layout
-
-\begin_layout Standard
-An example problem using gravity is described in Section 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "sec:Tutorial-3d-hex8-gravity"
-
-\end_inset
-
-.
-\end_layout
-
-\end_body
-\end_document
+#LyX 2.0 created this file. For more info see http://www.lyx.org/
+\lyxformat 413
+\begin_document
+\begin_header
+\textclass book
+\use_default_options false
+\maintain_unincluded_children false
+\language english
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman default
+\font_sans default
+\font_typewriter default
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100
+\font_tt_scale 100
+
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry true
+\use_amsmath 1
+\use_esint 0
+\use_mhchem 1
+\use_mathdots 1
+\cite_engine basic
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\use_refstyle 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\leftmargin 1in
+\topmargin 1in
+\rightmargin 1in
+\bottommargin 1in
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\quotes_language english
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Chapter
+\begin_inset CommandInset label
+LatexCommand label
+name "cha:boundary:interface:conditions"
+
+\end_inset
+
+Boundary and Interface Conditions
+\end_layout
+
+\begin_layout Section
+Assigning Boundary Conditions
+\end_layout
+
+\begin_layout Standard
+There are four basic steps in assigning a specific boundary condition to
+ a portion of the domain.
+\end_layout
+
+\begin_layout Enumerate
+Create sets of vertices in the mesh generation process for each boundary
+ condition.
+\end_layout
+
+\begin_layout Enumerate
+Define boundary condition groups corresponding to the vertex sets.
+\end_layout
+
+\begin_layout Enumerate
+Set the parameters for each boundary condition group using 
+\family typewriter
+.cfg
+\family default
+ or 
+\family typewriter
+.pml
+\family default
+ files and/or command line arguments.
+\end_layout
+
+\begin_layout Enumerate
+Specify the spatial variation in parameters for the boundary condition using
+ a spatial database file.
+\end_layout
+
+\begin_layout Subsection
+Creating Sets of Vertices
+\end_layout
+
+\begin_layout Standard
+The procedure for creating sets of vertices differs depending on the mesh
+ generator.
+ For meshes specified using the PyLith mesh ASCII format, the sets of vertices
+ are specified using groups (see Appendix 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sec:MeshIOAscii"
+
+\end_inset
+
+).
+ In CUBIT the groups of vertices are created using nodesets.
+ Similarly, in LaGriT, psets are used.
+ Note that we chose to associate boundary conditions with groups of vertices
+ because nearly every mesh generation package supports associating a string
+ or integer with groups of vertices.
+ Note also that we currently associate boundary conditions with string identifie
+rs, so even if the mesh generator uses integers, the name is specified as
+ the digits of the integer value.
+ Finally, note that every vertex set that ultimately is associated with
+ a boundary condition on a cell face (e.g., Neumann boundary conditions and
+ fault interface conditions) must correspond to a simply-connected surface.
+\end_layout
+
+\begin_layout Subsection
+Arrays of Boundary Condition Components
+\end_layout
+
+\begin_layout Standard
+A dynamic array of boundary condition components associates a name (string)
+ with each boundary condition.
+ This dynamic array of boundary conditions replaces the boundary condition
+ container in PyLith 1.0.
+ User-defined containers are no longer necessary, and the predefined containers
+ are no longer available (or necessary).
+ The default boundary condition for each component in the array is the Dirichlet
+Points object.
+ Other boundary conditions can be bound to the named items in the array
+ via a 
+\family typewriter
+.cfg
+\family default
+ file, 
+\family typewriter
+.pml
+\family default
+ file, or the command line.
+ The parameters for the boundary condition are set using the name of the
+ boundary condition.
+ An example of setting the array of boundary condition components and changing
+ the types of boundary conditions in a 
+\family typewriter
+.cfg
+\family default
+ file:
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.problem]
+\end_layout
+
+\begin_layout LyX-Code
+bc = [x_neg,x_pos,y_pos,z_neg] ; Array of boundary conditions
+\end_layout
+
+\begin_layout LyX-Code
+# Default boundary condition is DirichletPoints
+\end_layout
+
+\begin_layout LyX-Code
+# Keep default value for bc.x_neg
+\end_layout
+
+\begin_layout LyX-Code
+bc.x_pos = pylith.bc.DirichletBoundary ; change BC type to DirichletBoundary
+\end_layout
+
+\begin_layout LyX-Code
+bc.y_pos = pylith.bc.AbsorbingDampers ; change BC type to AbsorbingDampers
+\end_layout
+
+\begin_layout LyX-Code
+bc.z_neg = pylith.bc.Neumann ; change BC type to Neumann (traction)
+\end_layout
+
+\begin_layout Section
+\begin_inset CommandInset label
+LatexCommand label
+name "sec:Time:Dependent:BC"
+
+\end_inset
+
+Time-Dependent Boundary Conditions
+\end_layout
+
+\begin_layout Standard
+Several boundary conditions use a common formulation for the spatial and
+ temporal variation of the boundary condition parameters,
+\begin_inset Formula 
+\begin{equation}
+f(\vec{x})=f_{0}(\vec{x})+\dot{f}_{0}(\vec{x})(t-t_{0}(\vec{x}))+f_{1}(\vec{x})a(t-t_{1}(\vec{x})),
+\end{equation}
+
+\end_inset
+
+where 
+\begin_inset Formula $f(\vec{x})$
+\end_inset
+
+ may be a scalar or vector parameter, 
+\begin_inset Formula $f_{0}(\vec{x})$
+\end_inset
+
+ is a constant value, 
+\begin_inset Formula $\dot{f}_{0}(\vec{x})$
+\end_inset
+
+ is a constant rate of change in the value, 
+\begin_inset Formula $t_{0}(\vec{x})$
+\end_inset
+
+ is the onset time for the constant rate of change, 
+\begin_inset Formula $f_{1}(\vec{x})$
+\end_inset
+
+ is the amplitude for the temporal modulation, 
+\begin_inset Formula $a(t)$
+\end_inset
+
+ is the variation in amplitude with time, 
+\begin_inset Formula $t_{1}(\vec{x})$
+\end_inset
+
+ is the onset time for the temporal modulation, and 
+\begin_inset Formula $\vec{x}$
+\end_inset
+
+ is the position of a location in space.
+ This common formulation permits easy specification of a scalar or vector
+ with a constant value, constant rate of change of a value, and/or modulation
+ of a value in time.
+ One can specify just the initial value, just the rate of change of the
+ value (along with the corresponding onset time), or just the modulation
+ in amplitude (along with the corresponding temporal variation and onset
+ time), or any combination of the three.
+ The facilities associated with this formulation are:
+\end_layout
+
+\begin_layout Description
+db_initial Spatial database specifying the spatial variation in the initial
+ value (default is none).
+\end_layout
+
+\begin_layout Description
+db_rate Spatial database specifying rate of change in the value (default
+ is none).
+\end_layout
+
+\begin_layout Description
+db_change Spatial database specifying the amplitude of the temporal modulation
+ (default is none).
+\end_layout
+
+\begin_layout Description
+th_change Time history database specifying the temporal change in amplitude
+ (default is none).
+\end_layout
+
+\begin_layout Subsection
+Dirichlet Boundary Conditions
+\end_layout
+
+\begin_layout Standard
+Dirichlet boundary conditions in PyLith prescribe the displacement of a
+ subset of the vertices of the finite-element mesh.
+ While Dirichlet boundary conditions can be applied to any vertex, usually
+ they are applied to vertices on the lateral and bottom boundaries of the
+ domain.
+ There are two types of Dirichlet boundary conditions, DirichletBC and Dirichlet
+Boundary.
+ Both provide identical constraints on the solution, but DirichletBoundary
+ is limited to vertices of a simply-connected surface, which allows diagnostic
+ output of the prescribed displacements.
+ DirichletBC can be applied to a set of unconnected vertices.
+\end_layout
+
+\begin_layout Subsubsection
+Dirichlet Boundary Condition Parameters
+\end_layout
+
+\begin_layout Standard
+The properties and components common to both the DirichletPoints and DirichletBo
+undary boundary conditions are:
+\end_layout
+
+\begin_layout Description
+label Label of the group of vertices associated with the boundary condition.
+\end_layout
+
+\begin_layout Description
+bc_dof Array of degrees of freedom to be fixed (first degree of freedom
+ is 0).
+\end_layout
+
+\begin_layout Standard
+DirichletBoundary contains an additional component:
+\end_layout
+
+\begin_layout Description
+output Manager for output of displacements on boundary with specified displaceme
+nts.
+\end_layout
+
+\begin_layout Standard
+By default the output manager does not output any information.
+ The specified displacements and velocities can be output by including ``displac
+ements'' and ``velocities'' in the output manager's 
+\family typewriter
+vertex_info_fields
+\family default
+ array parameter.
+ An example of setting the Dirichlet boundary condition parameters in a
+ 
+\family typewriter
+.cfg
+\family default
+ file is:
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.problem]
+\end_layout
+
+\begin_layout LyX-Code
+bc = [mybc]
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+[pylithapp.problem.bc.mybc]
+\end_layout
+
+\begin_layout LyX-Code
+label = group A 
+\end_layout
+
+\begin_layout LyX-Code
+bc_dof = [2] ; fixed displacement in z direction
+\end_layout
+
+\begin_layout LyX-Code
+db_initial = spatialdata.spatialdb.SimpleDB
+\end_layout
+
+\begin_layout LyX-Code
+db_initial.iohandler.filename = disp_A.spatialdb
+\end_layout
+
+\begin_layout LyX-Code
+db_initial.query_type = nearest ; change query type to nearest point algorithm
+\end_layout
+
+\begin_layout LyX-Code
+db_rate = spatialdata.spatialdb.UniformDB
+\end_layout
+
+\begin_layout LyX-Code
+db_rate.values = [displacement-rate-z]
+\end_layout
+
+\begin_layout LyX-Code
+db_rate.data = [1.0e-06*m/s] ; velocity is 1.0e-06 m/s
+\end_layout
+
+\begin_layout Standard
+We have created an array with one boundary condition, mybc.
+ The group of vertices associated with the boundary condition is group A.
+ For the database associated with the constant displacement, we use a SimpleDB.
+ We set the filename and query type for the database.
+ For the rate of change of values, we use a UniformDB and specify the velocity
+ in the z-direction to be 1.0e-06 m/s.
+ See Section 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sec:spatial:databases"
+
+\end_inset
+
+ for a discussion of the different types of spatial databases available.
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float table
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\noindent
+\align center
+\begin_inset Caption
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "tab:dirichlet:output"
+
+\end_inset
+
+Fields available in output of DirichletBoundary boundary condition information.
+\end_layout
+
+\end_inset
+
+
+\begin_inset VSpace defskip
+\end_inset
+
+
+\begin_inset Tabular
+<lyxtabular version="3" rows="6" columns="3">
+<features tabularvalignment="middle">
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="3in">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Field Type
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Field
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Description
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+vertex_info_fields
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+displacement_initial
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Initial displacement field in global coordinate system
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+velocity
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Rate of change of displacement field in global coordinate system
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+velocity_start_time
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Onset time in seconds for rate of change in displacement field
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+displacement_change
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Amplitude of change in displacement field in global coordinate system
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+change_start_time
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Onset time in seconds for the amplitude change in the displacement field
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsubsection
+Dirichlet Boundary Condition Spatial Database Files
+\end_layout
+
+\begin_layout Standard
+The spatial database files for the Dirichlet boundary condition specify
+ the fixed displacements.
+ The spatial database file may contain displacements at more degrees of
+ freedom than those specified in the Dirichlet boundary condition settings
+ using the 
+\family typewriter
+bc_dof
+\family default
+ setting.
+ Only those listed in 
+\family typewriter
+bc_dof
+\family default
+ will be used.
+ This permits using the same spatial database file for multiple Dirichlet
+ boundary conditions with the same displacement field.
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float table
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\noindent
+\align center
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Values in the spatial databases used for Dirichlet boundary conditions.
+\end_layout
+
+\end_inset
+
+
+\begin_inset VSpace defskip
+\end_inset
+
+
+\begin_inset Tabular
+<lyxtabular version="3" rows="4" columns="2">
+<features tabularvalignment="middle">
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="4in">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Spatial database
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Name in Spatial Database
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+db_initial
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+displacement-x, displacement-y, displacement-z
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+db_rate
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+displacement-rate-x, displacement-rate-y, displacement-rate-z, rate-start-time
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+db_change
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+displacement-x, displacement-y, displacement-z, change-start-time
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Neumann Boundary Conditions
+\end_layout
+
+\begin_layout Standard
+Neumann boundary conditions are surface tractions applied over a subset
+ of the mesh.
+ As with the DirichletBoundary condition, each Neumann boundary condition
+ can only be applied to a simply-connected surface.
+ The surface over which the tractions are applied always has a spatial dimension
+ that is one less than the dimension of the finite-element mesh.
+ Traction values are computed at the integration points of each cell on
+ the surface, using values from a spatial database.
+ The tractions are integrated over each cell and assembled to obtain the
+ forces applied at the vertices.
+ See Section 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sub:Tutorial-twoquad4-traction"
+
+\end_inset
+
+ for a tutorial that uses Neumann boundary conditions.
+\end_layout
+
+\begin_layout Subsubsection
+Neumann Boundary Condition Parameters
+\end_layout
+
+\begin_layout Standard
+The Neumann boundary condition properties and components are:
+\end_layout
+
+\begin_layout Description
+label Name of the group of vertices defining the mesh boundary for the Neumann
+ boundary condition.
+\end_layout
+
+\begin_layout Description
+up_dir This is a 3-vector that provides a hint for the direction perpendicular
+ to the horizontal tangent direction that is not collinear with the direction
+ normal to the surface.
+ The default value is (0,0,1), which assumes that the z-axis is positive
+ upward.
+ This vector is only needed for three-dimensional problems where the positive
+ upward direction differs from the default.
+\end_layout
+
+\begin_layout Description
+output The output manager associated with diagnostic output (traction vector).
+\end_layout
+
+\begin_layout Description
+quadrature The quadrature object to be used for numerical integration.
+ Since we are integrating over a surface that is one dimension lower than
+ the problem domain, this would typically be set to something like 
+\family typewriter
+Quadrature2Din3D
+\family default
+ (for a three-dimensional problem).
+\end_layout
+
+\begin_layout Standard
+By default the output manager does not output any information.
+ The specified tractions can be output in global coordinates by including
+ ``tractions'' in the output manager's 
+\family typewriter
+cell_info_fields
+\family default
+ array parameter.
+ An example of setting these parameters in a 
+\family typewriter
+.cfg
+\family default
+ file is:
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.timedependent]
+\end_layout
+
+\begin_layout LyX-Code
+bc = [x_neg,x_pos,y_neg]
+\end_layout
+
+\begin_layout LyX-Code
+bc.x_pos = pylith.bc.Neumann ; Change BC type to Neumann
+\end_layout
+
+\begin_layout LyX-Code
+ 
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.timedependent.bc.x_pos]
+\end_layout
+
+\begin_layout LyX-Code
+label = x_pos ; Name of group of vertices for +x boundary
+\end_layout
+
+\begin_layout LyX-Code
+db_initial = spatialdata.spatialdb.SimpleDB
+\end_layout
+
+\begin_layout LyX-Code
+db_initial.label = Neumann BC +x edge
+\end_layout
+
+\begin_layout LyX-Code
+db_initial.iohandler.filename = axialtract.spatialdb
+\end_layout
+
+\begin_layout LyX-Code
+db_initial.query_type = nearest
+\end_layout
+
+\begin_layout LyX-Code
+quadrature.cell = pylith.feassemble.FIATLagrange
+\end_layout
+
+\begin_layout LyX-Code
+quadrature.cell.dimension = 1
+\end_layout
+
+\begin_layout LyX-Code
+quadrature.cell.quad_order = 2
+\end_layout
+
+\begin_layout Standard
+These settings correspond to the example problem described in Section 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sub:Tutorial-twoquad4-traction"
+
+\end_inset
+
+.
+ It is necessary to set the boundary condition type to 
+\family typewriter
+pylith.bc.Neumann
+\family default
+, since the default value is 
+\family typewriter
+DirichletBC
+\family default
+.
+ Constant tractions are used for this particular problem, so a quadrature
+ order of one would have been sufficient; however, for problems involving
+ more complex variations (e.g., a linear variation), a quadrature order of
+ two will provide more accurate results.
+ Note that there is no advantage to specifying an integration order higher
+ than two, since linear elements are being used for this problem.
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float table
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\noindent
+\align center
+\begin_inset Caption
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "tab:neumann:output"
+
+\end_inset
+
+Fields available in output of Neumann boundary condition information.
+\end_layout
+
+\end_inset
+
+
+\begin_inset VSpace defskip
+\end_inset
+
+
+\begin_inset Tabular
+<lyxtabular version="3" rows="6" columns="3">
+<features tabularvalignment="middle">
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="3in">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Field Type
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Field
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Description
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+cell_info_fields
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+tracton_initial
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Initial traction field in global coordinate system
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+traction_rate
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Rate of change of traction field in global coordinate system
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+rate_start_time
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Onset time in seconds for rate of change in traction field
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+traction_change
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Amplitude of change in traction field in global coordinate system
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+change_start_time
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Onset time in seconds for the amplitude change in the traction field
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsubsection
+Neumann Boundary Condition Spatial Database Files
+\end_layout
+
+\begin_layout Standard
+The spatial database files for the Neumann boundary condition specify the
+ applied tractions.
+ The number of traction components is equal to the spatial dimension for
+ the problem.
+ The tractions are specified in a local coordinate system for the boundary.
+ The names of the components of the traction vector are:
+\end_layout
+
+\begin_layout Description
+one-dimensional 
+\family typewriter
+normal
+\end_layout
+
+\begin_layout Description
+two-dimensional 
+\family typewriter
+shear
+\family default
+, 
+\family typewriter
+normal
+\end_layout
+
+\begin_layout Description
+three-dimensional 
+\family typewriter
+horiz-shear
+\family default
+, 
+\family typewriter
+vert-shear
+\family default
+, 
+\family typewriter
+normal
+\end_layout
+
+\begin_layout Standard
+Ambiguities in specifying the shear tractions in 3D problems are resolved
+ using the 
+\family typewriter
+up_dir
+\family default
+ parameter.
+ In the case of a horizontal surface, users will need to pick an alternative
+ vector, as the default 
+\family typewriter
+up_dir
+\family default
+ would coincide with the normal direction.
+ In this case, the orientation for the 
+\family typewriter
+vert-shear-traction
+\family default
+ component will correspond to whatever the user specifies for 
+\family typewriter
+up_dir
+\family default
+, rather than the actual vertical direction.
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float table
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\noindent
+\align center
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Values in the spatial databases used for Dirichlet boundary conditions in
+ three dimensions.
+ In one- and two-dimensional problems, the names of the components are slightly
+ different as described earlier in this section.
+\end_layout
+
+\end_inset
+
+
+\begin_inset VSpace defskip
+\end_inset
+
+
+\begin_inset Tabular
+<lyxtabular version="3" rows="4" columns="2">
+<features tabularvalignment="middle">
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="4in">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Spatial database
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Name in Spatial Database
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+db_initial
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+traction-shear-horiz, traction-shear-vert, traction-normal
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+db_rate
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+traction-rate-horiz-shear, traction-rate-vert-shear, traction-rate-normal,
+ rate-start-time
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+db_change
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+traction-horiz-shear, traction-vert-shear, traction-normal, change-start-time
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Point Force Boundary Conditions
+\end_layout
+
+\begin_layout Standard
+Point force boundary conditions in PyLith prescribe the application of point
+ forces to a subset of the vertices of the finite-element mesh.
+ While point force boundary conditions can be applied to any vertex, usually
+ they are applied to vertices on the lateral, top, and bottom boundaries
+ of the domain.
+\end_layout
+
+\begin_layout Subsubsection
+Point Force Parameters
+\end_layout
+
+\begin_layout Standard
+The properties and components common to both the DirichletBC and DirichletBounda
+ry boundary conditions are:
+\end_layout
+
+\begin_layout Description
+label Label of the group of vertices associated with the boundary condition.
+\end_layout
+
+\begin_layout Description
+bc_dof Array of degrees of freedom to which forces are applied (first degree
+ of freedom is 0).
+\end_layout
+
+\begin_layout Standard
+An example of setting the point force boundary condition parameters in a
+ 
+\family typewriter
+.cfg
+\family default
+ file is:
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.problem]
+\end_layout
+
+\begin_layout LyX-Code
+bc = [mybc]
+\end_layout
+
+\begin_layout LyX-Code
+bc.mybc = pylith.bc.PointForce
+\end_layout
+
+\begin_layout LyX-Code
+
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.problem.bc.mybc]
+\end_layout
+
+\begin_layout LyX-Code
+label = group A 
+\end_layout
+
+\begin_layout LyX-Code
+bc_dof = [2] ; fixed displacement in z direction
+\end_layout
+
+\begin_layout LyX-Code
+db_initial = spatialdata.spatialdb.SimpleDB
+\end_layout
+
+\begin_layout LyX-Code
+db_initial.iohandler.filename = force_A.spatialdb
+\end_layout
+
+\begin_layout LyX-Code
+db_initial.query_type = nearest ; change query type to nearest point algorithm
+\end_layout
+
+\begin_layout LyX-Code
+db_rate = spatialdata.spatialdb.UniformDB
+\end_layout
+
+\begin_layout LyX-Code
+db_rate.values = [force-rate-z]
+\end_layout
+
+\begin_layout LyX-Code
+db_rate.data = [1.0e+5*newton/s]
+\end_layout
+
+\begin_layout Standard
+We have created an array with one boundary condition, mybc.
+ The group of vertices associated with the boundary condition is group A.
+ For the database associated with the constant force, we use a SimpleDB.
+ We set the filename and query type for the database.
+ For the rate of change of values, we use a UniformDB and specify the rate
+ of change in the force to be 1.0e+5 Newton/s.
+ See Section 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sec:spatial:databases"
+
+\end_inset
+
+ for a discussion of the different types of spatial databases available.
+\end_layout
+
+\begin_layout Subsubsection
+Point Force Spatial Database Files
+\end_layout
+
+\begin_layout Standard
+The spatial database files for the Dirichlet boundary condition specify
+ the fixed displacements.
+ The spatial database file may contain displacements at more degrees of
+ freedom than those specified in the Dirichlet boundary condition settings
+ using the 
+\family typewriter
+bc_dof
+\family default
+ setting.
+ Only those listed in 
+\family typewriter
+bc_dof
+\family default
+ will be used.
+ This permits using the same spatial database file for multiple Dirichlet
+ boundary conditions with the same displacement field.
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float table
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\noindent
+\align center
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Values in the spatial databases used for point force boundary conditions.
+\end_layout
+
+\end_inset
+
+
+\begin_inset VSpace defskip
+\end_inset
+
+
+\begin_inset Tabular
+<lyxtabular version="3" rows="4" columns="2">
+<features tabularvalignment="middle">
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="4in">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Spatial database
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Name in Spatial Database
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+db_initial
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+force-x, force-y, force-z
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+db_rate
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+force-rate-x, force-rate-y, force-rate-z, rate-start-time
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+db_change
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+force-x, force-y, force-z, change-start-time
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+\begin_inset CommandInset label
+LatexCommand label
+name "sec:absorbing:boundaries"
+
+\end_inset
+
+Absorbing Boundary Conditions
+\end_layout
+
+\begin_layout Standard
+This boundary condition attempts to prevent seismic waves reflecting off
+ of a boundary by placing simple dashpots on the boundary.
+ Normally incident dilatational and shear waves are perfectly absorbed.
+ Waves incident at other angles are only partially absorbed.
+ This boundary condition is simpler than a perfectly matched layer (PML)
+ boundary condition but does not perform quite as well, especially for surface
+ waves.
+ If the waves arriving at the absorbing boundary are relatively small in
+ amplitude compared to the amplitudes of primary interest, this boundary
+ condition gives reasonable results.
+\end_layout
+
+\begin_layout Standard
+The Absorbing boundary condition properties and components are:
+\end_layout
+
+\begin_layout Description
+label Name of the group of vertices defining the mesh boundary for the absorbing
+ boundary condition.
+\end_layout
+
+\begin_layout Description
+up_dir This is a 3-vector that provides a hint for the direction perpendicular
+ to the horizontal tangent direction that is not collinear with the direction
+ normal to the surface.
+ The default value is (0,0,1), which assumes that the z-axis is positive
+ upward.
+ This vector is only needed for three-dimensional problems where the positive
+ upward direction differs from the default.
+\end_layout
+
+\begin_layout Description
+db The spatial database specifying the material properties for the seismic
+ velocities.
+\end_layout
+
+\begin_layout Description
+quadrature The quadrature object to be used for numerical integration.
+ Since we are integrating over a surface that is one dimension lower than
+ the problem domain, this would typically be set to something like 
+\family typewriter
+Quadrature2Din3D
+\family default
+ (for a three-dimensional problem).
+\end_layout
+
+\begin_layout Subsection
+Finite-Element Implementation of Absorbing Boundary
+\end_layout
+
+\begin_layout Standard
+Consider a plane wave propagating at a velocity 
+\begin_inset Formula $c$
+\end_inset
+
+.
+ We can write the displacement field as
+\begin_inset Formula 
+\begin{equation}
+\vec{u}(\vec{x},t)=\vec{u^{t}}(t-\frac{\vec{x}}{c}),
+\end{equation}
+
+\end_inset
+
+where 
+\begin_inset Formula $\vec{x}$
+\end_inset
+
+ is position, 
+\begin_inset Formula $t$
+\end_inset
+
+ is time, and 
+\begin_inset Formula $\vec{u^{t}}$
+\end_inset
+
+ is the shape of the propagating wave.
+ For an absorbing boundary we want the traction on the boundary to be equal
+ to the traction associated with the wave propagating out of the domain.
+ Starting with the expression for the traction on a boundary, 
+\begin_inset Formula $T_{i}=\sigma_{ij}n_{j},$
+\end_inset
+
+ and using the local coordinate system for the boundary 
+\begin_inset Formula $s_{h}s_{v}n,$
+\end_inset
+
+ where 
+\begin_inset Formula $\vec{n}$
+\end_inset
+
+ is the direction normal to the boundary, 
+\begin_inset Formula $\overrightarrow{s}_{h}$
+\end_inset
+
+ is the horizontal direction tangent to the boundary, and 
+\begin_inset Formula $\overrightarrow{s}_{v}$
+\end_inset
+
+ is the vertical direction tangent to the boundary, the tractions on the
+ boundary are
+\begin_inset Formula 
+\begin{gather}
+T_{s_{h}}=\sigma_{s_{h}n}\\
+T_{s_{v}}=\sigma_{s_{v}n}\\
+T_{n}=\sigma_{nn}.
+\end{gather}
+
+\end_inset
+
+In the case of a horizontal boundary, we can define an auxiliary direction
+ in order to assign unique tangential directions.
+ For a linear elastic isotropic material, 
+\begin_inset Formula $\sigma_{ij}=\lambda\epsilon_{kk}\delta_{ij}+2\mu\epsilon_{ij},$
+\end_inset
+
+ and we can write the tractions as 
+\begin_inset Formula 
+\begin{gather}
+T_{s_{h}}=2\mu\epsilon_{s_{h}n}\\
+T_{s_{v}}=2\epsilon_{s_{v}n}\\
+T_{n}=(\lambda+2\mu)\epsilon_{nn}+\lambda(\epsilon_{s_{h}s_{h}}+\epsilon_{s_{v}s_{v}}).
+\end{gather}
+
+\end_inset
+
+For infinitesimal strains, 
+\begin_inset Formula $\epsilon_{ij}=\frac{1}{2}(u_{i,j}+u_{j,i})$
+\end_inset
+
+ and we have
+\begin_inset Formula 
+\begin{gather}
+\epsilon_{s_{h}n}=\frac{1}{2}(u_{s_{h},n}+u_{n,s_{h}})\\
+\epsilon_{s_{v}n}=\frac{1}{2}(u_{s_{v},n}+u_{n,s_{v}})\\
+\epsilon_{nn}=u_{n,n}.
+\end{gather}
+
+\end_inset
+
+For our propagating plane wave, we recognize that
+\begin_inset Formula 
+\begin{equation}
+\frac{\partial\vec{u^{t}}(t-\frac{\vec{x}}{c})}{\partial x_{i}}=-\frac{1}{c}\frac{\partial\vec{u^{t}}(t-\frac{\vec{x}}{c})}{\partial t},
+\end{equation}
+
+\end_inset
+
+so that our expressions for the tractions become
+\begin_inset Formula 
+\begin{gather}
+T_{s_{h}}=-\frac{\mu}{c}\left(\frac{\partial u_{s_{h}}^{t}(t-\frac{\vec{x}}{c})}{\partial t}+\frac{\partial u_{n}^{t}(t-\frac{\vec{x}}{c})}{\partial t}\right),\\
+T_{s_{v}}=-\frac{\mu}{c}\left(\frac{\partial u_{s_{v}}^{t}(t-\frac{\vec{x}}{c})}{\partial t}+\frac{\partial u_{n}^{t}(t-\frac{\vec{x}}{c})}{\partial t}\right).
+\end{gather}
+
+\end_inset
+
+For the normal traction, consider a dilatational wave propagating normal
+ to the boundary at speed 
+\begin_inset Formula $v_{p}$
+\end_inset
+
+; in this case 
+\begin_inset Formula $u_{s_{h}}=u_{s_{v}}=0$
+\end_inset
+
+ and 
+\begin_inset Formula $c=v_{p}$
+\end_inset
+
+.
+ For the shear tractions, consider a shear wave propagating normal to the
+ boundary at speed 
+\begin_inset Formula $v_{s}$
+\end_inset
+
+; we can decompose this into one case where 
+\begin_inset Formula $u_{n}=u_{s_{v}}=0$
+\end_inset
+
+ and another case where 
+\begin_inset Formula $u_{n}=u_{s_{h}}=0$
+\end_inset
+
+, with 
+\begin_inset Formula $c=v_{s}$
+\end_inset
+
+ in both cases.
+ We also recognize that 
+\begin_inset Formula $\mu=\rho v_{s}^{2}$
+\end_inset
+
+ and 
+\begin_inset Formula $\lambda+2\mu=\rho v_{p}^{2}$
+\end_inset
+
+.
+ This leads to the following expressions for the tractions:
+\begin_inset Formula 
+\begin{gather}
+T_{s_{h}}=-\rho v_{s}\frac{\partial u_{s_{h}}^{t}(t-\frac{\vec{x}}{c})}{\partial t}\\
+T_{s_{v}}=-\rho v_{s}\frac{\partial u_{v}^{t}(t-\frac{\vec{x}}{c})}{\partial t}\\
+T_{n}=-\rho v_{p}\frac{\partial u_{n}^{t}(t-\frac{\vec{x}}{c})}{\partial t}
+\end{gather}
+
+\end_inset
+
+We write the weak form of the boundary condition as
+\begin_inset Formula 
+\[
+\int_{S_{T}}T_{i}\phi_{i}\, dS=\int_{S_{T}}-\rho c_{i}\frac{\partial u_{i}}{\partial t}\phi_{i}\, dS,
+\]
+
+\end_inset
+
+where 
+\begin_inset Formula $c_{i}$
+\end_inset
+
+ equals 
+\begin_inset Formula $v_{p}$
+\end_inset
+
+ for the normal traction and 
+\begin_inset Formula $v_{s}$
+\end_inset
+
+ for the shear tractions, and 
+\begin_inset Formula $\phi_{i}$
+\end_inset
+
+ is our weighting function.
+ We express the trial solution and weighting function as linear combinations
+ of basis functions,
+\begin_inset Formula 
+\begin{gather}
+u_{i}=\sum_{m}a_{i}^{m}N^{m},\\
+\phi_{i}=\sum_{n}c_{i}^{n}N^{n}.
+\end{gather}
+
+\end_inset
+
+Substituting into our integral over the absorbing boundaries yields
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{equation}
+\int_{S_{T}}T_{i}\phi_{i}\, dS=\int_{S_{T}}-\rho c_{i}\sum_{m}\dot{a}_{i}^{m}N^{m}\sum_{n}c_{i}^{n}N^{n}\, dS.
+\end{equation}
+
+\end_inset
+
+In the derivation of the governing equations, we recognized that the weighting
+ function is arbitrary, so we form the residual by setting the terms associated
+ with the coefficients 
+\begin_inset Formula $c_{i}^{n}$
+\end_inset
+
+ to zero,
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{equation}
+r_{i}^{n}=\sum_{\text{tract cells}}\sum_{\text{quad pts}}-\rho(x_{q})c_{i}(x_{q})\sum_{m}\dot{a}_{i}^{m}N^{m}(x_{q})N^{n}(x_{q})w_{q}|J_{cell}(x_{q})|,
+\end{equation}
+
+\end_inset
+
+ where 
+\begin_inset Formula $x_{q}$
+\end_inset
+
+ are the coordinates of the quadrature points, 
+\begin_inset Formula $w_{q}$
+\end_inset
+
+ are the weights of the quadrature points, and 
+\begin_inset Formula $|J_{cell}(x_{q})|$
+\end_inset
+
+ is the determinant of the Jacobian matrix evaluated at the quadrature points
+ associated with mapping the reference cell to the actual cell.
+\end_layout
+
+\begin_layout Standard
+The appearance of velocity in the expression for the residual means that
+ the absorbing dampers also contribute to the system Jacobian matrix.
+ Using the central difference method, the velocity is written in terms of
+ the displacements,
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{equation}
+\dot{u}_{i}(t)=\frac{1}{2\Delta t}(u_{i}(t+\Delta t)-u_{i}(t-\Delta t)).
+\end{equation}
+
+\end_inset
+
+Expressing the displacement at time 
+\begin_inset Formula $t+\Delta t$
+\end_inset
+
+ in terms of the displacement at time 
+\begin_inset Formula $t$
+\end_inset
+
+ (
+\begin_inset Formula $u_{i}(t)$
+\end_inset
+
+) and the increment in the displacement at time 
+\begin_inset Formula $t$
+\end_inset
+
+ (
+\begin_inset Formula $du_{i}(t)$
+\end_inset
+
+) leads to
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{equation}
+\dot{u}_{i}(t)=\frac{1}{2\Delta t}(du_{i}(t)+u_{i}(t)-u_{i}(t-\Delta t))
+\end{equation}
+
+\end_inset
+
+The terms contributing to the system Jacobian are associated with the increment
+ in the displacement at time 
+\begin_inset Formula $t$
+\end_inset
+
+.
+ Substituting into the governing equations and isolating the term associated
+ with the increment in the displacement at time t yields
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{equation}
+A_{ij}^{nm}=\sum_{\text{tract cells}}\sum_{\text{quad pts}}\delta_{ij}\frac{1}{2\Delta t}\rho(x_{q})v_{i}(x_{q})N^{m}(x_{q})N^{n}(x_{q})w_{q}|J_{cells}(x_{q})|,
+\end{equation}
+
+\end_inset
+
+where 
+\begin_inset Formula $A_{ij}^{mn}$
+\end_inset
+
+ is an 
+\begin_inset Formula $nd$
+\end_inset
+
+ by 
+\begin_inset Formula $md$
+\end_inset
+
+ matrix (
+\begin_inset Formula $d$
+\end_inset
+
+ is the dimension of the vector space), 
+\begin_inset Formula $m$
+\end_inset
+
+ and 
+\begin_inset Formula $n$
+\end_inset
+
+ refer to the basis functions and 
+\begin_inset Formula $i$
+\end_inset
+
+ and 
+\begin_inset Formula $j$
+\end_inset
+
+ are vector space components.
+\end_layout
+
+\begin_layout Section
+\begin_inset CommandInset label
+LatexCommand label
+name "sec:fault"
+
+\end_inset
+
+Fault Interface Conditions
+\end_layout
+
+\begin_layout Standard
+Fault interfaces are used to create dislocations (jumps in the displacement
+ field) in the model.
+ The dislocations arise from slip across a fault surface.
+ Both shear and tensile dislocations are supported.
+ For fault interfaces, dislocations in 1D correspond to fault-opening (and
+ closing), in 2D lateral-slip and fault opening, and in 3D lateral-slip,
+ reverse-slip, and fault opening.
+ PyLith supports kinematic (prescribed) slip and dynamic (spontaneous) rupture
+ simulations.
+\end_layout
+
+\begin_layout Subsection
+Conventions
+\end_layout
+
+\begin_layout Standard
+Slip corresponds to relative motion across a fault surface.
+ Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:fault:orientation"
+
+\end_inset
+
+ shows the orientation of the slip vector in 3D with respect to the fault
+ surface and coordinate axes.
+ PyLith automatically determines the orientation of the fault surface.
+ This alleviates the user from having to compute the strike, dip, and rake
+ angles over potentially complex, nonplanar fault surfaces.
+ Instead, the user specifies fault parameters in terms of lateral motion,
+ reverse motion, and fault opening as shown in Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:fault:slip:motions"
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float figure
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\align center
+\begin_inset Graphics
+	filename figs/faultOrientation.eps
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Orientation of a fault surface in 3D, where 
+\begin_inset Formula $\phi$
+\end_inset
+
+ denotes the angle of the fault strike, 
+\begin_inset Formula $\delta$
+\end_inset
+
+ denotes the angle of the fault dip, and 
+\begin_inset Formula $\lambda$
+\end_inset
+
+ the rake angle.
+ 
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:fault:orientation"
+
+\end_inset
+
+ 
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float figure
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\align center
+\begin_inset Graphics
+	filename figs/slipmotions.eps
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Sign conventions associated with fault slip.
+ Positive values are associated with left-lateral, reverse, and fault opening
+ motions.
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:fault:slip:motions"
+
+\end_inset
+
+ 
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Fault Implementation
+\end_layout
+
+\begin_layout Standard
+In order to create relative motion across the fault surface in the finite-elemen
+t mesh, additional degrees of freedom are added along with adjustment of
+ the topology of the mesh.
+ These additional degrees of freedom are associated with cohesive cells.
+ These zero-volume cells allow control of the relative motion between vertices
+ on the two sides of the fault.
+ PyLith automatically adds cohesive cells for each fault surface.
+ Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:fault:cohesive:quad4"
+
+\end_inset
+
+ illustrates the results of inserting a cohesive cell in a mesh consisting
+ of two quadrilateral cells.
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float figure
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\align center
+\begin_inset Graphics
+	filename figs/quad4cohesivekin.eps
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Example of cohesive cell inserted into a mesh of two quadrilateral cells.
+ The zero thickness cohesive cell (shown with dashed lines) controls slip
+ on the fault via the relative motion between vertices 3 and 7 and 2 and
+ 6.
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:fault:cohesive:quad4"
+
+\end_inset
+
+ 
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Fault Parameters
+\end_layout
+
+\begin_layout Standard
+The principal parameters for fault interface conditions are:
+\end_layout
+
+\begin_layout Description
+id This is an integer identifier for the fault surface.
+ It is used to specify the 
+\family typewriter
+material-id
+\family default
+ of the cohesive cells in the mesh.
+ Material identifiers must be unique so this value cannot be the same as
+ any of the material models or any other fault.
+\end_layout
+
+\begin_layout Description
+name Name or label for the fault.
+ This associates sets of vertices with the fault surface in addition to
+ being used in error and diagnostic reports.
+\end_layout
+
+\begin_layout Description
+up_dir Up-dir or up direction (used in 2D and 3D simulations).
+ In 2D the default in-plane slip is left-lateral, so we use the up-direction
+ to resolve the ambiguity in specifying reverse slip.
+ In 3D the up-direction is used to resolve the ambiguity in the along-strike
+ and dip-dir directions.
+ If the fault plane is horizontal, then the up-dir corresponds to the reverse-mo
+tion on the +z side of the fault.
+ The only requirement for this direction is that it not be collinear with
+ the fault normal direction.
+ The default value of [0, 0, 1] is appropriate for most 3D problems.
+\end_layout
+
+\begin_layout Description
+quadrature Quadrature object used in integrating fault quantities.
+\end_layout
+
+\begin_layout Description
+output Manager for output of diagnostic and data fields for the fault.
+\end_layout
+
+\begin_layout Standard
+By default the output manager outputs both diagnostic information (e.g., fault
+ normal direction) and the slip at each time step.
+ Tables 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "tab:fault:kin:output"
+
+\end_inset
+
+ and 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "tab:fault:dyn:output"
+
+\end_inset
+
+ list the fields available for output for a fault with kinematic (prescribed)
+ earthquake rupture and a fault with dynamic rupture, respectively.
+ The fault coordinate system is shown in Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:fault:slip:motions"
+
+\end_inset
+
+.
+ The vectors in the fault coordinate system can be transformed to the global
+ coordinate system using the direction vectors in the diagnostic output.
+ An example of setting these parameters in a 
+\family typewriter
+.cfg
+\family default
+ file is:
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.problem]
+\end_layout
+
+\begin_layout LyX-Code
+interfaces = [fault]
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+[pylithapp.problem.interfaces]
+\end_layout
+
+\begin_layout LyX-Code
+fault = pylith.faults.FaultCohesiveKin ; default
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+label = fault A ; Group of vertices defining the fault surface
+\end_layout
+
+\begin_layout LyX-Code
+id = 100 ; Value for material identifier associated with fault's cohesive
+ cells
+\end_layout
+
+\begin_layout LyX-Code
+up_dir = [0, 0, 1] ; default
+\end_layout
+
+\begin_layout LyX-Code
+quadrature.cell = pylith.feassemble.FIATLagrange
+\end_layout
+
+\begin_layout LyX-Code
+quadrature.cell.dimension = 2
+\end_layout
+
+\begin_layout Standard
+The group of vertices has the label ``fault A.'' We replicate the default
+ values for the fault ``up'' direction.
+ These settings apply to a 2D fault surface embedded within a 3D mesh, so
+ we use 2D Lagrange reference cells.
+ The spatial database for elastic properties is used to determine the approximat
+e shear modulus and condition the equations for faster convergence rates.
+\end_layout
+
+\begin_layout Subsection
+Kinematic Earthquake Rupture
+\end_layout
+
+\begin_layout Standard
+Kinematic earthquake ruptures use the FaultCohesiveKin object to specify
+ the slip as a function of time on the fault surface.
+ Slip may evolve simultaneously over the fault surface instantaneously in
+ a single time step (as is usually done in quasi-static simulations) or
+ propagate over the fault surface over hundreds and up to thousands of time
+ steps (as is usually done in a dynamic simulation).
+\end_layout
+
+\begin_layout Subsubsection
+Governing Equations
+\end_layout
+
+\begin_layout Standard
+The insertion of cohesive cells into the finite-element mesh has the effect
+ of decoupling the motion of the two sides of the fault surface.
+ In order to impose the desired relative motion, we must adjust the governing
+ equations.
+ PyLith employs Lagrange multiplier constraints to enforce the constraint
+ of the relative motion in the strong sense.
+ That is, we enforce the slip across the fault at each degree of freedom.
+\end_layout
+
+\begin_layout Standard
+In conventional implementations the additional degrees of freedom associated
+ with the Lagrange multipliers result in a complex implementation.
+ However, the use of Lagrange multiplier constraints with cohesive cells
+ provides for a simple formulation; we simply add the additional degrees
+ of freedom associated with the Lagrange multipliers to the cohesive cells
+ as shown in Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:fault:cohesive:quad4"
+
+\end_inset
+
+.
+ As a result, the fault implementation is completely confined to the cohesive
+ cell.
+ Furthermore, the Lagrange multiplier constraints correspond to forces required
+ to impose the relative motions, so they are related to the change in stress
+ on the fault surface associated with fault slip.
+ If we write the algebraic system of equations associated with elasticity
+ in the form
+\begin_inset Formula 
+\begin{equation}
+\underline{A}\overrightarrow{u}=\overrightarrow{b}\,,
+\end{equation}
+
+\end_inset
+
+then adding in the Lagrange multiplier constraints associated with fault
+ slip leads to a new system of equations of the form 
+\begin_inset Formula 
+\begin{equation}
+\left[\begin{array}{cc}
+\underline{A} & \underline{C}^{T}\\
+\underline{C} & 0
+\end{array}\right]\left[\begin{array}{c}
+\overrightarrow{u}\\
+\overrightarrow{l}
+\end{array}\right]=\left[\begin{array}{c}
+\overrightarrow{b}\\
+\overrightarrow{d}
+\end{array}\right]\,,\label{eq:fault:cohesive:lagrange}
+\end{equation}
+
+\end_inset
+
+where 
+\begin_inset Formula $\overrightarrow{l}$
+\end_inset
+
+ is the vector of Lagrange multipliers and 
+\begin_inset Formula $\underline{C}$
+\end_inset
+
+ is composed of rotation submatrices, 
+\begin_inset Formula $\underline{R}$
+\end_inset
+
+, associated with the direction cosines relating the relative displacements
+ across the fault to the vector of fault slip, 
+\begin_inset Formula $\overrightarrow{d}$
+\end_inset
+
+.
+ Note that by using the direction cosines to relate the relative motion
+ across the fault, the slip vector and Lagrange multipliers (forces required
+ to impose the slip) are in the local fault coordinate system (lateral motion,
+ reverse motion, and fault opening).
+ 
+\end_layout
+
+\begin_layout Paragraph
+Non-diagonal A
+\end_layout
+
+\begin_layout Standard
+The Lagrange multipliers contribute to both the system Jacobian matrix and
+ the residual.
+ Because we enforce the constraints in a strong sense, the terms do not
+ involve integrals over the fault surface.
+ The additional terms in the residual are
+\begin_inset Formula 
+\begin{gather}
+r_{i}^{n}=-C_{ji}^{pn}l_{j}^{p},\\
+r_{i}^{p}=d_{i}^{p}-C_{ij}^{pn}u_{j}^{n},
+\end{gather}
+
+\end_inset
+
+where 
+\begin_inset Formula $n$
+\end_inset
+
+ denotes a conventional degree of freedom and 
+\begin_inset Formula $p$
+\end_inset
+
+ denotes a degree of freedom associated with a Lagrange multiplier.
+ The additional terms in the system Jacobian matrix are simply the direction
+ cosines,
+\begin_inset Formula 
+\begin{gather}
+J_{ij}^{np}=C_{ji}^{pn},\\
+J_{ij}^{pn}=C_{ij}^{pn}.
+\end{gather}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Diagonal A
+\end_layout
+
+\begin_layout Standard
+When we use a lumped system Jacobian matrix, we cannot lump the terms associated
+ with the Lagrange multipliers.
+ Instead, we formulate the Jacobian ignoring the contributions from the
+ Lagrange multipliers, and then adjust the solution after the solve to account
+ for their presence.
+ Including the Lagrange multipliers in the general expression for the residual
+ at time 
+\begin_inset Formula $t+\Delta t$
+\end_inset
+
+, we have
+\begin_inset Formula 
+\begin{equation}
+r_{i}^{n}(t+\Delta t)=A_{ij}^{nm}(u_{j}^{m}(t)+du_{j}^{m}(t))+C_{ki}^{pn}(l_{k}^{p}(t)+dl_{k}^{p}(t)),
+\end{equation}
+
+\end_inset
+
+where we have written the displacements and Lagrange multipliers at time
+ 
+\begin_inset Formula $t+\Delta t$
+\end_inset
+
+ in terms of the values at time 
+\begin_inset Formula $t$
+\end_inset
+
+ and the increment from time 
+\begin_inset Formula $t$
+\end_inset
+
+ to 
+\begin_inset Formula $t+\Delta t$
+\end_inset
+
+.
+ When we solve the lumped system ignoring the Lagrange multipliers contributions
+ to the Jacobian, we formulate the residual assuming the values 
+\begin_inset Formula $du_{i}^{n}$
+\end_inset
+
+(t) and 
+\begin_inset Formula $dl_{k}^{p}(t)$
+\end_inset
+
+ are zero.
+ So our task is to determine the increment in the Lagrange multiplier, 
+\begin_inset Formula $dl_{k}^{p}$
+\end_inset
+
+, and the correction to the displacement increment, 
+\begin_inset Formula $du_{i}^{n}$
+\end_inset
+
+, and by setting the residual with all terms included to zero; thus, we
+ have
+\begin_inset Formula 
+\begin{gather}
+A_{ij}^{nm}(u_{j}^{m}(t)+du_{j}^{m}(t))+C_{ki}^{pn}(l_{k}^{p}(t)+dl_{k}^{p}(t))=0\text{ subject to}\\
+C_{ij}^{pn}(u_{j}^{n}(t)+du_{j}^{n}(t))=d_{i}^{p}.
+\end{gather}
+
+\end_inset
+
+Making use of the residual computed with 
+\begin_inset Formula $du_{i}^{n}(t)=0$
+\end_inset
+
+ and 
+\begin_inset Formula $dl_{k}^{p}(t)=0$
+\end_inset
+
+,
+\begin_inset Formula 
+\begin{gather}
+r_{i}^{n}+A_{ij}^{nm}du_{j}^{m}+C_{ki}^{pn}dl_{k}^{p}=0\text{ subject to}\\
+C_{ij}^{pn}(u_{j}^{n}(t)+du_{j}^{n}(t))=d_{i}^{p}.
+\end{gather}
+
+\end_inset
+
+Explicitly writing the equations for the vertices on the negative and positive
+ sides of the fault yields
+\begin_inset Formula 
+\begin{gather}
+r_{i}^{n-}+A_{ij}^{nm-}du_{j}^{m-}+R_{ki}^{pn}dl_{k}^{p}=0,\\
+r_{i}^{n+}+A_{ij}^{nm+}du_{j}^{m+}+R_{ki}^{pn}dl_{k}^{p}=0,\\
+R_{ij}^{pn}(u_{j}^{n+}+du_{j}^{n+}-u_{j}^{n-}-du_{j}^{n-})=d_{i}^{p}.
+\end{gather}
+
+\end_inset
+
+Solving the first two equations for 
+\begin_inset Formula $du_{j}^{m-}$
+\end_inset
+
+ and 
+\begin_inset Formula $du_{j}^{m+}$
+\end_inset
+
+ and combining them using the third equation leads to
+\begin_inset Formula 
+\begin{multline}
+R_{ij}^{pn}\left((A_{ij}^{nm+})^{-1}+(A_{ij}^{nm+})^{-1}\right)R_{ki}^{pn}dl_{k}^{p}=d_{i}^{p}-R_{ij}^{pn}(u_{j}^{n+}-u_{j}^{n-})\\
++R_{ij}^{pn}\left((A_{ij}^{nm+})^{-1}r_{i}^{n+}-(A_{ij}^{nm-})^{-1}r_{i}^{n-}\right).
+\end{multline}
+
+\end_inset
+
+We do not allow overlap between the fault interface and the absorbing boundary,
+ so 
+\begin_inset Formula $A_{ij}^{nm}$
+\end_inset
+
+ is the same for all components at a vertex.
+ As a result the matrix on the left hand side simplifies to
+\begin_inset Formula 
+\begin{equation}
+S_{ik}^{pn}=\delta_{ik}\left(\frac{1}{A^{nm+}}+\frac{1}{A^{nm-}}\right),
+\end{equation}
+
+\end_inset
+
+and
+\begin_inset Formula 
+\begin{equation}
+dl_{k}^{p}=(S_{ik}^{pn})^{-1}\left(d_{i}^{p}-R_{ij}^{pn}(u_{j}^{n+}-u_{j}^{n-})+R_{ij}^{pn}\left((A_{ij}^{nm+})^{-1}r_{i}^{n+}-(A_{ij}^{nm-})^{-1}r_{i}^{n-}\right)\right).
+\end{equation}
+
+\end_inset
+
+Now that we know the value of the increment in the Lagrange multiplier from
+ time 
+\begin_inset Formula $t$
+\end_inset
+
+ to time 
+\begin_inset Formula $t+\Delta t$
+\end_inset
+
+, we can correct the value for the displacement increment from time 
+\begin_inset Formula $t$
+\end_inset
+
+ to 
+\begin_inset Formula $t+\Delta t$
+\end_inset
+
+ using
+\begin_inset Formula 
+\begin{gather}
+\Delta du_{j}^{n-}=(A_{ij}^{nm-})^{-1}C_{ki}^{pn}dl_{k}^{p}\text{ and}\\
+\Delta du_{j}^{n+}=-(A_{ij}^{nm+})^{-1}C_{ki}^{pn}dl_{k}^{p}.
+\end{gather}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsubsection
+Arrays of Kinematic Rupture Components
+\end_layout
+
+\begin_layout Standard
+Multiple earthquake ruptures can be specified on a single fault surface.
+ This permits repeatedly rupturing the same portion of a fault or combining
+ earthquake rupture on one subset of the fault surface with steady aseismic
+ slip on another subset (the two subsets may overlap in both time and space).
+ A dynamic array of kinematic earthquake rupture components associates a
+ name (string) with each kinematic rupture.
+ The default dynamic array contains a single earthquake rupture, ``rupture''.
+ The 
+\family typewriter
+eq_srcs
+\family default
+ is the 
+\family typewriter
+FaultCohesiveKin
+\family default
+ facility for this dynamic array.
+ An example of setting the array of kinematic rupture components in a 
+\family typewriter
+.cfg
+\family default
+ file:
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.problem.interfaces.fault]
+\end_layout
+
+\begin_layout LyX-Code
+eq_srcs = [earthquake,creep]
+\end_layout
+
+\begin_layout Standard
+The output manager includes generic fault information (orientation) as well
+ as the final slip or slip rate (as in the case of the constant slip rate
+ slip time function) and slip initiation time for each kinematic rupture.
+ The name of the slip and slip initiation time vertex fields are of the
+ form 
+\family typewriter
+final_slip_NAME
+\family default
+ and 
+\family typewriter
+slip_time_NAME
+\family default
+, respectively, where 
+\family typewriter
+NAME
+\family default
+ refers to the name used in the dynamic array of kinematic ruptures, 
+\family typewriter
+eq_srcs
+\family default
+.
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float table
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\noindent
+\align center
+\begin_inset Caption
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "tab:fault:kin:output"
+
+\end_inset
+
+Fields available in output of fault information.
+\end_layout
+
+\end_inset
+
+
+\begin_inset VSpace defskip
+\end_inset
+
+
+\begin_inset Tabular
+<lyxtabular version="3" rows="8" columns="3">
+<features tabularvalignment="middle">
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="3.5in">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Field Type
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Field
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Description
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+vertex_info_fields
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+normal_dir
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Direction of fault normal in global coordinate system
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+strike_dir
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Direction of fault strike in global coordinate system
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+dip_dir
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Up-dip direction on hanging wall in global coordinate system
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+final_slip_NAME
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Vector of final slip (in fault coordinate system) in meters
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+slip_time
+\family default
+_
+\family typewriter
+\noun on
+NAME
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Time at which slip begins in seconds
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+vertex_data_fields
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+slip
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Slip vector at time step (in fault coordinate system) in meters
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+traction_change
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Change in fault tractions (in fault coordinate system) in Pa
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsubsection
+Kinematic Rupture Parameters
+\end_layout
+
+\begin_layout Standard
+The kinematic rupture parameters include the origin time and slip time function.
+ The slip initiation time in the slip time function is relative to the origin
+ time (default is 0).
+ This means that slip initiates at a point at a time corresponding to the
+ sum of the kinematic rupture's origin time and the slip initiation time
+ for that point.
+ An example of specifying the kinematic earthquake rupture properties and
+ components in a 
+\family typewriter
+.cfg
+\family default
+ file:
+\end_layout
+
+\begin_layout LyX-Code
+
+\size footnotesize
+[pylithapp.problem.interfaces.fault]
+\end_layout
+
+\begin_layout LyX-Code
+
+\size footnotesize
+eq_srcs = [earthquake,creep]
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+[pylithapp.problem.interfaces.fault.eq_srcs.earthquake]
+\end_layout
+
+\begin_layout LyX-Code
+
+\size footnotesize
+origin_time = 0.0*s ; default origin time
+\end_layout
+
+\begin_layout LyX-Code
+
+\size footnotesize
+slip_function = pylith.faults.StepSlipFn ; default slip time function
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+[pylithapp.problem.interfaces.fault.eq_srcs.creep]
+\end_layout
+
+\begin_layout LyX-Code
+
+\size footnotesize
+origin_time = 10.0*year ; start creep at 10.0 years
+\end_layout
+
+\begin_layout LyX-Code
+
+\size footnotesize
+slip_function = pylith.faults.ConstRateSlipFn ; switch to constant slip rate
+ slip function
+\end_layout
+
+\begin_layout Subsubsection
+Slip Time Function
+\end_layout
+
+\begin_layout Standard
+The current release of PyLith supports specification of the evolution of
+ fault slip using analytical expressions for the slip time history at each
+ point, where the parameters for the slip time function may vary over the
+ fault surface.
+ Currently, three slip time functions are available: (1) a step-function
+ for quasi-static modeling of earthquake rupture, (2) a constant slip rate
+ time function for modeling steady aseismic slip, and (3) the integral of
+ Brune's far-field time function 
+\begin_inset CommandInset citation
+LatexCommand cite
+key "Brune:1970"
+
+\end_inset
+
+ for modeling the dynamics of earthquake rupture.
+ Additional slip time functions will likely be available in future releases.
+ The default slip time function is the step-function slip function.
+\end_layout
+
+\begin_layout Paragraph
+Step-Function Slip Time Function
+\end_layout
+
+\begin_layout Standard
+This slip function prescribes a step in slip at a given time at a point:
+ 
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{gather}
+D(t)=\left\{ \begin{array}{cc}
+0 & 0\leq t<t_{r}\\
+D_{final} & t\ge t_{r}
+\end{array}\right.\,,
+\end{gather}
+
+\end_inset
+
+where 
+\begin_inset Formula $D(t)$
+\end_inset
+
+ is slip at time 
+\begin_inset Formula $t$
+\end_inset
+
+, 
+\begin_inset Formula $D_{final}$
+\end_inset
+
+ is the final slip, and 
+\begin_inset Formula $t_{r}$
+\end_inset
+
+ is the slip initiation time (time when rupture reaches the location).
+ The slip is specified independently for each of the components of slip,
+ and the slip and slip starting time may vary over the fault surface.
+\end_layout
+
+\begin_layout Description
+final_slip Spatial database of slip (
+\begin_inset Formula $D_{final})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Description
+slip_time Spatial database of slip initiation times (
+\begin_inset Formula $t_{r}$
+\end_inset
+
+).
+\end_layout
+
+\begin_layout Standard
+An example of setting these parameters in a 
+\family typewriter
+.cfg
+\family default
+ file is:
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.problem.interfaces.fault.eq_srcs.rupture]
+\end_layout
+
+\begin_layout LyX-Code
+slip_function = pylith.faults.StepSlipFn
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+[pylithapp.problem.interfaces.fault.eq_srcs.rupture.slip_function]
+\end_layout
+
+\begin_layout LyX-Code
+final_slip.iohandler.filename = final_slip.spatialdb
+\end_layout
+
+\begin_layout LyX-Code
+slip_time.iohandler.filename = sliptime.spatialdb
+\end_layout
+
+\begin_layout Standard
+The spatial database files for the slip time function specify the spatial
+ variation in the parameters for the slip time function, as shown in Table
+ 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "tab:step-function-db-params"
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float table
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\noindent
+\align center
+\begin_inset Caption
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "tab:step-function-db-params"
+
+\end_inset
+
+Values in spatial database used as parameters in the step function slip
+ time function.
+\end_layout
+
+\end_inset
+
+
+\begin_inset VSpace defskip
+\end_inset
+
+
+\begin_inset Tabular
+<lyxtabular version="3" rows="5" columns="3">
+<features tabularvalignment="middle">
+<column alignment="left" valignment="top" width="0">
+<column alignment="left" valignment="top" width="0">
+<column alignment="left" valignment="top" width="2.5in">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Spatial database
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Value
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Description
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+final_slip
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+left-lateral-slip
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Amount of left-lateral final slip in meters.
+ Use negative values for right-lateral slip.
+ Applies to faults in 2D and 3D only.
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+reverse-slip
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Amount of reverse slip in meters.
+ Use negative values for normal slip.
+ Applies to faults in 3D only.
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+fault-opening
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Amount of fault opening in meters.
+ Negative values imply penetration.
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+slip_time
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+slip-time
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Slip initiation time (
+\begin_inset Formula $t_{t})$
+\end_inset
+
+ in seconds.
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Constant Slip Rate Slip Time Function
+\end_layout
+
+\begin_layout Standard
+This slip function prescribes a constant slip rate for the evolution of
+ slip at a point: 
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{gather}
+D(t)=\left\{ \begin{array}{cc}
+0 & 0\leq t<t_{r}\\
+V(t-t_{r}) & t\ge t_{r}
+\end{array}\right.\,,
+\end{gather}
+
+\end_inset
+
+where 
+\begin_inset Formula $D(t)$
+\end_inset
+
+ is slip at time 
+\begin_inset Formula $t$
+\end_inset
+
+, 
+\begin_inset Formula $V$
+\end_inset
+
+ is the slip rate, and 
+\begin_inset Formula $t_{r}$
+\end_inset
+
+ is the slip initiation time (time when rupture reaches the location).
+ The slip rate is specified independently for each of the components of
+ slip, and the slip rate and slip starting time may vary over the fault
+ surface.
+\end_layout
+
+\begin_layout Description
+slip_rate Spatial database of slip rate (
+\begin_inset Formula $V)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Description
+slip_time Spatial database of slip initiation times (
+\begin_inset Formula $t_{r}$
+\end_inset
+
+).
+\end_layout
+
+\begin_layout Standard
+An example of setting these parameters in a 
+\family typewriter
+.cfg
+\family default
+ file is:
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.problem.interfaces.fault.eq_srcs.ruptures]
+\end_layout
+
+\begin_layout LyX-Code
+slip_function = pylith.faults.ConstRateSlipFn
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+[pylithapp.problem.interfaces.fault.eq_srcs.ruptures.slip_function]
+\end_layout
+
+\begin_layout LyX-Code
+slip_rate.iohandler.filename = slip_rate.spatialdb
+\end_layout
+
+\begin_layout LyX-Code
+slip_time.iohandler.filename = sliptime.spatialdb
+\end_layout
+
+\begin_layout Standard
+The spatial database files for the slip time function specify the spatial
+ variation in the parameters for the slip time function, as shown in Table
+ 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "tab:const-slip-rate-db-params"
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float table
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\noindent
+\align center
+\begin_inset Caption
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "tab:const-slip-rate-db-params"
+
+\end_inset
+
+Values in spatial database used as parameters in the constant slip rate
+ slip time function.
+\end_layout
+
+\end_inset
+
+
+\begin_inset VSpace defskip
+\end_inset
+
+
+\begin_inset Tabular
+<lyxtabular version="3" rows="5" columns="3">
+<features tabularvalignment="middle">
+<column alignment="left" valignment="top" width="0">
+<column alignment="left" valignment="top" width="0">
+<column alignment="left" valignment="top" width="2.5in">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Spatial database
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Value
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Description
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+slip_rate
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+left-lateral-slip
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Slip rate for left-lateral final slip in meters per second.
+ Use negative values for right-lateral slip.
+ Applies to faults in 2D and 3D only.
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+reverse-slip
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Slip rate for reverse slip in meters per second.
+ Use negative values for normal slip.
+ Applies to faults in 3D only.
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+fault-opening
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Slip rate for fault opening in meters per second.
+ Negative values imply penetration.
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+slip_time
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+slip-time
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Slip initiation time (
+\begin_inset Formula $t_{t})$
+\end_inset
+
+ in seconds.
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Brune Slip Time Function
+\end_layout
+
+\begin_layout Standard
+We use an integral of Brune's far-field time function 
+\begin_inset CommandInset citation
+LatexCommand cite
+key "Brune:1970"
+
+\end_inset
+
+ to describe the evolution in time of slip at a point: 
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{gather}
+D(t)=\left\{ \begin{array}{cc}
+0 & 0\leq t<t_{r}\\
+D_{final}\left(1-exp\left(-\frac{t-t_{r}}{t_{0}}\right)\left(1+\frac{t-t_{r}}{t_{0}}\right)\right) & t\ge t_{r}
+\end{array}\right.\,,\\
+t_{0}=0.6195t_{\mathit{rise}}\,,
+\end{gather}
+
+\end_inset
+
+where 
+\begin_inset Formula $D(t)$
+\end_inset
+
+ is slip at time 
+\begin_inset Formula $t$
+\end_inset
+
+, 
+\begin_inset Formula $D_{final}$
+\end_inset
+
+ is the final slip at the location, 
+\begin_inset Formula $t_{r}$
+\end_inset
+
+ is the slip initiation time (time when rupture reaches the location), and
+ 
+\begin_inset Formula $t_{\mathit{rise}}$
+\end_inset
+
+ is the rise time.
+\end_layout
+
+\begin_layout Description
+slip Spatial database of final slip distribution (
+\begin_inset Formula $D_{final})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Description
+slip_time Spatial database of slip initiation times (
+\begin_inset Formula $t_{r}$
+\end_inset
+
+).
+\end_layout
+
+\begin_layout Description
+rise_time Spatial database for rise time (
+\begin_inset Formula $t_{\mathit{rise}}$
+\end_inset
+
+).
+\end_layout
+
+\begin_layout Standard
+An example of setting these parameters in a 
+\family typewriter
+.cfg
+\family default
+ file is:
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.problem.interfaces.fault.eq_srcs.ruptures]
+\end_layout
+
+\begin_layout LyX-Code
+slip_function = pylith.faults.BruneSlipFn
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+[pylithapp.problem.interfaces.fault.eq_srcs.rupture.slip_function]
+\end_layout
+
+\begin_layout LyX-Code
+slip.iohandler.filename = finalslip.spatialdb
+\end_layout
+
+\begin_layout LyX-Code
+rise_time.iohandler.filename = risetime.spatialdb
+\end_layout
+
+\begin_layout LyX-Code
+slip_time.iohandler.filename = sliptime.spatialdb
+\end_layout
+
+\begin_layout Standard
+The spatial database files for the slip time function specify the spatial
+ variation in the parameters for the slip time function, as shown in Table
+ 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "tab:Brune-slip-db-params"
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float table
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\noindent
+\align center
+\begin_inset Caption
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "tab:Brune-slip-db-params"
+
+\end_inset
+
+Values in spatial database used as parameters in the Brune slip time function.
+\end_layout
+
+\end_inset
+
+
+\begin_inset VSpace defskip
+\end_inset
+
+
+\begin_inset Tabular
+<lyxtabular version="3" rows="6" columns="3">
+<features tabularvalignment="middle">
+<column alignment="left" valignment="top" width="0">
+<column alignment="left" valignment="top" width="0">
+<column alignment="left" valignment="top" width="2.5in">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Spatial database
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Value
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Description
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+slip
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+left-lateral-slip
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Amount of left-lateral final slip in meters.
+ Use negative values for right-lateral slip.
+ Applies to faults in 2D and 3D only.
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+reverse-slip
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Amount of reverse slip in meters.
+ Use negative values for normal slip.
+ Applies to faults in 3D only.
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+fault-opening
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Amount of fault opening in meters.
+ Negative values imply penetration.
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+rise_time
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+rise-time
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Rise time (
+\begin_inset Formula $t_{r})$
+\end_inset
+
+ in seconds.
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+slip_time
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+slip-time
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Slip initiation time (
+\begin_inset Formula $t_{t})$
+\end_inset
+
+ in meters.
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Liu-Cosine Slip Time Function
+\end_layout
+
+\begin_layout Standard
+This slip time function, proposed by Liu, Archuleta, and Hartzell for use
+ in ground-motion modeling
+\begin_inset CommandInset citation
+LatexCommand cite
+key "Liu:etal:2006"
+
+\end_inset
+
+, combines several cosine and sine functions together to create a slip time
+ history with a sharp rise and gradual termination with a finite duration
+ of slip.
+ The evolution of slip at a point follows: 
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{gather}
+D(t)=\left\{ \begin{array}{cc}
+D_{\mathit{final}}C_{n}\left(0.7t-0.7\frac{t_{1}}{\pi}\sin\frac{\pi t}{t_{1}}-1.2\frac{t_{1}}{\pi}\left(\cos\frac{\pi t}{2t_{1}}-1\right)\right) & 0\leq t<t_{1}\\
+D_{\mathit{final}}C_{n}\left(1.0t-0.7\frac{t1}{\pi}\sin\frac{\pi t}{t_{1}}+0.3\frac{t2}{\pi}\sin\frac{\pi(t-t1)}{t_{2}}+\frac{1.2}{\pi}t_{1}-0.3t_{1}\right) & t_{1}\leq t<2t_{1}\\
+D_{\mathit{final}}C_{n}\left(0.7-0.7\cos\frac{\pi t}{t_{1}}+0.6\sin\frac{\pi t}{2t_{1}}\right) & 2t_{1}\leq t\leq t_{0}
+\end{array}\right.\,,\\
+C_{n}=\frac{\pi}{1.4\pi t_{1}+1.2t_{1}+0.3\pi t_{2}},\\
+t_{0}=1.525t_{\mathit{rise}},\\
+t_{1}=0.13t_{0},\\
+t_{2}=t_{0}-t_{1},
+\end{gather}
+
+\end_inset
+
+where 
+\begin_inset Formula $D(t)$
+\end_inset
+
+ is slip at time 
+\begin_inset Formula $t$
+\end_inset
+
+, 
+\begin_inset Formula $D_{final}$
+\end_inset
+
+ is the final slip at the location, 
+\begin_inset Formula $t_{r}$
+\end_inset
+
+ is the slip initiation time (time when rupture reaches the location), and
+ 
+\begin_inset Formula $t_{\mathit{rise}}$
+\end_inset
+
+ is the rise time.
+\end_layout
+
+\begin_layout Description
+slip Spatial database of final slip distribution (
+\begin_inset Formula $D_{final})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Description
+slip_time Spatial database of slip initiation times (
+\begin_inset Formula $t_{r}$
+\end_inset
+
+).
+\end_layout
+
+\begin_layout Description
+rise_time Spatial database for rise time (
+\begin_inset Formula $t_{\mathit{rise}}$
+\end_inset
+
+).
+\end_layout
+
+\begin_layout Standard
+The spatial database files for the slip time function use the same parameters
+ for the slip time function as the Brune slip time function shown in Table
+ 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "tab:Brune-slip-db-params"
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Paragraph
+Time-History Slip Time Function
+\end_layout
+
+\begin_layout Standard
+This slip time function reads the slip time function from a data file, so
+ it can have an arbitrary shape.
+ The slip and slip initiation times are specified using spatial databases,
+ so the slip time function, in general, will use a normalized amplitude.
+\end_layout
+
+\begin_layout Description
+slip Spatial database of final slip distribution (
+\begin_inset Formula $D_{final})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Description
+slip_time Spatial database of slip initiation times (
+\begin_inset Formula $t_{r}$
+\end_inset
+
+).
+\end_layout
+
+\begin_layout Description
+time_history Temporal database for slip evolution.
+\end_layout
+
+\begin_layout Standard
+An example of setting these parameters in a 
+\family typewriter
+.cfg
+\family default
+ file is:
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.problem.interfaces.fault.eq_srcs.ruptures]
+\end_layout
+
+\begin_layout LyX-Code
+slip_function = pylith.faults.TimeHistorySlipFn
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+[pylithapp.problem.interfaces.fault.eq_srcs.rupture.slip_function]
+\end_layout
+
+\begin_layout LyX-Code
+slip.iohandler.filename = finalslip.spatialdb
+\end_layout
+
+\begin_layout LyX-Code
+slip_time.iohandler.filename = sliptime.spatialdb
+\end_layout
+
+\begin_layout LyX-Code
+time_history.iohandler.filename = myfunction.timedb
+\end_layout
+
+\begin_layout Standard
+The spatial database files for the slip time function specify the spatial
+ variation in the parameters for the slip time function, as shown in Table
+ 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "tab:Brune-slip-db-params-2"
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float table
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\noindent
+\align center
+\begin_inset Caption
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "tab:Brune-slip-db-params-2"
+
+\end_inset
+
+Values in spatial database used as parameters in the time history slip time
+ function.
+\end_layout
+
+\end_inset
+
+
+\begin_inset VSpace defskip
+\end_inset
+
+
+\begin_inset Tabular
+<lyxtabular version="3" rows="6" columns="3">
+<features tabularvalignment="middle">
+<column alignment="left" valignment="top" width="0">
+<column alignment="left" valignment="top" width="0">
+<column alignment="left" valignment="top" width="2.5in">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Spatial database
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Value
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Description
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+slip
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+left-lateral-slip
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Amount of left-lateral final slip in meters.
+ Use negative values for right-lateral slip.
+ Applies to faults in 2D and 3D only.
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+reverse-slip
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Amount of reverse slip in meters.
+ Use negative values for normal slip.
+ Applies to faults in 3D only.
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+fault-opening
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Amount of fault opening in meters.
+ Negative values imply penetration.
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+rise_time
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+rise-time
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Rise time (
+\begin_inset Formula $t_{r})$
+\end_inset
+
+ in seconds.
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+slip_time
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+slip-time
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Slip initiation time (
+\begin_inset Formula $t_{t})$
+\end_inset
+
+ in meters.
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Dynamic Earthquake Rupture
+\end_layout
+
+\begin_layout Standard
+Dynamic fault interfaces use the FaultCohesiveDyn object to specify a fault
+ constitutive model to govern the fault tractions (friction) and the resulting
+ slip.
+ When friction is large enough such that there is no sliding on the fault,
+ the fault is locked (slip is zero) and the Lagrange multipliers assume
+ their values just as they do in kinematic ruptures.
+ In this case, the Lagrange multipliers correspond to the forces necessary
+ to keep the slip zero.
+ When the driving forces exceed those allowed by friction, we reduce the
+ values of the Lagrange multipliers to those consistent with friction from
+ the fault constitutive model.
+ When we reduce the Lagrange multipliers, we must increment the slip accordingly
+ to maintain consistency in the algebraic system of equations.
+\end_layout
+
+\begin_layout Subsubsection
+Governing Equations
+\end_layout
+
+\begin_layout Standard
+The algebraic systems of equations for dynamic earthquake rupture are the
+ same as those for kinematic rupture
+\begin_inset Formula 
+\begin{equation}
+\left[\begin{array}{cc}
+\underline{A} & \underline{C}^{T}\\
+\underline{C} & 0
+\end{array}\right]\left[\begin{array}{c}
+\overrightarrow{u}\\
+\overrightarrow{l}
+\end{array}\right]=\left[\begin{array}{c}
+\overrightarrow{b}\\
+\overrightarrow{d}
+\end{array}\right].
+\end{equation}
+
+\end_inset
+
+Enforcing the limits imposed on the Lagrange multipliers by the fault constituti
+ve model requires determining the increment in slip for an increment in
+ the Lagrange multipliers.
+ The increment in the Lagrange multipliers is the difference between the
+ value computed for the current slip (either zero or the slip at the previous
+ time step) and the value computed from the fault constitutive model.
+ Starting from our system of algebraic equations,
+\begin_inset Formula 
+\begin{equation}
+A_{ij}^{nm}u_{j}^{m}+C_{ji}^{pn}l_{j}^{p}=b_{i}^{n},
+\end{equation}
+
+\end_inset
+
+we compute the sensitivity for the given loading and boundary conditions,
+\begin_inset Formula 
+\begin{equation}
+A_{ij}^{nm}\partial u_{j}^{m}=-C_{ji}^{pn}\partial l_{j}^{p}.
+\end{equation}
+
+\end_inset
+
+Computing the increment in the slip requires computing the increment in
+ the displacements.
+ Solving this equation rigorously would require inverting the system Jacobian,
+ which we do not want to do unless it is diagonal (as it is in the case
+ of the lumped formulations).
+ 
+\end_layout
+
+\begin_layout Paragraph
+Non-Diagonal A
+\end_layout
+
+\begin_layout Standard
+In general A is a sparse matrix with off-diagonal terms of the form
+\begin_inset Formula 
+\begin{equation}
+A=\left(\begin{array}{ccc}
+A_{0} & A_{1} & A_{2}\\
+A_{3} & A_{n-} & 0\\
+A_{4} & 0 & A_{n+}
+\end{array}\right),
+\end{equation}
+
+\end_inset
+
+where the degrees of freedom on either side of the fault are uncoupled.
+ We formulate two small linear systems involving just the degrees of freedom
+ associated with vertices on either the positive or negative sides of the
+ fault,
+\begin_inset Formula 
+\begin{gather}
+A_{ij}^{nm-}\partial u_{j}^{m-}=-R_{ij}^{pn}\partial l_{j}^{p},\\
+A_{ij}^{nm+}\partial u_{j}^{m+}=R_{ij}^{pn}\partial l_{j}^{p},
+\end{gather}
+
+\end_inset
+
+where we have replaced 
+\begin_inset Formula $\underline{C}$
+\end_inset
+
+ with 
+\begin_inset Formula $\underline{R}$
+\end_inset
+
+ to denote the explicit inclusion of the signs for the terms in 
+\begin_inset Formula $\underline{C}$
+\end_inset
+
+ associated with the positive (
+\begin_inset Formula $n^{+}$
+\end_inset
+
+) and negative (
+\begin_inset Formula $n^{-}$
+\end_inset
+
+) sides of the fault.
+ After solving these two linear systems of equations, we compute the increment
+ in slip using
+\begin_inset Formula 
+\begin{equation}
+\partial d_{i}^{p}=R_{ij}^{pn}(\partial u_{j}^{n+}-\partial u_{j}^{n-}).
+\end{equation}
+
+\end_inset
+
+The solution of these two linear systems gives the increment in slip assuming
+ all the degrees of freedom except those immediately adjacent to the fault
+ remain fixed.
+ In real applications where the deformation associated with fault slip is
+ localized around the fault, this provides good enough approximations so
+ that the nonlinear solver converges quickly.
+ In problems where deformation associated with slip on the fault is not
+ localized (as in the case in some of the example problems), the increment
+ in slip computed by solving these two linear systems is not a good approximatio
+n and the nonlinear solve requires a large number of iterations.
+\end_layout
+
+\begin_layout Standard
+We use the PETSc Krylov subspace solver (KSP) to solve these two linear
+ systems.
+ The PETSc settings for the KSP object are set in the same manner as the
+ main solver, except we use the prefix 
+\family typewriter
+friction_
+\family default
+ in all of the settings related to the KSP solver for these two linear systems.
+ For example, to use the recommended additive Schwarz preconditioner in
+ the friction sensitivity solves, the settings in a 
+\family typewriter
+.cfg
+\family default
+ file are:
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.petsc]
+\end_layout
+
+\begin_layout LyX-Code
+friction_pc_type = asm
+\end_layout
+
+\begin_layout Standard
+See the examples in Sections 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sec:Tutorial-3d-hex8-friction"
+
+\end_inset
+
+ and 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sec:tutorial:shearwave:quad4"
+
+\end_inset
+
+ for details.
+\end_layout
+
+\begin_layout Paragraph
+Diagonal A
+\end_layout
+
+\begin_layout Standard
+With a lumped Jacobian matrix, we can solve for the increment in slip directly,
+\begin_inset Formula 
+\begin{equation}
+\partial d_{i}^{p}=-C_{ij}^{pn}(A_{jk}^{nm})^{-1}C_{lk}^{pm}\partial l_{l}^{p}.
+\end{equation}
+
+\end_inset
+
+By not allowing the fault interface to overlap with the absorbing boundary,
+ the terms in 
+\begin_inset Formula $A$
+\end_inset
+
+ for a given vertex are identical and the expression on the right-hand side
+ reduces to
+\begin_inset Formula 
+\begin{equation}
+\partial d_{i}^{p}=-\left(\frac{1}{A^{n+}}+\frac{1}{A^{n-}}\right)\partial l_{i}^{p}.
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsubsection
+Dynamic Rupture Parameters
+\end_layout
+
+\begin_layout Standard
+The components of the FaultCohesiveDyn object include
+\end_layout
+
+\begin_layout Description
+open_free_surface If true, enforce traction free surface when the fault
+ opens, otherwise apply prescribed tractions even when the fault opens (default
+ is true); to mimic a dike opening, use false.
+\end_layout
+
+\begin_layout Description
+zero_tolerance Tolerance for detecting zero values (default is 1.0e-10);
+ should be larger than absolute tolerance in KSP solves.
+\end_layout
+
+\begin_layout Description
+traction_perturbation Prescribed tractions on fault surface (generally used
+ for nucleating earthquake ruptures; default is none).
+\end_layout
+
+\begin_layout Description
+friction Fault constitutive model.
+\end_layout
+
+\begin_layout Standard
+An example of specifying the dynamic earthquake rupture properties and component
+s in a 
+\family typewriter
+.cfg
+\family default
+ file:
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.problem.interfaces.fault]
+\end_layout
+
+\begin_layout LyX-Code
+open_free_surface = True ; default
+\end_layout
+
+\begin_layout LyX-Code
+traction_perturbation = pylith.faults.TractPerturbation ; not default
+\end_layout
+
+\begin_layout LyX-Code
+traction_perturbation.db_initial = spatialdata.spatialdb.SimpleDB
+\end_layout
+
+\begin_layout LyX-Code
+traction_perturbation.db_initial.iohandler.filename = tractions.spatialdb
+\end_layout
+
+\begin_layout LyX-Code
+friction = pylith.friction.StaticFriction
+\end_layout
+
+\begin_layout LyX-Code
+friction.db_properties = spatialdata.spatialdb.SimpleDB
+\end_layout
+
+\begin_layout LyX-Code
+friction.db_properties.iohandler.filename = friction.spatialdb
+\end_layout
+
+\begin_layout Standard
+The prescribed traction perturbation is specified using the same fault coordinat
+e system as the slip directions in the kinematic ruptures.
+ The perurbation has the same functional form as the time-dependent boundary
+ conditions (and same spatial databases).
+ Table 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "tab:fault:cohesive:dyn:prescribed:tractions"
+
+\end_inset
+
+ gives the values in the spatial database for the prescribed tractions.
+ Table 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "tab:fault:dyn:output"
+
+\end_inset
+
+ shows the fields available for output.
+ Additional fields are available depending on the fault constitutive model.
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float table
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\noindent
+\align center
+\begin_inset Caption
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "tab:fault:cohesive:dyn:prescribed:tractions"
+
+\end_inset
+
+Values in spatial databases for prescribed tractions.
+\end_layout
+
+\end_inset
+
+
+\begin_inset VSpace defskip
+\end_inset
+
+
+\begin_inset Tabular
+<lyxtabular version="3" rows="22" columns="4">
+<features tabularvalignment="middle">
+<column alignment="left" valignment="top" width="0">
+<column alignment="left" valignment="top" width="0">
+<column alignment="left" valignment="top" width="0">
+<column alignment="left" valignment="top" width="2.5in">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Spatial database
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Dimension
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Value
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Description
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+db_initial
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+1D
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+traction-normal
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Normal traction (tension is positive)
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+2D
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+traction-shear
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Left-lateral shear traction (reverse shear for dipping faults)
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+traction-normal
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Normal traction (tension is positive)
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+3D
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+traction-shear-leftlateral
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Left-lateral shear traction
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+traction-shear-updip
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Reverse shear traction
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+traction-normal
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Normal traction (tension is positive)
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+db_rate
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+1D
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+traction-rate-normal
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Rate of change of normal traction (tension is positive)
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+2D
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+traction-rate-shear
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Rate of change of left-lateral shear traction (reverse shear for dipping
+ faults)
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+traction-rate-normal
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Rate of change of normal traction (tension is positive)
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+3D
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+traction-rate-leftlateral
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Rate of change of left-lateral shear traction
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+traction-rate-shear-updip
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Rate of change of reverse shear traction
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+traction-rate-normal
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Rate of change of normal traction (tension is positive)
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+all
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+rate-start-time
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Time at which rate of change begins
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+db_change
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+1D
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+traction-normal
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Change in normal traction (tension is positive)
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+2D
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+traction-shear
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Change in left-lateral shear traction (reverse shear for dipping faults)
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+traction-normal
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Change in normal traction (tension is positive)
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+3D
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+traction-leftlateral
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Change in left-lateral shear traction
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+traction-shear-updip
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Change in reverse shear traction
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+traction-normal
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Change in normal traction (tension is positive)
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+all
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+change-start-time
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Time at which change begins
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+th_change
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+all
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+None
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Time history for change
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Float table
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\noindent
+\align center
+\begin_inset Caption
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "tab:fault:dyn:output"
+
+\end_inset
+
+Fields available in output of fault information.
+\end_layout
+
+\end_inset
+
+
+\begin_inset VSpace defskip
+\end_inset
+
+
+\begin_inset Tabular
+<lyxtabular version="3" rows="11" columns="3">
+<features tabularvalignment="middle">
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="3.5in">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Field Type
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Field
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Description
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+vertex_info_fields
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+normal_dir
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\align left
+Direction of fault normal in global coordinate system
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+strike_dir
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\align left
+Direction of fault strike in global coordinate system
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+dip_dir
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\align left
+Up-dip direction on hanging wall in global coordinate system
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+traction_initial
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\align left
+Initial tractions (if specified) in fault coordinate system
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+traction_rate
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\align left
+Rate of change in tractions (if specified) in fault coordinate system
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+rate_start_time
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\align left
+Time at which rate of change begins (if specified)
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+traction_change
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\align left
+Change in tractions (if specified) in fault coordinate system
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+change_start_time
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\align left
+Time at which change occurs (if specified)
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+vertex_data_fields
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+slip
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\align left
+Slip vector at time step (in fault coordinate system) in meters
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+traction
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\align left
+Fault tractions (in fault coordinate system) in Pa
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsubsection
+Fault Constitutive Models
+\begin_inset CommandInset label
+LatexCommand label
+name "sec:fault:constitutive:models"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+PyLith provides four fault constitutive models.
+ Future releases may contain additional models, and a template is provided
+ for you to construct your own (see Section 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sec:Extending:FaultConstitutiveModels"
+
+\end_inset
+
+).
+ The fault constitutive model implementations are independent of dimension
+ and work in both 2D and 3D.
+ In solving the governing equations, PyLith will use a scalar representation
+ of the shear traction in 2D and a vector representation of the shear traction
+ in 3D, with the shear traction resolved in the direction of current slip.
+ The fault constitutive models contain a common set of properties and components
+:
+\end_layout
+
+\begin_layout Description
+label Name of the friction model.
+\end_layout
+
+\begin_layout Description
+db_properties Spatial database of the friction model parameters (default
+ is SimpleDB).
+\end_layout
+
+\begin_layout Description
+db_initial_state Spatial database for initial state variables (default is
+ none).
+\end_layout
+
+\begin_layout Paragraph
+Static Friction
+\end_layout
+
+\begin_layout Standard
+The static friction model produces shear tractions proportional to the fault
+ normal traction plus a cohesive stress,
+\begin_inset Formula 
+\begin{equation}
+T_{f}=\begin{cases}
+T_{c}-\mu_{f}T_{n} & T_{n}\leq0\\
+0 & T_{n}>0
+\end{cases}.
+\end{equation}
+
+\end_inset
+
+The spatial database file for the static friction model properties specifies
+ the spatial variation of the parameters given in Table 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "tab:static:friction:properties"
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float table
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\noindent
+\align center
+\begin_inset Caption
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "tab:static:friction:properties"
+
+\end_inset
+
+Values in the spatial database for constant friction parameters.
+\end_layout
+
+\end_inset
+
+
+\begin_inset VSpace defskip
+\end_inset
+
+
+\begin_inset Tabular
+<lyxtabular version="3" rows="3" columns="2">
+<features tabularvalignment="middle">
+<column alignment="center" valignment="top" width="0">
+<column alignment="left" valignment="top" width="2.5in">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Value
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\align center
+
+\series bold
+Description
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+friction-coefficient
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\align center
+Coefficient of friction, 
+\begin_inset Formula $\mu_{f}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+cohesion
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\align center
+Cohesive stress, 
+\begin_inset Formula $T_{c}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+\begin_inset CommandInset label
+LatexCommand label
+name "sec:friction:slip:weakening"
+
+\end_inset
+
+Slip-Weakening Friction
+\end_layout
+
+\begin_layout Standard
+The linear slip-weakening friction model produces shear tractions equal
+ to the cohesive stress plus a contribution proportional to the fault normal
+ traction that decreases from a static value to a dynamic value as slip
+ progresses,
+\begin_inset Formula 
+\begin{equation}
+T_{f}=\begin{cases}
+T_{c}-(\mu_{s}-(\mu_{s}-\mu_{d})\frac{d}{d_{0}})T_{n} & d\leq d_{0}\text{ and }T_{n}\leq0\\
+T_{c}-\mu_{d}T_{n} & d>d_{0}\text{ and }T_{n}\leq0\\
+0 & T_{n}>0
+\end{cases}
+\end{equation}
+
+\end_inset
+
+The spatial database files for the slip-weakening friction model properties
+ and state variables specify the spatial variation of the fault constitutive
+ model parameters given in Table 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "tab:slip:weakening:properties:statevars"
+
+\end_inset
+
+.
+ As long as the fault is locked, the initial state variables are zero, so
+ specifying the initial state variables for slip-weakening friction is rare.
+ The slip-weakening friction also includes a parameter, 
+\family typewriter
+force_healing
+\family default
+, to control healing.
+ In quasi-static simulations, one usually wants slip confined to a single
+ time step (
+\family typewriter
+force_healing
+\family default
+ = True), whereas in a dynamic simulation slip occurs over many time steps
+ (
+\family typewriter
+force_healing
+\family default
+ = False; default behavior) and fault healing is often neglected.
+ The properties include:
+\end_layout
+
+\begin_layout Description
+force_healing Flag indicating whether healing (cumalative slip state variable
+ reset to zero) is forced after every time step.
+\end_layout
+
+\begin_layout Standard
+An example of setting the properties for the rate and state friction component
+ in a 
+\family typewriter
+.cfg
+\family default
+ file is:
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.problem.interfaces.fault]
+\end_layout
+
+\begin_layout LyX-Code
+friction = pylith.friction.SlipWeakening ; Change friction model from the
+ default
+\end_layout
+
+\begin_layout LyX-Code
+friction.force_healing = False ; default value
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float table
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\noindent
+\align center
+\begin_inset Caption
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "tab:slip:weakening:properties:statevars"
+
+\end_inset
+
+Values in spatial databases for slip-weakening friction.
+\end_layout
+
+\end_inset
+
+
+\begin_inset VSpace defskip
+\end_inset
+
+
+\begin_inset Tabular
+<lyxtabular version="3" rows="7" columns="3">
+<features tabularvalignment="middle">
+<column alignment="left" valignment="top" width="0">
+<column alignment="left" valignment="top" width="0">
+<column alignment="left" valignment="top" width="2.5in">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Spatial database
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Value
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Description
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+db_properties
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+static-coefficient
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Static coefficient of friction, 
+\begin_inset Formula $\mu_{s}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+dynamic-coefficient
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Dynamic coefficient of friction, 
+\begin_inset Formula $\mu_{d}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+slip-weakening-parameter
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Slip-weakening parameter, 
+\begin_inset Formula $d_{0}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+cohesion
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Cohesive stress, 
+\begin_inset Formula $T_{c}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+db_initial_state
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+cumulative-slip
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Cumulative slip, 
+\begin_inset Formula $d$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+previous-slip
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Slip at previous time step, 
+\begin_inset Formula $d(t-\Delta t)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Time-Weakening Friction
+\end_layout
+
+\begin_layout Standard
+The linear time-weakening friction model is analogous to the linear slip-weakeni
+ng friction model with time replacing slip.
+ It produces shear tractions equal to the cohesive stress plus a contribution
+ proportional to the fault normal traction that decreases from a static
+ value to a dynamic value as time progresses,
+\begin_inset Formula 
+\begin{equation}
+T_{f}=\begin{cases}
+T_{c}-(\mu_{s}-(\mu_{s}-\mu_{d})\frac{t}{t_{0}})T_{n} & t\leq t_{0}\text{ and }T_{n}\leq0\\
+T_{c}-\mu_{d}T_{n} & t>t_{0}\text{ and }T_{n}\leq0\\
+0 & T_{n}>0
+\end{cases}
+\end{equation}
+
+\end_inset
+
+The spatial database files for the time-weakening friction model properties
+ and state variables specify the spatial variation of the fault constitutive
+ model parameters given in Table 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "tab:time:weakening:properties:statevars"
+
+\end_inset
+
+.
+ As long as the fault is locked, the initial state variable is zero, so
+ specifying the initial state variable for time-weakening friction is rare.
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float table
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\noindent
+\align center
+\begin_inset Caption
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "tab:time:weakening:properties:statevars"
+
+\end_inset
+
+Values in spatial databases for time-weakening friction.
+\end_layout
+
+\end_inset
+
+
+\begin_inset VSpace defskip
+\end_inset
+
+
+\begin_inset Tabular
+<lyxtabular version="3" rows="6" columns="3">
+<features tabularvalignment="middle">
+<column alignment="left" valignment="top" width="0">
+<column alignment="left" valignment="top" width="0">
+<column alignment="left" valignment="top" width="2.5in">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Database
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Value
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Description
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+db_properties
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+static-coefficient
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Static coefficient of friction, 
+\begin_inset Formula $\mu_{s}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+dynamic-coefficient
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Dynamic coefficient of friction, 
+\begin_inset Formula $\mu_{d}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+time-weakening-parameter
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Time-weakening parameter, 
+\begin_inset Formula $t_{0}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+cohesion
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Cohesive stress, 
+\begin_inset Formula $T_{c}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+db_initial_state
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+elapsed-time
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Elasped time of slip, 
+\begin_inset Formula $t$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+\begin_inset CommandInset label
+LatexCommand label
+name "sec:friction:rate:state:ageing"
+
+\end_inset
+
+Rate- and State-Friction with Ageing Law
+\end_layout
+
+\begin_layout Standard
+The Dieterich-Ruina rate and state friction model produces shear tractions
+ equal to the cohesive stress plus a contribution proportional to the fault
+ normal traction that depends on a state variable,
+\begin_inset Formula 
+\begin{gather}
+T_{f}=\begin{cases}
+T_{c}-\mu_{f}T_{n} & T_{n}\leq0\\
+0 & T_{n}>0
+\end{cases}\\
+\mu_{f}=\begin{cases}
+\mu_{0}+a\ln\left(\frac{V}{V_{0}}\right)+b\ln\left(\frac{V_{0}\theta}{L}\right) & V\ge V_{\mathit{linear}}\\
+\mu_{0}+a\ln\left(\frac{V_{linear}}{V_{0}}\right)+b\ln\left(\frac{V_{0}\theta}{L}\right)-a\left(1-\frac{V}{V_{linear}}\right) & V<V_{linear}
+\end{cases}\\
+\frac{d\theta}{dt}=1-\frac{V\theta}{L}
+\end{gather}
+
+\end_inset
+
+where 
+\begin_inset Formula $V$
+\end_inset
+
+ is slip rate, 
+\begin_inset Formula $V_{linear}$
+\end_inset
+
+ is a cutoff for a linear slip rate dependence, 
+\begin_inset Formula $a$
+\end_inset
+
+ and 
+\begin_inset Formula $b$
+\end_inset
+
+ are coefficients, 
+\begin_inset Formula $L$
+\end_inset
+
+ is the characteristic slip distance, 
+\begin_inset Formula $\theta$
+\end_inset
+
+ is a state variable.
+ With an interative solver in quasi-static simulations with its small, but
+ nonzero residual tolerance we never encounter zero slip rates in quasi-static
+ simulations.
+ Instead we want to avoid significant variations in the coefficient of friction
+ for slip rates on the same order as our residual tolerance.
+ We regularize the rate and state friction model by imposing a linearization
+ of the variation of the coefficient of friction with slip rate when the
+ slip rate drops below a cutoff slip rate, 
+\begin_inset Formula $V_{linear}$
+\end_inset
+
+ (
+\family typewriter
+linear_slip_rate
+\family default
+ property with a default value of 1.0e-12).
+ Note that this is different than the popular inverse hyperbolic sine regulariza
+tion proposed by Ben-Zion and Rice 
+\begin_inset CommandInset citation
+LatexCommand cite
+key "BenZion:Rice:1997"
+
+\end_inset
+
+ to permit zero slip rates.
+ Following Kaneko 
+\shape italic
+et al.
+
+\shape default
+ 
+\begin_inset CommandInset citation
+LatexCommand cite
+key "Kaneko:etal:2008"
+
+\end_inset
+
+, we integrate the evolution equation for the state variable, keeping slip
+ rate constant, to get
+\begin_inset Formula 
+\begin{equation}
+\theta(t+\Delta t)=\theta(t)\exp\left(\frac{-V(t)\Delta t}{L}\right)+\frac{L}{V(t)}\left(1-\exp\left(-\frac{V(t)\Delta t}{L}\right)\right).
+\end{equation}
+
+\end_inset
+
+As the slip rate approaches zero, the first exponential term approaches
+ 1.
+ Using the first three terms of the Taylor series expansion of the second
+ exponential yields
+\begin_inset Formula 
+\begin{equation}
+\theta(t+\Delta t)=\begin{cases}
+\theta(t)\exp\left(-\frac{V(t)\Delta t}{L}\right)+\Delta t-\frac{1}{2}\frac{V(t)\Delta t^{2}}{L} & \frac{V(t)\Delta t}{L}<0.00001\\
+\theta(t)\exp\left(-\frac{V(t)\Delta t}{L}\right)+\frac{L}{V(t)}\left(1-\exp\left(-\frac{V(t)\Delta t}{L}\right)\right) & \frac{V(t)\Delta t}{L}\ge0.00001
+\end{cases}.
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+The properties include:
+\end_layout
+
+\begin_layout Description
+linear_slip_rate Nondimensional slip rate at which linearization occurs,
+ 
+\begin_inset Formula $V_{linear}$
+\end_inset
+
+.
+ In quasi-static simulations it should be about one order of magnitude larger
+ than absolute tolerance in solve.
+\end_layout
+
+\begin_layout Standard
+An example of setting the properties for the rate- and state-friction component
+ in a 
+\family typewriter
+.cfg
+\family default
+ file is:
+\end_layout
+
+\begin_layout LyX-Code
+
+\size footnotesize
+[pylithapp.problem.interfaces.fault]
+\end_layout
+
+\begin_layout LyX-Code
+
+\size footnotesize
+friction = pylith.friction.RateStateAgeing ; Change friction model from the
+ default
+\end_layout
+
+\begin_layout LyX-Code
+
+\size footnotesize
+friction.linear_slip_rate = 1.0e-12 ; default value
+\end_layout
+
+\begin_layout Standard
+The spatial database files for the rate- and state-friction model properties
+ and state variables specify the spatial variation of the fault constitutive
+ model parameters given in Table 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "tab:rate:state:ageing:properties:statevars"
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float table
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\noindent
+\align center
+\begin_inset Caption
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "tab:rate:state:ageing:properties:statevars"
+
+\end_inset
+
+Values in spatial databases for Dieterich-Ruina rate-state friction.
+\end_layout
+
+\end_inset
+
+
+\begin_inset VSpace defskip
+\end_inset
+
+
+\begin_inset Tabular
+<lyxtabular version="3" rows="8" columns="3">
+<features tabularvalignment="middle">
+<column alignment="left" valignment="top" width="0">
+<column alignment="left" valignment="top" width="0">
+<column alignment="left" valignment="top" width="2.5in">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Database
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Value
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Description
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+db_properties
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+reference-friction-coefficient
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Steady-state coefficient of friction at slip rate 
+\begin_inset Formula $V_{0}$
+\end_inset
+
+, 
+\begin_inset Formula $\mu_{s}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+reference-slip-rate
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Reference slip rate, 
+\begin_inset Formula $V_{0}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+characteristic-slip-distance
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Slip-weakening parameter, 
+\begin_inset Formula $L$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+constitutive-parameter-a
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Coefficient for the 
+\begin_inset Formula $\ln$
+\end_inset
+
+ slip rate term, 
+\begin_inset Formula $a$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+constitutive-parameter-b
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Coefficient for the 
+\begin_inset Formula $\ln$
+\end_inset
+
+ state variable term, 
+\begin_inset Formula $b$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+cohesion
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Cohesive stress, 
+\begin_inset Formula $T_{c}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+db_initial_state
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+state-variable
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+State variable, 
+\begin_inset Formula $\theta$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+\begin_inset CommandInset label
+LatexCommand label
+name "sec:fault:cohesive:impulses"
+
+\end_inset
+
+Slip Impulses for Green's Functions
+\end_layout
+
+\begin_layout Standard
+Computing static Green's functions using the GreensFns problem requires
+ a specialized fault implementation, FaultCohesiveImpulses, to set up the
+ slip impulses.
+ The parameters controlling the slip impulses include the components involved
+ (lateral, reverse, and/or fault opening) and the amplitude of the pulses
+ (e.g., selecting a subset of a fault or including a spatial variation).
+ The GreensFns properties and facilities include:
+\end_layout
+
+\begin_layout Description
+threshold Threshold for non-zero amplitude; impulses will only be generated
+ at locations on the fault where the amplitude exceeds this threshold.
+\end_layout
+
+\begin_layout Description
+impulse_dof Array of components associated with impulses, e.g., [0, 1] for
+ slip involving the lateral and up-dip components.
+\end_layout
+
+\begin_layout Description
+db_impulse_amplitude Spatial database for amplitude of slip impulse (scalar
+ field).
+ Default is SimpleDB.
+\end_layout
+
+\begin_layout Standard
+An example of setting the properties and facilities for FaultCohesiveImpulses
+ in a 
+\family typewriter
+.cfg
+\family default
+ file is:
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.problem.interfaces]
+\end_layout
+
+\begin_layout LyX-Code
+fault = pylith.faults.FaultCohesiveImpulses ; Change from the default
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+[pylithapp.problem.interfaces.fault]
+\end_layout
+
+\begin_layout LyX-Code
+threshold = 1.0e-6*m ; default
+\end_layout
+
+\begin_layout LyX-Code
+impulse_dof = [0] ; lateral slip-only
+\end_layout
+
+\begin_layout LyX-Code
+db_impulse_amplitude.iohandler.filename = myimpulse.spatialdb
+\end_layout
+
+\begin_layout LyX-Code
+db_impulse_amplitude.label = Impulse amplitude
+\end_layout
+
+\begin_layout Section
+Gravitational Body Forces
+\end_layout
+
+\begin_layout Standard
+Many problems in geophysics require the consideration of gravitational body
+ forces.
+ For example, it is often important to include the effects of the lithostatic
+ (overburden) pressure.
+ In future releases of PyLith that permit nonlinear bulk rheologies, body
+ forces will affect plastic yield criteria and the deformation field for
+ large deformation/finite strain problems.
+ As described in Chapter 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "cha:Governing-Equations"
+
+\end_inset
+
+, the body forces contribute to the residual,
+\begin_inset Formula 
+\begin{equation}
+r_{i}^{n}=\int_{V}f_{i}N^{n}\: dV.
+\end{equation}
+
+\end_inset
+
+For gravitational body forces, the body force per unit volume, 
+\begin_inset Formula $f_{i}$
+\end_inset
+
+, is given as the product of the mass density, 
+\begin_inset Formula $\rho$
+\end_inset
+
+, the scalar gravitational acceleration value, 
+\begin_inset Formula $g$
+\end_inset
+
+, and the gravitational acceleration orientation vector, 
+\begin_inset Formula $a_{i}$
+\end_inset
+
+:
+\begin_inset Formula 
+\begin{equation}
+f_{i}=\rho ga_{i}.
+\end{equation}
+
+\end_inset
+
+The mass density is a property of every material model, and is thus included
+ in the spatial database with the physical properties for each material.
+ The gravitational acceleration is assumed to be uniform and constant for
+ a given problem, with a default value of 9.80665 m/s
+\begin_inset Formula $^{\text{2}}$
+\end_inset
+
+.
+ The orientation vector will depend on the dimension of the problem as well
+ as the coordinate system being used.
+ The default orientation vector has components (0, 0, -1).
+ This is appropriate for three-dimensional problems where the gravity vector
+ is aligned with the negative z-axis, as would be the case in a geographic-proje
+cted coordinate system or a generic Cartesian coordinate system.
+ For cases in which the curvature of the earth should be considered, the
+ spatialdata package provides an earth-centered, earth-fixed (ECEF) coordinate
+ system and a local georeferenced Cartesian system; in each of these cases
+ the orientation vector is computed automatically, although this feature
+ has not been tested.
+ For problems in one or two dimensions where the orientation vector is constant,
+ the vector will need to be explicitly specified.
+ For example, in a two-dimensional problem, the vector might be specified
+ as (0, -1, 0).
+ The vector still has three components, although the extra component is
+ not used.
+\end_layout
+
+\begin_layout Standard
+Gravity is turned off by default.
+ To include gravitational effects in a simulation, you can turn it on as
+ follows:
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.timedependent]
+\end_layout
+
+\begin_layout LyX-Code
+use_gravity = true
+\end_layout
+
+\begin_layout Standard
+By simply adding this flag, the default gravity field values will be used
+ and a 
+\family typewriter
+gravity_field
+\family default
+ component will be assigned for the problem.
+ The default values may be changed by altering the properties of 
+\family typewriter
+gravity_field
+\family default
+:
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.timedependent.gravity_field]
+\end_layout
+
+\begin_layout LyX-Code
+acceleration = 100.0*m*s**-2
+\end_layout
+
+\begin_layout LyX-Code
+gravity_dir = [0, -1, 0]
+\end_layout
+
+\begin_layout Standard
+An example problem using gravity is described in Section 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sec:Tutorial-3d-hex8-gravity"
+
+\end_inset
+
+.
+\end_layout
+
+\end_body
+\end_document

Modified: short/3D/PyLith/branches/v1.7-trunk/doc/userguide/components.lyx
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/doc/userguide/components.lyx	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/doc/userguide/components.lyx	2012-08-21 18:24:25 UTC (rev 20616)
@@ -1,1506 +1,1510 @@
-#LyX 2.0 created this file. For more info see http://www.lyx.org/
-\lyxformat 413
-\begin_document
-\begin_header
-\textclass book
-\begin_preamble
-
-\end_preamble
-\use_default_options false
-\maintain_unincluded_children false
-\language english
-\language_package default
-\inputencoding latin1
-\fontencoding global
-\font_roman default
-\font_sans default
-\font_typewriter default
-\font_default_family default
-\use_non_tex_fonts false
-\font_sc false
-\font_osf false
-\font_sf_scale 100
-\font_tt_scale 100
-
-\graphics default
-\default_output_format default
-\output_sync 0
-\bibtex_command default
-\index_command default
-\paperfontsize default
-\spacing single
-\use_hyperref false
-\papersize default
-\use_geometry true
-\use_amsmath 1
-\use_esint 0
-\use_mhchem 1
-\use_mathdots 1
-\cite_engine basic
-\use_bibtopic false
-\use_indices false
-\paperorientation portrait
-\suppress_date false
-\use_refstyle 0
-\index Index
-\shortcut idx
-\color #008000
-\end_index
-\leftmargin 1in
-\topmargin 1in
-\rightmargin 1in
-\bottommargin 2in
-\secnumdepth 3
-\tocdepth 3
-\paragraph_separation indent
-\paragraph_indentation default
-\quotes_language english
-\papercolumns 1
-\papersides 2
-\paperpagestyle default
-\tracking_changes false
-\output_changes false
-\html_math_output 0
-\html_css_as_file 0
-\html_be_strict false
-\end_header
-
-\begin_body
-
-\begin_layout Chapter
-\begin_inset CommandInset label
-LatexCommand label
-name "cha:components"
-
-\end_inset
-
-PyLith and Spatialdata Components
-\end_layout
-
-\begin_layout Standard
-The name of the component is followed by the full path name and description.
- The full path name is used when setting a component to a facility in a
- 
-\family typewriter
-.cfg
-\family default
- file or with command line arguments.
-\end_layout
-
-\begin_layout Section
-Application components
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-PyLithApp pylith.apps.PyLithApp
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-PyLith application.
-\end_layout
-
-\begin_layout Subsection
-Problem components
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-TimeDependent pylith.problems.TimeDependent
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Time-dependent problem.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-GreensFns pylith.problems.GreensFns
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Static Green's function problem with slip impulses.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-Implicit pylith.problems.Implicit
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Implicit time stepping for static and quasi-static simulations with infinitesima
-l strains.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-ImplicitLgDeform pylith.problems.ImplicitLgDeform
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Implicit time stepping for static and quasi-static simulations including
- the effects of rigid body motion and small strains.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-Explicit pylith.problems.Explicit
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Explicit time stepping for dynamic simulations with infinitesimal strains.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-ExplicitLgDeform pylith.problems.ExplicitLgDeform
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Explicit time stepping for dynamic simulations including the effects of
- rigid body motion and small strains.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-ExplicitLumped pylith.problems.ExplicitLumped
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Explicit time stepping for dynamic simulations with a lumped system Jacobian
- matrix.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-ExplicitLumpedTri3 pylith.problems.ExplicitLumpedTri3
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Optimized elasticity formulation for linear triangular cells and one quadrature
- point for explicit time stepping in dynamic simulations.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-ExplicitLumpedTet4 pylith.problems.ExplicitLumpedTet4
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Optimized elasticity formulation for linear tetrahedral cells and one quadrature
- point for explicit time stepping in dynamic simulations.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-SolverLinear pylith.problems.SolverLinear
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Linear PETSc solver (KSP).
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-SolverNonlinear pylith.problems.SolverNonlinear
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Nonlinear PETSc solver (SNES).
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-SolverLumped pylith.problems.SolverLumped
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Built-in simple, optimized solver for solving systems with a lumped Jacobian.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-TimeStepUniform pylith.problems.TimeStepUniform
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Uniform time stepping.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-TimeStepAdapt pylith.problems.TimeStepAdapt
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Adaptive time stepping (time step selected based on estimated stable time
- step).
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-TimeStepUser pylith.problems.TimeStepUser
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-User defined time stepping (variable time step set by user).
-\end_layout
-
-\begin_layout Subsection
-Utility components
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-NullComponent pylith.utils.NullComponent
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Null component used to set a facility to an empty value.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-EventLogger pylith.utils.EventLogger
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-PETSc event logger.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-MemoryLogger pylith.perf.MemoryLogger
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Memory model logger for reporting memory use by Sieve.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-VTKDataReader pylith.utils.
-\family default
-VTKDataReader
-\begin_inset Newline newline
-\end_inset
-
-Data reader for VTK files, requires TVTK Enthought package available from
- 
-\family typewriter
-https://github.com/enthought/mayavi.
-\end_layout
-
-\begin_layout Subsection
-Topology components
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-Distributor pylith.topology.Distributor
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Distributor of mesh among processors in parallel simulations.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-JacobianViewer pylith.topology.JacobianViewer
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Viewer for writing Jacobian sparse matrix to file.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-MeshGenerator pylith.topology.MeshGenerator
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Mesh generator/importer.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-MeshImporter pylith.topology.MeshImporter
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Mesh importer/reader.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-MeshRefiner pylith.topology.Mesh
-\family default
-Refiner
-\begin_inset Newline newline
-\end_inset
-
-Default (null) mesh refinement object that does not refine the mesh.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-RefineUniform pylith.topology.
-\family default
-RefineUniform
-\begin_inset Newline newline
-\end_inset
-
-Uniform global mesh refinement.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-ReverseCuthillMcKee pylith.topology.
-\family default
-ReverseCuthillMcKee
-\begin_inset Newline newline
-\end_inset
-
-Object used to manage reordering cells and vertices using the reverse Cuthill-Mc
-Kee algorithm.
-\end_layout
-
-\begin_layout Subsection
-Material components
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-ElasticStrain1D pylith.materials.ElasticStrain1D
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Linearly elastic 1-D bulk constitutive model with 1-D strain (
-\begin_inset Formula $\epsilon_{yy}=\epsilon_{zz}=0$
-\end_inset
-
-).
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-ElasticStress1D pylith.materials.ElasticStress1D
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Linearly elastic 1-D bulk constitutive model with 1-D stress (
-\begin_inset Formula $\sigma_{yy}=\sigma_{zz}=0$
-\end_inset
-
-).
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-ElasticPlaneStrain pylith.materials.ElasticPlaneStrain
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Linearly elastic 2-D bulk constitutive model with plane strain (
-\begin_inset Formula $\epsilon_{zz}=0$
-\end_inset
-
-).
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-ElasticPlaneStress pylith.materials.ElasticPlaneStress
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Linearly elastic 2-D bulk constitutive model with plane stress (
-\begin_inset Formula $\sigma_{zz}=0$
-\end_inset
-
-).
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-ElasticIsotropic3D pylith.materials.ElasticIsotropic3D
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Linearly elastic 3-D bulk constitutive model.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-MaxwellIsotropic3D pylith.materials.MaxwellIsotropic3D
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Linear Maxwell viscoelastic bulk constitutive model.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-MaxwellPlaneStrain pylith.materials.MaxwellPlaneStrain
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Linear Maxwell viscoelastic bulk constitutive model for plane strain problems.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-GenMaxwellIsotropic3D pylith.materials.GenMaxwellIsotropic3D
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Generalized Maxwell viscoelastic bulk constitutive model.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-GenMaxwellPlaneStrain pylith.materials.GenMaxwellPlaneStrain
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Generalized Maxwell viscoelastic bulk constitutive model for plane strain
- problems.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-PowerLaw3D pylith.materials.PowerLaw3D
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Power-law viscoelastic bulk constitutive model.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-PowerLawPlaneStrain pylith.materials.PowerLawPlaneStrain
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Power-law viscoelastic bulk constitutive model for plane strain problems.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-DruckerPrage3D pylith.materials.DruckerPrager3D
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Drucker-Prager elastoplastic bulk constitutive model.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-DruckerPragePlaneStrain pylith.materials.DruckerPragerPlaneStrain
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Drucker-Prager elastoplastic bulk constitutive model for plane strain problems.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-Homogeneous pylith.materials.Homogeneous
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Container with a single bulk material.
-\end_layout
-
-\begin_layout Subsection
-Boundary condition components
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-AbsorbingDampers pylith.bc.AbsorbingDampers
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Absorbing boundary condition using simple dashpots.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-DirichletBC pylith.bc.DirichletBC
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Dirichlet (prescribed displacements) boundary condition for a set of points.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-DirichletBoundary pylith.bc.DirichletBoundary
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Dirichlet (prescribed displacements) boundary condition for a set of points
- associated with a boundary surface.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-Neumann pylith.bc.Neumann
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Neumann (traction) boundary conditions applied to a boundary surface.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-PointForce pylith.bc.PointForce
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Point forces applied to a set of vertices.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-ZeroDispDB pylith.bc.ZeroDispDB
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Specialized UniformDB with uniform zero displacements at all degrees of
- freedom.
-\end_layout
-
-\begin_layout Subsection
-Fault components
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-FaultCohesiveKin pylith.faults.FaultCohesiveKin
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Fault surface with kinematic (prescribed) slip implemented using cohesive
- elements.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-FaultCohesiveDyn pylith.faults.FaultCohesive
-\family default
-Dyn
-\begin_inset Newline newline
-\end_inset
-
-Fault surface with dynamic (friction) slip implemented using cohesive elements.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-FaultCohesiveImpulses pylith.faults.FaultCohesive
-\family default
-Impulses
-\begin_inset Newline newline
-\end_inset
-
-Fault surface with Green's functions slip impulses implemented using cohesive
- elements.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-EqKinSrc pylith.faults.EqKinSrc
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Kinematic (prescribed) slip earthquake rupture.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-SingleRupture pylith.faults.SingleRupture
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Container with one kinematic earthquake rupture.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-StepSlipFn pylith.faults.StepSlipFn
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Step function slip-time function.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-ConstRateSlipFn pylith.faults.ConstRateSlipFn
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Constant slip rate slip-time function.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-BruneSlipFn pylith.faults.BruneSlipFn
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Slip-time function where slip rate is equal to Brune's far-field slip function.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-LiuCosSlipFn pylith.faults.LiuCosSlipFn
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Slip-time function composed of three sine/cosine functions.
- Similar to Brune's far-field time function but with more abrupt termination
- of slip.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-TimeHistorySlipFn pylith.faults.TimeHistorySlipFn
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Slip-time function with a user-defined slip time function.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-TractPerturbation pylith.faults.TractPerturbation
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Prescribed traction perturbation applied to fault with constitituve model
- in additional to tractions from deformation (generally used to nucleate
- a rupture).
-\end_layout
-
-\begin_layout Subsection
-Friction components
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-StaticFriction pylith.friction.StaticFriction
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Static friction fault constitutive model.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-SlipWeakening pylith.friction.SlipWeakening
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Linear slip-weakening friction fault constitutive model.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-RateStateAgeing pylith.friction.RateStateAgeing
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Dieterich-Ruina rate and state friction with ageing law state variable evolution.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-TimeWeakening pylith.friction.TimeWeakening
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Linear time-weakening friction fault constitutive model.
-\end_layout
-
-\begin_layout Subsection
-Discretization components
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-FIATLagrange pylith.feassemble.FIATLagrange
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Finite-element basis functions and quadrature rules for a Lagrange reference
- finite-element cell (point, line, quadrilateral, or hexahedron) using FIAT.
- The basis functions are constructed from the tensor produce of 1-D Lagrange
- reference cells.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-FIATSimplex pylith.feassemble.FIATSimplex
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Finite-element basis functions and quadrature rules for a simplex finite-element
- cell (point, line, triangle, or tetrahedron) using FIAT.
-\end_layout
-
-\begin_layout Subsection
-Output components
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-MeshIOAscii pylith.meshio.MeshIOAscii
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Reader for simple mesh ASCII files.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-MeshIOCubit pylith.meshio.MeshIOCubit
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Reader for CUBIT Exodus files.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-MeshIOLagrit pylith.meshio.MeshIOLagrit
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Reader for LaGriT GMV/Pset files.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-OutputManagerMesh pylith.meshio.OutputManagerMesh
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Output manager for mesh information and data.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-OutputManagerSubMesh pylith.meshio.OutputManagerSubMesh
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Output manager for submesh information and data.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-OutputSoln pylith.meshio.OutputSoln
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Output manager for solution data.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-OutputSolnSubset pylith.meshio.OutputSolnSubset
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Output manager for solution data over a submesh.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-OutputSolnPoints pylith.meshio.OutputSoln
-\family default
-Points
-\begin_inset Newline newline
-\end_inset
-
-Output manager for solution data at arbitrary points in the domain.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-OutputDirichlet pylith.meshio.OutputDirichlet
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Output manager for Dirichlet boundary condition information over a submesh.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-OutputNeumann pylith.meshio.OutputNeumann
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Output manager for Neumann boundary condition information over a submesh.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-OutputFaultKin pylith.meshio.OutputFaultKin
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Output manager for fault with kinematic (prescribed) earthquake ruptures.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-OutputFaultDyn pylith.meshio.OutputFaultDyn
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Output manager for fault with dynamic (friction) earthquake ruptures.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-OutputFaultImpulses pylith.meshio.OutputFaultImpulses
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Output manager for fault with static slip impulses.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-OutputMatElastic pylith.meshio.OutputMatElastic
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Output manager for bulk constitutive models for elasticity.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-SingleOutput pylith.meshio.SingleOutput
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Container with single output manger.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-PointsList pylith.meshio.PointsList
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Manager for text file container points for 
-\family typewriter
-OutputSolnPoints
-\family default
-.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-DataWriterVTKMesh pylith.meshio.DataWriterVTKMesh
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Writer for output to VTK files with mesh information.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-DataWriterVTKSubMesh pylith.meshio.DataWriterVTKSubMesh
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Writer for output to VTK files with mesh information over a submesh.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-DataWriterVTKSubSubMesh pylith.meshio.DataWriterVTKSubSubMesh
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Writer for output to VTK files with submesh information over a submesh.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-DataWriterHDF5Mesh pylith.meshio.DataWriterHDF5Mesh
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Writer for output to HDF5 files with mesh information.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-DataWriterHDF5SubMesh pylith.meshio.DataWriterHDF5SubMesh
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Writer for output to HDF5 files with mesh information over a submesh.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-DataWriterHDF5ExtSubSubMesh pylith.meshio.DataWriterHDF5ExtSubSubMesh
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Writer for output to HDF5 files with submesh information over a submesh
- with datasets written to external raw binary files.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-DataWriterHDF5ExtMesh pylith.meshio.DataWriterHDF5ExtMesh
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Writer for output to HDF5 files with mesh information with datasets written
- to external raw binary files.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-DataWriterHDF5ExtSubMesh pylith.meshio.DataWriterHDF5ExtSubMesh
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Writer for output to HDF5 files with mesh information over a submesh with
- datasets written to external raw binary files.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-DataWriterHDF5SubSubMesh pylith.meshio.DataWriterHDF5SubSubMesh
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Writer for output to HDF5 files with submesh information over a submesh.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-CellFilterAvgMesh pylith.meshio.CellFilterAvgMesh
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Filter that averages information over quadrature points of mesh cells.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-CellFilterAvgSubMesh pylith.meshio.CellFilterAvgSubMesh
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Filter that averages information over quadrature points of submesh cells.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-VertexFilterVecNormMesh pylith.meshio.VertexFilterVecNormMesh
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Filter that computes magnitude of vectors for vertex field over mesh.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-VertexFilterVecNormSubMesh pylith.meshio.VertexFilterVecNormSubMesh
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Filter that computes magnitude of vectors for vertex field over submesh.
-\end_layout
-
-\begin_layout Section
-Spatialdata components
-\end_layout
-
-\begin_layout Subsection
-Coordinate system components
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-CSCart spatialdata.geocoords.CSCart
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Cartesian coordinate system (0-D, 1-D, 2-D, or 3-D).
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-CSGeo spatialdata.geocoords.CSGeo
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Geographic coordinate system.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-CSGeoProj spatialdata.geocoords.CSGeoProj
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Coordinate system associated with a geographic projection.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-CSGeoLocalCart spatialdata.geocoords.CSGeoLocalCart
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Local, georeferenced Cartesian coordinate system.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-Projector spatialdata.geocoords.Projector
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Geographic projection.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-Converter spatialdata.geocoords.Converter
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Converter for transforming coordinates of points from one coordinate system
- to another.
-\end_layout
-
-\begin_layout Subsection
-Spatial database components
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-UniformDB spatialdata.spatialdb.UniformDB
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Spatial database with uniform values.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-SimpleDB spatialdata.spatialdb.SimpleDB
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Simple spatial database that defines fields using a point cloud.
- Values are determined using a nearest neighbor search or linear interpolation
- in 0-D, 1-D, 2-D, or 3-D.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-SimpleIOAscii spatialdata.spatialdb.SimpleIOAscii
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Reader/writer for simple spatial database files.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-SCECCVMH spatialdata.spatialdb.SCECCVMH
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Spatial database interface to the SCEC CVM-H (seismic velocity model).
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-CompositeDB spatialdata.spatialdb.CompositeDB
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Spatial database composed from multiple other spatial databases.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-TimeHistory spatialdata.spatialdb.TimeHistory
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Time history for temporal variations of a parameter.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-GravityField spatialdata.spatialdb.GravityField
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Spatial database providing vector for body forces associated with gravity.
-\end_layout
-
-\begin_layout Subsection
-Nondimensionalization components
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-Nondimensional spatialdata.units.Nondimensional
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Nondimensionalization of length, time, and pressure scales.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-NondimensionalElasticDynamic spatialdata.units.NondimensionalElasticDynamic
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Nondimensionalization of scales for dynamic problems.
-\end_layout
-
-\begin_layout Description
-
-\family typewriter
-NondimensionalElasticQuasistatic spatialdata.units.NondimensionalElasticQuasistati
-c
-\family default
-
-\begin_inset Newline newline
-\end_inset
-
-Nondimensionalization of scales for quasi-static problems.
-\end_layout
-
-\end_body
-\end_document
+#LyX 2.0 created this file. For more info see http://www.lyx.org/
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+\use_hyperref false
+\papersize default
+\use_geometry true
+\use_amsmath 1
+\use_esint 0
+\use_mhchem 1
+\use_mathdots 1
+\cite_engine basic
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\use_refstyle 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\leftmargin 1in
+\topmargin 1in
+\rightmargin 1in
+\bottommargin 2in
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\quotes_language english
+\papercolumns 1
+\papersides 2
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Chapter
+\begin_inset CommandInset label
+LatexCommand label
+name "cha:components"
+
+\end_inset
+
+PyLith and Spatialdata Components
+\end_layout
+
+\begin_layout Standard
+The name of the component is followed by the full path name and description.
+ The full path name is used when setting a component to a facility in a
+ 
+\family typewriter
+.cfg
+\family default
+ file or with command line arguments.
+\end_layout
+
+\begin_layout Section
+Application components
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+PyLithApp pylith.apps.PyLithApp
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+PyLith application.
+\end_layout
+
+\begin_layout Subsection
+Problem Components
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+TimeDependent pylith.problems.TimeDependent
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Time-dependent problem.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+GreensFns pylith.problems.GreensFns
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Static Green's function problem with slip impulses.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+Implicit pylith.problems.Implicit
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Implicit time stepping for static and quasi-static simulations with infinitesima
+l strains.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+ImplicitLgDeform pylith.problems.ImplicitLgDeform
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Implicit time stepping for static and quasi-static simulations including
+ the effects of rigid body motion and small strains.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+Explicit pylith.problems.Explicit
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Explicit time stepping for dynamic simulations with infinitesimal strains.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+ExplicitLgDeform pylith.problems.ExplicitLgDeform
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Explicit time stepping for dynamic simulations including the effects of
+ rigid body motion and small strains.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+ExplicitLumped pylith.problems.ExplicitLumped
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Explicit time stepping for dynamic simulations with a lumped system Jacobian
+ matrix.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+ExplicitLumpedTri3 pylith.problems.ExplicitLumpedTri3
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Optimized elasticity formulation for linear triangular cells and one quadrature
+ point for explicit time stepping in dynamic simulations.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+ExplicitLumpedTet4 pylith.problems.ExplicitLumpedTet4
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Optimized elasticity formulation for linear tetrahedral cells and one quadrature
+ point for explicit time stepping in dynamic simulations.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+SolverLinear pylith.problems.SolverLinear
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Linear PETSc solver (KSP).
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+SolverNonlinear pylith.problems.SolverNonlinear
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Nonlinear PETSc solver (SNES).
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+SolverLumped pylith.problems.SolverLumped
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Built-in simple, optimized solver for solving systems with a lumped Jacobian.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+TimeStepUniform pylith.problems.TimeStepUniform
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Uniform time stepping.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+TimeStepAdapt pylith.problems.TimeStepAdapt
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Adaptive time stepping (time step selected based on estimated stable time
+ step).
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+TimeStepUser pylith.problems.TimeStepUser
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+User defined time stepping (variable time step set by user).
+\end_layout
+
+\begin_layout Subsection
+Utility Components
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+NullComponent pylith.utils.NullComponent
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Null component used to set a facility to an empty value.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+EventLogger pylith.utils.EventLogger
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+PETSc event logger.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+MemoryLogger pylith.perf.MemoryLogger
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Memory model logger for reporting memory use by Sieve.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+VTKDataReader pylith.utils.VTKDataReader
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Data reader for VTK files, requires TVTK Enthought package available from
+ 
+\family typewriter
+
+\begin_inset Newline newline
+\end_inset
+
+https://github.com/enthought/mayavi.
+\end_layout
+
+\begin_layout Subsection
+Topology Components
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+Distributor pylith.topology.Distributor
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Distributor of mesh among processors in parallel simulations.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+JacobianViewer pylith.topology.JacobianViewer
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Viewer for writing Jacobian sparse matrix to file.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+MeshGenerator pylith.topology.MeshGenerator
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Mesh generator/importer.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+MeshImporter pylith.topology.MeshImporter
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Mesh importer/reader.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+MeshRefiner pylith.topology.MeshRefiner
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Default (null) mesh refinement object that does not refine the mesh.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+RefineUniform pylith.topology.RefineUniform
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Uniform global mesh refinement.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+ReverseCuthillMcKee pylith.topology.ReverseCuthillMcKee
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Object used to manage reordering cells and vertices using the reverse Cuthill-Mc
+Kee algorithm.
+\end_layout
+
+\begin_layout Subsection
+Material Components
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+ElasticStrain1D pylith.materials.ElasticStrain1D
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Linearly elastic 1D bulk constitutive model with 1D strain (
+\begin_inset Formula $\epsilon_{yy}=\epsilon_{zz}=0$
+\end_inset
+
+).
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+ElasticStress1D pylith.materials.ElasticStress1D
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Linearly elastic 1D bulk constitutive model with 1D stress (
+\begin_inset Formula $\sigma_{yy}=\sigma_{zz}=0$
+\end_inset
+
+).
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+ElasticPlaneStrain pylith.materials.ElasticPlaneStrain
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Linearly elastic 2D bulk constitutive model with plane strain (
+\begin_inset Formula $\epsilon_{zz}=0$
+\end_inset
+
+).
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+ElasticPlaneStress pylith.materials.ElasticPlaneStress
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Linearly elastic 2D bulk constitutive model with plane stress (
+\begin_inset Formula $\sigma_{zz}=0$
+\end_inset
+
+).
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+ElasticIsotropic3D pylith.materials.ElasticIsotropic3D
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Linearly elastic 3D bulk constitutive model.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+MaxwellIsotropic3D pylith.materials.MaxwellIsotropic3D
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Linear Maxwell viscoelastic bulk constitutive model.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+MaxwellPlaneStrain pylith.materials.MaxwellPlaneStrain
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Linear Maxwell viscoelastic bulk constitutive model for plane strain problems.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+GenMaxwellIsotropic3D pylith.materials.GenMaxwellIsotropic3D
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Generalized Maxwell viscoelastic bulk constitutive model.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+GenMaxwellPlaneStrain pylith.materials.GenMaxwellPlaneStrain
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Generalized Maxwell viscoelastic bulk constitutive model for plane strain
+ problems.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+PowerLaw3D pylith.materials.PowerLaw3D
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Power-law viscoelastic bulk constitutive model.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+PowerLawPlaneStrain pylith.materials.PowerLawPlaneStrain
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Power-law viscoelastic bulk constitutive model for plane strain problems.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+DruckerPrage3D pylith.materials.DruckerPrager3D
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Drucker-Prager elastoplastic bulk constitutive model.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+DruckerPragePlaneStrain pylith.materials.DruckerPragerPlaneStrain
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Drucker-Prager elastoplastic bulk constitutive model for plane strain problems.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+Homogeneous pylith.materials.Homogeneous
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Container with a single bulk material.
+\end_layout
+
+\begin_layout Subsection
+Boundary Condition Components
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+AbsorbingDampers pylith.bc.AbsorbingDampers
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Absorbing boundary condition using simple dashpots.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+DirichletBC pylith.bc.DirichletBC
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Dirichlet (prescribed displacements) boundary condition for a set of points.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+DirichletBoundary pylith.bc.DirichletBoundary
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Dirichlet (prescribed displacements) boundary condition for a set of points
+ associated with a boundary surface.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+Neumann pylith.bc.Neumann
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Neumann (traction) boundary conditions applied to a boundary surface.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+PointForce pylith.bc.PointForce
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Point forces applied to a set of vertices.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+ZeroDispDB pylith.bc.ZeroDispDB
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Specialized UniformDB with uniform zero displacements at all degrees of
+ freedom.
+\end_layout
+
+\begin_layout Subsection
+Fault Components
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+FaultCohesiveKin pylith.faults.FaultCohesiveKin
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Fault surface with kinematic (prescribed) slip implemented using cohesive
+ elements.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+FaultCohesiveDyn pylith.faults.FaultCohesiveDyn
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Fault surface with dynamic (friction) slip implemented using cohesive elements.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+FaultCohesiveImpulses pylith.faults.FaultCohesiveImpulses
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Fault surface with Green's functions slip impulses implemented using cohesive
+ elements.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+EqKinSrc pylith.faults.EqKinSrc
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Kinematic (prescribed) slip earthquake rupture.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+SingleRupture pylith.faults.SingleRupture
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Container with one kinematic earthquake rupture.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+StepSlipFn pylith.faults.StepSlipFn
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Step function slip-time function.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+ConstRateSlipFn pylith.faults.ConstRateSlipFn
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Constant slip rate slip-time function.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+BruneSlipFn pylith.faults.BruneSlipFn
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Slip-time function where slip rate is equal to Brune's far-field slip function.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+LiuCosSlipFn pylith.faults.LiuCosSlipFn
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Slip-time function composed of three sine/cosine functions.
+ Similar to Brune's far-field time function but with more abrupt termination
+ of slip.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+TimeHistorySlipFn pylith.faults.TimeHistorySlipFn
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Slip-time function with a user-defined slip time function.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+TractPerturbation pylith.faults.TractPerturbation
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Prescribed traction perturbation applied to fault with constitituve model
+ in additional to tractions from deformation (generally used to nucleate
+ a rupture).
+\end_layout
+
+\begin_layout Subsection
+Friction Components
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+StaticFriction pylith.friction.StaticFriction
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Static friction fault constitutive model.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+SlipWeakening pylith.friction.SlipWeakening
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Linear slip-weakening friction fault constitutive model.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+RateStateAgeing pylith.friction.RateStateAgeing
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Dieterich-Ruina rate and state friction with ageing law state variable evolution.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+TimeWeakening pylith.friction.TimeWeakening
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Linear time-weakening friction fault constitutive model.
+\end_layout
+
+\begin_layout Subsection
+Discretization Components
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+FIATLagrange pylith.feassemble.FIATLagrange
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Finite-element basis functions and quadrature rules for a Lagrange reference
+ finite-element cell (point, line, quadrilateral, or hexahedron) using FIAT.
+ The basis functions are constructed from the tensor produce of 1D Lagrange
+ reference cells.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+FIATSimplex pylith.feassemble.FIATSimplex
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Finite-element basis functions and quadrature rules for a simplex finite-element
+ cell (point, line, triangle, or tetrahedron) using FIAT.
+\end_layout
+
+\begin_layout Subsection
+Output Components
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+MeshIOAscii pylith.meshio.MeshIOAscii
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Reader for simple mesh ASCII files.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+MeshIOCubit pylith.meshio.MeshIOCubit
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Reader for CUBIT Exodus files.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+MeshIOLagrit pylith.meshio.MeshIOLagrit
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Reader for LaGriT GMV/Pset files.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+OutputManagerMesh pylith.meshio.OutputManagerMesh
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Output manager for mesh information and data.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+OutputManagerSubMesh pylith.meshio.OutputManagerSubMesh
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Output manager for submesh information and data.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+OutputSoln pylith.meshio.OutputSoln
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Output manager for solution data.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+OutputSolnSubset pylith.meshio.OutputSolnSubset
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Output manager for solution data over a submesh.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+OutputSolnPoints pylith.meshio.OutputSolnPoints
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Output manager for solution data at arbitrary points in the domain.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+OutputDirichlet pylith.meshio.OutputDirichlet
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Output manager for Dirichlet boundary condition information over a submesh.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+OutputNeumann pylith.meshio.OutputNeumann
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Output manager for Neumann boundary condition information over a submesh.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+OutputFaultKin pylith.meshio.OutputFaultKin
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Output manager for fault with kinematic (prescribed) earthquake ruptures.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+OutputFaultDyn pylith.meshio.OutputFaultDyn
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Output manager for fault with dynamic (friction) earthquake ruptures.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+OutputFaultImpulses pylith.meshio.OutputFaultImpulses
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Output manager for fault with static slip impulses.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+OutputMatElastic pylith.meshio.OutputMatElastic
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Output manager for bulk constitutive models for elasticity.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+SingleOutput pylith.meshio.SingleOutput
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Container with single output manger.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+PointsList pylith.meshio.PointsList
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Manager for text file container points for 
+\family typewriter
+OutputSolnPoints
+\family default
+.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+DataWriterVTKMesh pylith.meshio.DataWriterVTKMesh
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Writer for output to VTK files with mesh information.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+DataWriterVTKSubMesh pylith.meshio.DataWriterVTKSubMesh
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Writer for output to VTK files with mesh information over a submesh.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+DataWriterVTKSubSubMesh pylith.meshio.DataWriterVTKSubSubMesh
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Writer for output to VTK files with submesh information over a submesh.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+DataWriterHDF5Mesh pylith.meshio.DataWriterHDF5Mesh
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Writer for output to HDF5 files with mesh information.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+DataWriterHDF5SubMesh pylith.meshio.DataWriterHDF5SubMesh
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Writer for output to HDF5 files with mesh information over a submesh.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+DataWriterHDF5ExtSubSubMesh pylith.meshio.DataWriterHDF5ExtSubSubMesh
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Writer for output to HDF5 files with submesh information over a submesh
+ with datasets written to external raw binary files.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+DataWriterHDF5ExtMesh pylith.meshio.DataWriterHDF5ExtMesh
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Writer for output to HDF5 files with mesh information with datasets written
+ to external raw binary files.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+DataWriterHDF5ExtSubMesh pylith.meshio.DataWriterHDF5ExtSubMesh
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Writer for output to HDF5 files with mesh information over a submesh with
+ datasets written to external raw binary files.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+DataWriterHDF5SubSubMesh pylith.meshio.DataWriterHDF5SubSubMesh
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Writer for output to HDF5 files with submesh information over a submesh.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+CellFilterAvgMesh pylith.meshio.CellFilterAvgMesh
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Filter that averages information over quadrature points of mesh cells.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+CellFilterAvgSubMesh pylith.meshio.CellFilterAvgSubMesh
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Filter that averages information over quadrature points of submesh cells.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+VertexFilterVecNormMesh pylith.meshio.VertexFilterVecNormMesh
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Filter that computes magnitude of vectors for vertex field over mesh.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+VertexFilterVecNormSubMesh pylith.meshio.VertexFilterVecNormSubMesh
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Filter that computes magnitude of vectors for vertex field over submesh.
+\end_layout
+
+\begin_layout Section
+Spatialdata Components
+\end_layout
+
+\begin_layout Subsection
+Coordinate System Components
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+CSCart spatialdata.geocoords.CSCart
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Cartesian coordinate system (0D, 1D, 2D, or 3D).
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+CSGeo spatialdata.geocoords.CSGeo
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Geographic coordinate system.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+CSGeoProj spatialdata.geocoords.CSGeoProj
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Coordinate system associated with a geographic projection.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+CSGeoLocalCart spatialdata.geocoords.CSGeoLocalCart
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Local, georeferenced Cartesian coordinate system.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+Projector spatialdata.geocoords.Projector
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Geographic projection.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+Converter spatialdata.geocoords.Converter
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Converter for transforming coordinates of points from one coordinate system
+ to another.
+\end_layout
+
+\begin_layout Subsection
+Spatial database Components
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+UniformDB spatialdata.spatialdb.UniformDB
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Spatial database with uniform values.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+SimpleDB spatialdata.spatialdb.SimpleDB
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Simple spatial database that defines fields using a point cloud.
+ Values are determined using a nearest neighbor search or linear interpolation
+ in 0D, 1D, 2D, or 3D.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+SimpleIOAscii spatialdata.spatialdb.SimpleIOAscii
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Reader/writer for simple spatial database files.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+SCECCVMH spatialdata.spatialdb.SCECCVMH
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Spatial database interface to the SCEC CVM-H (seismic velocity model).
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+CompositeDB spatialdata.spatialdb.CompositeDB
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Spatial database composed from multiple other spatial databases.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+TimeHistory spatialdata.spatialdb.TimeHistory
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Time history for temporal variations of a parameter.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+GravityField spatialdata.spatialdb.GravityField
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Spatial database providing vector for body forces associated with gravity.
+\end_layout
+
+\begin_layout Subsection
+Nondimensionalization components
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+Nondimensional spatialdata.units.Nondimensional
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Nondimensionalization of length, time, and pressure scales.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+NondimensionalElasticDynamic spatialdata.units.NondimensionalElasticDynamic
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Nondimensionalization of scales for dynamic problems.
+\end_layout
+
+\begin_layout Description
+
+\family typewriter
+NondimensionalElasticQuasistatic spatialdata.units.NondimensionalElasticQuasistati
+c
+\family default
+
+\begin_inset Newline newline
+\end_inset
+
+Nondimensionalization of scales for quasi-static problems.
+\end_layout
+
+\end_body
+\end_document

Modified: short/3D/PyLith/branches/v1.7-trunk/doc/userguide/extending/extending.lyx
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/doc/userguide/extending/extending.lyx	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/doc/userguide/extending/extending.lyx	2012-08-21 18:24:25 UTC (rev 20616)
@@ -200,8 +200,8 @@
 \end_layout
 
 \begin_layout Standard
-The UniformVelModel component provides uniform physical properties: P wave
- speed, S wave speed, and density.
+The UniformVelModel component provides uniform physical properties: P-wave
+ speed, S-wave speed, and density.
  Although this is a rather trivial specification of physical properties
  that could be easily done using a UniformDB, this example demonstrates
  how to create a user-defined component that matches the requirements of
@@ -225,7 +225,7 @@
  known names might be necessary.
  Are there other parameters that control the behavior of the component,
  such as a minimum shear wave speed? In our example the user supplies values
- for the P wave speed, S wave speed, and density.
+ for the P-wave speed, S-wave speed, and density.
  The user-supplied parameters become Pyre properties and facilities in the
  Python source file.
  Because our user supplied parameters are floating point values with dimensions,
@@ -270,7 +270,7 @@
 \family default
 .
  The C++ object also includes the accessor functions that allow us to set
- the P wave speed, S wave speed, and density values to the user-specified
+ the P-wave speed, S-wave speed, and density values to the user-specified
  values in the Python object.
  Additional information, such as a file name, parameters defined as data
  structures, etc., would be set via similar accessor functions.
@@ -386,9 +386,9 @@
  example must also use georeferenced coordinates.
  The dislocation example in the PyLith 
 \family typewriter
-examples/twocells/twotet4-geopro
+examples/twocells/twotet4-geoproj
 \family default
-j directory uses UTM zone 11 coordinates.
+ directory uses UTM zone 11 coordinates.
  The spatial database package will transform the coordinates between the
  two projections as defined in the UniformVelModel 
 \family typewriter
@@ -464,8 +464,7 @@
 \family default
 , is not particularly useful, but it illustrates the basic steps involved
  in creating a bulk constitutive model with state variables.
- The sources files are included with the main PyLith source code in the
- 
+ The source files are included with the main PyLith source code in the 
 \family typewriter
 templates/materials
 \family default
@@ -489,8 +488,8 @@
  The specification of the physical properties and state variables associated
  with the constitutive model is handled directly in the C++ code.
  As a result, the Python object for the constitutive model component is
- very simple and customization is limited to simply changing the names of
- objects and labels.
+ very simple, and customization is limited to simply changing the names
+ of objects and labels.
 \end_layout
 
 \begin_layout Standard
@@ -589,7 +588,7 @@
 \begin_layout Standard
 Because it is sometimes convenient to supply physical properties for a bulk
  constitutive model that are equivalent but different from the ones that
- appear in the constitutive equations (e.g., P wave speed, S wave speed, and
+ appear in the constitutive equations (e.g., P-wave speed, S-wave speed, and
  density rather then Lame's constants 
 \begin_inset Formula $\mu,$
 \end_inset
@@ -620,7 +619,7 @@
 \end_layout
 
 \begin_layout Standard
-The SWIG interface files for a bulk constitutive component are setup in
+The SWIG interface files for a bulk constitutive component are set up in
  the same manner as in the previous example of creating a customized spatial
  database component.
  The ``main'' SWIG interface file (
@@ -666,9 +665,12 @@
 \family typewriter
 Makefile.am
 \family default
- file as necessary, then generate the configure script, run configure, and
- then build and install the library and module (see the 
+ file as necessary, then generate the configure script, run 
 \family typewriter
+configure
+\family default
+, and then build and install the library and module (see the 
+\family typewriter
 README
 \family default
  file for detailed instructions).
@@ -782,8 +784,7 @@
 \family default
 , is not particularly useful, but it illustrates the basic steps involved
  in creating a fault constitutive model.
- The sources files are included with the main PyLith source code in the
- 
+ The source files are included with the main PyLith source code in the 
 \family typewriter
 templates/friction
 \family default
@@ -922,7 +923,7 @@
 \end_layout
 
 \begin_layout Standard
-The SWIG interface files for a fault constitutive component are setup in
+The SWIG interface files for a fault constitutive component are set up in
  the same manner as in the previous examples of creating a bulk constitutive
  model or a customized spatial database component.
  The ``main'' SWIG interface file (
@@ -968,9 +969,12 @@
 \family typewriter
 Makefile.am
 \family default
- file as necessary, then generate the configure script, run configure, and
- then build and install the library and module (see the 
+ file as necessary, then generate the configure script, run 
 \family typewriter
+configure
+\family default
+, and then build and install the library and module (see the 
+\family typewriter
 README
 \family default
  file for detailed instructions).
@@ -1016,8 +1020,12 @@
  You can try using it in any of the 2D or 3D examples.
  For the examples in 
 \family typewriter
-examples/bar_shearwave/quad4, in shearwave_staticfriction.cfg
+examples/bar_shearwave/quad4,
 \family default
+ in 
+\family typewriter
+shearwave_staticfriction.cfg
+\family default
  replace the line
 \end_layout
 

Modified: short/3D/PyLith/branches/v1.7-trunk/doc/userguide/fileformats/fileformats.lyx
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/doc/userguide/fileformats/fileformats.lyx	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/doc/userguide/fileformats/fileformats.lyx	2012-08-21 18:24:25 UTC (rev 20616)
@@ -1,1694 +1,1694 @@
-#LyX 2.0 created this file. For more info see http://www.lyx.org/
-\lyxformat 413
-\begin_document
-\begin_header
-\textclass book
-\begin_preamble
-
-\end_preamble
-\use_default_options false
-\maintain_unincluded_children false
-\language english
-\language_package default
-\inputencoding latin1
-\fontencoding global
-\font_roman default
-\font_sans default
-\font_typewriter default
-\font_default_family default
-\use_non_tex_fonts false
-\font_sc false
-\font_osf false
-\font_sf_scale 100
-\font_tt_scale 100
-
-\graphics default
-\default_output_format default
-\output_sync 0
-\bibtex_command default
-\index_command default
-\paperfontsize default
-\spacing single
-\use_hyperref false
-\papersize default
-\use_geometry true
-\use_amsmath 1
-\use_esint 0
-\use_mhchem 1
-\use_mathdots 1
-\cite_engine basic
-\use_bibtopic false
-\use_indices false
-\paperorientation portrait
-\suppress_date false
-\use_refstyle 0
-\index Index
-\shortcut idx
-\color #008000
-\end_index
-\leftmargin 1in
-\topmargin 1in
-\rightmargin 1in
-\bottommargin 2in
-\secnumdepth 3
-\tocdepth 3
-\paragraph_separation indent
-\paragraph_indentation default
-\quotes_language english
-\papercolumns 1
-\papersides 2
-\paperpagestyle default
-\tracking_changes false
-\output_changes false
-\html_math_output 0
-\html_css_as_file 0
-\html_be_strict false
-\end_header
-
-\begin_body
-
-\begin_layout Chapter
-\begin_inset CommandInset label
-LatexCommand label
-name "cha:File-Formats"
-
-\end_inset
-
-File Formats
-\end_layout
-
-\begin_layout Section
-\begin_inset CommandInset label
-LatexCommand label
-name "sec:MeshIOAscii"
-
-\end_inset
-
-PyLith Mesh ASCII Files
-\end_layout
-
-\begin_layout Standard
-PyLith mesh ASCII files allow quick specification of the mesh information
- for very small, simple meshes that are most easily written by hand.
- We do not recommend using this format for anything other than these very
- small, simple meshes.
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float figure
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\noindent
-\align center
-\begin_inset Graphics
-	filename figs/meshquad4.eps
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Plain Layout
-\begin_inset Caption
-
-\begin_layout Plain Layout
-Diagram of mesh specified in Figure 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "fig:meshioascii:format"
-
-\end_inset
-
-.
-\begin_inset CommandInset label
-LatexCommand label
-name "fig:meshioascii:diagram"
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout LyX-Code
-// This mesh file defines a finite-element mesh composed of two
-\end_layout
-
-\begin_layout LyX-Code
-// square cells of edge length 2.
-\end_layout
-
-\begin_layout LyX-Code
-
-\end_layout
-
-\begin_layout LyX-Code
-// Comments can appear almost anywhere in these files and are
-\end_layout
-
-\begin_layout LyX-Code
-// delimited with two slashes (//) just like in C++.
- All text and 
-\end_layout
-
-\begin_layout LyX-Code
-// whitespace after the delimiter on a given line is ignored.
-\end_layout
-
-\begin_layout LyX-Code
-mesh = { // begin specification of the mesh
-\end_layout
-
-\begin_layout LyX-Code
-  dimension = 2 // spatial dimension of the mesh
-\end_layout
-
-\begin_layout LyX-Code
-
-\end_layout
-
-\begin_layout LyX-Code
-  // Begin vertex and cell labels with 0.
- This is the default so 
-\end_layout
-
-\begin_layout LyX-Code
-  // this next line is optional
-\end_layout
-
-\begin_layout LyX-Code
-  use-index-zero = true
-\end_layout
-
-\begin_layout LyX-Code
-
-\end_layout
-
-\begin_layout LyX-Code
-  vertices = { // vertices or nodes of the finite-element cells
-\end_layout
-
-\begin_layout LyX-Code
-    dimension = 2 // spatial dimension of the vertex coordinates
-\end_layout
-
-\begin_layout LyX-Code
-    count = 6 // number of vertices in the mesh
-\end_layout
-
-\begin_layout LyX-Code
-    coordinates = { // list of vertex index and coordinates
-\end_layout
-
-\begin_layout LyX-Code
-      // the coordinates must coincide with the coordinate 
-\end_layout
-
-\begin_layout LyX-Code
-      // system specified in the Mesh object
-\end_layout
-
-\begin_layout LyX-Code
- 
-\end_layout
-
-\begin_layout LyX-Code
-      // exactly one vertex must appear on each line
-\end_layout
-
-\begin_layout LyX-Code
-      // (excluding whitespace)
-\end_layout
-
-\begin_layout LyX-Code
-      0    -2.0  -1.0
-\end_layout
-
-\begin_layout LyX-Code
-      1    -2.0  +1.0
-\end_layout
-
-\begin_layout LyX-Code
-      2     0.0  -1.0
-\end_layout
-
-\begin_layout LyX-Code
-      3     0.0  +1.0
-\end_layout
-
-\begin_layout LyX-Code
-      4    +2.0  -1.0
-\end_layout
-
-\begin_layout LyX-Code
-      5    +2.0  +1.0
-\end_layout
-
-\begin_layout LyX-Code
-    } // end of coordinates list
-\end_layout
-
-\begin_layout LyX-Code
-  } // end of vertices
-\end_layout
-
-\begin_layout LyX-Code
- 
-\end_layout
-
-\begin_layout LyX-Code
-  cells = { // finite-element cells
-\end_layout
-
-\begin_layout LyX-Code
-    count = 2 // number of cells in the mesh
-\end_layout
-
-\begin_layout LyX-Code
-    num-corners = 4 // number of vertices defining the cell
-\end_layout
-
-\begin_layout LyX-Code
-    simplices = { // list of vertices in each cell
-\end_layout
-
-\begin_layout LyX-Code
-      // see Section 4.2 for diagrams giving the order for each 
-\end_layout
-
-\begin_layout LyX-Code
-      // type of cell supported in PyLith
-\end_layout
-
-\begin_layout LyX-Code
- 
-\end_layout
-
-\begin_layout LyX-Code
-      // index of cell precedes the list of vertices for the cell 
-\end_layout
-
-\begin_layout LyX-Code
-
-\end_layout
-
-\begin_layout LyX-Code
-      // exactly one cell must appear on each line
-\end_layout
-
-\begin_layout LyX-Code
-      // (excluding whitespace)
-\end_layout
-
-\begin_layout LyX-Code
-      0    0  2  3  1
-\end_layout
-
-\begin_layout LyX-Code
-      1    4  5  3  2
-\end_layout
-
-\begin_layout LyX-Code
-    } // end of simplices list
-\end_layout
-
-\begin_layout LyX-Code
- 
-\end_layout
-
-\begin_layout LyX-Code
-    material-ids = { // associated each cell with a material model
-\end_layout
-
-\begin_layout LyX-Code
-      // the material id is specified using the index of the cell 
-\end_layout
-
-\begin_layout LyX-Code
-      // and then the corresponding material id
-\end_layout
-
-\begin_layout LyX-Code
-      0   0 // cell 0 has a material id of 0
-\end_layout
-
-\begin_layout LyX-Code
-      1   2 // cell 1 has a material id of 2
-\end_layout
-
-\begin_layout LyX-Code
-    } // end of material-ids list
-\end_layout
-
-\begin_layout LyX-Code
-  } // end of cells
-\end_layout
-
-\begin_layout LyX-Code
- 
-\end_layout
-
-\begin_layout LyX-Code
-  // This next section lists groups of vertices that can be used
-\end_layout
-
-\begin_layout LyX-Code
-  // in applying boundary conditions to portions of the domain
-\end_layout
-
-\begin_layout LyX-Code
-  group = { // start of a group
-\end_layout
-
-\begin_layout LyX-Code
-    // the name can have whitespace, so no comments are allowed
-\end_layout
-
-\begin_layout LyX-Code
-    // after the name
-\end_layout
-
-\begin_layout LyX-Code
-    name = face +y
-\end_layout
-
-\begin_layout LyX-Code
- 
-\end_layout
-
-\begin_layout LyX-Code
-    // Either groups of vertices or groups of cells can be     
-\end_layout
-
-\begin_layout LyX-Code
-    // specified, but currently PyLith only makes use of groups 
-\end_layout
-
-\begin_layout LyX-Code
-    // of vertices
-\end_layout
-
-\begin_layout LyX-Code
-    type = vertices // 'vertices' or 'cells'
-\end_layout
-
-\begin_layout LyX-Code
-    count = 2 // number of vertices in the group
-\end_layout
-
-\begin_layout LyX-Code
-    indices = { // list of vertex indices in the group
-\end_layout
-
-\begin_layout LyX-Code
-      // multiple vertices may appear on a line
-\end_layout
-
-\begin_layout LyX-Code
-      0  4 // this group contains vertices 0 and 4
-\end_layout
-
-\begin_layout LyX-Code
-    } // end of list of vertices
-\end_layout
-
-\begin_layout LyX-Code
-  } // end of group
-\end_layout
-
-\begin_layout LyX-Code
-  // additional groups can be listed here
-\end_layout
-
-\begin_layout Standard
-\begin_inset Float figure
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\begin_inset Caption
-
-\begin_layout Plain Layout
-Format of 
-\family typewriter
-PyLith
-\family default
- mesh ASCII files.
-\begin_inset CommandInset label
-LatexCommand label
-name "fig:meshioascii:format"
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-\begin_inset CommandInset label
-LatexCommand label
-name "sec:Spatialdata:SimpleIOAscii"
-
-\end_inset
-
-SimpleDB Spatial Database Files
-\end_layout
-
-\begin_layout Standard
-SimpleDB spatial database files contain a header describing the set of points
- and then the data with each line listing the coordinates of a point followed
- by the values of the fields for that point.
- 
-\end_layout
-
-\begin_layout LyX-Code
-// This spatial database specifies the distribution of slip on the
-\end_layout
-
-\begin_layout LyX-Code
-// fault surface.
- In this case we prescribe a piecewise linear, 
-\end_layout
-
-\begin_layout LyX-Code
-// depth dependent distribution of slip.
- The slip is 2.0 m 
-\end_layout
-
-\begin_layout LyX-Code
-// right-lateral with 0.25 m of reverse slip at the surface with
-\end_layout
-
-\begin_layout LyX-Code
-// a linear taper from 2.0 m to 0.0 m from -2 km to -4 km.
-\end_layout
-
-\begin_layout LyX-Code
- 
-\end_layout
-
-\begin_layout LyX-Code
-// Comments can appear almost anywhere in these files and are
-\end_layout
-
-\begin_layout LyX-Code
-// delimited with two slashes (//) just like in C++.
- All text and 
-\end_layout
-
-\begin_layout LyX-Code
-// whitespace after the delimiter on a given line is ignored.
-\end_layout
-
-\begin_layout LyX-Code
- 
-\end_layout
-
-\begin_layout LyX-Code
-// The next line is the magic header for spatial database files 
-\end_layout
-
-\begin_layout LyX-Code
-// in ASCII format.
-\end_layout
-
-\begin_layout LyX-Code
-#SPATIAL.ascii 1
-\end_layout
-
-\begin_layout LyX-Code
-SimpleDB { // start specifying the database parameters
-\end_layout
-
-\begin_layout LyX-Code
-  num-values = 3 // number of values in the database
-\end_layout
-
-\begin_layout LyX-Code
- 
-\end_layout
-
-\begin_layout LyX-Code
-  // Specify the names and the order of the values as they appear 
-\end_layout
-
-\begin_layout LyX-Code
-  // in the data.
- The names of the values must correspond to the 
-\end_layout
-
-\begin_layout LyX-Code
-  // names PyLith requests in querying the database.
-\end_layout
-
-\begin_layout LyX-Code
-  value-names =  left-lateral-slip  reverse-slip  fault-opening
-\end_layout
-
-\begin_layout LyX-Code
- 
-\end_layout
-
-\begin_layout LyX-Code
-  // Specify the units of the values in Python syntax (e.g., kg/m**3).
-\end_layout
-
-\begin_layout LyX-Code
-  value-units =  m  m  m
-\end_layout
-
-\begin_layout LyX-Code
-
-\end_layout
-
-\begin_layout LyX-Code
-  num-locs = 3 // Number of locations where values are given
-\end_layout
-
-\begin_layout LyX-Code
-  data-dim = 1 // Locations of data points form a line
-\end_layout
-
-\begin_layout LyX-Code
-  space-dim = 3 // Spatial dimension in which data resides
-\end_layout
-
-\begin_layout LyX-Code
- 
-\end_layout
-
-\begin_layout LyX-Code
-  // Specify the coordinate system associated with the 
-\end_layout
-
-\begin_layout LyX-Code
-  // coordinates of the locations where data is given
-\end_layout
-
-\begin_layout LyX-Code
-  cs-data = cartesian { // Use a Cartesian coordinate system
-\end_layout
-
-\begin_layout LyX-Code
-    to-meters = 1.0e+3 // Coordinates are in km
-\end_layout
-
-\begin_layout LyX-Code
- 
-\end_layout
-
-\begin_layout LyX-Code
-    // Specify the spatial dimension of the coordinate system
-\end_layout
-
-\begin_layout LyX-Code
-    // This value must match the one associated with the database
-\end_layout
-
-\begin_layout LyX-Code
-    space-dim = 3
-\end_layout
-
-\begin_layout LyX-Code
-  } // cs-data // end of coordinate system specification
-\end_layout
-
-\begin_layout LyX-Code
-} // end of SimpleDB specification
-\end_layout
-
-\begin_layout LyX-Code
- 
-\end_layout
-
-\begin_layout LyX-Code
-// The locations and values are listed after the parameters.
-\end_layout
-
-\begin_layout LyX-Code
-// Columns are coordinates of the points (1 column for each 
-\end_layout
-
-\begin_layout LyX-Code
-// spatial dimension) followed by the data values in the order 
-\end_layout
-
-\begin_layout LyX-Code
-// specified by the value-names field.
-\end_layout
-
-\begin_layout LyX-Code
-0.0  0.0  0.0    -2.00  0.25  0.00
-\end_layout
-
-\begin_layout LyX-Code
-0.0  0.0 -2.0    -2.00  0.00  0.00
-\end_layout
-
-\begin_layout LyX-Code
-0.0  0.0 -4.0     0.00  0.00  0.00
-\end_layout
-
-\begin_layout Standard
-\begin_inset Float figure
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\begin_inset Caption
-
-\begin_layout Plain Layout
-Format of spatial database files.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsection
-Spatial Database Coordinate Systems
-\end_layout
-
-\begin_layout Standard
-The spatial database files support four different types of coordinate systems.
- Conversions among the three geographic coordinate systems are supported
- in 3D.
- Conversions among the geographic and geographic projected coordinate systems
- are supported in 2D.
- In using the coordinate systems, we assume that you have installed the
- 
-\family typewriter
-proj
-\family default
- program in addition to the Proj.4 libraries, so that you can obtain a list
- of support projections, datums, and ellipsoids.
- Alternatively, refer to the documentation for the Proj.4 Cartographic Projection
-s library 
-\begin_inset Flex URL
-status collapsed
-
-\begin_layout Plain Layout
-
-trac.osgeo.org/proj
-\end_layout
-
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Subsubsection
-Cartesian
-\end_layout
-
-\begin_layout Standard
-This is a conventional Cartesian coordinate system.
- Conversions to other Cartesian coordinate systems are possible.
-\end_layout
-
-\begin_layout LyX-Code
-cs-data = cartesian {
-\end_layout
-
-\begin_layout LyX-Code
-  to-meters = 1.0e+3 // Locations in km
-\end_layout
-
-\begin_layout LyX-Code
-  space-dim = 2 // 1, 2, or 3 dimensions
-\end_layout
-
-\begin_layout LyX-Code
-}
-\end_layout
-
-\begin_layout Standard
-\begin_inset Float figure
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\begin_inset Caption
-
-\begin_layout Plain Layout
-Format for Cartesian coordinate systems in spatial database files.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsubsection
-Geographic
-\end_layout
-
-\begin_layout Standard
-This coordinate system is for geographic coordinates, such as longitude
- and latitude.
- Specification of the location in three-dimensions is supported.
- The vertical datum can be either the reference ellipsoid or mean sea level.
- The vertical coordinate is positive upwards.
-\end_layout
-
-\begin_layout LyX-Code
-cs-data = geographic {
-\end_layout
-
-\begin_layout LyX-Code
-  // Conversion factor to get to meters (only applies to vertical 
-\end_layout
-
-\begin_layout LyX-Code
-  // coordinate unless you are using a geocentric coordinate system).
-\end_layout
-
-\begin_layout LyX-Code
-  to-meters = 1.0e+3
-\end_layout
-
-\begin_layout LyX-Code
-  space-dim = 2 // 2 or 3 dimensions
-\end_layout
-
-\begin_layout LyX-Code
- 
-\end_layout
-
-\begin_layout LyX-Code
-  // Run 
-\begin_inset Quotes eld
-\end_inset
-
-proj -le
-\begin_inset Quotes erd
-\end_inset
-
- to see a list of available reference ellipsoids.
-\end_layout
-
-\begin_layout LyX-Code
-  // Comments are not allowed at the end of the next line.
-\end_layout
-
-\begin_layout LyX-Code
-  ellipsoid = WGS84
-\end_layout
-
-\begin_layout LyX-Code
- 
-\end_layout
-
-\begin_layout LyX-Code
-  // Run 
-\begin_inset Quotes eld
-\end_inset
-
-proj -ld
-\begin_inset Quotes erd
-\end_inset
-
- to see a list of available datums.
-\end_layout
-
-\begin_layout LyX-Code
-  // Comments are not allowed at the end of the next line.
-\end_layout
-
-\begin_layout LyX-Code
-  datum-horiz = WGS84
-\end_layout
-
-\begin_layout LyX-Code
- 
-\end_layout
-
-\begin_layout LyX-Code
-  // 
-\begin_inset Quotes eld
-\end_inset
-
-ellipsoid
-\begin_inset Quotes erd
-\end_inset
-
- or 
-\begin_inset Quotes eld
-\end_inset
-
-mean sea level
-\begin_inset Quotes erd
-\end_inset
-
-
-\end_layout
-
-\begin_layout LyX-Code
-  // Comments are not allowed at the end of the next line.
-\end_layout
-
-\begin_layout LyX-Code
-  datum-vert = ellipsoid
-\end_layout
-
-\begin_layout LyX-Code
- 
-\end_layout
-
-\begin_layout LyX-Code
-  // Use a geocentric coordinate system?
-\end_layout
-
-\begin_layout LyX-Code
-  is-geocentric = false // true or false
-\end_layout
-
-\begin_layout LyX-Code
-}
-\end_layout
-
-\begin_layout Standard
-\begin_inset Float figure
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\begin_inset Caption
-
-\begin_layout Plain Layout
-Format for geographic coordinate systems in spatial database files.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsubsection
-Geographic Projection
-\end_layout
-
-\begin_layout Standard
-This coordinate system applies to geographic projections.
- As in the geographic coordinate system, the vertical coordinate (if used)
- can be specified with respect to either the reference ellipsoid or mean
- sea level.
-\end_layout
-
-\begin_layout LyX-Code
-cs-data = geo-projected {
-\end_layout
-
-\begin_layout LyX-Code
-  to-meters = 1.0e+3 // Conversion factor to get to meters.
-\end_layout
-
-\begin_layout LyX-Code
-  space-dim = 2 // 2 or 3 dimensions
-\end_layout
-
-\begin_layout LyX-Code
-
-\end_layout
-
-\begin_layout LyX-Code
-  // Run 
-\begin_inset Quotes eld
-\end_inset
-
-proj -le
-\begin_inset Quotes erd
-\end_inset
-
- to see a list of available reference ellipsoids.
-\end_layout
-
-\begin_layout LyX-Code
-  // Comments are not allowed at the end of the next line.
-\end_layout
-
-\begin_layout LyX-Code
-  ellipsoid = WGS84
-\end_layout
-
-\begin_layout LyX-Code
- 
-\end_layout
-
-\begin_layout LyX-Code
-  // Run 
-\begin_inset Quotes eld
-\end_inset
-
-proj -ld
-\begin_inset Quotes erd
-\end_inset
-
- to see a list of available datums.
-\end_layout
-
-\begin_layout LyX-Code
-  // Comments are not allowed at the end of the next line.
-\end_layout
-
-\begin_layout LyX-Code
-  datum-horiz = WGS84
-\end_layout
-
-\begin_layout LyX-Code
- 
-\end_layout
-
-\begin_layout LyX-Code
-  // 
-\begin_inset Quotes eld
-\end_inset
-
-ellipsoid
-\begin_inset Quotes erd
-\end_inset
-
- or 
-\begin_inset Quotes eld
-\end_inset
-
-mean sea level
-\begin_inset Quotes erd
-\end_inset
-
-
-\end_layout
-
-\begin_layout LyX-Code
-  // Comments are not allowed at the end of the next line.
-\end_layout
-
-\begin_layout LyX-Code
-  datum-vert = ellipsoid
-\end_layout
-
-\begin_layout LyX-Code
- 
-\end_layout
-
-\begin_layout LyX-Code
-  // Run 
-\begin_inset Quotes eld
-\end_inset
-
-proj -lp
-\begin_inset Quotes erd
-\end_inset
-
- to see a list of available geographic 
-\end_layout
-
-\begin_layout LyX-Code
-  // projections.
-\end_layout
-
-\begin_layout LyX-Code
-    projector = projection {
-\end_layout
-
-\begin_layout LyX-Code
-    // Name of the projection.
- run 
-\begin_inset Quotes eld
-\end_inset
-
-proj -lp
-\begin_inset Quotes erd
-\end_inset
-
- to see a list of 
-\end_layout
-
-\begin_layout LyX-Code
-    // supported projections.
- Use the Universal Transverse Mercator
-\end_layout
-
-\begin_layout LyX-Code
-    // projection.
-\end_layout
-
-\begin_layout LyX-Code
-    projection = utm
-\end_layout
-
-\begin_layout LyX-Code
-    units = m // Units in the projection.
-\end_layout
-
-\begin_layout LyX-Code
- 
-\end_layout
-
-\begin_layout LyX-Code
-    // Provide a list of projection options; these are the command 
-\end_layout
-
-\begin_layout LyX-Code
-    // line arguments you would use with the proj program.
- Refer to
-\end_layout
-
-\begin_layout LyX-Code
-    // the Proj.4 library documentation for complete details.
-\end_layout
-
-\begin_layout LyX-Code
-    // Comments are not allowed at the end of the next line.
-\end_layout
-
-\begin_layout LyX-Code
-    proj-options = +zone=10
-\end_layout
-
-\begin_layout LyX-Code
-}
-\end_layout
-
-\begin_layout Standard
-\begin_inset Float figure
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\begin_inset Caption
-
-\begin_layout Plain Layout
-Format for geographic projection coordinate systems in spatial database
- files.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsubsection
-Geographic Local Cartesian
-\end_layout
-
-\begin_layout Standard
-This coordinate system is a geographically referenced, local 3D Cartesian
- coordinate system.
- This allows use of a conventional Cartesian coordinate system with accurate
- georeferencing.
- For example, one can construct a finite-element model in this coordinate
- system and use spatial databases in geographic coordinates.
- This coordinate system provides an alternative to using a geographic projection
- as the Cartesian grip.
- The advantage of this coordinate system is that it retains the curvature
- of the Earth, while a geographic projection does not.
-\end_layout
-
-\begin_layout LyX-Code
-cs-data = geo-local-cartesian {
-\end_layout
-
-\begin_layout LyX-Code
-  // Conversion factor to get to meters (only applies to vertical
-\end_layout
-
-\begin_layout LyX-Code
-  // coordinate unless you are using a geocentric coordinate system).
-\end_layout
-
-\begin_layout LyX-Code
-  to-meters = 1.0 // use meters
-\end_layout
-
-\begin_layout LyX-Code
-  space-dim = 2 // 2 or 3 dimensions
-\end_layout
-
-\begin_layout LyX-Code
- 
-\end_layout
-
-\begin_layout LyX-Code
-  // Run 
-\begin_inset Quotes eld
-\end_inset
-
-proj -le
-\begin_inset Quotes erd
-\end_inset
-
- to see a list of available reference ellipsoids.
-\end_layout
-
-\begin_layout LyX-Code
-  // Comments are not allowed at the end of the next line.
-\end_layout
-
-\begin_layout LyX-Code
-  ellipsoid = WGS84
-\end_layout
-
-\begin_layout LyX-Code
- 
-\end_layout
-
-\begin_layout LyX-Code
-  // Run 
-\begin_inset Quotes eld
-\end_inset
-
-proj -ld
-\begin_inset Quotes erd
-\end_inset
-
- to see a list of available datums.
-\end_layout
-
-\begin_layout LyX-Code
-  // Comments are not allowed at the end of the next line.
-\end_layout
-
-\begin_layout LyX-Code
-  datum-horiz = WGS84
-\end_layout
-
-\begin_layout LyX-Code
- 
-\end_layout
-
-\begin_layout LyX-Code
-  // 
-\begin_inset Quotes eld
-\end_inset
-
-ellipsoid
-\begin_inset Quotes erd
-\end_inset
-
- or 
-\begin_inset Quotes eld
-\end_inset
-
-mean sea level
-\begin_inset Quotes erd
-\end_inset
-
-
-\end_layout
-
-\begin_layout LyX-Code
-  // Comments are not allowed at the end of the next line.
-\end_layout
-
-\begin_layout LyX-Code
-  datum-vert = ellipsoid
-\end_layout
-
-\begin_layout LyX-Code
- 
-\end_layout
-
-\begin_layout LyX-Code
-  // Origin of the local Cartesian coordinate system.
- To avoid
-\end_layout
-
-\begin_layout LyX-Code
-  // round-off errors it is best to pick a location near the center of
-\end_layout
-
-\begin_layout LyX-Code
-  // the region of interest.
- An elevation on the surface of the Earth
-\end_layout
-
-\begin_layout LyX-Code
-  // in the middle of the region also works well (and makes the
-\end_layout
-
-\begin_layout LyX-Code
-  // vertical coordinate easy to interpret).
-\end_layout
-
-\begin_layout LyX-Code
-  origin-lon = -116.7094 // Longitude of the origin in decimal degrees
-\end_layout
-
-\begin_layout LyX-Code
-                         // (west is negative).
-\end_layout
-
-\begin_layout LyX-Code
-  origin-lat = 36.3874 // Latitude of the origin in decimal degrees 
-\end_layout
-
-\begin_layout LyX-Code
-                       // (north is positive).
-\end_layout
-
-\begin_layout LyX-Code
- 
-\end_layout
-
-\begin_layout LyX-Code
-  // Elevation with respect to the vertical datum.
- Units are the
-\end_layout
-
-\begin_layout LyX-Code
-  // same as the Cartesian coordinate system (in this case meters).
-\end_layout
-
-\begin_layout LyX-Code
-  origin-elev = 3.5
-\end_layout
-
-\begin_layout LyX-Code
-}
-\end_layout
-
-\begin_layout Standard
-\begin_inset Float figure
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\begin_inset Caption
-
-\begin_layout Plain Layout
-Format for the geographic local Cartesian coordinate system in spatial database
- files.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-\begin_inset CommandInset label
-LatexCommand label
-name "sec:Spatialdata:TimeHistoryIO"
-
-\end_inset
-
-Time History Database Files
-\end_layout
-
-\begin_layout Standard
-Time history database files contain a header describing the numer of points
- in the time history and the units for the time stamps followed by a list
- with pairs of time stamps and amplitude values.
- The amplitude at an arbitraty point in time is computed via interpolation
- of the values in the database.
- This means that the time history database must span the range of time values
- of interest.
- The points in the time history must also be ordered in time.
-\end_layout
-
-\begin_layout LyX-Code
-// This time history database specifies temporal variation in
-\end_layout
-
-\begin_layout LyX-Code
-// amplitude.
- In this case we prescribe a triangular slip time
-\end_layout
-
-\begin_layout LyX-Code
-// history.
- 
-\end_layout
-
-\begin_layout LyX-Code
- 
-\end_layout
-
-\begin_layout LyX-Code
-// Comments can appear almost anywhere in these files and are
-\end_layout
-
-\begin_layout LyX-Code
-// delimited with two slashes (//) just like in C++.
- All text and 
-\end_layout
-
-\begin_layout LyX-Code
-// whitespace after the delimiter on a given line is ignored.
-\end_layout
-
-\begin_layout LyX-Code
- 
-\end_layout
-
-\begin_layout LyX-Code
-// The next line is the magic header for spatial database files 
-\end_layout
-
-\begin_layout LyX-Code
-// in ASCII format.
-\end_layout
-
-\begin_layout LyX-Code
-#TIME HISTORY ascii
-\end_layout
-
-\begin_layout LyX-Code
-TimeHistory { // start specifying the database parameters
-\end_layout
-
-\begin_layout LyX-Code
-  num-points = 5 // number of points in time history
-\end_layout
-
-\begin_layout LyX-Code
- 
-\end_layout
-
-\begin_layout LyX-Code
-  // Specify the units used in the time stamps.
-\end_layout
-
-\begin_layout LyX-Code
-  time-units = year
-\end_layout
-
-\begin_layout LyX-Code
-
-\end_layout
-
-\begin_layout LyX-Code
-} // end of TimeHistory header
-\end_layout
-
-\begin_layout LyX-Code
- 
-\end_layout
-
-\begin_layout LyX-Code
-// The time history values are listed after the parameters.
-\end_layout
-
-\begin_layout LyX-Code
-// Columns time and amplitude where the amplitude values are unitless.
-\end_layout
-
-\begin_layout LyX-Code
- 0.0     0.00
-\end_layout
-
-\begin_layout LyX-Code
- 2.0     1.00
-\end_layout
-
-\begin_layout LyX-Code
- 6.0     4.00
-\end_layout
-
-\begin_layout LyX-Code
-10.0     2.00
-\end_layout
-
-\begin_layout LyX-Code
-11.0     0.00
-\end_layout
-
-\begin_layout Standard
-\begin_inset Float figure
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\begin_inset Caption
-
-\begin_layout Plain Layout
-Format of time history database files.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-\begin_inset CommandInset label
-LatexCommand label
-name "sec:FileFormat:TimeStepUser"
-
-\end_inset
-
-User-Specified Time Step File
-\end_layout
-
-\begin_layout Standard
-This file lists the time step sizes for nonuniform, user-specified time
- steps.
- The file's format is an ASCII file that includes the units for the time
- step sizes and then a list of the time steps.
- 
-\end_layout
-
-\begin_layout LyX-Code
-// This time step file specifies five time steps with the units in years.
-\end_layout
-
-\begin_layout LyX-Code
- 
-\end_layout
-
-\begin_layout LyX-Code
-// Comments can appear almost anywhere in these files and are
-\end_layout
-
-\begin_layout LyX-Code
-// delimited with two slashes (//) just like in C++.
- All text and 
-\end_layout
-
-\begin_layout LyX-Code
-// whitespace after the delimiter on a given line is ignored.
-\end_layout
-
-\begin_layout LyX-Code
-
-\end_layout
-
-\begin_layout LyX-Code
-// Units for the time steps
-\end_layout
-
-\begin_layout LyX-Code
-units = year
-\end_layout
-
-\begin_layout LyX-Code
-
-\end_layout
-
-\begin_layout LyX-Code
-1.0 // Comment
-\end_layout
-
-\begin_layout LyX-Code
-2.0
-\end_layout
-
-\begin_layout LyX-Code
-3.0
-\end_layout
-
-\begin_layout LyX-Code
-2.5
-\end_layout
-
-\begin_layout LyX-Code
-3.0
-\end_layout
-
-\begin_layout Standard
-\begin_inset Float figure
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\begin_inset Caption
-
-\begin_layout Plain Layout
-Format of user-specified time step files.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-\begin_inset CommandInset label
-LatexCommand label
-name "sec:FileFormat:PointsList"
-
-\end_inset
-
-Points List File
-\end_layout
-
-\begin_layout Standard
-This file lists the coordinates of the locations where output is requested
- for the 
-\family typewriter
-OutputSolnPoints
-\family default
- component.
- The coordinate system is specified in the 
-\family typewriter
-OutputSolnPoints
-\family default
- component.
- 
-\end_layout
-
-\begin_layout LyX-Code
-# This time step file specifies five time steps with the units in years.
-\end_layout
-
-\begin_layout LyX-Code
-#
-\end_layout
-
-\begin_layout LyX-Code
-# Comments are limited to complete lines.
- The default delimiter for comments
-\end_layout
-
-\begin_layout LyX-Code
-# is '#', which can be changed via parameters.
- Additionally, the delimiter 
-\end_layout
-
-\begin_layout LyX-Code
-# separating values can also be customized (default is whitespace).
-\end_layout
-
-\begin_layout LyX-Code
-1.0  -2.0   0.0
-\end_layout
-
-\begin_layout LyX-Code
-2.0  -4.0  -0.1
-\end_layout
-
-\begin_layout LyX-Code
-0.0  +2.0   0.0
-\end_layout
-
-\begin_layout LyX-Code
-2.5  -0.2  -0.2 
-\end_layout
-
-\begin_layout LyX-Code
-0.0   2.0  +0.2
-\end_layout
-
-\begin_layout Standard
-\begin_inset Float figure
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\begin_inset Caption
-
-\begin_layout Plain Layout
-Format of files with coordinates of points for output.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_body
-\end_document
+#LyX 2.0 created this file. For more info see http://www.lyx.org/
+\lyxformat 413
+\begin_document
+\begin_header
+\textclass book
+\begin_preamble
+
+\end_preamble
+\use_default_options false
+\maintain_unincluded_children false
+\language english
+\language_package default
+\inputencoding latin1
+\fontencoding global
+\font_roman default
+\font_sans default
+\font_typewriter default
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100
+\font_tt_scale 100
+
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry true
+\use_amsmath 1
+\use_esint 0
+\use_mhchem 1
+\use_mathdots 1
+\cite_engine basic
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\use_refstyle 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\leftmargin 1in
+\topmargin 1in
+\rightmargin 1in
+\bottommargin 2in
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\quotes_language english
+\papercolumns 1
+\papersides 2
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Chapter
+\begin_inset CommandInset label
+LatexCommand label
+name "cha:File-Formats"
+
+\end_inset
+
+File Formats
+\end_layout
+
+\begin_layout Section
+\begin_inset CommandInset label
+LatexCommand label
+name "sec:MeshIOAscii"
+
+\end_inset
+
+PyLith Mesh ASCII Files
+\end_layout
+
+\begin_layout Standard
+PyLith mesh ASCII files allow quick specification of the mesh information
+ for very small, simple meshes that are most easily written by hand.
+ We do not recommend using this format for anything other than these very
+ small, simple meshes.
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float figure
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\noindent
+\align center
+\begin_inset Graphics
+	filename figs/meshquad4.eps
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Diagram of mesh specified in Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:meshioascii:format"
+
+\end_inset
+
+.
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:meshioascii:diagram"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+// This mesh file defines a finite-element mesh composed of two
+\end_layout
+
+\begin_layout LyX-Code
+// square cells of edge length 2.
+\end_layout
+
+\begin_layout LyX-Code
+
+\end_layout
+
+\begin_layout LyX-Code
+// Comments can appear almost anywhere in these files and are
+\end_layout
+
+\begin_layout LyX-Code
+// delimited with two slashes (//) just like in C++.
+ All text and 
+\end_layout
+
+\begin_layout LyX-Code
+// whitespace after the delimiter on a given line is ignored.
+\end_layout
+
+\begin_layout LyX-Code
+mesh = { // begin specification of the mesh
+\end_layout
+
+\begin_layout LyX-Code
+  dimension = 2 // spatial dimension of the mesh
+\end_layout
+
+\begin_layout LyX-Code
+
+\end_layout
+
+\begin_layout LyX-Code
+  // Begin vertex and cell labels with 0.
+ This is the default so 
+\end_layout
+
+\begin_layout LyX-Code
+  // this next line is optional
+\end_layout
+
+\begin_layout LyX-Code
+  use-index-zero = true
+\end_layout
+
+\begin_layout LyX-Code
+
+\end_layout
+
+\begin_layout LyX-Code
+  vertices = { // vertices or nodes of the finite-element cells
+\end_layout
+
+\begin_layout LyX-Code
+    dimension = 2 // spatial dimension of the vertex coordinates
+\end_layout
+
+\begin_layout LyX-Code
+    count = 6 // number of vertices in the mesh
+\end_layout
+
+\begin_layout LyX-Code
+    coordinates = { // list of vertex index and coordinates
+\end_layout
+
+\begin_layout LyX-Code
+      // the coordinates must coincide with the coordinate 
+\end_layout
+
+\begin_layout LyX-Code
+      // system specified in the Mesh object
+\end_layout
+
+\begin_layout LyX-Code
+ 
+\end_layout
+
+\begin_layout LyX-Code
+      // exactly one vertex must appear on each line
+\end_layout
+
+\begin_layout LyX-Code
+      // (excluding whitespace)
+\end_layout
+
+\begin_layout LyX-Code
+      0    -2.0  -1.0
+\end_layout
+
+\begin_layout LyX-Code
+      1    -2.0  +1.0
+\end_layout
+
+\begin_layout LyX-Code
+      2     0.0  -1.0
+\end_layout
+
+\begin_layout LyX-Code
+      3     0.0  +1.0
+\end_layout
+
+\begin_layout LyX-Code
+      4    +2.0  -1.0
+\end_layout
+
+\begin_layout LyX-Code
+      5    +2.0  +1.0
+\end_layout
+
+\begin_layout LyX-Code
+    } // end of coordinates list
+\end_layout
+
+\begin_layout LyX-Code
+  } // end of vertices
+\end_layout
+
+\begin_layout LyX-Code
+ 
+\end_layout
+
+\begin_layout LyX-Code
+  cells = { // finite-element cells
+\end_layout
+
+\begin_layout LyX-Code
+    count = 2 // number of cells in the mesh
+\end_layout
+
+\begin_layout LyX-Code
+    num-corners = 4 // number of vertices defining the cell
+\end_layout
+
+\begin_layout LyX-Code
+    simplices = { // list of vertices in each cell
+\end_layout
+
+\begin_layout LyX-Code
+      // see Section 4.2 for diagrams giving the order for each 
+\end_layout
+
+\begin_layout LyX-Code
+      // type of cell supported in PyLith
+\end_layout
+
+\begin_layout LyX-Code
+ 
+\end_layout
+
+\begin_layout LyX-Code
+      // index of cell precedes the list of vertices for the cell 
+\end_layout
+
+\begin_layout LyX-Code
+
+\end_layout
+
+\begin_layout LyX-Code
+      // exactly one cell must appear on each line
+\end_layout
+
+\begin_layout LyX-Code
+      // (excluding whitespace)
+\end_layout
+
+\begin_layout LyX-Code
+      0    0  2  3  1
+\end_layout
+
+\begin_layout LyX-Code
+      1    4  5  3  2
+\end_layout
+
+\begin_layout LyX-Code
+    } // end of simplices list
+\end_layout
+
+\begin_layout LyX-Code
+ 
+\end_layout
+
+\begin_layout LyX-Code
+    material-ids = { // associated each cell with a material model
+\end_layout
+
+\begin_layout LyX-Code
+      // the material id is specified using the index of the cell 
+\end_layout
+
+\begin_layout LyX-Code
+      // and then the corresponding material id
+\end_layout
+
+\begin_layout LyX-Code
+      0   0 // cell 0 has a material id of 0
+\end_layout
+
+\begin_layout LyX-Code
+      1   2 // cell 1 has a material id of 2
+\end_layout
+
+\begin_layout LyX-Code
+    } // end of material-ids list
+\end_layout
+
+\begin_layout LyX-Code
+  } // end of cells
+\end_layout
+
+\begin_layout LyX-Code
+ 
+\end_layout
+
+\begin_layout LyX-Code
+  // This next section lists groups of vertices that can be used
+\end_layout
+
+\begin_layout LyX-Code
+  // in applying boundary conditions to portions of the domain
+\end_layout
+
+\begin_layout LyX-Code
+  group = { // start of a group
+\end_layout
+
+\begin_layout LyX-Code
+    // the name can have whitespace, so no comments are allowed
+\end_layout
+
+\begin_layout LyX-Code
+    // after the name
+\end_layout
+
+\begin_layout LyX-Code
+    name = face +y
+\end_layout
+
+\begin_layout LyX-Code
+ 
+\end_layout
+
+\begin_layout LyX-Code
+    // Either groups of vertices or groups of cells can be     
+\end_layout
+
+\begin_layout LyX-Code
+    // specified, but currently PyLith only makes use of groups 
+\end_layout
+
+\begin_layout LyX-Code
+    // of vertices
+\end_layout
+
+\begin_layout LyX-Code
+    type = vertices // 'vertices' or 'cells'
+\end_layout
+
+\begin_layout LyX-Code
+    count = 2 // number of vertices in the group
+\end_layout
+
+\begin_layout LyX-Code
+    indices = { // list of vertex indices in the group
+\end_layout
+
+\begin_layout LyX-Code
+      // multiple vertices may appear on a line
+\end_layout
+
+\begin_layout LyX-Code
+      0  4 // this group contains vertices 0 and 4
+\end_layout
+
+\begin_layout LyX-Code
+    } // end of list of vertices
+\end_layout
+
+\begin_layout LyX-Code
+  } // end of group
+\end_layout
+
+\begin_layout LyX-Code
+  // additional groups can be listed here
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Format of 
+\family typewriter
+PyLith
+\family default
+ mesh ASCII files.
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:meshioascii:format"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+\begin_inset CommandInset label
+LatexCommand label
+name "sec:Spatialdata:SimpleIOAscii"
+
+\end_inset
+
+SimpleDB Spatial Database Files
+\end_layout
+
+\begin_layout Standard
+SimpleDB spatial database files contain a header describing the set of points
+ and then the data with each line listing the coordinates of a point followed
+ by the values of the fields for that point.
+ 
+\end_layout
+
+\begin_layout LyX-Code
+// This spatial database specifies the distribution of slip on the
+\end_layout
+
+\begin_layout LyX-Code
+// fault surface.
+ In this case we prescribe a piecewise linear, 
+\end_layout
+
+\begin_layout LyX-Code
+// depth dependent distribution of slip.
+ The slip is 2.0 m 
+\end_layout
+
+\begin_layout LyX-Code
+// right-lateral with 0.25 m of reverse slip at the surface with
+\end_layout
+
+\begin_layout LyX-Code
+// a linear taper from 2.0 m to 0.0 m from -2 km to -4 km.
+\end_layout
+
+\begin_layout LyX-Code
+ 
+\end_layout
+
+\begin_layout LyX-Code
+// Comments can appear almost anywhere in these files and are
+\end_layout
+
+\begin_layout LyX-Code
+// delimited with two slashes (//) just like in C++.
+ All text and 
+\end_layout
+
+\begin_layout LyX-Code
+// whitespace after the delimiter on a given line is ignored.
+\end_layout
+
+\begin_layout LyX-Code
+ 
+\end_layout
+
+\begin_layout LyX-Code
+// The next line is the magic header for spatial database files 
+\end_layout
+
+\begin_layout LyX-Code
+// in ASCII format.
+\end_layout
+
+\begin_layout LyX-Code
+#SPATIAL.ascii 1
+\end_layout
+
+\begin_layout LyX-Code
+SimpleDB { // start specifying the database parameters
+\end_layout
+
+\begin_layout LyX-Code
+  num-values = 3 // number of values in the database
+\end_layout
+
+\begin_layout LyX-Code
+ 
+\end_layout
+
+\begin_layout LyX-Code
+  // Specify the names and the order of the values as they appear 
+\end_layout
+
+\begin_layout LyX-Code
+  // in the data.
+ The names of the values must correspond to the 
+\end_layout
+
+\begin_layout LyX-Code
+  // names PyLith requests in querying the database.
+\end_layout
+
+\begin_layout LyX-Code
+  value-names =  left-lateral-slip  reverse-slip  fault-opening
+\end_layout
+
+\begin_layout LyX-Code
+ 
+\end_layout
+
+\begin_layout LyX-Code
+  // Specify the units of the values in Python syntax (e.g., kg/m**3).
+\end_layout
+
+\begin_layout LyX-Code
+  value-units =  m  m  m
+\end_layout
+
+\begin_layout LyX-Code
+
+\end_layout
+
+\begin_layout LyX-Code
+  num-locs = 3 // Number of locations where values are given
+\end_layout
+
+\begin_layout LyX-Code
+  data-dim = 1 // Locations of data points form a line
+\end_layout
+
+\begin_layout LyX-Code
+  space-dim = 3 // Spatial dimension in which data resides
+\end_layout
+
+\begin_layout LyX-Code
+ 
+\end_layout
+
+\begin_layout LyX-Code
+  // Specify the coordinate system associated with the 
+\end_layout
+
+\begin_layout LyX-Code
+  // coordinates of the locations where data is given
+\end_layout
+
+\begin_layout LyX-Code
+  cs-data = cartesian { // Use a Cartesian coordinate system
+\end_layout
+
+\begin_layout LyX-Code
+    to-meters = 1.0e+3 // Coordinates are in km
+\end_layout
+
+\begin_layout LyX-Code
+ 
+\end_layout
+
+\begin_layout LyX-Code
+    // Specify the spatial dimension of the coordinate system
+\end_layout
+
+\begin_layout LyX-Code
+    // This value must match the one associated with the database
+\end_layout
+
+\begin_layout LyX-Code
+    space-dim = 3
+\end_layout
+
+\begin_layout LyX-Code
+  } // cs-data // end of coordinate system specification
+\end_layout
+
+\begin_layout LyX-Code
+} // end of SimpleDB specification
+\end_layout
+
+\begin_layout LyX-Code
+ 
+\end_layout
+
+\begin_layout LyX-Code
+// The locations and values are listed after the parameters.
+\end_layout
+
+\begin_layout LyX-Code
+// Columns are coordinates of the points (1 column for each 
+\end_layout
+
+\begin_layout LyX-Code
+// spatial dimension) followed by the data values in the order 
+\end_layout
+
+\begin_layout LyX-Code
+// specified by the value-names field.
+\end_layout
+
+\begin_layout LyX-Code
+0.0  0.0  0.0    -2.00  0.25  0.00
+\end_layout
+
+\begin_layout LyX-Code
+0.0  0.0 -2.0    -2.00  0.00  0.00
+\end_layout
+
+\begin_layout LyX-Code
+0.0  0.0 -4.0     0.00  0.00  0.00
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Format of spatial database files.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Spatial Database Coordinate Systems
+\end_layout
+
+\begin_layout Standard
+The spatial database files support four different types of coordinate systems.
+ Conversions among the three geographic coordinate systems are supported
+ in 3D.
+ Conversions among the geographic and geographic projected coordinate systems
+ are supported in 2D.
+ In using the coordinate systems, we assume that you have installed the
+ 
+\family typewriter
+proj
+\family default
+ program in addition to the Proj.4 libraries, so that you can obtain a list
+ of support projections, datums, and ellipsoids.
+ Alternatively, refer to the documentation for the Proj.4 Cartographic Projection
+s library 
+\begin_inset Flex URL
+status collapsed
+
+\begin_layout Plain Layout
+
+trac.osgeo.org/proj
+\end_layout
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsubsection
+Cartesian
+\end_layout
+
+\begin_layout Standard
+This is a conventional Cartesian coordinate system.
+ Conversions to other Cartesian coordinate systems are possible.
+\end_layout
+
+\begin_layout LyX-Code
+cs-data = cartesian {
+\end_layout
+
+\begin_layout LyX-Code
+  to-meters = 1.0e+3 // Locations in km
+\end_layout
+
+\begin_layout LyX-Code
+  space-dim = 2 // 1, 2, or 3 dimensions
+\end_layout
+
+\begin_layout LyX-Code
+}
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Format for Cartesian coordinate systems in spatial database files.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsubsection
+Geographic
+\end_layout
+
+\begin_layout Standard
+This coordinate system is for geographic coordinates, such as longitude
+ and latitude.
+ Specification of the location in three-dimensions is supported.
+ The vertical datum can be either the reference ellipsoid or mean sea level.
+ The vertical coordinate is positive upwards.
+\end_layout
+
+\begin_layout LyX-Code
+cs-data = geographic {
+\end_layout
+
+\begin_layout LyX-Code
+  // Conversion factor to get to meters (only applies to vertical 
+\end_layout
+
+\begin_layout LyX-Code
+  // coordinate unless you are using a geocentric coordinate system).
+\end_layout
+
+\begin_layout LyX-Code
+  to-meters = 1.0e+3
+\end_layout
+
+\begin_layout LyX-Code
+  space-dim = 2 // 2 or 3 dimensions
+\end_layout
+
+\begin_layout LyX-Code
+ 
+\end_layout
+
+\begin_layout LyX-Code
+  // Run 
+\begin_inset Quotes eld
+\end_inset
+
+proj -le
+\begin_inset Quotes erd
+\end_inset
+
+ to see a list of available reference ellipsoids.
+\end_layout
+
+\begin_layout LyX-Code
+  // Comments are not allowed at the end of the next line.
+\end_layout
+
+\begin_layout LyX-Code
+  ellipsoid = WGS84
+\end_layout
+
+\begin_layout LyX-Code
+ 
+\end_layout
+
+\begin_layout LyX-Code
+  // Run 
+\begin_inset Quotes eld
+\end_inset
+
+proj -ld
+\begin_inset Quotes erd
+\end_inset
+
+ to see a list of available datums.
+\end_layout
+
+\begin_layout LyX-Code
+  // Comments are not allowed at the end of the next line.
+\end_layout
+
+\begin_layout LyX-Code
+  datum-horiz = WGS84
+\end_layout
+
+\begin_layout LyX-Code
+ 
+\end_layout
+
+\begin_layout LyX-Code
+  // 
+\begin_inset Quotes eld
+\end_inset
+
+ellipsoid
+\begin_inset Quotes erd
+\end_inset
+
+ or 
+\begin_inset Quotes eld
+\end_inset
+
+mean sea level
+\begin_inset Quotes erd
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+  // Comments are not allowed at the end of the next line.
+\end_layout
+
+\begin_layout LyX-Code
+  datum-vert = ellipsoid
+\end_layout
+
+\begin_layout LyX-Code
+ 
+\end_layout
+
+\begin_layout LyX-Code
+  // Use a geocentric coordinate system?
+\end_layout
+
+\begin_layout LyX-Code
+  is-geocentric = false // true or false
+\end_layout
+
+\begin_layout LyX-Code
+}
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Format for geographic coordinate systems in spatial database files.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsubsection
+Geographic Projection
+\end_layout
+
+\begin_layout Standard
+This coordinate system applies to geographic projections.
+ As in the geographic coordinate system, the vertical coordinate (if used)
+ can be specified with respect to either the reference ellipsoid or mean
+ sea level.
+\end_layout
+
+\begin_layout LyX-Code
+cs-data = geo-projected {
+\end_layout
+
+\begin_layout LyX-Code
+  to-meters = 1.0e+3 // Conversion factor to get to meters.
+\end_layout
+
+\begin_layout LyX-Code
+  space-dim = 2 // 2 or 3 dimensions
+\end_layout
+
+\begin_layout LyX-Code
+
+\end_layout
+
+\begin_layout LyX-Code
+  // Run 
+\begin_inset Quotes eld
+\end_inset
+
+proj -le
+\begin_inset Quotes erd
+\end_inset
+
+ to see a list of available reference ellipsoids.
+\end_layout
+
+\begin_layout LyX-Code
+  // Comments are not allowed at the end of the next line.
+\end_layout
+
+\begin_layout LyX-Code
+  ellipsoid = WGS84
+\end_layout
+
+\begin_layout LyX-Code
+ 
+\end_layout
+
+\begin_layout LyX-Code
+  // Run 
+\begin_inset Quotes eld
+\end_inset
+
+proj -ld
+\begin_inset Quotes erd
+\end_inset
+
+ to see a list of available datums.
+\end_layout
+
+\begin_layout LyX-Code
+  // Comments are not allowed at the end of the next line.
+\end_layout
+
+\begin_layout LyX-Code
+  datum-horiz = WGS84
+\end_layout
+
+\begin_layout LyX-Code
+ 
+\end_layout
+
+\begin_layout LyX-Code
+  // 
+\begin_inset Quotes eld
+\end_inset
+
+ellipsoid
+\begin_inset Quotes erd
+\end_inset
+
+ or 
+\begin_inset Quotes eld
+\end_inset
+
+mean sea level
+\begin_inset Quotes erd
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+  // Comments are not allowed at the end of the next line.
+\end_layout
+
+\begin_layout LyX-Code
+  datum-vert = ellipsoid
+\end_layout
+
+\begin_layout LyX-Code
+ 
+\end_layout
+
+\begin_layout LyX-Code
+  // Run 
+\begin_inset Quotes eld
+\end_inset
+
+proj -lp
+\begin_inset Quotes erd
+\end_inset
+
+ to see a list of available geographic 
+\end_layout
+
+\begin_layout LyX-Code
+  // projections.
+\end_layout
+
+\begin_layout LyX-Code
+    projector = projection {
+\end_layout
+
+\begin_layout LyX-Code
+    // Name of the projection.
+ run 
+\begin_inset Quotes eld
+\end_inset
+
+proj -lp
+\begin_inset Quotes erd
+\end_inset
+
+ to see a list of 
+\end_layout
+
+\begin_layout LyX-Code
+    // supported projections.
+ Use the Universal Transverse Mercator
+\end_layout
+
+\begin_layout LyX-Code
+    // projection.
+\end_layout
+
+\begin_layout LyX-Code
+    projection = utm
+\end_layout
+
+\begin_layout LyX-Code
+    units = m // Units in the projection.
+\end_layout
+
+\begin_layout LyX-Code
+ 
+\end_layout
+
+\begin_layout LyX-Code
+    // Provide a list of projection options; these are the command 
+\end_layout
+
+\begin_layout LyX-Code
+    // line arguments you would use with the proj program.
+ Refer to
+\end_layout
+
+\begin_layout LyX-Code
+    // the Proj.4 library documentation for complete details.
+\end_layout
+
+\begin_layout LyX-Code
+    // Comments are not allowed at the end of the next line.
+\end_layout
+
+\begin_layout LyX-Code
+    proj-options = +zone=10
+\end_layout
+
+\begin_layout LyX-Code
+}
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Format for geographic projection coordinate systems in spatial database
+ files.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsubsection
+Geographic Local Cartesian
+\end_layout
+
+\begin_layout Standard
+This coordinate system is a geographically referenced, local 3D Cartesian
+ coordinate system.
+ This allows use of a conventional Cartesian coordinate system with accurate
+ georeferencing.
+ For example, one can construct a finite-element model in this coordinate
+ system and use spatial databases in geographic coordinates.
+ This coordinate system provides an alternative to using a geographic projection
+ as the Cartesian grip.
+ The advantage of this coordinate system is that it retains the curvature
+ of the Earth, while a geographic projection does not.
+\end_layout
+
+\begin_layout LyX-Code
+cs-data = geo-local-cartesian {
+\end_layout
+
+\begin_layout LyX-Code
+  // Conversion factor to get to meters (only applies to vertical
+\end_layout
+
+\begin_layout LyX-Code
+  // coordinate unless you are using a geocentric coordinate system).
+\end_layout
+
+\begin_layout LyX-Code
+  to-meters = 1.0 // use meters
+\end_layout
+
+\begin_layout LyX-Code
+  space-dim = 2 // 2 or 3 dimensions
+\end_layout
+
+\begin_layout LyX-Code
+ 
+\end_layout
+
+\begin_layout LyX-Code
+  // Run 
+\begin_inset Quotes eld
+\end_inset
+
+proj -le
+\begin_inset Quotes erd
+\end_inset
+
+ to see a list of available reference ellipsoids.
+\end_layout
+
+\begin_layout LyX-Code
+  // Comments are not allowed at the end of the next line.
+\end_layout
+
+\begin_layout LyX-Code
+  ellipsoid = WGS84
+\end_layout
+
+\begin_layout LyX-Code
+ 
+\end_layout
+
+\begin_layout LyX-Code
+  // Run 
+\begin_inset Quotes eld
+\end_inset
+
+proj -ld
+\begin_inset Quotes erd
+\end_inset
+
+ to see a list of available datums.
+\end_layout
+
+\begin_layout LyX-Code
+  // Comments are not allowed at the end of the next line.
+\end_layout
+
+\begin_layout LyX-Code
+  datum-horiz = WGS84
+\end_layout
+
+\begin_layout LyX-Code
+ 
+\end_layout
+
+\begin_layout LyX-Code
+  // 
+\begin_inset Quotes eld
+\end_inset
+
+ellipsoid
+\begin_inset Quotes erd
+\end_inset
+
+ or 
+\begin_inset Quotes eld
+\end_inset
+
+mean sea level
+\begin_inset Quotes erd
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+  // Comments are not allowed at the end of the next line.
+\end_layout
+
+\begin_layout LyX-Code
+  datum-vert = ellipsoid
+\end_layout
+
+\begin_layout LyX-Code
+ 
+\end_layout
+
+\begin_layout LyX-Code
+  // Origin of the local Cartesian coordinate system.
+ To avoid
+\end_layout
+
+\begin_layout LyX-Code
+  // round-off errors it is best to pick a location near the center of
+\end_layout
+
+\begin_layout LyX-Code
+  // the region of interest.
+ An elevation on the surface of the Earth
+\end_layout
+
+\begin_layout LyX-Code
+  // in the middle of the region also works well (and makes the
+\end_layout
+
+\begin_layout LyX-Code
+  // vertical coordinate easy to interpret).
+\end_layout
+
+\begin_layout LyX-Code
+  origin-lon = -116.7094 // Longitude of the origin in decimal degrees
+\end_layout
+
+\begin_layout LyX-Code
+                         // (west is negative).
+\end_layout
+
+\begin_layout LyX-Code
+  origin-lat = 36.3874 // Latitude of the origin in decimal degrees 
+\end_layout
+
+\begin_layout LyX-Code
+                       // (north is positive).
+\end_layout
+
+\begin_layout LyX-Code
+ 
+\end_layout
+
+\begin_layout LyX-Code
+  // Elevation with respect to the vertical datum.
+ Units are the
+\end_layout
+
+\begin_layout LyX-Code
+  // same as the Cartesian coordinate system (in this case meters).
+\end_layout
+
+\begin_layout LyX-Code
+  origin-elev = 3.5
+\end_layout
+
+\begin_layout LyX-Code
+}
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Format for the geographic local Cartesian coordinate system in spatial database
+ files.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+\begin_inset CommandInset label
+LatexCommand label
+name "sec:Spatialdata:TimeHistoryIO"
+
+\end_inset
+
+Time History Database Files
+\end_layout
+
+\begin_layout Standard
+Time history database files contain a header describing the number of points
+ in the time history and the units for the time stamps followed by a list
+ with pairs of time stamps and amplitude values.
+ The amplitude at an arbitrary point in time is computed via interpolation
+ of the values in the database.
+ This means that the time history database must span the range of time values
+ of interest.
+ The points in the time history must also be ordered in time.
+\end_layout
+
+\begin_layout LyX-Code
+// This time history database specifies temporal variation in
+\end_layout
+
+\begin_layout LyX-Code
+// amplitude.
+ In this case we prescribe a triangular slip time
+\end_layout
+
+\begin_layout LyX-Code
+// history.
+ 
+\end_layout
+
+\begin_layout LyX-Code
+ 
+\end_layout
+
+\begin_layout LyX-Code
+// Comments can appear almost anywhere in these files and are
+\end_layout
+
+\begin_layout LyX-Code
+// delimited with two slashes (//) just like in C++.
+ All text and 
+\end_layout
+
+\begin_layout LyX-Code
+// whitespace after the delimiter on a given line is ignored.
+\end_layout
+
+\begin_layout LyX-Code
+ 
+\end_layout
+
+\begin_layout LyX-Code
+// The next line is the magic header for spatial database files 
+\end_layout
+
+\begin_layout LyX-Code
+// in ASCII format.
+\end_layout
+
+\begin_layout LyX-Code
+#TIME HISTORY ascii
+\end_layout
+
+\begin_layout LyX-Code
+TimeHistory { // start specifying the database parameters
+\end_layout
+
+\begin_layout LyX-Code
+  num-points = 5 // number of points in time history
+\end_layout
+
+\begin_layout LyX-Code
+ 
+\end_layout
+
+\begin_layout LyX-Code
+  // Specify the units used in the time stamps.
+\end_layout
+
+\begin_layout LyX-Code
+  time-units = year
+\end_layout
+
+\begin_layout LyX-Code
+
+\end_layout
+
+\begin_layout LyX-Code
+} // end of TimeHistory header
+\end_layout
+
+\begin_layout LyX-Code
+ 
+\end_layout
+
+\begin_layout LyX-Code
+// The time history values are listed after the parameters.
+\end_layout
+
+\begin_layout LyX-Code
+// Columns time and amplitude where the amplitude values are unitless.
+\end_layout
+
+\begin_layout LyX-Code
+ 0.0     0.00
+\end_layout
+
+\begin_layout LyX-Code
+ 2.0     1.00
+\end_layout
+
+\begin_layout LyX-Code
+ 6.0     4.00
+\end_layout
+
+\begin_layout LyX-Code
+10.0     2.00
+\end_layout
+
+\begin_layout LyX-Code
+11.0     0.00
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Format of time history database files.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+\begin_inset CommandInset label
+LatexCommand label
+name "sec:FileFormat:TimeStepUser"
+
+\end_inset
+
+User-Specified Time-Step File
+\end_layout
+
+\begin_layout Standard
+This file lists the time-step sizes for nonuniform, user-specified time
+ steps.
+ The file's format is an ASCII file that includes the units for the time-step
+ sizes and then a list of the time steps.
+ 
+\end_layout
+
+\begin_layout LyX-Code
+// This time step file specifies five time steps with the units in years.
+\end_layout
+
+\begin_layout LyX-Code
+ 
+\end_layout
+
+\begin_layout LyX-Code
+// Comments can appear almost anywhere in these files and are
+\end_layout
+
+\begin_layout LyX-Code
+// delimited with two slashes (//) just like in C++.
+ All text and 
+\end_layout
+
+\begin_layout LyX-Code
+// whitespace after the delimiter on a given line is ignored.
+\end_layout
+
+\begin_layout LyX-Code
+
+\end_layout
+
+\begin_layout LyX-Code
+// Units for the time steps
+\end_layout
+
+\begin_layout LyX-Code
+units = year
+\end_layout
+
+\begin_layout LyX-Code
+
+\end_layout
+
+\begin_layout LyX-Code
+1.0 // Comment
+\end_layout
+
+\begin_layout LyX-Code
+2.0
+\end_layout
+
+\begin_layout LyX-Code
+3.0
+\end_layout
+
+\begin_layout LyX-Code
+2.5
+\end_layout
+
+\begin_layout LyX-Code
+3.0
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Format of user-specified time-step files.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+\begin_inset CommandInset label
+LatexCommand label
+name "sec:FileFormat:PointsList"
+
+\end_inset
+
+Points List File
+\end_layout
+
+\begin_layout Standard
+This file lists the coordinates of the locations where output is requested
+ for the 
+\family typewriter
+OutputSolnPoints
+\family default
+ component.
+ The coordinate system is specified in the 
+\family typewriter
+OutputSolnPoints
+\family default
+ component.
+ 
+\end_layout
+
+\begin_layout LyX-Code
+# This time step file specifies five time steps with the units in years.
+\end_layout
+
+\begin_layout LyX-Code
+#
+\end_layout
+
+\begin_layout LyX-Code
+# Comments are limited to complete lines.
+ The default delimiter for comments
+\end_layout
+
+\begin_layout LyX-Code
+# is '#', which can be changed via parameters.
+ Additionally, the delimiter 
+\end_layout
+
+\begin_layout LyX-Code
+# separating values can also be customized (default is whitespace).
+\end_layout
+
+\begin_layout LyX-Code
+1.0  -2.0   0.0
+\end_layout
+
+\begin_layout LyX-Code
+2.0  -4.0  -0.1
+\end_layout
+
+\begin_layout LyX-Code
+0.0  +2.0   0.0
+\end_layout
+
+\begin_layout LyX-Code
+2.5  -0.2  -0.2 
+\end_layout
+
+\begin_layout LyX-Code
+0.0   2.0  +0.2
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Format of files with coordinates of points for output.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document

Modified: short/3D/PyLith/branches/v1.7-trunk/doc/userguide/governingeqns/governingeqns.lyx
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/doc/userguide/governingeqns/governingeqns.lyx	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/doc/userguide/governingeqns/governingeqns.lyx	2012-08-21 18:24:25 UTC (rev 20616)
@@ -1,1859 +1,1859 @@
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-name "cha:Governing-Equations"
-
-\end_inset
-
-Governing Equations
-\end_layout
-
-\begin_layout Standard
-We present here a brief derivation of the equations for both quasi-static
- and dynamic computations.
- Since the general equations are the same (except for the absence of inertial
- terms in the quasi-static case), we first derive these equations.
- We then present solution methods for each specific case.
- In all of our derivations, we use the notation described in Table 
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-
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-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\rho$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Mass density scalar field
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-Derivation of Elasticity Equation
-\end_layout
-
-\begin_layout Subsection
-Index Notation
-\end_layout
-
-\begin_layout Standard
-Consider volume 
-\begin_inset Formula $V$
-\end_inset
-
- bounded by surface 
-\begin_inset Formula $S$
-\end_inset
-
-.
- Applying a Lagrangian description of the conservation of momentum gives
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula 
-\begin{equation}
-\frac{\partial}{\partial t}\int_{V}\rho\frac{\partial u_{i}}{\partial t}\, dV=\int_{V}f_{i}\, dV+\int_{S}T_{i}\, dS.\label{eqn:momentum:index}
-\end{equation}
-
-\end_inset
-
-The traction vector field is related to the stress tensor through
-\begin_inset Formula 
-\begin{equation}
-T_{i}=\sigma_{ij}n_{j},
-\end{equation}
-
-\end_inset
-
-where 
-\begin_inset Formula $n_{j}$
-\end_inset
-
- is the vector normal to 
-\begin_inset Formula $S$
-\end_inset
-
-.
- Substituting into equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eqn:momentum:index"
-
-\end_inset
-
- yields
-\begin_inset Formula 
-\begin{equation}
-\frac{\partial}{\partial t}\int_{V}\rho\frac{\partial u_{i}}{\partial t}\, dV=\int_{V}f_{i}\, dV+\int_{S}\sigma_{ij}n_{j}\, dS.
-\end{equation}
-
-\end_inset
-
-Applying the divergence theorem,
-\begin_inset Formula 
-\begin{equation}
-\int_{V}a_{i,j}\: dV=\int_{S}a_{j}n_{j}\: dS,
-\end{equation}
-
-\end_inset
-
-to the surface integral results in
-\begin_inset Formula 
-\begin{equation}
-\frac{\partial}{\partial t}\int_{V}\rho\frac{\partial u_{i}}{\partial t}\, dV=\int_{V}f_{i}\, dV+\int_{V}\sigma_{ij,j}\, dV,
-\end{equation}
-
-\end_inset
-
-which we can rewrite as
-\begin_inset Formula 
-\begin{equation}
-\int_{V}\left(\rho\frac{\partial^{2}u_{i}}{\partial t^{2}}-f_{i}-\sigma_{ij,j}\right)\, dV=0.
-\end{equation}
-
-\end_inset
-
-Because the volume 
-\begin_inset Formula $V$
-\end_inset
-
- is arbitrary, the integrand must be zero at every location in the volume,
- so that we end up with
-\begin_inset Formula 
-\begin{gather}
-\rho\frac{\partial^{2}u_{i}}{\partial t^{2}}-f_{i}-\sigma_{ij,j}=0\text{ in }V,\\
-\sigma_{ij}n_{j}=T_{i}\text{ on }S_{T}\text{,}\\
-u_{i}=u_{i}^{o}\text{ on }S_{u}\text{, and}\\
-R_{ki}(u_{i}^{+}-u_{i}^{-})=d_{k}\text{ on }S_{f}.
-\end{gather}
-
-\end_inset
-
-We specify tractions, 
-\begin_inset Formula $T_{i}$
-\end_inset
-
-, on surface 
-\begin_inset Formula $S_{f}$
-\end_inset
-
-, displacements, 
-\begin_inset Formula $u_{i}^{o}$
-\end_inset
-
-, on surface 
-\begin_inset Formula $S_{u}$
-\end_inset
-
-, and slip, 
-\begin_inset Formula $d_{k}$
-\end_inset
-
-, on fault surface 
-\begin_inset Formula $S_{f}$
-\end_inset
-
- (we will consider the case of fault constitutive models in Section 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "sec:fault"
-
-\end_inset
-
-).
- The rotation matrix 
-\begin_inset Formula $R_{ki}$
-\end_inset
-
- transforms vectors from the global coordinate system to the fault coordinate
- system.
- Note that since both 
-\begin_inset Formula $T_{i}$
-\end_inset
-
- and 
-\begin_inset Formula $u_{i}$
-\end_inset
-
- are vector quantities, there can be some spatial overlap of the surfaces
- 
-\begin_inset Formula $S_{T}$
-\end_inset
-
- and 
-\begin_inset Formula $S_{u}$
-\end_inset
-
-; however, the same degree of freedom cannot simultaneously have both types
- of boundary conditions.
-\end_layout
-
-\begin_layout Subsection
-Vector Notation
-\end_layout
-
-\begin_layout Standard
-Consider volume 
-\begin_inset Formula $V$
-\end_inset
-
- bounded by surface 
-\begin_inset Formula $S$
-\end_inset
-
-.
- Applying a Lagrangian description of the conservation of momentum gives
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula 
-\begin{equation}
-\frac{\partial}{\partial t}\int_{V}\rho\frac{\partial\vec{u}}{\partial t}\, dV=\int_{V}\overrightarrow{f}\, dV+\int_{S}\overrightarrow{T}\, dS.\label{eqn:momentum:vec}
-\end{equation}
-
-\end_inset
-
-The traction vector field is related to the stress tensor through
-\begin_inset Formula 
-\begin{equation}
-\overrightarrow{T}=\underline{\sigma}\cdot\overrightarrow{n},
-\end{equation}
-
-\end_inset
-
-where 
-\begin_inset Formula $\overrightarrow{n}$
-\end_inset
-
- is the vector normal to 
-\begin_inset Formula $S$
-\end_inset
-
-.
- Substituting into equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eqn:momentum:vec"
-
-\end_inset
-
- yields
-\begin_inset Formula 
-\begin{equation}
-\frac{\partial}{\partial t}\int_{V}\rho\frac{\partial\overrightarrow{u}}{\partial t}\, dV=\int_{V}\overrightarrow{f}\, dV+\int_{S}\underline{\sigma}\cdot\overrightarrow{n}\, dS.
-\end{equation}
-
-\end_inset
-
-Applying the divergence theorem,
-\begin_inset Formula 
-\begin{equation}
-\int_{V}\nabla\cdot\overrightarrow{a}\: dV=\int_{S}\overrightarrow{a}\cdot\overrightarrow{n}\: dS,
-\end{equation}
-
-\end_inset
-
-to the surface integral results in
-\begin_inset Formula 
-\begin{equation}
-\frac{\partial}{\partial t}\int_{V}\rho\frac{\partial\overrightarrow{u}}{\partial t}\, dV=\int_{V}\overrightarrow{f}\, dV+\int_{V}\nabla\cdot\underline{\sigma}\, dV,
-\end{equation}
-
-\end_inset
-
-which we can rewrite as
-\begin_inset Formula 
-\begin{equation}
-\int_{V}\left(\rho\frac{\partial^{2}\overrightarrow{u}}{\partial t^{2}}-\overrightarrow{f}-\nabla\cdot\overrightarrow{\sigma}\right)\, dV=\vec{0}.
-\end{equation}
-
-\end_inset
-
-Because the volume 
-\begin_inset Formula $V$
-\end_inset
-
- is arbitrary, the integrand must be the zero vector at every location in
- the volume, so that we end up with
-\begin_inset Formula 
-\begin{gather}
-\rho\frac{\partial^{2}\overrightarrow{u}}{\partial t^{2}}-\overrightarrow{f}-\nabla\cdot\overrightarrow{\sigma}=\vec{0}\text{ in }V,\\
-\underline{\sigma}\cdot\overrightarrow{n}=\overrightarrow{T}\text{ on }S_{T}\text{,}\\
-\overrightarrow{u}=\overrightarrow{u^{o}}\text{ on }S_{u},\text{ and}\\
-\underbar{R}\cdot(\vec{u^{+}}-\vec{u^{-}})=\vec{d}\text{ on }S_{f}.
-\end{gather}
-
-\end_inset
-
-We specify tractions, 
-\begin_inset Formula $\vec{T}$
-\end_inset
-
-, on surface 
-\begin_inset Formula $S_{f}$
-\end_inset
-
-, displacements, 
-\begin_inset Formula $\overrightarrow{u^{o}}$
-\end_inset
-
-, on surface 
-\begin_inset Formula $S_{u}$
-\end_inset
-
-, and slip, 
-\begin_inset Formula $\vec{d}$
-\end_inset
-
-, on fault surface 
-\begin_inset Formula $S_{f}$
-\end_inset
-
- (we will consider the case of fault constitutive models in Section 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "sec:fault"
-
-\end_inset
-
-).
- The rotation matrix 
-\begin_inset Formula $\underline{R}$
-\end_inset
-
- transforms vectors from the global coordinate system to the fault coordinate
- system.
- Note that since both 
-\begin_inset Formula $\overrightarrow{T}$
-\end_inset
-
- and 
-\begin_inset Formula $\overrightarrow{u}$
-\end_inset
-
- are vector quantities, there can be some spatial overlap of the surfaces
- 
-\begin_inset Formula $S_{T}$
-\end_inset
-
- and 
-\begin_inset Formula $S_{u}$
-\end_inset
-
-; however, the same degree of freedom cannot simultaneously have both types
- of boundary conditions.
-\end_layout
-
-\begin_layout Section
-Finite-Element Formulation of Elasticity Equation
-\end_layout
-
-\begin_layout Standard
-We formulate a set of algebraic equations using Galerkin's method.
- We consider (1) a trial solution, 
-\begin_inset Formula $\vec{u}$
-\end_inset
-
-, that is a piecewise differentiable vector field and satisfies the Dirichlet
- boundary conditions on 
-\begin_inset Formula $S_{u}$
-\end_inset
-
-, and (2) a weighting function, 
-\begin_inset Formula $\vec{\phi}$
-\end_inset
-
-, that is a piecewise differentiable vector field and is zero on 
-\begin_inset Formula $S_{u}$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Subsection
-Index Notation
-\end_layout
-
-\begin_layout Standard
-We start with the wave equation (strong form),
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula 
-\begin{gather}
-\sigma_{ij,j}+f_{i}=\rho\ddot{u_{i}}\text{ in }V,\\
-\sigma_{ij}n_{j}=T_{i}\text{ on }S_{T},\\
-u_{i}=u_{i}^{o}\text{ on }S_{u},\\
-R_{ki}(u_{i}^{+}-u_{i}^{-})=d_{k}\text{ on }S_{f},\text{ and}\\
-\sigma_{ij}=\sigma_{ji}\text{ (symmetric).}
-\end{gather}
-
-\end_inset
-
-We construct the weak form by computing the dot product of the wave equation
- and weighting function and setting the integral over the domain to zero:
-\begin_inset Formula 
-\begin{gather}
-\int_{V}\left(\sigma_{ij,j}+f_{i}-\rho\ddot{u}_{i}\right)\phi_{i}\, dV=0\text{, or }\\
-\int_{V}\sigma_{ij,j}\phi_{i}\: dV+\int_{V}f_{i}\phi_{i}\: dV-\int_{V}\rho\ddot{u}_{i}\phi_{i}\: dV=0.
-\end{gather}
-
-\end_inset
-
- Consider the divergence theorem applied to the dot product of the stress
- tensor and the weighting function, 
-\begin_inset Formula $\sigma_{ij}\phi_{i}$
-\end_inset
-
-,
-\begin_inset Formula 
-\begin{equation}
-\int_{V}(\sigma_{ij}\phi_{i})_{,j}\, dV=\int_{S}(\sigma_{ij}\phi_{i})n_{i}\, dS.
-\end{equation}
-
-\end_inset
-
-Expanding the left-hand side yields
-\begin_inset Formula 
-\begin{gather}
-\int_{V}\sigma_{ij,j}\phi_{i}\: dV+\int_{V}\sigma_{ij}\phi_{i,j}\: dV=\int_{S}\sigma_{ij}\phi_{i}n_{i}\: dS,\text{ or}\\
-\int_{V}\sigma_{ij,j}\phi_{i}\: dV=-\int_{V}\sigma_{ij}\phi_{i,j}\, dV+\int_{S}\sigma_{ij}\phi_{i}n_{i}\, dS.
-\end{gather}
-
-\end_inset
-
-Substituting into the weak form gives
-\begin_inset Formula 
-\begin{equation}
--\int_{V}\sigma_{ij}\phi_{i,j}\, dV+\int_{S}\sigma_{ij}\phi_{i}n_{i}\, dS+\int_{V}f_{i}\phi_{i}\, dV-\int_{V}\rho\ddot{u}_{i}\phi_{i}\, dV=0.
-\end{equation}
-
-\end_inset
-
-Turning our attention to the second term, we separate the integration over
- 
-\begin_inset Formula $S$
-\end_inset
-
- into integration over 
-\begin_inset Formula $S_{T}$
-\end_inset
-
- and 
-\begin_inset Formula $S_{u}$
-\end_inset
-
- (we will consider tractions over the fault surface, 
-\begin_inset Formula $S_{f}$
-\end_inset
-
-, associated with the fault constitutive model in Section 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "sec:fault"
-
-\end_inset
-
-),
-\begin_inset Formula 
-\begin{equation}
--\int_{V}\sigma_{ij}\phi_{i,j}\, dV+\int_{S_{T}}\sigma_{ij}\phi_{i}n_{i}\, dS+\int_{S_{u}}\sigma_{ij}\phi_{i}n_{i}\, dS+\int_{V}f_{i}\phi_{i}\, dV-\int_{V}\rho\ddot{u}_{i}\phi_{i}\, dV=0,
-\end{equation}
-
-\end_inset
-
-and recognize that
-\begin_inset Formula 
-\begin{gather}
-\sigma_{ij}n_{i}=T_{i}\text{ on }S_{T}\text{ and}\\
-\phi_{i}=0\text{ on }S_{u},
-\end{gather}
-
-\end_inset
-
-so that the equation reduces to
-\begin_inset Formula 
-\begin{equation}
--\int_{V}\sigma_{ij}\phi_{i,j}\: dV+\int_{S_{T}}T_{i}\phi_{i}\, dS+\int_{V}f_{i}\phi_{i}\, dV-\int_{V}\rho\ddot{u}_{i}\phi_{i}\, dV=0.\label{eq:elasticity:integral}
-\end{equation}
-
-\end_inset
-
-We express the trial solution and weighting function as linear combinations
- of basis functions,
-\begin_inset Formula 
-\begin{gather}
-u_{i}=\sum_{m}a_{i}^{m}N^{m},\\
-\phi_{i}=\sum_{n}c_{i}^{n}N^{n}.
-\end{gather}
-
-\end_inset
-
-Note that because the trial solution satisfies the Dirichlet boundary condition,
- the number of basis functions for 
-\begin_inset Formula $u$
-\end_inset
-
- is generally greater than the number of basis functions for 
-\begin_inset Formula $\phi$
-\end_inset
-
-, i.e., 
-\begin_inset Formula $m>n$
-\end_inset
-
-.
- Substituting in the expressions for the trial solution and weighting function
- yields
-\begin_inset Formula 
-\begin{gather}
--\int_{V}\sigma_{ij}\sum_{n}c_{i}^{n}N_{,j}^{n}\: dV+\int_{S_{T}}T_{i}\sum_{n}c_{i}^{n}N^{n}\, dS+\int_{V}f_{i}\sum_{n}c_{i}^{n}N^{n}\, dV-\int_{V}\rho\sum_{m}\ddot{a}_{i}^{m}N^{m}\sum_{n}c_{i}^{n}N^{n}\ dV=0,\text{ or}\\
-\sum_{n}c_{i}^{n}(-\int_{V}\sigma_{ij}N_{,j}^{n}\: dV+\int_{S_{T}}T_{i}N^{n}\, dS+\int_{V}f_{i}N^{n}\, dV-\int_{V}\rho\sum_{m}\ddot{a}_{i}^{m}N^{m}N^{n}\ dV)=0.
-\end{gather}
-
-\end_inset
-
- Because the weighting function is arbitrary, this equation must hold for
- all 
-\begin_inset Formula $c_{i}^{n}$
-\end_inset
-
-, so that the quantity in parenthesis is zero for each 
-\begin_inset Formula $c_{i}^{n}$
-\end_inset
-
-
-\begin_inset Formula 
-\begin{equation}
--\int_{V}\sigma_{ij}N_{,j}^{n}\: dV+\int_{S_{T}}T_{i}N^{n}\, dS+\int_{V}f_{i}N^{n}\, dV-\int_{V}\rho\sum_{m}\ddot{a}_{i}^{m}N^{m}N^{n}\ dV=\vec{0}.\label{eq:elasticity:integral:discretized}
-\end{equation}
-
-\end_inset
-
-We want to solve this equation for the unknown coefficients 
-\begin_inset Formula $a_{i}^{m}$
-\end_inset
-
- subject to
-\end_layout
-
-\begin_layout Standard
-
-\family roman
-\series medium
-\shape up
-\size normal
-\emph off
-\bar no
-\noun off
-\color none
-\begin_inset Formula 
-\begin{gather}
-u_{i}=u_{i}^{o}\text{ on }S_{u},\text{ and}\\
-R_{ki}(u_{i}^{+}-u_{i}^{-})=d_{k}\text{ on }S_{f},
-\end{gather}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsection
-Vector Notation
-\end_layout
-
-\begin_layout Standard
-We start with the wave equation (strong form),
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula 
-\begin{gather}
-\nabla\cdot\underline{\sigma}+\overrightarrow{f}=\rho\frac{\partial^{2}\overrightarrow{u}}{\partial t^{2}}\text{ in }V,\\
-\underline{\sigma}\cdot\overrightarrow{n}=\overrightarrow{T}\text{ on }S_{T},\\
-\overrightarrow{u}=\overrightarrow{u^{o}}\text{ on }S_{u},\\
-\underbar{R}\cdot(\overrightarrow{u^{+}}-\overrightarrow{u^{-}})=\vec{d}\text{ on }S_{f}\\
-\underline{\sigma}=\underline{\sigma}^{T}\text{ (symmetric).}
-\end{gather}
-
-\end_inset
-
-We construct the weak form by multiplying the wave equation by a weighting
- function and setting the integral over the domain to zero.
- The weighting function is a piecewise differential vector field, 
-\begin_inset Formula $\overrightarrow{\phi}$
-\end_inset
-
-, where 
-\begin_inset Formula $\overrightarrow{\phi}=0$
-\end_inset
-
- on 
-\begin_inset Formula $S_{u}.$
-\end_inset
-
- Hence our weak form is
-\begin_inset Formula 
-\begin{gather}
-\int_{V}\left(\nabla\cdot\underline{\sigma}+\overrightarrow{f}-\rho\frac{\partial^{2}\overrightarrow{u}}{\partial t^{2}}\right)\cdot\overrightarrow{\phi}\, dV=0\text{, or }\\
-\int_{V}(\nabla\cdot\underline{\sigma})\cdot\overrightarrow{\phi}\: dV+\int_{V}\overrightarrow{f}\cdot\overrightarrow{\phi}\: dV-\int_{V}\rho\frac{\partial^{2}\overrightarrow{u}}{\partial t^{2}}\cdot\overrightarrow{\phi}\: dV=0.
-\end{gather}
-
-\end_inset
-
- Consider the divergence theorem applied to the dot product of the stress
- tensor and the trial function, 
-\begin_inset Formula $\underline{\sigma}\cdot\overrightarrow{\phi}$
-\end_inset
-
-,
-\begin_inset Formula 
-\begin{equation}
-\int_{V}\nabla\cdot(\underline{\sigma}\cdot\overrightarrow{\phi})\, dV=\int_{S}(\underline{\sigma}\cdot\overrightarrow{\phi})\cdot\overrightarrow{n}\, dS.
-\end{equation}
-
-\end_inset
-
-Expanding the left-hand side yields
-\begin_inset Formula 
-\begin{equation}
-\int_{V}(\nabla\cdot\underline{\sigma})\cdot\overrightarrow{\phi}\: dV+\int_{V}\underline{\sigma}:\nabla\overrightarrow{\phi}\: dV=\int_{S}(\underline{\sigma}\cdot\overrightarrow{\phi})\cdot\overrightarrow{n}\: dS,\text{ or}
-\end{equation}
-
-\end_inset
-
-
-\begin_inset Formula 
-\begin{equation}
-\int_{V}(\nabla\cdot\underline{\sigma})\cdot\overrightarrow{\phi}\: dV=-\int_{V}\underline{\sigma}:\nabla\overrightarrow{\phi}\, dV+\int_{S}\underline{\sigma}\cdot\overrightarrow{n}\cdot\overrightarrow{\phi}\, dS.
-\end{equation}
-
-\end_inset
-
-Substituting into the weak form gives
-\begin_inset Formula 
-\begin{equation}
--\int_{V}\underline{\sigma}:\nabla\overrightarrow{\phi}\, dV+\int_{S}\underline{\sigma}\cdot\overrightarrow{n}\cdot\overrightarrow{\phi}\, dS+\int_{V}\overrightarrow{f}\cdot\overrightarrow{\phi}\, dV-\int_{V}\rho\frac{\partial^{2}\overrightarrow{u}}{\partial t^{2}}\cdot\overrightarrow{\phi}\, dV=0.
-\end{equation}
-
-\end_inset
-
-We separate the integration over 
-\begin_inset Formula $S$
-\end_inset
-
- into integration over 
-\begin_inset Formula $S_{T}$
-\end_inset
-
- and 
-\begin_inset Formula $S_{u}$
-\end_inset
-
-,
-\begin_inset Formula 
-\begin{multline}
--\int_{V}\underline{\sigma}:\nabla\overrightarrow{\phi}\, dV+\int_{S_{T}}\underline{\sigma}\cdot\overrightarrow{n}\cdot\overrightarrow{\phi}\, dS+\int_{S_{u}}\underline{\sigma}\cdot\overrightarrow{n}\cdot\overrightarrow{\phi}\, dS+\int_{V}\overrightarrow{f}\cdot\overrightarrow{\phi}\, dV-\int_{V}\rho\frac{\partial^{2}\overrightarrow{u}}{\partial t^{2}}\cdot\overrightarrow{\phi}\, dV=0,
-\end{multline}
-
-\end_inset
-
-and recognize that
-\begin_inset Formula 
-\begin{gather}
-\underline{\sigma}\cdot\overrightarrow{n}=\overrightarrow{T}\text{ on }S_{T}\text{ and}\\
-\overrightarrow{\phi}=0\text{ on }S_{u},
-\end{gather}
-
-\end_inset
-
-so that the equation reduces to
-\begin_inset Formula 
-\begin{equation}
--\int_{V}\underline{\sigma}:\nabla\overrightarrow{\phi}\: dV+\int_{S_{T}}\overrightarrow{T}\cdot\overrightarrow{\phi}\, dS+\int_{V}\overrightarrow{f}\cdot\overrightarrow{\phi}\, dV-\int_{V}\rho\frac{\partial^{2}\overrightarrow{u}}{\partial t^{2}}\cdot\overrightarrow{\phi}\, dV=0.
-\end{equation}
-
-\end_inset
-
-We express the trial solution and weighting function as linear combinations
- of basis functions,
-\begin_inset Formula 
-\begin{gather}
-\vec{u}=\sum_{m}\overrightarrow{a^{m}}N^{m},\\
-\vec{\phi}=\sum_{n}\overrightarrow{c^{n}}N^{n}.
-\end{gather}
-
-\end_inset
-
-Note that because the weighting function is zero on 
-\begin_inset Formula $S_{u}$
-\end_inset
-
-, the number of basis functions for 
-\begin_inset Formula $\vec{u}$
-\end_inset
-
- is generally greater than the number of basis functions for 
-\begin_inset Formula $\vec{\phi}$
-\end_inset
-
-, i.e., 
-\begin_inset Formula $m>n$
-\end_inset
-
-.
- Substituting in the expressions for the trial solution and weighting function
- yields
-\begin_inset Formula 
-\begin{multline}
--\int_{V}\underline{\sigma}:\sum_{n}\overrightarrow{c^{n}}\nabla N_{,}^{n}\, dV+\int_{S_{T}}\vec{T}\cdot\sum_{n}\overrightarrow{c^{n}}N^{n}\, dS+\int_{V}\vec{f}\cdot\sum_{n}\overrightarrow{c^{n}}N^{n}\, dV\\
--\int_{V}\rho\sum_{m}\frac{\partial^{2}\overrightarrow{a^{m}}}{\partial t^{2}}N^{m}\cdot\sum_{n}\overrightarrow{c^{n}}N^{n}\ dV=0.
-\end{multline}
-
-\end_inset
-
- Because the weighting function is arbitrary, this equation must hold for
- all 
-\begin_inset Formula $\overrightarrow{c^{n}}$
-\end_inset
-
-, so that
-\begin_inset Formula 
-\begin{equation}
--\int_{V}\underline{\sigma}:\nabla N^{n}\, dV+\int_{S_{T}}\vec{T}N^{n}\, dS+\int_{V}\vec{f}N^{n}\, dV-\int_{V}\rho\sum_{m}\frac{\partial^{2}\overrightarrow{a^{m}}}{\partial t^{2}}N^{m}N^{n}\, dV=\vec{0}.
-\end{equation}
-
-\end_inset
-
-We want to solve this equation for the unknown coefficients 
-\begin_inset Formula $\overrightarrow{a^{m}}$
-\end_inset
-
- subject to
-\end_layout
-
-\begin_layout Standard
-
-\family roman
-\series medium
-\shape up
-\size normal
-\emph off
-\bar no
-\noun off
-\color none
-\begin_inset Formula 
-\begin{gather}
-\vec{u}=u^{o}\overrightarrow{}\text{ on }S_{u},\text{ and}\\
-\underline{R}(\overrightarrow{u^{+}}-\overrightarrow{u^{-}})=\vec{d}\text{ on }S_{f},
-\end{gather}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-Solution Method for Quasi-Static Problems
-\end_layout
-
-\begin_layout Standard
-For brevity we outline the solution method for quasi-static problems using
- only index notation.
- In quasi-static problems we neglect the inertial terms, so equation 
-\begin_inset CommandInset ref
-LatexCommand eqref
-reference "eq:elasticity:integral:discretized"
-
-\end_inset
-
- reduces to
-\begin_inset Formula 
-\begin{equation}
--\int_{V}\sigma_{ij}N_{,j}^{n}\: dV+\int_{S_{T}}T_{i}N^{n}\, dS+\int_{V}f_{i}N^{n}\, dV=\vec{0}.
-\end{equation}
-
-\end_inset
-
-As a result, time-dependence only enters through the constitutive relationships
- and the loading conditions.
- We consider the deformation at time 
-\begin_inset Formula $t+\Delta t$
-\end_inset
-
-,
-\begin_inset Formula 
-\begin{equation}
--\int_{V}\sigma_{ij}(t+\Delta t)N_{,j}^{n}\: dV+\int_{S_{T}}T_{i}(t+\Delta t)N^{n}\, dS+\int_{V}f_{i}(t+\Delta t)N^{n}\, dV=\vec{0}.\label{eq:elasticity:integral:quasistatic}
-\end{equation}
-
-\end_inset
-
-We solve this equation through formulation of a linear algebraic system
- of equations (
-\begin_inset Formula $Au=b$
-\end_inset
-
-), involving the residual (
-\begin_inset Formula $r=b-Au$
-\end_inset
-
-) and Jacobian (
-\begin_inset Formula $A$
-\end_inset
-
-).
- The residual is simply
-\begin_inset Formula 
-\begin{equation}
-r_{i}^{n}=-\int_{V}\sigma_{ij}(t+\Delta t)N_{,j}^{n}\: dV+\int_{S_{T}}T_{i}(t+\Delta t)N^{n}\, dS+\int_{V}f_{i}(t+\Delta t)N^{n}\, dV.
-\end{equation}
-
-\end_inset
-
-We employ numerical quadrature in the finite-element discretization and
- replace the integrals with sums over the cells and quadrature points,
-\begin_inset Formula 
-\begin{multline}
-r_{i}^{n}=-\sum_{\text{vol cells}}\sum_{\text{quad pts}}\sigma_{ij}(x_{q},t+\Delta t)N_{,j}^{n}(x_{q})\: w_{q}|J_{cell}(x_{q})|+\sum_{\text{vol cells}}\sum_{\text{quad pt}s}f_{i}(x_{q},t+\Delta t)N^{n}(x_{q})\, w_{q}|J_{cell}(x_{q})|\\
-+\sum_{\text{tract cells}}\sum_{\text{quad pts}}T_{i}(x_{q},t+\Delta t)N^{n}(x_{q})\, w_{q}|J_{cell}(x_{q})|,
-\end{multline}
-
-\end_inset
-
-where 
-\begin_inset Formula $r_{i}^{n}$
-\end_inset
-
- is an 
-\begin_inset Formula $nd$
-\end_inset
-
- vector (
-\begin_inset Formula $d$
-\end_inset
-
- is the dimension of the vector space) and 
-\begin_inset Formula $i$
-\end_inset
-
- is a vector space component, 
-\begin_inset Formula $x_{q}$
-\end_inset
-
- are the coordinates of the quadrature points, 
-\begin_inset Formula $w_{q}$
-\end_inset
-
- are the weights of the quadrature points, and 
-\begin_inset Formula $|J_{cell}(x_{q})|$
-\end_inset
-
- is the determinant of the Jacobian matrix evaluated at the quadrature points
- associated with mapping the reference cell to the actual cell.
- The quadrature scheme for the integral over the tractions is one dimension
- lower than the one used in integrating the terms for the volume cells.
-\end_layout
-
-\begin_layout Standard
-In order to find the Jacobian of the system, we let
-\begin_inset Formula 
-\begin{equation}
-\sigma_{ij}(t+\Delta t)=\sigma_{ij}(t)+d\sigma_{ij}(t).
-\end{equation}
-
-\end_inset
-
-Isolating the term associated with the increment in stresses yields
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula 
-\begin{equation}
-\int_{V}d\sigma_{ij}(t)N_{j}^{n}\ dV=-\int_{V}\sigma_{ij}(t)N_{,j}^{n}\: dV+\int_{S_{T}}T_{i}(t+\Delta t)N^{n}\, dS+\int_{V}f_{i}(t+\Delta t)N^{n}\, dV
-\end{equation}
-
-\end_inset
-
-We associate the term on the left-hand-side with the action of the system
- Jacobian on the increment of the displacement field.
- We approximate the increment in stresses using linear elasticity and infinitesi
-mal strains,
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula 
-\begin{gather}
-d\sigma_{ij}(t)=C_{ijkl}(t)d\varepsilon_{kl}(t)\\
-d\sigma_{ij}(t)=\frac{1}{2}C_{ijkl}(t)(du_{k.l}(t)+du_{l,k}(t))\\
-d\sigma_{ij}(t)=\frac{1}{2}C_{ijkl}(t)(\sum_{m}da_{k,l}^{m}(t)N^{m}+\sum_{m}da_{l,k}^{m}(t)N^{m})
-\end{gather}
-
-\end_inset
-
-Now, 
-\begin_inset Formula $d\sigma_{ij}\phi_{i,j}$
-\end_inset
-
- is a scalar, so it is symmetric,
-\begin_inset Formula 
-\begin{equation}
-d\sigma_{ij}\phi_{i,j}=d\sigma_{ji}\phi_{j,i},
-\end{equation}
-
-\end_inset
-
-and we know that 
-\begin_inset Formula $d\sigma_{ij}$
-\end_inset
-
- is symmetric, so
-\begin_inset Formula 
-\begin{equation}
-d\sigma_{ij}\phi_{i,j}=d\sigma_{ij}\phi_{j,i},
-\end{equation}
-
-\end_inset
-
-which means
-\begin_inset Formula 
-\begin{equation}
-\phi_{i,j}=\phi_{j,i},
-\end{equation}
-
-\end_inset
-
-which we can write as
-\begin_inset Formula 
-\begin{equation}
-\phi_{i,j}=\frac{1}{2}(\phi_{i,j}+\phi_{j,i}).
-\end{equation}
-
-\end_inset
-
-In terms of the basis functions, we have
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula 
-\begin{equation}
-\sum_{n}c_{i}^{n}N_{,j}^{n}=\frac{1}{2}(\sum_{n}c_{i}^{n}N_{,j}^{n}+\sum_{n}c_{j}^{n}N_{,i}^{n}).
-\end{equation}
-
-\end_inset
-
-Combining these expressions for the increment in stresses and making use
- of the symmetry of the weighting functions, we find the system Jacobian
- is
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula 
-\begin{equation}
-A_{ij}^{nm}=\int_{V}\frac{1}{4}C_{ijkl}(N_{,l}^{m}+N_{,k}^{m})(N_{,j}^{n}+N_{,i}^{n})\ dV.
-\end{equation}
-
-\end_inset
-
-We employ numerical quadrature in the finite-element discretization and
- replace the integral with a sum over the cells and quadrature points,
-\begin_inset Formula 
-\begin{equation}
-A_{ij}^{nm}=\sum_{\text{vol cells}}\sum_{\text{quad pts}}\frac{1}{4}C_{ijkl}(N_{,l}^{m}(x_{q})+N_{,k}^{m}(x_{q}))(N_{,j}^{n}(x_{q})+N_{,i}^{n}(x_{q}))w_{q}|J_{cell}(x_{q}).
-\end{equation}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-Solution Method for Dynamic Problems
-\end_layout
-
-\begin_layout Standard
-For brevity we outline the solution method for dynamic problems using only
- index notation.
- Time-dependence enters through the constitutive relationships, loading
- conditions, and the inertial terms.
- We consider the deformation at time 
-\begin_inset Formula $t$
-\end_inset
-
-,
-\begin_inset Formula 
-\begin{equation}
--\int_{V}\sigma_{ij}(t)N_{,j}^{n}\: dV+\int_{S_{T}}T_{i}(t)N^{n}\, dS+\int_{V}f_{i}(t)N^{n}\, dV-\int_{V}\rho\sum_{m}\ddot{a}_{i}^{m}(t)N^{m}N^{n}\ dV=\vec{0}.\label{eq:elasticity:integral:dynamic:t}
-\end{equation}
-
-\end_inset
-
-We solve this equation through formulation of a linear algebraic system
- of equations (
-\begin_inset Formula $Au=b$
-\end_inset
-
-), involving the residual (
-\begin_inset Formula $r=b-Au$
-\end_inset
-
-) and Jacobian (
-\begin_inset Formula $A$
-\end_inset
-
-).
- The residual is simply
-\begin_inset Formula 
-\begin{equation}
-r_{i}^{n}=-\int_{V}\sigma_{ij}(t)N_{,j}^{n}\: dV+\int_{S_{T}}T_{i}(t)N^{n}\, dS+\int_{V}f_{i}(t)N^{n}\, dV-\int_{V}\rho\sum_{m}\ddot{a}_{i}^{m}(t)N^{m}N^{n}\ dV.
-\end{equation}
-
-\end_inset
-
-We employ numerical quadrature in the finite-element discretization and
- replace the integrals with sums over the cells and quadrature points,
-\begin_inset Formula 
-\begin{multline}
-r_{i}^{n}=-\sum_{\text{vol cells}}\sum_{\text{quad pts}}\sigma_{ij}(x_{q},t)N^{n}(x_{q})\: w_{q}|J_{cell}(x_{q})|+\sum_{\text{vol cells}}\sum_{\text{quad pt}s}f_{i}(x_{q},t)N^{n}(x_{q})\, w_{q}|J_{cell}(x_{q})|\\
-+\sum_{\text{tract cells}}\sum_{\text{quad pts}}T_{i}(x_{q},t)N^{n}(x_{q})\, w_{q}|J_{cell}(x_{q})|-\sum_{\text{vol cells}}\sum_{\text{quad pts}}\rho\sum_{m}\ddot{a}_{i}^{m}(t)N^{m}N^{n}\ w_{q|J_{cell}(x_{q})},
-\end{multline}
-
-\end_inset
-
-where 
-\begin_inset Formula $x_{q}$
-\end_inset
-
- are the coordinates of the quadrature points, 
-\begin_inset Formula $w_{q}$
-\end_inset
-
- are the weights of the quadrature points, and 
-\begin_inset Formula $|J_{cell}(x_{q})|$
-\end_inset
-
- is the determinant of the Jacobian matrix evaluated at the quadrature points
- associated with mapping the reference cell to the actual cell.
- The quadrature scheme for the integral over the tractions is one dimension
- lower than the one used in integrating the terms for the volume cells.
- 
-\end_layout
-
-\begin_layout Standard
-We find the system Jacobian matrix by making use of the temporal discretization
- and isolating the term for the increment in the displacement field at time
- 
-\begin_inset Formula $t$
-\end_inset
-
-.
- Using the central difference method to approximate the acceleration (and
- velocity),
-\begin_inset Formula 
-\begin{gather}
-\ddot{u}_{i}(t)=\frac{1}{\Delta t^{2}}\left(u_{i}(t+\Delta t)-2u_{i}(t)+u_{i}(t-\Delta t)\right)\\
-\dot{u}_{i}(t)=\frac{1}{2\Delta t}\left(u_{i}(t+\Delta t)-u_{i}(t-\Delta t)\right)
-\end{gather}
-
-\end_inset
-
-and writing the displacement at time 
-\begin_inset Formula $t+\Delta t$
-\end_inset
-
- in terms of the displacement at 
-\begin_inset Formula $t$
-\end_inset
-
- (for consistency with the displacement increment quasi-static formulation),
-\begin_inset Formula 
-\begin{gather}
-u_{i}(t+\Delta t)=u_{i}(t)+du_{i}(t),\\
-\ddot{u}_{i}(t)=\frac{1}{\Delta t^{2}}\left(du_{i}(t)-u_{i}(t)+u_{i}(t-\Delta t)\right),\\
-\dot{u}_{i}(t)=\frac{1}{2\Delta t}\left(du_{i}(t)+u_{i}(t)-u_{i}(t-\Delta t)\right).
-\end{gather}
-
-\end_inset
-
-Substituting into equation 
-\begin_inset CommandInset ref
-LatexCommand eqref
-reference "eq:elasticity:integral:dynamic:t"
-
-\end_inset
-
- yields
-\begin_inset Formula 
-\begin{multline}
-\frac{1}{\Delta t^{2}}\int_{V}\rho\sum_{m}da_{i}^{m}(t)N^{m}N^{n}\ dV=-\int_{V}\sigma_{ij}N_{,j}^{n}\: dV+\int_{S_{T}}T_{i}N^{n}\, dS+\int_{V}f_{i}N^{n}\, dV\\
--\frac{1}{\Delta t^{2}}\int_{V}\rho\sum_{m}(a_{i}^{m}(t)-a_{i}^{m}(t-\Delta t))N^{m}N^{n}\ dV.
-\end{multline}
-
-\end_inset
-
-Thus, the Jacobian for the system is
-\begin_inset Formula 
-\begin{equation}
-A_{ij}^{nm}=\delta_{ij}\frac{1}{\Delta t^{2}}\int_{V}\rho N^{m}N^{n}\ dV,
-\end{equation}
-
-\end_inset
-
-and using numerical quadrature in the finite-element discretization to replace
- the integrals with sums over the cells and quadrature points,
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula 
-\begin{equation}
-A_{ij}^{nm}=\delta_{ij}\frac{1}{\Delta t^{2}}\sum_{\text{vol cells}}\sum_{\text{quad pts}}\rho(x_{q})N^{m}(x_{q})N^{n}(x_{q}),
-\end{equation}
-
-\end_inset
-
-where 
-\begin_inset Formula $A_{ij}^{mn}$
-\end_inset
-
- is a 
-\begin_inset Formula $nd$
-\end_inset
-
- by 
-\begin_inset Formula $md$
-\end_inset
-
- matrix (
-\begin_inset Formula $d$
-\end_inset
-
- is the dimension of the vector space), 
-\begin_inset Formula $m$
-\end_inset
-
- and 
-\begin_inset Formula $n$
-\end_inset
-
- refer to the basis functions and 
-\begin_inset Formula $i$
-\end_inset
-
- and 
-\begin_inset Formula $j$
-\end_inset
-
- are vector space components.
- We consider the contributions associated with the fault in section 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "sec:fault"
-
-\end_inset
-
- and with absorbing boundaries is section 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "sec:absorbing:boundaries"
-
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Section
-Small Strain Formulation
-\begin_inset CommandInset label
-LatexCommand label
-name "sec:Small-Strain-Formulation"
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-In some crustal deformation problems sufficient deformation may occur that
- the assumptions associated with infinitesimal strains no longer hold.
- This is often the case for problems when one wants to include the effects
- of gravitational body forces and deformation on the overburden pressure.
- In such cases we want to account for both rigid body motion and small strains.
- The elasticity formulation in PyLith for small strains uses the Green-Lagrange
- strain tensor and the Second Piola-Kirchhoff stress tensor as is based
- on the one presented by Bathe 
-\begin_inset CommandInset citation
-LatexCommand cite
-key "Bathe:1995"
-
-\end_inset
-
-.
- The Green-Lagrange strain provides a measure of the strain relative to
- the original, undeformed configuration.
-\begin_inset Formula 
-\begin{gather}
-\varepsilon_{ij}=\frac{1}{2}(u_{i,j}+u_{j,i}+u_{k,i}u_{k,j}),\text{ or}\\
-\varepsilon_{ij}=X_{ji}X_{ij}-\delta_{ij},\text{ where}\\
-X_{ij}=x_{i,j}(t)=\frac{\partial}{\partial x_{j}}(x_{i}(0)+u_{i}(t)),
-\end{gather}
-
-\end_inset
-
-and 
-\begin_inset Formula $X_{ij}$
-\end_inset
-
- is the deformation tensor.
- The Second Piola-Kirchhoff stress tensor, 
-\begin_inset Formula $S_{ij}$
-\end_inset
-
-, is related to the Green-Lagrange strain tensor through the elasticity
- constants,
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula 
-\begin{equation}
-S_{ij}=C_{ijkl}\varepsilon_{kl},
-\end{equation}
-
-\end_inset
-
-in the same manner as in the infinitesimal strain formulation.
-\end_layout
-
-\begin_layout Standard
-The elasticity integral in the finite-element formulation includes additional
- terms when we account for small strains.
- Recognizing the similarity between the weighting function and an increment
- in strain in the infinitesimal formulation (many finite-element texts derive
- the finite-element formulation for elasticity using the Principle of Virtual
- Work), we replace 
-\begin_inset Formula $\int_{V}\sigma_{ij}\phi_{i,j}\: dV$
-\end_inset
-
- with 
-\begin_inset Formula $\int_{V}S_{ij}\delta\varepsilon_{ij}\: dV$
-\end_inset
-
- in equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:elasticity:integral"
-
-\end_inset
-
-, where 
-\begin_inset Formula $\delta\varepsilon_{ij}$
-\end_inset
-
- is the 
-\begin_inset Quotes eld
-\end_inset
-
-virtual
-\begin_inset Quotes erd
-\end_inset
-
- strain.
- Using the definition of the Green-Lagrangian strain, we have
-\begin_inset Formula 
-\begin{equation}
-\int_{V}S_{ij}\delta\varepsilon_{ij}\: dV=\int_{V}\frac{1}{2}S_{ij}(\delta u_{i,j}+\delta u_{j,i}+u_{k,i}\delta u_{k,j}+u_{k,j}\delta u_{k,i})\: dV.
-\end{equation}
-
-\end_inset
-
-Writing the displacements in terms of the basis functions and forcing the
- terms associated with the arbitrary weighting function (
-\begin_inset Quotes eld
-\end_inset
-
-virtual
-\begin_inset Quotes erd
-\end_inset
-
- strain) to zero yields the elastic term in the residual,
-\begin_inset Formula 
-\begin{equation}
-r_{i}^{n}=\int_{V}S_{ij}(N_{,i}^{n}+(\sum_{m}a_{k}^{m}N_{,j}^{m})N_{,i}^{n})\: dV.
-\end{equation}
-
-\end_inset
-
-Thus, we have one additional term (the second term) compared with the residual
- for infinitesimal strains.
- Just as in the infinitesimal formulation, we evaluate the integral over
- the volume using numerical quadrature with sums over the quadrature points
- of each cell.
-\end_layout
-
-\begin_layout Subsection
-Quasi-static Problems
-\end_layout
-
-\begin_layout Standard
-The system Jacobian for quasi-static problems includes terms associated
- with elasticity.
- For the small strain formulation, we write the elasticity term at time
- 
-\begin_inset Formula $t+\Delta t$
-\end_inset
-
- and consider the first terms of the Taylor series expansion,
-\begin_inset Formula 
-\begin{equation}
-\int_{v}S_{ij}(t+\Delta t)\delta\varepsilon_{ij}(t+\Delta t)\: dV=\int_{V}(S_{ij}(t)\delta\varepsilon_{ij}(t)+dS_{ij}(t)\delta\varepsilon_{ij}(t)+S_{ij}(t)d\delta\varepsilon_{ij}(t))\: dV.
-\end{equation}
-
-\end_inset
-
-We approximate the increment in the stress tensor using the elastic constants,
-\begin_inset Formula 
-\begin{equation}
-dS_{ij}=C_{ijkl}d\varepsilon_{kl},
-\end{equation}
-
-\end_inset
-
-and the increment in the 
-\begin_inset Quotes eld
-\end_inset
-
-virtual
-\begin_inset Quotes erd
-\end_inset
-
- strain via
-\begin_inset Formula 
-\begin{equation}
-d\delta\varepsilon_{ij}=\frac{1}{2}(du_{k,i}\delta u_{k,j}+du_{k,j}\delta u_{k,i}).
-\end{equation}
-
-\end_inset
-
-We associate the system Jacobian with the terms involving the increment
- in displacements.
- After substituting in the expressions for the increment in the stresses
- and the increment in the 
-\begin_inset Quotes eld
-\end_inset
-
-virtual
-\begin_inset Quotes erd
-\end_inset
-
- strains, we have
-\begin_inset Formula 
-\begin{equation}
-A_{ij}^{nm}=\int_{V}\frac{1}{4}C_{ijkl}(N_{,k}^{m}+(\sum_{r}a_{p}^{r}N_{,l}^{r})N_{,k}^{m})(N_{,i}^{n}+(\sum_{r}a_{p}^{r}N_{,j}^{r})N_{,i}^{n})+\frac{1}{2}S_{kl}N_{,l}^{m}N_{,l}^{n}\delta_{ij}\: dV.
-\end{equation}
-
-\end_inset
-
-The small strain formulation produces additional terms associated with the
- elastic constants and new a new term associated with the stress tensor.
-\end_layout
-
-\begin_layout Subsection
-Dynamic Problems
-\end_layout
-
-\begin_layout Standard
-The system Jacobian matrix in dynamic problems does not include any terms
- associated with elasticity, so the system Jacobian matrix in the small
- strain formulation matches the one used in the infinitesimal strain formulation.
-\end_layout
-
-\end_body
-\end_document
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+\begin_body
+
+\begin_layout Chapter
+\begin_inset CommandInset label
+LatexCommand label
+name "cha:Governing-Equations"
+
+\end_inset
+
+Governing Equations
+\end_layout
+
+\begin_layout Standard
+We present here a brief derivation of the equations for both quasi-static
+ and dynamic computations.
+ Since the general equations are the same (except for the absence of inertial
+ terms in the quasi-static case), we first derive these equations.
+ We then present solution methods for each specific case.
+ In all of our derivations, we use the notation described in Table 
+\begin_inset CommandInset ref
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+reference "tab:Mathematical-notation"
+
+\end_inset
+
+ for both index and vector notation.
+ When using index notation, we use the common convention that repeated indices
+ indicate summation over the range of the index.
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+\begin_layout Standard
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+
+\end_inset
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+Mathematical notation
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\noindent
+\align center
+\begin_inset VSpace defskip
+\end_inset
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+
+\begin_inset Tabular
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+
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+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $u_{i}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\overrightarrow{u}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Displacement vector field
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $d_{i}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\vec{{d}}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Fault slip vector field
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $f_{i}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\overrightarrow{f}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Body force vector field
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $T_{i}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\overrightarrow{T}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Traction vector field
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\sigma_{ij}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\underline{\sigma}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Stress tensor field
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $n_{i}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\overrightarrow{n}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Normal vector field
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\rho$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\rho$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Mass density scalar field
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Derivation of Elasticity Equation
+\end_layout
+
+\begin_layout Subsection
+Index Notation
+\end_layout
+
+\begin_layout Standard
+Consider volume 
+\begin_inset Formula $V$
+\end_inset
+
+ bounded by surface 
+\begin_inset Formula $S$
+\end_inset
+
+.
+ Applying a Lagrangian description of the conservation of momentum gives
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{equation}
+\frac{\partial}{\partial t}\int_{V}\rho\frac{\partial u_{i}}{\partial t}\, dV=\int_{V}f_{i}\, dV+\int_{S}T_{i}\, dS.\label{eqn:momentum:index}
+\end{equation}
+
+\end_inset
+
+The traction vector field is related to the stress tensor through
+\begin_inset Formula 
+\begin{equation}
+T_{i}=\sigma_{ij}n_{j},
+\end{equation}
+
+\end_inset
+
+where 
+\begin_inset Formula $n_{j}$
+\end_inset
+
+ is the vector normal to 
+\begin_inset Formula $S$
+\end_inset
+
+.
+ Substituting into equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eqn:momentum:index"
+
+\end_inset
+
+ yields
+\begin_inset Formula 
+\begin{equation}
+\frac{\partial}{\partial t}\int_{V}\rho\frac{\partial u_{i}}{\partial t}\, dV=\int_{V}f_{i}\, dV+\int_{S}\sigma_{ij}n_{j}\, dS.
+\end{equation}
+
+\end_inset
+
+Applying the divergence theorem,
+\begin_inset Formula 
+\begin{equation}
+\int_{V}a_{i,j}\: dV=\int_{S}a_{j}n_{j}\: dS,
+\end{equation}
+
+\end_inset
+
+to the surface integral results in
+\begin_inset Formula 
+\begin{equation}
+\frac{\partial}{\partial t}\int_{V}\rho\frac{\partial u_{i}}{\partial t}\, dV=\int_{V}f_{i}\, dV+\int_{V}\sigma_{ij,j}\, dV,
+\end{equation}
+
+\end_inset
+
+which we can rewrite as
+\begin_inset Formula 
+\begin{equation}
+\int_{V}\left(\rho\frac{\partial^{2}u_{i}}{\partial t^{2}}-f_{i}-\sigma_{ij,j}\right)\, dV=0.
+\end{equation}
+
+\end_inset
+
+Because the volume 
+\begin_inset Formula $V$
+\end_inset
+
+ is arbitrary, the integrand must be zero at every location in the volume,
+ so that we end up with
+\begin_inset Formula 
+\begin{gather}
+\rho\frac{\partial^{2}u_{i}}{\partial t^{2}}-f_{i}-\sigma_{ij,j}=0\text{ in }V,\\
+\sigma_{ij}n_{j}=T_{i}\text{ on }S_{T}\text{,}\\
+u_{i}=u_{i}^{o}\text{ on }S_{u}\text{, and}\\
+R_{ki}(u_{i}^{+}-u_{i}^{-})=d_{k}\text{ on }S_{f}.
+\end{gather}
+
+\end_inset
+
+We specify tractions, 
+\begin_inset Formula $T_{i}$
+\end_inset
+
+, on surface 
+\begin_inset Formula $S_{f}$
+\end_inset
+
+, displacements, 
+\begin_inset Formula $u_{i}^{o}$
+\end_inset
+
+, on surface 
+\begin_inset Formula $S_{u}$
+\end_inset
+
+, and slip, 
+\begin_inset Formula $d_{k}$
+\end_inset
+
+, on fault surface 
+\begin_inset Formula $S_{f}$
+\end_inset
+
+ (we will consider the case of fault constitutive models in Section 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sec:fault"
+
+\end_inset
+
+).
+ The rotation matrix 
+\begin_inset Formula $R_{ki}$
+\end_inset
+
+ transforms vectors from the global coordinate system to the fault coordinate
+ system.
+ Note that since both 
+\begin_inset Formula $T_{i}$
+\end_inset
+
+ and 
+\begin_inset Formula $u_{i}$
+\end_inset
+
+ are vector quantities, there can be some spatial overlap of the surfaces
+ 
+\begin_inset Formula $S_{T}$
+\end_inset
+
+ and 
+\begin_inset Formula $S_{u}$
+\end_inset
+
+; however, the same degree of freedom cannot simultaneously have both types
+ of boundary conditions.
+\end_layout
+
+\begin_layout Subsection
+Vector Notation
+\end_layout
+
+\begin_layout Standard
+Consider volume 
+\begin_inset Formula $V$
+\end_inset
+
+ bounded by surface 
+\begin_inset Formula $S$
+\end_inset
+
+.
+ Applying a Lagrangian description of the conservation of momentum gives
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{equation}
+\frac{\partial}{\partial t}\int_{V}\rho\frac{\partial\vec{u}}{\partial t}\, dV=\int_{V}\overrightarrow{f}\, dV+\int_{S}\overrightarrow{T}\, dS.\label{eqn:momentum:vec}
+\end{equation}
+
+\end_inset
+
+The traction vector field is related to the stress tensor through
+\begin_inset Formula 
+\begin{equation}
+\overrightarrow{T}=\underline{\sigma}\cdot\overrightarrow{n},
+\end{equation}
+
+\end_inset
+
+where 
+\begin_inset Formula $\overrightarrow{n}$
+\end_inset
+
+ is the vector normal to 
+\begin_inset Formula $S$
+\end_inset
+
+.
+ Substituting into equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eqn:momentum:vec"
+
+\end_inset
+
+ yields
+\begin_inset Formula 
+\begin{equation}
+\frac{\partial}{\partial t}\int_{V}\rho\frac{\partial\overrightarrow{u}}{\partial t}\, dV=\int_{V}\overrightarrow{f}\, dV+\int_{S}\underline{\sigma}\cdot\overrightarrow{n}\, dS.
+\end{equation}
+
+\end_inset
+
+Applying the divergence theorem,
+\begin_inset Formula 
+\begin{equation}
+\int_{V}\nabla\cdot\overrightarrow{a}\: dV=\int_{S}\overrightarrow{a}\cdot\overrightarrow{n}\: dS,
+\end{equation}
+
+\end_inset
+
+to the surface integral results in
+\begin_inset Formula 
+\begin{equation}
+\frac{\partial}{\partial t}\int_{V}\rho\frac{\partial\overrightarrow{u}}{\partial t}\, dV=\int_{V}\overrightarrow{f}\, dV+\int_{V}\nabla\cdot\underline{\sigma}\, dV,
+\end{equation}
+
+\end_inset
+
+which we can rewrite as
+\begin_inset Formula 
+\begin{equation}
+\int_{V}\left(\rho\frac{\partial^{2}\overrightarrow{u}}{\partial t^{2}}-\overrightarrow{f}-\nabla\cdot\overrightarrow{\sigma}\right)\, dV=\vec{0}.
+\end{equation}
+
+\end_inset
+
+Because the volume 
+\begin_inset Formula $V$
+\end_inset
+
+ is arbitrary, the integrand must be the zero vector at every location in
+ the volume, so that we end up with
+\begin_inset Formula 
+\begin{gather}
+\rho\frac{\partial^{2}\overrightarrow{u}}{\partial t^{2}}-\overrightarrow{f}-\nabla\cdot\overrightarrow{\sigma}=\vec{0}\text{ in }V,\\
+\underline{\sigma}\cdot\overrightarrow{n}=\overrightarrow{T}\text{ on }S_{T}\text{,}\\
+\overrightarrow{u}=\overrightarrow{u^{o}}\text{ on }S_{u},\text{ and}\\
+\underbar{R}\cdot(\vec{u^{+}}-\vec{u^{-}})=\vec{d}\text{ on }S_{f}.
+\end{gather}
+
+\end_inset
+
+We specify tractions, 
+\begin_inset Formula $\vec{T}$
+\end_inset
+
+, on surface 
+\begin_inset Formula $S_{f}$
+\end_inset
+
+, displacements, 
+\begin_inset Formula $\overrightarrow{u^{o}}$
+\end_inset
+
+, on surface 
+\begin_inset Formula $S_{u}$
+\end_inset
+
+, and slip, 
+\begin_inset Formula $\vec{d}$
+\end_inset
+
+, on fault surface 
+\begin_inset Formula $S_{f}$
+\end_inset
+
+ (we will consider the case of fault constitutive models in Section 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sec:fault"
+
+\end_inset
+
+).
+ The rotation matrix 
+\begin_inset Formula $\underline{R}$
+\end_inset
+
+ transforms vectors from the global coordinate system to the fault coordinate
+ system.
+ Note that since both 
+\begin_inset Formula $\overrightarrow{T}$
+\end_inset
+
+ and 
+\begin_inset Formula $\overrightarrow{u}$
+\end_inset
+
+ are vector quantities, there can be some spatial overlap of the surfaces
+ 
+\begin_inset Formula $S_{T}$
+\end_inset
+
+ and 
+\begin_inset Formula $S_{u}$
+\end_inset
+
+; however, the same degree of freedom cannot simultaneously have both types
+ of boundary conditions.
+\end_layout
+
+\begin_layout Section
+Finite-Element Formulation of Elasticity Equation
+\end_layout
+
+\begin_layout Standard
+We formulate a set of algebraic equations using Galerkin's method.
+ We consider (1) a trial solution, 
+\begin_inset Formula $\vec{u}$
+\end_inset
+
+, that is a piecewise differentiable vector field and satisfies the Dirichlet
+ boundary conditions on 
+\begin_inset Formula $S_{u}$
+\end_inset
+
+, and (2) a weighting function, 
+\begin_inset Formula $\vec{\phi}$
+\end_inset
+
+, that is a piecewise differentiable vector field and is zero on 
+\begin_inset Formula $S_{u}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsection
+Index Notation
+\end_layout
+
+\begin_layout Standard
+We start with the wave equation (strong form),
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{gather}
+\sigma_{ij,j}+f_{i}=\rho\ddot{u_{i}}\text{ in }V,\\
+\sigma_{ij}n_{j}=T_{i}\text{ on }S_{T},\\
+u_{i}=u_{i}^{o}\text{ on }S_{u},\\
+R_{ki}(u_{i}^{+}-u_{i}^{-})=d_{k}\text{ on }S_{f},\text{ and}\\
+\sigma_{ij}=\sigma_{ji}\text{ (symmetric).}
+\end{gather}
+
+\end_inset
+
+We construct the weak form by computing the dot product of the wave equation
+ and weighting function and setting the integral over the domain to zero:
+\begin_inset Formula 
+\begin{gather}
+\int_{V}\left(\sigma_{ij,j}+f_{i}-\rho\ddot{u}_{i}\right)\phi_{i}\, dV=0\text{, or }\\
+\int_{V}\sigma_{ij,j}\phi_{i}\: dV+\int_{V}f_{i}\phi_{i}\: dV-\int_{V}\rho\ddot{u}_{i}\phi_{i}\: dV=0.
+\end{gather}
+
+\end_inset
+
+ Consider the divergence theorem applied to the dot product of the stress
+ tensor and the weighting function, 
+\begin_inset Formula $\sigma_{ij}\phi_{i}$
+\end_inset
+
+,
+\begin_inset Formula 
+\begin{equation}
+\int_{V}(\sigma_{ij}\phi_{i})_{,j}\, dV=\int_{S}(\sigma_{ij}\phi_{i})n_{i}\, dS.
+\end{equation}
+
+\end_inset
+
+Expanding the left-hand side yields
+\begin_inset Formula 
+\begin{gather}
+\int_{V}\sigma_{ij,j}\phi_{i}\: dV+\int_{V}\sigma_{ij}\phi_{i,j}\: dV=\int_{S}\sigma_{ij}\phi_{i}n_{i}\: dS,\text{ or}\\
+\int_{V}\sigma_{ij,j}\phi_{i}\: dV=-\int_{V}\sigma_{ij}\phi_{i,j}\, dV+\int_{S}\sigma_{ij}\phi_{i}n_{i}\, dS.
+\end{gather}
+
+\end_inset
+
+Substituting into the weak form gives
+\begin_inset Formula 
+\begin{equation}
+-\int_{V}\sigma_{ij}\phi_{i,j}\, dV+\int_{S}\sigma_{ij}\phi_{i}n_{i}\, dS+\int_{V}f_{i}\phi_{i}\, dV-\int_{V}\rho\ddot{u}_{i}\phi_{i}\, dV=0.
+\end{equation}
+
+\end_inset
+
+Turning our attention to the second term, we separate the integration over
+ 
+\begin_inset Formula $S$
+\end_inset
+
+ into integration over 
+\begin_inset Formula $S_{T}$
+\end_inset
+
+ and 
+\begin_inset Formula $S_{u}$
+\end_inset
+
+ (we will consider tractions over the fault surface, 
+\begin_inset Formula $S_{f}$
+\end_inset
+
+, associated with the fault constitutive model in Section 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sec:fault"
+
+\end_inset
+
+),
+\begin_inset Formula 
+\begin{equation}
+-\int_{V}\sigma_{ij}\phi_{i,j}\, dV+\int_{S_{T}}\sigma_{ij}\phi_{i}n_{i}\, dS+\int_{S_{u}}\sigma_{ij}\phi_{i}n_{i}\, dS+\int_{V}f_{i}\phi_{i}\, dV-\int_{V}\rho\ddot{u}_{i}\phi_{i}\, dV=0,
+\end{equation}
+
+\end_inset
+
+and recognize that
+\begin_inset Formula 
+\begin{gather}
+\sigma_{ij}n_{i}=T_{i}\text{ on }S_{T}\text{ and}\\
+\phi_{i}=0\text{ on }S_{u},
+\end{gather}
+
+\end_inset
+
+so that the equation reduces to
+\begin_inset Formula 
+\begin{equation}
+-\int_{V}\sigma_{ij}\phi_{i,j}\: dV+\int_{S_{T}}T_{i}\phi_{i}\, dS+\int_{V}f_{i}\phi_{i}\, dV-\int_{V}\rho\ddot{u}_{i}\phi_{i}\, dV=0.\label{eq:elasticity:integral}
+\end{equation}
+
+\end_inset
+
+We express the trial solution and weighting function as linear combinations
+ of basis functions,
+\begin_inset Formula 
+\begin{gather}
+u_{i}=\sum_{m}a_{i}^{m}N^{m},\\
+\phi_{i}=\sum_{n}c_{i}^{n}N^{n}.
+\end{gather}
+
+\end_inset
+
+Note that because the trial solution satisfies the Dirichlet boundary condition,
+ the number of basis functions for 
+\begin_inset Formula $u$
+\end_inset
+
+ is generally greater than the number of basis functions for 
+\begin_inset Formula $\phi$
+\end_inset
+
+, i.e., 
+\begin_inset Formula $m>n$
+\end_inset
+
+.
+ Substituting in the expressions for the trial solution and weighting function
+ yields
+\begin_inset Formula 
+\begin{gather}
+-\int_{V}\sigma_{ij}\sum_{n}c_{i}^{n}N_{,j}^{n}\: dV+\int_{S_{T}}T_{i}\sum_{n}c_{i}^{n}N^{n}\, dS+\int_{V}f_{i}\sum_{n}c_{i}^{n}N^{n}\, dV-\int_{V}\rho\sum_{m}\ddot{a}_{i}^{m}N^{m}\sum_{n}c_{i}^{n}N^{n}\ dV=0,\text{ or}\\
+\sum_{n}c_{i}^{n}(-\int_{V}\sigma_{ij}N_{,j}^{n}\: dV+\int_{S_{T}}T_{i}N^{n}\, dS+\int_{V}f_{i}N^{n}\, dV-\int_{V}\rho\sum_{m}\ddot{a}_{i}^{m}N^{m}N^{n}\ dV)=0.
+\end{gather}
+
+\end_inset
+
+ Because the weighting function is arbitrary, this equation must hold for
+ all 
+\begin_inset Formula $c_{i}^{n}$
+\end_inset
+
+, so that the quantity in parenthesis is zero for each 
+\begin_inset Formula $c_{i}^{n}$
+\end_inset
+
+
+\begin_inset Formula 
+\begin{equation}
+-\int_{V}\sigma_{ij}N_{,j}^{n}\: dV+\int_{S_{T}}T_{i}N^{n}\, dS+\int_{V}f_{i}N^{n}\, dV-\int_{V}\rho\sum_{m}\ddot{a}_{i}^{m}N^{m}N^{n}\ dV=\vec{0}.\label{eq:elasticity:integral:discretized}
+\end{equation}
+
+\end_inset
+
+We want to solve this equation for the unknown coefficients 
+\begin_inset Formula $a_{i}^{m}$
+\end_inset
+
+ subject to
+\end_layout
+
+\begin_layout Standard
+
+\family roman
+\series medium
+\shape up
+\size normal
+\emph off
+\bar no
+\noun off
+\color none
+\begin_inset Formula 
+\begin{gather}
+u_{i}=u_{i}^{o}\text{ on }S_{u},\text{ and}\\
+R_{ki}(u_{i}^{+}-u_{i}^{-})=d_{k}\text{ on }S_{f},
+\end{gather}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Vector Notation
+\end_layout
+
+\begin_layout Standard
+We start with the wave equation (strong form),
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{gather}
+\nabla\cdot\underline{\sigma}+\overrightarrow{f}=\rho\frac{\partial^{2}\overrightarrow{u}}{\partial t^{2}}\text{ in }V,\\
+\underline{\sigma}\cdot\overrightarrow{n}=\overrightarrow{T}\text{ on }S_{T},\\
+\overrightarrow{u}=\overrightarrow{u^{o}}\text{ on }S_{u},\\
+\underbar{R}\cdot(\overrightarrow{u^{+}}-\overrightarrow{u^{-}})=\vec{d}\text{ on }S_{f}\\
+\underline{\sigma}=\underline{\sigma}^{T}\text{ (symmetric).}
+\end{gather}
+
+\end_inset
+
+We construct the weak form by multiplying the wave equation by a weighting
+ function and setting the integral over the domain to zero.
+ The weighting function is a piecewise differential vector field, 
+\begin_inset Formula $\overrightarrow{\phi}$
+\end_inset
+
+, where 
+\begin_inset Formula $\overrightarrow{\phi}=0$
+\end_inset
+
+ on 
+\begin_inset Formula $S_{u}.$
+\end_inset
+
+ Hence our weak form is
+\begin_inset Formula 
+\begin{gather}
+\int_{V}\left(\nabla\cdot\underline{\sigma}+\overrightarrow{f}-\rho\frac{\partial^{2}\overrightarrow{u}}{\partial t^{2}}\right)\cdot\overrightarrow{\phi}\, dV=0\text{, or }\\
+\int_{V}(\nabla\cdot\underline{\sigma})\cdot\overrightarrow{\phi}\: dV+\int_{V}\overrightarrow{f}\cdot\overrightarrow{\phi}\: dV-\int_{V}\rho\frac{\partial^{2}\overrightarrow{u}}{\partial t^{2}}\cdot\overrightarrow{\phi}\: dV=0.
+\end{gather}
+
+\end_inset
+
+ Consider the divergence theorem applied to the dot product of the stress
+ tensor and the trial function, 
+\begin_inset Formula $\underline{\sigma}\cdot\overrightarrow{\phi}$
+\end_inset
+
+,
+\begin_inset Formula 
+\begin{equation}
+\int_{V}\nabla\cdot(\underline{\sigma}\cdot\overrightarrow{\phi})\, dV=\int_{S}(\underline{\sigma}\cdot\overrightarrow{\phi})\cdot\overrightarrow{n}\, dS.
+\end{equation}
+
+\end_inset
+
+Expanding the left-hand side yields
+\begin_inset Formula 
+\begin{equation}
+\int_{V}(\nabla\cdot\underline{\sigma})\cdot\overrightarrow{\phi}\: dV+\int_{V}\underline{\sigma}:\nabla\overrightarrow{\phi}\: dV=\int_{S}(\underline{\sigma}\cdot\overrightarrow{\phi})\cdot\overrightarrow{n}\: dS,\text{ or}
+\end{equation}
+
+\end_inset
+
+
+\begin_inset Formula 
+\begin{equation}
+\int_{V}(\nabla\cdot\underline{\sigma})\cdot\overrightarrow{\phi}\: dV=-\int_{V}\underline{\sigma}:\nabla\overrightarrow{\phi}\, dV+\int_{S}\underline{\sigma}\cdot\overrightarrow{n}\cdot\overrightarrow{\phi}\, dS.
+\end{equation}
+
+\end_inset
+
+Substituting into the weak form gives
+\begin_inset Formula 
+\begin{equation}
+-\int_{V}\underline{\sigma}:\nabla\overrightarrow{\phi}\, dV+\int_{S}\underline{\sigma}\cdot\overrightarrow{n}\cdot\overrightarrow{\phi}\, dS+\int_{V}\overrightarrow{f}\cdot\overrightarrow{\phi}\, dV-\int_{V}\rho\frac{\partial^{2}\overrightarrow{u}}{\partial t^{2}}\cdot\overrightarrow{\phi}\, dV=0.
+\end{equation}
+
+\end_inset
+
+We separate the integration over 
+\begin_inset Formula $S$
+\end_inset
+
+ into integration over 
+\begin_inset Formula $S_{T}$
+\end_inset
+
+ and 
+\begin_inset Formula $S_{u}$
+\end_inset
+
+,
+\begin_inset Formula 
+\begin{multline}
+-\int_{V}\underline{\sigma}:\nabla\overrightarrow{\phi}\, dV+\int_{S_{T}}\underline{\sigma}\cdot\overrightarrow{n}\cdot\overrightarrow{\phi}\, dS+\int_{S_{u}}\underline{\sigma}\cdot\overrightarrow{n}\cdot\overrightarrow{\phi}\, dS+\int_{V}\overrightarrow{f}\cdot\overrightarrow{\phi}\, dV-\int_{V}\rho\frac{\partial^{2}\overrightarrow{u}}{\partial t^{2}}\cdot\overrightarrow{\phi}\, dV=0,
+\end{multline}
+
+\end_inset
+
+and recognize that
+\begin_inset Formula 
+\begin{gather}
+\underline{\sigma}\cdot\overrightarrow{n}=\overrightarrow{T}\text{ on }S_{T}\text{ and}\\
+\overrightarrow{\phi}=0\text{ on }S_{u},
+\end{gather}
+
+\end_inset
+
+so that the equation reduces to
+\begin_inset Formula 
+\begin{equation}
+-\int_{V}\underline{\sigma}:\nabla\overrightarrow{\phi}\: dV+\int_{S_{T}}\overrightarrow{T}\cdot\overrightarrow{\phi}\, dS+\int_{V}\overrightarrow{f}\cdot\overrightarrow{\phi}\, dV-\int_{V}\rho\frac{\partial^{2}\overrightarrow{u}}{\partial t^{2}}\cdot\overrightarrow{\phi}\, dV=0.
+\end{equation}
+
+\end_inset
+
+We express the trial solution and weighting function as linear combinations
+ of basis functions,
+\begin_inset Formula 
+\begin{gather}
+\vec{u}=\sum_{m}\overrightarrow{a^{m}}N^{m},\\
+\vec{\phi}=\sum_{n}\overrightarrow{c^{n}}N^{n}.
+\end{gather}
+
+\end_inset
+
+Note that because the weighting function is zero on 
+\begin_inset Formula $S_{u}$
+\end_inset
+
+, the number of basis functions for 
+\begin_inset Formula $\vec{u}$
+\end_inset
+
+ is generally greater than the number of basis functions for 
+\begin_inset Formula $\vec{\phi}$
+\end_inset
+
+, i.e., 
+\begin_inset Formula $m>n$
+\end_inset
+
+.
+ Substituting in the expressions for the trial solution and weighting function
+ yields
+\begin_inset Formula 
+\begin{multline}
+-\int_{V}\underline{\sigma}:\sum_{n}\overrightarrow{c^{n}}\nabla N_{,}^{n}\, dV+\int_{S_{T}}\vec{T}\cdot\sum_{n}\overrightarrow{c^{n}}N^{n}\, dS+\int_{V}\vec{f}\cdot\sum_{n}\overrightarrow{c^{n}}N^{n}\, dV\\
+-\int_{V}\rho\sum_{m}\frac{\partial^{2}\overrightarrow{a^{m}}}{\partial t^{2}}N^{m}\cdot\sum_{n}\overrightarrow{c^{n}}N^{n}\ dV=0.
+\end{multline}
+
+\end_inset
+
+ Because the weighting function is arbitrary, this equation must hold for
+ all 
+\begin_inset Formula $\overrightarrow{c^{n}}$
+\end_inset
+
+, so that
+\begin_inset Formula 
+\begin{equation}
+-\int_{V}\underline{\sigma}:\nabla N^{n}\, dV+\int_{S_{T}}\vec{T}N^{n}\, dS+\int_{V}\vec{f}N^{n}\, dV-\int_{V}\rho\sum_{m}\frac{\partial^{2}\overrightarrow{a^{m}}}{\partial t^{2}}N^{m}N^{n}\, dV=\vec{0}.
+\end{equation}
+
+\end_inset
+
+We want to solve this equation for the unknown coefficients 
+\begin_inset Formula $\overrightarrow{a^{m}}$
+\end_inset
+
+ subject to
+\end_layout
+
+\begin_layout Standard
+
+\family roman
+\series medium
+\shape up
+\size normal
+\emph off
+\bar no
+\noun off
+\color none
+\begin_inset Formula 
+\begin{gather}
+\vec{u}=u^{o}\overrightarrow{}\text{ on }S_{u},\text{ and}\\
+\underline{R}(\overrightarrow{u^{+}}-\overrightarrow{u^{-}})=\vec{d}\text{ on }S_{f},
+\end{gather}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Solution Method for Quasi-Static Problems
+\end_layout
+
+\begin_layout Standard
+For brevity we outline the solution method for quasi-static problems using
+ only index notation.
+ In quasi-static problems we neglect the inertial terms, so equation 
+\begin_inset CommandInset ref
+LatexCommand eqref
+reference "eq:elasticity:integral:discretized"
+
+\end_inset
+
+ reduces to
+\begin_inset Formula 
+\begin{equation}
+-\int_{V}\sigma_{ij}N_{,j}^{n}\: dV+\int_{S_{T}}T_{i}N^{n}\, dS+\int_{V}f_{i}N^{n}\, dV=\vec{0}.
+\end{equation}
+
+\end_inset
+
+As a result, time-dependence only enters through the constitutive relationships
+ and the loading conditions.
+ We consider the deformation at time 
+\begin_inset Formula $t+\Delta t$
+\end_inset
+
+,
+\begin_inset Formula 
+\begin{equation}
+-\int_{V}\sigma_{ij}(t+\Delta t)N_{,j}^{n}\: dV+\int_{S_{T}}T_{i}(t+\Delta t)N^{n}\, dS+\int_{V}f_{i}(t+\Delta t)N^{n}\, dV=\vec{0}.\label{eq:elasticity:integral:quasistatic}
+\end{equation}
+
+\end_inset
+
+We solve this equation through formulation of a linear algebraic system
+ of equations (
+\begin_inset Formula $Au=b$
+\end_inset
+
+), involving the residual (
+\begin_inset Formula $r=b-Au$
+\end_inset
+
+) and Jacobian (
+\begin_inset Formula $A$
+\end_inset
+
+).
+ The residual is simply
+\begin_inset Formula 
+\begin{equation}
+r_{i}^{n}=-\int_{V}\sigma_{ij}(t+\Delta t)N_{,j}^{n}\: dV+\int_{S_{T}}T_{i}(t+\Delta t)N^{n}\, dS+\int_{V}f_{i}(t+\Delta t)N^{n}\, dV.
+\end{equation}
+
+\end_inset
+
+We employ numerical quadrature in the finite-element discretization and
+ replace the integrals with sums over the cells and quadrature points,
+\begin_inset Formula 
+\begin{multline}
+r_{i}^{n}=-\sum_{\text{vol cells}}\sum_{\text{quad pts}}\sigma_{ij}(x_{q},t+\Delta t)N_{,j}^{n}(x_{q})\: w_{q}|J_{cell}(x_{q})|+\sum_{\text{vol cells}}\sum_{\text{quad pt}s}f_{i}(x_{q},t+\Delta t)N^{n}(x_{q})\, w_{q}|J_{cell}(x_{q})|\\
++\sum_{\text{tract cells}}\sum_{\text{quad pts}}T_{i}(x_{q},t+\Delta t)N^{n}(x_{q})\, w_{q}|J_{cell}(x_{q})|,
+\end{multline}
+
+\end_inset
+
+where 
+\begin_inset Formula $r_{i}^{n}$
+\end_inset
+
+ is an 
+\begin_inset Formula $nd$
+\end_inset
+
+ vector (
+\begin_inset Formula $d$
+\end_inset
+
+ is the dimension of the vector space) and 
+\begin_inset Formula $i$
+\end_inset
+
+ is a vector space component, 
+\begin_inset Formula $x_{q}$
+\end_inset
+
+ are the coordinates of the quadrature points, 
+\begin_inset Formula $w_{q}$
+\end_inset
+
+ are the weights of the quadrature points, and 
+\begin_inset Formula $|J_{cell}(x_{q})|$
+\end_inset
+
+ is the determinant of the Jacobian matrix evaluated at the quadrature points
+ associated with mapping the reference cell to the actual cell.
+ The quadrature scheme for the integral over the tractions is one dimension
+ lower than the one used in integrating the terms for the volume cells.
+\end_layout
+
+\begin_layout Standard
+In order to find the Jacobian of the system, we let
+\begin_inset Formula 
+\begin{equation}
+\sigma_{ij}(t+\Delta t)=\sigma_{ij}(t)+d\sigma_{ij}(t).
+\end{equation}
+
+\end_inset
+
+Isolating the term associated with the increment in stresses yields
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{equation}
+\int_{V}d\sigma_{ij}(t)N_{j}^{n}\ dV=-\int_{V}\sigma_{ij}(t)N_{,j}^{n}\: dV+\int_{S_{T}}T_{i}(t+\Delta t)N^{n}\, dS+\int_{V}f_{i}(t+\Delta t)N^{n}\, dV
+\end{equation}
+
+\end_inset
+
+We associate the term on the left-hand-side with the action of the system
+ Jacobian on the increment of the displacement field.
+ We approximate the increment in stresses using linear elasticity and infinitesi
+mal strains,
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{gather}
+d\sigma_{ij}(t)=C_{ijkl}(t)d\varepsilon_{kl}(t)\\
+d\sigma_{ij}(t)=\frac{1}{2}C_{ijkl}(t)(du_{k.l}(t)+du_{l,k}(t))\\
+d\sigma_{ij}(t)=\frac{1}{2}C_{ijkl}(t)(\sum_{m}da_{k,l}^{m}(t)N^{m}+\sum_{m}da_{l,k}^{m}(t)N^{m})
+\end{gather}
+
+\end_inset
+
+Now, 
+\begin_inset Formula $d\sigma_{ij}\phi_{i,j}$
+\end_inset
+
+ is a scalar, so it is symmetric,
+\begin_inset Formula 
+\begin{equation}
+d\sigma_{ij}\phi_{i,j}=d\sigma_{ji}\phi_{j,i},
+\end{equation}
+
+\end_inset
+
+and we know that 
+\begin_inset Formula $d\sigma_{ij}$
+\end_inset
+
+ is symmetric, so
+\begin_inset Formula 
+\begin{equation}
+d\sigma_{ij}\phi_{i,j}=d\sigma_{ij}\phi_{j,i},
+\end{equation}
+
+\end_inset
+
+which means
+\begin_inset Formula 
+\begin{equation}
+\phi_{i,j}=\phi_{j,i},
+\end{equation}
+
+\end_inset
+
+which we can write as
+\begin_inset Formula 
+\begin{equation}
+\phi_{i,j}=\frac{1}{2}(\phi_{i,j}+\phi_{j,i}).
+\end{equation}
+
+\end_inset
+
+In terms of the basis functions, we have
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{equation}
+\sum_{n}c_{i}^{n}N_{,j}^{n}=\frac{1}{2}(\sum_{n}c_{i}^{n}N_{,j}^{n}+\sum_{n}c_{j}^{n}N_{,i}^{n}).
+\end{equation}
+
+\end_inset
+
+Combining these expressions for the increment in stresses and making use
+ of the symmetry of the weighting functions, we find the system Jacobian
+ is
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{equation}
+A_{ij}^{nm}=\int_{V}\frac{1}{4}C_{ijkl}(N_{,l}^{m}+N_{,k}^{m})(N_{,j}^{n}+N_{,i}^{n})\ dV.
+\end{equation}
+
+\end_inset
+
+We employ numerical quadrature in the finite-element discretization and
+ replace the integral with a sum over the cells and quadrature points,
+\begin_inset Formula 
+\begin{equation}
+A_{ij}^{nm}=\sum_{\text{vol cells}}\sum_{\text{quad pts}}\frac{1}{4}C_{ijkl}(N_{,l}^{m}(x_{q})+N_{,k}^{m}(x_{q}))(N_{,j}^{n}(x_{q})+N_{,i}^{n}(x_{q}))w_{q}|J_{cell}(x_{q}).
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Solution Method for Dynamic Problems
+\end_layout
+
+\begin_layout Standard
+For brevity we outline the solution method for dynamic problems using only
+ index notation.
+ Time-dependence enters through the constitutive relationships, loading
+ conditions, and the inertial terms.
+ We consider the deformation at time 
+\begin_inset Formula $t$
+\end_inset
+
+,
+\begin_inset Formula 
+\begin{equation}
+-\int_{V}\sigma_{ij}(t)N_{,j}^{n}\: dV+\int_{S_{T}}T_{i}(t)N^{n}\, dS+\int_{V}f_{i}(t)N^{n}\, dV-\int_{V}\rho\sum_{m}\ddot{a}_{i}^{m}(t)N^{m}N^{n}\ dV=\vec{0}.\label{eq:elasticity:integral:dynamic:t}
+\end{equation}
+
+\end_inset
+
+We solve this equation through formulation of a linear algebraic system
+ of equations (
+\begin_inset Formula $Au=b$
+\end_inset
+
+), involving the residual (
+\begin_inset Formula $r=b-Au$
+\end_inset
+
+) and Jacobian (
+\begin_inset Formula $A$
+\end_inset
+
+).
+ The residual is simply
+\begin_inset Formula 
+\begin{equation}
+r_{i}^{n}=-\int_{V}\sigma_{ij}(t)N_{,j}^{n}\: dV+\int_{S_{T}}T_{i}(t)N^{n}\, dS+\int_{V}f_{i}(t)N^{n}\, dV-\int_{V}\rho\sum_{m}\ddot{a}_{i}^{m}(t)N^{m}N^{n}\ dV.
+\end{equation}
+
+\end_inset
+
+We employ numerical quadrature in the finite-element discretization and
+ replace the integrals with sums over the cells and quadrature points,
+\begin_inset Formula 
+\begin{multline}
+r_{i}^{n}=-\sum_{\text{vol cells}}\sum_{\text{quad pts}}\sigma_{ij}(x_{q},t)N^{n}(x_{q})\: w_{q}|J_{cell}(x_{q})|+\sum_{\text{vol cells}}\sum_{\text{quad pt}s}f_{i}(x_{q},t)N^{n}(x_{q})\, w_{q}|J_{cell}(x_{q})|\\
++\sum_{\text{tract cells}}\sum_{\text{quad pts}}T_{i}(x_{q},t)N^{n}(x_{q})\, w_{q}|J_{cell}(x_{q})|-\sum_{\text{vol cells}}\sum_{\text{quad pts}}\rho\sum_{m}\ddot{a}_{i}^{m}(t)N^{m}N^{n}\ w_{q|J_{cell}(x_{q})},
+\end{multline}
+
+\end_inset
+
+where 
+\begin_inset Formula $x_{q}$
+\end_inset
+
+ are the coordinates of the quadrature points, 
+\begin_inset Formula $w_{q}$
+\end_inset
+
+ are the weights of the quadrature points, and 
+\begin_inset Formula $|J_{cell}(x_{q})|$
+\end_inset
+
+ is the determinant of the Jacobian matrix evaluated at the quadrature points
+ associated with mapping the reference cell to the actual cell.
+ The quadrature scheme for the integral over the tractions is one dimension
+ lower than the one used in integrating the terms for the volume cells.
+ 
+\end_layout
+
+\begin_layout Standard
+We find the system Jacobian matrix by making use of the temporal discretization
+ and isolating the term for the increment in the displacement field at time
+ 
+\begin_inset Formula $t$
+\end_inset
+
+.
+ Using the central difference method to approximate the acceleration (and
+ velocity),
+\begin_inset Formula 
+\begin{gather}
+\ddot{u}_{i}(t)=\frac{1}{\Delta t^{2}}\left(u_{i}(t+\Delta t)-2u_{i}(t)+u_{i}(t-\Delta t)\right)\\
+\dot{u}_{i}(t)=\frac{1}{2\Delta t}\left(u_{i}(t+\Delta t)-u_{i}(t-\Delta t)\right)
+\end{gather}
+
+\end_inset
+
+and writing the displacement at time 
+\begin_inset Formula $t+\Delta t$
+\end_inset
+
+ in terms of the displacement at 
+\begin_inset Formula $t$
+\end_inset
+
+ (for consistency with the displacement increment quasi-static formulation),
+\begin_inset Formula 
+\begin{gather}
+u_{i}(t+\Delta t)=u_{i}(t)+du_{i}(t),\\
+\ddot{u}_{i}(t)=\frac{1}{\Delta t^{2}}\left(du_{i}(t)-u_{i}(t)+u_{i}(t-\Delta t)\right),\\
+\dot{u}_{i}(t)=\frac{1}{2\Delta t}\left(du_{i}(t)+u_{i}(t)-u_{i}(t-\Delta t)\right).
+\end{gather}
+
+\end_inset
+
+Substituting into equation 
+\begin_inset CommandInset ref
+LatexCommand eqref
+reference "eq:elasticity:integral:dynamic:t"
+
+\end_inset
+
+ yields
+\begin_inset Formula 
+\begin{multline}
+\frac{1}{\Delta t^{2}}\int_{V}\rho\sum_{m}da_{i}^{m}(t)N^{m}N^{n}\ dV=-\int_{V}\sigma_{ij}N_{,j}^{n}\: dV+\int_{S_{T}}T_{i}N^{n}\, dS+\int_{V}f_{i}N^{n}\, dV\\
+-\frac{1}{\Delta t^{2}}\int_{V}\rho\sum_{m}(a_{i}^{m}(t)-a_{i}^{m}(t-\Delta t))N^{m}N^{n}\ dV.
+\end{multline}
+
+\end_inset
+
+Thus, the Jacobian for the system is
+\begin_inset Formula 
+\begin{equation}
+A_{ij}^{nm}=\delta_{ij}\frac{1}{\Delta t^{2}}\int_{V}\rho N^{m}N^{n}\ dV,
+\end{equation}
+
+\end_inset
+
+and using numerical quadrature in the finite-element discretization to replace
+ the integrals with sums over the cells and quadrature points,
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{equation}
+A_{ij}^{nm}=\delta_{ij}\frac{1}{\Delta t^{2}}\sum_{\text{vol cells}}\sum_{\text{quad pts}}\rho(x_{q})N^{m}(x_{q})N^{n}(x_{q}),
+\end{equation}
+
+\end_inset
+
+where 
+\begin_inset Formula $A_{ij}^{mn}$
+\end_inset
+
+ is a 
+\begin_inset Formula $nd$
+\end_inset
+
+ by 
+\begin_inset Formula $md$
+\end_inset
+
+ matrix (
+\begin_inset Formula $d$
+\end_inset
+
+ is the dimension of the vector space), 
+\begin_inset Formula $m$
+\end_inset
+
+ and 
+\begin_inset Formula $n$
+\end_inset
+
+ refer to the basis functions and 
+\begin_inset Formula $i$
+\end_inset
+
+ and 
+\begin_inset Formula $j$
+\end_inset
+
+ are vector space components.
+ We consider the contributions associated with the fault in section 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sec:fault"
+
+\end_inset
+
+ and with absorbing boundaries is section 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sec:absorbing:boundaries"
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Small Strain Formulation
+\begin_inset CommandInset label
+LatexCommand label
+name "sec:Small-Strain-Formulation"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+In some crustal deformation problems sufficient deformation may occur that
+ the assumptions associated with infinitesimal strains no longer hold.
+ This is often the case for problems when one wants to include the effects
+ of gravitational body forces and deformation on the overburden pressure.
+ In such cases we want to account for both rigid body motion and small strains.
+ The elasticity formulation in PyLith for small strains uses the Green-Lagrange
+ strain tensor and the Second Piola-Kirchhoff stress tensor as is based
+ on the one presented by Bathe 
+\begin_inset CommandInset citation
+LatexCommand cite
+key "Bathe:1995"
+
+\end_inset
+
+.
+ The Green-Lagrange strain provides a measure of the strain relative to
+ the original, undeformed configuration.
+\begin_inset Formula 
+\begin{gather}
+\varepsilon_{ij}=\frac{1}{2}(u_{i,j}+u_{j,i}+u_{k,i}u_{k,j}),\text{ or}\\
+\varepsilon_{ij}=X_{ji}X_{ij}-\delta_{ij},\text{ where}\\
+X_{ij}=x_{i,j}(t)=\frac{\partial}{\partial x_{j}}(x_{i}(0)+u_{i}(t)),
+\end{gather}
+
+\end_inset
+
+and 
+\begin_inset Formula $X_{ij}$
+\end_inset
+
+ is the deformation tensor.
+ The Second Piola-Kirchhoff stress tensor, 
+\begin_inset Formula $S_{ij}$
+\end_inset
+
+, is related to the Green-Lagrange strain tensor through the elasticity
+ constants,
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{equation}
+S_{ij}=C_{ijkl}\varepsilon_{kl},
+\end{equation}
+
+\end_inset
+
+in the same manner as in the infinitesimal strain formulation.
+\end_layout
+
+\begin_layout Standard
+The elasticity integral in the finite-element formulation includes additional
+ terms when we account for small strains.
+ Recognizing the similarity between the weighting function and an increment
+ in strain in the infinitesimal formulation (many finite-element texts derive
+ the finite-element formulation for elasticity using the Principle of Virtual
+ Work), we replace 
+\begin_inset Formula $\int_{V}\sigma_{ij}\phi_{i,j}\: dV$
+\end_inset
+
+ with 
+\begin_inset Formula $\int_{V}S_{ij}\delta\varepsilon_{ij}\: dV$
+\end_inset
+
+ in equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:elasticity:integral"
+
+\end_inset
+
+, where 
+\begin_inset Formula $\delta\varepsilon_{ij}$
+\end_inset
+
+ is the 
+\begin_inset Quotes eld
+\end_inset
+
+virtual
+\begin_inset Quotes erd
+\end_inset
+
+ strain.
+ Using the definition of the Green-Lagrangian strain, we have
+\begin_inset Formula 
+\begin{equation}
+\int_{V}S_{ij}\delta\varepsilon_{ij}\: dV=\int_{V}\frac{1}{2}S_{ij}(\delta u_{i,j}+\delta u_{j,i}+u_{k,i}\delta u_{k,j}+u_{k,j}\delta u_{k,i})\: dV.
+\end{equation}
+
+\end_inset
+
+Writing the displacements in terms of the basis functions and forcing the
+ terms associated with the arbitrary weighting function (
+\begin_inset Quotes eld
+\end_inset
+
+virtual
+\begin_inset Quotes erd
+\end_inset
+
+ strain) to zero yields the elastic term in the residual,
+\begin_inset Formula 
+\begin{equation}
+r_{i}^{n}=\int_{V}S_{ij}(N_{,i}^{n}+(\sum_{m}a_{k}^{m}N_{,j}^{m})N_{,i}^{n})\: dV.
+\end{equation}
+
+\end_inset
+
+Thus, we have one additional term (the second term) compared with the residual
+ for infinitesimal strains.
+ Just as in the infinitesimal formulation, we evaluate the integral over
+ the volume using numerical quadrature with sums over the quadrature points
+ of each cell.
+\end_layout
+
+\begin_layout Subsection
+Quasi-static Problems
+\end_layout
+
+\begin_layout Standard
+The system Jacobian for quasi-static problems includes terms associated
+ with elasticity.
+ For the small strain formulation, we write the elasticity term at time
+ 
+\begin_inset Formula $t+\Delta t$
+\end_inset
+
+ and consider the first terms of the Taylor series expansion,
+\begin_inset Formula 
+\begin{equation}
+\int_{v}S_{ij}(t+\Delta t)\delta\varepsilon_{ij}(t+\Delta t)\: dV=\int_{V}(S_{ij}(t)\delta\varepsilon_{ij}(t)+dS_{ij}(t)\delta\varepsilon_{ij}(t)+S_{ij}(t)d\delta\varepsilon_{ij}(t))\: dV.
+\end{equation}
+
+\end_inset
+
+We approximate the increment in the stress tensor using the elastic constants,
+\begin_inset Formula 
+\begin{equation}
+dS_{ij}=C_{ijkl}d\varepsilon_{kl},
+\end{equation}
+
+\end_inset
+
+and the increment in the 
+\begin_inset Quotes eld
+\end_inset
+
+virtual
+\begin_inset Quotes erd
+\end_inset
+
+ strain via
+\begin_inset Formula 
+\begin{equation}
+d\delta\varepsilon_{ij}=\frac{1}{2}(du_{k,i}\delta u_{k,j}+du_{k,j}\delta u_{k,i}).
+\end{equation}
+
+\end_inset
+
+We associate the system Jacobian with the terms involving the increment
+ in displacements.
+ After substituting in the expressions for the increment in the stresses
+ and the increment in the 
+\begin_inset Quotes eld
+\end_inset
+
+virtual
+\begin_inset Quotes erd
+\end_inset
+
+ strains, we have
+\begin_inset Formula 
+\begin{equation}
+A_{ij}^{nm}=\int_{V}\frac{1}{4}C_{ijkl}(N_{,k}^{m}+(\sum_{r}a_{p}^{r}N_{,l}^{r})N_{,k}^{m})(N_{,i}^{n}+(\sum_{r}a_{p}^{r}N_{,j}^{r})N_{,i}^{n})+\frac{1}{2}S_{kl}N_{,l}^{m}N_{,l}^{n}\delta_{ij}\: dV.
+\end{equation}
+
+\end_inset
+
+The small strain formulation produces additional terms associated with the
+ elastic constants and a new term associated with the stress tensor.
+\end_layout
+
+\begin_layout Subsection
+Dynamic Problems
+\end_layout
+
+\begin_layout Standard
+The system Jacobian matrix in dynamic problems does not include any terms
+ associated with elasticity, so the system Jacobian matrix in the small
+ strain formulation matches the one used in the infinitesimal strain formulation.
+\end_layout
+
+\end_body
+\end_document

Modified: short/3D/PyLith/branches/v1.7-trunk/doc/userguide/intro/intro.lyx
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/doc/userguide/intro/intro.lyx	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/doc/userguide/intro/intro.lyx	2012-08-21 18:24:25 UTC (rev 20616)
@@ -1,542 +1,542 @@
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-
-\begin_body
-
-\begin_layout Chapter
-Introduction
-\end_layout
-
-\begin_layout Section
-Overview
-\end_layout
-
-\begin_layout Standard
-PyLith is a multi-scale simulation software package for earthquake physics.
- It is portable, scalable software for simulation of crustal deformation
- across spatial scales ranging from meters to hundreds of kilometers and
- temporal scales ranging from milliseconds to thousands of years.
-\end_layout
-
-\begin_layout Section
-History
-\end_layout
-
-\begin_layout Standard
-PyLith 1.0 was the first version to allow the solution of both implicit (quasi-st
-atic) and explicit (dynamic) problems and was a complete rewrite of the
- original PyLith (version 0.8).
- PyLith 1.0 combines the functionality of EqSim 
-\begin_inset CommandInset citation
-LatexCommand cite
-key "Aagaard:etal:2001a,Aagaard:etal:2001b"
-
-\end_inset
-
- and PyLith 0.8.
- PyLith 0.8 was a direct descendant of LithoMop and was the first version
- that ran in parallel, as well as providing several other improvements over
- LithoMop.
- LithoMop was the product of major reengineering of Tecton, a finite-element
- code for simulating static and quasi-static crustal deformation.
- The major new features present in LithoMop included dynamic memory allocation
- and the use of the Pyre simulation framework and PETSc solvers.
- EqSim was written by Brad Aagaard to solve problems in earthquake dynamics,
- including rupture propagation and seismic wave propagation.
-\end_layout
-
-\begin_layout Standard
-The release of PyLith 1.0 has been followed by additional releases that expand
- the number of features as well as improve performance.
- The PyLith 1.x series of releases allows the solution of both quasi-static
- and dynamic problems in one, two, or three dimensions.
- The code runs in either serial or parallel, and the design allows for relativel
-y easy scripting using the Python programming language.
- Material properties and values for boundary and fault conditions are specified
- using spatial databases, which permit easy prescription of complex spatial
- variations of properties and parameters.
- Simulation parameters are generally specified through the use of simple
- ASCII files or the command line.
- At present, mesh information may be provided using a simple ASCII file
- (PyLith mesh ASCII format) or imported from CUBIT or LaGriT, two widely-used
- meshing packages.
- The elements currently available include a linear bar in 1-D, linear triangles
- and quadrilaterals in 2-D, and linear tetrahedra and hexahedra in 3-D.
- Materials presently available include isotropic elastic, linear Maxwell
- viscoelastic, generalized Maxwell viscoelastic, power-law viscoelastic,
- and Drucker-Prager elastoplastic.
- Boundary conditions include Dirichlet (prescribed displacements and velocities)
-, Neumann (traction), point forces, and absorbing boundaries.
- Cohesive elements are used to implement slip across interior surfaces (faults)
- with both kinematically-specified fault slip and slip governed by fault
- constitutive models.
- PyLith also includes an interface for computing static Green's functions
- for fault slip.
-\end_layout
-
-\begin_layout Standard
-PyLith is under active development and we expect a number of additions and
- improvements in the near future.
- Likely enhancements will include additional bulk and fault constitutive
- models, coupled quasi-static and dynamic simulations for earthquake cycle
- modeling, and coupling between elasticity, heat flow, and/or fluid flow.
-\end_layout
-
-\begin_layout Section
-PyLith Workflow
-\end_layout
-
-\begin_layout Standard
-PyLith is one component in the process of investigating problems in tectonics
- (Figure 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "fig:Workflow-summary"
-
-\end_inset
-
-).
- Given a geological problem of interest, a scientist must first provide
- a geometrical representation of the desired structure.
- Once the structure has been defined, a computational mesh must be created.
- PyLith presently provides three mesh importing options: CUBIT Exodus format,
- LaGriT GMV and Pset files, and PyLith mesh ASCII format.
- The modeling of the physical processes of interest is performed by a code
- such as PyLith.
- Present output consists of VTK or HDF5/Xdmf files which can be used by
- a number of visualization codes (e.g., ParaView, Visit, MayaVi, and Matlab).
- 
-\end_layout
-
-\begin_layout Standard
-\begin_inset Float figure
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\align center
-\begin_inset Graphics
-	filename figs/workflow.eps
-	scale 67
-	keepAspectRatio
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Plain Layout
-\begin_inset Caption
-
-\begin_layout Plain Layout
-\begin_inset CommandInset label
-LatexCommand label
-name "fig:Workflow-summary"
-
-\end_inset
-
-Workflow involved in going from geologic structure to problem analysis.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-PyLith Design
-\end_layout
-
-\begin_layout Standard
-PyLith is separated into modules to encapsulate behavior and facilitate
- use across multiple applications.
- This allows expert users to replace functionality of a wide variety of
- components without recompiling or polluting the main code.
- PyLith employs external packages (see Figure 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "fig:pylith-dependencies"
-
-\end_inset
-
-) to reduce development time and enhance computational efficiency; for example,
- PyLith 0.8 ran two times faster when the PETSc linear solver was used.
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float figure
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\align center
-\begin_inset Graphics
-	filename figs/packages.eps
-	scale 40
-
-\end_inset
-
-
-\begin_inset Caption
-
-\begin_layout Plain Layout
-\begin_inset CommandInset label
-LatexCommand label
-name "fig:pylith-dependencies"
-
-\end_inset
-
-PyLith dependencies.
- PyLith makes direct use of several other packages, some of which have their
- own dependencies.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-PyLith is written in two programming languages.
- High-level code is written in Python; this rich, expressive interpreted
- language with dynamic typing reduces development time and permits flexible
- addition of user-contributed modules.
- This high-level code makes use of Pyre, a science-neutral simulation framework
- developed at Caltech, to link the modules together at runtime and gather
- user-input.
- Low-level code is written in C++, providing fast execution while still
- allowing an object-oriented implementation.
- This low-level code relies on PETSc to perform operations on matrices and
- vectors in parallel.
- We also make extensive use of two Python packages.
- SWIG is a package that simplifies the task of adding C++ extensions to
- Python code, and FIAT provides tabulated basis functions and numerical
- quadrature points.
- 
-\end_layout
-
-\begin_layout Standard
-In writing PyLith 1.0, the code was designed to be object-oriented and modular.
- Each type of module is accessed through a specified interface (set of functions
-).
- This permits adding, replacing, and rewriting modules without affecting
- other parts of the code.
- This code structure simplifies code maintenance and development.
- Extending the set of code features is also easier, since developers can
- create new modules derived from the existing ones.
-\end_layout
-
-\begin_layout Standard
-The new code design leverages Pyre, Sieve, and PETSc much more extensively
- than the previous version.
-  Pyre is used to glue together the various modules used to construct a
- simulation and specify the parameters.
- Sieve is used for all finite-element  storage and manipulation and handles
- the creation of the PETSc matrices and vectors.
-  As a result, most of the PyLith source code pertains to implementing the
- geodynamics, such as bulk rheology, boundary conditions, and slip on faults.
- 
-\end_layout
-
-\begin_layout Standard
-PyLith also uses FIAT to tabulate the finite-element basis functions  at
- the numerical integration (quadrature) points.
- Nemesis allows PyLith to run Python using the Message Passing Interface
- (MPI) for parallel processing.
- Additional, indirect dependencies (see Figure 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "fig:pylith-dependencies"
-
-\end_inset
-
-) include numpy (efficient operations on numerical arrays in Python), Proj.4
- (geographic projections), and SWIG (calling C++ functions from Python).
-\end_layout
-
-\begin_layout Standard
-During development, tests were constructed for nearly every module function.
- These unit tests are distributed with the source code.
- These tests are run throughout the development cycle to expose bugs and
- isolate their origin.
- As additional changes are made to the code, the tests are rerun to help
- prevent introduction of new bugs.
- A number of simple, full-scale tests, such as axial compression and extension,
- simple shear, and slip on through-going faults, have been used to test
- the code.
- Additionally, we have run the Southern California Earthquake Center crustal
- deformation and several of the spontaneous rupture benchmarks for strike-slip
- and reverse-slip to determine the relative local and global error (see
- Chapter 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "cha:Benchmarks"
-
-\end_inset
-
-).
-\end_layout
-
-\begin_layout Subsection
-Pyre
-\end_layout
-
-\begin_layout Standard
-Pyre is an object-oriented environment capable of specifying and launching
- numerical simulations on multiple platforms, including Beowulf-class parallel
- computers and grid computing systems.
- Pyre allows the binding of multiple components such as solid and fluid
- models used in Earth science simulations, and different meshers.
- The Pyre framework enables the elegant setup, modification and launching
- of massively parallel solver applications.
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float figure
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\align center
-\begin_inset Graphics
-	filename figs/pyre_overview.png
-	width 4in
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Plain Layout
-\begin_inset Caption
-
-\begin_layout Plain Layout
-\begin_inset CommandInset label
-LatexCommand label
-name "fig:Pyre:Architecture"
-
-\end_inset
-
-Pyre Architecture.
- The integration framework is a set of cooperating abstract services.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Pyre is a framework, a combination of software and design philosophy that
- promotes the reuse of code.
- In their canonical software design book, 
-\emph on
-Design Patterns
-\emph default
-, Erich Gamma 
-\shape italic
-et al
-\shape default
-.
- condense the concept of a framework concept down to, ``When you use a framework
-, you reuse the main body and write the code it calls.'' In the context of
- frameworks and object-oriented programming, Pyre can be thought of as a
- collection of classes and the way their instances interact.
- Programming applications based on Pyre will look similar to those written
- in any other object-oriented language.
- The Pyre framework contains a subset of parts that make up the overall
- framework.
- Each of those parts is designed to solve a specific problem.
-\end_layout
-
-\begin_layout Standard
-The framework approach to computation offers many advantages.
- It permits the exchange of codes and promotes the reuse of standardized
- software while preserving efficiency.
- Frameworks are also an efficient way to handle changes in computer architecture.
- They present programmers and scientists with a unified and well-defined
- task and allow for shared costs of the housekeeping aspects of software
- development.
- They provide greater institutional continuity to model development than
- piecemeal approaches.
-\end_layout
-
-\begin_layout Standard
-The Pyre framework incorporates features aimed at enabling the scientific
- non-expert to perform tasks easily without hindering the expert.
- Target features for end users allow complete and intuitive simulation specifica
-tion, reasonable defaults, consistency checks of input, good diagnostics,
- easy access to remote facilities, and status monitoring.
- Target features for developers include easy access to user input, a shorter
- development cycle, and good debugging support.
-\end_layout
-
-\begin_layout Subsection
-Sieve and PETSc
-\end_layout
-
-\begin_layout Standard
-PyLith 1.x makes use of a set of data structures and routines in PETSc called
- 
-\family typewriter
-Sieve
-\family default
-, which is still under active development.
- 
-\family typewriter
-Sieve
-\family default
- provides data structures and routines for for representing and manipulating
- computational meshes, and it greatly simplifies finite-element computations.
-
-\family typewriter
- Sieve
-\family default
- represents the topology of the domain.
- Zero volume elements are inserted along all fault surfaces to implement
- kinematic (prescribed) or dynamic (constitutive model) implementations
- of fault slip.
- Material properties and other parameters are represented as sections (scalar
- and vector fields) over the mesh, and values for a vertex or cell can be
- retrieved by restricting the section to the vertex or cell.
- For each problem, functions are provided to calculate the residual and
- its Jacobian.
- All numerical integration is done in these functions, and parallel assembly
- is accomplished using the restrict/update paradigm of the 
-\family typewriter
-Sieve
-\family default
- framework.
- We assemble into PETSc linear algebra objects and then call PETSc solvers.
- The solution is mapped back into a section, which can be output in VTK
- format.
-\end_layout
-
-\begin_layout Standard
-PETSc 
-\begin_inset Flex URL
-status collapsed
-
-\begin_layout Plain Layout
-
-www-unix.mcs.anl.gov/petsc/petsc-as
-\end_layout
-
-\end_inset
-
-, the Portable, Extensible Toolkit for Scientific computation, provides
- a suite of routines for parallel, numerical solution of partial differential
- equations for linear and nonlinear systems with large, sparse systems of
- equations.
- PETSc includes solvers that implement a variety of Newton and Krylov subspace
- methods.
- It can also interface with many external packages, including ESSL, MUMPS,
- Matlab, ParMETIS, PVODE, and Hypre, thereby providing additional solvers
- and interaction with other software packages.
-\end_layout
-
-\begin_layout Standard
-PETSc includes interfaces for FORTRAN 77/90, C, C++, and Python for nearly
- all of the routines, and PETSc can be installed on most Unix systems.
- PETSc can be built with user-supplied, highly optimized linear algebra
- routines (e.g., ATLAS and commercial versions of BLAS/LAPACK), thereby improving
- application performance.
- Users can use PETSc parallel matrices, vectors, and other data structures
- for most parallel operations, eliminating the need for explicit calls to
- Message Passing Interface (MPI) routines.
- Many settings and options can be controlled with PETSc-specific command-line
- arguments, including selection of preconditions, solvers, and generation
- of performance logs.
-\end_layout
-
-\end_body
-\end_document
+#LyX 2.0 created this file. For more info see http://www.lyx.org/
+\lyxformat 413
+\begin_document
+\begin_header
+\textclass book
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+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Chapter
+Introduction
+\end_layout
+
+\begin_layout Section
+Overview
+\end_layout
+
+\begin_layout Standard
+PyLith is a multi-scale simulation software package for earthquake physics.
+ It is portable, scalable software for simulation of crustal deformation
+ across spatial scales ranging from meters to hundreds of kilometers and
+ temporal scales ranging from milliseconds to thousands of years.
+\end_layout
+
+\begin_layout Section
+History
+\end_layout
+
+\begin_layout Standard
+PyLith 1.0 was the first version to allow the solution of both implicit (quasi-st
+atic) and explicit (dynamic) problems and was a complete rewrite of the
+ original PyLith (version 0.8).
+ PyLith 1.0 combines the functionality of EqSim 
+\begin_inset CommandInset citation
+LatexCommand cite
+key "Aagaard:etal:2001a,Aagaard:etal:2001b"
+
+\end_inset
+
+ and PyLith 0.8.
+ PyLith 0.8 was a direct descendant of LithoMop and was the first version
+ that ran in parallel, as well as providing several other improvements over
+ LithoMop.
+ LithoMop was the product of major reengineering of Tecton, a finite-element
+ code for simulating static and quasi-static crustal deformation.
+ The major new features present in LithoMop included dynamic memory allocation
+ and the use of the Pyre simulation framework and PETSc solvers.
+ EqSim was written by Brad Aagaard to solve problems in earthquake dynamics,
+ including rupture propagation and seismic wave propagation.
+\end_layout
+
+\begin_layout Standard
+The release of PyLith 1.0 has been followed by additional releases that expand
+ the number of features as well as improve performance.
+ The PyLith 1.x series of releases allows the solution of both quasi-static
+ and dynamic problems in one, two, or three dimensions.
+ The code runs in either serial or parallel, and the design allows for relativel
+y easy scripting using the Python programming language.
+ Material properties and values for boundary and fault conditions are specified
+ using spatial databases, which permit easy prescription of complex spatial
+ variations of properties and parameters.
+ Simulation parameters are generally specified through the use of simple
+ ASCII files or the command line.
+ At present, mesh information may be provided using a simple ASCII file
+ (PyLith mesh ASCII format) or imported from CUBIT or LaGriT, two widely-used
+ meshing packages.
+ The elements currently available include a linear bar in 1D, linear triangles
+ and quadrilaterals in 2D, and linear tetrahedra and hexahedra in 3D.
+ Materials presently available include isotropic elastic, linear Maxwell
+ viscoelastic, generalized Maxwell viscoelastic, power-law viscoelastic,
+ and Drucker-Prager elastoplastic.
+ Boundary conditions include Dirichlet (prescribed displacements and velocities)
+, Neumann (traction), point forces, and absorbing boundaries.
+ Cohesive elements are used to implement slip across interior surfaces (faults)
+ with both kinematically-specified fault slip and slip governed by fault
+ constitutive models.
+ PyLith also includes an interface for computing static Green's functions
+ for fault slip.
+\end_layout
+
+\begin_layout Standard
+PyLith is under active development and we expect a number of additions and
+ improvements in the near future.
+ Likely enhancements will include additional bulk and fault constitutive
+ models, coupled quasi-static and dynamic simulations for earthquake cycle
+ modeling, and coupling between elasticity, heat flow, and/or fluid flow.
+\end_layout
+
+\begin_layout Section
+PyLith Workflow
+\end_layout
+
+\begin_layout Standard
+PyLith is one component in the process of investigating problems in tectonics
+ (Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:Workflow-summary"
+
+\end_inset
+
+).
+ Given a geological problem of interest, a scientist must first provide
+ a geometrical representation of the desired structure.
+ Once the structure has been defined, a computational mesh must be created.
+ PyLith presently provides three mesh importing options: CUBIT Exodus format,
+ LaGriT GMV and Pset files, and PyLith mesh ASCII format.
+ The modeling of the physical processes of interest is performed by a code
+ such as PyLith.
+ Present output consists of VTK or HDF5/Xdmf files which can be used by
+ a number of visualization codes (e.g., ParaView, Visit, MayaVi, and Matlab).
+ 
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\align center
+\begin_inset Graphics
+	filename figs/workflow.eps
+	scale 67
+	keepAspectRatio
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:Workflow-summary"
+
+\end_inset
+
+Workflow involved in going from geologic structure to problem analysis.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+PyLith Design
+\end_layout
+
+\begin_layout Standard
+PyLith is separated into modules to encapsulate behavior and facilitate
+ use across multiple applications.
+ This allows expert users to replace functionality of a wide variety of
+ components without recompiling or polluting the main code.
+ PyLith employs external packages (see Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:pylith-dependencies"
+
+\end_inset
+
+) to reduce development time and enhance computational efficiency; for example,
+ PyLith 0.8 ran two times faster when the PETSc linear solver was used.
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float figure
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\align center
+\begin_inset Graphics
+	filename figs/packages.eps
+	scale 40
+
+\end_inset
+
+
+\begin_inset Caption
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:pylith-dependencies"
+
+\end_inset
+
+PyLith dependencies.
+ PyLith makes direct use of several other packages, some of which have their
+ own dependencies.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+PyLith is written in two programming languages.
+ High-level code is written in Python; this rich, expressive interpreted
+ language with dynamic typing reduces development time and permits flexible
+ addition of user-contributed modules.
+ This high-level code makes use of Pyre, a science-neutral simulation framework
+ developed at Caltech, to link the modules together at runtime and gather
+ user-input.
+ Low-level code is written in C++, providing fast execution while still
+ allowing an object-oriented implementation.
+ This low-level code relies on PETSc to perform operations on matrices and
+ vectors in parallel.
+ We also make extensive use of two Python packages.
+ SWIG is a package that simplifies the task of adding C++ extensions to
+ Python code, and FIAT provides tabulated basis functions and numerical
+ quadrature points.
+ 
+\end_layout
+
+\begin_layout Standard
+In writing PyLith 1.0, the code was designed to be object-oriented and modular.
+ Each type of module is accessed through a specified interface (set of functions
+).
+ This permits adding, replacing, and rewriting modules without affecting
+ other parts of the code.
+ This code structure simplifies code maintenance and development.
+ Extending the set of code features is also easier, since developers can
+ create new modules derived from the existing ones.
+\end_layout
+
+\begin_layout Standard
+The new code design leverages Pyre, Sieve, and PETSc much more extensively
+ than the previous version.
+  Pyre is used to glue together the various modules used to construct a
+ simulation and specify the parameters.
+ Sieve is used for all finite-element  storage and manipulation and handles
+ the creation of the PETSc matrices and vectors.
+  As a result, most of the PyLith source code pertains to implementing the
+ geodynamics, such as bulk rheology, boundary conditions, and slip on faults.
+ 
+\end_layout
+
+\begin_layout Standard
+PyLith also uses FIAT to tabulate the finite-element basis functions  at
+ the numerical integration (quadrature) points.
+ Nemesis allows PyLith to run Python using the Message Passing Interface
+ (MPI) for parallel processing.
+ Additional, indirect dependencies (see Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:pylith-dependencies"
+
+\end_inset
+
+) include numpy (efficient operations on numerical arrays in Python), Proj.4
+ (geographic projections), and SWIG (calling C++ functions from Python).
+\end_layout
+
+\begin_layout Standard
+During development, tests were constructed for nearly every module function.
+ These unit tests are distributed with the source code.
+ These tests are run throughout the development cycle to expose bugs and
+ isolate their origin.
+ As additional changes are made to the code, the tests are rerun to help
+ prevent introduction of new bugs.
+ A number of simple, full-scale tests, such as axial compression and extension,
+ simple shear, and slip on through-going faults, have been used to test
+ the code.
+ Additionally, we have run the Southern California Earthquake Center crustal
+ deformation and several of the spontaneous rupture benchmarks for strike-slip
+ and reverse-slip to determine the relative local and global error (see
+ Chapter 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "cha:Benchmarks"
+
+\end_inset
+
+).
+\end_layout
+
+\begin_layout Subsection
+Pyre
+\end_layout
+
+\begin_layout Standard
+Pyre is an object-oriented environment capable of specifying and launching
+ numerical simulations on multiple platforms, including Beowulf-class parallel
+ computers and grid computing systems.
+ Pyre allows the binding of multiple components such as solid and fluid
+ models used in Earth science simulations, and different meshers.
+ The Pyre framework enables the elegant setup, modification and launching
+ of massively parallel solver applications.
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float figure
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\align center
+\begin_inset Graphics
+	filename figs/pyre_overview.png
+	width 4in
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:Pyre:Architecture"
+
+\end_inset
+
+Pyre Architecture.
+ The integration framework is a set of cooperating abstract services.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Pyre is a framework, a combination of software and design philosophy that
+ promotes the reuse of code.
+ In their canonical software design book, 
+\emph on
+Design Patterns
+\emph default
+, Erich Gamma 
+\shape italic
+et al
+\shape default
+.
+ condense the concept of a framework concept down to, ``When you use a framework
+, you reuse the main body and write the code it calls.'' In the context of
+ frameworks and object-oriented programming, Pyre can be thought of as a
+ collection of classes and the way their instances interact.
+ Programming applications based on Pyre will look similar to those written
+ in any other object-oriented language.
+ The Pyre framework contains a subset of parts that make up the overall
+ framework.
+ Each of those parts is designed to solve a specific problem.
+\end_layout
+
+\begin_layout Standard
+The framework approach to computation offers many advantages.
+ It permits the exchange of codes and promotes the reuse of standardized
+ software while preserving efficiency.
+ Frameworks are also an efficient way to handle changes in computer architecture.
+ They present programmers and scientists with a unified and well-defined
+ task and allow for shared costs of the housekeeping aspects of software
+ development.
+ They provide greater institutional continuity to model development than
+ piecemeal approaches.
+\end_layout
+
+\begin_layout Standard
+The Pyre framework incorporates features aimed at enabling the scientific
+ non-expert to perform tasks easily without hindering the expert.
+ Target features for end users allow complete and intuitive simulation specifica
+tion, reasonable defaults, consistency checks of input, good diagnostics,
+ easy access to remote facilities, and status monitoring.
+ Target features for developers include easy access to user input, a shorter
+ development cycle, and good debugging support.
+\end_layout
+
+\begin_layout Subsection
+Sieve and PETSc
+\end_layout
+
+\begin_layout Standard
+PyLith 1.x makes use of a set of data structures and routines in PETSc called
+ 
+\family typewriter
+Sieve
+\family default
+, which is still under active development.
+ 
+\family typewriter
+Sieve
+\family default
+ provides data structures and routines for for representing and manipulating
+ computational meshes, and it greatly simplifies finite-element computations.
+
+\family typewriter
+ Sieve
+\family default
+ represents the topology of the domain.
+ Zero volume elements are inserted along all fault surfaces to implement
+ kinematic (prescribed) or dynamic (constitutive model) implementations
+ of fault slip.
+ Material properties and other parameters are represented as sections (scalar
+ and vector fields) over the mesh, and values for a vertex or cell can be
+ retrieved by restricting the section to the vertex or cell.
+ For each problem, functions are provided to calculate the residual and
+ its Jacobian.
+ All numerical integration is done in these functions, and parallel assembly
+ is accomplished using the restrict/update paradigm of the 
+\family typewriter
+Sieve
+\family default
+ framework.
+ We assemble into PETSc linear algebra objects and then call PETSc solvers.
+ The solution is mapped back into a section, which can be output in VTK
+ format.
+\end_layout
+
+\begin_layout Standard
+PETSc 
+\begin_inset Flex URL
+status collapsed
+
+\begin_layout Plain Layout
+
+www-unix.mcs.anl.gov/petsc/petsc-as
+\end_layout
+
+\end_inset
+
+, the Portable, Extensible Toolkit for Scientific computation, provides
+ a suite of routines for parallel, numerical solution of partial differential
+ equations for linear and nonlinear systems with large, sparse systems of
+ equations.
+ PETSc includes solvers that implement a variety of Newton and Krylov subspace
+ methods.
+ It can also interface with many external packages, including ESSL, MUMPS,
+ Matlab, ParMETIS, PVODE, and Hypre, thereby providing additional solvers
+ and interaction with other software packages.
+\end_layout
+
+\begin_layout Standard
+PETSc includes interfaces for FORTRAN 77/90, C, C++, and Python for nearly
+ all of the routines, and PETSc can be installed on most Unix systems.
+ PETSc can be built with user-supplied, highly optimized linear algebra
+ routines (e.g., ATLAS and commercial versions of BLAS/LAPACK), thereby improving
+ application performance.
+ Users can use PETSc parallel matrices, vectors, and other data structures
+ for most parallel operations, eliminating the need for explicit calls to
+ Message Passing Interface (MPI) routines.
+ Many settings and options can be controlled with PETSc-specific command-line
+ arguments, including selection of preconditions, solvers, and generation
+ of performance logs.
+\end_layout
+
+\end_body
+\end_document

Modified: short/3D/PyLith/branches/v1.7-trunk/doc/userguide/materials/altformulations.lyx
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/doc/userguide/materials/altformulations.lyx	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/doc/userguide/materials/altformulations.lyx	2012-08-21 18:24:25 UTC (rev 20616)
@@ -1,320 +1,321 @@
-#LyX 2.0 created this file. For more info see http://www.lyx.org/
-\lyxformat 413
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-\end_header
-
-\begin_body
-
-\begin_layout Chapter
-\begin_inset CommandInset label
-LatexCommand label
-name "cha:Alternative-Formulations"
-
-\end_inset
-
-Alternative Material Model Formulations
-\end_layout
-
-\begin_layout Section
-\begin_inset CommandInset label
-LatexCommand label
-name "sec:ViscoelasticFormulations"
-
-\end_inset
-
-Viscoelastic Formulations
-\end_layout
-
-\begin_layout Standard
-The viscoelastic formulations presently used in PyLith are described in
- 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "sec:Viscoelastic-Materials"
-
-\end_inset
-
-.
- In some cases there are alternative formulations that may be used in future
- versions of PyLith, and those are described here.
-\end_layout
-
-\begin_layout Subsection
-\begin_inset CommandInset label
-LatexCommand label
-name "sub:Effective-Stress-Formulation-Maxwell"
-
-\end_inset
-
-Effective Stress Formulation for a Linear Maxwell Viscoelastic Material
-\end_layout
-
-\begin_layout Standard
-An alternative technique for solving the equations for a Maxwell viscoelastic
- material is based on the effective stress formulation described in 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "sub:Effective-Stress-Formulations-Viscoelastic"
-
-\end_inset
-
-.
- A linear Maxwell viscoelastic material may be characterized by the same
- elastic parameters as an isotropic elastic material (
-\begin_inset Formula $E$
-\end_inset
-
- and 
-\begin_inset Formula $\nu$
-\end_inset
-
-), as well as the viscosity, 
-\begin_inset Formula $\eta$
-\end_inset
-
-.
- The creep strain increment is
-\begin_inset Formula 
-\begin{gather}
-\underline{\Delta e}^{C}=\frac{\Delta t\phantom{}^{\tau}\underline{S}}{2\eta}\,\,.\label{eq:D1}
-\end{gather}
-
-\end_inset
-
-Therefore,
-\begin_inset Formula 
-\begin{gather}
-\Delta\overline{e}^{C}=\frac{\Delta t\sqrt{^{\tau}J_{2}^{\prime}}}{\sqrt{3\eta}}=\frac{\Delta t\phantom{}^{\tau}\overline{\sigma}}{3\eta}\,,\,\mathrm{and}\,^{\tau}\gamma=\frac{1}{2\eta}\,\,.\label{eq:D2}
-\end{gather}
-
-\end_inset
-
-Substituting Equations 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:46"
-
-\end_inset
-
-, 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:D1"
-
-\end_inset
-
-, and 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:D2"
-
-\end_inset
-
- into 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:43"
-
-\end_inset
-
-, we obtain
-\begin_inset Formula 
-\begin{gather}
-^{t+\Delta t}\underline{S}=\frac{1}{a_{E}}\left\{ ^{t+\Delta t}\underline{e}^{\prime}-\frac{\Delta t}{2\eta}\left[(1-\alpha)^{t}\underline{S}+\alpha\phantom{}^{t+\Delta t}\underline{S}\right]\right\} +\underline{S}^{I}\,\,.\label{eq:D3}
-\end{gather}
-
-\end_inset
-
-Solving for 
-\begin_inset Formula $^{t+\Delta t}\underline{S}$
-\end_inset
-
-,
-\begin_inset Formula 
-\begin{gather}
-^{t+\Delta t}\underline{S}=\frac{1}{a_{E}+\frac{\alpha\Delta t}{2\eta}}\left[^{t+\Delta t}\underline{e}^{\prime}-\frac{\Delta t}{2\eta}(1-\alpha)^{t}\underline{S}+\frac{1+\mathrm{\nu}}{E}\underline{S}^{I}\right]\,\,.\label{eq:D4}
-\end{gather}
-
-\end_inset
-
-In this case it is possible to solve directly for the deviatoric stresses,
- and the effective stress function approach is not needed.
- To obtain the total stress, we simply use
-\begin_inset Formula 
-\begin{gather}
-^{t+\Delta t}\sigma_{ij}=\phantom{}^{t+\Delta t}S_{ij}+\frac{\mathit{1}}{a_{m}}\left(\,^{t+\Delta t}\theta-\theta^{I}\right)\delta_{ij}+P^{I}\delta_{ij}\,\,.\label{eq:D5}
-\end{gather}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-To compute the viscoelastic tangent material matrix relating stress and
- strain, we need to compute the first term in 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:58"
-
-\end_inset
-
-.
- From Equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:D4"
-
-\end_inset
-
-, we have
-\begin_inset Formula 
-\begin{gather}
-\frac{\partial\phantom{}^{t+\Delta t}S_{i}}{\partial\phantom{}^{t+\Delta t}e_{k}^{\prime}}=\frac{\delta_{ik}}{a_{E}+\frac{\alpha\Delta t}{2\eta}}\,\,.\label{eq:D12}
-\end{gather}
-
-\end_inset
-
-Using this, along with 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:58"
-
-\end_inset
-
-, 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:59"
-
-\end_inset
-
-, and 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:60"
-
-\end_inset
-
-, the final material matrix relating stress and tensor strain is
-\begin_inset Formula 
-\begin{gather}
-C_{ij}^{VE}=\frac{1}{3a_{m}}\left[\begin{array}{cccccc}
-1 & 1 & 1 & 0 & 0 & 0\\
-1 & 1 & 1 & 0 & 0 & 0\\
-1 & 1 & 1 & 0 & 0 & 0\\
-0 & 0 & 0 & 0 & 0 & 0\\
-0 & 0 & 0 & 0 & 0 & 0\\
-0 & 0 & 0 & 0 & 0 & 0
-\end{array}\right]+\frac{1}{3\left(a_{E}+\frac{\alpha\Delta t}{2\eta}\right)}\left[\begin{array}{cccccc}
-2 & -1 & -1 & 0 & 0 & 0\\
--1 & 2 & -1 & 0 & 0 & 0\\
--1 & -1 & 2 & 0 & 0 & 0\\
-0 & 0 & 0 & 3 & 0 & 0\\
-0 & 0 & 0 & 0 & 3 & 0\\
-0 & 0 & 0 & 0 & 0 & 3
-\end{array}\right]\,.\label{eq:D13}
-\end{gather}
-
-\end_inset
-
-Note that the coefficient of the second matrix approaches 
-\begin_inset Formula $E/3(1+\nu)=1/3a_{E}$
-\end_inset
-
- as 
-\begin_inset Formula $\eta$
-\end_inset
-
- goes to infinity.
- To check the results we make sure that the regular elastic constitutive
- matrix is obtained for selected terms in the case where 
-\begin_inset Formula $\eta$
-\end_inset
-
- goes to infinity.
-\begin_inset Formula 
-\begin{gather}
-C_{11}^{E}=\frac{E(1-\nu)}{(1+\nu)(1-2\nu)}\,\,\nonumber \\
-C_{12}^{E}=\frac{E\nu}{(1+\nu)(1-2\nu)}\,.\label{eq:D14}\\
-C_{44}^{E}=\frac{E}{1+\nu}\,\,\nonumber 
-\end{gather}
-
-\end_inset
-
-This is consistent with the regular elasticity matrix, and Equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:D13"
-
-\end_inset
-
- should thus be used when forming the stiffness matrix.
- We do not presently use this formulation, but it may be included in future
- versions.
-\end_layout
-
-\end_body
-\end_document
+#LyX 2.0 created this file. For more info see http://www.lyx.org/
+\lyxformat 413
+\begin_document
+\begin_header
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+
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+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Chapter
+\begin_inset CommandInset label
+LatexCommand label
+name "cha:Alternative-Formulations"
+
+\end_inset
+
+Alternative Material Model Formulations
+\end_layout
+
+\begin_layout Section
+\begin_inset CommandInset label
+LatexCommand label
+name "sec:ViscoelasticFormulations"
+
+\end_inset
+
+Viscoelastic Formulations
+\end_layout
+
+\begin_layout Standard
+The viscoelastic formulations presently used in PyLith are described in
+ Section 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sec:Viscoelastic-Materials"
+
+\end_inset
+
+.
+ In some cases there are alternative formulations that may be used in future
+ versions of PyLith, and those are described here.
+\end_layout
+
+\begin_layout Subsection
+\begin_inset CommandInset label
+LatexCommand label
+name "sub:Effective-Stress-Formulation-Maxwell"
+
+\end_inset
+
+Effective Stress Formulation for a Linear Maxwell Viscoelastic Material
+\end_layout
+
+\begin_layout Standard
+An alternative technique for solving the equations for a Maxwell viscoelastic
+ material is based on the effective stress formulation described in Section
+ 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sub:Effective-Stress-Formulations-Viscoelastic"
+
+\end_inset
+
+.
+ A linear Maxwell viscoelastic material may be characterized by the same
+ elastic parameters as an isotropic elastic material (
+\begin_inset Formula $E$
+\end_inset
+
+ and 
+\begin_inset Formula $\nu$
+\end_inset
+
+), as well as the viscosity, 
+\begin_inset Formula $\eta$
+\end_inset
+
+.
+ The creep strain increment is
+\begin_inset Formula 
+\begin{gather}
+\underline{\Delta e}^{C}=\frac{\Delta t\phantom{}^{\tau}\underline{S}}{2\eta}\,\,.\label{eq:D1}
+\end{gather}
+
+\end_inset
+
+Therefore,
+\begin_inset Formula 
+\begin{gather}
+\Delta\overline{e}^{C}=\frac{\Delta t\sqrt{^{\tau}J_{2}^{\prime}}}{\sqrt{3\eta}}=\frac{\Delta t\phantom{}^{\tau}\overline{\sigma}}{3\eta}\,,\,\mathrm{and}\,^{\tau}\gamma=\frac{1}{2\eta}\,\,.\label{eq:D2}
+\end{gather}
+
+\end_inset
+
+Substituting Equations 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:46"
+
+\end_inset
+
+, 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:D1"
+
+\end_inset
+
+, and 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:D2"
+
+\end_inset
+
+ into 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:43"
+
+\end_inset
+
+, we obtain
+\begin_inset Formula 
+\begin{gather}
+^{t+\Delta t}\underline{S}=\frac{1}{a_{E}}\left\{ ^{t+\Delta t}\underline{e}^{\prime}-\frac{\Delta t}{2\eta}\left[(1-\alpha)^{t}\underline{S}+\alpha\phantom{}^{t+\Delta t}\underline{S}\right]\right\} +\underline{S}^{I}\,\,.\label{eq:D3}
+\end{gather}
+
+\end_inset
+
+Solving for 
+\begin_inset Formula $^{t+\Delta t}\underline{S}$
+\end_inset
+
+,
+\begin_inset Formula 
+\begin{gather}
+^{t+\Delta t}\underline{S}=\frac{1}{a_{E}+\frac{\alpha\Delta t}{2\eta}}\left[^{t+\Delta t}\underline{e}^{\prime}-\frac{\Delta t}{2\eta}(1-\alpha)^{t}\underline{S}+\frac{1+\mathrm{\nu}}{E}\underline{S}^{I}\right]\,\,.\label{eq:D4}
+\end{gather}
+
+\end_inset
+
+In this case it is possible to solve directly for the deviatoric stresses,
+ and the effective stress function approach is not needed.
+ To obtain the total stress, we simply use
+\begin_inset Formula 
+\begin{gather}
+^{t+\Delta t}\sigma_{ij}=\phantom{}^{t+\Delta t}S_{ij}+\frac{\mathit{1}}{a_{m}}\left(\,^{t+\Delta t}\theta-\theta^{I}\right)\delta_{ij}+P^{I}\delta_{ij}\,\,.\label{eq:D5}
+\end{gather}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+To compute the viscoelastic tangent material matrix relating stress and
+ strain, we need to compute the first term in Equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:58"
+
+\end_inset
+
+.
+ From Equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:D4"
+
+\end_inset
+
+, we have
+\begin_inset Formula 
+\begin{gather}
+\frac{\partial\phantom{}^{t+\Delta t}S_{i}}{\partial\phantom{}^{t+\Delta t}e_{k}^{\prime}}=\frac{\delta_{ik}}{a_{E}+\frac{\alpha\Delta t}{2\eta}}\,\,.\label{eq:D12}
+\end{gather}
+
+\end_inset
+
+Using this, along with Equations 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:58"
+
+\end_inset
+
+, 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:59"
+
+\end_inset
+
+, and 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:60"
+
+\end_inset
+
+, the final material matrix relating stress and tensor strain is
+\begin_inset Formula 
+\begin{gather}
+C_{ij}^{VE}=\frac{1}{3a_{m}}\left[\begin{array}{cccccc}
+1 & 1 & 1 & 0 & 0 & 0\\
+1 & 1 & 1 & 0 & 0 & 0\\
+1 & 1 & 1 & 0 & 0 & 0\\
+0 & 0 & 0 & 0 & 0 & 0\\
+0 & 0 & 0 & 0 & 0 & 0\\
+0 & 0 & 0 & 0 & 0 & 0
+\end{array}\right]+\frac{1}{3\left(a_{E}+\frac{\alpha\Delta t}{2\eta}\right)}\left[\begin{array}{cccccc}
+2 & -1 & -1 & 0 & 0 & 0\\
+-1 & 2 & -1 & 0 & 0 & 0\\
+-1 & -1 & 2 & 0 & 0 & 0\\
+0 & 0 & 0 & 3 & 0 & 0\\
+0 & 0 & 0 & 0 & 3 & 0\\
+0 & 0 & 0 & 0 & 0 & 3
+\end{array}\right]\,.\label{eq:D13}
+\end{gather}
+
+\end_inset
+
+Note that the coefficient of the second matrix approaches 
+\begin_inset Formula $E/3(1+\nu)=1/3a_{E}$
+\end_inset
+
+ as 
+\begin_inset Formula $\eta$
+\end_inset
+
+ goes to infinity.
+ To check the results we make sure that the regular elastic constitutive
+ matrix is obtained for selected terms in the case where 
+\begin_inset Formula $\eta$
+\end_inset
+
+ goes to infinity.
+\begin_inset Formula 
+\begin{gather}
+C_{11}^{E}=\frac{E(1-\nu)}{(1+\nu)(1-2\nu)}\,\,\nonumber \\
+C_{12}^{E}=\frac{E\nu}{(1+\nu)(1-2\nu)}\,.\label{eq:D14}\\
+C_{44}^{E}=\frac{E}{1+\nu}\,\,\nonumber 
+\end{gather}
+
+\end_inset
+
+This is consistent with the regular elasticity matrix, and Equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:D13"
+
+\end_inset
+
+ should thus be used when forming the stiffness matrix.
+ We do not presently use this formulation, but it may be included in future
+ versions.
+\end_layout
+
+\end_body
+\end_document

Modified: short/3D/PyLith/branches/v1.7-trunk/doc/userguide/materials/materials.lyx
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/doc/userguide/materials/materials.lyx	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/doc/userguide/materials/materials.lyx	2012-08-21 18:24:25 UTC (rev 20616)
@@ -1,6177 +1,6161 @@
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-
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-
-\end_inset
-
-Material Models
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status collapsed
-
-\begin_layout Plain Layout
-
-
-\backslash
-raggedbottom
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-Specifying Material Properties
-\end_layout
-
-\begin_layout Standard
-Associating material properties with a given cell involves several steps.
- 
-\end_layout
-
-\begin_layout Enumerate
-In the mesh generation process, assign a material identifier to each cell.
-\end_layout
-
-\begin_layout Enumerate
-Define material property groups corresponding to each material identifier.
-\end_layout
-
-\begin_layout Enumerate
-Set the parameters for each material group using 
-\family typewriter
-.cfg
-\family default
- or 
-\family typewriter
-.pml
-\family default
- files and/or command-line arguments.
-\end_layout
-
-\begin_layout Enumerate
-Specify the spatial variation in material property parameters using a spatial
- database file.
-\end_layout
-
-\begin_layout Subsection
-Setting the Material Identifier
-\end_layout
-
-\begin_layout Standard
-Each cell in the finite-element mesh must have a material identifier.
- This integer value is associated with a bulk material model.
- The parameters of the material model need not be uniform for cells with
- the same material identifier.
- The bulk constitutive model and numerical integration (quadrature) scheme
- will, however, be the same for all cells with the same material identifier
- value.
- The material identifier is set during the mesh generation process.
- The procedure for assigning this integer value to a cell depends on the
- mesh generator.
- For example, in the PyLith mesh ASCII format, the identifiers are listed
- in the cells group using the material-id data; in CUBIT materials are defined
- using blocks; in LaGriT materials are defined by the attribute 
-\family typewriter
-imt1
-\family default
- and the mregion command.
-\end_layout
-
-\begin_layout Subsection
-Material Property Groups
-\end_layout
-
-\begin_layout Standard
-The material property group associates a material model (label for the material,
- a bulk constitutive model, and parameters for the constitutive model) with
- a material identifier.
- In previous versions of PyLith it was necessary to specify containers that
- defined the number of groups and associated information for each group.
- This was necessary because previous versions of Pyre did not support dynamic
- arrays of components, and it was necessary to predefine these arrays.
- More recent versions of Pythia do support this, however, and it is now
- possible to define material property groups using a 
-\family typewriter
-.cfg
-\family default
- file, a 
-\family typewriter
-.pml
-\family default
- file, or on the command-line.
- User-defined containers are no longer necessary, and the predefined containers
- are no longer available (or necessary).
- If a set of material groups is not specified, a single material model is
- used for the entire problem.
- See Sections 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "sec:Tutorial-3d-hex8"
-
-\end_inset
-
- and 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "sec:Tutorial-3d-tet4"
-
-\end_inset
-
-for examples 
-\family typewriter
-that
-\family default
- demonstrate how to specify more than one material model.
-\end_layout
-
-\begin_layout Subsection
-\begin_inset CommandInset label
-LatexCommand label
-name "sec:material:parameters"
-
-\end_inset
-
-Material Parameters
-\end_layout
-
-\begin_layout Standard
-For each material group, there is a single component defining the material
- model to be used.
- The default material model is 
-\family typewriter
-elasticisotropic3d
-\family default
-.
- For each material model, the available components are:
-\end_layout
-
-\begin_layout Description
-db_properties Spatial database specifying the spatial variation in the parameter
-s of the bulk constitutive model (default is a SimpleDB).
-\end_layout
-
-\begin_layout Description
-db_initial_state Spatial database specifying the spatial variation in the
- initial variables (default is none).
-\end_layout
-
-\begin_layout Description
-output The output manager used for outputting material information.
-\end_layout
-
-\begin_layout Description
-quadrature Numerical integration scheme used in integrating fields over
- each cell.
-\end_layout
-
-\begin_layout Standard
-The properties for each material group are:
-\end_layout
-
-\begin_layout Description
-id This is the material identifier that matches the integer value assigned
- to each cell in the mesh generation process.
-\end_layout
-
-\begin_layout Description
-label Name or label for the material.
- This is used in error and diagnostic reports.
-\end_layout
-
-\begin_layout Standard
-An example of setting these parameters in a 
-\family typewriter
-.cfg
-\family default
- file for a problem with two material groups is:
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.timedependent]
-\end_layout
-
-\begin_layout LyX-Code
-materials = [elastic,viscoelastic]
-\end_layout
-
-\begin_layout LyX-Code
- 
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.timedependent.materials.elastic]
-\end_layout
-
-\begin_layout LyX-Code
-label = Elastic material
-\end_layout
-
-\begin_layout LyX-Code
-id = 1
-\end_layout
-
-\begin_layout LyX-Code
-db_properties.iohandler.filename = mat_elastic.spatialdb
-\end_layout
-
-\begin_layout LyX-Code
-quadrature.cell = pylith.feassemble.FIATLagrange
-\end_layout
-
-\begin_layout LyX-Code
-quadrature.cell.dimension = 3
-\end_layout
-
-\begin_layout LyX-Code
- 
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.timedependent.materials.viscoelastic]
-\end_layout
-
-\begin_layout LyX-Code
-label = Viscoelastic material
-\end_layout
-
-\begin_layout LyX-Code
-id = 2
-\end_layout
-
-\begin_layout LyX-Code
-db_properties.iohandler.filename = mat_viscoelastic.spatialdb
-\end_layout
-
-\begin_layout LyX-Code
-quadrature.cell = pylith.feassemble.FIATLagrange
-\end_layout
-
-\begin_layout LyX-Code
-quadrature.cell.dimension = 3
-\end_layout
-
-\begin_layout Standard
-These settings correspond to the the problem in 
-\family typewriter
-Section 
-\family default
-
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "sec:Tutorial-3d-hex8"
-
-\end_inset
-
-.
- The parameters for the bulk constitutive models are specified using the
- spatial databases 
-\family typewriter
-mat_elastic.spatialdb
-\family default
- and 
-\family typewriter
-mat_viscoelastic.spatialdb
-\family default
-.
- Refer to the discussion of each material model to find the parameters that
- must be specified in the spatial database.
- Appendix 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "sec:Spatialdata:SimpleIOAscii"
-
-\end_inset
-
- describes the format of the SimpleDB spatial database files.
- In a more realistic problem, a different spatial database, and possibly
- a different material model, would be used for each material group.
-\end_layout
-
-\begin_layout Standard
-By default the output manager for a material does not output any information.
- This behavior may be changed by specifying output files for each material
- group:
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.timedependent.materials.elastic.output]
-\end_layout
-
-\begin_layout LyX-Code
-cell_filter = pylith.meshio.CellFilterAvgMesh
-\end_layout
-
-\begin_layout LyX-Code
-writer.filename = dislocation-elastic.vtk
-\end_layout
-
-\begin_layout LyX-Code
- 
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.timedependent.materials.viscoelastic.output]
-\end_layout
-
-\begin_layout LyX-Code
-cell_filter = pylith.meshio.CellFilterAvgMesh
-\end_layout
-
-\begin_layout LyX-Code
-writer.filename = dislocation-viscoelastic.vtk
-\end_layout
-
-\begin_layout Standard
-These settings again correspond to the problem in 
-\family typewriter
-Section 
-\family default
-
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "sec:Tutorial-3d-hex8"
-
-\end_inset
-
-.
- The specification of a state variable base filename (
-\family typewriter
-writer.filename
-\family default
- settings) will cause two files to be created for each material group: an
- info file, which describes the material properties used in the model, and
- a state variables file, which contains the state variable information.
- Note that the material properties described by the info file are the properties
- used internally by PyLith.
- They do not necessarily correspond to properties specified by the user.
- If the problem has more than one time step, a state variable output file
- will be created for each requested time step.
- We have requested that the values be averaged over each cell.
- Otherwise, output would be produced for each quadrature point, which can
- cause problems with some visualization packages.
- For this example problem, the material is three-dimensional isotropic elastic,
- and is thus described by only three material properties (
-\begin_inset Formula $\lambda$
-\end_inset
-
-, 
-\begin_inset Formula $\mu$
-\end_inset
-
-, 
-\begin_inset Formula $\rho$
-\end_inset
-
-), as described below.
- These properties are output by default.
- Other material models require additional parameters, and if users want
- these to be output, they must be specified.
- Similarly, other material models require state variables in addition to
- the default stress and strain variables that are used by all material models.
- Additional output may be requested for a material model, as in this example
- (
-\family typewriter
-see Section 
-\family default
-
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "sec:Tutorial-Two-hexahedra"
-
-\end_inset
-
-):
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.timedependent.materials.material.output]
-\end_layout
-
-\begin_layout LyX-Code
-cell_data_fields = [total_strain,viscous_strain,stress]
-\end_layout
-
-\begin_layout LyX-Code
-cell_info_fields = [mu,lambda,density,maxwell_time]
-\end_layout
-
-\begin_layout Standard
-The properties and state variables available for output in each material
- model are listed in Table 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "tab:material-model-output"
-
-\end_inset
-
-.
- The order of the state variables in the output arrays is given in Table
- 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "tab:material-model-statevars"
-
-\end_inset
-
-.
- For the generalized Maxwell model, values of 
-\family typewriter
-shear_ratio
-\family default
- and 
-\family typewriter
-maxwell_time
-\family default
- are given for each Maxwell element in the model (there are presently three,
- as described below).
- Similarly, there are three sets of 
-\family typewriter
-viscous_strain
-\family default
- values for the generalized Maxwell model.
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float table
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\align center
-\begin_inset Caption
-
-\begin_layout Plain Layout
-\begin_inset CommandInset label
-LatexCommand label
-name "tab:material-model-output"
-
-\end_inset
-
-Properties and state variables available for output for existing material
- models.
- Physical properties are available for output as 
-\family typewriter
-cell_info_fields
-\family default
- and state variables are available for output as 
-\family typewriter
-cell_data_fields
-\family default
-.
-\end_layout
-
-\end_inset
-
-
-\begin_inset Tabular
-<lyxtabular version="3" rows="6" columns="4">
-<features tabularvalignment="middle">
-<column alignment="center" valignment="top" width="1.5in">
-<column alignment="center" valignment="top" width="1.8in">
-<column alignment="center" valignment="top" width="1.5in">
-<column alignment="center" valignment="top" width="1in">
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Model
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Physical Properties
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-State Variables
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Requires nonlinear solver?
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Elastic
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-mu, lambda, density
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-total_strain, stress
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-No
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Maxwell Viscoelastic
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-mu, lambda, density, maxwell_time
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-total_strain,
-\end_layout
-
-\begin_layout Plain Layout
-
-\family typewriter
-stress, viscous_strain
-\end_layout
-
-\end_inset
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-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
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-\begin_layout Plain Layout
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-
-\end_inset
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-\begin_inset Text
-
-\begin_layout Plain Layout
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-\end_layout
-
-\end_inset
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-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-mu, lambda, density,
-\end_layout
-
-\begin_layout Plain Layout
-
-\family typewriter
-shear_ratio,
-\end_layout
-
-\begin_layout Plain Layout
-
-\family typewriter
-maxwell_time
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-total_strain, stress, viscous_strain_1,
-\begin_inset Newline newline
-\end_inset
-
-viscous_strain_2,
-\begin_inset Newline newline
-\end_inset
-
-viscous_strain_3
-\end_layout
-
-\end_inset
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-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-No
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-
-\end_inset
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-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Power-law Viscoelastic
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-mu, lambda, density,
-\end_layout
-
-\begin_layout Plain Layout
-
-\family typewriter
-reference_strain_rate, reference_stress,
-\end_layout
-
-\begin_layout Plain Layout
-
-\family typewriter
-power_law_exponent
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-total_strain, stress, viscous_strain
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Yes
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Drucker-Prager Elastoplastic
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-mu, lambda, density, 
-\family default
-alpha_yield,
-\family typewriter
- beta, alpha_flow 
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-total_strain, stress, plastic_strain
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Yes
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\begin_inset Float table
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\align center
-\begin_inset Caption
-
-\begin_layout Plain Layout
-\begin_inset CommandInset label
-LatexCommand label
-name "tab:material-model-statevars"
-
-\end_inset
-
-Order of components in tensor state-variables for material models.
-\end_layout
-
-\end_inset
-
-
-\begin_inset Tabular
-<lyxtabular version="3" rows="5" columns="4">
-<features tabularvalignment="middle">
-<column alignment="center" valignment="top" width="1.25in">
-<column alignment="center" valignment="top" width="0.5in">
-<column alignment="center" valignment="top" width="1.25in">
-<column alignment="center" valignment="top" width="2.25in">
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-State Variable
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-1D
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
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-\begin_layout Plain Layout
-
-\series bold
-2D
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-3D
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-total_strain
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\epsilon_{xx}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\epsilon_{xx}$
-\end_inset
-
-, 
-\begin_inset Formula $\epsilon_{yy}$
-\end_inset
-
-, 
-\begin_inset Formula $\epsilon_{xy}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\epsilon_{xx}$
-\end_inset
-
-, 
-\begin_inset Formula $\epsilon_{yy}$
-\end_inset
-
-, 
-\begin_inset Formula $\epsilon_{zz}$
-\end_inset
-
-, 
-\begin_inset Formula $\epsilon_{xy}$
-\end_inset
-
-, 
-\begin_inset Formula $\epsilon_{yz}$
-\end_inset
-
-, 
-\begin_inset Formula $\epsilon_{xz}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-stress
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\sigma_{xx}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
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-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\sigma_{xx}$
-\end_inset
-
-, 
-\begin_inset Formula $\sigma_{yy}$
-\end_inset
-
-, 
-\begin_inset Formula $\sigma_{xy}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\sigma_{xx}$
-\end_inset
-
-, 
-\begin_inset Formula $\sigma_{yy}$
-\end_inset
-
-, 
-\begin_inset Formula $\sigma_{zz}$
-\end_inset
-
-, 
-\begin_inset Formula $\sigma_{xy}$
-\end_inset
-
-, 
-\begin_inset Formula $\sigma_{yz}$
-\end_inset
-
-, 
-\begin_inset Formula $\sigma_{xz}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-viscous_strain, plastic_strain
-\end_layout
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-\end_inset
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-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
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-\begin_layout Plain Layout
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-\begin_inset Formula $\epsilon_{xx}$
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-
-, 
-\begin_inset Formula $\epsilon_{yy}$
-\end_inset
-
-, 
-\begin_inset Formula $\epsilon_{zz}$
-\end_inset
-
-, 
-\begin_inset Formula $\epsilon_{xy}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\epsilon_{xx}$
-\end_inset
-
-, 
-\begin_inset Formula $\epsilon_{yy}$
-\end_inset
-
-, 
-\begin_inset Formula $\epsilon_{zz}$
-\end_inset
-
-, 
-\begin_inset Formula $\epsilon_{xy}$
-\end_inset
-
-, 
-\begin_inset Formula $\epsilon_{yz}$
-\end_inset
-
-, 
-\begin_inset Formula $\epsilon_{xz}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-stress4
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\sigma_{xx}$
-\end_inset
-
-, 
-\begin_inset Formula $\sigma_{yy}$
-\end_inset
-
-, 
-\begin_inset Formula $\sigma_{zz}$
-\end_inset
-
-, 
-\begin_inset Formula $\sigma_{xy}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\begin_inset Float table
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\align center
-\begin_inset Caption
-
-\begin_layout Plain Layout
-\begin_inset CommandInset label
-LatexCommand label
-name "tab:material:output:components"
-
-\end_inset
-
-Order of components in tensor values in Xdmf files for 3-D material models.
- All nine tensor components are available even though the tensor is symmetric
- with six independent components.
-\end_layout
-
-\end_inset
-
-
-\begin_inset Tabular
-<lyxtabular version="3" rows="7" columns="2">
-<features tabularvalignment="middle">
-<column alignment="center" valignment="top" width="1.25in">
-<column alignment="center" valignment="top" width="2.25in">
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Index
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Component
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-0
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\epsilon_{xx}$
-\end_inset
-
-, 
-\begin_inset Formula $\sigma_{xx}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-1, 3
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\epsilon_{xy}$
-\end_inset
-
-, 
-\begin_inset Formula $\sigma_{xy}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-2, 6
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\epsilon_{xz}$
-\end_inset
-
-, 
-\begin_inset Formula $\sigma_{xz}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-4
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\epsilon_{yy}$
-\end_inset
-
-, 
-\begin_inset Formula $\sigma_{yy}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-5, 7
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\epsilon_{yz}$
-\end_inset
-
-, 
-\begin_inset Formula $\sigma_{yz}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-8
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\epsilon_{zz}$
-\end_inset
-
-, 
-\begin_inset Formula $\sigma_{zz}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-Elastic Material Models
-\end_layout
-
-\begin_layout Standard
-The generalized form of Hooke's law relating stress and strain for linear
- elastic materials is
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula 
-\begin{gather}
-\sigma_{ij}=C_{ijkl}\left(\epsilon_{kl}-\epsilon_{kl}^{I}\right)+\sigma_{ij}^{I}\,,\label{eq:1}
-\end{gather}
-
-\end_inset
-
-where we have included both initial strains and initial stresses, denoted
- with the superscript 
-\shape slanted
-I
-\shape default
-.
- Due to symmetry considerations, however, the 81 components of the elasticity
- matrix are reduced to 21 independent components for the most general case
- of anisotropic elasticity.
- Representing the stress and strain in terms of vectors, the constitutive
- relation may be written
-\begin_inset Formula 
-\begin{gather}
-\overrightarrow{\sigma}=\underline{C}\left(\vec{\epsilon}-\vec{\epsilon}^{I}\right)+\vec{\sigma}^{I},\label{eq:2}
-\end{gather}
-
-\end_inset
-
-where
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula 
-\begin{gather}
-\underline{C}=\left[\begin{array}{cccccc}
-C_{1111} & C_{1122} & C_{1133} & C_{1112} & C_{1123} & C_{1113}\\
-C_{1122} & C_{2222} & C_{2233} & C_{2212} & C_{2223} & C_{2213}\\
-C_{1133} & C_{2233} & C_{3333} & C_{3312} & C_{3323} & C_{3313}\\
-C_{1112} & C_{2212} & C_{3312} & C_{1212} & C_{1223} & C_{1213}\\
-C_{1123} & C_{2223} & C_{3323} & C_{1223} & C_{2323} & C_{2313}\\
-C_{1113} & C_{2213} & C_{3313} & C_{1213} & C_{2313} & C_{1313}
-\end{array}\right]\:.\label{eq:3}
-\end{gather}
-
-\end_inset
-
-For the case of isotropic elasticity, the number of independent components
- reduces to two, and the model can be characterized by two parameters, Lame's
- constants 
-\begin_inset Formula $\mu$
-\end_inset
-
- and 
-\begin_inset Formula $\lambda$
-\end_inset
-
-.
- Lame's constants are related to the density (
-\begin_inset Formula $\rho$
-\end_inset
-
-), shear wave speed (
-\begin_inset Formula $v_{s}$
-\end_inset
-
-), and compressional wave speed (
-\begin_inset Formula $v_{p}$
-\end_inset
-
-) via
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula 
-\begin{equation}
-\begin{aligned}\mu= & \rho v_{s}^{2}\\
-\lambda= & \rho v_{p}^{2}-2\mu
-\end{aligned}
-\label{eq:4}
-\end{equation}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float table
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\noindent
-\align center
-\begin_inset Caption
-
-\begin_layout Plain Layout
-Values in spatial database used as parameters in the elastic material constituti
-ve models.
-\end_layout
-
-\end_inset
-
-
-\begin_inset Tabular
-<lyxtabular version="3" rows="4" columns="2">
-<features tabularvalignment="middle">
-<column alignment="center" valignment="middle" width="0.85in">
-<column alignment="center" valignment="middle" width="2.47in">
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Parameter
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\align center
-
-\series bold
-Name in Spatial Database
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $v_{p}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-Vp
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $v_{s}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-Vs
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\rho$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-density
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsection
-1D Elastic Material Models
-\end_layout
-
-\begin_layout Standard
-In 1D we can write Hooke's law as 
-\begin_inset Formula $\sigma_{11}=C_{1111}\left(\epsilon_{11}-\epsilon_{11}^{I}\right)+\sigma_{11}^{I}$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Subsubsection
-Elastic 
-\begin_inset Quotes eld
-\end_inset
-
-Linear
-\begin_inset Quotes erd
-\end_inset
-
- Strain
-\end_layout
-
-\begin_layout Standard
-For purely 1D axial deformation 
-\begin_inset Formula $C_{1111}=\lambda+2\mu$
-\end_inset
-
-, so we have
-\begin_inset Formula 
-\begin{equation}
-\sigma_{11}=(\lambda+2\mu)\left(\epsilon_{11}-\epsilon_{11}^{I}\right)+\sigma_{11}^{I},\label{eq:5}
-\end{equation}
-
-\end_inset
-
-with
-\begin_inset Formula 
-\begin{gather}
-\sigma_{22}=\sigma_{33}=\lambda\left(\epsilon_{11}-\epsilon_{11}^{I}\right)+\sigma_{22}^{I},\nonumber \\
-\sigma_{12}=\sigma_{23}=\sigma_{13}=0.\label{eq:6}
-\end{gather}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsubsection
-Elastic 
-\begin_inset Quotes eld
-\end_inset
-
-Linear
-\begin_inset Quotes erd
-\end_inset
-
- Stress
-\end_layout
-
-\begin_layout Standard
-For deformation where the tractions are confined to the axial direction,
- 
-\begin_inset Formula $C_{1111}=\frac{\mu(3\lambda+2\mu)}{\lambda+\mu}$
-\end_inset
-
-, so we have
-\begin_inset Formula 
-\begin{equation}
-\sigma_{11}=\frac{\mu(3\lambda+2\mu)}{\lambda+\mu}\left(\epsilon_{11}-\epsilon_{11}^{I}\right)+\sigma_{11}^{I},\label{eq:7}
-\end{equation}
-
-\end_inset
-
-with
-\begin_inset Formula 
-\begin{gather}
-\epsilon_{22}=\epsilon_{33}=-\frac{\lambda}{2(\lambda+\mu)}\epsilon_{11}+\epsilon_{22}^{I},\label{eq:8}\\
-\sigma_{22}=\sigma_{33}=\sigma_{12}=\sigma_{23}=\sigma_{13}=0.\nonumber 
-\end{gather}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsection
-2D Elastic Material Models
-\end_layout
-
-\begin_layout Standard
-In 2D we can write Hooke's law as
-\begin_inset Formula 
-\begin{gather}
-\left[\begin{array}{c}
-\sigma_{11}\\
-\sigma_{22}\\
-\sigma_{12}
-\end{array}\right]=\left[\begin{array}{ccc}
-C_{1111} & C_{1122} & C_{1112}\\
-C_{1122} & C_{2222} & C_{2212}\\
-C_{1112} & C_{2212} & C_{1212}
-\end{array}\right]\left[\begin{array}{c}
-\epsilon_{11}-\epsilon_{11}^{I}\\
-\epsilon_{22}-\epsilon_{22}^{I}\\
-\epsilon_{12}-\epsilon_{12}^{I}
-\end{array}\right]+\left[\begin{array}{c}
-\sigma_{11}^{I}\\
-\sigma_{22}^{I}\\
-\sigma_{12}^{I}
-\end{array}\right]\:.\label{eq:9}
-\end{gather}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsubsection
-Elastic Plane Strain
-\end_layout
-
-\begin_layout Standard
-If the gradient in deformation with respect to the 
-\begin_inset Formula $x_{3}$
-\end_inset
-
- axis is zero, then 
-\begin_inset Formula $\epsilon_{33}=\epsilon_{13}=\epsilon_{23}=0$
-\end_inset
-
- and plane strain conditions apply, so we have 
-\begin_inset Formula 
-\begin{gather}
-\left[\begin{array}{c}
-\sigma_{11}\\
-\sigma_{22}\\
-\sigma_{12}
-\end{array}\right]=\left[\begin{array}{ccc}
-\lambda+2\mu & \lambda & 0\\
-\lambda & \lambda+2\mu & 0\\
-0 & 0 & 2\mu
-\end{array}\right]\left[\begin{array}{c}
-\epsilon_{11}-\epsilon_{11}^{I}\\
-\epsilon_{22}-\epsilon_{22}^{I}\\
-\epsilon_{12}-\epsilon_{12}^{I}
-\end{array}\right]+\left[\begin{array}{c}
-\sigma_{11}^{I}\\
-\sigma_{22}^{I}\\
-\sigma_{12}^{I}
-\end{array}\right]\:.\label{eq:10}
-\end{gather}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsubsection
-Elastic Plane Stress
-\end_layout
-
-\begin_layout Standard
-If the 
-\begin_inset Formula $x_{1}x_{2}$
-\end_inset
-
- plane is traction free, then 
-\begin_inset Formula $\sigma_{33}=\sigma_{13}=\sigma_{23}=0$
-\end_inset
-
- and plane stress conditions apply, so we have
-\begin_inset Formula 
-\begin{gather}
-\left[\begin{array}{c}
-\sigma_{11}\\
-\sigma_{22}\\
-\sigma_{12}
-\end{array}\right]=\left[\begin{array}{ccc}
-\frac{4\mu(\lambda+\mu)}{\lambda+2\mu} & \frac{2\mu\lambda}{\lambda+2\mu} & 0\\
-\frac{2\mu\lambda}{\lambda+2} & \frac{4\mu(\lambda+\mu)}{\lambda+2\mu} & 0\\
-0 & 0 & 2\mu
-\end{array}\right]\left[\begin{array}{c}
-\epsilon_{11}-\epsilon_{11}^{I}\\
-\epsilon_{22}-\epsilon_{22}^{I}\\
-\epsilon_{12}-\epsilon_{12}^{I}
-\end{array}\right]+\left[\begin{array}{c}
-\sigma_{11}^{I}\\
-\sigma_{22}^{I}\\
-\sigma_{12}^{I}
-\end{array}\right]\:,\label{eq:11}
-\end{gather}
-
-\end_inset
-
-where
-\begin_inset Formula 
-\begin{equation}
-\begin{gathered}\epsilon_{33}=-\frac{\lambda}{\lambda+2\mu}(\epsilon_{11}+\epsilon_{22})+\epsilon_{33}^{I}\\
-\epsilon_{13}=\epsilon_{23}=0\,.
-\end{gathered}
-\label{eq:12}
-\end{equation}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsection
-3D Elastic Material Models
-\end_layout
-
-\begin_layout Subsubsection
-Isotropic
-\end_layout
-
-\begin_layout Standard
-For this case the stress-strain matrix, 
-\begin_inset Formula $\underline{C}$
-\end_inset
-
-, becomes
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula 
-\begin{gather}
-\underline{C}=\left[\begin{array}{cccccc}
-\lambda+2\mu & \lambda & \lambda & 0 & 0 & 0\\
-\lambda & \lambda+2\mu & \lambda & 0 & 0 & 0\\
-\lambda & \lambda & \lambda+2\mu & 0 & 0 & 0\\
-0 & 0 & 0 & 2\mu & 0 & 0\\
-0 & 0 & 0 & 0 & 2\mu & 0\\
-0 & 0 & 0 & 0 & 0 & 2\mu
-\end{array}\right]\:.\label{eq:13}
-\end{gather}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-\begin_inset CommandInset label
-LatexCommand label
-name "sec:Viscoelastic-Materials"
-
-\end_inset
-
-Viscoelastic Materials
-\end_layout
-
-\begin_layout Standard
-At present, there are six viscoelastic material models available in PyLith
- (Table 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "tab:Viscoelastic-models-available"
-
-\end_inset
-
- and Figure 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "fig:Material-models"
-
-\end_inset
-
-).
- Future code versions may include alternative formulations for the various
- material models (Appendix 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "cha:Alternative-Formulations"
-
-\end_inset
-
-), so that users may use the most efficient formulation for a particular
- problem.
- Note that both 2D and 3D viscoelastic models are described, but we present
- below only the 3D formulations.
- The 2D formulations are easily obtained from the plane strain definition.
- The one aspect of the 2D formulations that is different is the specification
- of initial stresses.
- Since 2D models only have three tensor components, it is not possible to
- specify the normal stress in the out-of-plane direction (
-\begin_inset Formula $\sigma_{33}$
-\end_inset
-
-), which is generally nonzero, using the same method as the other tensor
- components.
- To allow for the specification of this initial stress component, an additional
- state variable corresponding to 
-\begin_inset Formula $\sigma_{33}^{I}$
-\end_inset
-
- is provided (
-\family typewriter
-stress_zz_initial
-\family default
-).
- Note that this is not an issue for the 2D elastic models, since this initial
- stress component is not needed.
- For the PowerLawPlaneStrain model, all four of the stress components are
- needed, so a 4-component stress state variable (
-\family typewriter
-stress4
-\family default
-) is provided in addition to the normal 3-component 
-\family typewriter
-stress
-\family default
- state variable (see Table 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "tab:material-model-output"
-
-\end_inset
-
-).
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float table
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\noindent
-\align center
-\begin_inset Caption
-
-\begin_layout Plain Layout
-\begin_inset CommandInset label
-LatexCommand label
-name "tab:Viscoelastic-models-available"
-
-\end_inset
-
-Available viscoelastic materials for PyLith.
-\end_layout
-
-\end_inset
-
-
-\begin_inset Tabular
-<lyxtabular version="3" rows="7" columns="2">
-<features tabularvalignment="middle">
-<column alignment="left" valignment="top" width="2.85in">
-<column alignment="center" valignment="top" width="2.47in">
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Model Name
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Description
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-MaxwellPlaneStrain
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Plane strain Maxwell material with linear viscous rheology
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-GenMaxwellPlaneStrain
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Plane strain generalized Maxwell material (3 Maxwell models in parallel)
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-PowerLawPlaneStrain
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Plane strain Maxwell material with power-law viscous rheology
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-MaxwellIsotropic3D
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Isotropic Maxwell material with linear viscous rheology
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-GenMaxwellIsotropic3D
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Generalized model consisting of 3 Maxwell models in parallel
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-PowerLaw3D
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Isotropic Maxwell material with power-law viscous rheology
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float figure
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\align center
-\begin_inset Graphics
-	filename figs/pylith-materials.eps
-	scale 75
-
-\end_inset
-
-
-\begin_inset Caption
-
-\begin_layout Plain Layout
-\begin_inset CommandInset label
-LatexCommand label
-name "fig:Material-models"
-
-\end_inset
-
-Spring-dashpot 1D representations of the available 3D elastic and 2D/3D
- viscoelastic material models for PyLith.
- The top model is a linear elastic model, the middle model is a Maxwell
- model, and the bottom model is a generalized Maxwell model.
- For the generalized Maxwell model, 
-\begin_inset Formula $\lambda$
-\end_inset
-
- and 
-\begin_inset Formula $\mu_{tot}$
-\end_inset
-
- are specified for the entire model, and then the ratio 
-\begin_inset Formula $\mu_{i}/\mu_{tot}$
-\end_inset
-
- is specified for each Maxwell model.
- For the power-law model, the linear dashpot in the Maxwell model is replaced
- by a nonlinear dashpot obeying a power-law.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsection
-Definitions
-\end_layout
-
-\begin_layout Standard
-In the following sections, we use a combination of vector and index notation
- (our notation conventions are shown in Table 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "tab:Mathematical-notation"
-
-\end_inset
-
-).
- When using index notation, we use the common convention where repeated
- indices indicate summation over the range of the index.
- We also make frequent use of the scalar inner product.
- The scalar inner product of two second-order tensors may be written
-\begin_inset Formula 
-\begin{gather}
-\underline{a}\cdot\underline{b}=a_{ij}b_{ij}\,.\label{eq:14}
-\end{gather}
-
-\end_inset
-
-Although the general constitutive relations are formulated in terms of the
- stress and strain, we frequently make use of the deviatoric stress and
- strain in our formulation.
- We first define the mean stress, 
-\begin_inset Formula $P$
-\end_inset
-
-, and mean strain, 
-\begin_inset Formula $\theta$
-\end_inset
-
-:
-\begin_inset Formula 
-\begin{gather}
-P=\frac{\sigma_{ii}}{3}\,,\,\,\,\,\theta=\frac{\epsilon_{ii}}{3}\,,\label{eq:15}
-\end{gather}
-
-\end_inset
-
-where the 
-\begin_inset Formula $\sigma_{ii}$
-\end_inset
-
- and 
-\begin_inset Formula $\epsilon_{ii}$
-\end_inset
-
- represent the trace of the stress and strain tensors, respectively.
- We then define the deviatoric components of stress and strain as
-\begin_inset Formula 
-\begin{gather}
-S_{ij}=\sigma_{ij}-P\delta_{ij}\,,\,\,\,\, e_{ij}=\epsilon_{ij}-\theta\delta_{ij}\,,\label{eq:16}
-\end{gather}
-
-\end_inset
-
-where 
-\begin_inset Formula $\delta_{ij}$
-\end_inset
-
- is the Kronecker delta.
- Using the deviatoric components, we define the effective stress, 
-\begin_inset Formula $\overline{\sigma}$
-\end_inset
-
-, the second deviatoric stress invariant, 
-\begin_inset Formula $J_{2}^{\prime}$
-\end_inset
-
-, the effective deviatoric strain, 
-\begin_inset Formula $\overline{e}$
-\end_inset
-
-, and the second deviatoric strain invariant, 
-\begin_inset Formula $L_{2}^{\prime}$
-\end_inset
-
-, as
-\begin_inset Formula 
-\begin{gather}
-\overline{\sigma}=\sqrt{\frac{3}{2}\underline{S}\cdot\underline{S}}\,\,\nonumber \\
-J_{2}^{\prime}=\frac{1}{2}\underline{S}\cdot\underline{S}\,.\label{eq:17}\\
-\overline{e}=\sqrt{\frac{2}{3}\underline{e}\cdot\underline{e}}\,\,\nonumber \\
-L_{2}^{\prime}=\frac{1}{2}\underline{e}\cdot\underline{e}\,\,\nonumber 
-\end{gather}
-
-\end_inset
-
-Due to the symmetry of the stress and strain tensors, it is sometimes convenient
- to represent them as vectors:
-\begin_inset Formula 
-\begin{gather}
-\overrightarrow{\sigma^{T}}=\left[\begin{array}{cccccc}
-\sigma_{11} & \sigma_{22} & \sigma_{33} & \sigma_{12} & \sigma_{23} & \sigma_{31}\end{array}\right]\label{eq:18}\\
-\overrightarrow{\epsilon^{T}}=\left[\begin{array}{cccccc}
-\epsilon_{11} & \epsilon_{22} & \epsilon_{33} & \epsilon_{12} & \epsilon_{23} & \epsilon_{31}\end{array}\right]\:.\nonumber 
-\end{gather}
-
-\end_inset
-
-Note that when taking the scalar inner product of two tensors represented
- as vectors, it is necessary to double the products representing off-diagonal
- terms.
-\end_layout
-
-\begin_layout Standard
-For quantities evaluated over a specific time period, we represent the initial
- time as a prefixed subscript and the end time as a prefixed superscript.
- In cases where the initial time does not appear, it is understood to be
- 
-\begin_inset Formula $-\infty$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float table
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\noindent
-\align center
-\begin_inset Caption
-
-\begin_layout Plain Layout
-\begin_inset CommandInset label
-LatexCommand label
-name "tab:Mathematical-notation"
-
-\end_inset
-
-Mathematical notation used in this section.
-\end_layout
-
-\end_inset
-
-
-\begin_inset Tabular
-<lyxtabular version="3" rows="3" columns="3">
-<features tabularvalignment="middle">
-<column alignment="center" valignment="top" width="0">
-<column alignment="center" valignment="top" width="0">
-<column alignment="center" valignment="top" width="0">
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Index notation
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Vector notation
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Description
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $a_{i}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\overrightarrow{a}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Vector field a
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $a_{ij}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\underline{a}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-Second order tensor field a
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsection
-Linear Viscoelastic Models
-\end_layout
-
-\begin_layout Standard
-Linear viscoelastic models are obtained by various combinations of a linear
- elastic spring and a linear viscous dashpot in series or parallel.
- The simplest example is probably the linear Maxwell model, which consists
- of a spring in series with a dashpot, as shown in Figure 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "fig:Material-models"
-
-\end_inset
-
-.
- For a one-dimensional model, the response is given by
-\begin_inset Formula 
-\begin{equation}
-\frac{d\epsilon_{Total}}{dt}=\frac{d\epsilon_{D}}{dt}+\frac{d\epsilon_{S}}{dt}=\frac{\sigma}{\eta}+\frac{1}{E}\frac{d\sigma}{dt}\:,
-\end{equation}
-
-\end_inset
-
-where 
-\begin_inset Formula $\epsilon_{Total}$
-\end_inset
-
- is the total strain, 
-\begin_inset Formula $\epsilon_{D}$
-\end_inset
-
- is the strain in the dashpot, 
-\begin_inset Formula $\epsilon_{S}$
-\end_inset
-
- is the strain in the spring, 
-\begin_inset Formula $\sigma$
-\end_inset
-
- is the stress, 
-\begin_inset Formula $\eta$
-\end_inset
-
- is the viscosity of the dashpot, and 
-\begin_inset Formula $E$
-\end_inset
-
- is the spring constant.
- When a Maxwell material is subjected to constant strain, the stresses relax
- exponentially with time.
- When a Maxwell material is subjected to a constant stress, there is an
- immediate elastic strain, corresponding to the response of the spring,
- and a viscous strain that increases linearly with time.
- Since the strain response is unbounded, the Maxwell model actually represents
- a fluid.
-\end_layout
-
-\begin_layout Standard
-Another simple model is the Kelvin-Voigt model, which consists of a spring
- in parallel with a dashpot.
- In this case, the one-dimensional response is given by
-\begin_inset Formula 
-\begin{equation}
-\sigma\left(t\right)=E\epsilon\left(t\right)+\eta\frac{d\epsilon\left(t\right)}{dt}\:.
-\end{equation}
-
-\end_inset
-
-As opposed to the Maxwell model, which represents a fluid, the Kelvin-Voigt
- model represents a solid undergoing reversible, viscoelastic strain.
- If the material is subjected to a constant stress, it deforms at a decreasing
- rate, gradually approaching the strain that would occur for a purely elastic
- material.
- When the stress is released, the material gradually relaxes back to its
- undeformed state.
-\end_layout
-
-\begin_layout Standard
-The most general form of linear viscoelastic model is the generalized Maxwell
- model, which consists of a spring in parallel with a number of Maxwell
- models (see Figure 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "fig:Material-models"
-
-\end_inset
-
-).
- Using this model, it is possible to represent a number of simpler viscoelastic
- models.
- For example, a simple Maxwell model is obtained by setting the elastic
- constants of all springs to zero, with the exception of the spring contained
- in the first Maxwell model (
-\begin_inset Formula $\mu_{1}$
-\end_inset
-
-).
- Similarly, the Kelvin-Voigt model may be obtained by setting the elastic
- constants 
-\begin_inset Formula $\mu_{2}=\mu_{3}=0$
-\end_inset
-
-, and setting 
-\begin_inset Formula $\mu_{1}=\infty$
-\end_inset
-
- (or a very large number).
-\end_layout
-
-\begin_layout Subsection
-Formulation for Generalized Maxwell Models
-\begin_inset CommandInset label
-LatexCommand label
-name "sub:Formulation-for-Gen-Max"
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-As described above, the generalized Maxwell viscoelastic model consists
- of a number of Maxwell linear viscoelastic models in parallel with a spring,
- as shown in Figure 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "fig:Material-models"
-
-\end_inset
-
-.
- PyLith includes the specific case of a spring in parallel with three Maxwell
- models.
- As described in the previous paragraph, a number of common material models
- may be obtained from this model by setting the shear moduli of various
- springs to zero or infinity (or a large number), such as the Maxwell model,
- the Kelvin model, and the standard linear solid.
- We follow formulations similar to those used by Zienkiewicz and Taylor
- 
-\begin_inset CommandInset citation
-LatexCommand cite
-key "Zienkiewicz:Taylor:2000"
-
-\end_inset
-
- and Taylor 
-\begin_inset CommandInset citation
-LatexCommand cite
-key "Taylor:2003"
-
-\end_inset
-
-.
- In this formulation, we specify the total shear modulus of the model (
-\begin_inset Formula $\mu_{tot}$
-\end_inset
-
-) and Lame's constant (
-\begin_inset Formula $\lambda$
-\end_inset
-
-).
- We then provide the fractional shear modulus for each Maxwell element spring
- in the model.
- It is not necessary to specify the fractional modulus for 
-\begin_inset Formula $\mu_{0}$
-\end_inset
-
-, since this is obtained by subtracting the sum of the other ratios from
- 1.
- Note that the sum of all these fractions must equal 1.
- We use a similar formulation for our linear Maxwell viscoelastic model,
- but in that case 
-\begin_inset Formula $\mu_{0}$
-\end_inset
-
- is always zero and we only use a single Maxwell model.
- The parameters defining the standard Maxwell model are shown in Table 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "tab:linearMaxwell"
-
-\end_inset
-
-, and those defining the generalized Maxwell model are shown in Table 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "tab:genMaxwell"
-
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-As for all our viscoelastic models, the volumetric strain is completely
- elastic, and the viscoelastic deformation may be expressed purely in terms
- of the deviatoric components:
-\begin_inset Formula 
-\begin{equation}
-\underline{S}=2\mu_{tot}\left[\mu_{0}\underline{e}+\sum_{i=1}^{N}\mu_{i}\underline{q}^{i}-\underline{e}^{I}\right]+\underline{S}^{I}\,;\; P=3K\left(\theta-\theta^{I}\right)+P^{I}\,,\label{eq:19}
-\end{equation}
-
-\end_inset
-
-where 
-\shape slanted
-K
-\shape default
- is the bulk modulus, 
-\begin_inset Formula $N$
-\end_inset
-
- is the number of Maxwell models, and the variable 
-\begin_inset Formula $\underline{q}^{i}$
-\end_inset
-
- follows the evolution equations
-\begin_inset Formula 
-\begin{equation}
-\underline{\dot{q}}^{i}+\frac{1}{\lambda_{i}}\underline{q}^{i}=\underline{\dot{e}}.\label{eq:20}
-\end{equation}
-
-\end_inset
-
-The 
-\begin_inset Formula $\lambda_{i}$
-\end_inset
-
- are the relaxation times for each Maxwell model.
-\end_layout
-
-\begin_layout Standard
-An alternative to the differential equation form above is an integral equation
- form expressed in terms of the relaxation modulus function.
- This function is defined in terms of an idealized experiment in which,
- at time labeled zero (
-\begin_inset Formula $t=0$
-\end_inset
-
-), a specimen is subjected to a constant strain, 
-\begin_inset Formula $\underline{e}_{0}$
-\end_inset
-
-, and the stress response, 
-\begin_inset Formula $\underline{S}\left(t\right)$
-\end_inset
-
-, is measured.
- For a linear material we obtain:
-\begin_inset Formula 
-\begin{equation}
-\underline{S}\left(t\right)=2\mu\left(t\right)\left(\underline{e}_{0}-\underline{e}^{I}\right)+\underline{S}^{I}\,,\label{eq:21}
-\end{equation}
-
-\end_inset
-
-where 
-\begin_inset Formula $\mu\left(t\right)$
-\end_inset
-
- is the shear relaxation modulus function.
- Using linearity and superposition for an arbitrary state of strain yields
- an integral equation:
-\begin_inset Formula 
-\begin{equation}
-\underline{S}\left(t\right)=\intop_{-\infty}^{t}\mu\left(t-\tau\right)\underline{\dot{e}}\, d\tau\,.\label{eq:22}
-\end{equation}
-
-\end_inset
-
-If we assume the modulus function in Prony series form we obtain
-\begin_inset Formula 
-\begin{equation}
-\mu\left(t\right)=\mu_{tot}\left(\mu_{0}+\sum_{i=1}^{N}\mu_{i}\exp\frac{-t}{\lambda_{i}}\right)\,,\label{eq:23}
-\end{equation}
-
-\end_inset
-
-where
-\begin_inset Formula 
-\begin{equation}
-\mu_{0}+\sum_{i=1}^{N}\mu_{i}=1\,.\label{eq:24}
-\end{equation}
-
-\end_inset
-
-With the form in Equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:23"
-
-\end_inset
-
-, the integral equation form is identical to the differential equation form.
-\end_layout
-
-\begin_layout Standard
-If we assume the material is undisturbed until a strain is suddenly applied
- at time zero, we can divide the integral into
-\begin_inset Formula 
-\begin{equation}
-\intop_{-\infty}^{t}\left(\cdot\right)\, d\tau=\intop_{-\infty}^{0^{-}}\left(\cdot\right)\, d\tau+\intop_{0^{-}}^{0^{+}}\left(\cdot\right)\, d\tau+\intop_{0^{+}}^{t}\left(\cdot\right)\, d\tau\,.\label{eq:27}
-\end{equation}
-
-\end_inset
-
-The first term is zero, the second term includes a jump term associated
- with 
-\begin_inset Formula $\underline{e}_{0}$
-\end_inset
-
- at time zero, and the last term covers the subsequent history of strain.
- Applying this separation to Equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:22"
-
-\end_inset
-
-,
-\begin_inset Formula 
-\begin{equation}
-\underline{S}\left(t\right)=2\mu\left(t\right)\left(\underline{e}_{0}-\underline{e}^{I}\right)+\underline{S}^{I}+2\int_{0}^{t}\mu\left(t-\tau\right)\underline{\dot{e}}\left(\tau\right)\, d\tau\,,\label{eq:28}
-\end{equation}
-
-\end_inset
-
-where we have left the sign off of the lower limit on the integral.
-\end_layout
-
-\begin_layout Standard
-Substituting Equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:23"
-
-\end_inset
-
- into 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:28"
-
-\end_inset
-
-, we obtain
-\begin_inset Formula 
-\begin{equation}
-\underline{S}\left(t\right)=2\mu_{tot}\left\{ \mu_{0}\underline{e}\left(t\right)+\sum_{i=1}^{N}\left[\mu_{i}\exp\frac{-t}{\lambda_{i}}\left(\underline{e}_{0}+\intop_{0}^{t}\exp\frac{t}{\lambda_{i}}\underline{\dot{e}}\left(\tau\right)\, d\tau\right)\right]-\underline{e}^{I}\right\} +\underline{S}^{I}\,.\label{eq:29}
-\end{equation}
-
-\end_inset
-
-We then split each integral into two ranges: from 0 to 
-\begin_inset Formula $t_{n}$
-\end_inset
-
-, and from 
-\begin_inset Formula $t_{n}$
-\end_inset
-
- to 
-\begin_inset Formula $t$
-\end_inset
-
-, and define each integral as
-\begin_inset Formula 
-\begin{equation}
-\underline{i}_{i}^{1}\left(t\right)=\intop_{0}^{t}\exp\frac{\tau}{\lambda_{i}}\underline{\dot{e}}\left(\tau\right)\, d\tau\,.\label{eq:30}
-\end{equation}
-
-\end_inset
-
-The integral then becomes
-\begin_inset Formula 
-\begin{equation}
-\underline{i}_{i}^{1}\left(t\right)=\underline{i}_{i}^{1}\left(t_{n}\right)+\intop_{t_{n}}^{t}\exp\frac{\tau}{\lambda_{i}}\underline{\dot{e}}\left(\tau\right)\, d\tau\,.\label{eq:31}
-\end{equation}
-
-\end_inset
-
-Including the negative exponential multiplier:
-\begin_inset Formula 
-\begin{equation}
-\underline{h}_{i}^{1}\left(t\right)=\exp\frac{-t}{\lambda_{i}}\underline{i}_{i}^{1}\,.\label{eq:32}
-\end{equation}
-
-\end_inset
-
-Then
-\begin_inset Formula 
-\begin{equation}
-\underline{h}_{i}^{1}\left(t\right)=\exp\frac{-\Delta t}{\lambda_{i}}\underline{h}_{i}^{1}\left(t_{n}\right)+\Delta\underline{h}_{i}\,,\label{eq:33}
-\end{equation}
-
-\end_inset
-
-where
-\begin_inset Formula 
-\begin{equation}
-\Delta\underline{h}_{i}=\exp\frac{-t}{\lambda_{i}}\intop_{t_{n}}^{t}\exp\frac{\tau}{\lambda_{i}}\underline{\dot{e}}\left(\tau\right)\, d\tau\,.\label{eq:34}
-\end{equation}
-
-\end_inset
-
-Approximating the strain rate as constant over each time step, the solution
- may be found as
-\begin_inset Formula 
-\begin{equation}
-\Delta\underline{h}_{i}=\frac{\lambda_{i}}{\Delta t}\left(1-\exp\frac{-\Delta t}{\lambda_{i}}\right)\left(\underline{e}-\underline{e}_{n}\right)=\Delta h_{i}\left(\underline{e}-\underline{e}_{n}\right)\,.\label{eq:35}
-\end{equation}
-
-\end_inset
-
-The approximation is singular for zero time steps, but a series expansion
- may be used for small time-step sizes:
-\begin_inset Formula 
-\begin{equation}
-\Delta h_{i}\approx1-\frac{1}{2}\left(\frac{\Delta t}{\lambda_{i}}\right)+\frac{1}{3!}\left(\frac{\Delta t}{\lambda_{i}}\right)^{2}-\frac{1}{4!}\left(\frac{\Delta t}{\lambda_{i}}\right)^{3}+\cdots\,.\label{eq:36}
-\end{equation}
-
-\end_inset
-
-This converges with only a few terms.
- With this formulation, the constitutive relation now has the simple form:
-\begin_inset Formula 
-\begin{equation}
-\underline{S}\left(t\right)=2\mu_{tot}\left(\mu_{0}\underline{e}\left(t\right)+\sum_{i=1}^{N}\mu_{i}\underline{h}_{i}^{1}\left(t\right)-\underline{e}^{I}\right)+\underline{S}^{I}\,.\label{eq:37}
-\end{equation}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-We need to compute the tangent constitutive matrix when forming the stiffness
- matrix.
- In addition to the volumetric contribution to the tangent constitutive
- matrix, we require the deviatoric part:
-\begin_inset Formula 
-\begin{equation}
-\frac{\partial\underline{S}}{\partial\underline{\epsilon}}=\frac{\partial\underline{S}}{\partial\underline{e}}\frac{\partial\underline{e}}{\partial\underline{\epsilon}}\,,\label{eq:38}
-\end{equation}
-
-\end_inset
-
-where the second derivative on the right may be easily deduced from Equation
- 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:16"
-
-\end_inset
-
-.
- The other derivative is given by
-\begin_inset Formula 
-\begin{equation}
-\frac{\partial\underline{S}}{\partial\underline{e}}=2\mu_{tot}\left[\mu_{0}\underline{I}+\sum_{i=1}^{N}\mu_{i}\frac{\partial\underline{h}_{i}^{1}}{\partial\underline{e}}\right]\,,\label{eq:39}
-\end{equation}
-
-\end_inset
-
-where 
-\begin_inset Formula $\underline{I}$
-\end_inset
-
- is the identity matrix.
- From Equations 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:33"
-
-\end_inset
-
- through 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:35"
-
-\end_inset
-
-, the derivative inside the brackets is
-\begin_inset Formula 
-\begin{equation}
-\frac{\partial\underline{h}_{i}^{1}}{\partial\underline{e}}=\Delta h_{i}\left(\Delta t\right)\underline{I}\,.\label{eq:40}
-\end{equation}
-
-\end_inset
-
-The complete deviatoric tangent relation is then
-\begin_inset Formula 
-\begin{equation}
-\frac{\partial\underline{S}}{\partial\underline{\epsilon}}=2\mu_{tot}\left[\mu_{0}+\sum_{i=1}^{N}\mu_{i}\Delta h_{i}\left(\Delta t\right)\right]\frac{\partial\underline{e}}{\partial\underline{\epsilon}}\,.\label{eq:41}
-\end{equation}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-We use this formulation for both our Maxwell and generalized Maxwell viscoelasti
-c models.
- For the Maxwell model, 
-\begin_inset Formula $\mu_{0}=0$
-\end_inset
-
- and 
-\begin_inset Formula $N=1$
-\end_inset
-
-.
- For the generalized Maxwell model, 
-\begin_inset Formula $N=3.$
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float table
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\noindent
-\align center
-\begin_inset Caption
-
-\begin_layout Plain Layout
-\begin_inset CommandInset label
-LatexCommand label
-name "tab:linearMaxwell"
-
-\end_inset
-
-Values in spatial database used as parameters in the linear Maxwell viscoelastic
- material constitutive model.
-\end_layout
-
-\end_inset
-
-
-\begin_inset Tabular
-<lyxtabular version="3" rows="5" columns="2">
-<features tabularvalignment="middle">
-<column alignment="center" valignment="middle" width="0.85in">
-<column alignment="center" valignment="middle" width="2.47in">
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Parameter
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\align center
-
-\series bold
-Name in Spatial Database
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $v_{p}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-Vp
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $v_{s}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-Vs
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\rho$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-density
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\eta$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-viscosity
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float table
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\noindent
-\align center
-\begin_inset Caption
-
-\begin_layout Plain Layout
-\begin_inset CommandInset label
-LatexCommand label
-name "tab:genMaxwell"
-
-\end_inset
-
-Values in spatial database used as parameters in the generalized linear
- Maxwell viscoelastic material constitutive model.
-\end_layout
-
-\end_inset
-
-
-\begin_inset Tabular
-<lyxtabular version="3" rows="10" columns="2">
-<features tabularvalignment="middle">
-<column alignment="center" valignment="middle" width="0.85in">
-<column alignment="center" valignment="middle" width="2.47in">
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Parameter
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\align center
-
-\series bold
-Name in Spatial Database
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $v_{p}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-Vp
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $v_{s}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-Vs
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\rho$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-density
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\mu_{1}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-shear-ratio-1
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\mu_{2}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-shear-ratio-2
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\mu_{3}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-shear-ratio-3
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\eta_{1}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-viscosity-1
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\eta_{2}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-viscosity-2
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\eta_{3}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-viscosity-3
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsection
-\begin_inset CommandInset label
-LatexCommand label
-name "sub:Effective-Stress-Formulations-Viscoelastic"
-
-\end_inset
-
-Effective Stress Formulations for Viscoelastic Materials
-\end_layout
-
-\begin_layout Standard
-As an alternative to the approach outlined above, an effective stress function
- formulation 
-\begin_inset CommandInset citation
-LatexCommand cite
-key "Kojic:Bathe:1987"
-
-\end_inset
-
- may be employed for both a linear Maxwell model and a power-law Maxwell
- model.
- Note that this formulation is not presently employed for linear viscoelastic
- models (see Appendix 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "cha:Alternative-Formulations"
-
-\end_inset
-
-), but it is used for power-law viscoelastic materials.
- For the viscoelastic materials considered here, the viscous volumetric
- strains are zero (incompressible flow), and it is convenient to separate
- the general stress-strain relationship at time 
-\begin_inset Formula $t+\Delta t$
-\end_inset
-
- into deviatoric and volumetric parts:
-\begin_inset Formula 
-\begin{gather}
-\phantom{}{}^{t+\Delta t}\underline{S}=\frac{E}{1+\nu}\left(^{t+\Delta t}\underline{e}-\phantom{}^{t+\Delta t}\underline{e}^{C}-\underline{e}^{I}\right)+\underline{S}^{I}=\frac{1}{a_{E}}\left(^{t+\Delta t}\underline{e}-\phantom{}^{t+\Delta t}\underline{e}^{C}-\underline{e}^{I}\right)\label{eq:42}\\
-^{t+\Delta t}P=\frac{E}{1-2\nu}\left(^{t+\Delta t}\theta-\theta^{I}\right)+P^{I}=\frac{1}{a_{m}}\left(^{t+\Delta t}\theta-\theta^{I}\right)\:,\nonumber 
-\end{gather}
-
-\end_inset
-
-where 
-\begin_inset Formula $^{t+\Delta t}\underline{e}$
-\end_inset
-
- is the total deviatoric strain, 
-\begin_inset Formula $^{t+\Delta t}\underline{e}^{C}$
-\end_inset
-
- is the total viscous strain, 
-\begin_inset Formula $\underline{e}^{I}$
-\end_inset
-
- is the initial deviatoric strain, 
-\begin_inset Formula $^{t+\Delta t}P$
-\end_inset
-
- is the pressure, 
-\begin_inset Formula $^{t+\Delta t}\theta$
-\end_inset
-
- is the mean strain evaluated at time 
-\begin_inset Formula $t+\Delta t$
-\end_inset
-
- , and 
-\begin_inset Formula $\theta^{I}$
-\end_inset
-
- is the initial mean strain.
- The initial deviatoric stress and initial pressure are given by 
-\begin_inset Formula $\underline{S}^{I}$
-\end_inset
-
- and 
-\begin_inset Formula $P^{I}$
-\end_inset
-
-, respectively.
- The topmost equation in Equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:42"
-
-\end_inset
-
- may also be written as
-\begin_inset Formula 
-\begin{gather}
-^{t+\Delta t}\underline{S}=\frac{1}{a_{E}}(^{t+\Delta t}\underline{e}^{\prime}-\underline{\Delta e}^{C})+\underline{S}^{I}\,,\label{eq:43}
-\end{gather}
-
-\end_inset
-
-where
-\begin_inset Formula 
-\begin{gather}
-^{t+\Delta t}\underline{e}^{\prime}=\phantom{}^{t+\Delta t}\underline{e}-\phantom{}^{t}\underline{e}^{C}-\underline{e}^{I}\,\,,\,\,\,\underline{\Delta e}^{C}=\phantom{}^{t+\Delta t}\underline{e}^{C}-\phantom{}^{t}\underline{e}^{C}\,.\label{eq:44}
-\end{gather}
-
-\end_inset
-
-The creep strain increment is approximated using
-\begin_inset Formula 
-\begin{gather}
-\underline{\Delta e}^{C}=\Delta t\phantom{}^{\tau}\gamma\phantom{}^{\tau}\underline{S}\,,\label{eq:45}
-\end{gather}
-
-\end_inset
-
-where, using the 
-\begin_inset Formula $\alpha$
-\end_inset
-
--method of time integration,
-\begin_inset Formula 
-\begin{gather}
-^{\tau}\underline{S}=(1-\alpha)_{I}^{t}\underline{S}+\alpha\phantom{}_{I}^{t+\Delta t}\underline{S}+\underline{S}^{I}=(1-\alpha)^{t}\underline{S}+\alpha\phantom{}^{t+\Delta t}\underline{S}\,\,,\label{eq:46}
-\end{gather}
-
-\end_inset
-
-and
-\begin_inset Formula 
-\begin{gather}
-^{\tau}\gamma=\frac{3\Delta\overline{e}^{C}}{2\Delta t\phantom{}^{\tau}\overline{\sigma}}\,\,,\label{eq:47}
-\end{gather}
-
-\end_inset
-
-where
-\begin_inset Formula 
-\begin{gather}
-\Delta\overline{e}^{C}=\sqrt{\frac{2}{3}\underline{\Delta e}^{C}\cdot\underline{\Delta e}^{C}}\label{eq:48}
-\end{gather}
-
-\end_inset
-
-and
-\begin_inset Formula 
-\begin{gather}
-^{\tau}\overline{\sigma}=(1-\alpha)_{I}^{t}\overline{\sigma}+\alpha\phantom{}_{I}^{t+\Delta t}\overline{\sigma}+\overline{\sigma}^{I}=\sqrt{3\phantom{}^{\tau}J_{2}^{\prime}}\,\,.\label{eq:49}
-\end{gather}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-To form the global stiffness matrix, it is necessary to provide a relationship
- for the viscoelastic tangent material matrix relating stress and strain.
- If we use vectors composed of the stresses and tensor strains, this relationshi
-p is
-\begin_inset Formula 
-\begin{gather}
-\underline{C}^{VE}=\frac{\partial\phantom{}^{t+\Delta t}\overrightarrow{\sigma}}{\partial\phantom{}^{t+\Delta t}\overrightarrow{\epsilon}}\,\,.\label{eq:55}
-\end{gather}
-
-\end_inset
-
-In terms of the vectors, we have
-\begin_inset Formula 
-\begin{gather}
-^{t+\Delta t}\sigma_{i}=\phantom{}^{t+\Delta t}S_{i}+\phantom{}^{t+\Delta t}P\,\,;\,\,\, i=1,2,3\label{eq:56}\\
-^{t+\Delta t}\sigma_{i}=\phantom{}^{t+\Delta t}S_{i}\,;\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, i=4,5,6\nonumber 
-\end{gather}
-
-\end_inset
-
-Therefore,
-\begin_inset Formula 
-\begin{gather}
-C_{ij}^{VE}=C_{ij}^{\prime}+\frac{1}{3a_{m}}\,;\,\,1\leq i,j\leq3\,\,.\label{eq:57}\\
-C_{ij}^{VE}=C_{ij}^{\prime}\,;\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\textrm{otherwise}\nonumber 
-\end{gather}
-
-\end_inset
-
-Using the chain rule,
-\begin_inset Formula 
-\begin{gather}
-C_{ij}^{\prime}=\frac{\partial\phantom{}^{t+\Delta t}S_{i}}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}=\frac{\partial\phantom{}^{t+\Delta t}S_{i}}{\partial\phantom{}^{t+\Delta t}e_{k}^{\prime}}\frac{\partial\phantom{}^{t+\Delta t}e_{k}^{\prime}}{\partial\phantom{}^{t+\Delta t}e_{l}}\frac{\partial\phantom{}^{t+\Delta t}e_{l}}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}\,\,.\label{eq:58}
-\end{gather}
-
-\end_inset
-
-From Equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:44"
-
-\end_inset
-
-, we obtain
-\begin_inset Formula 
-\begin{gather}
-\frac{\partial\phantom{}^{t+\Delta t}e_{k}^{\prime}}{\partial\phantom{}^{t+\Delta t}e_{l}}=\delta_{kl}\,\,,\label{eq:59}
-\end{gather}
-
-\end_inset
-
-and from Equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:16"
-
-\end_inset
-
-:
-\begin_inset Formula 
-\begin{gather}
-\frac{\partial\phantom{}^{t+\Delta t}e_{l}}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}=\frac{1}{3}\left[\begin{array}{ccc}
-2 & -1 & -1\\
--1 & 2 & -1\\
--1 & -1 & 2
-\end{array}\right];\,\,1\leq l,j\leq3\label{eq:60}\\
-\frac{\partial\phantom{}^{t+\Delta t}e_{l}}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}=\delta_{lj}\,\,;\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\textrm{otherwise.}\nonumber 
-\end{gather}
-
-\end_inset
-
-The first term of Equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:58"
-
-\end_inset
-
- depends on the particular constitutive relationship, and the complete tangent
- matrix may then be obtained from Equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:57"
-
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Subsubsection
-Power-Law Maxwell Viscoelastic Material
-\begin_inset CommandInset label
-LatexCommand label
-name "sub:Power-Law-Maxwell-Viscoelastic"
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Laboratory results on rock rheology are typically performed using a triaxial
- experiment, and the creep data are fit to a power-law equation of the form
- (e.g., 
-\begin_inset CommandInset citation
-LatexCommand cite
-key "Kirby:Kronenberg:1987"
-
-\end_inset
-
-):
-\begin_inset Formula 
-\begin{equation}
-\dot{\epsilon}_{11}^{C}=A_{E}\exp\left(\frac{-Q}{RT}\right)\left(\sigma_{1}-\sigma_{3}\right)^{n}=A_{E}\exp\left(\frac{-Q}{RT}\right)\sigma_{d}^{n}\:,\label{eq:64}
-\end{equation}
-
-\end_inset
-
-where 
-\begin_inset Formula $\dot{\epsilon}_{11}^{C}$
-\end_inset
-
- is the strain rate in the direction of the maximum principal stress 
-\begin_inset Formula $\left(\sigma_{1}\right)$
-\end_inset
-
-, 
-\begin_inset Formula $A_{E}$
-\end_inset
-
- is the experimentally-derived pre-exponential constant, 
-\begin_inset Formula $Q$
-\end_inset
-
- is the activation enthalpy, 
-\begin_inset Formula $R$
-\end_inset
-
- is the universal gas constant, 
-\begin_inset Formula $T$
-\end_inset
-
- is the absolute temperature, 
-\begin_inset Formula $n$
-\end_inset
-
- is the power-law exponent, 
-\begin_inset Formula $\sigma_{3}\:\left(=\sigma_{2}\right)$
-\end_inset
-
- is equal to the confining pressure, and 
-\begin_inset Formula $\sigma_{d}$
-\end_inset
-
- is the differential stress.
- To properly formulate the flow law, it must be generalized so that the
- results are not influenced by the experiment type or the choice of coordinate
- systems (e.g., 
-\begin_inset CommandInset citation
-LatexCommand cite
-key "Paterson:1994"
-
-\end_inset
-
-).
- The flow law may then be generalized in terms of the deviatoric stress
- and strain rate invariants:
-\begin_inset Formula 
-\begin{equation}
-\sqrt{\dot{L}_{2}^{\prime C}}=A_{M}\exp\left(\frac{-Q}{RT}\right)\sqrt{J_{2}^{\prime}}^{n}\:,\label{eq:65}
-\end{equation}
-
-\end_inset
-
-where 
-\begin_inset Formula $A_{M}$
-\end_inset
-
- is now a pre-exponential constant used in the formulation for modeling.
- In practice, it is necessary to compute each strain rate component using
- the flow law.
- This is accomplished using:
-\begin_inset Formula 
-\begin{equation}
-\dot{e}_{ij}^{C}=A_{M}\exp\left(\frac{-Q}{RT}\right)\sqrt{J_{2}^{\prime}}^{n-1}S_{ij}\:.\label{eq:66}
-\end{equation}
-
-\end_inset
-
-Note that Equations 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:65"
-
-\end_inset
-
- and 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:66"
-
-\end_inset
-
- are consistent, since Equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:65"
-
-\end_inset
-
- may be obtained from Equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:66"
-
-\end_inset
-
- by taking the scalar inner product of both sides, multiplying by 1/2, and
- taking the square root.
-\end_layout
-
-\begin_layout Standard
-In a triaxial experiment with confining pressure 
-\begin_inset Formula $P_{c}$
-\end_inset
-
-, we have
-\begin_inset Formula 
-\begin{gather}
-\sigma_{2}=\sigma_{3}=P_{c}\nonumber \\
-\sigma_{1}=\sigma_{1}^{app}\label{eq:67}\\
-P=\frac{\sigma_{1}+2P_{c}}{3}\:,\nonumber 
-\end{gather}
-
-\end_inset
-
-where 
-\begin_inset Formula $\sigma_{1}^{app}$
-\end_inset
-
- is the applied load.
- The deviatoric stresses are then:
-\begin_inset Formula 
-\begin{gather}
-S_{1}=\frac{2}{3}\left(\sigma_{1}-P_{c}\right)\nonumber \\
-S_{2}=S_{3}=-\frac{1}{3}\left(\sigma_{1}-P_{c}\right)\:.\label{eq:68}
-\end{gather}
-
-\end_inset
-
-This gives
-\begin_inset Formula 
-\begin{gather}
-S_{1}=\frac{2}{3}\left(\sigma_{1}-\sigma_{3}\right)=\frac{2}{3}\sigma_{d}\nonumber \\
-S_{2}=S_{3}=-\frac{1}{3}\left(\sigma_{1}-\sigma_{3}\right)=-\frac{1}{3}\sigma_{d}\:.\label{eq:69}
-\end{gather}
-
-\end_inset
-
-In terms of the second deviatoric stress invariant, we then have
-\begin_inset Formula 
-\begin{equation}
-\sqrt{J_{2}^{\prime}}=\frac{\sigma_{d}}{\sqrt{3}}\:.\label{eq:70}
-\end{equation}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Under the assumption that the creep measured in the laboratory experiments
- is incompressible, we have
-\begin_inset Formula 
-\begin{gather}
-\dot{e}_{11}^{C}=\dot{\epsilon}_{11}\nonumber \\
-\dot{e}_{22}^{C}=\dot{e}_{33}^{C}=-\frac{1}{2}\dot{\epsilon}_{11}\:.\label{eq:71}
-\end{gather}
-
-\end_inset
-
-In terms of the second deviatoric strain rate invariant we then have
-\begin_inset Formula 
-\begin{equation}
-\sqrt{\dot{L}_{2}^{\prime C}}=\frac{\sqrt{3}}{2}\dot{\epsilon}_{11}\:.\label{eq:72}
-\end{equation}
-
-\end_inset
-
-Substituting Equations 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:70"
-
-\end_inset
-
- and 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:72"
-
-\end_inset
-
- into Equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:64"
-
-\end_inset
-
-, we obtain
-\begin_inset Formula 
-\begin{equation}
-\sqrt{\dot{L}_{2}^{\prime C}}=A_{E}\frac{\sqrt{3}^{n+1}}{2}\exp\left(\frac{-Q}{RT}\right)\sqrt{J_{2}^{\prime}}^{n}\:,\label{eq:73}
-\end{equation}
-
-\end_inset
-
-and therefore,
-\begin_inset Formula 
-\begin{equation}
-A_{M}=\frac{\sqrt{3}^{n+1}}{2}A_{E}\:.\label{eq:74}
-\end{equation}
-
-\end_inset
-
-When the exponential factor is included, we define a new parameter:
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula 
-\begin{equation}
-A_{T}=A_{M}\exp\left(\frac{-Q}{RT}\right)=\frac{\sqrt{3}^{n+1}}{2}A_{E}\exp\left(\frac{-Q}{RT}\right)\:.\label{eq:75}
-\end{equation}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-There is a problem with the usage of parameters 
-\begin_inset Formula $A_{E}$
-\end_inset
-
-, 
-\begin_inset Formula $A_{M}$
-\end_inset
-
-, and 
-\begin_inset Formula $A_{T}$
-\end_inset
-
-.
- Since the dimensions of these parameters are dependent on the value of
- the power-law exponent, they are not really constants.
- In addition to being logically inconsistent, this presents problems when
- specifying parameters for PyLith, since the power-law exponent must be
- known before the units can be determined.
- An alternative way of writing the flow rule is (e.g., 
-\begin_inset CommandInset citation
-LatexCommand cite
-key "Prentice:1968"
-
-\end_inset
-
-): 
-\begin_inset Formula 
-\begin{equation}
-\frac{\sqrt{\dot{L}_{2}^{\prime C}}}{\dot{e}_{0}}=\left(\frac{\sqrt{J_{2}^{\prime}}}{S_{0}}\right)^{n},\label{eq:76}
-\end{equation}
-
-\end_inset
-
-where 
-\begin_inset Formula $\dot{e}_{0}$
-\end_inset
-
- and 
-\begin_inset Formula $S_{0}$
-\end_inset
-
- are reference values for the strain rate and deviatoric stress.
- This means that
-\begin_inset Formula 
-\begin{equation}
-\frac{\dot{e}_{0}}{S_{0}^{n}}=A_{T}\:.\label{eq:77}
-\end{equation}
-
-\end_inset
-
-Users must therefore specify three parameters for a power-law material.
- The properties 
-\family typewriter
-reference-strain-rate
-\family default
-, 
-\family typewriter
-reference-stress
-\family default
-, and 
-\family typewriter
-power-law-exponent
-\family default
- in Table 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "tab:powerLaw"
-
-\end_inset
-
- refer to 
-\begin_inset Formula $\dot{e}_{0}$
-\end_inset
-
-, 
-\begin_inset Formula $S_{0}$
-\end_inset
-
-, and 
-\begin_inset Formula $n$
-\end_inset
-
-, respectively.
- To specify the power-law properties for PyLith using laboratory results,
- the user must first compute 
-\begin_inset Formula $A_{T}$
-\end_inset
-
- using Equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:75"
-
-\end_inset
-
-.
- Then, values for 
-\begin_inset Formula $\dot{e}_{0}$
-\end_inset
-
- and 
-\begin_inset Formula $S_{0}$
-\end_inset
-
- must be provided.
- The simplest method is probably to assume a reasonable value for the reference
- strain rate, and then compute 
-\begin_inset Formula $S_{0}$
-\end_inset
-
- as
-\begin_inset Formula 
-\begin{equation}
-S_{0}=\left(\frac{\dot{e}_{0}}{A_{T}}\right)^{\frac{1}{n}}\:.\label{eq:78}
-\end{equation}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-A utility code (
-\family typewriter
-powerlaw_gendb.py
-\family default
-) is provided to convert laboratory results to the properties used by PyLith.
- To use the code, users must specify the spatial variation of 
-\begin_inset Formula $A_{E}$
-\end_inset
-
-, 
-\begin_inset Formula $Q$
-\end_inset
-
-, 
-\begin_inset Formula $n$
-\end_inset
-
-, and 
-\begin_inset Formula $T$
-\end_inset
-
-.
- An additional parameter is given to define the units of 
-\begin_inset Formula $A_{E}$
-\end_inset
-
-.
- The user then specifies either a reference stress or a reference strain
- rate, and a database suitable for PyLith is generated.
- This utility is described more fully in Section 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "sub:Tutorial-Step08-Power-law"
-
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-The flow law in component form is 
-\begin_inset Formula 
-\begin{equation}
-\dot{e}_{ij}^{C}=\frac{\dot{e}_{0}\sqrt{J_{2}^{\prime}}^{n-1}S_{ij}}{S_{0}^{n}}\:,\label{eq:79}
-\end{equation}
-
-\end_inset
-
-and the creep strain increment is approximated as
-\begin_inset Formula 
-\begin{gather}
-\underline{\Delta e}^{C}\approx\frac{\Delta t\dot{e}_{0}\sqrt{^{\tau}J_{2}^{\prime}}^{n-1}\,^{\tau}\underline{S}}{S_{0}^{n}}=\frac{\Delta t\dot{e}_{0}\phantom{}^{\tau}\overline{\sigma}^{n-1}\,^{\tau}\underline{S}}{\sqrt{3}S_{0}^{n}}\,.\label{eq:80}
-\end{gather}
-
-\end_inset
-
- Therefore,
-\begin_inset Formula 
-\begin{gather}
-\Delta\bar{e}^{C}\approx\frac{2\Delta t\dot{e}_{0}\sqrt{^{\tau}J_{2}^{\prime}}^{n}}{\sqrt{3}S_{0}^{n}}=\frac{2\Delta t\dot{e}_{0}\phantom{}^{\tau}\overline{\sigma}^{n}}{\sqrt{3}^{n+1}S_{0}^{n}}\,,\,\textrm{and}\,^{\tau}\gamma=\frac{\dot{e}_{0}\sqrt{^{\tau}J_{2}^{\prime}}^{n-1}}{S_{0}^{n}}\,.\label{eq:81}
-\end{gather}
-
-\end_inset
-
-substituting Equations 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:46"
-
-\end_inset
-
-, 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:80"
-
-\end_inset
-
-, and 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:81"
-
-\end_inset
-
- into 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:43"
-
-\end_inset
-
-, we obtain:
-\begin_inset Formula 
-\begin{gather}
-^{t+\Delta t}\underline{S}=\frac{1}{a_{E}}\left\{ ^{t+\Delta t}\underline{e}^{\prime}-\Delta t\phantom{}^{\tau}\gamma\left[\left(1-\alpha\right)^{t}\underline{S}+\alpha{}^{t+\Delta t}\underline{S}\right]\right\} +\underline{S}^{I}\,,\label{eq:82}
-\end{gather}
-
-\end_inset
-
-which may be rewritten:
-\begin_inset Formula 
-\begin{gather}
-^{t+\Delta t}\underline{S}\left(a_{E}+\alpha\Delta t\phantom{}^{\tau}\gamma\right)={}^{t+\Delta t}\underline{e}^{\prime}-\Delta t\phantom{}^{\tau}\gamma\left(1-\alpha\right)^{t}\underline{S}+a_{E}\underline{S}^{I}\,.\label{eq:83}
-\end{gather}
-
-\end_inset
-
-Taking the scalar inner product of both sides we obtain:
-\begin_inset Formula 
-\begin{gather}
-a^{2}\,\,{}^{t+\Delta t}J_{2}^{\prime}-b+c\phantom{}^{\tau}\gamma-d^{2}\,^{\tau}\gamma^{2}=F=0\,,\label{eq:84}
-\end{gather}
-
-\end_inset
-
-where
-\begin_inset Formula 
-\begin{gather}
-a=a_{E}+\alpha\Delta t\phantom{}^{\tau}\gamma\,\,\nonumber \\
-b=\frac{1}{2}{}^{t+\Delta t}\underline{e}^{\prime}\cdot{}^{t+\Delta t}\underline{e}^{\prime}+a_{E}{}^{t+\Delta t}\underline{e}^{\prime}\cdot\underline{S}^{I}+a_{E}^{2}\,^{I}J_{2}^{\prime}\,.\label{eq:85}\\
-c=\Delta t\left(1-\alpha\right){}^{t+\Delta t}\underline{e}^{\prime}\cdot^{t}\underline{S}+\Delta t\left(1-\alpha\right)a_{E}\,^{t}\underline{S}\cdot\underline{S}^{I}\,\,\nonumber \\
-d=\Delta t\left(1-\alpha\right)\sqrt{^{t}J_{2}^{\prime}}\,\,\nonumber 
-\end{gather}
-
-\end_inset
-
-Equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:84"
-
-\end_inset
-
- is a function of a single unknown -- the square root of the second deviatoric
- stress invariant at time 
-\begin_inset Formula $t+\Delta t$
-\end_inset
-
- -- and may be solved by bisection or by Newton's method.
- Once this parameter has been found, the deviatoric stresses for the current
- time step may be found from Equations 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:49"
-
-\end_inset
-
-, 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:81"
-
-\end_inset
-
-, and 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:82"
-
-\end_inset
-
-, and the total stresses may be found by combining the deviatoric and volumetric
- components from Equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:42"
-
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-To compute the tangent stress-strain relation, we need to compute the first
- term in Equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:58"
-
-\end_inset
-
-.
- We begin by rewriting Equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:83"
-
-\end_inset
-
- as
-\begin_inset Formula 
-\begin{gather}
-F=^{t+\Delta t}S_{i}\left(a_{E}+\alpha\Delta t\phantom{}^{\tau}\gamma\right)-\phantom{}^{t+\Delta t}e_{i}^{\prime}+\Delta t\phantom{}^{\tau}\gamma\left(1-\alpha\right)^{t}S_{i}-a_{E}S_{i}^{I}=0\:.\label{eq:86}
-\end{gather}
-
-\end_inset
-
-The derivative of this function with respect to 
-\begin_inset Formula $^{t+\Delta t}e_{k}^{\prime\prime}$
-\end_inset
-
- is
-\begin_inset Formula 
-\begin{gather}
-\frac{\partial F}{\partial\phantom{}^{t+\Delta t}e_{k}^{\prime}}=-\delta_{ik}\:,\label{eq:87}
-\end{gather}
-
-\end_inset
-
-and the derivative with respect to 
-\begin_inset Formula $^{t+\Delta t}S_{i}$
-\end_inset
-
- is
-\begin_inset Formula 
-\begin{gather}
-\frac{\partial F}{\partial\phantom{}^{t+\Delta t}S_{i}}=a_{E}+\alpha\Delta t\phantom{}^{\tau}\gamma+\frac{\partial\phantom{}^{\tau}\gamma}{\partial\phantom{}^{t+\Delta t}S_{i}}\Delta t\left[\alpha\phantom{}^{t+\Delta t}S_{i}+\left(1-\alpha\right)^{t}S_{i}\right]\:.\label{eq:88}
-\end{gather}
-
-\end_inset
-
-From Equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:81"
-
-\end_inset
-
- and Equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:49"
-
-\end_inset
-
-,
-\begin_inset Formula 
-\begin{gather}
-^{\tau}\gamma=\frac{\dot{e}_{0}}{S_{0}^{n}}\left[\alpha\sqrt{^{t+\Delta t}J_{2}^{\prime}}+\left(1-\alpha\right)\sqrt{^{t}J_{2}^{\prime}}\right]^{n-1}\:.\label{eq:89}
-\end{gather}
-
-\end_inset
-
-Then
-\begin_inset Formula 
-\begin{gather}
-\frac{\partial\phantom{}^{\tau}\gamma}{\partial{}^{t+\Delta t}S_{i}}=\frac{\partial\phantom{}^{\tau}\gamma}{\partial\sqrt{^{t+\Delta t}J_{2}^{\prime}}}\frac{\partial\sqrt{^{t+\Delta t}J_{2}^{\prime}}}{\partial\phantom{}^{t+\Delta t}S_{l}}\label{eq:90}\\
-=\frac{\dot{e}_{0}\alpha\left(n-1\right)\sqrt{^{\tau}J_{2}^{\prime}}^{n-2}{}^{t+\Delta t}T_{i}}{2S_{0}^{n}}\,,\nonumber 
-\end{gather}
-
-\end_inset
-
-Where
-\begin_inset Formula 
-\begin{gather}
-^{t+\Delta t}T_{i}=\phantom{}^{t+\Delta t}S_{i}\:;\:\:1\leq i\leq3\label{eq:91}\\
-^{t+\Delta t}T_{i}=2\phantom{}^{t+\Delta t}S_{i}\:;\:\:\textrm{otherwise.}\nonumber 
-\end{gather}
-
-\end_inset
-
-Then using Equations 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:87"
-
-\end_inset
-
-, 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:88"
-
-\end_inset
-
-, 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:90"
-
-\end_inset
-
-, and the quotient rule for derivatives of an implicit function,
-\begin_inset Formula 
-\begin{gather}
-\frac{\partial\phantom{}^{t+\Delta t}S_{i}}{\partial{}^{t+\Delta t}e_{k}^{\prime}}=\frac{\delta_{ik}}{a_{E}+\alpha\Delta t\left[^{\tau}\gamma+\frac{\dot{e}_{0}{}^{\tau}S_{i}\left(n-1\right){}^{t+\Delta t}T_{i}\sqrt{^{\tau}J_{2}^{\prime}}^{n-2}}{2\sqrt{^{t+\Delta t}J_{2}^{\prime}}S_{0}^{n}}\right]}\,.\label{eq:92}
-\end{gather}
-
-\end_inset
-
-Note that for a linear material 
-\begin_inset Formula $\left(n=1\right)$
-\end_inset
-
-, this equation is identical to the linear formulation in Section 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "sub:Effective-Stress-Formulation-Maxwell"
-
-\end_inset
-
- (making the appropriate substitution for 
-\begin_inset Formula $^{\tau}\gamma$
-\end_inset
-
-).
- Then, using Equations 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:57"
-
-\end_inset
-
- through 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:60"
-
-\end_inset
-
-,
-\begin_inset Formula 
-\begin{gather}
-C_{ij}^{VE}=\frac{1}{3a_{m}}\left[\begin{array}{cccccc}
-1 & 1 & 1 & 0 & 0 & 0\\
-1 & 1 & 1 & 0 & 0 & 0\\
-1 & 1 & 1 & 0 & 0 & 0\\
-0 & 0 & 0 & 0 & 0 & 0\\
-0 & 0 & 0 & 0 & 0 & 0\\
-0 & 0 & 0 & 0 & 0 & 0
-\end{array}\right]+\frac{1}{3}\frac{\partial{}^{t+\Delta t}S_{i}}{\partial{}^{t+\Delta t}e_{k}^{\prime}}\left[\begin{array}{cccccc}
-2 & -1 & -1 & 0 & 0 & 0\\
--1 & 2 & -1 & 0 & 0 & 0\\
--1 & -1 & 2 & 0 & 0 & 0\\
-0 & 0 & 0 & 3 & 0 & 0\\
-0 & 0 & 0 & 0 & 3 & 0\\
-0 & 0 & 0 & 0 & 0 & 3
-\end{array}\right]\,.\label{eq:93}
-\end{gather}
-
-\end_inset
-
-Note that if there are no deviatoric stresses at the beginning and end of
- a time step (or if 
-\begin_inset Formula $\nicefrac{\dot{e}_{0}}{S_{0}^{n}}$
-\end_inset
-
- approaches zero), Equations 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:92"
-
-\end_inset
-
- and 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:93"
-
-\end_inset
-
- reduce to the elastic constitutive matrix, as expected.
-\end_layout
-
-\begin_layout Standard
-To compute the zero of the effective stress function using Newton's method,
- we require the derivative of Equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:84"
-
-\end_inset
-
-, which may be written:
-\begin_inset Formula 
-\begin{gather}
-\frac{\partial F}{\partial\sqrt{^{t+\Delta t}J_{2}^{\prime}}}=2a^{2}\sqrt{^{t+\Delta t}J_{2}^{\prime}}+\frac{\dot{e}_{0}\alpha\left(n-1\right)\sqrt{^{\tau}J_{2}^{\prime}}^{n-2}}{S_{0}^{n}}\left(2a\alpha\Delta t{}^{t+\Delta t}J_{2}^{\prime}+c-2d^{2}\,^{\tau}\gamma\right)\,.\label{eq:94}
-\end{gather}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float table
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\noindent
-\align center
-\begin_inset Caption
-
-\begin_layout Plain Layout
-\begin_inset CommandInset label
-LatexCommand label
-name "tab:powerLaw"
-
-\end_inset
-
-Values in spatial database used as parameters in the nonlinear power-law
- viscoelastic material constitutive model.
-\end_layout
-
-\end_inset
-
-
-\begin_inset Tabular
-<lyxtabular version="3" rows="7" columns="2">
-<features tabularvalignment="middle">
-<column alignment="center" valignment="middle" width="0.85in">
-<column alignment="center" valignment="middle" width="2.47in">
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Parameter
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\align center
-
-\series bold
-Name in Spatial Database
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $v_{p}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-Vp
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $v_{s}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-Vs
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\rho$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-density
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\dot{e}_{0}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-reference-strain-rate
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $S_{0}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-reference-stress
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $n$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-power-law-exponent
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-Elastoplastic Materials
-\end_layout
-
-\begin_layout Standard
-PyLith presently contains just a single elastoplastic material that implements
- the Drucker-Prager yield criterion.
- Future releases of PyLith may contain additional elastoplastic materials,
- such as Drucker-Prager with hardening/softening.
-\end_layout
-
-\begin_layout Subsection
-General Elastoplasticity Formulation
-\end_layout
-
-\begin_layout Standard
-The elastoplasticity formulation in PyLith is based on an additive decomposition
- of the total strain into elastic and plastic parts:
-\begin_inset Formula 
-\begin{equation}
-d\epsilon_{ij}=d\epsilon_{ij}^{E}+d\epsilon_{ij}^{P}\:.\label{eq:95}
-\end{equation}
-
-\end_inset
-
-The stress increment is then given by
-\begin_inset Formula 
-\begin{equation}
-d\sigma_{ij}=C_{ijrs}^{E}\left(d\epsilon_{rs}-d\epsilon_{rs}^{P}\right)\:,\label{eq:96}
-\end{equation}
-
-\end_inset
-
-where 
-\begin_inset Formula $C_{ijrs}^{E}$
-\end_inset
-
- are the components of the elastic constitutive tensor.
- To completely specify an elastoplastic problem, three components are needed.
- We first require a yield condition, which specifies the state of stress
- at which plastic flow initiates.
- This is generally given in the form:
-\begin_inset Formula 
-\begin{equation}
-f\left(\underline{\sigma},k\right)=0\:,\label{eq:97}
-\end{equation}
-
-\end_inset
-
-where 
-\shape italic
-k
-\shape default
- is an internal state parameter.
- It is then necessary to specify a flow rule, which describes the relationship
- between plastic strain and stress.
- The flow rule is given in the form:
-\begin_inset Formula 
-\begin{equation}
-g\left(\underline{\sigma},k\right)=0\:.\label{eq:98}
-\end{equation}
-
-\end_inset
-
-The plastic strain increment is then given as
-\begin_inset Formula 
-\begin{equation}
-d\epsilon_{ij}^{P}=d\lambda\frac{\partial g}{\partial\sigma_{ij}}\:,\label{eq:99}
-\end{equation}
-
-\end_inset
-
-where 
-\begin_inset Formula $d\lambda$
-\end_inset
-
- is the scalar plastic multiplier.
- When the flow rule is identical to the yield criterion (
-\begin_inset Formula $f\equiv g$
-\end_inset
-
-), the plasticity is described as associated.
- Otherwise, it is non-associated.
- The final component needed is a hardening hypothesis, which describes how
- the yield condition and flow rule are modified during plastic flow.
- When the yield condition and flow rule remain constant during plastic flow
- (e.g., no hardening), the material is referred to as perfectly plastic.
-\end_layout
-
-\begin_layout Standard
-To perform the solution, the yield condition (Equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:97"
-
-\end_inset
-
-) is first evaluated under the assumption of elastic behavior.
- If 
-\begin_inset Formula $^{t+\Delta t}f<0$
-\end_inset
-
-, the material behavior is elastic and no plastic flow occurs.
- Otherwise, the behavior is plastic and a plastic strain increment must
- be computed to return the stress state to the yield envelope.
- This procedure is known as an elastic predictor-plastic corrector algorithm.
-\end_layout
-
-\begin_layout Subsection
-Drucker-Prager Elastoplastic Material
-\end_layout
-
-\begin_layout Standard
-PyLith includes an elastoplastic implementation of the Drucker-Prager yield
- criterion 
-\begin_inset CommandInset citation
-LatexCommand cite
-key "Drucker:Prager:1952"
-
-\end_inset
-
-.
- This criterion was originally devised to model plastic deformation of soils,
- and it has also been used to model rock deformation.
- It is intended to be a smooth approximation of the Mohr-Coulomb yield criterion.
- The implementation used in PyLith includes non-associated plastic flow,
- which allows control over the unreasonable amounts of dilatation that are
- sometimes predicted by the associated model.
- The model is described by the following yield condition:
-\begin_inset Formula 
-\begin{equation}
-f\left(\underline{\sigma},k\right)=\alpha_{f}I_{1}+\sqrt{J_{2}^{\prime}}-\beta\:,\label{eq:100}
-\end{equation}
-
-\end_inset
-
-and a flow rule given by:
-\begin_inset Formula 
-\begin{equation}
-g\left(\underline{\sigma},k\right)=\sqrt{J_{2}^{\prime}}+\alpha_{g}I_{1}\:.\label{eq:101}
-\end{equation}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-The yield surface represents a circular cone in principal stress space,
- and the parameters can be related to the friction angle, 
-\begin_inset Formula $\phi$
-\end_inset
-
-, and the cohesion, 
-\begin_inset Formula $\bar{c}$
-\end_inset
-
-, of the Mohr-Coulomb model.
- The yield surface in Haigh-Westergaard space (
-\begin_inset Formula $\zeta=\frac{1}{\sqrt{3}}I_{1},p=\sqrt{2J_{2}},\cos(3\theta)=\frac{3\sqrt{3}}{2}\frac{J_{3}}{J_{2}^{3/2}}$
-\end_inset
-
-) is
-\begin_inset Formula 
-\begin{equation}
-\left(\sqrt{3}\sin\left(\theta+\frac{\pi}{3}\right)-\sin\phi\cos\left(\theta+\frac{\pi}{3}\right)\right)p-\sqrt{2}\sin\phi\zeta=\sqrt{6}\overline{c}\cos\theta.\label{eq:drucker:prager:haigh:westergaard}
-\end{equation}
-
-\end_inset
-
-The yield surface can be fit to the Mohr-Coulomb model in several different
- ways.
- The yield surface can touch the outer apices (
-\begin_inset Formula $\theta=\pi/3$
-\end_inset
-
-) of the Mohr-Coulomb model (inscribed version), the inner apices (
-\begin_inset Formula $\theta=0$
-\end_inset
-
-) of the Mohr-Coulomb model (circumscribed version), or halfway between
- the two (
-\begin_inset Formula $\theta=pi/6,$
-\end_inset
-
-middle version).
- Substituting these values for 
-\begin_inset Formula $\theta$
-\end_inset
-
- into equation (
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:drucker:prager:haigh:westergaard"
-
-\end_inset
-
-) and casting it into the same form as equation (
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:101"
-
-\end_inset
-
-) yields the values of 
-\begin_inset Formula $\alpha_{f}$
-\end_inset
-
-, 
-\begin_inset Formula $\beta$
-\end_inset
-
-, and 
-\begin_inset Formula $\alpha_{g}$
-\end_inset
-
- given in Table 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "tab:fit_mohr_coulomb"
-
-\end_inset
-
-, where 
-\begin_inset Formula $\phi_{0}$
-\end_inset
-
- refers to the initial friction angle.
- Similarly, the flow rule can be related to the dilatation angle, 
-\begin_inset Formula $\psi$
-\end_inset
-
-, of a Mohr-Coulomb model.
- It is also possible for the Mohr-Coulomb parameters to be functions of
- the internal state parameter, 
-\begin_inset Formula $k$
-\end_inset
-
-.
- In PyLith, the fit to the Mohr-Coulomb yield surface and flow rule is controlle
-d by the 
-\family typewriter
-fit_mohr_coulomb
-\family default
- property.
- 
-\end_layout
-
-\begin_layout Standard
-\begin_inset Float table
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\begin_inset Caption
-
-\begin_layout Plain Layout
-\begin_inset CommandInset label
-LatexCommand label
-name "tab:fit_mohr_coulomb"
-
-\end_inset
-
-Options for fitting the Drucker-Prager plastic parameters to a Mohr-Coulomb
- model using 
-\family typewriter
-fit_mohr_coulomb
-\family default
-.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Plain Layout
-\align center
-\begin_inset Tabular
-<lyxtabular version="3" rows="4" columns="4">
-<features tabularvalignment="middle">
-<column alignment="center" valignment="top" width="0">
-<column alignment="center" valignment="top" width="0">
-<column alignment="center" valignment="top" width="0">
-<column alignment="center" valignment="top" width="0">
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Parameter Value
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\alpha_{f}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\beta$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\alpha_{g}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-inscribed
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\frac{2\sin\phi\left(k\right)}{\sqrt{3}\left(3-\sin\phi\left(k\right)\right)}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\frac{6\bar{c}\left(k\right)\cos\phi_{0}}{\sqrt{3}\left(3-\sin\phi_{0}\right)}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\frac{2\sin\psi(k)}{\sqrt{3}\left(3-\sin\psi\left(k\right)\right)}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-middle
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\frac{\sin\phi\left(k\right)}{3}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\bar{c}\left(k\right)\cos\left(\phi_{0}\right)$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\frac{\sin\psi\left(k\right)}{3}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-circumscribed
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\frac{2\sin\phi\left(k\right)}{\sqrt{3}\left(3+\sin\phi\left(k\right)\right)}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\frac{6\bar{c}\left(k\right)\cos\phi_{0}}{\sqrt{3}\left(3+\sin\phi_{0}\right)}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\frac{2\sin\psi(k)}{\sqrt{3}\left(3+\sin\psi\left(k\right)\right)}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-As for the viscoelastic models, it is convenient to separate the deformation
- into deviatoric and volumetric parts:
-\begin_inset Formula 
-\begin{gather}
-^{t+\Delta t}S_{ij}=\frac{1}{a_{E}}\left(^{t+\Delta t}e_{ij}-\phantom{}^{t+\Delta t}e_{ij}^{P}-e_{ij}^{I}\right)+S_{ij}^{I}=\frac{1}{a_{E}}\left(^{t+\Delta t}e_{ij}^{\prime}-\Delta e_{ij}^{P}\right)+S_{ij}^{I}\label{eq:105}\\
-^{t+\Delta t}P=\frac{1}{a_{m}}\left(^{t+\Delta t}\theta-\phantom{}^{t+\Delta t}\theta^{P}-\theta^{I}\right)+P^{I}=\frac{1}{a_{m}}\left(^{t+\Delta t}\theta^{\prime}-\Delta\theta^{P}\right)+P^{I}\:,\nonumber 
-\end{gather}
-
-\end_inset
-
-where
-\begin_inset Formula 
-\begin{gather}
-^{t+\Delta t}e_{ij}^{\prime}=\phantom{}^{t+\Delta t}e_{ij}-\phantom{}^{t}e_{ij}^{P}-e_{ij}^{I}\nonumber \\
-\Delta e_{ij}^{P}=\phantom{}^{t+\Delta t}e_{ij}^{P}-\phantom{}^{t}e_{ij}^{P}\nonumber \\
-^{t+\Delta t}\theta^{\prime}=\phantom{}^{t+\Delta t}\theta-\phantom{}^{t}\theta^{P}-\theta^{I}\nonumber \\
-\Delta\theta^{P}=\phantom{}^{t+\Delta t}\theta^{P}-\phantom{}^{t}\theta^{P}\:.\label{eq:106}
-\end{gather}
-
-\end_inset
-
-Since the plasticity is pressure-dependent, there are volumetric plastic
- strains, unlike the viscous strains in the previous section.
- From Equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:99"
-
-\end_inset
-
-, the plastic strain increment is
-\begin_inset Formula 
-\begin{equation}
-\Delta\epsilon_{ij}^{P}=\lambda\frac{\partial\phantom{}^{t+\Delta t}g}{\partial\phantom{}^{t+\Delta t}\sigma_{ij}}=\lambda\alpha_{g}\delta_{ij}+\lambda\frac{^{t+\Delta t}S_{ij}}{2\sqrt{^{t+\Delta t}J_{2}^{\prime}}}\:.\label{eq:107}
-\end{equation}
-
-\end_inset
-
-The volumetric part is
-\begin_inset Formula 
-\begin{equation}
-\Delta\theta^{P}=\frac{1}{3}\Delta\epsilon_{ii}^{P}=\lambda\alpha_{g}\:,\label{eq:108}
-\end{equation}
-
-\end_inset
-
-and the deviatoric part is
-\begin_inset Formula 
-\begin{equation}
-\Delta e_{ij}^{P}=\Delta\epsilon_{ij}^{P}-\Delta\epsilon_{m}^{P}\delta_{ij}=\lambda\frac{^{t+\Delta t}S_{ij}}{2\sqrt{^{t+\Delta t}J_{2}^{\prime}}}\:.\label{eq:109}
-\end{equation}
-
-\end_inset
-
-The problem is reduced to solving for 
-\begin_inset Formula $\lambda$
-\end_inset
-
-.
- The procedure is different depending on whether hardening is included.
-\end_layout
-
-\begin_layout Subsubsection
-Drucker-Prager Elastoplastic With No Hardening (Perfectly Plastic)
-\end_layout
-
-\begin_layout Standard
-When there is no hardening (perfect plasticity), the Drucker-Prager elastoplasti
-c model may be parameterized with just three parameters, in addition to
- the normal elasticity parameters.
- The parameters 
-\family typewriter
-friction-angle
-\family default
-, 
-\family typewriter
-cohesion
-\family default
-, and 
-\family typewriter
-dilatation-angle
-\family default
- in Table 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "tab:druckerPrager"
-
-\end_inset
-
- refer respectively to 
-\begin_inset Formula $\phi$
-\end_inset
-
-, 
-\begin_inset Formula $\bar{c}$
-\end_inset
-
-, and 
-\begin_inset Formula $\psi$
-\end_inset
-
- in Table 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "tab:fit_mohr_coulomb"
-
-\end_inset
-
-.
- These are then converted to the properties 
-\begin_inset Formula $\alpha_{f}$
-\end_inset
-
- (
-\family typewriter
-alpha-yield
-\family default
-), 
-\begin_inset Formula $\beta$
-\end_inset
-
- (
-\family typewriter
-beta
-\family default
-), and 
-\begin_inset Formula $\alpha_{g}$
-\end_inset
-
- (
-\family typewriter
-alpha-flow
-\family default
-), as shown in Table 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "tab:material-model-output"
-
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-For perfect plasticity the yield and flow functions do not vary, and we
- can solve for 
-\begin_inset Formula $\lambda$
-\end_inset
-
- by substituting Equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:109"
-
-\end_inset
-
- into Equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:105"
-
-\end_inset
-
- and taking the scalar product of both sides:
-\begin_inset Formula 
-\begin{equation}
-\lambda=\sqrt{2}\,\phantom{}^{t+\Delta t}d-2a_{E}\sqrt{^{t+\Delta t}J_{2}^{\prime}}\:,\label{eq:110}
-\end{equation}
-
-\end_inset
-
-where
-\begin_inset Formula 
-\begin{equation}
-^{t+\Delta t}d^{2}=2a_{E}^{2}J_{2}^{\prime I}+2a_{E}S_{ij}^{I}\,\phantom{}^{t+\Delta t}e_{ij}^{\prime}+\phantom{}^{t+\Delta t}e_{ij}^{\prime}\,\phantom{}^{t+\Delta t}e_{ij}^{\prime}\:.\label{eq:111}
-\end{equation}
-
-\end_inset
-
-The second deviatoric stress invariant is therefore
-\begin_inset Formula 
-\begin{equation}
-\sqrt{^{t+\Delta t}J_{2}^{\prime}}=\frac{\sqrt{2}\,\phantom{}^{t+\Delta t}d-\lambda}{2a_{E}}\:,\label{eq:112}
-\end{equation}
-
-\end_inset
-
-and the pressure is computed from Equations 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:105"
-
-\end_inset
-
- and 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:108"
-
-\end_inset
-
- as:
-\begin_inset Formula 
-\begin{equation}
-^{t+\Delta t}P=\frac{^{t+\Delta t}I_{1}}{3}=\frac{1}{a_{m}}\left(^{t+\Delta t}\theta^{\prime}-\lambda\alpha_{g}\right)+P^{I}\:.\label{eq:113}
-\end{equation}
-
-\end_inset
-
-We then use the yield condition (
-\begin_inset Formula $^{t+\Delta t}f=0$
-\end_inset
-
-) and substitute for the stress invariants at 
-\begin_inset Formula $t+\Delta t$
-\end_inset
-
- to obtain:
-\begin_inset Formula 
-\begin{equation}
-\lambda=\frac{2a_{E}a_{m}\left(\frac{3\alpha_{f}}{a_{m}}\phantom{}^{t+\Delta t}\theta^{\prime}+\frac{^{t+\Delta t}d}{\sqrt{2}a_{E}}-\beta\bar{c}\right)}{6\alpha_{f}\alpha_{g}a_{E}+a_{m}}\:.\label{eq:114}
-\end{equation}
-
-\end_inset
-
-Since 
-\begin_inset Formula $\lambda$
-\end_inset
-
- is now known, we can substitute 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:112"
-
-\end_inset
-
- into 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:109"
-
-\end_inset
-
- to obtain
-\begin_inset Formula 
-\begin{equation}
-^{t+\Delta t}S_{ij}=\frac{\Delta e_{ij}^{P}\left(\sqrt{2}\,\phantom{\,}^{t+\Delta t}d-\lambda\right)}{\lambda a_{E}}\:.\label{eq:115}
-\end{equation}
-
-\end_inset
-
-Substituting this into Equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:105"
-
-\end_inset
-
-, we obtain the deviatoric plastic strain increment:
-\begin_inset Formula 
-\begin{equation}
-\Delta e_{ij}^{P}=\frac{\lambda}{\sqrt{2}\,\phantom{}^{t+\Delta t}d}\left(^{t+\Delta t}e_{ij}^{\prime}+a_{E}S_{ij}^{I}\right)\:.\label{eq:116}
-\end{equation}
-
-\end_inset
-
-We then use Equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:108"
-
-\end_inset
-
- and the second line of Equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:105"
-
-\end_inset
-
- to obtain the volumetric plastic strains and the pressure, and we use 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:116"
-
-\end_inset
-
- and the first line of Equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:105"
-
-\end_inset
-
- to obtain the deviatoric plastic strains and the deviatoric stresses.
-\end_layout
-
-\begin_layout Standard
-In certain cases where the mean stress is tensile, it is possible that the
- flow rule will not allow the stresses to project back to the yield surface,
- since they would project beyond the tip of the cone.
- Although this stress state is not likely to be encountered for quasi-static
- tectonic problems, it can occur for dynamic problems.
- One simple solution is to redefine the plastic multiplier, 
-\begin_inset Formula $\lambda$
-\end_inset
-
-.
- We do this by taking the smaller of the values yielded by Equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:114"
-
-\end_inset
-
- or by the following relation:
-\begin_inset Formula 
-\begin{equation}
-\lambda=\sqrt{2}\,\phantom{}^{t+\Delta t}d\:.\label{eq:127}
-\end{equation}
-
-\end_inset
-
-This is equivalent to setting the second deviatoric stress invariant to
- zero in Equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:110"
-
-\end_inset
-
-.
- By default, PyLith does not allow such tensile yield, since this would
- generally represent an error in problem setup for tectonic problems; however,
- for cases where such behavior is necessary, the material flag 
-\family typewriter
-allow_tensile_yield
-\family default
- may be set to 
-\family typewriter
-True
-\family default
-.
-\end_layout
-
-\begin_layout Standard
-To compute the elastoplastic tangent matrix we begin by writing Equation
- 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:105"
-
-\end_inset
-
- as a single expression in terms of stress and strain vectors:
-\begin_inset Formula 
-\begin{equation}
-^{t+\Delta t}\sigma_{i}=\frac{1}{a_{E}}\left(^{t+\Delta t}e_{i}^{\prime}-\Delta e_{i}^{P}\right)+S_{i}^{I}+\frac{R_{i}}{a_{m}}\left(^{t+\Delta t}\theta^{\prime}-\Delta\theta^{P}\right)+P^{I}\label{eq:117}
-\end{equation}
-
-\end_inset
-
-where
-\begin_inset Formula 
-\begin{gather}
-R_{i}=1\:;\; i=1,2,3\label{eq:118}\\
-R_{i}=0\:;\; i=4,5,6\:.\nonumber 
-\end{gather}
-
-\end_inset
-
-The elastoplastic tangent matrix is then given by
-\begin_inset Formula 
-\begin{equation}
-C_{ij}^{EP}=\frac{\partial\phantom{}^{t+\Delta t}\sigma_{i}}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}=\frac{1}{a_{E}}\left(\frac{\partial\phantom{}^{t+\Delta t}e_{i}^{\prime}}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}-\frac{\partial\Delta e_{i}^{P}}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}\right)+\frac{R_{i}}{a_{m}}\left(\frac{\partial\phantom{}^{t+\Delta t}\theta^{\prime}}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}-\frac{\partial\Delta\theta^{P}}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}\right)\:.\label{eq:119}
-\end{equation}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-From Equations 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:16"
-
-\end_inset
-
- and 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:106"
-
-\end_inset
-
-, we have
-\begin_inset Formula 
-\begin{equation}
-\frac{\partial\phantom{}^{t+\Delta t}e_{i}^{\prime}}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}=\frac{1}{3}\left[\begin{array}{cccccc}
-2 & -1 & -1 & 0 & 0 & 0\\
--1 & 2 & -1 & 0 & 0 & 0\\
--1 & -1 & 2 & 0 & 0 & 0\\
-0 & 0 & 0 & 3 & 0 & 0\\
-0 & 0 & 0 & 0 & 3 & 0\\
-0 & 0 & 0 & 0 & 0 & 3
-\end{array}\right]\:,\label{eq:120}
-\end{equation}
-
-\end_inset
-
-and from Equations 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:15"
-
-\end_inset
-
- and 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:106"
-
-\end_inset
-
- we have
-\begin_inset Formula 
-\begin{equation}
-\frac{\partial\phantom{}^{t+\Delta t}\theta^{\prime}}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}=\frac{R_{j}}{3}\:.\label{eq:121}
-\end{equation}
-
-\end_inset
-
-From Equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:116"
-
-\end_inset
-
- we have
-\begin_inset Formula 
-\begin{equation}
-\frac{\partial\Delta e_{i}^{P}}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}=\frac{1}{\sqrt{2}\,\phantom{}^{t+\Delta t}d}\left[\left(^{t+\Delta t}e_{i}^{\prime}+a_{E}S_{i}^{I}\right)\left(\frac{\partial\lambda}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}-\frac{\lambda}{\phantom{}^{t+\Delta t}d}\frac{\partial\phantom{}^{t+\Delta t}d}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}\right)+\lambda\frac{\partial\phantom{}^{t+\Delta t}e_{i}^{\prime}}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}\right]\:.\label{eq:122}
-\end{equation}
-
-\end_inset
-
-The derivative of 
-\begin_inset Formula $^{t+\Delta t}d$
-\end_inset
-
- is
-\begin_inset Formula 
-\begin{equation}
-\frac{\partial\phantom{}^{t+\Delta t}d}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}=\frac{a_{E}T_{j}^{I}+\phantom{}^{t+\Delta t}E_{j}}{\phantom{}^{t+\Delta t}d}\:,\label{eq:123}
-\end{equation}
-
-\end_inset
-
-where
-\begin_inset Formula 
-\begin{align}
-T_{j}^{I} & =S_{j}^{I}\;\mathrm{and}\;\phantom{}^{t+\Delta t}E_{j}=\phantom{}^{t+\Delta t}e_{j}^{\prime}\:;\; j=1,2,3\nonumber \\
-T_{j}^{I} & =2S_{j}^{I}\;\mathrm{and}\;\phantom{}^{t+\Delta t}E_{j}=2\phantom{}^{t+\Delta t}e_{j}^{\prime}\:;\; j=4,5,6\:.\label{eq:124}
-\end{align}
-
-\end_inset
-
-The derivative of 
-\begin_inset Formula $^{t+\Delta t}\lambda$
-\end_inset
-
- is a function of derivatives already computed:
-\begin_inset Formula 
-\begin{align}
-\frac{\partial\lambda}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}} & =\frac{2a_{E}a_{m}}{6\alpha_{f}\alpha_{g}a_{E}+a_{m}}\left(\frac{3\alpha_{f}}{a_{m}}\frac{\partial\phantom{}^{t+\Delta t}\theta^{\prime}}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}+\frac{1}{\sqrt{2}a_{E}}\frac{\partial\phantom{}^{t+\Delta t}d}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}\right)\nonumber \\
- & =\frac{2a_{E}a_{m}}{6\alpha_{f}\alpha_{g}a_{E}+a_{m}}\left(\frac{\alpha_{f}R_{j}}{a_{m}}+\frac{a_{E}T_{j}^{I}+\phantom{}^{t+\Delta t}E_{j}}{\sqrt{2}a_{E}\phantom{}^{t+\Delta t}d}\right)\:.\label{eq:125}
-\end{align}
-
-\end_inset
-
-Finally, from Equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:108"
-
-\end_inset
-
-, the derivative of the volumetric plastic strain increment is:
-\begin_inset Formula 
-\begin{equation}
-\frac{\partial\Delta\theta^{P}}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}=\alpha_{g}\frac{\partial\lambda}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}\:.\label{eq:126}
-\end{equation}
-
-\end_inset
-
-
-\begin_inset Float table
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\align center
-\begin_inset Caption
-
-\begin_layout Plain Layout
-\begin_inset CommandInset label
-LatexCommand label
-name "tab:druckerPrager"
-
-\end_inset
-
-Values in spatial database used as parameters in the Drucker-Prager elastoplasti
-c model with perfect plasticity.
-\end_layout
-
-\end_inset
-
-
-\begin_inset Tabular
-<lyxtabular version="3" rows="7" columns="2">
-<features tabularvalignment="middle">
-<column alignment="center" valignment="top" width="0">
-<column alignment="center" valignment="top" width="0">
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Parameter
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Name in Spatial Database
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $v_{p}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-vp
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $v_{s}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-vs
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\rho$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-density
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\phi$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-friction-angle
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\bar{c}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-cohesion
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\psi$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-dilatation-angle
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-In addition to the properties available for every material, the properties
- for the Drucker-Prager model also includes:
-\end_layout
-
-\begin_layout Description
-fit_mohr_coulomb Fit to the yield surface to the Mohr-Coulomb model (default
- is a inscribed).
-\end_layout
-
-\begin_layout Description
-allow_tensile_yield If true, allow yield beyond tensile strength; otherwise
- an error message will occur when the model fails beyond the tensile strength
- (default is false).
-\end_layout
-
-\begin_layout Standard
-An example of setting these parameters in a 
-\family typewriter
-.cfg
-\family default
- file is:
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.timedependent]
-\end_layout
-
-\begin_layout LyX-Code
-materials = [plastic]
-\end_layout
-
-\begin_layout LyX-Code
-materials.plastic = pylith.materials.DruckerPrager3D
-\end_layout
-
-\begin_layout LyX-Code
-
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.timedependent.materials.plastic]
-\end_layout
-
-\begin_layout LyX-Code
-fit_mohr_coulomb = inscribed ; default
-\end_layout
-
-\begin_layout LyX-Code
-allow_tensile_yield = False ; default
-\end_layout
-
-\begin_layout Section
-Initial State Variables
-\end_layout
-
-\begin_layout Standard
-In many problems of interest, the state variables describing a material
- model may already have nonzero values prior to the application of any boundary
- conditions.
- For problems in geophysics, the most common example is a problem that includes
- the effects of gravitational body forces.
- In the real earth, rocks were emplaced and formed under the influence of
- gravity.
- When performing numerical simulations, however, it is not possible to represent
- the entire time history of rock emplacement.
- Instead, gravity must be ``turned on'' at the beginning of the simulation.
- Unfortunately, this results in unrealistic amounts of deformation at the
- beginning of a simulation.
- An alternative is to provide initial state variables for the region under
- consideration.
- This allows the specification of a set of state variables that is consistent
- with the prior application of gravitational body forces.
- In a more general sense, initial values for state variables may be used
- to provide values that are consistent with any set of conditions that occurred
- prior to the beginning of a simulation.
- The current release of PyLith allows the specification of initial stresses,
- strains, and state variables for all materials; however, the initial state
- variables are not presently used.
- The reason for this is that the present version of PyLith always performs
- an elastic solution prior to performing the time-dependent solution, and
- the state variables are not used in the elastic solution.
- In future versions of PyLith, the solution of the elastic problem will
- be optional, so that initial state variables may be used.
-\end_layout
-
-\begin_layout Subsection
-Specification of Initial State Variables
-\end_layout
-
-\begin_layout Standard
-Since state variables are specific to a given material, initial values for
- state variables are specified as part of the material description.
- By default, initial state values are not used.
- To override this behavior, specify a spatial database for the initial stress,
- strain, and/or state variables as in the example from the tutorial in Section
- 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "sec:Tutorial-3d-hex8"
-
-\end_inset
-
-:
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.timedependent.materials.elastic]
-\end_layout
-
-\begin_layout LyX-Code
-db_initial_stress = spatialdata.spatialdb.SimpleDB
-\end_layout
-
-\begin_layout LyX-Code
-db_initial_stress.iohandler.filename = initial_stress.spatialdb
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float table
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\noindent
-\align center
-\begin_inset Caption
-
-\begin_layout Plain Layout
-Values in spatial database for initial state variables for 3D problems.
- 2D problems use only the relevant values.
- Note that initial stress and strain are available for all material models.
- Some models have additional state variables (Table 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "tab:material-model-output"
-
-\end_inset
-
-) and initial values for these may also be provided.
-\end_layout
-
-\end_inset
-
-
-\begin_inset Tabular
-<lyxtabular version="3" rows="3" columns="2">
-<features tabularvalignment="middle">
-<column alignment="center" valignment="middle" width="0.85in">
-<column alignment="center" valignment="middle" width="2.47in">
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-State Variable
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\align center
-
-\series bold
-Values in Spatial Database
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-initial stress
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-stress-xx, stress-yy, stress-zz, stress-xy, stress-yz, stress-xz
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-initial strain
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-strain-xx, strain-yy, strain-zz, strain-xy, strain-yz, strain-xz
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_body
-\end_document
+#LyX 2.0 created this file. For more info see http://www.lyx.org/
+\lyxformat 413
+\begin_document
+\begin_header
+\textclass book
+\begin_preamble
+\newcommand\Prefix[3]{\vphantom{#3}#1#2#3}
+\end_preamble
+\use_default_options false
+\maintain_unincluded_children false
+\language english
+\language_package default
+\inputencoding latin1
+\fontencoding global
+\font_roman default
+\font_sans default
+\font_typewriter default
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100
+\font_tt_scale 100
+
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry true
+\use_amsmath 1
+\use_esint 0
+\use_mhchem 1
+\use_mathdots 1
+\cite_engine basic
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\use_refstyle 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\leftmargin 1in
+\topmargin 1in
+\rightmargin 1in
+\bottommargin 2in
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\quotes_language english
+\papercolumns 1
+\papersides 2
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Chapter
+\begin_inset CommandInset label
+LatexCommand label
+name "cha:material:models"
+
+\end_inset
+
+Material Models
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status collapsed
+
+\begin_layout Plain Layout
+
+
+\backslash
+raggedbottom
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Specifying Material Properties
+\end_layout
+
+\begin_layout Standard
+Associating material properties with a given cell involves several steps.
+ 
+\end_layout
+
+\begin_layout Enumerate
+In the mesh generation process, assign a material identifier to each cell.
+\end_layout
+
+\begin_layout Enumerate
+Define material property groups corresponding to each material identifier.
+\end_layout
+
+\begin_layout Enumerate
+Set the parameters for each material group using 
+\family typewriter
+.cfg
+\family default
+ or 
+\family typewriter
+.pml
+\family default
+ files and/or command-line arguments.
+\end_layout
+
+\begin_layout Enumerate
+Specify the spatial variation in material property parameters using a spatial
+ database file.
+\end_layout
+
+\begin_layout Subsection
+Setting the Material Identifier
+\end_layout
+
+\begin_layout Standard
+Each cell in the finite-element mesh must have a material identifier.
+ This integer value is associated with a bulk material model.
+ The parameters of the material model need not be uniform for cells with
+ the same material identifier.
+ The bulk constitutive model and numerical integration (quadrature) scheme
+ will, however, be the same for all cells with the same material identifier
+ value.
+ The material identifier is set during the mesh generation process.
+ The procedure for assigning this integer value to a cell depends on the
+ mesh generator.
+ For example, in the PyLith mesh ASCII format, the identifiers are listed
+ in the cells group using the material-id data; in CUBIT materials are defined
+ using blocks; in LaGriT materials are defined by the attribute 
+\family typewriter
+imt1
+\family default
+ and the mregion command.
+\end_layout
+
+\begin_layout Subsection
+Material Property Groups
+\end_layout
+
+\begin_layout Standard
+The material property group associates a material model (label for the material,
+ a bulk constitutive model, and parameters for the constitutive model) with
+ a material identifier.
+ In previous versions of PyLith it was necessary to specify containers that
+ defined the number of groups and associated information for each group.
+ This was necessary because previous versions of Pyre did not support dynamic
+ arrays of components, and it was necessary to predefine these arrays.
+ More recent versions of Pythia do support this, however, and it is now
+ possible to define material property groups using a 
+\family typewriter
+.cfg
+\family default
+ file, a 
+\family typewriter
+.pml
+\family default
+ file, or on the command-line.
+ User-defined containers are no longer necessary, and the predefined containers
+ are no longer available (or necessary).
+ If a set of material groups is not specified, a single material model is
+ used for the entire problem.
+ See Sections 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sec:Tutorial-3d-hex8"
+
+\end_inset
+
+ and 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sec:Tutorial-3d-tet4"
+
+\end_inset
+
+ for examples that demonstrate how to specify more than one material model.
+\end_layout
+
+\begin_layout Subsection
+\begin_inset CommandInset label
+LatexCommand label
+name "sec:material:parameters"
+
+\end_inset
+
+Material Parameters
+\end_layout
+
+\begin_layout Standard
+For each material group, there is a single component defining the material
+ model to be used.
+ The default material model is 
+\family typewriter
+elasticisotropic3d
+\family default
+.
+ For each material model, the available components are:
+\end_layout
+
+\begin_layout Description
+db_properties Spatial database specifying the spatial variation in the parameter
+s of the bulk constitutive model (default is a SimpleDB).
+\end_layout
+
+\begin_layout Description
+db_initial_state Spatial database specifying the spatial variation in the
+ initial variables (default is none).
+\end_layout
+
+\begin_layout Description
+output The output manager used for outputting material information.
+\end_layout
+
+\begin_layout Description
+quadrature Numerical integration scheme used in integrating fields over
+ each cell.
+\end_layout
+
+\begin_layout Standard
+The properties for each material group are:
+\end_layout
+
+\begin_layout Description
+id This is the material identifier that matches the integer value assigned
+ to each cell in the mesh generation process.
+\end_layout
+
+\begin_layout Description
+label Name or label for the material.
+ This is used in error and diagnostic reports.
+\end_layout
+
+\begin_layout Standard
+An example of setting these parameters in a 
+\family typewriter
+.cfg
+\family default
+ file for a problem with two material groups is:
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.timedependent]
+\end_layout
+
+\begin_layout LyX-Code
+materials = [elastic,viscoelastic]
+\end_layout
+
+\begin_layout LyX-Code
+ 
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.timedependent.materials.elastic]
+\end_layout
+
+\begin_layout LyX-Code
+label = Elastic material
+\end_layout
+
+\begin_layout LyX-Code
+id = 1
+\end_layout
+
+\begin_layout LyX-Code
+db_properties.iohandler.filename = mat_elastic.spatialdb
+\end_layout
+
+\begin_layout LyX-Code
+quadrature.cell = pylith.feassemble.FIATLagrange
+\end_layout
+
+\begin_layout LyX-Code
+quadrature.cell.dimension = 3
+\end_layout
+
+\begin_layout LyX-Code
+ 
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.timedependent.materials.viscoelastic]
+\end_layout
+
+\begin_layout LyX-Code
+label = Viscoelastic material
+\end_layout
+
+\begin_layout LyX-Code
+id = 2
+\end_layout
+
+\begin_layout LyX-Code
+db_properties.iohandler.filename = mat_viscoelastic.spatialdb
+\end_layout
+
+\begin_layout LyX-Code
+quadrature.cell = pylith.feassemble.FIATLagrange
+\end_layout
+
+\begin_layout LyX-Code
+quadrature.cell.dimension = 3
+\end_layout
+
+\begin_layout Standard
+These settings correspond to the the problem in Section 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sec:Tutorial-3d-hex8"
+
+\end_inset
+
+.
+ The parameters for the bulk constitutive models are specified using the
+ spatial databases 
+\family typewriter
+mat_elastic.spatialdb
+\family default
+ and 
+\family typewriter
+mat_viscoelastic.spatialdb
+\family default
+.
+ Refer to the discussion of each material model to find the parameters that
+ must be specified in the spatial database.
+ Appendix 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sec:Spatialdata:SimpleIOAscii"
+
+\end_inset
+
+ describes the format of the SimpleDB spatial database files.
+ In a more realistic problem, a different spatial database, and possibly
+ a different material model, would be used for each material group.
+\end_layout
+
+\begin_layout Standard
+By default the output manager for a material does not output any information.
+ This behavior may be changed by specifying output files for each material
+ group:
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.timedependent.materials.elastic.output]
+\end_layout
+
+\begin_layout LyX-Code
+cell_filter = pylith.meshio.CellFilterAvgMesh
+\end_layout
+
+\begin_layout LyX-Code
+writer.filename = dislocation-elastic.vtk
+\end_layout
+
+\begin_layout LyX-Code
+ 
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.timedependent.materials.viscoelastic.output]
+\end_layout
+
+\begin_layout LyX-Code
+cell_filter = pylith.meshio.CellFilterAvgMesh
+\end_layout
+
+\begin_layout LyX-Code
+writer.filename = dislocation-viscoelastic.vtk
+\end_layout
+
+\begin_layout Standard
+These settings again correspond to the problem in Section 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sec:Tutorial-3d-hex8"
+
+\end_inset
+
+.
+ The specification of a state variable base filename (
+\family typewriter
+writer.filename
+\family default
+ settings) will cause two files to be created for each material group: an
+ info file, which describes the material properties used in the model, and
+ a state variables file, which contains the state variable information.
+ Note that the material properties described by the info file are the properties
+ used internally by PyLith.
+ They do not necessarily correspond to properties specified by the user.
+ If the problem has more than one time step, a state variable output file
+ will be created for each requested time step.
+ We have requested that the values be averaged over each cell.
+ Otherwise, output would be produced for each quadrature point, which can
+ cause problems with some visualization packages.
+ For this example problem, the material is three-dimensional isotropic elastic,
+ and is thus described by only three material properties (
+\begin_inset Formula $\lambda$
+\end_inset
+
+, 
+\begin_inset Formula $\mu$
+\end_inset
+
+, 
+\begin_inset Formula $\rho$
+\end_inset
+
+), as described below.
+ These properties are output by default.
+ Other material models require additional parameters, and if users want
+ these to be output, they must be specified.
+ Similarly, other material models require state variables in addition to
+ the default stress and strain variables that are used by all material models.
+ Additional output may be requested for a material model, as in this example
+ (see Section 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sec:Tutorial-Two-hexahedra"
+
+\end_inset
+
+):
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.timedependent.materials.material.output]
+\end_layout
+
+\begin_layout LyX-Code
+cell_data_fields = [total_strain,viscous_strain,stress]
+\end_layout
+
+\begin_layout LyX-Code
+cell_info_fields = [mu,lambda,density,maxwell_time]
+\end_layout
+
+\begin_layout Standard
+The properties and state variables available for output in each material
+ model are listed in Table 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "tab:material-model-output"
+
+\end_inset
+
+.
+ The order of the state variables in the output arrays is given in Table
+ 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "tab:material-model-statevars"
+
+\end_inset
+
+.
+ For the generalized Maxwell model, values of 
+\family typewriter
+shear_ratio
+\family default
+ and 
+\family typewriter
+maxwell_time
+\family default
+ are given for each Maxwell element in the model (there are presently three,
+ as described below).
+ Similarly, there are three sets of 
+\family typewriter
+viscous_strain
+\family default
+ values for the generalized Maxwell model.
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float table
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\align center
+\begin_inset Caption
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "tab:material-model-output"
+
+\end_inset
+
+Properties and state variables available for output for existing material
+ models.
+ Physical properties are available for output as 
+\family typewriter
+cell_info_fields
+\family default
+ and state variables are available for output as 
+\family typewriter
+cell_data_fields
+\family default
+.
+\end_layout
+
+\end_inset
+
+
+\begin_inset Tabular
+<lyxtabular version="3" rows="6" columns="4">
+<features tabularvalignment="middle">
+<column alignment="center" valignment="top" width="1.5in">
+<column alignment="center" valignment="top" width="1.8in">
+<column alignment="center" valignment="top" width="1.5in">
+<column alignment="center" valignment="top" width="1in">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Model
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Physical Properties
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+State Variables
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Requires nonlinear solver?
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Elastic
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+mu, lambda, density
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+total_strain, stress
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+No
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Maxwell Viscoelastic
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+mu, lambda, density, maxwell_time
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+total_strain,
+\end_layout
+
+\begin_layout Plain Layout
+
+\family typewriter
+stress, viscous_strain
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+No
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Generalized Maxwell Viscoelastic
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+mu, lambda, density,
+\end_layout
+
+\begin_layout Plain Layout
+
+\family typewriter
+shear_ratio,
+\end_layout
+
+\begin_layout Plain Layout
+
+\family typewriter
+maxwell_time
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+total_strain, stress, viscous_strain_1,
+\begin_inset Newline newline
+\end_inset
+
+viscous_strain_2,
+\begin_inset Newline newline
+\end_inset
+
+viscous_strain_3
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+No
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Power-law Viscoelastic
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+mu, lambda, density,
+\end_layout
+
+\begin_layout Plain Layout
+
+\family typewriter
+reference_strain_rate, reference_stress,
+\end_layout
+
+\begin_layout Plain Layout
+
+\family typewriter
+power_law_exponent
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+total_strain, stress, viscous_strain
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Yes
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Drucker-Prager Elastoplastic
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+mu, lambda, density, alpha_yield
+\family default
+,
+\family typewriter
+ beta, alpha_flow 
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+total_strain, stress, plastic_strain
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Yes
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Float table
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\align center
+\begin_inset Caption
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "tab:material-model-statevars"
+
+\end_inset
+
+Order of components in tensor state-variables for material models.
+\end_layout
+
+\end_inset
+
+
+\begin_inset Tabular
+<lyxtabular version="3" rows="5" columns="4">
+<features tabularvalignment="middle">
+<column alignment="center" valignment="top" width="1.25in">
+<column alignment="center" valignment="top" width="0.5in">
+<column alignment="center" valignment="top" width="1.25in">
+<column alignment="center" valignment="top" width="2.25in">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+State Variable
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+1D
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+2D
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+3D
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+total_strain
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\epsilon_{xx}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\epsilon_{xx}$
+\end_inset
+
+, 
+\begin_inset Formula $\epsilon_{yy}$
+\end_inset
+
+, 
+\begin_inset Formula $\epsilon_{xy}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\epsilon_{xx}$
+\end_inset
+
+, 
+\begin_inset Formula $\epsilon_{yy}$
+\end_inset
+
+, 
+\begin_inset Formula $\epsilon_{zz}$
+\end_inset
+
+, 
+\begin_inset Formula $\epsilon_{xy}$
+\end_inset
+
+, 
+\begin_inset Formula $\epsilon_{yz}$
+\end_inset
+
+, 
+\begin_inset Formula $\epsilon_{xz}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+stress
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\sigma_{xx}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\sigma_{xx}$
+\end_inset
+
+, 
+\begin_inset Formula $\sigma_{yy}$
+\end_inset
+
+, 
+\begin_inset Formula $\sigma_{xy}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\sigma_{xx}$
+\end_inset
+
+, 
+\begin_inset Formula $\sigma_{yy}$
+\end_inset
+
+, 
+\begin_inset Formula $\sigma_{zz}$
+\end_inset
+
+, 
+\begin_inset Formula $\sigma_{xy}$
+\end_inset
+
+, 
+\begin_inset Formula $\sigma_{yz}$
+\end_inset
+
+, 
+\begin_inset Formula $\sigma_{xz}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+viscous_strain, plastic_strain
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\epsilon_{xx}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\epsilon_{xx}$
+\end_inset
+
+, 
+\begin_inset Formula $\epsilon_{yy}$
+\end_inset
+
+, 
+\begin_inset Formula $\epsilon_{zz}$
+\end_inset
+
+, 
+\begin_inset Formula $\epsilon_{xy}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\epsilon_{xx}$
+\end_inset
+
+, 
+\begin_inset Formula $\epsilon_{yy}$
+\end_inset
+
+, 
+\begin_inset Formula $\epsilon_{zz}$
+\end_inset
+
+, 
+\begin_inset Formula $\epsilon_{xy}$
+\end_inset
+
+, 
+\begin_inset Formula $\epsilon_{yz}$
+\end_inset
+
+, 
+\begin_inset Formula $\epsilon_{xz}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+stress4
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\sigma_{xx}$
+\end_inset
+
+, 
+\begin_inset Formula $\sigma_{yy}$
+\end_inset
+
+, 
+\begin_inset Formula $\sigma_{zz}$
+\end_inset
+
+, 
+\begin_inset Formula $\sigma_{xy}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Float table
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\align center
+\begin_inset Caption
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "tab:material:output:components"
+
+\end_inset
+
+Order of components in tensor values in Xdmf files for 3D material models.
+ All nine tensor components are available even though the tensor is symmetric
+ with six independent components.
+\end_layout
+
+\end_inset
+
+
+\begin_inset Tabular
+<lyxtabular version="3" rows="7" columns="2">
+<features tabularvalignment="middle">
+<column alignment="center" valignment="top" width="1.25in">
+<column alignment="center" valignment="top" width="2.25in">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Index
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Component
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+0
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\epsilon_{xx}$
+\end_inset
+
+, 
+\begin_inset Formula $\sigma_{xx}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+1, 3
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\epsilon_{xy}$
+\end_inset
+
+, 
+\begin_inset Formula $\sigma_{xy}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+2, 6
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\epsilon_{xz}$
+\end_inset
+
+, 
+\begin_inset Formula $\sigma_{xz}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+4
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\epsilon_{yy}$
+\end_inset
+
+, 
+\begin_inset Formula $\sigma_{yy}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+5, 7
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\epsilon_{yz}$
+\end_inset
+
+, 
+\begin_inset Formula $\sigma_{yz}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+8
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\epsilon_{zz}$
+\end_inset
+
+, 
+\begin_inset Formula $\sigma_{zz}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Elastic Material Models
+\end_layout
+
+\begin_layout Standard
+The generalized form of Hooke's law relating stress and strain for linear
+ elastic materials is
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{gather}
+\sigma_{ij}=C_{ijkl}\left(\epsilon_{kl}-\epsilon_{kl}^{I}\right)+\sigma_{ij}^{I}\,,\label{eq:1}
+\end{gather}
+
+\end_inset
+
+where we have included both initial strains and initial stresses, denoted
+ with the superscript 
+\shape slanted
+I
+\shape default
+.
+ Due to symmetry considerations, however, the 81 components of the elasticity
+ matrix are reduced to 21 independent components for the most general case
+ of anisotropic elasticity.
+ Representing the stress and strain in terms of vectors, the constitutive
+ relation may be written
+\begin_inset Formula 
+\begin{gather}
+\overrightarrow{\sigma}=\underline{C}\left(\vec{\epsilon}-\vec{\epsilon}^{I}\right)+\vec{\sigma}^{I},\label{eq:2}
+\end{gather}
+
+\end_inset
+
+where
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{gather}
+\underline{C}=\left[\begin{array}{cccccc}
+C_{1111} & C_{1122} & C_{1133} & C_{1112} & C_{1123} & C_{1113}\\
+C_{1122} & C_{2222} & C_{2233} & C_{2212} & C_{2223} & C_{2213}\\
+C_{1133} & C_{2233} & C_{3333} & C_{3312} & C_{3323} & C_{3313}\\
+C_{1112} & C_{2212} & C_{3312} & C_{1212} & C_{1223} & C_{1213}\\
+C_{1123} & C_{2223} & C_{3323} & C_{1223} & C_{2323} & C_{2313}\\
+C_{1113} & C_{2213} & C_{3313} & C_{1213} & C_{2313} & C_{1313}
+\end{array}\right]\:.\label{eq:3}
+\end{gather}
+
+\end_inset
+
+For the case of isotropic elasticity, the number of independent components
+ reduces to two, and the model can be characterized by two parameters, Lame's
+ constants 
+\begin_inset Formula $\mu$
+\end_inset
+
+ and 
+\begin_inset Formula $\lambda$
+\end_inset
+
+.
+ Lame's constants are related to the density (
+\begin_inset Formula $\rho$
+\end_inset
+
+), shear wave speed (
+\begin_inset Formula $v_{s}$
+\end_inset
+
+), and compressional wave speed (
+\begin_inset Formula $v_{p}$
+\end_inset
+
+) via
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{equation}
+\begin{aligned}\mu= & \rho v_{s}^{2}\\
+\lambda= & \rho v_{p}^{2}-2\mu
+\end{aligned}
+\label{eq:4}
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float table
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\noindent
+\align center
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Values in spatial database used as parameters in the elastic material constituti
+ve models.
+\end_layout
+
+\end_inset
+
+
+\begin_inset Tabular
+<lyxtabular version="3" rows="4" columns="2">
+<features tabularvalignment="middle">
+<column alignment="center" valignment="middle" width="0.85in">
+<column alignment="center" valignment="middle" width="2.47in">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Parameter
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\align center
+
+\series bold
+Name in Spatial Database
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $v_{p}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+Vp
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $v_{s}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+Vs
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\rho$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+density
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+1D Elastic Material Models
+\end_layout
+
+\begin_layout Standard
+In 1D we can write Hooke's law as 
+\begin_inset Formula $\sigma_{11}=C_{1111}\left(\epsilon_{11}-\epsilon_{11}^{I}\right)+\sigma_{11}^{I}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsubsection
+Elastic 
+\begin_inset Quotes eld
+\end_inset
+
+Linear
+\begin_inset Quotes erd
+\end_inset
+
+ Strain
+\end_layout
+
+\begin_layout Standard
+For purely 1D axial deformation 
+\begin_inset Formula $C_{1111}=\lambda+2\mu$
+\end_inset
+
+, so we have
+\begin_inset Formula 
+\begin{equation}
+\sigma_{11}=(\lambda+2\mu)\left(\epsilon_{11}-\epsilon_{11}^{I}\right)+\sigma_{11}^{I},\label{eq:5}
+\end{equation}
+
+\end_inset
+
+with
+\begin_inset Formula 
+\begin{gather}
+\sigma_{22}=\sigma_{33}=\lambda\left(\epsilon_{11}-\epsilon_{11}^{I}\right)+\sigma_{22}^{I},\nonumber \\
+\sigma_{12}=\sigma_{23}=\sigma_{13}=0.\label{eq:6}
+\end{gather}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsubsection
+Elastic 
+\begin_inset Quotes eld
+\end_inset
+
+Linear
+\begin_inset Quotes erd
+\end_inset
+
+ Stress
+\end_layout
+
+\begin_layout Standard
+For deformation where the tractions are confined to the axial direction,
+ 
+\begin_inset Formula $C_{1111}=\frac{\mu(3\lambda+2\mu)}{\lambda+\mu}$
+\end_inset
+
+, so we have
+\begin_inset Formula 
+\begin{equation}
+\sigma_{11}=\frac{\mu(3\lambda+2\mu)}{\lambda+\mu}\left(\epsilon_{11}-\epsilon_{11}^{I}\right)+\sigma_{11}^{I},\label{eq:7}
+\end{equation}
+
+\end_inset
+
+with
+\begin_inset Formula 
+\begin{gather}
+\epsilon_{22}=\epsilon_{33}=-\frac{\lambda}{2(\lambda+\mu)}\epsilon_{11}+\epsilon_{22}^{I},\label{eq:8}\\
+\sigma_{22}=\sigma_{33}=\sigma_{12}=\sigma_{23}=\sigma_{13}=0.\nonumber 
+\end{gather}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+2D Elastic Material Models
+\end_layout
+
+\begin_layout Standard
+In 2D we can write Hooke's law as
+\begin_inset Formula 
+\begin{gather}
+\left[\begin{array}{c}
+\sigma_{11}\\
+\sigma_{22}\\
+\sigma_{12}
+\end{array}\right]=\left[\begin{array}{ccc}
+C_{1111} & C_{1122} & C_{1112}\\
+C_{1122} & C_{2222} & C_{2212}\\
+C_{1112} & C_{2212} & C_{1212}
+\end{array}\right]\left[\begin{array}{c}
+\epsilon_{11}-\epsilon_{11}^{I}\\
+\epsilon_{22}-\epsilon_{22}^{I}\\
+\epsilon_{12}-\epsilon_{12}^{I}
+\end{array}\right]+\left[\begin{array}{c}
+\sigma_{11}^{I}\\
+\sigma_{22}^{I}\\
+\sigma_{12}^{I}
+\end{array}\right]\:.\label{eq:9}
+\end{gather}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsubsection
+Elastic Plane Strain
+\end_layout
+
+\begin_layout Standard
+If the gradient in deformation with respect to the 
+\begin_inset Formula $x_{3}$
+\end_inset
+
+ axis is zero, then 
+\begin_inset Formula $\epsilon_{33}=\epsilon_{13}=\epsilon_{23}=0$
+\end_inset
+
+ and plane strain conditions apply, so we have 
+\begin_inset Formula 
+\begin{gather}
+\left[\begin{array}{c}
+\sigma_{11}\\
+\sigma_{22}\\
+\sigma_{12}
+\end{array}\right]=\left[\begin{array}{ccc}
+\lambda+2\mu & \lambda & 0\\
+\lambda & \lambda+2\mu & 0\\
+0 & 0 & 2\mu
+\end{array}\right]\left[\begin{array}{c}
+\epsilon_{11}-\epsilon_{11}^{I}\\
+\epsilon_{22}-\epsilon_{22}^{I}\\
+\epsilon_{12}-\epsilon_{12}^{I}
+\end{array}\right]+\left[\begin{array}{c}
+\sigma_{11}^{I}\\
+\sigma_{22}^{I}\\
+\sigma_{12}^{I}
+\end{array}\right]\:.\label{eq:10}
+\end{gather}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsubsection
+Elastic Plane Stress
+\end_layout
+
+\begin_layout Standard
+If the 
+\begin_inset Formula $x_{1}x_{2}$
+\end_inset
+
+ plane is traction free, then 
+\begin_inset Formula $\sigma_{33}=\sigma_{13}=\sigma_{23}=0$
+\end_inset
+
+ and plane stress conditions apply, so we have
+\begin_inset Formula 
+\begin{gather}
+\left[\begin{array}{c}
+\sigma_{11}\\
+\sigma_{22}\\
+\sigma_{12}
+\end{array}\right]=\left[\begin{array}{ccc}
+\frac{4\mu(\lambda+\mu)}{\lambda+2\mu} & \frac{2\mu\lambda}{\lambda+2\mu} & 0\\
+\frac{2\mu\lambda}{\lambda+2} & \frac{4\mu(\lambda+\mu)}{\lambda+2\mu} & 0\\
+0 & 0 & 2\mu
+\end{array}\right]\left[\begin{array}{c}
+\epsilon_{11}-\epsilon_{11}^{I}\\
+\epsilon_{22}-\epsilon_{22}^{I}\\
+\epsilon_{12}-\epsilon_{12}^{I}
+\end{array}\right]+\left[\begin{array}{c}
+\sigma_{11}^{I}\\
+\sigma_{22}^{I}\\
+\sigma_{12}^{I}
+\end{array}\right]\:,\label{eq:11}
+\end{gather}
+
+\end_inset
+
+where
+\begin_inset Formula 
+\begin{equation}
+\begin{gathered}\epsilon_{33}=-\frac{\lambda}{\lambda+2\mu}(\epsilon_{11}+\epsilon_{22})+\epsilon_{33}^{I}\\
+\epsilon_{13}=\epsilon_{23}=0\,.
+\end{gathered}
+\label{eq:12}
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+3D Elastic Material Models
+\end_layout
+
+\begin_layout Subsubsection
+Isotropic
+\end_layout
+
+\begin_layout Standard
+For this case the stress-strain matrix, 
+\begin_inset Formula $\underline{C}$
+\end_inset
+
+, becomes
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{gather}
+\underline{C}=\left[\begin{array}{cccccc}
+\lambda+2\mu & \lambda & \lambda & 0 & 0 & 0\\
+\lambda & \lambda+2\mu & \lambda & 0 & 0 & 0\\
+\lambda & \lambda & \lambda+2\mu & 0 & 0 & 0\\
+0 & 0 & 0 & 2\mu & 0 & 0\\
+0 & 0 & 0 & 0 & 2\mu & 0\\
+0 & 0 & 0 & 0 & 0 & 2\mu
+\end{array}\right]\:.\label{eq:13}
+\end{gather}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+\begin_inset CommandInset label
+LatexCommand label
+name "sec:Viscoelastic-Materials"
+
+\end_inset
+
+Viscoelastic Materials
+\end_layout
+
+\begin_layout Standard
+At present, there are six viscoelastic material models available in PyLith
+ (Table 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "tab:Viscoelastic-models-available"
+
+\end_inset
+
+ and Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:Material-models"
+
+\end_inset
+
+).
+ Future code versions may include alternative formulations for the various
+ material models (Appendix 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "cha:Alternative-Formulations"
+
+\end_inset
+
+), so that users may use the most efficient formulation for a particular
+ problem.
+ Note that both 2D and 3D viscoelastic models are described, but we present
+ below only the 3D formulations.
+ The 2D formulations are easily obtained from the plane strain definition.
+ The one aspect of the 2D formulations that is different is the specification
+ of initial stresses.
+ Since 2D models only have three tensor components, it is not possible to
+ specify the normal stress in the out-of-plane direction (
+\begin_inset Formula $\sigma_{33}$
+\end_inset
+
+), which is generally nonzero, using the same method as the other tensor
+ components.
+ To allow for the specification of this initial stress component, an additional
+ state variable corresponding to 
+\begin_inset Formula $\sigma_{33}^{I}$
+\end_inset
+
+ is provided (
+\family typewriter
+stress_zz_initial
+\family default
+).
+ Note that this is not an issue for the 2D elastic models, since this initial
+ stress component is not needed.
+ For the PowerLawPlaneStrain model, all four of the stress components are
+ needed, so a 4-component stress state variable (
+\family typewriter
+stress4
+\family default
+) is provided in addition to the normal 3-component 
+\family typewriter
+stress
+\family default
+ state variable (see Table 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "tab:material-model-output"
+
+\end_inset
+
+).
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float table
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\noindent
+\align center
+\begin_inset Caption
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "tab:Viscoelastic-models-available"
+
+\end_inset
+
+Available viscoelastic materials for PyLith.
+\end_layout
+
+\end_inset
+
+
+\begin_inset Tabular
+<lyxtabular version="3" rows="7" columns="2">
+<features tabularvalignment="middle">
+<column alignment="left" valignment="top" width="2.85in">
+<column alignment="center" valignment="top" width="2.47in">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Model Name
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Description
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+MaxwellPlaneStrain
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Plane strain Maxwell material with linear viscous rheology
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+GenMaxwellPlaneStrain
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Plane strain generalized Maxwell material (3 Maxwell models in parallel)
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+PowerLawPlaneStrain
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Plane strain Maxwell material with power-law viscous rheology
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+MaxwellIsotropic3D
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Isotropic Maxwell material with linear viscous rheology
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+GenMaxwellIsotropic3D
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Generalized model consisting of 3 Maxwell models in parallel
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+PowerLaw3D
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Isotropic Maxwell material with power-law viscous rheology
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float figure
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\align center
+\begin_inset Graphics
+	filename figs/pylith-materials.eps
+	scale 75
+
+\end_inset
+
+
+\begin_inset Caption
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:Material-models"
+
+\end_inset
+
+Spring-dashpot 1D representations of the available 3D elastic and 2D/3D
+ viscoelastic material models for PyLith.
+ The top model is a linear elastic model, the middle model is a Maxwell
+ model, and the bottom model is a generalized Maxwell model.
+ For the generalized Maxwell model, 
+\begin_inset Formula $\lambda$
+\end_inset
+
+ and 
+\begin_inset Formula $\mu_{tot}$
+\end_inset
+
+ are specified for the entire model, and then the ratio 
+\begin_inset Formula $\mu_{i}/\mu_{tot}$
+\end_inset
+
+ is specified for each Maxwell model.
+ For the power-law model, the linear dashpot in the Maxwell model is replaced
+ by a nonlinear dashpot obeying a power-law.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Definitions
+\end_layout
+
+\begin_layout Standard
+In the following sections, we use a combination of vector and index notation
+ (our notation conventions are shown in Table 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "tab:Mathematical-notation"
+
+\end_inset
+
+).
+ When using index notation, we use the common convention where repeated
+ indices indicate summation over the range of the index.
+ We also make frequent use of the scalar inner product.
+ The scalar inner product of two second-order tensors may be written
+\begin_inset Formula 
+\begin{gather}
+\underline{a}\cdot\underline{b}=a_{ij}b_{ij}\,.\label{eq:14}
+\end{gather}
+
+\end_inset
+
+Although the general constitutive relations are formulated in terms of the
+ stress and strain, we frequently make use of the deviatoric stress and
+ strain in our formulation.
+ We first define the mean stress, 
+\begin_inset Formula $P$
+\end_inset
+
+, and mean strain, 
+\begin_inset Formula $\theta$
+\end_inset
+
+:
+\begin_inset Formula 
+\begin{gather}
+P=\frac{\sigma_{ii}}{3}\,,\,\,\,\,\theta=\frac{\epsilon_{ii}}{3}\,,\label{eq:15}
+\end{gather}
+
+\end_inset
+
+where the 
+\begin_inset Formula $\sigma_{ii}$
+\end_inset
+
+ and 
+\begin_inset Formula $\epsilon_{ii}$
+\end_inset
+
+ represent the trace of the stress and strain tensors, respectively.
+ We then define the deviatoric components of stress and strain as
+\begin_inset Formula 
+\begin{gather}
+S_{ij}=\sigma_{ij}-P\delta_{ij}\,,\,\,\,\, e_{ij}=\epsilon_{ij}-\theta\delta_{ij}\,,\label{eq:16}
+\end{gather}
+
+\end_inset
+
+where 
+\begin_inset Formula $\delta_{ij}$
+\end_inset
+
+ is the Kronecker delta.
+ Using the deviatoric components, we define the effective stress, 
+\begin_inset Formula $\overline{\sigma}$
+\end_inset
+
+, the second deviatoric stress invariant, 
+\begin_inset Formula $J_{2}^{\prime}$
+\end_inset
+
+, the effective deviatoric strain, 
+\begin_inset Formula $\overline{e}$
+\end_inset
+
+, and the second deviatoric strain invariant, 
+\begin_inset Formula $L_{2}^{\prime}$
+\end_inset
+
+, as
+\begin_inset Formula 
+\begin{gather}
+\overline{\sigma}=\sqrt{\frac{3}{2}\underline{S}\cdot\underline{S}}\,\,\nonumber \\
+J_{2}^{\prime}=\frac{1}{2}\underline{S}\cdot\underline{S}\,.\label{eq:17}\\
+\overline{e}=\sqrt{\frac{2}{3}\underline{e}\cdot\underline{e}}\,\,\nonumber \\
+L_{2}^{\prime}=\frac{1}{2}\underline{e}\cdot\underline{e}\,\,\nonumber 
+\end{gather}
+
+\end_inset
+
+Due to the symmetry of the stress and strain tensors, it is sometimes convenient
+ to represent them as vectors:
+\begin_inset Formula 
+\begin{gather}
+\overrightarrow{\sigma^{T}}=\left[\begin{array}{cccccc}
+\sigma_{11} & \sigma_{22} & \sigma_{33} & \sigma_{12} & \sigma_{23} & \sigma_{31}\end{array}\right]\label{eq:18}\\
+\overrightarrow{\epsilon^{T}}=\left[\begin{array}{cccccc}
+\epsilon_{11} & \epsilon_{22} & \epsilon_{33} & \epsilon_{12} & \epsilon_{23} & \epsilon_{31}\end{array}\right]\:.\nonumber 
+\end{gather}
+
+\end_inset
+
+Note that when taking the scalar inner product of two tensors represented
+ as vectors, it is necessary to double the products representing off-diagonal
+ terms.
+\end_layout
+
+\begin_layout Standard
+For quantities evaluated over a specific time period, we represent the initial
+ time as a prefixed subscript and the end time as a prefixed superscript.
+ In cases where the initial time does not appear, it is understood to be
+ 
+\begin_inset Formula $-\infty$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float table
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\noindent
+\align center
+\begin_inset Caption
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "tab:Mathematical-notation"
+
+\end_inset
+
+Mathematical notation used in this section.
+\end_layout
+
+\end_inset
+
+
+\begin_inset Tabular
+<lyxtabular version="3" rows="3" columns="3">
+<features tabularvalignment="middle">
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="0">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Index notation
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Vector notation
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Description
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $a_{i}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\overrightarrow{a}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Vector field a
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $a_{ij}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\underline{a}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Second order tensor field a
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Linear Viscoelastic Models
+\end_layout
+
+\begin_layout Standard
+Linear viscoelastic models are obtained by various combinations of a linear
+ elastic spring and a linear viscous dashpot in series or parallel.
+ The simplest example is probably the linear Maxwell model, which consists
+ of a spring in series with a dashpot, as shown in Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:Material-models"
+
+\end_inset
+
+.
+ For a one-dimensional model, the response is given by
+\begin_inset Formula 
+\begin{equation}
+\frac{d\epsilon_{Total}}{dt}=\frac{d\epsilon_{D}}{dt}+\frac{d\epsilon_{S}}{dt}=\frac{\sigma}{\eta}+\frac{1}{E}\frac{d\sigma}{dt}\:,
+\end{equation}
+
+\end_inset
+
+where 
+\begin_inset Formula $\epsilon_{Total}$
+\end_inset
+
+ is the total strain, 
+\begin_inset Formula $\epsilon_{D}$
+\end_inset
+
+ is the strain in the dashpot, 
+\begin_inset Formula $\epsilon_{S}$
+\end_inset
+
+ is the strain in the spring, 
+\begin_inset Formula $\sigma$
+\end_inset
+
+ is the stress, 
+\begin_inset Formula $\eta$
+\end_inset
+
+ is the viscosity of the dashpot, and 
+\begin_inset Formula $E$
+\end_inset
+
+ is the spring constant.
+ When a Maxwell material is subjected to constant strain, the stresses relax
+ exponentially with time.
+ When a Maxwell material is subjected to a constant stress, there is an
+ immediate elastic strain, corresponding to the response of the spring,
+ and a viscous strain that increases linearly with time.
+ Since the strain response is unbounded, the Maxwell model actually represents
+ a fluid.
+\end_layout
+
+\begin_layout Standard
+Another simple model is the Kelvin-Voigt model, which consists of a spring
+ in parallel with a dashpot.
+ In this case, the one-dimensional response is given by
+\begin_inset Formula 
+\begin{equation}
+\sigma\left(t\right)=E\epsilon\left(t\right)+\eta\frac{d\epsilon\left(t\right)}{dt}\:.
+\end{equation}
+
+\end_inset
+
+As opposed to the Maxwell model, which represents a fluid, the Kelvin-Voigt
+ model represents a solid undergoing reversible, viscoelastic strain.
+ If the material is subjected to a constant stress, it deforms at a decreasing
+ rate, gradually approaching the strain that would occur for a purely elastic
+ material.
+ When the stress is released, the material gradually relaxes back to its
+ undeformed state.
+\end_layout
+
+\begin_layout Standard
+The most general form of linear viscoelastic model is the generalized Maxwell
+ model, which consists of a spring in parallel with a number of Maxwell
+ models (see Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:Material-models"
+
+\end_inset
+
+).
+ Using this model, it is possible to represent a number of simpler viscoelastic
+ models.
+ For example, a simple Maxwell model is obtained by setting the elastic
+ constants of all springs to zero, with the exception of the spring contained
+ in the first Maxwell model (
+\begin_inset Formula $\mu_{1}$
+\end_inset
+
+).
+ Similarly, the Kelvin-Voigt model may be obtained by setting the elastic
+ constants 
+\begin_inset Formula $\mu_{2}=\mu_{3}=0$
+\end_inset
+
+, and setting 
+\begin_inset Formula $\mu_{1}=\infty$
+\end_inset
+
+ (or a very large number).
+\end_layout
+
+\begin_layout Subsection
+Formulation for Generalized Maxwell Models
+\begin_inset CommandInset label
+LatexCommand label
+name "sub:Formulation-for-Gen-Max"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+As described above, the generalized Maxwell viscoelastic model consists
+ of a number of Maxwell linear viscoelastic models in parallel with a spring,
+ as shown in Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:Material-models"
+
+\end_inset
+
+.
+ PyLith includes the specific case of a spring in parallel with three Maxwell
+ models.
+ As described in the previous paragraph, a number of common material models
+ may be obtained from this model by setting the shear moduli of various
+ springs to zero or infinity (or a large number), such as the Maxwell model,
+ the Kelvin model, and the standard linear solid.
+ We follow formulations similar to those used by Zienkiewicz and Taylor
+ 
+\begin_inset CommandInset citation
+LatexCommand cite
+key "Zienkiewicz:Taylor:2000"
+
+\end_inset
+
+ and Taylor 
+\begin_inset CommandInset citation
+LatexCommand cite
+key "Taylor:2003"
+
+\end_inset
+
+.
+ In this formulation, we specify the total shear modulus of the model (
+\begin_inset Formula $\mu_{tot}$
+\end_inset
+
+) and Lame's constant (
+\begin_inset Formula $\lambda$
+\end_inset
+
+).
+ We then provide the fractional shear modulus for each Maxwell element spring
+ in the model.
+ It is not necessary to specify the fractional modulus for 
+\begin_inset Formula $\mu_{0}$
+\end_inset
+
+, since this is obtained by subtracting the sum of the other ratios from
+ 1.
+ Note that the sum of all these fractions must equal 1.
+ We use a similar formulation for our linear Maxwell viscoelastic model,
+ but in that case 
+\begin_inset Formula $\mu_{0}$
+\end_inset
+
+ is always zero and we only use a single Maxwell model.
+ The parameters defining the standard Maxwell model are shown in Table 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "tab:linearMaxwell"
+
+\end_inset
+
+, and those defining the generalized Maxwell model are shown in Table 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "tab:genMaxwell"
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+As for all our viscoelastic models, the volumetric strain is completely
+ elastic, and the viscoelastic deformation may be expressed purely in terms
+ of the deviatoric components:
+\begin_inset Formula 
+\begin{equation}
+\underline{S}=2\mu_{tot}\left[\mu_{0}\underline{e}+\sum_{i=1}^{N}\mu_{i}\underline{q}^{i}-\underline{e}^{I}\right]+\underline{S}^{I}\,;\; P=3K\left(\theta-\theta^{I}\right)+P^{I}\,,\label{eq:19}
+\end{equation}
+
+\end_inset
+
+where 
+\shape slanted
+K
+\shape default
+ is the bulk modulus, 
+\begin_inset Formula $N$
+\end_inset
+
+ is the number of Maxwell models, and the variable 
+\begin_inset Formula $\underline{q}^{i}$
+\end_inset
+
+ follows the evolution equations
+\begin_inset Formula 
+\begin{equation}
+\underline{\dot{q}}^{i}+\frac{1}{\lambda_{i}}\underline{q}^{i}=\underline{\dot{e}}.\label{eq:20}
+\end{equation}
+
+\end_inset
+
+The 
+\begin_inset Formula $\lambda_{i}$
+\end_inset
+
+ are the relaxation times for each Maxwell model.
+\end_layout
+
+\begin_layout Standard
+An alternative to the differential equation form above is an integral equation
+ form expressed in terms of the relaxation modulus function.
+ This function is defined in terms of an idealized experiment in which,
+ at time labeled zero (
+\begin_inset Formula $t=0$
+\end_inset
+
+), a specimen is subjected to a constant strain, 
+\begin_inset Formula $\underline{e}_{0}$
+\end_inset
+
+, and the stress response, 
+\begin_inset Formula $\underline{S}\left(t\right)$
+\end_inset
+
+, is measured.
+ For a linear material we obtain:
+\begin_inset Formula 
+\begin{equation}
+\underline{S}\left(t\right)=2\mu\left(t\right)\left(\underline{e}_{0}-\underline{e}^{I}\right)+\underline{S}^{I}\,,\label{eq:21}
+\end{equation}
+
+\end_inset
+
+where 
+\begin_inset Formula $\mu\left(t\right)$
+\end_inset
+
+ is the shear relaxation modulus function.
+ Using linearity and superposition for an arbitrary state of strain yields
+ an integral equation:
+\begin_inset Formula 
+\begin{equation}
+\underline{S}\left(t\right)=\intop_{-\infty}^{t}\mu\left(t-\tau\right)\underline{\dot{e}}\, d\tau\,.\label{eq:22}
+\end{equation}
+
+\end_inset
+
+If we assume the modulus function in Prony series form we obtain
+\begin_inset Formula 
+\begin{equation}
+\mu\left(t\right)=\mu_{tot}\left(\mu_{0}+\sum_{i=1}^{N}\mu_{i}\exp\frac{-t}{\lambda_{i}}\right)\,,\label{eq:23}
+\end{equation}
+
+\end_inset
+
+where
+\begin_inset Formula 
+\begin{equation}
+\mu_{0}+\sum_{i=1}^{N}\mu_{i}=1\,.\label{eq:24}
+\end{equation}
+
+\end_inset
+
+With the form in Equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:23"
+
+\end_inset
+
+, the integral equation form is identical to the differential equation form.
+\end_layout
+
+\begin_layout Standard
+If we assume the material is undisturbed until a strain is suddenly applied
+ at time zero, we can divide the integral into
+\begin_inset Formula 
+\begin{equation}
+\intop_{-\infty}^{t}\left(\cdot\right)\, d\tau=\intop_{-\infty}^{0^{-}}\left(\cdot\right)\, d\tau+\intop_{0^{-}}^{0^{+}}\left(\cdot\right)\, d\tau+\intop_{0^{+}}^{t}\left(\cdot\right)\, d\tau\,.\label{eq:27}
+\end{equation}
+
+\end_inset
+
+The first term is zero, the second term includes a jump term associated
+ with 
+\begin_inset Formula $\underline{e}_{0}$
+\end_inset
+
+ at time zero, and the last term covers the subsequent history of strain.
+ Applying this separation to Equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:22"
+
+\end_inset
+
+,
+\begin_inset Formula 
+\begin{equation}
+\underline{S}\left(t\right)=2\mu\left(t\right)\left(\underline{e}_{0}-\underline{e}^{I}\right)+\underline{S}^{I}+2\int_{0}^{t}\mu\left(t-\tau\right)\underline{\dot{e}}\left(\tau\right)\, d\tau\,,\label{eq:28}
+\end{equation}
+
+\end_inset
+
+where we have left the sign off of the lower limit on the integral.
+\end_layout
+
+\begin_layout Standard
+Substituting Equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:23"
+
+\end_inset
+
+ into 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:28"
+
+\end_inset
+
+, we obtain
+\begin_inset Formula 
+\begin{equation}
+\underline{S}\left(t\right)=2\mu_{tot}\left\{ \mu_{0}\underline{e}\left(t\right)+\sum_{i=1}^{N}\left[\mu_{i}\exp\frac{-t}{\lambda_{i}}\left(\underline{e}_{0}+\intop_{0}^{t}\exp\frac{t}{\lambda_{i}}\underline{\dot{e}}\left(\tau\right)\, d\tau\right)\right]-\underline{e}^{I}\right\} +\underline{S}^{I}\,.\label{eq:29}
+\end{equation}
+
+\end_inset
+
+We then split each integral into two ranges: from 0 to 
+\begin_inset Formula $t_{n}$
+\end_inset
+
+, and from 
+\begin_inset Formula $t_{n}$
+\end_inset
+
+ to 
+\begin_inset Formula $t$
+\end_inset
+
+, and define each integral as
+\begin_inset Formula 
+\begin{equation}
+\underline{i}_{i}^{1}\left(t\right)=\intop_{0}^{t}\exp\frac{\tau}{\lambda_{i}}\underline{\dot{e}}\left(\tau\right)\, d\tau\,.\label{eq:30}
+\end{equation}
+
+\end_inset
+
+The integral then becomes
+\begin_inset Formula 
+\begin{equation}
+\underline{i}_{i}^{1}\left(t\right)=\underline{i}_{i}^{1}\left(t_{n}\right)+\intop_{t_{n}}^{t}\exp\frac{\tau}{\lambda_{i}}\underline{\dot{e}}\left(\tau\right)\, d\tau\,.\label{eq:31}
+\end{equation}
+
+\end_inset
+
+Including the negative exponential multiplier:
+\begin_inset Formula 
+\begin{equation}
+\underline{h}_{i}^{1}\left(t\right)=\exp\frac{-t}{\lambda_{i}}\underline{i}_{i}^{1}\,.\label{eq:32}
+\end{equation}
+
+\end_inset
+
+Then
+\begin_inset Formula 
+\begin{equation}
+\underline{h}_{i}^{1}\left(t\right)=\exp\frac{-\Delta t}{\lambda_{i}}\underline{h}_{i}^{1}\left(t_{n}\right)+\Delta\underline{h}_{i}\,,\label{eq:33}
+\end{equation}
+
+\end_inset
+
+where
+\begin_inset Formula 
+\begin{equation}
+\Delta\underline{h}_{i}=\exp\frac{-t}{\lambda_{i}}\intop_{t_{n}}^{t}\exp\frac{\tau}{\lambda_{i}}\underline{\dot{e}}\left(\tau\right)\, d\tau\,.\label{eq:34}
+\end{equation}
+
+\end_inset
+
+Approximating the strain rate as constant over each time step, the solution
+ may be found as
+\begin_inset Formula 
+\begin{equation}
+\Delta\underline{h}_{i}=\frac{\lambda_{i}}{\Delta t}\left(1-\exp\frac{-\Delta t}{\lambda_{i}}\right)\left(\underline{e}-\underline{e}_{n}\right)=\Delta h_{i}\left(\underline{e}-\underline{e}_{n}\right)\,.\label{eq:35}
+\end{equation}
+
+\end_inset
+
+The approximation is singular for zero time steps, but a series expansion
+ may be used for small time-step sizes:
+\begin_inset Formula 
+\begin{equation}
+\Delta h_{i}\approx1-\frac{1}{2}\left(\frac{\Delta t}{\lambda_{i}}\right)+\frac{1}{3!}\left(\frac{\Delta t}{\lambda_{i}}\right)^{2}-\frac{1}{4!}\left(\frac{\Delta t}{\lambda_{i}}\right)^{3}+\cdots\,.\label{eq:36}
+\end{equation}
+
+\end_inset
+
+This converges with only a few terms.
+ With this formulation, the constitutive relation now has the simple form:
+\begin_inset Formula 
+\begin{equation}
+\underline{S}\left(t\right)=2\mu_{tot}\left(\mu_{0}\underline{e}\left(t\right)+\sum_{i=1}^{N}\mu_{i}\underline{h}_{i}^{1}\left(t\right)-\underline{e}^{I}\right)+\underline{S}^{I}\,.\label{eq:37}
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+We need to compute the tangent constitutive matrix when forming the stiffness
+ matrix.
+ In addition to the volumetric contribution to the tangent constitutive
+ matrix, we require the deviatoric part:
+\begin_inset Formula 
+\begin{equation}
+\frac{\partial\underline{S}}{\partial\underline{\epsilon}}=\frac{\partial\underline{S}}{\partial\underline{e}}\frac{\partial\underline{e}}{\partial\underline{\epsilon}}\,,\label{eq:38}
+\end{equation}
+
+\end_inset
+
+where the second derivative on the right may be easily deduced from Equation
+ 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:16"
+
+\end_inset
+
+.
+ The other derivative is given by
+\begin_inset Formula 
+\begin{equation}
+\frac{\partial\underline{S}}{\partial\underline{e}}=2\mu_{tot}\left[\mu_{0}\underline{I}+\sum_{i=1}^{N}\mu_{i}\frac{\partial\underline{h}_{i}^{1}}{\partial\underline{e}}\right]\,,\label{eq:39}
+\end{equation}
+
+\end_inset
+
+where 
+\begin_inset Formula $\underline{I}$
+\end_inset
+
+ is the identity matrix.
+ From Equations 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:33"
+
+\end_inset
+
+ through 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:35"
+
+\end_inset
+
+, the derivative inside the brackets is
+\begin_inset Formula 
+\begin{equation}
+\frac{\partial\underline{h}_{i}^{1}}{\partial\underline{e}}=\Delta h_{i}\left(\Delta t\right)\underline{I}\,.\label{eq:40}
+\end{equation}
+
+\end_inset
+
+The complete deviatoric tangent relation is then
+\begin_inset Formula 
+\begin{equation}
+\frac{\partial\underline{S}}{\partial\underline{\epsilon}}=2\mu_{tot}\left[\mu_{0}+\sum_{i=1}^{N}\mu_{i}\Delta h_{i}\left(\Delta t\right)\right]\frac{\partial\underline{e}}{\partial\underline{\epsilon}}\,.\label{eq:41}
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+We use this formulation for both our Maxwell and generalized Maxwell viscoelasti
+c models.
+ For the Maxwell model, 
+\begin_inset Formula $\mu_{0}=0$
+\end_inset
+
+ and 
+\begin_inset Formula $N=1$
+\end_inset
+
+.
+ For the generalized Maxwell model, 
+\begin_inset Formula $N=3.$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float table
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\noindent
+\align center
+\begin_inset Caption
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "tab:linearMaxwell"
+
+\end_inset
+
+Values in spatial database used as parameters in the linear Maxwell viscoelastic
+ material constitutive model.
+\end_layout
+
+\end_inset
+
+
+\begin_inset Tabular
+<lyxtabular version="3" rows="5" columns="2">
+<features tabularvalignment="middle">
+<column alignment="center" valignment="middle" width="0.85in">
+<column alignment="center" valignment="middle" width="2.47in">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Parameter
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\align center
+
+\series bold
+Name in Spatial Database
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $v_{p}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+Vp
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $v_{s}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+Vs
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\rho$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+density
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\eta$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+viscosity
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float table
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\noindent
+\align center
+\begin_inset Caption
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "tab:genMaxwell"
+
+\end_inset
+
+Values in spatial database used as parameters in the generalized linear
+ Maxwell viscoelastic material constitutive model.
+\end_layout
+
+\end_inset
+
+
+\begin_inset Tabular
+<lyxtabular version="3" rows="10" columns="2">
+<features tabularvalignment="middle">
+<column alignment="center" valignment="middle" width="0.85in">
+<column alignment="center" valignment="middle" width="2.47in">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Parameter
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\align center
+
+\series bold
+Name in Spatial Database
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $v_{p}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+Vp
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $v_{s}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+Vs
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\rho$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+density
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\mu_{1}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+shear-ratio-1
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\mu_{2}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+shear-ratio-2
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\mu_{3}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+shear-ratio-3
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\eta_{1}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+viscosity-1
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\eta_{2}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+viscosity-2
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\eta_{3}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+viscosity-3
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+\begin_inset CommandInset label
+LatexCommand label
+name "sub:Effective-Stress-Formulations-Viscoelastic"
+
+\end_inset
+
+Effective Stress Formulations for Viscoelastic Materials
+\end_layout
+
+\begin_layout Standard
+As an alternative to the approach outlined above, an effective stress function
+ formulation 
+\begin_inset CommandInset citation
+LatexCommand cite
+key "Kojic:Bathe:1987"
+
+\end_inset
+
+ may be employed for both a linear Maxwell model and a power-law Maxwell
+ model.
+ Note that this formulation is not presently employed for linear viscoelastic
+ models (see Appendix 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "cha:Alternative-Formulations"
+
+\end_inset
+
+), but it is used for power-law viscoelastic materials.
+ For the viscoelastic materials considered here, the viscous volumetric
+ strains are zero (incompressible flow), and it is convenient to separate
+ the general stress-strain relationship at time 
+\begin_inset Formula $t+\Delta t$
+\end_inset
+
+ into deviatoric and volumetric parts:
+\begin_inset Formula 
+\begin{gather}
+\phantom{}{}^{t+\Delta t}\underline{S}=\frac{E}{1+\nu}\left(^{t+\Delta t}\underline{e}-\phantom{}^{t+\Delta t}\underline{e}^{C}-\underline{e}^{I}\right)+\underline{S}^{I}=\frac{1}{a_{E}}\left(^{t+\Delta t}\underline{e}-\phantom{}^{t+\Delta t}\underline{e}^{C}-\underline{e}^{I}\right)\label{eq:42}\\
+^{t+\Delta t}P=\frac{E}{1-2\nu}\left(^{t+\Delta t}\theta-\theta^{I}\right)+P^{I}=\frac{1}{a_{m}}\left(^{t+\Delta t}\theta-\theta^{I}\right)\:,\nonumber 
+\end{gather}
+
+\end_inset
+
+where 
+\begin_inset Formula $^{t+\Delta t}\underline{e}$
+\end_inset
+
+ is the total deviatoric strain, 
+\begin_inset Formula $^{t+\Delta t}\underline{e}^{C}$
+\end_inset
+
+ is the total viscous strain, 
+\begin_inset Formula $\underline{e}^{I}$
+\end_inset
+
+ is the initial deviatoric strain, 
+\begin_inset Formula $^{t+\Delta t}P$
+\end_inset
+
+ is the pressure, 
+\begin_inset Formula $^{t+\Delta t}\theta$
+\end_inset
+
+ is the mean strain evaluated at time 
+\begin_inset Formula $t+\Delta t$
+\end_inset
+
+ , and 
+\begin_inset Formula $\theta^{I}$
+\end_inset
+
+ is the initial mean strain.
+ The initial deviatoric stress and initial pressure are given by 
+\begin_inset Formula $\underline{S}^{I}$
+\end_inset
+
+ and 
+\begin_inset Formula $P^{I}$
+\end_inset
+
+, respectively.
+ The topmost equation in Equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:42"
+
+\end_inset
+
+ may also be written as
+\begin_inset Formula 
+\begin{gather}
+^{t+\Delta t}\underline{S}=\frac{1}{a_{E}}(^{t+\Delta t}\underline{e}^{\prime}-\underline{\Delta e}^{C})+\underline{S}^{I}\,,\label{eq:43}
+\end{gather}
+
+\end_inset
+
+where
+\begin_inset Formula 
+\begin{gather}
+^{t+\Delta t}\underline{e}^{\prime}=\phantom{}^{t+\Delta t}\underline{e}-\phantom{}^{t}\underline{e}^{C}-\underline{e}^{I}\,\,,\,\,\,\underline{\Delta e}^{C}=\phantom{}^{t+\Delta t}\underline{e}^{C}-\phantom{}^{t}\underline{e}^{C}\,.\label{eq:44}
+\end{gather}
+
+\end_inset
+
+The creep strain increment is approximated using
+\begin_inset Formula 
+\begin{gather}
+\underline{\Delta e}^{C}=\Delta t\phantom{}^{\tau}\gamma\phantom{}^{\tau}\underline{S}\,,\label{eq:45}
+\end{gather}
+
+\end_inset
+
+where, using the 
+\begin_inset Formula $\alpha$
+\end_inset
+
+-method of time integration,
+\begin_inset Formula 
+\begin{gather}
+^{\tau}\underline{S}=(1-\alpha)_{I}^{t}\underline{S}+\alpha\phantom{}_{I}^{t+\Delta t}\underline{S}+\underline{S}^{I}=(1-\alpha)^{t}\underline{S}+\alpha\phantom{}^{t+\Delta t}\underline{S}\,\,,\label{eq:46}
+\end{gather}
+
+\end_inset
+
+and
+\begin_inset Formula 
+\begin{gather}
+^{\tau}\gamma=\frac{3\Delta\overline{e}^{C}}{2\Delta t\phantom{}^{\tau}\overline{\sigma}}\,\,,\label{eq:47}
+\end{gather}
+
+\end_inset
+
+where
+\begin_inset Formula 
+\begin{gather}
+\Delta\overline{e}^{C}=\sqrt{\frac{2}{3}\underline{\Delta e}^{C}\cdot\underline{\Delta e}^{C}}\label{eq:48}
+\end{gather}
+
+\end_inset
+
+and
+\begin_inset Formula 
+\begin{gather}
+^{\tau}\overline{\sigma}=(1-\alpha)_{I}^{t}\overline{\sigma}+\alpha\phantom{}_{I}^{t+\Delta t}\overline{\sigma}+\overline{\sigma}^{I}=\sqrt{3\phantom{}^{\tau}J_{2}^{\prime}}\,\,.\label{eq:49}
+\end{gather}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+To form the global stiffness matrix, it is necessary to provide a relationship
+ for the viscoelastic tangent material matrix relating stress and strain.
+ If we use vectors composed of the stresses and tensor strains, this relationshi
+p is
+\begin_inset Formula 
+\begin{gather}
+\underline{C}^{VE}=\frac{\partial\phantom{}^{t+\Delta t}\overrightarrow{\sigma}}{\partial\phantom{}^{t+\Delta t}\overrightarrow{\epsilon}}\,\,.\label{eq:55}
+\end{gather}
+
+\end_inset
+
+In terms of the vectors, we have
+\begin_inset Formula 
+\begin{gather}
+^{t+\Delta t}\sigma_{i}=\phantom{}^{t+\Delta t}S_{i}+\phantom{}^{t+\Delta t}P\,\,;\,\,\, i=1,2,3\label{eq:56}\\
+^{t+\Delta t}\sigma_{i}=\phantom{}^{t+\Delta t}S_{i}\,;\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, i=4,5,6\nonumber 
+\end{gather}
+
+\end_inset
+
+Therefore,
+\begin_inset Formula 
+\begin{gather}
+C_{ij}^{VE}=C_{ij}^{\prime}+\frac{1}{3a_{m}}\,;\,\,1\leq i,j\leq3\,\,.\label{eq:57}\\
+C_{ij}^{VE}=C_{ij}^{\prime}\,;\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\textrm{otherwise}\nonumber 
+\end{gather}
+
+\end_inset
+
+Using the chain rule,
+\begin_inset Formula 
+\begin{gather}
+C_{ij}^{\prime}=\frac{\partial\phantom{}^{t+\Delta t}S_{i}}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}=\frac{\partial\phantom{}^{t+\Delta t}S_{i}}{\partial\phantom{}^{t+\Delta t}e_{k}^{\prime}}\frac{\partial\phantom{}^{t+\Delta t}e_{k}^{\prime}}{\partial\phantom{}^{t+\Delta t}e_{l}}\frac{\partial\phantom{}^{t+\Delta t}e_{l}}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}\,\,.\label{eq:58}
+\end{gather}
+
+\end_inset
+
+From Equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:44"
+
+\end_inset
+
+, we obtain
+\begin_inset Formula 
+\begin{gather}
+\frac{\partial\phantom{}^{t+\Delta t}e_{k}^{\prime}}{\partial\phantom{}^{t+\Delta t}e_{l}}=\delta_{kl}\,\,,\label{eq:59}
+\end{gather}
+
+\end_inset
+
+and from Equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:16"
+
+\end_inset
+
+:
+\begin_inset Formula 
+\begin{gather}
+\frac{\partial\phantom{}^{t+\Delta t}e_{l}}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}=\frac{1}{3}\left[\begin{array}{ccc}
+2 & -1 & -1\\
+-1 & 2 & -1\\
+-1 & -1 & 2
+\end{array}\right];\,\,1\leq l,j\leq3\label{eq:60}\\
+\frac{\partial\phantom{}^{t+\Delta t}e_{l}}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}=\delta_{lj}\,\,;\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\textrm{otherwise.}\nonumber 
+\end{gather}
+
+\end_inset
+
+The first term of Equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:58"
+
+\end_inset
+
+ depends on the particular constitutive relationship, and the complete tangent
+ matrix may then be obtained from Equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:57"
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsubsection
+Power-Law Maxwell Viscoelastic Material
+\begin_inset CommandInset label
+LatexCommand label
+name "sub:Power-Law-Maxwell-Viscoelastic"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Laboratory results on rock rheology are typically performed using a triaxial
+ experiment, and the creep data are fit to a power-law equation of the form
+ (e.g., 
+\begin_inset CommandInset citation
+LatexCommand cite
+key "Kirby:Kronenberg:1987"
+
+\end_inset
+
+):
+\begin_inset Formula 
+\begin{equation}
+\dot{\epsilon}_{11}^{C}=A_{E}\exp\left(\frac{-Q}{RT}\right)\left(\sigma_{1}-\sigma_{3}\right)^{n}=A_{E}\exp\left(\frac{-Q}{RT}\right)\sigma_{d}^{n}\:,\label{eq:64}
+\end{equation}
+
+\end_inset
+
+where 
+\begin_inset Formula $\dot{\epsilon}_{11}^{C}$
+\end_inset
+
+ is the strain rate in the direction of the maximum principal stress 
+\begin_inset Formula $\left(\sigma_{1}\right)$
+\end_inset
+
+, 
+\begin_inset Formula $A_{E}$
+\end_inset
+
+ is the experimentally-derived pre-exponential constant, 
+\begin_inset Formula $Q$
+\end_inset
+
+ is the activation enthalpy, 
+\begin_inset Formula $R$
+\end_inset
+
+ is the universal gas constant, 
+\begin_inset Formula $T$
+\end_inset
+
+ is the absolute temperature, 
+\begin_inset Formula $n$
+\end_inset
+
+ is the power-law exponent, 
+\begin_inset Formula $\sigma_{3}\:\left(=\sigma_{2}\right)$
+\end_inset
+
+ is equal to the confining pressure, and 
+\begin_inset Formula $\sigma_{d}$
+\end_inset
+
+ is the differential stress.
+ To properly formulate the flow law, it must be generalized so that the
+ results are not influenced by the experiment type or the choice of coordinate
+ systems (e.g., 
+\begin_inset CommandInset citation
+LatexCommand cite
+key "Paterson:1994"
+
+\end_inset
+
+).
+ The flow law may then be generalized in terms of the deviatoric stress
+ and strain rate invariants:
+\begin_inset Formula 
+\begin{equation}
+\sqrt{\dot{L}_{2}^{\prime C}}=A_{M}\exp\left(\frac{-Q}{RT}\right)\sqrt{J_{2}^{\prime}}^{n}\:,\label{eq:65}
+\end{equation}
+
+\end_inset
+
+where 
+\begin_inset Formula $A_{M}$
+\end_inset
+
+ is now a pre-exponential constant used in the formulation for modeling.
+ In practice, it is necessary to compute each strain rate component using
+ the flow law.
+ This is accomplished using:
+\begin_inset Formula 
+\begin{equation}
+\dot{e}_{ij}^{C}=A_{M}\exp\left(\frac{-Q}{RT}\right)\sqrt{J_{2}^{\prime}}^{n-1}S_{ij}\:.\label{eq:66}
+\end{equation}
+
+\end_inset
+
+Note that Equations 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:65"
+
+\end_inset
+
+ and 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:66"
+
+\end_inset
+
+ are consistent, since Equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:65"
+
+\end_inset
+
+ may be obtained from Equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:66"
+
+\end_inset
+
+ by taking the scalar inner product of both sides, multiplying by 1/2, and
+ taking the square root.
+\end_layout
+
+\begin_layout Standard
+In a triaxial experiment with confining pressure 
+\begin_inset Formula $P_{c}$
+\end_inset
+
+, we have
+\begin_inset Formula 
+\begin{gather}
+\sigma_{2}=\sigma_{3}=P_{c}\nonumber \\
+\sigma_{1}=\sigma_{1}^{app}\label{eq:67}\\
+P=\frac{\sigma_{1}+2P_{c}}{3}\:,\nonumber 
+\end{gather}
+
+\end_inset
+
+where 
+\begin_inset Formula $\sigma_{1}^{app}$
+\end_inset
+
+ is the applied load.
+ The deviatoric stresses are then:
+\begin_inset Formula 
+\begin{gather}
+S_{1}=\frac{2}{3}\left(\sigma_{1}-P_{c}\right)\nonumber \\
+S_{2}=S_{3}=-\frac{1}{3}\left(\sigma_{1}-P_{c}\right)\:.\label{eq:68}
+\end{gather}
+
+\end_inset
+
+This gives
+\begin_inset Formula 
+\begin{gather}
+S_{1}=\frac{2}{3}\left(\sigma_{1}-\sigma_{3}\right)=\frac{2}{3}\sigma_{d}\nonumber \\
+S_{2}=S_{3}=-\frac{1}{3}\left(\sigma_{1}-\sigma_{3}\right)=-\frac{1}{3}\sigma_{d}\:.\label{eq:69}
+\end{gather}
+
+\end_inset
+
+In terms of the second deviatoric stress invariant, we then have
+\begin_inset Formula 
+\begin{equation}
+\sqrt{J_{2}^{\prime}}=\frac{\sigma_{d}}{\sqrt{3}}\:.\label{eq:70}
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Under the assumption that the creep measured in the laboratory experiments
+ is incompressible, we have
+\begin_inset Formula 
+\begin{gather}
+\dot{e}_{11}^{C}=\dot{\epsilon}_{11}\nonumber \\
+\dot{e}_{22}^{C}=\dot{e}_{33}^{C}=-\frac{1}{2}\dot{\epsilon}_{11}\:.\label{eq:71}
+\end{gather}
+
+\end_inset
+
+In terms of the second deviatoric strain rate invariant we then have
+\begin_inset Formula 
+\begin{equation}
+\sqrt{\dot{L}_{2}^{\prime C}}=\frac{\sqrt{3}}{2}\dot{\epsilon}_{11}\:.\label{eq:72}
+\end{equation}
+
+\end_inset
+
+Substituting Equations 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:70"
+
+\end_inset
+
+ and 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:72"
+
+\end_inset
+
+ into Equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:64"
+
+\end_inset
+
+, we obtain
+\begin_inset Formula 
+\begin{equation}
+\sqrt{\dot{L}_{2}^{\prime C}}=A_{E}\frac{\sqrt{3}^{n+1}}{2}\exp\left(\frac{-Q}{RT}\right)\sqrt{J_{2}^{\prime}}^{n}\:,\label{eq:73}
+\end{equation}
+
+\end_inset
+
+and therefore,
+\begin_inset Formula 
+\begin{equation}
+A_{M}=\frac{\sqrt{3}^{n+1}}{2}A_{E}\:.\label{eq:74}
+\end{equation}
+
+\end_inset
+
+When the exponential factor is included, we define a new parameter:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{equation}
+A_{T}=A_{M}\exp\left(\frac{-Q}{RT}\right)=\frac{\sqrt{3}^{n+1}}{2}A_{E}\exp\left(\frac{-Q}{RT}\right)\:.\label{eq:75}
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+There is a problem with the usage of parameters 
+\begin_inset Formula $A_{E}$
+\end_inset
+
+, 
+\begin_inset Formula $A_{M}$
+\end_inset
+
+, and 
+\begin_inset Formula $A_{T}$
+\end_inset
+
+.
+ Since the dimensions of these parameters are dependent on the value of
+ the power-law exponent, they are not really constants.
+ In addition to being logically inconsistent, this presents problems when
+ specifying parameters for PyLith, since the power-law exponent must be
+ known before the units can be determined.
+ An alternative way of writing the flow rule is (e.g., 
+\begin_inset CommandInset citation
+LatexCommand cite
+key "Prentice:1968"
+
+\end_inset
+
+): 
+\begin_inset Formula 
+\begin{equation}
+\frac{\sqrt{\dot{L}_{2}^{\prime C}}}{\dot{e}_{0}}=\left(\frac{\sqrt{J_{2}^{\prime}}}{S_{0}}\right)^{n},\label{eq:76}
+\end{equation}
+
+\end_inset
+
+where 
+\begin_inset Formula $\dot{e}_{0}$
+\end_inset
+
+ and 
+\begin_inset Formula $S_{0}$
+\end_inset
+
+ are reference values for the strain rate and deviatoric stress.
+ This means that
+\begin_inset Formula 
+\begin{equation}
+\frac{\dot{e}_{0}}{S_{0}^{n}}=A_{T}\:.\label{eq:77}
+\end{equation}
+
+\end_inset
+
+Users must therefore specify three parameters for a power-law material.
+ The properties 
+\family typewriter
+reference-strain-rate
+\family default
+, 
+\family typewriter
+reference-stress
+\family default
+, and 
+\family typewriter
+power-law-exponent
+\family default
+ in Table 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "tab:powerLaw"
+
+\end_inset
+
+ refer to 
+\begin_inset Formula $\dot{e}_{0}$
+\end_inset
+
+, 
+\begin_inset Formula $S_{0}$
+\end_inset
+
+, and 
+\begin_inset Formula $n$
+\end_inset
+
+, respectively.
+ To specify the power-law properties for PyLith using laboratory results,
+ the user must first compute 
+\begin_inset Formula $A_{T}$
+\end_inset
+
+ using Equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:75"
+
+\end_inset
+
+.
+ Then, values for 
+\begin_inset Formula $\dot{e}_{0}$
+\end_inset
+
+ and 
+\begin_inset Formula $S_{0}$
+\end_inset
+
+ must be provided.
+ The simplest method is probably to assume a reasonable value for the reference
+ strain rate, and then compute 
+\begin_inset Formula $S_{0}$
+\end_inset
+
+ as
+\begin_inset Formula 
+\begin{equation}
+S_{0}=\left(\frac{\dot{e}_{0}}{A_{T}}\right)^{\frac{1}{n}}\:.\label{eq:78}
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+A utility code (
+\family typewriter
+powerlaw_gendb.py
+\family default
+) is provided to convert laboratory results to the properties used by PyLith.
+ To use the code, users must specify the spatial variation of 
+\begin_inset Formula $A_{E}$
+\end_inset
+
+, 
+\begin_inset Formula $Q$
+\end_inset
+
+, 
+\begin_inset Formula $n$
+\end_inset
+
+, and 
+\begin_inset Formula $T$
+\end_inset
+
+.
+ An additional parameter is given to define the units of 
+\begin_inset Formula $A_{E}$
+\end_inset
+
+.
+ The user then specifies either a reference stress or a reference strain
+ rate, and a database suitable for PyLith is generated.
+ This utility is described more fully in Section 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sub:Tutorial-Step08-Power-law"
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+The flow law in component form is 
+\begin_inset Formula 
+\begin{equation}
+\dot{e}_{ij}^{C}=\frac{\dot{e}_{0}\sqrt{J_{2}^{\prime}}^{n-1}S_{ij}}{S_{0}^{n}}\:,\label{eq:79}
+\end{equation}
+
+\end_inset
+
+and the creep strain increment is approximated as
+\begin_inset Formula 
+\begin{gather}
+\underline{\Delta e}^{C}\approx\frac{\Delta t\dot{e}_{0}\sqrt{^{\tau}J_{2}^{\prime}}^{n-1}\,^{\tau}\underline{S}}{S_{0}^{n}}=\frac{\Delta t\dot{e}_{0}\phantom{}^{\tau}\overline{\sigma}^{n-1}\,^{\tau}\underline{S}}{\sqrt{3}S_{0}^{n}}\,.\label{eq:80}
+\end{gather}
+
+\end_inset
+
+ Therefore,
+\begin_inset Formula 
+\begin{gather}
+\Delta\bar{e}^{C}\approx\frac{2\Delta t\dot{e}_{0}\sqrt{^{\tau}J_{2}^{\prime}}^{n}}{\sqrt{3}S_{0}^{n}}=\frac{2\Delta t\dot{e}_{0}\phantom{}^{\tau}\overline{\sigma}^{n}}{\sqrt{3}^{n+1}S_{0}^{n}}\,,\,\textrm{and}\,^{\tau}\gamma=\frac{\dot{e}_{0}\sqrt{^{\tau}J_{2}^{\prime}}^{n-1}}{S_{0}^{n}}\,.\label{eq:81}
+\end{gather}
+
+\end_inset
+
+substituting Equations 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:46"
+
+\end_inset
+
+, 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:80"
+
+\end_inset
+
+, and 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:81"
+
+\end_inset
+
+ into 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:43"
+
+\end_inset
+
+, we obtain:
+\begin_inset Formula 
+\begin{gather}
+^{t+\Delta t}\underline{S}=\frac{1}{a_{E}}\left\{ ^{t+\Delta t}\underline{e}^{\prime}-\Delta t\phantom{}^{\tau}\gamma\left[\left(1-\alpha\right)^{t}\underline{S}+\alpha{}^{t+\Delta t}\underline{S}\right]\right\} +\underline{S}^{I}\,,\label{eq:82}
+\end{gather}
+
+\end_inset
+
+which may be rewritten:
+\begin_inset Formula 
+\begin{gather}
+^{t+\Delta t}\underline{S}\left(a_{E}+\alpha\Delta t\phantom{}^{\tau}\gamma\right)={}^{t+\Delta t}\underline{e}^{\prime}-\Delta t\phantom{}^{\tau}\gamma\left(1-\alpha\right)^{t}\underline{S}+a_{E}\underline{S}^{I}\,.\label{eq:83}
+\end{gather}
+
+\end_inset
+
+Taking the scalar inner product of both sides we obtain:
+\begin_inset Formula 
+\begin{gather}
+a^{2}\,\,{}^{t+\Delta t}J_{2}^{\prime}-b+c\phantom{}^{\tau}\gamma-d^{2}\,^{\tau}\gamma^{2}=F=0\,,\label{eq:84}
+\end{gather}
+
+\end_inset
+
+where
+\begin_inset Formula 
+\begin{gather}
+a=a_{E}+\alpha\Delta t\phantom{}^{\tau}\gamma\,\,\nonumber \\
+b=\frac{1}{2}{}^{t+\Delta t}\underline{e}^{\prime}\cdot{}^{t+\Delta t}\underline{e}^{\prime}+a_{E}{}^{t+\Delta t}\underline{e}^{\prime}\cdot\underline{S}^{I}+a_{E}^{2}\,^{I}J_{2}^{\prime}\,.\label{eq:85}\\
+c=\Delta t\left(1-\alpha\right){}^{t+\Delta t}\underline{e}^{\prime}\cdot^{t}\underline{S}+\Delta t\left(1-\alpha\right)a_{E}\,^{t}\underline{S}\cdot\underline{S}^{I}\,\,\nonumber \\
+d=\Delta t\left(1-\alpha\right)\sqrt{^{t}J_{2}^{\prime}}\,\,\nonumber 
+\end{gather}
+
+\end_inset
+
+Equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:84"
+
+\end_inset
+
+ is a function of a single unknown -- the square root of the second deviatoric
+ stress invariant at time 
+\begin_inset Formula $t+\Delta t$
+\end_inset
+
+ -- and may be solved by bisection or by Newton's method.
+ Once this parameter has been found, the deviatoric stresses for the current
+ time step may be found from Equations 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:49"
+
+\end_inset
+
+, 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:81"
+
+\end_inset
+
+, and 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:82"
+
+\end_inset
+
+, and the total stresses may be found by combining the deviatoric and volumetric
+ components from Equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:42"
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+To compute the tangent stress-strain relation, we need to compute the first
+ term in Equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:58"
+
+\end_inset
+
+.
+ We begin by rewriting Equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:83"
+
+\end_inset
+
+ as
+\begin_inset Formula 
+\begin{gather}
+F=^{t+\Delta t}S_{i}\left(a_{E}+\alpha\Delta t\phantom{}^{\tau}\gamma\right)-\phantom{}^{t+\Delta t}e_{i}^{\prime}+\Delta t\phantom{}^{\tau}\gamma\left(1-\alpha\right)^{t}S_{i}-a_{E}S_{i}^{I}=0\:.\label{eq:86}
+\end{gather}
+
+\end_inset
+
+The derivative of this function with respect to 
+\begin_inset Formula $^{t+\Delta t}e_{k}^{\prime\prime}$
+\end_inset
+
+ is
+\begin_inset Formula 
+\begin{gather}
+\frac{\partial F}{\partial\phantom{}^{t+\Delta t}e_{k}^{\prime}}=-\delta_{ik}\:,\label{eq:87}
+\end{gather}
+
+\end_inset
+
+and the derivative with respect to 
+\begin_inset Formula $^{t+\Delta t}S_{i}$
+\end_inset
+
+ is
+\begin_inset Formula 
+\begin{gather}
+\frac{\partial F}{\partial\phantom{}^{t+\Delta t}S_{i}}=a_{E}+\alpha\Delta t\phantom{}^{\tau}\gamma+\frac{\partial\phantom{}^{\tau}\gamma}{\partial\phantom{}^{t+\Delta t}S_{i}}\Delta t\left[\alpha\phantom{}^{t+\Delta t}S_{i}+\left(1-\alpha\right)^{t}S_{i}\right]\:.\label{eq:88}
+\end{gather}
+
+\end_inset
+
+From Equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:81"
+
+\end_inset
+
+ and Equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:49"
+
+\end_inset
+
+,
+\begin_inset Formula 
+\begin{gather}
+^{\tau}\gamma=\frac{\dot{e}_{0}}{S_{0}^{n}}\left[\alpha\sqrt{^{t+\Delta t}J_{2}^{\prime}}+\left(1-\alpha\right)\sqrt{^{t}J_{2}^{\prime}}\right]^{n-1}\:.\label{eq:89}
+\end{gather}
+
+\end_inset
+
+Then
+\begin_inset Formula 
+\begin{gather}
+\frac{\partial\phantom{}^{\tau}\gamma}{\partial{}^{t+\Delta t}S_{i}}=\frac{\partial\phantom{}^{\tau}\gamma}{\partial\sqrt{^{t+\Delta t}J_{2}^{\prime}}}\frac{\partial\sqrt{^{t+\Delta t}J_{2}^{\prime}}}{\partial\phantom{}^{t+\Delta t}S_{l}}\label{eq:90}\\
+=\frac{\dot{e}_{0}\alpha\left(n-1\right)\sqrt{^{\tau}J_{2}^{\prime}}^{n-2}{}^{t+\Delta t}T_{i}}{2S_{0}^{n}}\,,\nonumber 
+\end{gather}
+
+\end_inset
+
+where
+\begin_inset Formula 
+\begin{gather}
+^{t+\Delta t}T_{i}=\phantom{}^{t+\Delta t}S_{i}\:;\:\:1\leq i\leq3\label{eq:91}\\
+^{t+\Delta t}T_{i}=2\phantom{}^{t+\Delta t}S_{i}\:;\:\:\textrm{otherwise.}\nonumber 
+\end{gather}
+
+\end_inset
+
+Then using Equations 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:87"
+
+\end_inset
+
+, 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:88"
+
+\end_inset
+
+, 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:90"
+
+\end_inset
+
+, and the quotient rule for derivatives of an implicit function,
+\begin_inset Formula 
+\begin{gather}
+\frac{\partial\phantom{}^{t+\Delta t}S_{i}}{\partial{}^{t+\Delta t}e_{k}^{\prime}}=\frac{\delta_{ik}}{a_{E}+\alpha\Delta t\left[^{\tau}\gamma+\frac{\dot{e}_{0}{}^{\tau}S_{i}\left(n-1\right){}^{t+\Delta t}T_{i}\sqrt{^{\tau}J_{2}^{\prime}}^{n-2}}{2\sqrt{^{t+\Delta t}J_{2}^{\prime}}S_{0}^{n}}\right]}\,.\label{eq:92}
+\end{gather}
+
+\end_inset
+
+Note that for a linear material 
+\begin_inset Formula $\left(n=1\right)$
+\end_inset
+
+, this equation is identical to the linear formulation in Section 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sub:Effective-Stress-Formulation-Maxwell"
+
+\end_inset
+
+ (making the appropriate substitution for 
+\begin_inset Formula $^{\tau}\gamma$
+\end_inset
+
+).
+ Then, using Equations 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:57"
+
+\end_inset
+
+ through 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:60"
+
+\end_inset
+
+,
+\begin_inset Formula 
+\begin{gather}
+C_{ij}^{VE}=\frac{1}{3a_{m}}\left[\begin{array}{cccccc}
+1 & 1 & 1 & 0 & 0 & 0\\
+1 & 1 & 1 & 0 & 0 & 0\\
+1 & 1 & 1 & 0 & 0 & 0\\
+0 & 0 & 0 & 0 & 0 & 0\\
+0 & 0 & 0 & 0 & 0 & 0\\
+0 & 0 & 0 & 0 & 0 & 0
+\end{array}\right]+\frac{1}{3}\frac{\partial{}^{t+\Delta t}S_{i}}{\partial{}^{t+\Delta t}e_{k}^{\prime}}\left[\begin{array}{cccccc}
+2 & -1 & -1 & 0 & 0 & 0\\
+-1 & 2 & -1 & 0 & 0 & 0\\
+-1 & -1 & 2 & 0 & 0 & 0\\
+0 & 0 & 0 & 3 & 0 & 0\\
+0 & 0 & 0 & 0 & 3 & 0\\
+0 & 0 & 0 & 0 & 0 & 3
+\end{array}\right]\,.\label{eq:93}
+\end{gather}
+
+\end_inset
+
+Note that if there are no deviatoric stresses at the beginning and end of
+ a time step (or if 
+\begin_inset Formula $\nicefrac{\dot{e}_{0}}{S_{0}^{n}}$
+\end_inset
+
+ approaches zero), Equations 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:92"
+
+\end_inset
+
+ and 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:93"
+
+\end_inset
+
+ reduce to the elastic constitutive matrix, as expected.
+\end_layout
+
+\begin_layout Standard
+To compute the zero of the effective stress function using Newton's method,
+ we require the derivative of Equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:84"
+
+\end_inset
+
+, which may be written:
+\begin_inset Formula 
+\begin{gather}
+\frac{\partial F}{\partial\sqrt{^{t+\Delta t}J_{2}^{\prime}}}=2a^{2}\sqrt{^{t+\Delta t}J_{2}^{\prime}}+\frac{\dot{e}_{0}\alpha\left(n-1\right)\sqrt{^{\tau}J_{2}^{\prime}}^{n-2}}{S_{0}^{n}}\left(2a\alpha\Delta t{}^{t+\Delta t}J_{2}^{\prime}+c-2d^{2}\,^{\tau}\gamma\right)\,.\label{eq:94}
+\end{gather}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float table
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\noindent
+\align center
+\begin_inset Caption
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "tab:powerLaw"
+
+\end_inset
+
+Values in spatial database used as parameters in the nonlinear power-law
+ viscoelastic material constitutive model.
+\end_layout
+
+\end_inset
+
+
+\begin_inset Tabular
+<lyxtabular version="3" rows="7" columns="2">
+<features tabularvalignment="middle">
+<column alignment="center" valignment="middle" width="0.85in">
+<column alignment="center" valignment="middle" width="2.47in">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Parameter
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\align center
+
+\series bold
+Name in Spatial Database
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $v_{p}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+Vp
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $v_{s}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+Vs
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\rho$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+density
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\dot{e}_{0}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+reference-strain-rate
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $S_{0}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+reference-stress
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $n$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+power-law-exponent
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Elastoplastic Materials
+\end_layout
+
+\begin_layout Standard
+PyLith presently contains just a single elastoplastic material that implements
+ the Drucker-Prager yield criterion.
+ Future releases of PyLith may contain additional elastoplastic materials,
+ such as Drucker-Prager with hardening/softening.
+\end_layout
+
+\begin_layout Subsection
+General Elastoplasticity Formulation
+\end_layout
+
+\begin_layout Standard
+The elastoplasticity formulation in PyLith is based on an additive decomposition
+ of the total strain into elastic and plastic parts:
+\begin_inset Formula 
+\begin{equation}
+d\epsilon_{ij}=d\epsilon_{ij}^{E}+d\epsilon_{ij}^{P}\:.\label{eq:95}
+\end{equation}
+
+\end_inset
+
+The stress increment is then given by
+\begin_inset Formula 
+\begin{equation}
+d\sigma_{ij}=C_{ijrs}^{E}\left(d\epsilon_{rs}-d\epsilon_{rs}^{P}\right)\:,\label{eq:96}
+\end{equation}
+
+\end_inset
+
+where 
+\begin_inset Formula $C_{ijrs}^{E}$
+\end_inset
+
+ are the components of the elastic constitutive tensor.
+ To completely specify an elastoplastic problem, three components are needed.
+ We first require a yield condition, which specifies the state of stress
+ at which plastic flow initiates.
+ This is generally given in the form:
+\begin_inset Formula 
+\begin{equation}
+f\left(\underline{\sigma},k\right)=0\:,\label{eq:97}
+\end{equation}
+
+\end_inset
+
+where 
+\shape italic
+k
+\shape default
+ is an internal state parameter.
+ It is then necessary to specify a flow rule, which describes the relationship
+ between plastic strain and stress.
+ The flow rule is given in the form:
+\begin_inset Formula 
+\begin{equation}
+g\left(\underline{\sigma},k\right)=0\:.\label{eq:98}
+\end{equation}
+
+\end_inset
+
+The plastic strain increment is then given as
+\begin_inset Formula 
+\begin{equation}
+d\epsilon_{ij}^{P}=d\lambda\frac{\partial g}{\partial\sigma_{ij}}\:,\label{eq:99}
+\end{equation}
+
+\end_inset
+
+where 
+\begin_inset Formula $d\lambda$
+\end_inset
+
+ is the scalar plastic multiplier.
+ When the flow rule is identical to the yield criterion (
+\begin_inset Formula $f\equiv g$
+\end_inset
+
+), the plasticity is described as associated.
+ Otherwise, it is non-associated.
+ The final component needed is a hardening hypothesis, which describes how
+ the yield condition and flow rule are modified during plastic flow.
+ When the yield condition and flow rule remain constant during plastic flow
+ (e.g., no hardening), the material is referred to as perfectly plastic.
+\end_layout
+
+\begin_layout Standard
+To perform the solution, the yield condition (Equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:97"
+
+\end_inset
+
+) is first evaluated under the assumption of elastic behavior.
+ If 
+\begin_inset Formula $^{t+\Delta t}f<0$
+\end_inset
+
+, the material behavior is elastic and no plastic flow occurs.
+ Otherwise, the behavior is plastic and a plastic strain increment must
+ be computed to return the stress state to the yield envelope.
+ This procedure is known as an elastic predictor-plastic corrector algorithm.
+\end_layout
+
+\begin_layout Subsection
+Drucker-Prager Elastoplastic Material
+\end_layout
+
+\begin_layout Standard
+PyLith includes an elastoplastic implementation of the Drucker-Prager yield
+ criterion 
+\begin_inset CommandInset citation
+LatexCommand cite
+key "Drucker:Prager:1952"
+
+\end_inset
+
+.
+ This criterion was originally devised to model plastic deformation of soils,
+ and it has also been used to model rock deformation.
+ It is intended to be a smooth approximation of the Mohr-Coulomb yield criterion.
+ The implementation used in PyLith includes non-associated plastic flow,
+ which allows control over the unreasonable amounts of dilatation that are
+ sometimes predicted by the associated model.
+ The model is described by the following yield condition:
+\begin_inset Formula 
+\begin{equation}
+f\left(\underline{\sigma},k\right)=\alpha_{f}I_{1}+\sqrt{J_{2}^{\prime}}-\beta\:,\label{eq:100}
+\end{equation}
+
+\end_inset
+
+and a flow rule given by:
+\begin_inset Formula 
+\begin{equation}
+g\left(\underline{\sigma},k\right)=\sqrt{J_{2}^{\prime}}+\alpha_{g}I_{1}\:.\label{eq:101}
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+The yield surface represents a circular cone in principal stress space,
+ and the parameters can be related to the friction angle, 
+\begin_inset Formula $\phi$
+\end_inset
+
+, and the cohesion, 
+\begin_inset Formula $\bar{c}$
+\end_inset
+
+, of the Mohr-Coulomb model.
+ The yield surface in Haigh-Westergaard space (
+\begin_inset Formula $\zeta=\frac{1}{\sqrt{3}}I_{1},p=\sqrt{2J_{2}},\cos(3\theta)=\frac{3\sqrt{3}}{2}\frac{J_{3}}{J_{2}^{3/2}}$
+\end_inset
+
+) is
+\begin_inset Formula 
+\begin{equation}
+\left(\sqrt{3}\sin\left(\theta+\frac{\pi}{3}\right)-\sin\phi\cos\left(\theta+\frac{\pi}{3}\right)\right)p-\sqrt{2}\sin\phi\zeta=\sqrt{6}\overline{c}\cos\theta.\label{eq:drucker:prager:haigh:westergaard}
+\end{equation}
+
+\end_inset
+
+The yield surface can be fit to the Mohr-Coulomb model in several different
+ ways.
+ The yield surface can touch the outer apices (
+\begin_inset Formula $\theta=\pi/3$
+\end_inset
+
+) of the Mohr-Coulomb model (inscribed version), the inner apices (
+\begin_inset Formula $\theta=0$
+\end_inset
+
+) of the Mohr-Coulomb model (circumscribed version), or halfway between
+ the two (
+\begin_inset Formula $\theta=pi/6,$
+\end_inset
+
+middle version).
+ Substituting these values for 
+\begin_inset Formula $\theta$
+\end_inset
+
+ into Equation (
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:drucker:prager:haigh:westergaard"
+
+\end_inset
+
+) and casting it into the same form as Equation (
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:101"
+
+\end_inset
+
+) yields the values of 
+\begin_inset Formula $\alpha_{f}$
+\end_inset
+
+, 
+\begin_inset Formula $\beta$
+\end_inset
+
+, and 
+\begin_inset Formula $\alpha_{g}$
+\end_inset
+
+ given in Table 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "tab:fit_mohr_coulomb"
+
+\end_inset
+
+, where 
+\begin_inset Formula $\phi_{0}$
+\end_inset
+
+ refers to the initial friction angle.
+ Similarly, the flow rule can be related to the dilatation angle, 
+\begin_inset Formula $\psi$
+\end_inset
+
+, of a Mohr-Coulomb model.
+ It is also possible for the Mohr-Coulomb parameters to be functions of
+ the internal state parameter, 
+\begin_inset Formula $k$
+\end_inset
+
+.
+ In PyLith, the fit to the Mohr-Coulomb yield surface and flow rule is controlle
+d by the 
+\family typewriter
+fit_mohr_coulomb
+\family default
+ property.
+ 
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float table
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\begin_inset Caption
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "tab:fit_mohr_coulomb"
+
+\end_inset
+
+Options for fitting the Drucker-Prager plastic parameters to a Mohr-Coulomb
+ model using 
+\family typewriter
+fit_mohr_coulomb
+\family default
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\align center
+\begin_inset Tabular
+<lyxtabular version="3" rows="4" columns="4">
+<features tabularvalignment="middle">
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="0">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Parameter Value
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\alpha_{f}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\beta$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\alpha_{g}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+inscribed
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\frac{2\sin\phi\left(k\right)}{\sqrt{3}\left(3-\sin\phi\left(k\right)\right)}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\frac{6\bar{c}\left(k\right)\cos\phi_{0}}{\sqrt{3}\left(3-\sin\phi_{0}\right)}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\frac{2\sin\psi(k)}{\sqrt{3}\left(3-\sin\psi\left(k\right)\right)}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+middle
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\frac{\sin\phi\left(k\right)}{3}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\bar{c}\left(k\right)\cos\left(\phi_{0}\right)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\frac{\sin\psi\left(k\right)}{3}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+circumscribed
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\frac{2\sin\phi\left(k\right)}{\sqrt{3}\left(3+\sin\phi\left(k\right)\right)}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\frac{6\bar{c}\left(k\right)\cos\phi_{0}}{\sqrt{3}\left(3+\sin\phi_{0}\right)}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\frac{2\sin\psi(k)}{\sqrt{3}\left(3+\sin\psi\left(k\right)\right)}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+As for the viscoelastic models, it is convenient to separate the deformation
+ into deviatoric and volumetric parts:
+\begin_inset Formula 
+\begin{gather}
+^{t+\Delta t}S_{ij}=\frac{1}{a_{E}}\left(^{t+\Delta t}e_{ij}-\phantom{}^{t+\Delta t}e_{ij}^{P}-e_{ij}^{I}\right)+S_{ij}^{I}=\frac{1}{a_{E}}\left(^{t+\Delta t}e_{ij}^{\prime}-\Delta e_{ij}^{P}\right)+S_{ij}^{I}\label{eq:105}\\
+^{t+\Delta t}P=\frac{1}{a_{m}}\left(^{t+\Delta t}\theta-\phantom{}^{t+\Delta t}\theta^{P}-\theta^{I}\right)+P^{I}=\frac{1}{a_{m}}\left(^{t+\Delta t}\theta^{\prime}-\Delta\theta^{P}\right)+P^{I}\:,\nonumber 
+\end{gather}
+
+\end_inset
+
+where
+\begin_inset Formula 
+\begin{gather}
+^{t+\Delta t}e_{ij}^{\prime}=\phantom{}^{t+\Delta t}e_{ij}-\phantom{}^{t}e_{ij}^{P}-e_{ij}^{I}\nonumber \\
+\Delta e_{ij}^{P}=\phantom{}^{t+\Delta t}e_{ij}^{P}-\phantom{}^{t}e_{ij}^{P}\nonumber \\
+^{t+\Delta t}\theta^{\prime}=\phantom{}^{t+\Delta t}\theta-\phantom{}^{t}\theta^{P}-\theta^{I}\nonumber \\
+\Delta\theta^{P}=\phantom{}^{t+\Delta t}\theta^{P}-\phantom{}^{t}\theta^{P}\:.\label{eq:106}
+\end{gather}
+
+\end_inset
+
+Since the plasticity is pressure-dependent, there are volumetric plastic
+ strains, unlike the viscous strains in the previous section.
+ From Equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:99"
+
+\end_inset
+
+, the plastic strain increment is
+\begin_inset Formula 
+\begin{equation}
+\Delta\epsilon_{ij}^{P}=\lambda\frac{\partial\phantom{}^{t+\Delta t}g}{\partial\phantom{}^{t+\Delta t}\sigma_{ij}}=\lambda\alpha_{g}\delta_{ij}+\lambda\frac{^{t+\Delta t}S_{ij}}{2\sqrt{^{t+\Delta t}J_{2}^{\prime}}}\:.\label{eq:107}
+\end{equation}
+
+\end_inset
+
+The volumetric part is
+\begin_inset Formula 
+\begin{equation}
+\Delta\theta^{P}=\frac{1}{3}\Delta\epsilon_{ii}^{P}=\lambda\alpha_{g}\:,\label{eq:108}
+\end{equation}
+
+\end_inset
+
+and the deviatoric part is
+\begin_inset Formula 
+\begin{equation}
+\Delta e_{ij}^{P}=\Delta\epsilon_{ij}^{P}-\Delta\epsilon_{m}^{P}\delta_{ij}=\lambda\frac{^{t+\Delta t}S_{ij}}{2\sqrt{^{t+\Delta t}J_{2}^{\prime}}}\:.\label{eq:109}
+\end{equation}
+
+\end_inset
+
+The problem is reduced to solving for 
+\begin_inset Formula $\lambda$
+\end_inset
+
+.
+ The procedure is different depending on whether hardening is included.
+\end_layout
+
+\begin_layout Subsubsection
+Drucker-Prager Elastoplastic With No Hardening (Perfectly Plastic)
+\end_layout
+
+\begin_layout Standard
+When there is no hardening (perfect plasticity), the Drucker-Prager elastoplasti
+c model may be parameterized with just three parameters, in addition to
+ the normal elasticity parameters.
+ The parameters 
+\family typewriter
+friction-angle
+\family default
+, 
+\family typewriter
+cohesion
+\family default
+, and 
+\family typewriter
+dilatation-angle
+\family default
+ in Table 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "tab:druckerPrager"
+
+\end_inset
+
+ refer respectively to 
+\begin_inset Formula $\phi$
+\end_inset
+
+, 
+\begin_inset Formula $\bar{c}$
+\end_inset
+
+, and 
+\begin_inset Formula $\psi$
+\end_inset
+
+ in Table 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "tab:fit_mohr_coulomb"
+
+\end_inset
+
+.
+ These are then converted to the properties 
+\begin_inset Formula $\alpha_{f}$
+\end_inset
+
+ (
+\family typewriter
+alpha-yield
+\family default
+), 
+\begin_inset Formula $\beta$
+\end_inset
+
+ (
+\family typewriter
+beta
+\family default
+), and 
+\begin_inset Formula $\alpha_{g}$
+\end_inset
+
+ (
+\family typewriter
+alpha-flow
+\family default
+), as shown in Table 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "tab:material-model-output"
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+For perfect plasticity the yield and flow functions do not vary, and we
+ can solve for 
+\begin_inset Formula $\lambda$
+\end_inset
+
+ by substituting Equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:109"
+
+\end_inset
+
+ into Equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:105"
+
+\end_inset
+
+ and taking the scalar product of both sides:
+\begin_inset Formula 
+\begin{equation}
+\lambda=\sqrt{2}\,\phantom{}^{t+\Delta t}d-2a_{E}\sqrt{^{t+\Delta t}J_{2}^{\prime}}\:,\label{eq:110}
+\end{equation}
+
+\end_inset
+
+where
+\begin_inset Formula 
+\begin{equation}
+^{t+\Delta t}d^{2}=2a_{E}^{2}J_{2}^{\prime I}+2a_{E}S_{ij}^{I}\,\phantom{}^{t+\Delta t}e_{ij}^{\prime}+\phantom{}^{t+\Delta t}e_{ij}^{\prime}\,\phantom{}^{t+\Delta t}e_{ij}^{\prime}\:.\label{eq:111}
+\end{equation}
+
+\end_inset
+
+The second deviatoric stress invariant is therefore
+\begin_inset Formula 
+\begin{equation}
+\sqrt{^{t+\Delta t}J_{2}^{\prime}}=\frac{\sqrt{2}\,\phantom{}^{t+\Delta t}d-\lambda}{2a_{E}}\:,\label{eq:112}
+\end{equation}
+
+\end_inset
+
+and the pressure is computed from Equations 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:105"
+
+\end_inset
+
+ and 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:108"
+
+\end_inset
+
+ as:
+\begin_inset Formula 
+\begin{equation}
+^{t+\Delta t}P=\frac{^{t+\Delta t}I_{1}}{3}=\frac{1}{a_{m}}\left(^{t+\Delta t}\theta^{\prime}-\lambda\alpha_{g}\right)+P^{I}\:.\label{eq:113}
+\end{equation}
+
+\end_inset
+
+We then use the yield condition (
+\begin_inset Formula $^{t+\Delta t}f=0$
+\end_inset
+
+) and substitute for the stress invariants at 
+\begin_inset Formula $t+\Delta t$
+\end_inset
+
+ to obtain:
+\begin_inset Formula 
+\begin{equation}
+\lambda=\frac{2a_{E}a_{m}\left(\frac{3\alpha_{f}}{a_{m}}\phantom{}^{t+\Delta t}\theta^{\prime}+\frac{^{t+\Delta t}d}{\sqrt{2}a_{E}}-\beta\bar{c}\right)}{6\alpha_{f}\alpha_{g}a_{E}+a_{m}}\:.\label{eq:114}
+\end{equation}
+
+\end_inset
+
+Since 
+\begin_inset Formula $\lambda$
+\end_inset
+
+ is now known, we can substitute 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:112"
+
+\end_inset
+
+ into 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:109"
+
+\end_inset
+
+ to obtain
+\begin_inset Formula 
+\begin{equation}
+^{t+\Delta t}S_{ij}=\frac{\Delta e_{ij}^{P}\left(\sqrt{2}\,\phantom{\,}^{t+\Delta t}d-\lambda\right)}{\lambda a_{E}}\:.\label{eq:115}
+\end{equation}
+
+\end_inset
+
+Substituting this into Equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:105"
+
+\end_inset
+
+, we obtain the deviatoric plastic strain increment:
+\begin_inset Formula 
+\begin{equation}
+\Delta e_{ij}^{P}=\frac{\lambda}{\sqrt{2}\,\phantom{}^{t+\Delta t}d}\left(^{t+\Delta t}e_{ij}^{\prime}+a_{E}S_{ij}^{I}\right)\:.\label{eq:116}
+\end{equation}
+
+\end_inset
+
+We then use Equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:108"
+
+\end_inset
+
+ and the second line of Equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:105"
+
+\end_inset
+
+ to obtain the volumetric plastic strains and the pressure, and we use 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:116"
+
+\end_inset
+
+ and the first line of Equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:105"
+
+\end_inset
+
+ to obtain the deviatoric plastic strains and the deviatoric stresses.
+\end_layout
+
+\begin_layout Standard
+In certain cases where the mean stress is tensile, it is possible that the
+ flow rule will not allow the stresses to project back to the yield surface,
+ since they would project beyond the tip of the cone.
+ Although this stress state is not likely to be encountered for quasi-static
+ tectonic problems, it can occur for dynamic problems.
+ One simple solution is to redefine the plastic multiplier, 
+\begin_inset Formula $\lambda$
+\end_inset
+
+.
+ We do this by taking the smaller of the values yielded by Equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:114"
+
+\end_inset
+
+ or by the following relation:
+\begin_inset Formula 
+\begin{equation}
+\lambda=\sqrt{2}\,\phantom{}^{t+\Delta t}d\:.\label{eq:127}
+\end{equation}
+
+\end_inset
+
+This is equivalent to setting the second deviatoric stress invariant to
+ zero in Equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:110"
+
+\end_inset
+
+.
+ By default, PyLith does not allow such tensile yield, since this would
+ generally represent an error in problem setup for tectonic problems; however,
+ for cases where such behavior is necessary, the material flag 
+\family typewriter
+allow_tensile_yield
+\family default
+ may be set to 
+\family typewriter
+True
+\family default
+.
+\end_layout
+
+\begin_layout Standard
+To compute the elastoplastic tangent matrix we begin by writing Equation
+ 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:105"
+
+\end_inset
+
+ as a single expression in terms of stress and strain vectors:
+\begin_inset Formula 
+\begin{equation}
+^{t+\Delta t}\sigma_{i}=\frac{1}{a_{E}}\left(^{t+\Delta t}e_{i}^{\prime}-\Delta e_{i}^{P}\right)+S_{i}^{I}+\frac{R_{i}}{a_{m}}\left(^{t+\Delta t}\theta^{\prime}-\Delta\theta^{P}\right)+P^{I}\label{eq:117}
+\end{equation}
+
+\end_inset
+
+where
+\begin_inset Formula 
+\begin{gather}
+R_{i}=1\:;\; i=1,2,3\label{eq:118}\\
+R_{i}=0\:;\; i=4,5,6\:.\nonumber 
+\end{gather}
+
+\end_inset
+
+The elastoplastic tangent matrix is then given by
+\begin_inset Formula 
+\begin{equation}
+C_{ij}^{EP}=\frac{\partial\phantom{}^{t+\Delta t}\sigma_{i}}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}=\frac{1}{a_{E}}\left(\frac{\partial\phantom{}^{t+\Delta t}e_{i}^{\prime}}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}-\frac{\partial\Delta e_{i}^{P}}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}\right)+\frac{R_{i}}{a_{m}}\left(\frac{\partial\phantom{}^{t+\Delta t}\theta^{\prime}}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}-\frac{\partial\Delta\theta^{P}}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}\right)\:.\label{eq:119}
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+From Equations 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:16"
+
+\end_inset
+
+ and 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:106"
+
+\end_inset
+
+, we have
+\begin_inset Formula 
+\begin{equation}
+\frac{\partial\phantom{}^{t+\Delta t}e_{i}^{\prime}}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}=\frac{1}{3}\left[\begin{array}{cccccc}
+2 & -1 & -1 & 0 & 0 & 0\\
+-1 & 2 & -1 & 0 & 0 & 0\\
+-1 & -1 & 2 & 0 & 0 & 0\\
+0 & 0 & 0 & 3 & 0 & 0\\
+0 & 0 & 0 & 0 & 3 & 0\\
+0 & 0 & 0 & 0 & 0 & 3
+\end{array}\right]\:,\label{eq:120}
+\end{equation}
+
+\end_inset
+
+and from Equations 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:15"
+
+\end_inset
+
+ and 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:106"
+
+\end_inset
+
+ we have
+\begin_inset Formula 
+\begin{equation}
+\frac{\partial\phantom{}^{t+\Delta t}\theta^{\prime}}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}=\frac{R_{j}}{3}\:.\label{eq:121}
+\end{equation}
+
+\end_inset
+
+From Equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:116"
+
+\end_inset
+
+ we have
+\begin_inset Formula 
+\begin{equation}
+\frac{\partial\Delta e_{i}^{P}}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}=\frac{1}{\sqrt{2}\,\phantom{}^{t+\Delta t}d}\left[\left(^{t+\Delta t}e_{i}^{\prime}+a_{E}S_{i}^{I}\right)\left(\frac{\partial\lambda}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}-\frac{\lambda}{\phantom{}^{t+\Delta t}d}\frac{\partial\phantom{}^{t+\Delta t}d}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}\right)+\lambda\frac{\partial\phantom{}^{t+\Delta t}e_{i}^{\prime}}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}\right]\:.\label{eq:122}
+\end{equation}
+
+\end_inset
+
+The derivative of 
+\begin_inset Formula $^{t+\Delta t}d$
+\end_inset
+
+ is
+\begin_inset Formula 
+\begin{equation}
+\frac{\partial\phantom{}^{t+\Delta t}d}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}=\frac{a_{E}T_{j}^{I}+\phantom{}^{t+\Delta t}E_{j}}{\phantom{}^{t+\Delta t}d}\:,\label{eq:123}
+\end{equation}
+
+\end_inset
+
+where
+\begin_inset Formula 
+\begin{align}
+T_{j}^{I} & =S_{j}^{I}\;\mathrm{and}\;\phantom{}^{t+\Delta t}E_{j}=\phantom{}^{t+\Delta t}e_{j}^{\prime}\:;\; j=1,2,3\nonumber \\
+T_{j}^{I} & =2S_{j}^{I}\;\mathrm{and}\;\phantom{}^{t+\Delta t}E_{j}=2\phantom{}^{t+\Delta t}e_{j}^{\prime}\:;\; j=4,5,6\:.\label{eq:124}
+\end{align}
+
+\end_inset
+
+The derivative of 
+\begin_inset Formula $^{t+\Delta t}\lambda$
+\end_inset
+
+ is a function of derivatives already computed:
+\begin_inset Formula 
+\begin{align}
+\frac{\partial\lambda}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}} & =\frac{2a_{E}a_{m}}{6\alpha_{f}\alpha_{g}a_{E}+a_{m}}\left(\frac{3\alpha_{f}}{a_{m}}\frac{\partial\phantom{}^{t+\Delta t}\theta^{\prime}}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}+\frac{1}{\sqrt{2}a_{E}}\frac{\partial\phantom{}^{t+\Delta t}d}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}\right)\nonumber \\
+ & =\frac{2a_{E}a_{m}}{6\alpha_{f}\alpha_{g}a_{E}+a_{m}}\left(\frac{\alpha_{f}R_{j}}{a_{m}}+\frac{a_{E}T_{j}^{I}+\phantom{}^{t+\Delta t}E_{j}}{\sqrt{2}a_{E}\phantom{}^{t+\Delta t}d}\right)\:.\label{eq:125}
+\end{align}
+
+\end_inset
+
+Finally, from Equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:108"
+
+\end_inset
+
+, the derivative of the volumetric plastic strain increment is:
+\begin_inset Formula 
+\begin{equation}
+\frac{\partial\Delta\theta^{P}}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}=\alpha_{g}\frac{\partial\lambda}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}\:.\label{eq:126}
+\end{equation}
+
+\end_inset
+
+
+\begin_inset Float table
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\align center
+\begin_inset Caption
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "tab:druckerPrager"
+
+\end_inset
+
+Values in spatial database used as parameters in the Drucker-Prager elastoplasti
+c model with perfect plasticity.
+\end_layout
+
+\end_inset
+
+
+\begin_inset Tabular
+<lyxtabular version="3" rows="7" columns="2">
+<features tabularvalignment="middle">
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="0">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Parameter
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Name in Spatial Database
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $v_{p}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+vp
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $v_{s}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+vs
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\rho$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+density
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\phi$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+friction-angle
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\bar{c}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+cohesion
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\psi$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+dilatation-angle
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+In addition to the properties available for every material, the properties
+ for the Drucker-Prager model also includes:
+\end_layout
+
+\begin_layout Description
+fit_mohr_coulomb Fit to the yield surface to the Mohr-Coulomb model (default
+ is inscribed).
+\end_layout
+
+\begin_layout Description
+allow_tensile_yield If true, allow yield beyond tensile strength; otherwise
+ an error message will occur when the model fails beyond the tensile strength
+ (default is false).
+\end_layout
+
+\begin_layout Standard
+An example of setting these parameters in a 
+\family typewriter
+.cfg
+\family default
+ file is:
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.timedependent]
+\end_layout
+
+\begin_layout LyX-Code
+materials = [plastic]
+\end_layout
+
+\begin_layout LyX-Code
+materials.plastic = pylith.materials.DruckerPrager3D
+\end_layout
+
+\begin_layout LyX-Code
+
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.timedependent.materials.plastic]
+\end_layout
+
+\begin_layout LyX-Code
+fit_mohr_coulomb = inscribed ; default
+\end_layout
+
+\begin_layout LyX-Code
+allow_tensile_yield = False ; default
+\end_layout
+
+\begin_layout Section
+Initial State Variables
+\end_layout
+
+\begin_layout Standard
+In many problems of interest, the state variables describing a material
+ model may already have nonzero values prior to the application of any boundary
+ conditions.
+ For problems in geophysics, the most common example is a problem that includes
+ the effects of gravitational body forces.
+ In the real earth, rocks were emplaced and formed under the influence of
+ gravity.
+ When performing numerical simulations, however, it is not possible to represent
+ the entire time history of rock emplacement.
+ Instead, gravity must be ``turned on'' at the beginning of the simulation.
+ Unfortunately, this results in unrealistic amounts of deformation at the
+ beginning of a simulation.
+ An alternative is to provide initial state variables for the region under
+ consideration.
+ This allows the specification of a set of state variables that is consistent
+ with the prior application of gravitational body forces.
+ In a more general sense, initial values for state variables may be used
+ to provide values that are consistent with any set of conditions that occurred
+ prior to the beginning of a simulation.
+ The current release of PyLith allows the specification of initial stresses,
+ strains, and state variables for all materials; however, the initial state
+ variables are not presently used.
+ The reason for this is that the present version of PyLith always performs
+ an elastic solution prior to performing the time-dependent solution, and
+ the state variables are not used in the elastic solution.
+ In future versions of PyLith, the solution of the elastic problem will
+ be optional, so that initial state variables may be used.
+\end_layout
+
+\begin_layout Subsection
+Specification of Initial State Variables
+\end_layout
+
+\begin_layout Standard
+Since state variables are specific to a given material, initial values for
+ state variables are specified as part of the material description.
+ By default, initial state values are not used.
+ To override this behavior, specify a spatial database for the initial stress,
+ strain, and/or state variables as in the example from the tutorial in Section
+ 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sec:Tutorial-3d-hex8"
+
+\end_inset
+
+:
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.timedependent.materials.elastic]
+\end_layout
+
+\begin_layout LyX-Code
+db_initial_stress = spatialdata.spatialdb.SimpleDB
+\end_layout
+
+\begin_layout LyX-Code
+db_initial_stress.iohandler.filename = initial_stress.spatialdb
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float table
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\noindent
+\align center
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Values in spatial database for initial state variables for 3D problems.
+ 2D problems use only the relevant values.
+ Note that initial stress and strain are available for all material models.
+ Some models have additional state variables (Table 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "tab:material-model-output"
+
+\end_inset
+
+) and initial values for these may also be provided.
+\end_layout
+
+\end_inset
+
+
+\begin_inset Tabular
+<lyxtabular version="3" rows="3" columns="2">
+<features tabularvalignment="middle">
+<column alignment="center" valignment="middle" width="0.85in">
+<column alignment="center" valignment="middle" width="2.47in">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+State Variable
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\align center
+
+\series bold
+Values in Spatial Database
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+initial stress
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+stress-xx, stress-yy, stress-zz, stress-xy, stress-yz, stress-xz
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+initial strain
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+strain-xx, strain-yy, strain-zz, strain-xy, strain-yz, strain-xz
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document

Modified: short/3D/PyLith/branches/v1.7-trunk/doc/userguide/preface.lyx
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/doc/userguide/preface.lyx	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/doc/userguide/preface.lyx	2012-08-21 18:24:25 UTC (rev 20616)
@@ -1,280 +1,280 @@
-#LyX 2.0 created this file. For more info see http://www.lyx.org/
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-\begin_layout Chapter
-Preface
-\end_layout
-
-\begin_layout Section
-About This Document
-\end_layout
-
-\begin_layout Standard
-This document is organized into two parts.
- The first part begins with an introduction to PyLith and discusses the
- types of problems that PyLith can solve and how to run the software; the
- second part provides appendices and references.
-\end_layout
-
-\begin_layout Section
-Who Will Use This Documentation
-\end_layout
-
-\begin_layout Standard
-This documentation is aimed at two categories of users: scientists who prefer
- to use prepackaged and specialized analysis tools, and experienced computationa
-l Earth scientists.
- Of the latter, there are likely to be two classes of users: those who just
- run models, and those who modify the source code.
- Users who modify the source are likely to have familiarity with scripting,
- software installation, and programming, but are not necessarily professional
- programmers.
-\end_layout
-
-\begin_layout Section
-Citation
-\end_layout
-
-\begin_layout Standard
-The Computational Infrastructure for Geodynamics (CIG) 
-\begin_inset Flex URL
-status collapsed
-
-\begin_layout Plain Layout
-
-geodynamics.org
-\end_layout
-
-\end_inset
-
- is making this source code available to you at no cost in hopes that the
- software will enhance your research in geophysics.
- A number of individuals have contributed a significant portion of their
- careers toward the development of this software.
- It is essential that you recognize these individuals in the normal scientific
- practice by citing the appropriate peer-reviewed papers and making appropriate
- acknowledgements in talks and publications.
- At this time there are no refereed journal articles discussing the design
- and implementation of PyLith, so we request that you cite four AGU meeting
- abstracts associated with PyLith:
-\end_layout
-
-\begin_layout Itemize
-Williams, C.A., B.
- Aagaard, M.G.
- Knepley (2005), Development of software for studying earthquakes across
- multiple spatial and temporal scales by coupling quasi-static and dynamic
- simulations, 
-\emph on
-Eos Trans.
- AGU, 86
-\emph default
-(52), Fall Meet.
- Suppl., Abstract S53A-1072.
-\end_layout
-
-\begin_layout Itemize
-Williams, C.A.
- (2006), Development of a package for modeling stress in the lithosphere,
- 
-\emph on
-Eos Trans.
- AGU, 87
-\emph default
-(36), Jt.
- Assem.
- Suppl., Abstract T24A-01 Invited.
-\end_layout
-
-\begin_layout Itemize
-Aagaard, B., C.
- Williams, M.
- Knepley (2007), PyLith: A finite-element code for modeling quasi-static
- and dynamic crustal deformation, 
-\emph on
-Eos Trans.
- AGU, 8
-\emph default
-8(52), Fall Meet.
- Suppl., Abstract T21B-0592.
-\end_layout
-
-\begin_layout Itemize
-Aagaard, B., C.
- Williams, M.
- Knepley (2008), PyLith: A finite-element code for modeling quasi-static
- and dynamic crustal deformation, 
-\emph on
-Eos Trans.
- AGU, 89
-\emph default
-(53), Fall Meet.
- Suppl., Abstract T41A-1925.
-\end_layout
-
-\begin_layout Standard
-To cite this manual, use:
-\end_layout
-
-\begin_layout Itemize
-Aagaard, B., S.
- Kientz, M.
- Knepley, S.
- Somala, L.
- Strand, and C.
- Williams (2011), 
-\emph on
-PyLith User Manual, Version 1.7.1.
-
-\emph default
- Davis, CA: Computational Infrastructure of Geodynamics.
-\begin_inset Newline newline
-\end_inset
-
-URL: geodynamics.org/cig/software/pylith/pylith_manual-1.7.pdf
-\end_layout
-
-\begin_layout Section
-Support
-\end_layout
-
-\begin_layout Standard
-Current PyLith development is supported by the CIG, and internal GNS Science
- 
-\begin_inset Flex URL
-status open
-
-\begin_layout Plain Layout
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-www.gns.cri.nz
-\end_layout
-
-\end_inset
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-\begin_inset Flex URL
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-\end_layout
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-\end_inset
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- Pyre development was funded by the Department of Energy's 
-\begin_inset Flex URL
-status collapsed
-
-\begin_layout Plain Layout
-
-www.doe.gov/engine/content.do
-\end_layout
-
-\end_inset
-
- Advanced Simulation and Computing program and the National Science Foundation's
- Information Technology Research (ITR) program.
-\end_layout
-
-\begin_layout Standard
-This material is based upon work supported by the National Science Foundation
- under Grants No.
- 0313238, 0745391, and EAR-0949446.
- Any opinions, findings, and conclusions or recommendations expressed in
- this material are those of the author(s) and do not necessarily reflect
- the views of the National Science Foundation.
-\end_layout
-
-\begin_layout Section
-Request for Comments
-\end_layout
-
-\begin_layout Standard
-Your suggestions and corrections can only improve this documentation.
- Please report any errors, inaccuracies, or typos to the CIG Short-Term
- Tectonics email list 
-\begin_inset Flex URL
-status open
-
-\begin_layout Plain Layout
-
-cig-short at geodynamics.org
-\end_layout
-
-\end_inset
-
-.
- 
-\end_layout
-
-\end_body
-\end_document
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+\lyxformat 413
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+\begin_header
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+\begin_layout Chapter
+Preface
+\end_layout
+
+\begin_layout Section
+About This Document
+\end_layout
+
+\begin_layout Standard
+This document is organized into two parts.
+ The first part begins with an introduction to PyLith and discusses the
+ types of problems that PyLith can solve and how to run the software; the
+ second part provides appendices and references.
+\end_layout
+
+\begin_layout Section
+Who Will Use This Documentation
+\end_layout
+
+\begin_layout Standard
+This documentation is aimed at two categories of users: scientists who prefer
+ to use prepackaged and specialized analysis tools, and experienced computationa
+l Earth scientists.
+ Of the latter, there are likely to be two classes of users: those who just
+ run models, and those who modify the source code.
+ Users who modify the source are likely to have familiarity with scripting,
+ software installation, and programming, but are not necessarily professional
+ programmers.
+\end_layout
+
+\begin_layout Section
+Citation
+\end_layout
+
+\begin_layout Standard
+The Computational Infrastructure for Geodynamics (CIG) 
+\begin_inset Flex URL
+status collapsed
+
+\begin_layout Plain Layout
+
+geodynamics.org
+\end_layout
+
+\end_inset
+
+ is making this source code available to you at no cost in hopes that the
+ software will enhance your research in geophysics.
+ A number of individuals have contributed a significant portion of their
+ careers toward the development of this software.
+ It is essential that you recognize these individuals in the normal scientific
+ practice by citing the appropriate peer-reviewed papers and making appropriate
+ acknowledgements in talks and publications.
+ At this time there are no refereed journal articles discussing the design
+ and implementation of PyLith, so we request that you cite four AGU meeting
+ abstracts associated with PyLith:
+\end_layout
+
+\begin_layout Itemize
+Williams, C.A., B.
+ Aagaard, M.G.
+ Knepley (2005), Development of software for studying earthquakes across
+ multiple spatial and temporal scales by coupling quasi-static and dynamic
+ simulations, 
+\emph on
+Eos Trans.
+ AGU, 86
+\emph default
+(52), Fall Meet.
+ Suppl., Abstract S53A-1072.
+\end_layout
+
+\begin_layout Itemize
+Williams, C.A.
+ (2006), Development of a package for modeling stress in the lithosphere,
+ 
+\emph on
+Eos Trans.
+ AGU, 87
+\emph default
+(36), Jt.
+ Assem.
+ Suppl., Abstract T24A-01 Invited.
+\end_layout
+
+\begin_layout Itemize
+Aagaard, B., C.
+ Williams, M.
+ Knepley (2007), PyLith: A finite-element code for modeling quasi-static
+ and dynamic crustal deformation, 
+\emph on
+Eos Trans.
+ AGU, 8
+\emph default
+8(52), Fall Meet.
+ Suppl., Abstract T21B-0592.
+\end_layout
+
+\begin_layout Itemize
+Aagaard, B., C.
+ Williams, M.
+ Knepley (2008), PyLith: A finite-element code for modeling quasi-static
+ and dynamic crustal deformation, 
+\emph on
+Eos Trans.
+ AGU, 89
+\emph default
+(53), Fall Meet.
+ Suppl., Abstract T41A-1925.
+\end_layout
+
+\begin_layout Standard
+To cite this manual, use:
+\end_layout
+
+\begin_layout Itemize
+Aagaard, B., S.
+ Kientz, M.
+ Knepley, S.
+ Somala, L.
+ Strand, and C.
+ Williams (2011), 
+\emph on
+PyLith User Manual, Version 1.7.1.
+
+\emph default
+ Davis, CA: Computational Infrastructure of Geodynamics.
+\begin_inset Newline newline
+\end_inset
+
+URL: geodynamics.org/cig/software/pylith/pylith_manual-1.7.1.pdf
+\end_layout
+
+\begin_layout Section
+Support
+\end_layout
+
+\begin_layout Standard
+Current PyLith development is supported by the CIG, and internal GNS Science
+ 
+\begin_inset Flex URL
+status open
+
+\begin_layout Plain Layout
+
+www.gns.cri.nz
+\end_layout
+
+\end_inset
+
+ and U.S.
+ Geological Survey 
+\begin_inset Flex URL
+status collapsed
+
+\begin_layout Plain Layout
+
+www.usgs.gov
+\end_layout
+
+\end_inset
+
+ funding.
+ Pyre development was funded by the Department of Energy's 
+\begin_inset Flex URL
+status collapsed
+
+\begin_layout Plain Layout
+
+www.doe.gov/engine/content.do
+\end_layout
+
+\end_inset
+
+ Advanced Simulation and Computing program and the National Science Foundation's
+ Information Technology Research (ITR) program.
+\end_layout
+
+\begin_layout Standard
+This material is based upon work supported by the National Science Foundation
+ under Grants No.
+ 0313238, 0745391, and EAR-0949446.
+ Any opinions, findings, and conclusions or recommendations expressed in
+ this material are those of the author(s) and do not necessarily reflect
+ the views of the National Science Foundation.
+\end_layout
+
+\begin_layout Section
+Request for Comments
+\end_layout
+
+\begin_layout Standard
+Your suggestions and corrections can only improve this documentation.
+ Please report any errors, inaccuracies, or typos to the CIG Short-Term
+ Tectonics email list 
+\begin_inset Flex URL
+status open
+
+\begin_layout Plain Layout
+
+cig-short at geodynamics.org
+\end_layout
+
+\end_inset
+
+.
+ 
+\end_layout
+
+\end_body
+\end_document

Modified: short/3D/PyLith/branches/v1.7-trunk/doc/userguide/runpylith/runpylith.lyx
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/doc/userguide/runpylith/runpylith.lyx	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/doc/userguide/runpylith/runpylith.lyx	2012-08-21 18:24:25 UTC (rev 20616)
@@ -2309,6 +2309,8 @@
 \begin_inset Text
 
 \begin_layout Plain Layout
+
+\family typewriter
 ksp_type
 \end_layout
 
@@ -2756,6 +2758,7 @@
 \begin_layout Plain Layout
 
 \family typewriter
+\size footnotesize
 fs_pc_type
 \end_layout
 
@@ -2789,6 +2792,7 @@
 \begin_layout Plain Layout
 
 \family typewriter
+\size footnotesize
 fs_pc_fieldsplit_real_diagonal
 \end_layout
 
@@ -2822,6 +2826,7 @@
 \begin_layout Plain Layout
 
 \family typewriter
+\size footnotesize
 fs_pc_fieldsplit_type
 \end_layout
 
@@ -2856,6 +2861,7 @@
 \begin_layout Plain Layout
 
 \family typewriter
+\size footnotesize
 fs_fieldsplit_0_pc_type
 \end_layout
 
@@ -2889,6 +2895,7 @@
 \begin_layout Plain Layout
 
 \family typewriter
+\size footnotesize
 fs_fieldsplit_1_pc_type
 \end_layout
 
@@ -2923,6 +2930,7 @@
 \begin_layout Plain Layout
 
 \family typewriter
+\size footnotesize
 fs_fieldsplit_0_ksp_type
 \end_layout
 
@@ -2956,6 +2964,7 @@
 \begin_layout Plain Layout
 
 \family typewriter
+\size footnotesize
 fs_fieldsplit_1_ksp_type
 \end_layout
 
@@ -3185,7 +3194,7 @@
 \family default
 \shape default
 \color none
- in loading or deformation is applied, because we the time-stepping formulation
+ in loading or deformation is applied, because the time-stepping formulation
  is implemented using the increment in displacement.
 \end_layout
 
@@ -3280,27 +3289,39 @@
 \end_layout
 
 \begin_layout LyX-Code
+
+\size footnotesize
 [pylithapp.timedependent.formulation]
 \end_layout
 
 \begin_layout LyX-Code
+
+\size footnotesize
 time_step = pylith.problems.TimeStepUniform
 \end_layout
 
 \begin_layout LyX-Code
+
+\size footnotesize
 solver = pylith.problems.SolverLinear ; Nonlinear solver is pylith.problems.SolverNo
 nlinear
 \end_layout
 
 \begin_layout LyX-Code
+
+\size footnotesize
 output = [domain,ground_surface]
 \end_layout
 
 \begin_layout LyX-Code
+
+\size footnotesize
 matrix_type = sbaij ; To use a non-symmetric sparse matrix, set it to aij
 \end_layout
 
 \begin_layout LyX-Code
+
+\size footnotesize
 view_jacobian = false
 \end_layout
 
@@ -4069,26 +4090,26 @@
 
 \begin_layout LyX-Code
 
-\size small
+\size footnotesize
 [pylithapp.timedependent.materials.material]
 \end_layout
 
 \begin_layout LyX-Code
 
-\size small
+\size footnotesize
 db_properties = spatialdata.spatialdb.UniformDB ; Set the db to a UniformDB
 \end_layout
 
 \begin_layout LyX-Code
 
-\size small
+\size footnotesize
 db_properties.values = [vp,vs,density] ; Set the names of the values in the
  database
 \end_layout
 
 \begin_layout LyX-Code
 
-\size small
+\size footnotesize
 db_properties.data = [5773.5*m/s, 3333.3*m/s, 2700.0*kg/m**3] ; Set the values
  in the database
 \end_layout
@@ -4741,6 +4762,10 @@
 ).
  In most cases users will want to use the 
 \family typewriter
+
+\begin_inset Newline linebreak
+\end_inset
+
 DataWriterHDF5Mesh
 \family default
 .
@@ -5044,7 +5069,7 @@
 \family typewriter
 .dat
 \family default
-).
+.
  The HDF5 files include relative paths to the external data files, so these
  files can also be moved, but they, too, must be kept together in the same
  directory.
@@ -5128,7 +5153,7 @@
 
 \end_inset
 
-for a table of component values for tensor output.
+ for a table of component values for tensor output.
 \end_layout
 
 \begin_layout Subsubsection
@@ -5305,7 +5330,7 @@
 \family typewriter
 --help-properties
 \family default
- command line arguments when running PyLith to check the parameters to make
+ command-line arguments when running PyLith to check the parameters to make
  sure PyLith is using the parameters you intended.
 \end_layout
 
@@ -5324,13 +5349,17 @@
 \family default
 , 
 \family typewriter
+
+\begin_inset Newline newline
+\end_inset
+
 --petsc.ksp_converged_reason
 \family default
 , and 
 \family typewriter
 --petsc.snes_converged_reason
 \family default
- command line arguments (or set them in a parameter file) to view PyLith
+ command-line arguments (or set them in a parameter file) to view PyLith
  performance and monitor the convergence.
 \end_layout
 
@@ -5451,12 +5480,21 @@
  The workaround is:
 \end_layout
 
+\begin_deeper
 \begin_layout Enumerate
-Go to the lib/python2.6/site-packages directory.
+Go to the 
+\family typewriter
+lib/python2.6/site-packages
+\family default
+ directory.
 \end_layout
 
 \begin_layout Enumerate
-Unzip merlin-1.7-py2.6.egg (if it is a file and not a directory).
+Unzip 
+\family typewriter
+merlin-1.7-py2.6.egg
+\family default
+ (if it is a file and not a directory).
 \end_layout
 
 \begin_layout Enumerate
@@ -5464,15 +5502,28 @@
 \end_layout
 
 \begin_layout Enumerate
-Edit __init__.py.
+Edit 
+\family typewriter
+__init__.py
+\family default
+.
  Replace line 308 plat = get_platform() with plat = "linux-i686"
 \end_layout
 
 \begin_layout Enumerate
-If merlin-1.7-py2.6.egg is a file, rezip merlin.
- Go to the site-packages directory and enter "zip -r merlin-1.7-py2.6.egg merlin".
+If 
+\family typewriter
+merlin-1.7-py2.6.egg
+\family default
+ is a file, rezip merlin.
+ Go to the site-packages directory and enter "
+\family typewriter
+zip -r merlin-1.7-py2.6.egg merlin
+\family default
+".
 \end_layout
 
+\end_deeper
 \begin_layout Itemize
 
 \family typewriter
@@ -5497,7 +5548,7 @@
  such as the case of a problem without Dirichlet boundary conditions on
  any boundary.
  If this arises when using the split fields and algebraic multigrid precondition
-ing and no additional Dirichlet boundary conditions are desired, then the
+ing, and no additional Dirichlet boundary conditions are desired, then the
  workaround is to revert to using the Additive Schwarz preconditioning without
  split fields as discussed in Section 
 \begin_inset CommandInset ref
@@ -5517,8 +5568,8 @@
 \begin_layout Quote
 This often indicates that PyLith is using incompatible versions of libraries.
  This can result from changing your environment variables after configuring
- or installing PyLith (when building from source) or errors in setting the
- environment variables (PATH, LD_LIBRARY_PATH, and PYTHONPATH).
+ or installing PyLith (when building from source) or from errors in setting
+ the environment variables (PATH, LD_LIBRARY_PATH, and PYTHONPATH).
  If the former case, simply reconfigure and rebuild PyLith.
  In the latter case, check your environment variables (order matters!) to
  make sure PyLith finds the desired directories before system directories.
@@ -5554,7 +5605,7 @@
 \family typewriter
 --petsc.start_in_debugger
 \family default
- command line argument:
+ command-line argument:
 \end_layout
 
 \begin_layout LyX-Code
@@ -5592,7 +5643,7 @@
 
 \begin_layout LyX-Code
 
-\size small
+\size footnotesize
 mpirun -np 1 /path/to/valgrind --log-file=valgrind-log  /path/to/mpinemesis
  --pyre-start [..lots of junk..]
 \end_layout

Modified: short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/3dhex8/3dhex8.lyx
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/3dhex8/3dhex8.lyx	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/3dhex8/3dhex8.lyx	2012-08-21 18:24:25 UTC (rev 20616)
@@ -1,634 +1,638 @@
-#LyX 2.0 created this file. For more info see http://www.lyx.org/
-\lyxformat 413
-\begin_document
-\begin_header
-\textclass book
-\begin_preamble
-
-\end_preamble
-\use_default_options false
-\maintain_unincluded_children false
-\language english
-\language_package default
-\inputencoding latin1
-\fontencoding global
-\font_roman default
-\font_sans default
-\font_typewriter default
-\font_default_family default
-\use_non_tex_fonts false
-\font_sc false
-\font_osf false
-\font_sf_scale 100
-\font_tt_scale 100
-
-\graphics default
-\default_output_format default
-\output_sync 0
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-\use_bibtopic false
-\use_indices false
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-\papersides 1
-\paperpagestyle default
-\tracking_changes false
-\output_changes false
-\html_math_output 0
-\html_css_as_file 0
-\html_be_strict false
-\end_header
-
-\begin_body
-
-\begin_layout Section
-\begin_inset CommandInset label
-LatexCommand label
-name "sec:Tutorial-3d-hex8"
-
-\end_inset
-
-Tutorials Using Hexahedral Mesh Created by CUBIT
-\end_layout
-
-\begin_layout Standard
-PyLith features discussed in this set of tutorials:
-\end_layout
-
-\begin_layout Itemize
-Static solution
-\end_layout
-
-\begin_layout Itemize
-Quasi-static solution
-\end_layout
-
-\begin_layout Itemize
-CUBIT mesh format
-\end_layout
-
-\begin_layout Itemize
-Trilinear hexahedral cells
-\end_layout
-
-\begin_layout Itemize
-VTK output
-\end_layout
-
-\begin_layout Itemize
-HDF5 output
-\end_layout
-
-\begin_layout Itemize
-Dirichlet displacement and velocity boundary conditions
-\end_layout
-
-\begin_layout Itemize
-Neumann traction boundary conditions and time-varying tractions
-\end_layout
-
-\begin_layout Itemize
-ZeroDispDB spatial database
-\end_layout
-
-\begin_layout Itemize
-SimpleDB spatial database
-\end_layout
-
-\begin_layout Itemize
-UniformDB spatial database
-\end_layout
-
-\begin_layout Itemize
-Static fault rupture
-\end_layout
-
-\begin_layout Itemize
-Multiple kinematic fault ruptures
-\end_layout
-
-\begin_layout Itemize
-Specifying more than one material
-\end_layout
-
-\begin_layout Itemize
-Nonlinear solver
-\end_layout
-
-\begin_layout Itemize
-Linearly elastic isotropic material
-\end_layout
-
-\begin_layout Itemize
-Maxwell linear viscoelastic material
-\end_layout
-
-\begin_layout Itemize
-Generalized Maxwell linear viscoelastic material
-\end_layout
-
-\begin_layout Itemize
-Power-law viscoelastic material
-\end_layout
-
-\begin_layout Itemize
-Drucker-Prager elastoplastic material
-\end_layout
-
-\begin_layout Itemize
-Adaptive time stepping
-\end_layout
-
-\begin_layout Itemize
-Static fault friction
-\end_layout
-
-\begin_layout Itemize
-Slip-weakening fault friction
-\end_layout
-
-\begin_layout Itemize
-Rate-and-state fault friction
-\end_layout
-
-\begin_layout Itemize
-Gravitational body forces
-\end_layout
-
-\begin_layout Itemize
-Initial stresses
-\end_layout
-
-\begin_layout Itemize
-Finite strain
-\end_layout
-
-\begin_layout Standard
-All of the files necessary to run the examples are contained in the directory
- 
-\family typewriter
-examples/3d/hex8
-\family default
-.
-\end_layout
-
-\begin_layout Subsection
-Overview
-\end_layout
-
-\begin_layout Standard
-This tutorial is meant to demonstrate most of the important features of
- PyLith as a quasi-static finite-element code, using a sequence of example
- problems.
- All problems use the same 3D hexahedral mesh generated using the CUBIT
- 
-\begin_inset Flex URL
-status collapsed
-
-\begin_layout Plain Layout
-
-cubit.sandia.gov
-\end_layout
-
-\end_inset
-
- mesh generation package.
- Each example builds on the previous examples, as we demonstrate new features.
- As in the other examples, the files include extensive comments.
- We start with the generation of the mesh, which is composed of 144 trilinear
- hexahedra.
- The tutorial demonstrates the usage of the CUBIT mesh generation package
- to create a mesh, as well as describing how to use a CUBIT-generated mesh
- in PyLith.
- Following the discussion of how to generate the mesh, we discuss the 
-\family typewriter
-pylithapp.cfg
-\family default
- file, which contains information common to all the simulations.
- We group the examples into four sections, each pertaining to a particular
- set of PyLith features.
- We suggest users go through each of these sections in order as the complexity
- increases at each step.
- 
-\end_layout
-
-\begin_layout Subsection
-Mesh Generation and Description
-\end_layout
-
-\begin_layout Standard
-The mesh for these examples is a simple rectangular solid (Figure 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "fig:3dhex8-mesh"
-
-\end_inset
-
-).
- Although it would be possible to generate this mesh by hand, it is much
- simpler to use a mesh generation package, and we use CUBIT for this example.
- We provide documented journal files in 
-\family typewriter
-examples/3d/hex8/mesh.
-
-\family default
- Dissection of these journal files should provide some insight into how
- to use CUBIT with PyLith.
- For more detailed information on using CUBIT, refer to the CUBIT documentation
- 
-\begin_inset Flex URL
-status collapsed
-
-\begin_layout Plain Layout
-
-cubit.sandia.gov
-\end_layout
-
-\end_inset
-
-.
- If you have CUBIT installed on your machine, you can use the journal files
- to create your own mesh.
- Otherwise, you can use the mesh that has already been created.
-\end_layout
-
-\begin_layout Standard
-If you are using CUBIT to generate your own mesh, there are two ways to
- use the journal files.
- The simplest method is to go to the
-\family sans
- Tools
-\family default
- menu, select 
-\family sans
-Play Journal File
-\family default
-, and then select the file 
-\family typewriter
-mesh_hex8_1000m.jou
-\family default
-.
- This will run the commands in that file as well as the commands in 
-\family typewriter
-geometry.jou
-\family default
-, which is referenced from 
-\family typewriter
-mesh_hex8_1000m.jou
-\family default
-.
- Prior to doing this, you should set your directory to the one containing
- the journal files.
- This method will create the mesh, but you will gain very little insight
- into what is being done.
- A more informative approach is to input each journal command into the CUBIT
- command window directly.
- That way, you will see what each command does.
- The first command in 
-\family typewriter
-mesh_hex8_1000m.jou
-\family default
- is 
-\family typewriter
-`playback
-\family default
- 
-\family typewriter
-geometry.jou
-\family default
-', so you should start with the commands in 
-\family typewriter
-geometry.jou
-\family default
-.
- The first three commands, which define the block shape, are
-\end_layout
-
-\begin_layout LyX-Code
-reset
-\end_layout
-
-\begin_layout LyX-Code
-brick x 6000 y 6000 z 4000
-\end_layout
-
-\begin_layout LyX-Code
-volume 1 move x 0 y 0 z -2000
-\end_layout
-
-\begin_layout Standard
-Continuing through the remainder of the commands in 
-\family typewriter
-geometry.jou
-\family default
-, and then using the additional commands in 
-\family typewriter
-mesh_hex8_1000m.jou
-\family default
-, you will eventually end up with the file 
-\family typewriter
-box_hex8_1000m.exo
-\family default
-, which contains all of the mesh information.
- This information is similar to that included in PyLith mesh ASCII format,
- but the information is contained in an Exodus file, which is a specialized
- netCDF file.
- If you have the 
-\family typewriter
-ncdump
-\family default
- command available, you can see what is in the file by typing:
-\end_layout
-
-\begin_layout LyX-Code
-ncdump box_hex8_1000m.exo
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float figure
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\align center
-\begin_inset Graphics
-	filename figs/hex8-mesh.jpg
-	lyxscale 50
-	scale 33
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Plain Layout
-\begin_inset Caption
-
-\begin_layout Plain Layout
-Mesh composed of trilinear hexahedral cells generated by CUBIT used for
- the suite of example problems.
- The different colors represent the two different materials.
-\begin_inset CommandInset label
-LatexCommand label
-name "fig:3dhex8-mesh"
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsection
-Additional Common Information
-\end_layout
-
-\begin_layout Standard
-In addition to the mesh, the example problems share other information.
- As in previous examples, we place this information in 
-\family typewriter
-pylithapp.cfg
-\family default
-.
- Since these examples use a mesh from CUBIT, in this file we set the importer
- to 
-\family typewriter
-MeshIOCubit
-\family default
-:
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.mesh_generator]
-\end_layout
-
-\begin_layout LyX-Code
-reader = pylith.meshio.MeshIOCubit
-\begin_inset Newline newline
-\end_inset
-
-
-\begin_inset Newline newline
-\end_inset
-
-
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.mesh_generator.reader]
-\end_layout
-
-\begin_layout LyX-Code
-filename = mesh/box_hex8_1000m.exo
-\end_layout
-
-\begin_layout Standard
-This example differs from some earlier examples, because we specify two
- material groups:
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.timedependent]
-\end_layout
-
-\begin_layout LyX-Code
-materials = [upper_crust,lower_crust]
-\begin_inset Newline newline
-\end_inset
-
-
-\begin_inset Newline newline
-\end_inset
-
-
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.timedependent.materials.upper_crust]
-\end_layout
-
-\begin_layout LyX-Code
-label = Upper crust material
-\end_layout
-
-\begin_layout LyX-Code
-id = 1
-\end_layout
-
-\begin_layout LyX-Code
-db.iohandler.filename = spatialdb/mat_elastic.spatialdb
-\end_layout
-
-\begin_layout LyX-Code
-quadrature.cell = pylith.feassemble.FIATLagrange
-\end_layout
-
-\begin_layout LyX-Code
-quadrature.cell.dimension = 3
-\begin_inset Newline newline
-\end_inset
-
-
-\begin_inset Newline newline
-\end_inset
-
-
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.timedependent.materials.lower_crust]
-\end_layout
-
-\begin_layout LyX-Code
-label = Lower crust material
-\end_layout
-
-\begin_layout LyX-Code
-id = 2
-\end_layout
-
-\begin_layout LyX-Code
-db.iohandler.filename = spatialdb/mat_elastic.spatialdb
-\end_layout
-
-\begin_layout LyX-Code
-quadrature.cell = pylith.feassemble.FIATLagrange
-\end_layout
-
-\begin_layout LyX-Code
-quadrature.cell.dimension = 3
-\end_layout
-
-\begin_layout Standard
-The two material groups correspond to the two different colored regions
- in Figure 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "fig:3dhex8-mesh"
-
-\end_inset
-
-.
- Using two material groups allows us to specify different material types
- or material variations for the upper crust and lower crust, if desired.
- For now, we retain the default 
-\family typewriter
-ElasticIsotropic3D
-\family default
- material type for both materials.
- This behavior will be overridden by example-specific
-\family typewriter
-.cfg
-\family default
- files in some of the examples.
- Although the material groups are specified in 
-\family typewriter
-pylithapp.cfg
-\family default
-, the physical properties for the material models are given in 
-\family typewriter
-spatialdb/mat_elastic.spatialdb
-\family default
-.
- This spatial database provides values at a single point, resulting in uniform
- properties within the material.
-\end_layout
-
-\begin_layout Subsection
-Example Problems
-\end_layout
-
-\begin_layout Standard
-The example problems are divided into categories that roughly correspond
- to simple static problems, quasi-static problems, problems involving fault
- friction, and problems where gravity is used.
- For the most part, each successive example involves just adding or changing
- a few parameters from the previous example.
- For this reason, it is advisable to go through each example in order, starting
- with the simplest (static problems).
-\end_layout
-
-\begin_layout Standard
-\begin_inset CommandInset include
-LatexCommand input
-filename "static/static.lyx"
-
-\end_inset
-
-
-\begin_inset CommandInset include
-LatexCommand input
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-
-\end_inset
-
-
-\begin_inset CommandInset include
-LatexCommand input
-filename "friction/friction.lyx"
-
-\end_inset
-
-
-\begin_inset CommandInset include
-LatexCommand input
-filename "gravity/gravity.lyx"
-
-\end_inset
-
-
-\begin_inset CommandInset include
-LatexCommand input
-filename "surfload/surfload.lyx"
-
-\end_inset
-
-
-\begin_inset CommandInset include
-LatexCommand input
-filename "dike/dike.lyx"
-
-\end_inset
-
-
-\end_layout
-
-\end_body
-\end_document
+#LyX 2.0 created this file. For more info see http://www.lyx.org/
+\lyxformat 413
+\begin_document
+\begin_header
+\textclass book
+\begin_preamble
+
+\end_preamble
+\use_default_options false
+\maintain_unincluded_children false
+\language english
+\language_package default
+\inputencoding latin1
+\fontencoding global
+\font_roman default
+\font_sans default
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+\font_sc false
+\font_osf false
+\font_sf_scale 100
+\font_tt_scale 100
+
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry true
+\use_amsmath 0
+\use_esint 0
+\use_mhchem 1
+\use_mathdots 1
+\cite_engine basic
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\use_refstyle 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\leftmargin 1in
+\topmargin 1in
+\rightmargin 1in
+\bottommargin 2in
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\quotes_language english
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Section
+\begin_inset CommandInset label
+LatexCommand label
+name "sec:Tutorial-3d-hex8"
+
+\end_inset
+
+Tutorials Using Hexahedral Mesh Created by CUBIT
+\end_layout
+
+\begin_layout Standard
+PyLith features discussed in this set of tutorials:
+\end_layout
+
+\begin_layout Itemize
+Static solution
+\end_layout
+
+\begin_layout Itemize
+Quasi-static solution
+\end_layout
+
+\begin_layout Itemize
+CUBIT mesh format
+\end_layout
+
+\begin_layout Itemize
+Trilinear hexahedral cells
+\end_layout
+
+\begin_layout Itemize
+VTK output
+\end_layout
+
+\begin_layout Itemize
+HDF5 output
+\end_layout
+
+\begin_layout Itemize
+Dirichlet displacement and velocity boundary conditions
+\end_layout
+
+\begin_layout Itemize
+Neumann traction boundary conditions and time-varying tractions
+\end_layout
+
+\begin_layout Itemize
+ZeroDispDB spatial database
+\end_layout
+
+\begin_layout Itemize
+SimpleDB spatial database
+\end_layout
+
+\begin_layout Itemize
+UniformDB spatial database
+\end_layout
+
+\begin_layout Itemize
+Static fault rupture
+\end_layout
+
+\begin_layout Itemize
+Multiple kinematic fault ruptures
+\end_layout
+
+\begin_layout Itemize
+Specifying more than one material
+\end_layout
+
+\begin_layout Itemize
+Nonlinear solver
+\end_layout
+
+\begin_layout Itemize
+Linearly elastic isotropic material
+\end_layout
+
+\begin_layout Itemize
+Maxwell linear viscoelastic material
+\end_layout
+
+\begin_layout Itemize
+Generalized Maxwell linear viscoelastic material
+\end_layout
+
+\begin_layout Itemize
+Power-law viscoelastic material
+\end_layout
+
+\begin_layout Itemize
+Drucker-Prager elastoplastic material
+\end_layout
+
+\begin_layout Itemize
+Adaptive time stepping
+\end_layout
+
+\begin_layout Itemize
+Static fault friction
+\end_layout
+
+\begin_layout Itemize
+Slip-weakening fault friction
+\end_layout
+
+\begin_layout Itemize
+Rate-and-state fault friction
+\end_layout
+
+\begin_layout Itemize
+Gravitational body forces
+\end_layout
+
+\begin_layout Itemize
+Initial stresses
+\end_layout
+
+\begin_layout Itemize
+Finite strain
+\end_layout
+
+\begin_layout Standard
+All of the files necessary to run the examples are contained in the directory
+ 
+\family typewriter
+examples/3d/hex8
+\family default
+.
+\end_layout
+
+\begin_layout Subsection
+Overview
+\end_layout
+
+\begin_layout Standard
+This tutorial is meant to demonstrate most of the important features of
+ PyLith as a quasi-static finite-element code, using a sequence of example
+ problems.
+ All problems use the same 3D hexahedral mesh generated using the CUBIT
+ 
+\begin_inset Flex URL
+status collapsed
+
+\begin_layout Plain Layout
+
+cubit.sandia.gov
+\end_layout
+
+\end_inset
+
+ mesh generation package.
+ Each example builds on the previous examples, as we demonstrate new features.
+ As in the other examples, the files include extensive comments.
+ We start with the generation of the mesh, which is composed of 144 trilinear
+ hexahedra.
+ The tutorial demonstrates the usage of the CUBIT mesh generation package
+ to create a mesh, as well as describing how to use a CUBIT-generated mesh
+ in PyLith.
+ Following the discussion of how to generate the mesh, we discuss the 
+\family typewriter
+pylithapp.cfg
+\family default
+ file, which contains information common to all the simulations.
+ We group the examples into four sections, each pertaining to a particular
+ set of PyLith features.
+ We suggest users go through each of these sections in order as the complexity
+ increases at each step.
+ 
+\end_layout
+
+\begin_layout Subsection
+Mesh Generation and Description
+\end_layout
+
+\begin_layout Standard
+The mesh for these examples is a simple rectangular solid (Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:3dhex8-mesh"
+
+\end_inset
+
+).
+ Although it would be possible to generate this mesh by hand, it is much
+ simpler to use a mesh generation package, and we use CUBIT for this example.
+ We provide documented journal files in 
+\family typewriter
+examples/3d/hex8/mesh.
+
+\family default
+ Dissection of these journal files should provide some insight into how
+ to use CUBIT with PyLith.
+ For more detailed information on using CUBIT, refer to the CUBIT documentation
+ 
+\begin_inset Flex URL
+status collapsed
+
+\begin_layout Plain Layout
+
+cubit.sandia.gov
+\end_layout
+
+\end_inset
+
+.
+ If you have CUBIT installed on your machine, you can use the journal files
+ to create your own mesh.
+ Otherwise, you can use the mesh that has already been created.
+\end_layout
+
+\begin_layout Standard
+If you are using CUBIT to generate your own mesh, there are two ways to
+ use the journal files.
+ The simplest method is to go to the
+\family sans
+ Tools
+\family default
+ menu, select 
+\family sans
+Play Journal File
+\family default
+, and then select the file 
+\family typewriter
+mesh_hex8_1000m.jou
+\family default
+.
+ This will run the commands in that file as well as the commands in 
+\family typewriter
+geometry.jou
+\family default
+, which is referenced from 
+\family typewriter
+mesh_hex8_1000m.jou
+\family default
+.
+ Prior to doing this, you should set your directory to the one containing
+ the journal files.
+ This method will create the mesh, but you will gain very little insight
+ into what is being done.
+ A more informative approach is to input each journal command into the CUBIT
+ command window directly.
+ That way, you will see what each command does.
+ The first command in 
+\family typewriter
+mesh_hex8_1000m.jou
+\family default
+ is 
+\family typewriter
+`playback
+\family default
+ 
+\family typewriter
+geometry.jou
+\family default
+', so you should start with the commands in 
+\family typewriter
+geometry.jou
+\family default
+.
+ The first three commands, which define the block shape, are
+\end_layout
+
+\begin_layout LyX-Code
+reset
+\end_layout
+
+\begin_layout LyX-Code
+brick x 6000 y 6000 z 4000
+\end_layout
+
+\begin_layout LyX-Code
+volume 1 move x 0 y 0 z -2000
+\end_layout
+
+\begin_layout Standard
+Continuing through the remainder of the commands in 
+\family typewriter
+geometry.jou
+\family default
+, and then using the additional commands in 
+\family typewriter
+mesh_hex8_1000m.jou
+\family default
+, you will eventually end up with the file 
+\family typewriter
+box_hex8_1000m.exo
+\family default
+, which contains all of the mesh information.
+ This information is similar to that included in PyLith mesh ASCII format,
+ but the information is contained in an Exodus file, which is a specialized
+ netCDF file.
+ If you have the 
+\family typewriter
+ncdump
+\family default
+ command available, you can see what is in the file by typing:
+\end_layout
+
+\begin_layout LyX-Code
+ncdump box_hex8_1000m.exo
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float figure
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\align center
+\begin_inset Graphics
+	filename figs/hex8-mesh.jpg
+	lyxscale 50
+	scale 33
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Mesh composed of trilinear hexahedral cells generated by CUBIT used for
+ the suite of example problems.
+ The different colors represent the two different materials.
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:3dhex8-mesh"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Additional Common Information
+\end_layout
+
+\begin_layout Standard
+In addition to the mesh, the example problems share other information.
+ As in previous examples, we place this information in 
+\family typewriter
+pylithapp.cfg
+\family default
+.
+ Since these examples use a mesh from CUBIT, in this file we set the importer
+ to 
+\family typewriter
+MeshIOCubit
+\family default
+:
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.mesh_generator]
+\end_layout
+
+\begin_layout LyX-Code
+reader = pylith.meshio.MeshIOCubit
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.mesh_generator.reader]
+\end_layout
+
+\begin_layout LyX-Code
+filename = mesh/box_hex8_1000m.exo
+\end_layout
+
+\begin_layout Standard
+This example differs from some earlier examples, because we specify two
+ material groups:
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.timedependent]
+\end_layout
+
+\begin_layout LyX-Code
+materials = [upper_crust,lower_crust]
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.timedependent.materials.upper_crust]
+\end_layout
+
+\begin_layout LyX-Code
+label = Upper crust material
+\end_layout
+
+\begin_layout LyX-Code
+id = 1
+\end_layout
+
+\begin_layout LyX-Code
+db.iohandler.filename = spatialdb/mat_elastic.spatialdb
+\end_layout
+
+\begin_layout LyX-Code
+quadrature.cell = pylith.feassemble.FIATLagrange
+\end_layout
+
+\begin_layout LyX-Code
+quadrature.cell.dimension = 3
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.timedependent.materials.lower_crust]
+\end_layout
+
+\begin_layout LyX-Code
+label = Lower crust material
+\end_layout
+
+\begin_layout LyX-Code
+id = 2
+\end_layout
+
+\begin_layout LyX-Code
+db.iohandler.filename = spatialdb/mat_elastic.spatialdb
+\end_layout
+
+\begin_layout LyX-Code
+quadrature.cell = pylith.feassemble.FIATLagrange
+\end_layout
+
+\begin_layout LyX-Code
+quadrature.cell.dimension = 3
+\end_layout
+
+\begin_layout Standard
+The two material groups correspond to the two different colored regions
+ in Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:3dhex8-mesh"
+
+\end_inset
+
+.
+ Using two material groups allows us to specify different material types
+ or material variations for the upper crust and lower crust, if desired.
+ For now, we retain the default 
+\family typewriter
+ElasticIsotropic3D
+\family default
+ material type for both materials.
+ This behavior will be overridden by example-specific
+\family typewriter
+.cfg
+\family default
+ files in some of the examples.
+ Although the material groups are specified in 
+\family typewriter
+pylithapp.cfg
+\family default
+, the physical properties for the material models are given in 
+\family typewriter
+spatialdb/
+\begin_inset Newline linebreak
+\end_inset
+
+mat_elastic.spatialdb
+\family default
+.
+ This spatial database provides values at a single point, resulting in uniform
+ properties within the material.
+\end_layout
+
+\begin_layout Subsection
+Example Problems
+\end_layout
+
+\begin_layout Standard
+The example problems are divided into categories that roughly correspond
+ to simple static problems, quasi-static problems, problems involving fault
+ friction, and problems where gravity is used.
+ For the most part, each successive example involves just adding or changing
+ a few parameters from the previous example.
+ For this reason, it is advisable to go through each example in order, starting
+ with the simplest (static problems).
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "static/static.lyx"
+
+\end_inset
+
+
+\begin_inset CommandInset include
+LatexCommand input
+filename "quasistatic/quasistatic.lyx"
+
+\end_inset
+
+
+\begin_inset CommandInset include
+LatexCommand input
+filename "friction/friction.lyx"
+
+\end_inset
+
+
+\begin_inset CommandInset include
+LatexCommand input
+filename "gravity/gravity.lyx"
+
+\end_inset
+
+
+\begin_inset CommandInset include
+LatexCommand input
+filename "surfload/surfload.lyx"
+
+\end_inset
+
+
+\begin_inset CommandInset include
+LatexCommand input
+filename "dike/dike.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document

Modified: short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/3dhex8/dike/dike.lyx
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/3dhex8/dike/dike.lyx	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/3dhex8/dike/dike.lyx	2012-08-21 18:24:25 UTC (rev 20616)
@@ -261,11 +261,7 @@
 \begin_inset Caption
 
 \begin_layout Plain Layout
-Displacement magnitude for example 
-\family typewriter
-step20
-\family default
- visualized using ParaView.
+Displacement magnitude for example step20 visualized using ParaView.
  
 \begin_inset CommandInset label
 LatexCommand label

Modified: short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/3dhex8/friction/friction.lyx
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/3dhex8/friction/friction.lyx	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/3dhex8/friction/friction.lyx	2012-08-21 18:24:25 UTC (rev 20616)
@@ -183,17 +183,10 @@
 \family typewriter
 step10.cfg
 \family default
- file defines a problem that is identical to example 
-\family typewriter
-step01
-\family default
-, except for the presence of a vertical fault with static friction.
+ file defines a problem that is identical to example step01, except for
+ the presence of a vertical fault with static friction.
  In this case, the applied displacements are insufficient to cause the fault
- to slip, so the solution is identical to that in example 
-\family typewriter
-step01
-\family default
-.
+ to slip, so the solution is identical to that in example step01.
  As in previous examples involving faults, we must first provide an array
  defining the fault interfaces:
 \end_layout
@@ -1043,51 +1036,75 @@
 \end_layout
 
 \begin_layout LyX-Code
+
+\size small
 [pylithapp.timedependent.interfaces.fault]
 \end_layout
 
 \begin_layout LyX-Code
+
+\size small
 # Set rate-and-state parameters using a UniformDB.
  Set the parameters as
 \end_layout
 
 \begin_layout LyX-Code
+
+\size small
 # follows:
 \end_layout
 
 \begin_layout LyX-Code
+
+\size small
 # reference coefficient of friction: 0.6
 \end_layout
 
 \begin_layout LyX-Code
+
+\size small
 # reference slip rate: 1.0e-06 m/s
 \end_layout
 
 \begin_layout LyX-Code
+
+\size small
 # slip-weakening parameter: 0.037 m
 \end_layout
 
 \begin_layout LyX-Code
+
+\size small
 # a: 0.0125
 \end_layout
 
 \begin_layout LyX-Code
+
+\size small
 # b: 0.0172
 \end_layout
 
 \begin_layout LyX-Code
+
+\size small
 # cohesion: 0 Pa
 \end_layout
 
 \begin_layout LyX-Code
+
+\size small
 friction.db_properties = spatialdata.spatialdb.UniformDB
 \end_layout
 
 \begin_layout LyX-Code
-friction.db_properties.label = Rate Stete Ageing
+
+\size small
+friction.db_properties.label = Rate State Ageing
 \end_layout
 
 \begin_layout LyX-Code
+
+\size small
 friction.db_properties.values = [reference-friction-coefficient,reference-slip-rat
 e,
 \begin_inset Newline newline
@@ -1098,6 +1115,8 @@
 \end_layout
 
 \begin_layout LyX-Code
+
+\size small
 friction.db_properties.data = [0.6,1.0e-6*m/s,0.0370*m,0.0125,0.0172,0.0*Pa]
 \end_layout
 
@@ -1137,9 +1156,9 @@
 \family default
  (all with a prefix of 
 \family typewriter
-step1
+step14
 \family default
-4).
+).
  Results using ParaView are shown in Figure 
 \begin_inset CommandInset ref
 LatexCommand ref

Modified: short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/3dhex8/gravity/gravity.lyx
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/3dhex8/gravity/gravity.lyx	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/3dhex8/gravity/gravity.lyx	2012-08-21 18:24:25 UTC (rev 20616)
@@ -1,731 +1,731 @@
-#LyX 2.0 created this file. For more info see http://www.lyx.org/
-\lyxformat 413
-\begin_document
-\begin_header
-\textclass book
-\begin_preamble
-
-\end_preamble
-\use_default_options false
-\maintain_unincluded_children false
-\language english
-\language_package default
-\inputencoding latin1
-\fontencoding global
-\font_roman default
-\font_sans default
-\font_typewriter default
-\font_default_family default
-\use_non_tex_fonts false
-\font_sc false
-\font_osf false
-\font_sf_scale 100
-\font_tt_scale 100
-
-\graphics default
-\default_output_format default
-\output_sync 0
-\bibtex_command default
-\index_command default
-\paperfontsize default
-\spacing single
-\use_hyperref false
-\papersize default
-\use_geometry true
-\use_amsmath 0
-\use_esint 0
-\use_mhchem 1
-\use_mathdots 1
-\cite_engine basic
-\use_bibtopic false
-\use_indices false
-\paperorientation portrait
-\suppress_date false
-\use_refstyle 0
-\index Index
-\shortcut idx
-\color #008000
-\end_index
-\leftmargin 1in
-\topmargin 1in
-\rightmargin 1in
-\bottommargin 2in
-\secnumdepth 3
-\tocdepth 3
-\paragraph_separation indent
-\paragraph_indentation default
-\quotes_language english
-\papercolumns 1
-\papersides 1
-\paperpagestyle default
-\tracking_changes false
-\output_changes false
-\html_math_output 0
-\html_css_as_file 0
-\html_be_strict false
-\end_header
-
-\begin_body
-
-\begin_layout Subsection
-\begin_inset CommandInset label
-LatexCommand label
-name "sec:Tutorial-3d-hex8-gravity"
-
-\end_inset
-
-Gravitational Body Force Examples
-\end_layout
-
-\begin_layout Standard
-PyLith features discussed in this tutorial:
-\end_layout
-
-\begin_layout Itemize
-Gravitational body forces
-\end_layout
-
-\begin_layout Itemize
-Initial stresses
-\end_layout
-
-\begin_layout Itemize
-Finite strain
-\end_layout
-
-\begin_layout Itemize
-Generalized Maxwell linear viscoelastic material
-\end_layout
-
-\begin_layout Subsubsection
-Overview
-\end_layout
-
-\begin_layout Standard
-This set of examples describes a set of problems for PyLith involving gravitatio
-nal body forces.
- All of the examples are quasi-static and run for a time period of 200 years.
- These examples also demonstrate the use of a generalized Maxwell viscoelastic
- material, which is used for the lower crust in all examples.
- The final example (step17) demonstrates the usage of a finite strain formulatio
-n, which automatically invokes the nonlinear solver.
- All of the examples are contained in the directory 
-\family typewriter
-examples/3d/hex8
-\family default
-, and the corresponding 
-\family typewriter
-.cfg
-\family default
- files are 
-\family typewriter
-step15.cfg
-\family default
-, 
-\family typewriter
-step16.cfg
-\family default
-, and 
-\family typewriter
-step17.cfg
-\family default
-.
- Each example may be run as follows:
-\end_layout
-
-\begin_layout LyX-Code
-pylith stepXX.cfg
-\end_layout
-
-\begin_layout Standard
-This will cause PyLith to read the default parameters in 
-\family typewriter
-pylithapp.cfg
-\family default
-, and then override or augment them with the additional parameters in the
- 
-\family typewriter
-stepXX.cfg
-\family default
- file.
- Each 
-\family typewriter
-.cfg
-\family default
- file is extensively documented, to provide detailed information on the
- various parameters.
-\end_layout
-
-\begin_layout Subsubsection
-Step15 - Gravitational Body Forces
-\end_layout
-
-\begin_layout Standard
-The 
-\family typewriter
-step15.cfg
-\family default
- file defines a problem with extremely simple Dirichlet boundary conditions.
- On the positive and negative x-faces, the positive and negative y-faces,
- and the negative z-face, the displacements normal to the face are set to
- zero.
- Because all of the materials in the example have the same density, the
- elastic solution for loading via gravitational body forces is
-\begin_inset Formula 
-\begin{equation}
-\sigma_{zz}=\rho gh;\:\sigma_{xx}=\sigma_{yy}=\frac{\nu\rho gh}{1-\nu}\:.\label{eq:1-1}
-\end{equation}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-We first set the gravity field, which by default has values of 9.80655 
-\begin_inset Formula $\unitfrac{m}{s^{2}}$
-\end_inset
-
- for acceleration and 
-\begin_inset Formula $\left[0,0,-1\right]$
-\end_inset
-
- for direction:
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.timedependent]
-\end_layout
-
-\begin_layout LyX-Code
-# Set gravity field (default is None)
-\end_layout
-
-\begin_layout LyX-Code
-gravity_field = spatialdata.spatialdb.GravityField
-\end_layout
-
-\begin_layout Standard
-We use adaptive time stepping, set the simulation time to 200 years, and
- specify a maximum time step size of 10 years:
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.timedependent.implicit]
-\end_layout
-
-\begin_layout LyX-Code
-# Change time stepping algorithm from uniform time step, to adaptive
-\end_layout
-
-\begin_layout LyX-Code
-# time stepping.
-\end_layout
-
-\begin_layout LyX-Code
-time_step = pylith.problems.TimeStepAdapt
-\begin_inset Newline newline
-\end_inset
-
-
-\begin_inset Newline newline
-\end_inset
-
-
-\end_layout
-
-\begin_layout LyX-Code
-# Change the total simulation time to 200 years, and set the maximum time
-\end_layout
-
-\begin_layout LyX-Code
-# step size to 10 years.
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.timedependent.implicit.time_step]
-\end_layout
-
-\begin_layout LyX-Code
-total_time = 200.0*year
-\end_layout
-
-\begin_layout LyX-Code
-max_dt = 10.0*year
-\end_layout
-
-\begin_layout LyX-Code
-stability_factor = 1.0 ; use time step equal to stable value from materials
-\end_layout
-
-\begin_layout Standard
-We use a generalized Maxwell model for the lower crust (see Section 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "sub:Formulation-for-Gen-Max"
-
-\end_inset
-
-), and use a 
-\family typewriter
-SimpleDB
-\family default
- to provide the properties.
- We also request the relevant properties and state variables for output:
-\end_layout
-
-\begin_layout LyX-Code
-# Change material type of lower crust to generalized Maxwell viscoelastic.
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.timedependent]
-\end_layout
-
-\begin_layout LyX-Code
-materials.lower_crust = pylith.materials.GenMaxwellIsotropic3D
-\end_layout
-
-\begin_layout LyX-Code
-# Provide a spatial database from which to obtain property values.
-\end_layout
-
-\begin_layout LyX-Code
-# Since there are additional properties and state variables for the
-\end_layout
-
-\begin_layout LyX-Code
-# generalized Maxwell model, we explicitly request that they be output.
-\end_layout
-
-\begin_layout LyX-Code
-# Properties are named in cell_info_fields and state variables are named
- in
-\end_layout
-
-\begin_layout LyX-Code
-# cell_data_fields.
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.timedependent.materials.lower_crust]
-\end_layout
-
-\begin_layout LyX-Code
-db_properties.iohandler.filename = spatialdb/mat_genmaxwell.spatialdb
-\end_layout
-
-\begin_layout LyX-Code
-output.cell_info_fields = [density,mu,lambda,shear_ratio,maxwell_time]
-\end_layout
-
-\begin_layout LyX-Code
-output.cell_data_fields = [total_strain,stress,viscous_strain_1,viscous_strain_2,
-\begin_inset Newline newline
-\end_inset
-
-viscous_strain_3]
-\end_layout
-
-\begin_layout Standard
-The boundary conditions for this example are trivial, so we are able to
- use the default 
-\family typewriter
-ZeroDispDB
-\family default
- for all faces.
- When we have run the simulation, the output VTK files will be contained
- in 
-\family typewriter
-examples/3d/hex8/output
-\family default
- (all with a prefix of 
-\family typewriter
-step15
-\family default
-).
- Results using ParaView are shown in Figure 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "fig:step15-displ-t200"
-
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-\begin_inset Float figure
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\align center
-\begin_inset Graphics
-	filename figs/step15-displ-t200.jpg
-	lyxscale 50
-	width 10cm
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Plain Layout
-\begin_inset Caption
-
-\begin_layout Plain Layout
-Displacement field for example step15 at t = 200 years visualized using
- ParaView.
- The z-component of the displacement field is shown with the color contours,
- and the vectors show the computed displacements.
-\begin_inset CommandInset label
-LatexCommand label
-name "fig:step15-displ-t200"
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsubsection
-Step16 - Gravitational Body Forces with Initial Stresses
-\end_layout
-
-\begin_layout Standard
-The 
-\family typewriter
-step16.cfg
-\family default
- file defines a problem that is identical to example step15, except that
- initial stresses are used to prevent the initial large displacements due
- to 'turning on' gravity.
- Since all normal stress components are given an initial stress of 
-\begin_inset Formula $\rho gh$
-\end_inset
-
-, the initial stress state is lithostatic, which is an appropriate condition
- for many tectonic problems in the absence of tectonic stresses (e.g., McGarr
- 
-\begin_inset CommandInset citation
-LatexCommand cite
-key "McGarr:1988"
-
-\end_inset
-
-).
- When compared to example step15, this example should maintain a lithostatic
- state of stress for the entire simulation, and displacements should remain
- essentially zero.
-\end_layout
-
-\begin_layout Standard
-We set the gravity field, as in example step15, and we again use adaptive
- time stepping with a generalized Maxwell rheology for the lower crust.
- We provide values for the initial stress for both the upper and lower crust.
- Since the materials have the same density, we are able to use the same
- 
-\family typewriter
-SimpleDB
-\family default
- with a linear variation for both (see file 
-\family typewriter
-examples/3d/hex8/spatialdb/initial_stress.spatialdb
-\family default
-):
-\end_layout
-
-\begin_layout LyX-Code
-# We must specify initial stresses for each material.
-\end_layout
-
-\begin_layout LyX-Code
-# We provide a filename for the spatial database that gives the stresses,
-\end_layout
-
-\begin_layout LyX-Code
-# and we change the query_type from the default 'nearest' to 'linear'.
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.timedependent.materials.upper_crust]
-\end_layout
-
-\begin_layout LyX-Code
-db_initial_stress = spatialdata.spatialdb.SimpleDB
-\end_layout
-
-\begin_layout LyX-Code
-db_initial_stress.iohandler.filename = spatialdb/initial_stress.spatialdb
-\end_layout
-
-\begin_layout LyX-Code
-db_initial_stress.query_type = linear
-\begin_inset Newline newline
-\end_inset
-
-
-\begin_inset Newline newline
-\end_inset
-
-
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.timedependent.materials.lower_crust]
-\end_layout
-
-\begin_layout LyX-Code
-db_initial_stress = spatialdata.spatialdb.SimpleDB
-\end_layout
-
-\begin_layout LyX-Code
-db_initial_stress.iohandler.filename = spatialdb/initial_stress.spatialdb
-\end_layout
-
-\begin_layout LyX-Code
-db_initial_stress.query_type = linear
-\end_layout
-
-\begin_layout Standard
-Note that we use a 
-\family typewriter
-linear
-\family default
- 
-\family typewriter
-query_type
-\family default
- rather than the default type of 
-\family typewriter
-nearest
-\family default
-, so that a linear interpolation is performed along the z-direction.
- When we have run the simulation, the output VTK files will be contained
- in 
-\family typewriter
-examples/3d/hex8/output
-\family default
- (all with a prefix of 
-\family typewriter
-step16
-\family default
-).
- Results using ParaView are shown in Figure 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "fig:step16-stress_xx-t200"
-
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-\begin_inset Float figure
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\align center
-\begin_inset Graphics
-	filename figs/step16-stress_xx-t200.jpg
-	lyxscale 50
-	width 10cm
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Plain Layout
-\begin_inset Caption
-
-\begin_layout Plain Layout
-Stress field (xx-component) for example step16 at t = 200 years visualized
- using ParaView.
- Note that for this example, Stress_xx = Stress_yy = Stress_zz, and there
- is no vertical displacement throughout the simulation.
- Also note that the stresses appear as 4 layers since we have used 
-\family typewriter
-CellFilterAvgMesh
-\family default
- for material output.
-\begin_inset CommandInset label
-LatexCommand label
-name "fig:step16-stress_xx-t200"
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsubsection
-Step17 - Gravitational Body Forces with Small Strain
-\end_layout
-
-\begin_layout Standard
-The 
-\family typewriter
-step17.cfg
-\family default
- file defines a problem that is identical to example step15, except that
- we now use a small strain formulation (see Section 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "sec:Small-Strain-Formulation"
-
-\end_inset
-
-).
- All of the problems up to this point have assumed infinitesimal strain,
- meaning that the change in shape of the domain during deformation is not
- taken into account.
- In many problems it is important to consider the change in shape of the
- domain.
- This is particularly important in many problems involving gravitational
- body forces, since a change in shape of the domain results in a different
- stress field.
- By examining the stress and deformation fields for this example in comparison
- with those of example step15, we can see what effect the infinitesimal
- strain approximation has on our solution.
-\end_layout
-
-\begin_layout Standard
-We set the gravity field, as in example step15 and again use adaptive time
- stepping withs a generalized Maxwell rheology for the lower crust.
- The only change is that we change the problem formulation from the default
- 
-\family typewriter
-Implicit
-\family default
- to 
-\family typewriter
-ImplicitLgDeform
-\family default
-.
- Since the large deformation formulation is nonlinear, PyLith automatically
- switches the solver from the default 
-\family typewriter
-SolverLinear
-\family default
- to 
-\family typewriter
-SolverNonlinear
-\family default
-.
- It is thus only necessary to change the formulation:
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.timedependent]
-\end_layout
-
-\begin_layout LyX-Code
-# Set the formulation for finite strain.
- The default solver will
-\end_layout
-
-\begin_layout LyX-Code
-# automatically be switched to the nonlinear solver.
-\end_layout
-
-\begin_layout LyX-Code
-formulation = pylith.problems.ImplicitLgDeform
-\end_layout
-
-\begin_layout Standard
-When we have run the simulation, the output VTK files will be contained
- in 
-\family typewriter
-examples/3d/hex8/output
-\family default
- (all with a prefix of 
-\family typewriter
-step17
-\family default
-).
- Results using ParaView are shown in Figure 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "fig:step17-disp-t200"
-
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-\begin_inset Float figure
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\align center
-\begin_inset Graphics
-	filename figs/step17-displ-t200.jpg
-	lyxscale 50
-	width 10cm
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Plain Layout
-\begin_inset Caption
-
-\begin_layout Plain Layout
-Displacement field for example step17 at t = 200 years visualized using
- ParaView.
- The z-component of the displacement field is shown with the color contours,
- and the vectors show the computed displacements.
- Note the larger displacements compared with example step15.
-\begin_inset CommandInset label
-LatexCommand label
-name "fig:step17-disp-t200"
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_body
-\end_document
+#LyX 2.0 created this file. For more info see http://www.lyx.org/
+\lyxformat 413
+\begin_document
+\begin_header
+\textclass book
+\begin_preamble
+
+\end_preamble
+\use_default_options false
+\maintain_unincluded_children false
+\language english
+\language_package default
+\inputencoding latin1
+\fontencoding global
+\font_roman default
+\font_sans default
+\font_typewriter default
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100
+\font_tt_scale 100
+
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry true
+\use_amsmath 0
+\use_esint 0
+\use_mhchem 1
+\use_mathdots 1
+\cite_engine basic
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\use_refstyle 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\leftmargin 1in
+\topmargin 1in
+\rightmargin 1in
+\bottommargin 2in
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\quotes_language english
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Subsection
+\begin_inset CommandInset label
+LatexCommand label
+name "sec:Tutorial-3d-hex8-gravity"
+
+\end_inset
+
+Gravitational Body Force Examples
+\end_layout
+
+\begin_layout Standard
+PyLith features discussed in this tutorial:
+\end_layout
+
+\begin_layout Itemize
+Gravitational body forces
+\end_layout
+
+\begin_layout Itemize
+Initial stresses
+\end_layout
+
+\begin_layout Itemize
+Finite strain
+\end_layout
+
+\begin_layout Itemize
+Generalized Maxwell linear viscoelastic material
+\end_layout
+
+\begin_layout Subsubsection
+Overview
+\end_layout
+
+\begin_layout Standard
+This set of examples describes a set of problems for PyLith involving gravitatio
+nal body forces.
+ All of the examples are quasi-static and run for a time period of 200 years.
+ These examples also demonstrate the use of a generalized Maxwell viscoelastic
+ material, which is used for the lower crust in all examples.
+ The final example (step17) demonstrates the usage of a finite strain formulatio
+n, which automatically invokes the nonlinear solver.
+ All of the examples are contained in the directory 
+\family typewriter
+examples/3d/hex8
+\family default
+, and the corresponding 
+\family typewriter
+.cfg
+\family default
+ files are 
+\family typewriter
+step15.cfg
+\family default
+, 
+\family typewriter
+step16.cfg
+\family default
+, and 
+\family typewriter
+step17.cfg
+\family default
+.
+ Each example may be run as follows:
+\end_layout
+
+\begin_layout LyX-Code
+pylith stepXX.cfg
+\end_layout
+
+\begin_layout Standard
+This will cause PyLith to read the default parameters in 
+\family typewriter
+pylithapp.cfg
+\family default
+, and then override or augment them with the additional parameters in the
+ 
+\family typewriter
+stepXX.cfg
+\family default
+ file.
+ Each 
+\family typewriter
+.cfg
+\family default
+ file is extensively documented, to provide detailed information on the
+ various parameters.
+\end_layout
+
+\begin_layout Subsubsection
+Step15 - Gravitational Body Forces
+\end_layout
+
+\begin_layout Standard
+The 
+\family typewriter
+step15.cfg
+\family default
+ file defines a problem with extremely simple Dirichlet boundary conditions.
+ On the positive and negative x-faces, the positive and negative y-faces,
+ and the negative z-face, the displacements normal to the face are set to
+ zero.
+ Because all of the materials in the example have the same density, the
+ elastic solution for loading via gravitational body forces is
+\begin_inset Formula 
+\begin{equation}
+\sigma_{zz}=\rho gh;\:\sigma_{xx}=\sigma_{yy}=\frac{\nu\rho gh}{1-\nu}\:.\label{eq:1-1}
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+We first set the gravity field, which by default has values of 9.80655 
+\begin_inset Formula $\unitfrac{m}{s^{2}}$
+\end_inset
+
+ for acceleration and 
+\begin_inset Formula $\left[0,0,-1\right]$
+\end_inset
+
+ for direction:
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.timedependent]
+\end_layout
+
+\begin_layout LyX-Code
+# Set gravity field (default is None)
+\end_layout
+
+\begin_layout LyX-Code
+gravity_field = spatialdata.spatialdb.GravityField
+\end_layout
+
+\begin_layout Standard
+We use adaptive time stepping, set the simulation time to 200 years, and
+ specify a maximum time step size of 10 years:
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.timedependent.implicit]
+\end_layout
+
+\begin_layout LyX-Code
+# Change time stepping algorithm from uniform time step, to adaptive
+\end_layout
+
+\begin_layout LyX-Code
+# time stepping.
+\end_layout
+
+\begin_layout LyX-Code
+time_step = pylith.problems.TimeStepAdapt
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+# Change the total simulation time to 200 years, and set the maximum time
+\end_layout
+
+\begin_layout LyX-Code
+# step size to 10 years.
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.timedependent.implicit.time_step]
+\end_layout
+
+\begin_layout LyX-Code
+total_time = 200.0*year
+\end_layout
+
+\begin_layout LyX-Code
+max_dt = 10.0*year
+\end_layout
+
+\begin_layout LyX-Code
+stability_factor = 1.0 ; use time step equal to stable value from materials
+\end_layout
+
+\begin_layout Standard
+We use a generalized Maxwell model for the lower crust (see Section 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sub:Formulation-for-Gen-Max"
+
+\end_inset
+
+), and use a 
+\family typewriter
+SimpleDB
+\family default
+ to provide the properties.
+ We also request the relevant properties and state variables for output:
+\end_layout
+
+\begin_layout LyX-Code
+# Change material type of lower crust to generalized Maxwell viscoelastic.
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.timedependent]
+\end_layout
+
+\begin_layout LyX-Code
+materials.lower_crust = pylith.materials.GenMaxwellIsotropic3D
+\end_layout
+
+\begin_layout LyX-Code
+# Provide a spatial database from which to obtain property values.
+\end_layout
+
+\begin_layout LyX-Code
+# Since there are additional properties and state variables for the
+\end_layout
+
+\begin_layout LyX-Code
+# generalized Maxwell model, we explicitly request that they be output.
+\end_layout
+
+\begin_layout LyX-Code
+# Properties are named in cell_info_fields and state variables are named
+ in
+\end_layout
+
+\begin_layout LyX-Code
+# cell_data_fields.
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.timedependent.materials.lower_crust]
+\end_layout
+
+\begin_layout LyX-Code
+db_properties.iohandler.filename = spatialdb/mat_genmaxwell.spatialdb
+\end_layout
+
+\begin_layout LyX-Code
+output.cell_info_fields = [density,mu,lambda,shear_ratio,maxwell_time]
+\end_layout
+
+\begin_layout LyX-Code
+output.cell_data_fields = [total_strain,stress,viscous_strain_1,viscous_strain_2,
+\begin_inset Newline newline
+\end_inset
+
+viscous_strain_3]
+\end_layout
+
+\begin_layout Standard
+The boundary conditions for this example are trivial, so we are able to
+ use the default 
+\family typewriter
+ZeroDispDB
+\family default
+ for all faces.
+ When we have run the simulation, the output VTK files will be contained
+ in 
+\family typewriter
+examples/3d/hex8/output
+\family default
+ (all with a prefix of 
+\family typewriter
+step15
+\family default
+).
+ Results using ParaView are shown in Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:step15-displ-t200"
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\align center
+\begin_inset Graphics
+	filename figs/step15-displ-t200.jpg
+	lyxscale 50
+	width 10cm
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Displacement field for example step15 at t = 200 years visualized using
+ ParaView.
+ The z-component of the displacement field is shown with the color contours,
+ and the vectors show the computed displacements.
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:step15-displ-t200"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsubsection
+Step16 - Gravitational Body Forces with Initial Stresses
+\end_layout
+
+\begin_layout Standard
+The 
+\family typewriter
+step16.cfg
+\family default
+ file defines a problem that is identical to example step15, except that
+ initial stresses are used to prevent the initial large displacements due
+ to 'turning on' gravity.
+ Since all normal stress components are given an initial stress of 
+\begin_inset Formula $\rho gh$
+\end_inset
+
+, the initial stress state is lithostatic, which is an appropriate condition
+ for many tectonic problems in the absence of tectonic stresses (e.g., McGarr
+ 
+\begin_inset CommandInset citation
+LatexCommand cite
+key "McGarr:1988"
+
+\end_inset
+
+).
+ When compared to example step15, this example should maintain a lithostatic
+ state of stress for the entire simulation, and displacements should remain
+ essentially zero.
+\end_layout
+
+\begin_layout Standard
+We set the gravity field, as in example step15, and we again use adaptive
+ time stepping with a generalized Maxwell rheology for the lower crust.
+ We provide values for the initial stress for both the upper and lower crust.
+ Since the materials have the same density, we are able to use the same
+ 
+\family typewriter
+SimpleDB
+\family default
+ with a linear variation for both (see file 
+\family typewriter
+examples/3d/hex8/spatialdb/initial_stress.spatialdb
+\family default
+):
+\end_layout
+
+\begin_layout LyX-Code
+# We must specify initial stresses for each material.
+\end_layout
+
+\begin_layout LyX-Code
+# We provide a filename for the spatial database that gives the stresses,
+\end_layout
+
+\begin_layout LyX-Code
+# and we change the query_type from the default 'nearest' to 'linear'.
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.timedependent.materials.upper_crust]
+\end_layout
+
+\begin_layout LyX-Code
+db_initial_stress = spatialdata.spatialdb.SimpleDB
+\end_layout
+
+\begin_layout LyX-Code
+db_initial_stress.iohandler.filename = spatialdb/initial_stress.spatialdb
+\end_layout
+
+\begin_layout LyX-Code
+db_initial_stress.query_type = linear
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.timedependent.materials.lower_crust]
+\end_layout
+
+\begin_layout LyX-Code
+db_initial_stress = spatialdata.spatialdb.SimpleDB
+\end_layout
+
+\begin_layout LyX-Code
+db_initial_stress.iohandler.filename = spatialdb/initial_stress.spatialdb
+\end_layout
+
+\begin_layout LyX-Code
+db_initial_stress.query_type = linear
+\end_layout
+
+\begin_layout Standard
+Note that we use a 
+\family typewriter
+linear
+\family default
+ 
+\family typewriter
+query_type
+\family default
+ rather than the default type of 
+\family typewriter
+nearest
+\family default
+, so that a linear interpolation is performed along the z-direction.
+ When we have run the simulation, the output VTK files will be contained
+ in 
+\family typewriter
+examples/3d/hex8/output
+\family default
+ (all with a prefix of 
+\family typewriter
+step16
+\family default
+).
+ Results using ParaView are shown in Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:step16-stress_xx-t200"
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\align center
+\begin_inset Graphics
+	filename figs/step16-stress_xx-t200.jpg
+	lyxscale 50
+	width 10cm
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Stress field (xx-component) for example step16 at t = 200 years visualized
+ using ParaView.
+ Note that for this example, Stress_xx = Stress_yy = Stress_zz, and there
+ is no vertical displacement throughout the simulation.
+ Also note that the stresses appear as four layers since we have used 
+\family typewriter
+CellFilterAvgMesh
+\family default
+ for material output.
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:step16-stress_xx-t200"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsubsection
+Step17 - Gravitational Body Forces with Small Strain
+\end_layout
+
+\begin_layout Standard
+The 
+\family typewriter
+step17.cfg
+\family default
+ file defines a problem that is identical to example step15, except that
+ we now use a small strain formulation (see Section 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sec:Small-Strain-Formulation"
+
+\end_inset
+
+).
+ All of the problems up to this point have assumed infinitesimal strain,
+ meaning that the change in shape of the domain during deformation is not
+ taken into account.
+ In many problems it is important to consider the change in shape of the
+ domain.
+ This is particularly important in many problems involving gravitational
+ body forces, since a change in shape of the domain results in a different
+ stress field.
+ By examining the stress and deformation fields for this example in comparison
+ with those of example step15, we can see what effect the infinitesimal
+ strain approximation has on our solution.
+\end_layout
+
+\begin_layout Standard
+We set the gravity field, as in example step15 and again use adaptive time
+ stepping withs a generalized Maxwell rheology for the lower crust.
+ The only change is that we change the problem formulation from the default
+ 
+\family typewriter
+Implicit
+\family default
+ to 
+\family typewriter
+ImplicitLgDeform
+\family default
+.
+ Since the large deformation formulation is nonlinear, PyLith automatically
+ switches the solver from the default 
+\family typewriter
+SolverLinear
+\family default
+ to 
+\family typewriter
+SolverNonlinear
+\family default
+.
+ It is thus only necessary to change the formulation:
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.timedependent]
+\end_layout
+
+\begin_layout LyX-Code
+# Set the formulation for finite strain.
+ The default solver will
+\end_layout
+
+\begin_layout LyX-Code
+# automatically be switched to the nonlinear solver.
+\end_layout
+
+\begin_layout LyX-Code
+formulation = pylith.problems.ImplicitLgDeform
+\end_layout
+
+\begin_layout Standard
+When we have run the simulation, the output VTK files will be contained
+ in 
+\family typewriter
+examples/3d/hex8/output
+\family default
+ (all with a prefix of 
+\family typewriter
+step17
+\family default
+).
+ Results using ParaView are shown in Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:step17-disp-t200"
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\align center
+\begin_inset Graphics
+	filename figs/step17-displ-t200.jpg
+	lyxscale 50
+	width 10cm
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Displacement field for example step17 at t = 200 years visualized using
+ ParaView.
+ The z-component of the displacement field is shown with the color contours,
+ and the vectors show the computed displacements.
+ Note the larger displacements compared with example step15.
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:step17-disp-t200"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document

Modified: short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/3dhex8/quasistatic/quasistatic.lyx
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/3dhex8/quasistatic/quasistatic.lyx	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/3dhex8/quasistatic/quasistatic.lyx	2012-08-21 18:24:25 UTC (rev 20616)
@@ -756,12 +756,11 @@
 \end_layout
 
 \begin_layout LyX-Code
-db_rate.values = [traction-rate-shear-horiz,traction-rate-shear-vert,traction-rat
-e-normal,
+db_rate.values = [traction-rate-shear-horiz,traction-rate-shear-vert,
 \begin_inset Newline newline
 \end_inset
 
-rate-start-time]
+traction-rate-normal,rate-start-time]
 \end_layout
 
 \begin_layout LyX-Code
@@ -1483,13 +1482,9 @@
 \end_layout
 
 \begin_layout Standard
-In 
-\family typewriter
-step07
-\family default
- we add velocity boundary conditions in the positive and negative y-directions
- on the positive and negative x-faces, so that the external boundaries keep
- pace with the average fault slip.
+In step07 we add velocity boundary conditions in the positive and negative
+ y-directions on the positive and negative x-faces, so that the external
+ boundaries keep pace with the average fault slip.
  This problem is nearly identical to the strike-slip fault model of Savage
  and Prescott 
 \begin_inset CommandInset citation
@@ -1756,12 +1751,8 @@
 step07
 \family default
 ).
- As for example 
+ As for example step06, make sure to open the 
 \family typewriter
-step06
-\family default
-, make sure to open the 
-\family typewriter
 .xmf
 \family default
  files rather than the 
@@ -1796,11 +1787,7 @@
 
 \begin_layout Plain Layout
 Displacement field (color contours) and velocity field (vectors) for example
- 
-\family typewriter
-step07
-\family default
- at t = 300 years visualized using ParaView.
+ step07 at t = 300 years visualized using ParaView.
  The mesh has been distorted by the computed displacements (magnified by
  500), and the vectors show the computed velocities.
 \begin_inset CommandInset label
@@ -1809,7 +1796,7 @@
 
 \end_inset
 
-.
+
 \end_layout
 
 \end_inset
@@ -1839,11 +1826,8 @@
 \family typewriter
 step08.cfg
 \family default
- file defines a problem that is identical to example 
-\family typewriter
-step07
-\family default
-, except the the lower crust is composed of a power-law viscoelastic material.
+ file defines a problem that is identical to example step07, except the
+ the lower crust is composed of a power-law viscoelastic material.
  Since the material behavior is now nonlinear, we must use the nonlinear
  solver:
 \end_layout
@@ -1931,7 +1915,11 @@
  The user must provide a spatial database defining the spatial distribution
  of laboratory-derived parameters (contained in 
 \family typewriter
-powerlaw_params.spatialdb
+powerlaw_params.
+\begin_inset Newline linebreak
+\end_inset
+
+spatialdb
 \family default
 ), another spatial database defining the temperature field in degrees K
  (contained in 
@@ -1992,8 +1980,11 @@
 \end_layout
 
 \begin_layout Standard
-This code will automatically read the parameters in powerlaw_gendb.cfg in
- creating the file
+This code will automatically read the parameters in 
+\family typewriter
+powerlaw_gendb.cfg
+\family default
+ in creating the file
 \begin_inset Newline newline
 \end_inset
 
@@ -2077,39 +2068,57 @@
 \end_layout
 
 \begin_layout LyX-Code
+
+\size small
 # Provide the values to be obtained from each database and the database
 \end_layout
 
 \begin_layout LyX-Code
+
+\size small
 # name.
 \end_layout
 
 \begin_layout LyX-Code
+
+\size small
 [pylithapp.timedependent.materials.lower_crust.db_properties]
 \end_layout
 
 \begin_layout LyX-Code
+
+\size small
 values_A = [density,vs,vp]   ; Elastic properties.
 \end_layout
 
 \begin_layout LyX-Code
+
+\size small
 db_A.label = Elastic properties
 \end_layout
 
 \begin_layout LyX-Code
+
+\size small
 db_A.iohandler.filename = spatialdb/mat_elastic.spatialdb
 \end_layout
 
 \begin_layout LyX-Code
+
+\size small
 values_B = [reference-stress,reference-strain-rate,power-law-exponent] 
   ; Power-law properties.
 \end_layout
 
 \begin_layout LyX-Code
+
+\size small
 db_B.label = Power-law properties
 \end_layout
 
 \begin_layout LyX-Code
+
+\size small
 db_B.iohandler.filename = spatialdb/mat_powerlaw.spatialdb
 \end_layout
 
@@ -2202,11 +2211,7 @@
 
 \begin_layout Plain Layout
 The XY-component of strain (color contours) and displacement field (vectors)
- for example 
-\family typewriter
-step08
-\family default
- at t = 150 years visualized using ParaView.
+ for example step08 at t = 150 years visualized using ParaView.
  For this visualization, we loaded both the 
 \family typewriter
 step08-lower_crust.xmf
@@ -2227,7 +2232,7 @@
 
 \end_inset
 
-.
+
 \end_layout
 
 \end_inset
@@ -2248,11 +2253,8 @@
 \begin_layout Standard
 In this example we use a Drucker-Prager elastoplastic rheology in the lower
  crust.
- As in example 
-\family typewriter
-step08
-\family default
-, the material behavior is nonlinear so we again use the nonlinear solver.
+ As in example step08, the material behavior is nonlinear so we again use
+ the nonlinear solver.
  The material is elastoplastic, there is no inherent time-dependent response
  and the stable time-step size for the material depends on the loading condition
 s.
@@ -2363,12 +2365,8 @@
 \end_layout
 
 \begin_layout Standard
-As for the 
-\family typewriter
-step08
-\family default
- example, we first define the properties that come from each spatial database
- and then provide the database filename:
+As for the step08 example, we first define the properties that come from
+ each spatial database and then provide the database filename:
 \end_layout
 
 \begin_layout LyX-Code
@@ -2484,11 +2482,7 @@
 
 \begin_layout Plain Layout
 The XY-component of strain (color contours) and displacement field (vectors)
- for example 
-\family typewriter
-step09
-\family default
- at t = 150 years visualized using ParaView.
+ for example step09 at t = 150 years visualized using ParaView.
  For this visualization, we loaded both the 
 \family typewriter
 step09-lower_crust.xmf
@@ -2509,7 +2503,7 @@
 
 \end_inset
 
-.
+
 \end_layout
 
 \end_inset

Modified: short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/3dhex8/static/static.lyx
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/3dhex8/static/static.lyx	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/3dhex8/static/static.lyx	2012-08-21 18:24:25 UTC (rev 20616)
@@ -1,869 +1,872 @@
-#LyX 2.0 created this file. For more info see http://www.lyx.org/
-\lyxformat 413
-\begin_document
-\begin_header
-\textclass book
-\begin_preamble
-
-\end_preamble
-\use_default_options false
-\maintain_unincluded_children false
-\language english
-\language_package default
-\inputencoding latin1
-\fontencoding global
-\font_roman default
-\font_sans default
-\font_typewriter default
-\font_default_family default
-\use_non_tex_fonts false
-\font_sc false
-\font_osf false
-\font_sf_scale 100
-\font_tt_scale 100
-
-\graphics default
-\default_output_format default
-\output_sync 0
-\bibtex_command default
-\index_command default
-\paperfontsize default
-\spacing single
-\use_hyperref false
-\papersize default
-\use_geometry true
-\use_amsmath 0
-\use_esint 0
-\use_mhchem 1
-\use_mathdots 1
-\cite_engine basic
-\use_bibtopic false
-\use_indices false
-\paperorientation portrait
-\suppress_date false
-\use_refstyle 0
-\index Index
-\shortcut idx
-\color #008000
-\end_index
-\leftmargin 1in
-\topmargin 1in
-\rightmargin 1in
-\bottommargin 2in
-\secnumdepth 3
-\tocdepth 3
-\paragraph_separation indent
-\paragraph_indentation default
-\quotes_language english
-\papercolumns 1
-\papersides 1
-\paperpagestyle default
-\tracking_changes false
-\output_changes false
-\html_math_output 0
-\html_css_as_file 0
-\html_be_strict false
-\end_header
-
-\begin_body
-
-\begin_layout Subsection
-\begin_inset CommandInset label
-LatexCommand label
-name "sec:Tutorial-3d-hex8-static"
-
-\end_inset
-
-Static Examples
-\end_layout
-
-\begin_layout Standard
-PyLith features discussed in this tutorial:
-\end_layout
-
-\begin_layout Itemize
-Static solution
-\end_layout
-
-\begin_layout Itemize
-VTK output
-\end_layout
-
-\begin_layout Itemize
-Dirichlet displacement boundary conditions
-\end_layout
-
-\begin_layout Itemize
-Neumann traction boundary conditions
-\end_layout
-
-\begin_layout Itemize
-ZeroDispDB spatial database
-\end_layout
-
-\begin_layout Itemize
-SimpleDB spatial database
-\end_layout
-
-\begin_layout Itemize
-UniformDB spatial database
-\end_layout
-
-\begin_layout Itemize
-Static fault rupture
-\end_layout
-
-\begin_layout Itemize
-Specifying more than one material
-\end_layout
-
-\begin_layout Itemize
-Linearly elastic isotropic material
-\end_layout
-
-\begin_layout Subsubsection
-Overview
-\end_layout
-
-\begin_layout Standard
-This set of examples describe the simplest class of problems for PyLith.
- The problems are all purely elastic, and there is no time-dependence.
- This set of elastostatic examples primarily demonstrates the application
- of different types of boundary conditions in PyLith, as well as demonstrating
- the use of a kinematic fault for a static problem.
- All of the examples are contained in the directory 
-\family typewriter
-examples/3d/hex8
-\family default
-, and the corresponding 
-\family typewriter
-.cfg
-\family default
- files are 
-\family typewriter
-step01.cfg
-\family default
-, 
-\family typewriter
-step02.cfg
-\family default
-, and 
-\family typewriter
-step03.cfg
-\family default
-.
- Each example may be run as follows:
-\end_layout
-
-\begin_layout LyX-Code
-pylith stepXX.cfg
-\end_layout
-
-\begin_layout Standard
-This will cause PyLith to read the default parameters in 
-\family typewriter
-pylithapp.cfg
-\family default
-, and then override or augment them with the additional parameters in the
- 
-\family typewriter
-stepXX.cfg
-\family default
- file.
- Each 
-\family typewriter
-.cfg
-\family default
- file is extensively documented to provide detailed information on the various
- parameters.
-\end_layout
-
-\begin_layout Subsubsection
-Step01 - Pure Dirichlet Boundary Conditions
-\end_layout
-
-\begin_layout Standard
-The 
-\family typewriter
-step01.cfg
-\family default
- file defines a problem with pure Dirichlet (displacement) boundary conditions
- corresponding to compression in the x-direction and shear in the y-direction.
- The bottom (minimum z) boundary is held fixed in the z-direction.
- On the positive and negative x-faces, compressional displacements of 1
- m are applied in the x-direction and shear displacements yielding a left-latera
-l sense of shear are applied in the y-direction.
- In this example and in subsequent examples we would like to output the
- displacement solution over a subset of the domain corresponding to the
- ground surface.
- To do this, we first set the output to an array of two output managers
- as follows:
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.timedependent.implicit]
-\end_layout
-
-\begin_layout LyX-Code
-# Set the output to an array of 2 output managers.
-\end_layout
-
-\begin_layout LyX-Code
-# We will output the solution over the domain and the ground surface.
-\end_layout
-
-\begin_layout LyX-Code
-output = [domain,subdomain]
-\end_layout
-
-\begin_layout Standard
-We then define the subdomain output manager to correspond to a subset of
- the domain:
-\end_layout
-
-\begin_layout LyX-Code
-# Set subdomain component to OutputSolnSubset (boundary of the domain).
-\end_layout
-
-\begin_layout LyX-Code
-output.subdomain = pylith.meshio.OutputSolnSubset
-\end_layout
-
-\begin_layout Standard
-Later (in the output section at the end of the file), we specify the nodeset
- that corresponds to the desired output:
-\end_layout
-
-\begin_layout LyX-Code
-# Give basename for VTK domain output of solution over ground surface.
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.problem.formulation.output.subdomain]
-\end_layout
-
-\begin_layout LyX-Code
-# Name of nodeset for ground surface.
-\end_layout
-
-\begin_layout LyX-Code
-label = face_zpos
-\end_layout
-
-\begin_layout LyX-Code
-writer.filename = output/step01-groundsurf.vtk
-\end_layout
-
-\begin_layout Standard
-For the boundary conditions, we must describe which degrees of freedom are
- being constrained (
-\family typewriter
-bc_dof
-\family default
-), we must provide a the label associated with the CUBIT nodeset associated
- with the BC, and we must specify the type of spatial database is being
- used to describe the boundary conditions.
- For the x-faces, we use a 
-\family typewriter
-SimpleDB
-\family default
- to provide the displacements on the x-faces:
-\end_layout
-
-\begin_layout LyX-Code
-# +x face
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.timedependent.bc.x_pos]
-\end_layout
-
-\begin_layout LyX-Code
-bc_dof = [0, 1]
-\end_layout
-
-\begin_layout LyX-Code
-label = face_xpos
-\end_layout
-
-\begin_layout LyX-Code
-db_initial = spatialdata.spatialdb.SimpleDB
-\end_layout
-
-\begin_layout LyX-Code
-db_initial.label = Dirichlet BC on +x
-\end_layout
-
-\begin_layout LyX-Code
-db_initial.iohandler.filename = spatialdb/fixeddisp_axial_shear.spatialdb
-\begin_inset Newline newline
-\end_inset
-
-
-\begin_inset Newline newline
-\end_inset
-
-
-\end_layout
-
-\begin_layout LyX-Code
-# -x face
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.timedependent.bc.x_neg]
-\end_layout
-
-\begin_layout LyX-Code
-bc_dof = [0, 1]
-\end_layout
-
-\begin_layout LyX-Code
-label = face_xneg
-\end_layout
-
-\begin_layout LyX-Code
-db_initial = spatialdata.spatialdb.SimpleDB
-\end_layout
-
-\begin_layout LyX-Code
-db_initial.label = Dirichlet BC on -x
-\end_layout
-
-\begin_layout LyX-Code
-db_initial.iohandler.filename = spatialdb/fixeddisp_axial_shear.spatialdb
-\end_layout
-
-\begin_layout Standard
-For a 
-\family typewriter
-SimpleDB
-\family default
-, we must provide a filename.
- The default spatial database for 
-\family typewriter
-db_initial
-\family default
- is 
-\family typewriter
-ZeroDispBC
-\family default
-, which automatically applies zero displacements to all vertices in the
- nodeset, and no filename is required (or needed).
- This is what we use for the bottom of the mesh:
-\end_layout
-
-\begin_layout LyX-Code
-# -z face
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.timedependent.bc.z_neg]
-\end_layout
-
-\begin_layout LyX-Code
-bc_dof = [2]
-\end_layout
-
-\begin_layout LyX-Code
-label = face_zneg
-\end_layout
-
-\begin_layout LyX-Code
-db_initial.label = Dirichlet BC on -z
-\end_layout
-
-\begin_layout Standard
-When we have run the simulation, the output VTK files will be contained
- in 
-\family typewriter
-examples/3d/hex8/output
-\family default
- (all with a prefix of 
-\family typewriter
-step01
-\family default
-).
- Results using ParaView are shown in Figure 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "fig:step01-displ"
-
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-\begin_inset Float figure
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\align center
-\begin_inset Graphics
-	filename figs/step01-displ.jpg
-	width 10cm
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Plain Layout
-\begin_inset Caption
-
-\begin_layout Plain Layout
-Displacement field for example step01 visualized using ParaView.
- The mesh has been distorted by the computed displacements (magnified by
- 500), and the vectors show the computed displacements.
-\begin_inset CommandInset label
-LatexCommand label
-name "fig:step01-displ"
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsubsection
-Step02 - Dirichlet and Neumann Boundary Conditions
-\end_layout
-
-\begin_layout Standard
-The 
-\family typewriter
-step02.cfg
-\family default
- file defines a problem with Dirichlet (displacement) boundary conditions
- corresponding to zero x and y-displacements applied on the negative x-face
- and Neumann (traction) boundary conditions corresponding to normal compression
- and horizontal shear applied on the positive x-face.
- The bottom (negative z) boundary is held fixed in the z-direction.
- The problem is similar to example step01, except that 1 MPa of normal compressi
-on and 1 MPa of shear (in a left-lateral sense) are applied on the positive
- x-face, and the negative x-face is pinned in both the x and y-directions.
-\end_layout
-
-\begin_layout Standard
-For the boundary conditions, we must first change the boundary condition
- type for the positive x-face from the default Dirichlet to Neumann:
-\end_layout
-
-\begin_layout LyX-Code
-# +x face -- first change bc type to Neumann
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.timedependent.bc]
-\end_layout
-
-\begin_layout LyX-Code
-x_pos = pylith.bc.Neumann 
-\end_layout
-
-\begin_layout Standard
-We use a 
-\family typewriter
-SimpleDB
-\family default
- to describe the traction boundary conditions.
- When applying traction boundary conditions over a surface, it is also necessary
- to specify integration information for the surface.
- Since this is a three-dimensional problem, the dimension of the surface
- is 2.
- Since the cells being used are trilinear hexahedra, the cell type is 
-\family typewriter
-FIATLagrange
-\family default
- and we use an integration order of 2.
- A lower integration order would not provide sufficient accuracy while a
- higher integration order would offer no benefit (while requiring more computati
-on time and storage):
-\end_layout
-
-\begin_layout LyX-Code
-# +x face
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.timedependent.bc.x_pos]
-\end_layout
-
-\begin_layout LyX-Code
-label = face_xpos
-\end_layout
-
-\begin_layout LyX-Code
-db_initial = spatialdata.spatialdb.SimpleDB
-\end_layout
-
-\begin_layout LyX-Code
-db_initial.label = Neumann BC on +x
-\end_layout
-
-\begin_layout LyX-Code
-db_initial.iohandler.filename = spatialdb/tractions_axial_shear.spatialdb
-\begin_inset Newline newline
-\end_inset
-
-
-\begin_inset Newline newline
-\end_inset
-
-
-\end_layout
-
-\begin_layout LyX-Code
-# We must specify quadrature information for the cell faces.
-\end_layout
-
-\begin_layout LyX-Code
-quadrature.cell = pylith.feassemble.FIATLagrange
-\end_layout
-
-\begin_layout LyX-Code
-quadrature.cell.dimension = 2
-\end_layout
-
-\begin_layout LyX-Code
-quadrature.cell.quad_order = 2 
-\end_layout
-
-\begin_layout Standard
-The boundary conditions on the negative x-face are simpler than they were
- in example step01 (zero displacements in the x and y-directions), so we
- can use the default 
-\family typewriter
-ZeroDispBC
-\family default
-:
-\end_layout
-
-\begin_layout LyX-Code
-# -x face
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.timedependent.bc.x_neg]
-\end_layout
-
-\begin_layout LyX-Code
-bc_dof = [0, 1] 
-\end_layout
-
-\begin_layout LyX-Code
-label = face_xneg
-\end_layout
-
-\begin_layout LyX-Code
-db_initial.label = Dirichlet BC on -x 
-\end_layout
-
-\begin_layout Standard
-The boundary conditions on the negative z-face are supplied in the same
- manner as for example step01.
- When we have run the simulation, the output VTK files will be contained
- in 
-\family typewriter
-examples/3d/hex8/output
-\family default
- (all with a prefix of 
-\family typewriter
-step02
-\family default
-).
- Results using ParaView are shown in Figure 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "fig:step02-displ"
-
-\end_inset
-
-.
-\begin_inset Float figure
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\align center
-\begin_inset Graphics
-	filename figs/step02-displ.jpg
-	lyxscale 50
-	width 10cm
-
-\end_inset
-
-
-\begin_inset Caption
-
-\begin_layout Plain Layout
-Displacement field for example step02 visualized using ParaView.
- The mesh has been distorted by the computed displacements (magnified by
- 500), and the vectors show the computed displacements.
-\begin_inset CommandInset label
-LatexCommand label
-name "fig:step02-displ"
-
-\end_inset
-
-.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsubsection
-Step03 - Dirichlet Boundary Conditions with Kinematic Fault Slip
-\end_layout
-
-\begin_layout Standard
-The 
-\family typewriter
-step03.cfg
-\family default
- file describes a problem with Dirichlet (displacement) boundary conditions
- corresponding to zero x and y-displacements applied on the negative and
- positive x-faces and a vertical fault with a combination of left-lateral
- and updip motion.
- The left-lateral component of fault slip has a constant value of 2 m in
- the upper crust, and then decreases linearly to zero at the base of the
- model.
- The reverse slip component has a value of 0.25 m at the surface, and then
- decreases linearly to zero at 2 km depth.
-\end_layout
-
-\begin_layout Standard
-Due to the simplicity of the boundary conditions, we are able to use the
- default 
-\family typewriter
-ZeroDispBC
-\family default
- for the positive and negative x-faces, as well as the negative z-face.
- To use a fault, we must first define a fault interface.
- We do this by providing an array containing a single interface:
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.timedependent]
-\end_layout
-
-\begin_layout LyX-Code
-# Set interfaces to an array of 1 fault: 'fault'.
-\end_layout
-
-\begin_layout LyX-Code
-interfaces = [fault] 
-\end_layout
-
-\begin_layout Standard
-For this example we specify the fault slip, so we set the interface type
- to 
-\family typewriter
-FaultCohesiveKin
-\family default
-:
-\end_layout
-
-\begin_layout LyX-Code
-# Set the type of fault interface condition.
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.timedependent.interfaces]
-\end_layout
-
-\begin_layout LyX-Code
-fault = pylith.faults.FaultCohesiveKin 
-\end_layout
-
-\begin_layout Standard
-We must then identify the nodeset corresponding to the fault, and provide
- integration information for the fault surface:
-\end_layout
-
-\begin_layout LyX-Code
-# Set the parameters for the fault interface condition.
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.timedependent.interfaces.fault]
-\end_layout
-
-\begin_layout LyX-Code
-# The label corresponds to the name of the nodeset in CUBIT.
-\end_layout
-
-\begin_layout LyX-Code
-label = fault
-\begin_inset Newline newline
-\end_inset
-
-
-\begin_inset Newline newline
-\end_inset
-
-
-\end_layout
-
-\begin_layout LyX-Code
-# We must define the quadrature information for fault cells.
-\end_layout
-
-\begin_layout LyX-Code
-# The fault cells are 2D (surface).
-\end_layout
-
-\begin_layout LyX-Code
-quadrature.cell = pylith.feassemble.FIATLagrange
-\end_layout
-
-\begin_layout LyX-Code
-quadrature.cell.dimension = 2 
-\end_layout
-
-\begin_layout Standard
-We retain the default 
-\family typewriter
-StepSlipFn
-\family default
- since we want static fault slip.
- Finally, we use one 
-\family typewriter
-SimpleDB
-\family default
- to define the spatial variation of fault slip, and another 
-\family typewriter
-SimpleDB
-\family default
- to define the spatial variation in slip initiation times (the start time
- is 0.0 everywhere since this is a static problem):
-\end_layout
-
-\begin_layout LyX-Code
-# The slip time and final slip are defined in spatial databases.
- [pylithapp.timedependent.interfaces.fault.eq_srcs.rupture.slip_function]
-\end_layout
-
-\begin_layout LyX-Code
-slip.iohandler.filename = spatialdb/finalslip.spatialdb
-\end_layout
-
-\begin_layout LyX-Code
-slip.query_type = linear
-\end_layout
-
-\begin_layout LyX-Code
-slip_time.iohandler.filename = spatialdb/sliptime.spatialdb 
-\end_layout
-
-\begin_layout Standard
-Since the problem now contains a fault, we can request that fault infomation
- is also output:
-\end_layout
-
-\begin_layout LyX-Code
-# Give basename for VTK fault output.
-\end_layout
-
-\begin_layout LyX-Code
-[pylithapp.problem.interfaces.fault.output]
-\end_layout
-
-\begin_layout LyX-Code
-writer.filename = output/step03-fault.vtk 
-\end_layout
-
-\begin_layout Standard
-This will result in two extra files being produced.
- The first file (
-\family typewriter
-step03-fault_info.vtk
-\family default
-) contains information such as the normal directions to the fault surface,
- the applied fault slip, and the fault slip times.
- The second file (
-\family typewriter
-step03-fault_t0000000.vtk
-\family default
-) contains the cumulative fault slip for the time step and the change in
- tractions on the fault surface due to the slip.
- When we have run the simulation, the output VTK files will be contained
- in 
-\family typewriter
-examples/3d/hex8/output
-\family default
- (all with a prefix of 
-\family typewriter
-step03
-\family default
-).
- Results using ParaView are shown in Figure 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "fig:step03-displ"
-
-\end_inset
-
-.
-\begin_inset Float figure
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\align center
-\begin_inset Graphics
-	filename figs/step03-displ.jpg
-	lyxscale 50
-	width 10cm
-
-\end_inset
-
-
-\begin_inset Caption
-
-\begin_layout Plain Layout
-Displacement field for example step03 visualized using ParaView.
- The mesh has been distorted by the computed displacements (magnified by
- 500), and the vectors show the computed displacements.
-\begin_inset CommandInset label
-LatexCommand label
-name "fig:step03-displ"
-
-\end_inset
-
-.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_body
-\end_document
+#LyX 2.0 created this file. For more info see http://www.lyx.org/
+\lyxformat 413
+\begin_document
+\begin_header
+\textclass book
+\begin_preamble
+
+\end_preamble
+\use_default_options false
+\maintain_unincluded_children false
+\language english
+\language_package default
+\inputencoding latin1
+\fontencoding global
+\font_roman default
+\font_sans default
+\font_typewriter default
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100
+\font_tt_scale 100
+
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry true
+\use_amsmath 0
+\use_esint 0
+\use_mhchem 1
+\use_mathdots 1
+\cite_engine basic
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\use_refstyle 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\leftmargin 1in
+\topmargin 1in
+\rightmargin 1in
+\bottommargin 2in
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\quotes_language english
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Subsection
+\begin_inset CommandInset label
+LatexCommand label
+name "sec:Tutorial-3d-hex8-static"
+
+\end_inset
+
+Static Examples
+\end_layout
+
+\begin_layout Standard
+PyLith features discussed in this tutorial:
+\end_layout
+
+\begin_layout Itemize
+Static solution
+\end_layout
+
+\begin_layout Itemize
+VTK output
+\end_layout
+
+\begin_layout Itemize
+Dirichlet displacement boundary conditions
+\end_layout
+
+\begin_layout Itemize
+Neumann traction boundary conditions
+\end_layout
+
+\begin_layout Itemize
+ZeroDispDB spatial database
+\end_layout
+
+\begin_layout Itemize
+SimpleDB spatial database
+\end_layout
+
+\begin_layout Itemize
+UniformDB spatial database
+\end_layout
+
+\begin_layout Itemize
+Static fault rupture
+\end_layout
+
+\begin_layout Itemize
+Specifying more than one material
+\end_layout
+
+\begin_layout Itemize
+Linearly elastic isotropic material
+\end_layout
+
+\begin_layout Subsubsection
+Overview
+\end_layout
+
+\begin_layout Standard
+This set of examples describe the simplest class of problems for PyLith.
+ The problems are all purely elastic, and there is no time-dependence.
+ This set of elastostatic examples primarily demonstrates the application
+ of different types of boundary conditions in PyLith, as well as demonstrating
+ the use of a kinematic fault for a static problem.
+ All of the examples are contained in the directory 
+\family typewriter
+examples/3d/hex8
+\family default
+, and the corresponding 
+\family typewriter
+.cfg
+\family default
+ files are 
+\family typewriter
+step01.cfg
+\family default
+, 
+\family typewriter
+step02.cfg
+\family default
+, and 
+\family typewriter
+step03.cfg
+\family default
+.
+ Each example may be run as follows:
+\end_layout
+
+\begin_layout LyX-Code
+pylith stepXX.cfg
+\end_layout
+
+\begin_layout Standard
+This will cause PyLith to read the default parameters in 
+\family typewriter
+pylithapp.cfg
+\family default
+, and then override or augment them with the additional parameters in the
+ 
+\family typewriter
+stepXX.cfg
+\family default
+ file.
+ Each 
+\family typewriter
+.cfg
+\family default
+ file is extensively documented to provide detailed information on the various
+ parameters.
+\end_layout
+
+\begin_layout Subsubsection
+Step01 - Pure Dirichlet Boundary Conditions
+\end_layout
+
+\begin_layout Standard
+The 
+\family typewriter
+step01.cfg
+\family default
+ file defines a problem with pure Dirichlet (displacement) boundary conditions
+ corresponding to compression in the x-direction and shear in the y-direction.
+ The bottom (minimum z) boundary is held fixed in the z-direction.
+ On the positive and negative x-faces, compressional displacements of 1
+ m are applied in the x-direction and shear displacements yielding a left-latera
+l sense of shear are applied in the y-direction.
+ In this example and in subsequent examples we would like to output the
+ displacement solution over a subset of the domain corresponding to the
+ ground surface.
+ To do this, we first set the output to an array of two output managers
+ as follows:
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.timedependent.implicit]
+\end_layout
+
+\begin_layout LyX-Code
+# Set the output to an array of 2 output managers.
+\end_layout
+
+\begin_layout LyX-Code
+# We will output the solution over the domain and the ground surface.
+\end_layout
+
+\begin_layout LyX-Code
+output = [domain,subdomain]
+\end_layout
+
+\begin_layout Standard
+We then define the subdomain output manager to correspond to a subset of
+ the domain:
+\end_layout
+
+\begin_layout LyX-Code
+# Set subdomain component to OutputSolnSubset (boundary of the domain).
+\end_layout
+
+\begin_layout LyX-Code
+output.subdomain = pylith.meshio.OutputSolnSubset
+\end_layout
+
+\begin_layout Standard
+Later (in the output section at the end of the file), we specify the nodeset
+ that corresponds to the desired output:
+\end_layout
+
+\begin_layout LyX-Code
+# Give basename for VTK domain output of solution over ground surface.
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.problem.formulation.output.subdomain]
+\end_layout
+
+\begin_layout LyX-Code
+# Name of nodeset for ground surface.
+\end_layout
+
+\begin_layout LyX-Code
+label = face_zpos
+\end_layout
+
+\begin_layout LyX-Code
+writer.filename = output/step01-groundsurf.vtk
+\end_layout
+
+\begin_layout Standard
+For the boundary conditions, we must describe which degrees of freedom are
+ being constrained (
+\family typewriter
+bc_dof
+\family default
+), we must provide a the label associated with the CUBIT nodeset associated
+ with the BC, and we must specify the type of spatial database is being
+ used to describe the boundary conditions.
+ For the x-faces, we use a 
+\family typewriter
+SimpleDB
+\family default
+ to provide the displacements on the x-faces:
+\end_layout
+
+\begin_layout LyX-Code
+# +x face
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.timedependent.bc.x_pos]
+\end_layout
+
+\begin_layout LyX-Code
+bc_dof = [0, 1]
+\end_layout
+
+\begin_layout LyX-Code
+label = face_xpos
+\end_layout
+
+\begin_layout LyX-Code
+db_initial = spatialdata.spatialdb.SimpleDB
+\end_layout
+
+\begin_layout LyX-Code
+db_initial.label = Dirichlet BC on +x
+\end_layout
+
+\begin_layout LyX-Code
+db_initial.iohandler.filename = spatialdb/fixeddisp_axial_shear.spatialdb
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+# -x face
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.timedependent.bc.x_neg]
+\end_layout
+
+\begin_layout LyX-Code
+bc_dof = [0, 1]
+\end_layout
+
+\begin_layout LyX-Code
+label = face_xneg
+\end_layout
+
+\begin_layout LyX-Code
+db_initial = spatialdata.spatialdb.SimpleDB
+\end_layout
+
+\begin_layout LyX-Code
+db_initial.label = Dirichlet BC on -x
+\end_layout
+
+\begin_layout LyX-Code
+db_initial.iohandler.filename = spatialdb/fixeddisp_axial_shear.spatialdb
+\end_layout
+
+\begin_layout Standard
+For a 
+\family typewriter
+SimpleDB
+\family default
+, we must provide a filename.
+ The default spatial database for 
+\family typewriter
+db_initial
+\family default
+ is 
+\family typewriter
+ZeroDispBC
+\family default
+, which automatically applies zero displacements to all vertices in the
+ nodeset, and no filename is required (or needed).
+ This is what we use for the bottom of the mesh:
+\end_layout
+
+\begin_layout LyX-Code
+# -z face
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.timedependent.bc.z_neg]
+\end_layout
+
+\begin_layout LyX-Code
+bc_dof = [2]
+\end_layout
+
+\begin_layout LyX-Code
+label = face_zneg
+\end_layout
+
+\begin_layout LyX-Code
+db_initial.label = Dirichlet BC on -z
+\end_layout
+
+\begin_layout Standard
+When we have run the simulation, the output VTK files will be contained
+ in 
+\family typewriter
+examples/3d/hex8/output
+\family default
+ (all with a prefix of 
+\family typewriter
+step01
+\family default
+).
+ Results using ParaView are shown in Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:step01-displ"
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\align center
+\begin_inset Graphics
+	filename figs/step01-displ.jpg
+	width 10cm
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Displacement field for example step01 visualized using ParaView.
+ The mesh has been distorted by the computed displacements (magnified by
+ 500), and the vectors show the computed displacements.
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:step01-displ"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsubsection
+Step02 - Dirichlet and Neumann Boundary Conditions
+\end_layout
+
+\begin_layout Standard
+The 
+\family typewriter
+step02.cfg
+\family default
+ file defines a problem with Dirichlet (displacement) boundary conditions
+ corresponding to zero x and y-displacements applied on the negative x-face
+ and Neumann (traction) boundary conditions corresponding to normal compression
+ and horizontal shear applied on the positive x-face.
+ The bottom (negative z) boundary is held fixed in the z-direction.
+ The problem is similar to example step01, except that 1 MPa of normal compressi
+on and 1 MPa of shear (in a left-lateral sense) are applied on the positive
+ x-face, and the negative x-face is pinned in both the x and y-directions.
+\end_layout
+
+\begin_layout Standard
+For the boundary conditions, we must first change the boundary condition
+ type for the positive x-face from the default Dirichlet to Neumann:
+\end_layout
+
+\begin_layout LyX-Code
+# +x face -- first change bc type to Neumann
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.timedependent.bc]
+\end_layout
+
+\begin_layout LyX-Code
+x_pos = pylith.bc.Neumann 
+\end_layout
+
+\begin_layout Standard
+We use a 
+\family typewriter
+SimpleDB
+\family default
+ to describe the traction boundary conditions.
+ When applying traction boundary conditions over a surface, it is also necessary
+ to specify integration information for the surface.
+ Since this is a three-dimensional problem, the dimension of the surface
+ is 2.
+ Since the cells being used are trilinear hexahedra, the cell type is 
+\family typewriter
+FIATLagrange
+\family default
+ and we use an integration order of 2.
+ A lower integration order would not provide sufficient accuracy while a
+ higher integration order would offer no benefit (while requiring more computati
+on time and storage):
+\end_layout
+
+\begin_layout LyX-Code
+# +x face
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.timedependent.bc.x_pos]
+\end_layout
+
+\begin_layout LyX-Code
+label = face_xpos
+\end_layout
+
+\begin_layout LyX-Code
+db_initial = spatialdata.spatialdb.SimpleDB
+\end_layout
+
+\begin_layout LyX-Code
+db_initial.label = Neumann BC on +x
+\end_layout
+
+\begin_layout LyX-Code
+db_initial.iohandler.filename = spatialdb/tractions_axial_shear.spatialdb
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+# We must specify quadrature information for the cell faces.
+\end_layout
+
+\begin_layout LyX-Code
+quadrature.cell = pylith.feassemble.FIATLagrange
+\end_layout
+
+\begin_layout LyX-Code
+quadrature.cell.dimension = 2
+\end_layout
+
+\begin_layout LyX-Code
+quadrature.cell.quad_order = 2 
+\end_layout
+
+\begin_layout Standard
+The boundary conditions on the negative x-face are simpler than they were
+ in example step01 (zero displacements in the x and y-directions), so we
+ can use the default 
+\family typewriter
+ZeroDispBC
+\family default
+:
+\end_layout
+
+\begin_layout LyX-Code
+# -x face
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.timedependent.bc.x_neg]
+\end_layout
+
+\begin_layout LyX-Code
+bc_dof = [0, 1] 
+\end_layout
+
+\begin_layout LyX-Code
+label = face_xneg
+\end_layout
+
+\begin_layout LyX-Code
+db_initial.label = Dirichlet BC on -x 
+\end_layout
+
+\begin_layout Standard
+The boundary conditions on the negative z-face are supplied in the same
+ manner as for example step01.
+ When we have run the simulation, the output VTK files will be contained
+ in 
+\family typewriter
+examples/3d/hex8/output
+\family default
+ (all with a prefix of 
+\family typewriter
+step02
+\family default
+).
+ Results using ParaView are shown in Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:step02-displ"
+
+\end_inset
+
+.
+\begin_inset Float figure
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\align center
+\begin_inset Graphics
+	filename figs/step02-displ.jpg
+	lyxscale 50
+	width 10cm
+
+\end_inset
+
+
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Displacement field for example step02 visualized using ParaView.
+ The mesh has been distorted by the computed displacements (magnified by
+ 500), and the vectors show the computed displacements.
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:step02-displ"
+
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsubsection
+Step03 - Dirichlet Boundary Conditions with Kinematic Fault Slip
+\end_layout
+
+\begin_layout Standard
+The 
+\family typewriter
+step03.cfg
+\family default
+ file describes a problem with Dirichlet (displacement) boundary conditions
+ corresponding to zero x and y-displacements applied on the negative and
+ positive x-faces and a vertical fault with a combination of left-lateral
+ and updip motion.
+ The left-lateral component of fault slip has a constant value of 2 m in
+ the upper crust, and then decreases linearly to zero at the base of the
+ model.
+ The reverse slip component has a value of 0.25 m at the surface, and then
+ decreases linearly to zero at 2 km depth.
+\end_layout
+
+\begin_layout Standard
+Due to the simplicity of the boundary conditions, we are able to use the
+ default 
+\family typewriter
+ZeroDispBC
+\family default
+ for the positive and negative x-faces, as well as the negative z-face.
+ To use a fault, we must first define a fault interface.
+ We do this by providing an array containing a single interface:
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.timedependent]
+\end_layout
+
+\begin_layout LyX-Code
+# Set interfaces to an array of 1 fault: 'fault'.
+\end_layout
+
+\begin_layout LyX-Code
+interfaces = [fault] 
+\end_layout
+
+\begin_layout Standard
+For this example we specify the fault slip, so we set the interface type
+ to 
+\family typewriter
+FaultCohesiveKin
+\family default
+:
+\end_layout
+
+\begin_layout LyX-Code
+# Set the type of fault interface condition.
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.timedependent.interfaces]
+\end_layout
+
+\begin_layout LyX-Code
+fault = pylith.faults.FaultCohesiveKin 
+\end_layout
+
+\begin_layout Standard
+We must then identify the nodeset corresponding to the fault, and provide
+ integration information for the fault surface:
+\end_layout
+
+\begin_layout LyX-Code
+# Set the parameters for the fault interface condition.
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.timedependent.interfaces.fault]
+\end_layout
+
+\begin_layout LyX-Code
+# The label corresponds to the name of the nodeset in CUBIT.
+\end_layout
+
+\begin_layout LyX-Code
+label = fault
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+# We must define the quadrature information for fault cells.
+\end_layout
+
+\begin_layout LyX-Code
+# The fault cells are 2D (surface).
+\end_layout
+
+\begin_layout LyX-Code
+quadrature.cell = pylith.feassemble.FIATLagrange
+\end_layout
+
+\begin_layout LyX-Code
+quadrature.cell.dimension = 2 
+\end_layout
+
+\begin_layout Standard
+We retain the default 
+\family typewriter
+StepSlipFn
+\family default
+ since we want static fault slip.
+ Finally, we use one 
+\family typewriter
+SimpleDB
+\family default
+ to define the spatial variation of fault slip, and another 
+\family typewriter
+SimpleDB
+\family default
+ to define the spatial variation in slip initiation times (the start time
+ is 0.0 everywhere since this is a static problem):
+\end_layout
+
+\begin_layout LyX-Code
+# The slip time and final slip are defined in spatial databases.
+\begin_inset Newline newline
+\end_inset
+
+[pylithapp.timedependent.interfaces.fault.eq_srcs.rupture.slip_function]
+\begin_inset Newline newline
+\end_inset
+
+slip.iohandler.filename = spatialdb/finalslip.spatialdb
+\end_layout
+
+\begin_layout LyX-Code
+slip.query_type = linear
+\end_layout
+
+\begin_layout LyX-Code
+slip_time.iohandler.filename = spatialdb/sliptime.spatialdb 
+\end_layout
+
+\begin_layout Standard
+Since the problem now contains a fault, we can request that fault infomation
+ is also output:
+\end_layout
+
+\begin_layout LyX-Code
+# Give basename for VTK fault output.
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.problem.interfaces.fault.output]
+\end_layout
+
+\begin_layout LyX-Code
+writer.filename = output/step03-fault.vtk 
+\end_layout
+
+\begin_layout Standard
+This will result in two extra files being produced.
+ The first file (
+\family typewriter
+step03-fault_info.vtk
+\family default
+) contains information such as the normal directions to the fault surface,
+ the applied fault slip, and the fault slip times.
+ The second file (
+\family typewriter
+step03-fault_t0000000.vtk
+\family default
+) contains the cumulative fault slip for the time step and the change in
+ tractions on the fault surface due to the slip.
+ When we have run the simulation, the output VTK files will be contained
+ in 
+\family typewriter
+examples/3d/hex8/output
+\family default
+ (all with a prefix of 
+\family typewriter
+step03
+\family default
+).
+ Results using ParaView are shown in Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:step03-displ"
+
+\end_inset
+
+.
+\begin_inset Float figure
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\align center
+\begin_inset Graphics
+	filename figs/step03-displ.jpg
+	lyxscale 50
+	width 10cm
+
+\end_inset
+
+
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Displacement field for example step03 visualized using ParaView.
+ The mesh has been distorted by the computed displacements (magnified by
+ 500), and the vectors show the computed displacements.
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:step03-displ"
+
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document

Modified: short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/3dhex8/surfload/surfload.lyx
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/3dhex8/surfload/surfload.lyx	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/3dhex8/surfload/surfload.lyx	2012-08-21 18:24:25 UTC (rev 20616)
@@ -331,9 +331,9 @@
 \family default
  (all with a prefix of 
 \family typewriter
-step1
+step18
 \family default
-8).
+).
  Results using ParaView are shown in Figure 
 \begin_inset CommandInset ref
 LatexCommand ref
@@ -471,9 +471,9 @@
 \family default
  (all with a prefix of 
 \family typewriter
-step1
+step19
 \family default
-9).
+).
  Results using ParaView are shown in Figure 
 \begin_inset CommandInset ref
 LatexCommand ref
@@ -529,13 +529,10 @@
 \begin_inset Caption
 
 \begin_layout Plain Layout
-Stress field (zz-component) for example 
+Stress field (zz-component) for example step19 at t = 200 years visualized
+ using ParaView.
+ The stresses appear as four layers since we have used 
 \family typewriter
-step19
-\family default
- at t = 200 years visualized using ParaView.
- The stresses appear as 4 layers since we have used 
-\family typewriter
 CellFilterAvgMesh
 \family default
  for material output.

Modified: short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/3dtet4/3dtet4.lyx
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/3dtet4/3dtet4.lyx	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/3dtet4/3dtet4.lyx	2012-08-21 18:24:25 UTC (rev 20616)
@@ -218,6 +218,10 @@
 \family default
 examples directory (
 \family typewriter
+
+\begin_inset Newline linebreak
+\end_inset
+
 examples/3d/tet4
 \family default
 ), start LaGriT, and then type:
@@ -419,9 +423,9 @@
 \begin_layout Standard
 Since these examples use a mesh from LaGriT, we set the importer to 
 \family typewriter
-MeshIO
+MeshIOLagrit
 \family default
-Lagrit:
+:
 \end_layout
 
 \begin_layout LyX-Code
@@ -691,7 +695,11 @@
 \begin_layout Description
 pylithapp.problem.formulation.output.subdomain Gives the label of the nodeset
  defining the subdomain and gives the base filename for VTK output over
- the subdomain corresponding to the ground surface (
+ the subdomain corresponding to the ground surface 
+\begin_inset Newline linebreak
+\end_inset
+
+(
 \family typewriter
 step01-groundsurf.vtk
 \family default
@@ -745,36 +753,20 @@
 \begin_layout Standard
 The files containing common information (
 \family typewriter
-\size small
-tet4_1000m_binary.gmv
+tet4_1000m_binary.gmv, tet4_1000m_binary.pset, 
+\begin_inset Newline linebreak
+\end_inset
+
+pylithapp.cfg, mat_elastic.spatialdb, and mat_viscoelastic.spatialdb
 \family default
-, 
-\family typewriter
-tet4_1000m_binary.pset
-\family default
-, 
-\family typewriter
-pylithapp.cfg
-\family default
-, 
-\family typewriter
-mat_elastic.spatialdb
-\family default
-\size default
-, and 
-\family typewriter
-mat_viscoelastic.spatialdb
-\family default
 ) along with the problem-specific files (
 \family typewriter
-\size small
-step01.cfg
+step01.cfg 
 \family default
- and 
+and
 \family typewriter
-fixeddisp_shear.spatialdb
+ fixeddisp_shear.spatialdb
 \family default
-\size default
 ) provide a complete description of the problem, and we can then run this
  example by typing
 \end_layout
@@ -800,6 +792,10 @@
  mesh vertices.
  The second file is named 
 \family typewriter
+
+\begin_inset Newline linebreak
+\end_inset
+
 step01-statevars-elastic_t0000000.vtk
 \family default
 .
@@ -1576,7 +1572,11 @@
 \begin_layout Description
 pylithapp.problem.formulation.output.subdomain Gives the label of the nodeset
  defining the subdomain and gives the base filename for VTK output over
- the subdomain corresponding to the ground surface (
+ the subdomain corresponding to the ground surface 
+\begin_inset Newline linebreak
+\end_inset
+
+(
 \family typewriter
 step03-groundsurf.vtk
 \family default
@@ -1585,7 +1585,11 @@
 
 \begin_layout Description
 pylithapp.timedependent.interfaces.fault.output.writer Gives the base filename
- for cohesive cell output files (
+ for cohesive cell output files 
+\begin_inset Newline linebreak
+\end_inset
+
+(
 \family typewriter
 step03-fault.vtk
 \family default
@@ -1712,6 +1716,10 @@
  mesh vertices.
  The second file is named 
 \family typewriter
+
+\begin_inset Newline linebreak
+\end_inset
+
 step03-statevars-elastic_t0000000.vtk
 \family default
 .

Modified: short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/greensfns2d/greensfns2d.lyx
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/greensfns2d/greensfns2d.lyx	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/greensfns2d/greensfns2d.lyx	2012-08-21 18:24:25 UTC (rev 20616)
@@ -902,9 +902,15 @@
 \end_layout
 
 \begin_layout LyX-Code
-../invert_slip.py --impulses=output/greensfns-fault.h5 --responses=output/greensfns-
-points.h5 --data=output/eqsim-points.h5 --penalty=penalty_params.txt --output=outpu
-t/slip_inverted.txt
+../invert_slip.py --impulses=output/greensfns-fault.h5
+\begin_inset Newline newline
+\end_inset
+
+--responses=output/greensfns-points.h5 --data=output/eqsim-points.h5
+\begin_inset Newline newline
+\end_inset
+
+--penalty=penalty_params.txt --output=output/slip_inverted.txt
 \end_layout
 
 \begin_layout Standard
@@ -935,6 +941,8 @@
 \end_layout
 
 \begin_layout LyX-Code
+
+\size small
 plot_invresults.py --solution=output/eqsim-fault.h5 --predicted=output/slip_invert
 ed.txt
 \end_layout

Modified: short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/shearwave/hex8.lyx
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/shearwave/hex8.lyx	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/shearwave/hex8.lyx	2012-08-21 18:24:25 UTC (rev 20616)
@@ -74,7 +74,7 @@
 
 \end_inset
 
-3-D Bar Discretized with Hexahedra
+3D Bar Discretized with Hexahedra
 \end_layout
 
 \begin_layout Standard

Modified: short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/shearwave/quad4.lyx
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/shearwave/quad4.lyx	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/shearwave/quad4.lyx	2012-08-21 18:24:25 UTC (rev 20616)
@@ -601,7 +601,7 @@
 \end_inset
 
 ).
- The evolution of the state variable uses the aging law.
+ The evolution of the state variable uses the ageing law.
  We set the fault constitutive model via
 \end_layout
 

Modified: short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/subduction/subduction.lyx
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/subduction/subduction.lyx	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/subduction/subduction.lyx	2012-08-21 18:24:25 UTC (rev 20616)
@@ -1058,7 +1058,11 @@
  consistent with a coefficient of friction of 0.6.
  The 
 \family typewriter
-afterslip_tractions.spatialdb
+afterslip_tractions.
+\begin_inset Newline linebreak
+\end_inset
+
+spatialdb
 \family default
  file is provided, so you do not need to run the Python script 
 \family typewriter
@@ -1170,8 +1174,8 @@
 
 , which was created using ParaView, displays the magnitude of the displacement
  field with the original configuration.
- Slip occurs down-dip from the coseismic slip as well as three areas with
- sharp gradients in slip, including the trench.
+ Slip occurs down-dip from the coseismic slip as well as in three areas
+ with sharp gradients in slip, including the trench.
  The location of the afterslip can be shifted by changing the spatial variation
  of the coseismic slip and background stress field.
 \end_layout

Modified: short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/twohex8/twohex8.lyx
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/twohex8/twohex8.lyx	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/twohex8/twohex8.lyx	2012-08-21 18:24:25 UTC (rev 20616)
@@ -237,54 +237,80 @@
 \end_layout
 
 \begin_layout LyX-Code
+
+\size small
 [pylithapp.timedependent.materials]
 \end_layout
 
 \begin_layout LyX-Code
+
+\size small
 material = pylith.materials.MaxwellIsotropic3D
 \end_layout
 
 \begin_layout LyX-Code
+
+\size small
  
 \end_layout
 
 \begin_layout LyX-Code
+
+\size small
 [pylithapp.timedependent.materials.material]
 \end_layout
 
 \begin_layout LyX-Code
+
+\size small
 label = viscoelastic material
 \end_layout
 
 \begin_layout LyX-Code
+
+\size small
 id = 1
 \end_layout
 
 \begin_layout LyX-Code
+
+\size small
  
 \end_layout
 
 \begin_layout LyX-Code
+
+\size small
 db = spatialdata.spatialdb.UniformDB
 \end_layout
 
 \begin_layout LyX-Code
+
+\size small
 db.values = [vp,vs,density,viscosity]
 \end_layout
 
 \begin_layout LyX-Code
+
+\size small
 db.data = [5773.502691896258*m/s, 3333.333333333333*m/s, 2700.0*kg/m**3, 1.0e18*Pa*s]
 \end_layout
 
 \begin_layout LyX-Code
+
+\size small
  
 \end_layout
 
 \begin_layout LyX-Code
+
+\size small
 quadrature.cell = pylith.feassemble.FIATLagrange
 \end_layout
 
 \begin_layout LyX-Code
+
+\size small
 quadrature.cell.dimension = 3
 \end_layout
 
@@ -763,28 +789,23 @@
 .
  The files containing common information (
 \family typewriter
+twohex8.mesh,
+\family default
 \size small
-twohex8.mesh
-\family default
-, 
+ 
 \family typewriter
+
+\begin_inset Newline linebreak
+\end_inset
+
+
+\size default
 pylithapp.cfg
 \family default
-\size default
 ) along with the problem-specific files (
 \family typewriter
-\size small
-dislocation.cfg
+dislocation.cfg, dislocation_slip.spatialdb, dislocation_sliptime.spatialdb
 \family default
-, 
-\family typewriter
-dislocation_slip.spatialdb
-\family default
-, 
-\family typewriter
-dislocation_sliptime.spatialdb
-\family default
-\size default
 ) provide a complete description of the problem, and we can then run this
  example by typing
 \end_layout

Modified: short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/twoquad4/twoquad4.lyx
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/twoquad4/twoquad4.lyx	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/twoquad4/twoquad4.lyx	2012-08-21 18:24:25 UTC (rev 20616)
@@ -445,9 +445,9 @@
  left side of the mesh, defining which degrees of freedom are being constrained
  (y only), giving the label (
 \family typewriter
-x_
+x_pos
 \family default
-pos, defined in 
+, defined in 
 \family typewriter
 twoquad4.mesh
 \family default
@@ -680,6 +680,10 @@
 \family default
 , 
 \family typewriter
+
+\begin_inset Newline linebreak
+\end_inset
+
 dislocation_sliptime.spatialdb
 \family default
 \size default
@@ -695,6 +699,10 @@
 The addition of a fault results in two additional output files (as in the
  two triangle fault example), 
 \family typewriter
+
+\begin_inset Newline linebreak
+\end_inset
+
 dislocation-fault_t0000000.vtk
 \family default
  and 

Modified: short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/twotet4/twotet4.lyx
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/twotet4/twotet4.lyx	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/twotet4/twotet4.lyx	2012-08-21 18:24:25 UTC (rev 20616)
@@ -470,18 +470,12 @@
 \family default
 ) along with the problem-specific files (
 \family typewriter
-\size small
-dislocation.cfg
-\family default
-, 
-\family typewriter
-dislocation_slip.spatialdb
-\family default
-, 
-\family typewriter
+dislocation.cfg, dislocation_slip.spatialdb, 
+\begin_inset Newline linebreak
+\end_inset
+
 dislocation_sliptime.spatialdb
 \family default
-\size default
 ) provide a complete description of the problem, and we can then run this
  example by typing
 \end_layout

Modified: short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/twotet4-geoproj/twotet4-geoproj.lyx
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/twotet4-geoproj/twotet4-geoproj.lyx	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/twotet4-geoproj/twotet4-geoproj.lyx	2012-08-21 18:24:25 UTC (rev 20616)
@@ -463,6 +463,10 @@
  Note that we vary the amount of fault slip for each vertex with this example,
  as described in 
 \family typewriter
+
+\begin_inset Newline linebreak
+\end_inset
+
 dislocation_slip.spatialdb
 \family default
 .
@@ -497,18 +501,8 @@
 \family default
 ) along with the problem-specific files (
 \family typewriter
-\size small
-dislocation.cfg
+dislocation.cfg, dislocation_slip.spatialdb, dislocation_sliptime.spatialdb
 \family default
-, 
-\family typewriter
-dislocation_slip.spatialdb
-\family default
-, 
-\family typewriter
-dislocation_sliptime.spatialdb
-\family default
-\size default
 ) provide a complete description of the problem, and we can then run this
  example by typing
 \end_layout

Modified: short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/twotri3/twotri3.lyx
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/twotri3/twotri3.lyx	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/doc/userguide/tutorials/twotri3/twotri3.lyx	2012-08-21 18:24:25 UTC (rev 20616)
@@ -360,7 +360,11 @@
 
 \begin_layout Description
 pylithapp.timedependent.materials.material.output Gives the base filename for
- state variable output files (
+ state variable output files 
+\begin_inset Newline linebreak
+\end_inset
+
+(
 \family typewriter
 axialdisp-statevars.vtk
 \family default
@@ -562,9 +566,9 @@
  left side of the mesh, defining which degrees of freedom are being constrained
  (y only), giving the label (
 \family typewriter
-x_
+x_pos
 \family default
-pos, defined in 
+, defined in 
 \family typewriter
 twotri3.mesh
 \family default
@@ -719,7 +723,11 @@
 
 \begin_layout Description
 pylithapp.timedependent.interfaces.fault.output.writer Gives the base filename
- for cohesive cell output files (
+ for cohesive cell output files 
+\begin_inset Newline linebreak
+\end_inset
+
+(
 \family typewriter
 dislocation-fault.vtk
 \family default
@@ -783,6 +791,10 @@
 \family default
 , 
 \family typewriter
+
+\begin_inset Newline linebreak
+\end_inset
+
 dislocation_sliptime.spatialdb
 \family default
 \size default

Modified: short/3D/PyLith/branches/v1.7-trunk/doc/userguide/userguide.lyx
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/doc/userguide/userguide.lyx	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/doc/userguide/userguide.lyx	2012-08-21 18:24:25 UTC (rev 20616)
@@ -270,12 +270,12 @@
 
 \begin_layout Standard
 \begin_inset CommandInset include
-LatexCommand input
+LatexCommand include
 filename "boundaryconditions/boundaryconditions.lyx"
 
 \end_inset
 
- 
+
 \begin_inset CommandInset include
 LatexCommand input
 filename "tutorials/tutorials.lyx"
@@ -475,9 +475,14 @@
 Ben-Zion, Y.
  and J.R.
  Rice (1997), Dynamic simulations of slip on a smooth fault in an elastic
- solid, J.
- Geophys.
- Res., 102, 17,771–17,784.
+ solid, 
+\emph on
+Journal of Geophysical Research
+\shape italic
+\emph default
+, 102
+\shape default
+, 17,771–17,784.
 \end_layout
 
 \begin_layout Bibliography
@@ -562,9 +567,9 @@
 Bull.
  Seismol.
  Soc.
- Am.
+ Am., 96
 \shape default
-, 96, 2118-2130.
+, 2118-2130.
 \end_layout
 
 \begin_layout Bibliography
@@ -584,7 +589,11 @@
 \shape italic
 Journal of Geophysical Research
 \shape default
-, 113, B09317, doi:10.1029/2007JB005553.
+,
+\shape italic
+ 113
+\shape default
+, B09317, doi:10.1029/2007JB005553.
 \end_layout
 
 \begin_layout Bibliography
@@ -620,9 +629,13 @@
  and X.X.
  Ni (2001), Numerical instability at the edge of a dynamic fracture, 
 \emph on
-Geophysical Journal International, 
+Geophysical Journal International,
+\shape italic
+ 
 \emph default
-147(3), 1-6, doi: 10.1046/j.1365-246x.2001.01567.x.
+147
+\shape default
+(3), 1-6, doi: 10.1046/j.1365-246x.2001.01567.x.
 \end_layout
 
 \begin_layout Bibliography

Modified: short/3D/PyLith/branches/v1.7-trunk/examples/3d/hex8/step14.cfg
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/examples/3d/hex8/step14.cfg	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/examples/3d/hex8/step14.cfg	2012-08-21 18:24:25 UTC (rev 20616)
@@ -125,7 +125,7 @@
 # Use the rate-and-state aging friction model.
 friction = pylith.friction.RateStateAgeing
 friction.label = Rate and state
-friction.min_slip_rate = 1.0e-9
+friction.linear_slip_rate = 1.0e-9
 
 # We must define the quadrature information for fault cells.
 # The fault cells are 2D (surface).
@@ -141,7 +141,7 @@
 # b: 0.08
 # cohesion: 0 Pa
 friction.db_properties = spatialdata.spatialdb.UniformDB
-friction.db_properties.label = Rate Stete Ageing
+friction.db_properties.label = Rate State Ageing
 friction.db_properties.values = [reference-friction-coefficient,reference-slip-rate,characteristic-slip-distance,constitutive-parameter-a,constitutive-parameter-b,cohesion]
 friction.db_properties.data = [0.4,2.0e-11*m/s,0.05*m,0.002,0.08,0.0*Pa]
 

Modified: short/3D/PyLith/branches/v1.7-trunk/libsrc/pylith/feassemble/ElasticityExplicit.cc
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/libsrc/pylith/feassemble/ElasticityExplicit.cc	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/libsrc/pylith/feassemble/ElasticityExplicit.cc	2012-08-21 18:24:25 UTC (rev 20616)
@@ -223,7 +223,7 @@
 				    _normalizer->densityScale());
 
   const PylithScalar dt = _dt;
-  assert(_normViscosity > 0.0);
+  assert(_normViscosity >= 0.0);
   assert(dt > 0);
   const PylithScalar viscosity = dt*_normViscosity;
 
@@ -495,7 +495,7 @@
             _normalizer->densityScale());
 
   const PylithScalar dt = _dt;
-  assert(_normViscosity > 0.0);
+  assert(_normViscosity >= 0.0);
   assert(dt > 0);
   const PylithScalar viscosity = dt*_normViscosity;
 

Modified: short/3D/PyLith/branches/v1.7-trunk/libsrc/pylith/feassemble/ElasticityExplicitLgDeform.cc
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/libsrc/pylith/feassemble/ElasticityExplicitLgDeform.cc	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/libsrc/pylith/feassemble/ElasticityExplicitLgDeform.cc	2012-08-21 18:24:25 UTC (rev 20616)
@@ -185,17 +185,14 @@
   assert(!accSection.isNull());
   RestrictVisitor accVisitor(*accSection, accCell.size(), &accCell[0]);
 
-#if 0 // Numerical damping not yet implemented
   scalar_array velCell(numBasis*spaceDim);
   const ALE::Obj<RealSection>& velSection = 
     fields->get("velocity(t)").section();
   assert(!velSection.isNull());
   RestrictVisitor velVisitor(*velSection, velCell.size(), &velCell[0]);
 
-  scalar_array dispAdjCell(numBasis*spaceDim);
-#endif
-
   scalar_array dispCell(numBasis*spaceDim);
+  scalar_array dispAdjCell(numBasis*spaceDim);
   const ALE::Obj<RealSection>& dispSection = fields->get("disp(t)").section();
   assert(!dispSection.isNull());
   RestrictVisitor dispVisitor(*dispSection, dispCell.size(), &dispCell[0]);
@@ -217,7 +214,7 @@
 				    _normalizer->densityScale());
 
   const PylithScalar dt = _dt;
-  assert(_normViscosity > 0.0);
+  assert(_normViscosity >= 0.0);
   assert(dt > 0);
   const PylithScalar viscosity = dt*_normViscosity;
 
@@ -311,21 +308,17 @@
     } // for
     PetscLogFlops(numQuadPts*(2+numBasis*(1+numBasis*(2*spaceDim))));
 
-#if 0 // Numerical damping not yet implemented. Is small strain
-      // formulation compatible with numerical damping?
-
     // Numerical damping. Compute displacements adjusted by velocity
     // times normalized viscosity.
     dispAdjCell = dispCell + viscosity * velCell;
-#endif
 
     // Compute B(transpose) * sigma, first computing strains
-    _calcDeformation(&deformCell, basisDeriv, coordinatesCell, dispCell,
+    _calcDeformation(&deformCell, basisDeriv, coordinatesCell, dispAdjCell,
 		     numBasis, numQuadPts, spaceDim);
     calcTotalStrainFn(&strainCell, deformCell, numQuadPts);
     const scalar_array& stressCell = _material->calcStress(strainCell, true);
 
-    CALL_MEMBER_FN(*this, elasticityResidualFn)(stressCell, dispCell);
+    CALL_MEMBER_FN(*this, elasticityResidualFn)(stressCell, dispAdjCell);
 
     // Assemble cell contribution into field
     residualVisitor.clear();
@@ -426,17 +419,14 @@
   assert(!accSection.isNull());
   RestrictVisitor accVisitor(*accSection, accCell.size(), &accCell[0]);
 
-#if 0 // Numerical damping not yet implemented
   scalar_array velCell(numBasis*spaceDim);
   const ALE::Obj<RealSection>& velSection = 
     fields->get("velocity(t)").section();
   assert(!velSection.isNull());
   RestrictVisitor velVisitor(*velSection, velCell.size(), &velCell[0]);
 
-  scalar_array dispAdjCell(numBasis*spaceDim);
-#endif
-
   scalar_array dispCell(numBasis*spaceDim);
+  scalar_array dispAdjCell(numBasis*spaceDim);
   const ALE::Obj<RealSection>& dispSection = fields->get("disp(t)").section();
   assert(!dispSection.isNull());
   RestrictVisitor dispVisitor(*dispSection, dispCell.size(), &dispCell[0]);
@@ -457,6 +447,11 @@
     _normalizer->pressureScale() / (_normalizer->lengthScale() *
 				    _normalizer->densityScale());
 
+  const PylithScalar dt = _dt;
+  assert(_normViscosity >= 0.0);
+  assert(dt > 0);
+  const PylithScalar viscosity = dt*_normViscosity;
+
   // Get parameters used in integration.
   scalar_array valuesIJ(numBasis);
 
@@ -552,20 +547,17 @@
 	  accCell[iBasis*spaceDim+iDim];
     PetscLogFlops(numQuadPts*(4+numBasis*3));
 
-#if 0 // Numerical damping not yet implemented. Is small strain
-      // formulation compatible with numerical damping?
-
     // Numerical damping. Compute displacements adjusted by velocity
     // times normalized viscosity.
     dispAdjCell = dispCell + viscosity * velCell;
-#endif
+
     // Compute B(transpose) * sigma, first computing strains
-    _calcDeformation(&deformCell, basisDeriv, coordinatesCell, dispCell,
+    _calcDeformation(&deformCell, basisDeriv, coordinatesCell, dispAdjCell,
 		     numBasis, numQuadPts, spaceDim);
     calcTotalStrainFn(&strainCell, deformCell, numQuadPts);
     const scalar_array& stressCell = _material->calcStress(strainCell, true);
 
-    CALL_MEMBER_FN(*this, elasticityResidualFn)(stressCell, dispCell);
+    CALL_MEMBER_FN(*this, elasticityResidualFn)(stressCell, dispAdjCell);
     
     // Assemble cell contribution into field
     residualVisitor.clear();

Modified: short/3D/PyLith/branches/v1.7-trunk/libsrc/pylith/friction/RateStateAgeing.cc
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/libsrc/pylith/friction/RateStateAgeing.cc	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/libsrc/pylith/friction/RateStateAgeing.cc	2012-08-21 18:24:25 UTC (rev 20616)
@@ -122,7 +122,7 @@
 				    _RateStateAgeing::numStateVars,
 				    _RateStateAgeing::dbStateVars,
 				    _RateStateAgeing::numDBStateVars)),
-  _minSlipRate(1.0e-12)
+  _linearSlipRate(1.0e-12)
 { // constructor
 } // constructor
 
@@ -133,20 +133,21 @@
 } // destructor
 
 // ----------------------------------------------------------------------
-// Set floor for slip rate used in computing friction. Used to
+// Set nondimensional slip rate below which friction varies
+//  linearly with slip rate.
 void
-pylith::friction::RateStateAgeing::minSlipRate(const PylithScalar value)
-{ // minSlipRate
+pylith::friction::RateStateAgeing::linearSlipRate(const PylithScalar value)
+{ // linearSlipRate
   if (value < 0.0) {
     std::ostringstream msg;
-    msg << "Minimum slip rate (" << value << ") for rate state friction model "
+    msg << "Nondimensional linear slip rate threshold (" << value << ") for rate state friction model "
 	<< label() << " must be nonnegative.";
     throw std::runtime_error(msg.str());
   } // if
+  
+  _linearSlipRate = value;
+} // linearSlipRate
 
-  _minSlipRate = value;
-} // minSlipRate
-
 // ----------------------------------------------------------------------
 // Compute properties from values in spatial database.
 void
@@ -310,7 +311,7 @@
   if (normalTraction <= 0.0) {
     // if fault is in compression
 
-    const PylithScalar slipRateLinear = _minSlipRate;
+    const PylithScalar slipRateLinear = _linearSlipRate;
 
     const PylithScalar f0 = properties[p_coef];
     const PylithScalar a = properties[p_a];
@@ -375,7 +376,7 @@
 
   // Since regulatized friction -> 0 as slipRate -> 0, limit slip
   // rate to some minimum value
-  const PylithScalar slipRateEff = std::max(_minSlipRate, slipRate);
+  const PylithScalar slipRateEff = std::max(_linearSlipRate, slipRate);
 
   const PylithScalar dt = _dt;
   const PylithScalar thetaTVertex = stateVars[s_state];

Modified: short/3D/PyLith/branches/v1.7-trunk/libsrc/pylith/friction/RateStateAgeing.hh
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/libsrc/pylith/friction/RateStateAgeing.hh	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/libsrc/pylith/friction/RateStateAgeing.hh	2012-08-21 18:24:25 UTC (rev 20616)
@@ -20,10 +20,11 @@
  *
  * @brief C++ Rate and State fault constitutive model with ageing law.
  *
- * Implementation comes from "Kaneko, Y., N. Lapusta, and J.-P. Ampuero 
- * (2008), Spectral element modeling of spontaneous earthquake rupture on
- * rate and state faults: Effect of velocity-strengthening friction at 
- * shallow depths, J. Geophys. Res., 113, B09317" 
+ * Implementation of evolving state variable comes from "Kaneko, Y.,
+ * N. Lapusta, and J.-P. Ampuero (2008), Spectral element modeling of
+ * spontaneous earthquake rupture on rate and state faults: Effect of
+ * velocity-strengthening friction at shallow depths,
+ * J. Geophys. Res., 113, B09317"
  *
  * Regularized Rate & State equation : Eqn(15) of Kaneko et. al. (2008)
  *
@@ -58,12 +59,12 @@
   /// Destructor.
   ~RateStateAgeing(void);
 
-  /** Set floor for slip rate used in computing friction. Used to
-   *  avoid zero friction at zero slip rate.
+  /** Set nondimensional slip rate below which friction varies
+   *  linearly with slip rate.
    *
-   * @param value Floor for slip rate.
+   * @param value Nondimensional slip rate.
    */
-  void minSlipRate(const PylithScalar value);
+  void linearSlipRate(const PylithScalar value);
 
   // PROTECTED METHODS //////////////////////////////////////////////////
 protected :
@@ -167,7 +168,7 @@
 private :
 
   /// Floor for slip rate used in friction calculation.
-  PylithScalar _minSlipRate;
+  PylithScalar _linearSlipRate;
 
   /// Indices for properties in section and spatial database.
   static const int p_coef;

Modified: short/3D/PyLith/branches/v1.7-trunk/modulesrc/friction/RateStateAgeing.i
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/modulesrc/friction/RateStateAgeing.i	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/modulesrc/friction/RateStateAgeing.i	2012-08-21 18:24:25 UTC (rev 20616)
@@ -36,12 +36,12 @@
       /// Destructor.
       ~RateStateAgeing(void);
 
-      /** Set floor for slip rate used in computing friction. Used to
-       *  avoid zero friction at zero slip rate.
+      /** Set nondimensional slip rate below which friction varies
+       *  linearly with slip rate.
        *
-       * @param value Floor for slip rate.
+       * @param value Nondimensional slip rate.
        */
-      void minSlipRate(const PylithScalar value);
+      void linearSlipRate(const PylithScalar value);
 
       // PROTECTED METHODS //////////////////////////////////////////////
     protected :

Modified: short/3D/PyLith/branches/v1.7-trunk/modulesrc/mpi/Makefile.am
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/modulesrc/mpi/Makefile.am	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/modulesrc/mpi/Makefile.am	2012-08-21 18:24:25 UTC (rev 20616)
@@ -26,6 +26,7 @@
 swig_sources = \
 	mpi.i \
 	mpi_comm.i \
+	mpi_reduce.i \
 	mpi_error.i
 
 swig_generated = \

Modified: short/3D/PyLith/branches/v1.7-trunk/modulesrc/mpi/mpi.i
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/modulesrc/mpi/mpi.i	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/modulesrc/mpi/mpi.i	2012-08-21 18:24:25 UTC (rev 20616)
@@ -29,6 +29,7 @@
 // Interfaces
 %include "mpi_comm.i"
 %include "mpi_error.i"
+%include "mpi_reduce.i"
 
 
 // End of file

Copied: short/3D/PyLith/branches/v1.7-trunk/modulesrc/mpi/mpi_reduce.i (from rev 20615, short/3D/PyLith/branches/v1.7-stable/modulesrc/mpi/mpi_reduce.i)
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/modulesrc/mpi/mpi_reduce.i	                        (rev 0)
+++ short/3D/PyLith/branches/v1.7-trunk/modulesrc/mpi/mpi_reduce.i	2012-08-21 18:24:25 UTC (rev 20616)
@@ -0,0 +1,78 @@
+// -*- C++ -*-
+//
+// ======================================================================
+//
+// Brad T. Aagaard, U.S. Geological Survey
+// Charles A. Williams, GNS Science
+// Matthew G. Knepley, University of Chicago
+//
+// This code was developed as part of the Computational Infrastructure
+// for Geodynamics (http://geodynamics.org).
+//
+// Copyright (c) 2010-2012 University of California, Davis
+//
+// See COPYING for license information.
+//
+// ======================================================================
+//
+
+// ----------------------------------------------------------------------
+// MPI_SUM
+%inline %{
+  MPI_Op*
+  mpi_sum(void) {
+    return new MPI_Op(MPI_SUM);
+  } // mpi_sum
+%}
+
+
+// ----------------------------------------------------------------------
+// MPI_MIN
+%inline %{
+  MPI_Op*
+  mpi_min(void) {
+    return new MPI_Op(MPI_MIN);
+  } // mpi_min
+%}
+
+
+// ----------------------------------------------------------------------
+// MPI_AX
+%inline %{
+  MPI_Op*
+  mpi_max(void) {
+    return new MPI_Op(MPI_MAX);
+  } // mpi_max
+%}
+
+
+// ----------------------------------------------------------------------
+// allreduce_scalar_double
+%inline %{
+  double
+    allreduce_scalar_double(double value,
+			    MPI_Op* op,
+			    MPI_Comm* comm) {
+    double result = 0.0;
+    MPI_Allreduce(&value, &result, 1, MPI_DOUBLE, *op, *comm);
+    return result;
+  } // allreduce_scalar_double
+%}
+
+
+// ----------------------------------------------------------------------
+// allreduce_scalar_int
+%inline %{
+  int
+    allreduce_scalar_int(int value,
+			 MPI_Op* op,
+			 MPI_Comm* comm) {
+    int result = 0;
+    MPI_Allreduce(&value, &result, 1, MPI_INT, *op, *comm);
+    return result;
+  } // allreduce_int
+%}
+
+
+// End of file
+

Modified: short/3D/PyLith/branches/v1.7-trunk/pylith/apps/PyLithApp.py
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/pylith/apps/PyLithApp.py	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/pylith/apps/PyLithApp.py	2012-08-21 18:24:25 UTC (rev 20616)
@@ -39,7 +39,7 @@
     ## Python object for managing PyLithApp facilities and properties.
     ##
     ## \b Properties
-    ## @li None
+    ## @li \b initialize_only Stop simulation after initializing problem.
     ##
     ## \b Facilities
     ## @li \b mesher Generates or imports the computational mesh.
@@ -48,6 +48,9 @@
 
     import pyre.inventory
 
+    initializeOnly = pyre.inventory.bool("initialize_only", default=False)
+    initializeOnly.meta['tip'] = "Stop simulation after initializing problem."
+
     from pylith.topology.MeshImporter import MeshImporter
     mesher = pyre.inventory.facility("mesh_generator", family="mesh_generator",
                                      factory=MeshImporter)
@@ -115,6 +118,10 @@
 
     self._eventLogger.stagePop()
 
+    # If initializing only, stop before running problem
+    if self.initializeOnly:
+      return
+
     # Run problem
     self.problem.run(self)
     self._debug.log(resourceUsageString())
@@ -141,6 +148,7 @@
     Setup members using inventory.
     """
     PetscApplication._configure(self)
+    self.initializeOnly = self.inventory.initializeOnly
     self.mesher = self.inventory.mesher
     self.problem = self.inventory.problem
     self.perfLogger = self.inventory.perfLogger

Modified: short/3D/PyLith/branches/v1.7-trunk/pylith/friction/RateStateAgeing.py
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/pylith/friction/RateStateAgeing.py	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/pylith/friction/RateStateAgeing.py	2012-08-21 18:24:25 UTC (rev 20616)
@@ -44,18 +44,18 @@
     ## Python object for managing RateStateAgeing facilities and properties.
     ##
     ## \b Properties
-    ## @li \b min_slip_rate Floor for nondimensional slip rate used in 
-    ##   friction calculation.
+    ## @li \b linear_slip_rate Nondimensional slip rate below which friction 
+    ## varies linearly with slip rate.
     ##
     ## \b Facilities
     ## @li None
 
     import pyre.inventory
 
-    minSlipRate = pyre.inventory.float("min_slip_rate", default=1.0e-12,
-                                       validator=pyre.inventory.greaterEqual(0.0))
-    minSlipRate.meta['tip'] = "Floor for nondimensional slip rate used in "\
-        "friction calculation."
+    linearSlipRate = pyre.inventory.float("linear_slip_rate", default=1.0e-12,
+                                          validator=pyre.inventory.greaterEqual(0.0))
+    linearSlipRate.meta['tip'] = "Nondimensional slip rate below which friction " \
+        "varies linearly with slip rate."
 
   # PUBLIC METHODS /////////////////////////////////////////////////////
 
@@ -88,7 +88,7 @@
     """
     try:
       FrictionModel._configure(self)
-      ModuleRateStateAgeing.minSlipRate(self, self.inventory.minSlipRate)
+      ModuleRateStateAgeing.linearSlipRate(self, self.inventory.linearSlipRate)
     except ValueError, err:
       aliases = ", ".join(self.aliases)
       raise ValueError("Error while configuring friction model "

Modified: short/3D/PyLith/branches/v1.7-trunk/pylith/problems/TimeStep.py
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/pylith/problems/TimeStep.py	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/pylith/problems/TimeStep.py	2012-08-21 18:24:25 UTC (rev 20616)
@@ -186,7 +186,9 @@
       dt = integrator.stableTimeStep(mesh)
       if dt < dtStable:
         dtStable = dt
-
+    import pylith.mpi.mpi as mpi
+    comm = mesh.getComm()
+    dtStableAll = mpi.allreduce_scalar_double(dtStable, mpi.mpi_min(), comm.handle)
     return dtStable
 
 

Modified: short/3D/PyLith/branches/v1.7-trunk/tests/2d/frictionslide/ratestate_stable.cfg
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/tests/2d/frictionslide/ratestate_stable.cfg	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/tests/2d/frictionslide/ratestate_stable.cfg	2012-08-21 18:24:25 UTC (rev 20616)
@@ -12,7 +12,7 @@
 [pylithapp.timedependent.interfaces.fault]
 friction = pylith.friction.RateStateAgeing
 friction.label = Rate state ageing
-friction.min_slip_rate = 1.0e-8
+friction.linear_slip_rate = 1.0e-8
 
 # Set the friction model parameters.
 #  reference coefficient of friction: 0.6

Modified: short/3D/PyLith/branches/v1.7-trunk/tests/2d/frictionslide/ratestate_weak.cfg
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/tests/2d/frictionslide/ratestate_weak.cfg	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/tests/2d/frictionslide/ratestate_weak.cfg	2012-08-21 18:24:25 UTC (rev 20616)
@@ -12,7 +12,7 @@
 [pylithapp.timedependent.interfaces.fault]
 friction = pylith.friction.RateStateAgeing
 friction.label = Rate state ageing
-friction.min_slip_rate = 1.0e-9
+friction.linear_slip_rate = 1.0e-9
 
 # Set the friction model parameters.
 #  reference coefficient of friction: 0.6

Modified: short/3D/PyLith/branches/v1.7-trunk/unittests/libtests/friction/TestRateStateAgeing.cc
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/unittests/libtests/friction/TestRateStateAgeing.cc	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/unittests/libtests/friction/TestRateStateAgeing.cc	2012-08-21 18:24:25 UTC (rev 20616)
@@ -38,18 +38,18 @@
 } // setUp
 
 // ----------------------------------------------------------------------
-// Test floor for minimum slip rate.
+// Test cutoff for linear slip rate.
 void
-pylith::friction::TestRateStateAgeing::testMinSlipRate(void)
-{ // testMinSlipRate
+pylith::friction::TestRateStateAgeing::testLinearSlipRate(void)
+{ // testLinearSlipRate
   RateStateAgeing model;
 
-  CPPUNIT_ASSERT_EQUAL(PylithScalar(1.0e-12), model._minSlipRate); // default
+  CPPUNIT_ASSERT_EQUAL(PylithScalar(1.0e-12), model._linearSlipRate); // default
 
   const PylithScalar value = 1.0e-20;
-  model.minSlipRate(value);
-  CPPUNIT_ASSERT_EQUAL(value, model._minSlipRate);
-} // testMinSlipRate
+  model.linearSlipRate(value);
+  CPPUNIT_ASSERT_EQUAL(value, model._linearSlipRate);
+} // testLinearSlipRate
   
 // ----------------------------------------------------------------------
 // Test properties metadata.

Modified: short/3D/PyLith/branches/v1.7-trunk/unittests/libtests/friction/TestRateStateAgeing.hh
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/unittests/libtests/friction/TestRateStateAgeing.hh	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/unittests/libtests/friction/TestRateStateAgeing.hh	2012-08-21 18:24:25 UTC (rev 20616)
@@ -43,7 +43,7 @@
   // CPPUNIT TEST SUITE /////////////////////////////////////////////////
   CPPUNIT_TEST_SUITE( TestRateStateAgeing );
 
-  CPPUNIT_TEST( testMinSlipRate );
+  CPPUNIT_TEST( testLinearSlipRate );
   CPPUNIT_TEST( testPropertiesMetadata );
   CPPUNIT_TEST( testStateVarsMetadata );
   CPPUNIT_TEST( testDBToProperties );
@@ -64,8 +64,8 @@
   /// Setup testing data.
   void setUp(void);
 
-  /// Test floor for minimum slip rate.
-  void testMinSlipRate(void);
+  /// Test cutoff for linear slip rate.
+  void testLinearSlipRate(void);
   
   /// Test properties metadata.
   void testPropertiesMetadata(void);

Modified: short/3D/PyLith/branches/v1.7-trunk/unittests/pytests/mpi/Makefile.am
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/unittests/pytests/mpi/Makefile.am	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/unittests/pytests/mpi/Makefile.am	2012-08-21 18:24:25 UTC (rev 20616)
@@ -25,7 +25,8 @@
 TESTS_ENVIRONMENT = $(PYTHON)
 
 noinst_PYTHON = \
-	TestCommunicator.py
+	TestCommunicator.py \
+	TestReduce.py
 
 
 # End of file 

Copied: short/3D/PyLith/branches/v1.7-trunk/unittests/pytests/mpi/TestReduce.py (from rev 20615, short/3D/PyLith/branches/v1.7-stable/unittests/pytests/mpi/TestReduce.py)
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/unittests/pytests/mpi/TestReduce.py	                        (rev 0)
+++ short/3D/PyLith/branches/v1.7-trunk/unittests/pytests/mpi/TestReduce.py	2012-08-21 18:24:25 UTC (rev 20616)
@@ -0,0 +1,66 @@
+#!/usr/bin/env python
+#
+# ======================================================================
+#
+# Brad T. Aagaard, U.S. Geological Survey
+# Charles A. Williams, GNS Science
+# Matthew G. Knepley, University of Chicago
+#
+# This code was developed as part of the Computational Infrastructure
+# for Geodynamics (http://geodynamics.org).
+#
+# Copyright (c) 2010-2012 University of California, Davis
+#
+# See COPYING for license information.
+#
+# ======================================================================
+#
+
+## @file unittests/pytests/utils/TestReduce.py
+
+## @brief Unit testing of MPI reduce functions.
+
+import unittest
+
+import pylith.mpi.mpi as mpi
+
+# ----------------------------------------------------------------------
+class TestReduce(unittest.TestCase):
+  """
+  Unit testing of MPI reduce functions.
+  """
+  
+
+  def test_allreduce_scalar_double(self):
+    """
+    Test allreduce_double().
+    """
+    value = 2.0
+    result = mpi.allreduce_scalar_double(value, mpi.mpi_sum(), mpi.petsc_comm_world())
+    self.assertEqual(value, result)
+
+    result = mpi.allreduce_scalar_double(value, mpi.mpi_min(), mpi.petsc_comm_self())
+    self.assertEqual(value, result)
+
+    result = mpi.allreduce_scalar_double(value, mpi.mpi_max(), mpi.petsc_comm_world())
+    self.assertEqual(value, result)
+    return
+
+
+  def test_allreduce_scalar_int(self):
+    """
+    Test allreduce_int().
+    """
+    value = 3
+    result = mpi.allreduce_scalar_int(value, mpi.mpi_sum(), mpi.petsc_comm_world())
+    self.assertEqual(value, result)
+
+    result = mpi.allreduce_scalar_int(value, mpi.mpi_min(), mpi.petsc_comm_self())
+    self.assertEqual(value, result)
+
+    result = mpi.allreduce_scalar_int(value, mpi.mpi_max(), mpi.petsc_comm_world())
+    self.assertEqual(value, result)
+    return
+
+
+# End of file 

Modified: short/3D/PyLith/branches/v1.7-trunk/unittests/pytests/mpi/testmpi.py
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/unittests/pytests/mpi/testmpi.py	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/unittests/pytests/mpi/testmpi.py	2012-08-21 18:24:25 UTC (rev 20616)
@@ -65,6 +65,9 @@
     from TestCommunicator import TestCommunicator
     suite.addTest(unittest.makeSuite(TestCommunicator))
 
+    from TestReduce import TestReduce
+    suite.addTest(unittest.makeSuite(TestReduce))
+
     return suite
 
 

Modified: short/3D/PyLith/branches/v1.7-trunk/unittests/pytests/problems/TestTimeStepAdapt.py
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/unittests/pytests/problems/TestTimeStepAdapt.py	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/unittests/pytests/problems/TestTimeStepAdapt.py	2012-08-21 18:24:25 UTC (rev 20616)
@@ -91,8 +91,12 @@
     tstep.adaptSkip = 2
     integrators = [Integrator(2.0),
                    Integrator(0.5)]
-    mesh = None
 
+    from pylith.topology.Mesh import Mesh
+    from pylith.mpi.Communicator import petsc_comm_world
+    mesh = Mesh()
+    mesh.setComm(petsc_comm_world())
+
     # Set time step
     dt = 0.5 / 2.0
     self.assertEqual(dt, tstep.timeStep(mesh, integrators))
@@ -145,7 +149,12 @@
     
     integrators = [Integrator(3.0),
                    Integrator(2.4)]
-    mesh = None
+
+    from pylith.topology.Mesh import Mesh
+    from pylith.mpi.Communicator import petsc_comm_world
+    mesh = Mesh()
+    mesh.setComm(petsc_comm_world())
+
     dt = 2.4 / 2.0
     tstep.timeStep(mesh, integrators)
     self.assertEqual(dt, tstep.currentStep())

Modified: short/3D/PyLith/branches/v1.7-trunk/unittests/pytests/problems/TestTimeStepUniform.py
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/unittests/pytests/problems/TestTimeStepUniform.py	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/unittests/pytests/problems/TestTimeStepUniform.py	2012-08-21 18:24:25 UTC (rev 20616)
@@ -83,8 +83,12 @@
 
     integrators = [Integrator(4.0),
                    Integrator(8.0)]
-    mesh = None
 
+    from pylith.topology.Mesh import Mesh
+    from pylith.mpi.Communicator import petsc_comm_world
+    mesh = Mesh()
+    mesh.setComm(petsc_comm_world())
+
     self.assertEqual(1.0, tstep.timeStep(mesh, integrators))
 
     tstep.dtN = 0.5

Modified: short/3D/PyLith/branches/v1.7-trunk/unittests/pytests/problems/TestTimeStepUser.py
===================================================================
--- short/3D/PyLith/branches/v1.7-trunk/unittests/pytests/problems/TestTimeStepUser.py	2012-08-21 17:00:25 UTC (rev 20615)
+++ short/3D/PyLith/branches/v1.7-trunk/unittests/pytests/problems/TestTimeStepUser.py	2012-08-21 18:24:25 UTC (rev 20616)
@@ -108,8 +108,12 @@
 
     integrators = [Integrator(40.0),
                    Integrator(80.0)]
-    mesh = None
 
+    from pylith.topology.Mesh import Mesh
+    from pylith.mpi.Communicator import petsc_comm_world
+    mesh = Mesh()
+    #mesh.setComm(petsc_comm_world())
+
     self.assertEqual(step1, tstep.timeStep(mesh, integrators))
     self.assertEqual(step2, tstep.timeStep(mesh, integrators))
     self.assertEqual(step3, tstep.timeStep(mesh, integrators))
@@ -144,8 +148,12 @@
 
     integrators = [Integrator(4.0),
                    Integrator(8.0)]
-    mesh = None
 
+    from pylith.topology.Mesh import Mesh
+    from pylith.mpi.Communicator import petsc_comm_world
+    mesh = Mesh()
+    #mesh.setComm(petsc_comm_world())
+
     tstep.timeStep(mesh, integrators)
     stepE = 1.0 / 0.5 # Nondimensionalize
     self.assertEqual(stepE, tstep.currentStep())

