[cig-commits] commit: Fixed some typos. Added references.
Mercurial
hg at geodynamics.org
Wed Aug 31 08:44:36 PDT 2011
changeset: 58:0a787949f1ae
tag: tip
user: Brad Aagaard <baagaard at usgs.gov>
date: Wed Aug 31 08:44:33 2011 -0700
files: faultRup.tex pylith.bib references.bib
description:
Fixed some typos. Added references.
diff -r d15fdc196d9a -r 0a787949f1ae faultRup.tex
--- a/faultRup.tex Tue Aug 30 17:15:25 2011 -0700
+++ b/faultRup.tex Wed Aug 31 08:44:33 2011 -0700
@@ -335,11 +335,12 @@ solution. We solve these equations usin
solution. We solve these equations using the Portable, Extensible
Toolkit for Scientific Computation (PETSc), which provides a suite of
tools for solving linear systems of algebraic equations with parallel
-processing \cite{PETSc}. In solving the system, we compute the
-residual (i.e., $\vec{r} = \vec{b} - \tensor{A} \vec{u}$ and the
-Jacobian of the system ($A$). In our case the solution is $\vec{u} =
-\left( \begin{smallmatrix} \vec{u}_m \\ \vec{l}_m \end{smallmatrix} \right)$,
-and the residual is simply the left hand sides of
+processing \cite{PETSc:web:page,PETSc:manual,PETSC:efficient}. In
+solving the system, we compute the residual (i.e., $\vec{r} = \vec{b}
+- \tensor{A} \vec{u}$ and the Jacobian of the system ($A$). In our
+case the solution is $\vec{u} = \left( \begin{smallmatrix} \vec{u}_m
+ \\ \vec{l}_m \end{smallmatrix} \right)$, and the residual is
+simply the left hand sides of
equations~(\ref{eqn:quasi-static:residual:domain})
and~(\ref{eqn:quasi-static:residual:fault}). That is,
\begin{equation}
@@ -386,7 +387,7 @@ Following a similar procedure, we find t
Following a similar procedure, we find the portion of the Jacobian
associated with equation~(\ref{eqn:quasi-static:residual:fault}) is
\begin{equation}
- \tensor{L} = \int_{S_f} \tensor{N}_p (-\tensor{N}_{m^+}+\tensor{N}_{m^-}) \, dS.
+ \tensor{L} = \int_{S_f} \tensor{N}_p^T (-\tensor{N}_{m^+}+\tensor{N}_{m^-}) \, dS.
\end{equation}
Thus, the Jacobian of the entire system has the form,
\begin{equation}
@@ -488,8 +489,8 @@ the change in the tractions on the fault
the change in the tractions on the fault surfaces corresponding to the
slip. PyLith includes a variety of slip-time histories, including a
step function, constant slip rate, the integral of Brune's far-field
-time function, a sine-cosine function developed by
-\citeN{Liu:etal:200?}, and a user-defined time history. These are
+time function \cite{Brune:1970}, a sine-cosine function developed by
+\citeN{Liu:etal:2006}, and a user-defined time history. These are
discussed in detail in the PyLith manual
\cite{PyLith:manual:1.6.2}. PyLith allows multiple earthquake ruptures
to be superimposed on each other, thereby permitting rather complex
@@ -525,8 +526,8 @@ of the general form of a linear system o
\vec{u} = \vec{b}$), our subset of equations has the form
\begin{equation}
\begin{pmatrix}
- \tensor{K}_{m^+m^+} & 0 & \tensor{L}_p \\
- 0 & \tensor{K}_{m^-m^-} & -\tensor{L}_p \\
+ \tensor{K}_{m^+m^+} & 0 & \tensor{L}_p^T \\
+ 0 & \tensor{K}_{m^-m^-} & -\tensor{L}_p^T \\
\tensor{L}_p & -\tensor{L}_p & 0
\end{pmatrix}
\begin{pmatrix}
@@ -550,17 +551,18 @@ corresponding to a perturbation in the L
corresponding to a perturbation in the Lagrange multipliers $\partial
\vec{l}_p$:
\begin{gather}
- \tensor{K}_{m^+m^+} \vec{u}_{m^+} = - \tensor{L}_p \partial \vec{l}_p, \\
- \tensor{K}_{m^-m^-} \vec{u}_{m^-} = - \tensor{L}_p \partial \vec{l}_p, \\
- \partial \vec{d} = \tensor{R} \left( \vec{u}_{m^+} - \vec{u}_{m^-} \right).
+ \tensor{K}_{m^+m^+} \partial \vec{u}_{m^+} = - \tensor{L}_p^T \partial \vec{l}_p, \\
+ \tensor{K}_{m^-m^-} \partial \vec{u}_{m^-} = - \tensor{L}_p^T \partial \vec{l}_p, \\
+ \partial \vec{d} = \tensor{R} \left( \partial \vec{u}_{m^+} - \partial \vec{u}_{m^-} \right).
\end{gather}
This estimate is exact if all other degrees of freedom are constrained
(which is generally only the case for simple toy problems), quite good
if the deformation associated with the fault slip is confined close to
-the fault (which occurs in most simulations because the fault slip is
-limited to the middle of the domain), and poor if the deformation
-associated with fault slip extends to the boundaries (which occurs in
-races cases in which the domain has a through-going fault).
+the fault (which occurs in most simulations because the deformation
+from fault slip is limited to the middle of the domain), and poor if
+the deformation associated with fault slip extends to the boundaries
+(which occurs in the rare cases in which the domain has a through-going
+fault).
PyLith includes several commonly used fault constitutive models, all
of which specify the shear traction on the fault $T_f$ as a function
diff -r d15fdc196d9a -r 0a787949f1ae pylith.bib
--- a/pylith.bib Tue Aug 30 17:15:25 2011 -0700
+++ b/pylith.bib Wed Aug 31 08:44:33 2011 -0700
@@ -31,28 +31,3 @@
month = may,
}
- at Article{Liu:etal:2006,
- author = {Liu, P. and Archuleta, R.~J. and Hartzell, S.~H.},
- title = {Prediction of Broadband Ground-Motion Time
- Histories: {Hybrid} Low/High- Frequency Method with
- Correlated Random Source Parameters},
- journal = BSSA,
- year = {2006},
- volume = {96},
- number = {6},
- pages = {2118--2130},
- month = dec,
- doi = {10.1785/0120060036},
-}
-
- at article{Brune:1970,
- author = {Brune, James N.},
- title = {Tectonic stress and spectra of seismic shear waves
- from earthquakes},
- journal = JGR,
- year = 1970,
- volume = 75,
- month = sep # {~10},
- pages = {4997--5009},
-}
-
diff -r d15fdc196d9a -r 0a787949f1ae references.bib
--- a/references.bib Tue Aug 30 17:15:25 2011 -0700
+++ b/references.bib Wed Aug 31 08:44:33 2011 -0700
@@ -613,3 +613,79 @@
be far more aperiodic than has been suggested. },
}
+ at Article{Liu:etal:2006,
+ author = {Liu, P. and Archuleta, R.~J. and Hartzell, S.~H.},
+ title = {Prediction of Broadband Ground-Motion Time
+ Histories: {Hybrid} Low/High- Frequency Method with
+ Correlated Random Source Parameters},
+ journal = BSSA,
+ year = {2006},
+ volume = {96},
+ number = {6},
+ pages = {2118--2130},
+ month = dec,
+ doi = {10.1785/0120060036},
+}
+
+ at article{Brune:1970,
+ author = {Brune, James N.},
+ title = {Tectonic stress and spectra of seismic shear waves
+ from earthquakes},
+ journal = JGR,
+ year = 1970,
+ volume = 75,
+ month = sep # {~10},
+ pages = {4997--5009},
+}
+
+
+
+
+ at Book{Zienkiewicz:Taylor:2005,
+ author = {Ziekiewicz, O.~C. and Taylor, R.~L. and Zhu, J.~Z.},
+ title = {The Finite-Element Method: Its Basis and Fundamentals},
+ publisher = {Butterworth-Heinemann},
+ year = {2005},
+ edition = {6th},
+}
+
+ at Misc{PETSc:web:page,
+ Author = "Balay, S. and Brown, J. and Buschelman, K. and Gropp,
+ W.~D. and Kaushik, D. and Knepley, M.~G. and
+ McInnes, L.~C. and Smith, B.~F. and Zhang, H.",
+ Title = "{PETSc} {Web} page",
+ Note = "http://www.mcs.anl.gov/petsc",
+ Year = 2011,
+}
+
+ at TechReport{PETSc:manual,
+ Author = "Balay, S. and Brown, J. and Buschelman, K. and Gropp,
+ W.~D. and Kaushik, D. and Knepley, M.~G. and
+ McInnes, L.~C. and Smith, B.~F. and Zhang, H.",
+ Title = "{PETSc} Users Manual",
+ Number = "ANL-95/11 - Revision 3.1",
+ Institution = "Argonne National Laboratory",
+ Year = 2010,
+}
+
+ at InProceedings{PETSc:efficient,
+ Author = "Balay, S. and Gropp, W.~D. and McInnes, L.~C. and
+ Smith, B.~F. ",
+ Title = "Efficient Management of Parallelism in Object
+ Oriented Numerical Software Libraries",
+ Booktitle = "Modern Software Tools in Scientific Computing",
+ Editor = "E. Arge and A. M. Bruaset and H. P. Langtangen",
+ Pages = "163--202",
+ Publisher = "Birkh{\"{a}}user Press",
+ Year = 1997,
+}
+
+ at Manual{PyLith:manual:1.6.2,
+ title = {PyLith User Manual, Version 1.6.2},
+ author = {Aagaard, B. and Kientz, S. and Knepley, M. and
+ Somala, S. and Strand, L. and Williams, C.},
+ organization = {Computational Infrastructure for Geodynamics (CIG)},
+ address = {University of California, Davis},
+ year = 2011,
+ note = {http://www.geodynamics.org/cig/software/pylith/pylith\_manual-1.6.2.pdf}
+}
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