[cig-commits] commit: Finished section on dynamic rupture benchmarks. Added first draft of abstract.
Mercurial
hg at geodynamics.org
Wed Apr 25 14:13:21 PDT 2012
changeset: 109:b8d8625c5901
user: Brad Aagaard <baagaard at usgs.gov>
date: Wed Apr 25 14:02:44 2012 -0700
files: faultRup.tex references.bib
description:
Finished section on dynamic rupture benchmarks. Added first draft of abstract.
diff -r 01f064fce659 -r b8d8625c5901 faultRup.tex
--- a/faultRup.tex Tue Apr 24 17:15:57 2012 -0700
+++ b/faultRup.tex Wed Apr 25 14:02:44 2012 -0700
@@ -52,8 +52,8 @@
Deformation}
\authors{B. T. Aagaard,\altaffilmark{1}
- Matthew G. Knepley,\altaffilmark{2}
- and Charles A. Williams\altaffilmark{3}}
+ M. G. Knepley,\altaffilmark{2}
+ and C. A. Williams\altaffilmark{3}}
\altaffiltext{1}{Earthquake Science Center, U.S. Geological Survey,
Menlo Park, California, USA.}
@@ -65,7 +65,24 @@
% ------------------------------------------------------------------
\begin{abstract}
- \brad{ADD ABSTRACT HERE}
+ We employ a domain decomposition approach with Lagrange multipliers
+ to implement fault slip in a finite-element code, PyLith, for use in
+ both quasi-static and dynamic crustal deformation applications. This
+ integrated approach to solving both quasi-static and dynamic
+ simulations leverages common finite-element data structures and
+ implementations of various boundary conditions, discretization
+ schemes, and bulk and fault rheologies. We have developed a custom
+ preconditioner for the Lagrange multiplier portion of the system of
+ equations that provides excellent scalability with problem size
+ compared to conventional Additive Schwarz methods. We demonstrate
+ application of this approach using benchmarks for both quasi-static
+ viscoelastic deformation and spontaneous dynamic rupture propagation
+ that verify the numerical implementation of these features in
+ PyLith. Future work will focus on linking the quasi-static and
+ dynamic simulations together to capture both the slow strain
+ accumulation and post-seismic relaxation at long time scales and the
+ dynamic rupture propagation and radiation of seismic waves at short
+ time scales.
\end{abstract}
% ------------------------------------------------------------------
@@ -112,14 +129,14 @@ attempted to examine a broader space-tim
attempted to examine a broader space-time window in order to remove
simplifying assumptions and more accurately capture the complex
interactions over the earthquake cycle. For example,
-\cite{Duan:Oglesby:2005} simulated multiple earthquake cycles on a
+\citet{Duan:Oglesby:2005} simulated multiple earthquake cycles on a
fault with a bend in order to capture the spatial variation in the
stress field around the bend, which they found to have a strong role
in determining whether a rupture would propagate past the bend. By
spinning up the model over many earthquake cycles, they obtained a
much more realistic stress field immediately prior to rupture compared
with assuming a simple stress field or calculating the stress field
-from a static analysis. \cite{Chen:Lapusta:2009} examined the
+from a static analysis. \citet{Chen:Lapusta:2009} examined the
behavior of small repeating earthquakes by modeling a stable sliding
region (friction increases with slip rate) surrounding an unstable
sliding region (friction decreases with slip rate). They found that
@@ -130,7 +147,7 @@ possible if they did not explicitly mode
possible if they did not explicitly model the interseismic
deformation.
-\cite{Kaneko:etal:????} have developed some of the most sophisticated
+\citet{Kaneko:etal:2011} have developed some of the most sophisticated
earthquake cycle models. Using spectral element simulations that
capture the dynamic rupture propagation as well as the interseismic
deformation, they examined the effects of low-rigidity layers and a
@@ -173,7 +190,7 @@ boundary conditions, with interseismic d
boundary conditions, with interseismic deformation usually driven by
Dirichlet (displacement/velocity) or Neumann (traction) boundary
conditions and rupture propagation simulations using absorbing
-boundaries to truncate the laterl and bottom boundaries of the
+boundaries to truncate the lateral and bottom boundaries of the
domains. However, these features constitute a small fraction of the
code. The primary difference between the two types of simulations is
the time integration scheme, with an implicit scheme used in the
@@ -189,7 +206,7 @@ needs of the community with limited reso
needs of the community with limited resources generated further
incentives for designing PyLith to leverage common infrastructure for
simulating quasi-static and dynamic deformation. Maintaining two
-seperate code bases would require a considerably greater development
+separate code bases would require a considerably greater development
effort.
With the goal of modeling the entire earthquake cycle with as few
@@ -522,8 +539,10 @@ so that the upper portion of the Jacobia
\frac{1}{\Delta t^2} \int_{V} \rho \mathbf{N}_m^T\ \cdot \mathbf{N}_n \, dV.
\end{equation}
+\brad{Add a description of the numerical damping implementation}
+
% ------------------------------------------------------------------
-% Taking advantage of the commonality
+\subsection{Leveraging Common Components}
Many of the advantages of designing PyLith to handle both quasi-static
and dynamic simulations should now be apparent. There are only minor
@@ -549,7 +568,7 @@ and the Lagrange multipliers, which corr
and the Lagrange multipliers, which correspond to fault tractions. We
expect the displacements to be generally on the order of mm to m
whereas the fault tractions will be on the order of 10$^6$ Pa. Thus,
-if we use dimensioned quantities in SI units, then we would expec the
+if we use dimensioned quantities in SI units, then we would expect the
solution to include terms that differ by up to nine orders of
magnitude. This results in a rather ill-conditioned system. We avoid
this ill-conditioning by nondimensionalizing all of the quantities
@@ -573,7 +592,7 @@ histories, including a step function, a
histories, including a step function, a linear ramp (constant slip
rate), the integral of Brune's far-field time function
\citep{Brune:1970}, a sine-cosine function developed by
-\cite{Liu:etal:2006}, and a user-defined time history. These are
+\citet{Liu:etal:2006}, and a user-defined time history. These are
discussed in detail in the PyLith manual
\citep{PyLith:manual:1.6.2}. PyLith allows specification of the slip
initiation time independently at each location as well as
@@ -604,7 +623,7 @@ multipliers that satisfy the fault const
multipliers that satisfy the fault constitutive model. On the other
hand, if the Lagrange multipliers do not exceed the fault tractions
allowed by the fault constitutive model, then the increment in fault
-slip remains zero, and no adjustements to the solution are necessary.
+slip remains zero, and no adjustments to the solution are necessary.
In iterating to find the fault slip and Lagrange multipliers that
satisfy the fault constitutive model, we employ the following
@@ -753,7 +772,7 @@ to the Lagrange multiplier constraint. T
to the Lagrange multiplier constraint. The Lagrange multiplier vertex
lies on an edge between the vertex on $S_{f^+}$ and the vertex on
$S_{f^-}$. The fault faces are organized as a Sieve, and each face has
-the two cells it is associated with as descendents. Because the cells
+the two cells it is associated with as descendants. Because the cells
are consistently oriented, the first cell attached to each face is on
the negative side of the fault, i.e., $S_{f^-}$. We
replace the vertices on the fault face of each second cell, which is
@@ -839,7 +858,7 @@ The PCFIELDSPLIT \citep{PETSc:manual} pr
The PCFIELDSPLIT \citep{PETSc:manual} preconditioner in PETSc allows
the user to define sets of unknowns which correspond to different
fields in the physical problem. This scheme is flexible enough to
-accomodate an arbitrary number of fields, mixed discretizations,
+accommodate an arbitrary number of fields, mixed discretizations,
fields defined over a subset of the mesh, etc. Once these fields are
defined, a substantial range of preconditioners can be assembled using
only PyLith options for PETSc. Table~\ref{tab:solver:options} shows
@@ -892,13 +911,13 @@ displacements along the $x$, $y$, and $z
displacements along the $x$, $y$, and $z$ axes. It is known that the
vector Laplacian is spectrally equivalent to this
operator~\citep{AskMarkAdams}\matt{Ask Mark Adams}, and each component
-is efficiently preconditioned by Algebraic Multigrid (AMG), such as
-the ML library \citep{ML}. AMG mimics the action of traditional
-geometric multgrid, but it generates coarse level operators and
-interpolation matrices using only the system matrix, treated as a
-weighted graph, rather than a separate description of the problem
-geometry, such as a mesh. We use PCFIELDSPLIT to split the elastic
-block and separately apply AMG to each component.
+is efficiently preconditioned by algebraic multigrid (AMG) methods,
+such as the ML library \citep{ML:users:guide}. AMG mimics the action of
+traditional geometric multgrid, but it generates coarse level
+operators and interpolation matrices using only the system matrix,
+treated as a weighted graph, rather than a separate description of the
+problem geometry, such as a mesh. We use PCFIELDSPLIT to split the
+elastic block and separately apply AMG to each component.
We now turn our attention to evaluating the fault portion of the
preconditioning matrix associated with the Lagrange multipliers since
@@ -938,9 +957,9 @@ required to solve a problem with prescri
required to solve a problem with prescribed slip on three faults
\brad{Setup solver test for weak scaling computations or use
strike-slip quasi-static benchmark? Add figure describing
- problem. Add problm description to PyLith manual and cite manual for
+ problem. Add problem description to PyLith manual and cite manual for
details?} to a relative tolerance of $10^{-8}$. It clearly shows
-the surperiority of our custom fault preconditioner. It also reveals
+the superiority of our custom fault preconditioner. It also reveals
that while is scales fairly well, the custom preconditioner does have
a dependence on problem size.
@@ -1146,13 +1165,13 @@ along the buried edges.
along the buried edges.
We generate both hexahedral meshes and tetrahedral meshes using CUBIT
-\citep{cubit} and construct meshes so that the problem size (number of
-DOF) for the two different cell types (hexahedra and
-tetahedra) are nearly the same. The suite of simulations examine
-problem sizes increasing by about a factor of two from $1.78\times
-10^5$ DOF to $2.14\times 10^7$ DOF. The
-corresponding discretization sizes are 1520 m to 392 m for the
-hexahedral meshes and 1744 m to 456 m for the tetrahedral meshes.
+(available from http://cubit.sandia.gov) and construct meshes so that
+the problem size (number of DOF) for the two different cell types
+(hexahedra and tetrahedra) are nearly the same. The suite of
+simulations examine problem sizes increasing by about a factor of two
+from $1.78\times 10^5$ DOF to $2.14\times 10^7$ DOF. The corresponding
+discretization sizes are 1520 m to 392 m for the hexahedral meshes and
+1744 m to 456 m for the tetrahedral meshes.
Figure~\ref{fig:solvertest:mesh} shows the 1744 m resolution
tetrahedral mesh. As we will see in
Section~\ref{src:verification:quasi-static}, the hexahedral mesh for a
@@ -1163,13 +1182,13 @@ pair of meshes are significantly larger
We characterize preconditioner performance in terms of the number of
iterations required for the residual to reach a convergence tolerance
-and the sensitivity of the number of interations to the problem
+and the sensitivity of the number of iterations to the problem
size. An ideal preconditioner would yield a small, constant number of
iterations independent of problem size. However, for complex problems
such as elasticity with fault slip and potentially nonuniform physical
properties, ideal preconditioners may not exist. Hence, we seek a
preconditioner that provides a minimal increase in the number of
-interations as the problem size increases, so that we can efficiently
+iterations as the problem size increases, so that we can efficiently
simulate quasi-static crustal deformation related to faulting and
post-seismic and interseismic deformation.
@@ -1206,16 +1225,16 @@ each core. Ideally, the time for the var
each core. Ideally, the time for the various stages of the simulation
should be independent of the number of processors/cores. For this
performance benchmark we use the entire suite of hexahedral and
-tetrahedral meshes described ealier that range in size from
+tetrahedral meshes described earlier that range in size from
$1.78\times 10^5$ DOF to $2.14\times 10^7$ DOF. In each of these
simulations, we employ the field split algebraic multigrid
preconditioner with multiplicative composition and the custom fault
block preconditioner. We ran the simulations on a Beowulf cluster
comprised of 24 compute nodes connected by QDR Infiniband, where each
-compute node consists of two quad-core Intel Xeron E5620 processors
+compute node consists of two quad-core Intel Xeon E5620 processors
with 24 GB RAM. Simulations run on eight or fewer cores were run on a
single compute node. Thus, in addition to algorithm bottlenecks,
-runtime performance is potentially impeeded by core/memory affinity,
+runtime performance is potentially impeded by core/memory affinity,
memory bandwidth, and communication among compute nodes.
\brad{Update this after tuning solver}%
@@ -1235,21 +1254,23 @@ In developing PyLith we verify the numer
In developing PyLith we verify the numerical implementation using a
number of techniques. We employ unit testing to verify correct
implementation of nearly all of the individual routines. Having a test
-for most object methods or functions isolates most bugs at their
-origin during code development and prevents new bugs from occurring as
-code is modified or optimized. We also rely on full-scale benchmarks
-to verify that the code properly solves the numerical problem. In this
-section we focus on two benchmarks that test two different
-applications: quasi-static relaxation of a Maxwell viscoelastic
-material subjected to multiple earthquake cycles involving slip and
-steady creep on a vertical strike-slip fault
-\citep{Savage:Prescott:????} and supeshear spontaneous dynamic rupture
-of a 60 degree dipping normal fault in a Drucker-Prager elastoplastic
-medium. This second benchmark corresponds
-to benchmark TPV13 in the suite of spontaneous dynamic rupture
-benchmarks constructed by the Southern California Earthquake Center
-(SCEC) and the United States Geological Survey
-\citep{Harris:etal:SRL:2009}.
+for most object methods or functions isolates bugs at their origin
+during code development and prevents new bugs from occurring as code
+is modified or optimized. We also rely on full-scale benchmarks to
+verify that the code properly solves the numerical problem. These
+benchmark problems include quasi-static strike-slip and reverse
+viscoelastic simulations and many of problems in the suite of
+spontaneous dynamic rupture benchmarks developed by the Southern
+California Earthquake Center (SCEC) and the United States Geological
+Survey \citep{Harris:etal:SRL:2009}. In this section we focus on two
+benchmarks that test different scientific applications: quasi-static
+relaxation of a Maxwell viscoelastic material subjected to multiple
+earthquake cycles involving slip and steady creep on a vertical
+strike-slip fault \citep{Savage:Prescott:1978} and supershear
+spontaneous dynamic rupture of a 60 degree dipping normal fault in a
+Drucker-Prager elastoplastic medium. This second benchmark corresponds
+to benchmark TPV13 in the SCEC suite of spontaneous dynamic rupture
+benchmarks.
\subsection{Quasi-static}
\label{sec:verification:quasi-static}
@@ -1269,7 +1290,7 @@ benchmarks constructed by the Southern C
\end{itemize}
\brad{QUESTIONS FOR CHARLES: The hex8 versus tet4 comparison isn't very
useful as both give essentially identical results. Would looking at
- another quanitity (displacement along a vertical profile, or stress)
+ another quantity (displacement along a vertical profile, or stress)
provide a more stringent test? Is there an easy way to adjust the
parameters to give something more numerically challenging
(higher/lower viscosity, coarser mesh)?}
@@ -1286,14 +1307,14 @@ depth-dependent initial stress field. Fi
depth-dependent initial stress field. Figure~\ref{fig:tpv13:geometry}
show the geometry of the benchmark problem and the size of the domain
we used in our verification test. The benchmark includes both 2-D
-(TPV13-2D is a vertical slice through the fault centerline with plane
+(TPV13-2D is a vertical slice through the fault center-line with plane
strain conditions) and 3-D versions (TPV13). This benchmark specifies
a spatial resolution of 100 m on the fault surface. In order to
examine the effects of cell type and discretization size we consider
both triangular and quadrilateral discretizations with resolutions on
the fault of 50 m, 100 m, and 200 m for TPV13-2D and 100 m and 200 m
for TPV13. We gradually coarsen the mesh with distance from the fault
-by increasing the discretization size at a geometrirc rate of
+by increasing the discretization size at a geometric rate of
2\%. This provides high resolution at the fault surface to resolve the
small scale features of the rupture process with less resolution at
the edges of the boundary where the solution is much
@@ -1304,7 +1325,7 @@ nucleation region. Figure~\ref{fig:tpv13
nucleation region. Figure~\ref{fig:tpv13-2d:stress:slip} illustrates
the depth dependence of the stress field in terms of the fault
tractions and Table~\ref{tab:tpv13:parameters} summarizes the
-benchmark parameters. \cite{Harris:etal:SRL:2011} provides a more
+benchmark parameters. \citet{Harris:etal:SRL:2011} provides a more
complete description with all of the details available from
\url{http://scecdata.usc.edu/cvws/cgi-bin/cvws.cgi}. An unfortunate
feature of this, and many other benchmarks in the SCEC Spontaneous
@@ -1329,50 +1350,64 @@ the normal faulting component of fault s
the normal faulting component of fault slip rate time histories. For a
resolution of 200 m on the fault, the solution contains some
high-frequency oscillation due to insufficient resolution of the
-cohesive zone \citep{Rice:????}. The finer meshes provide sufficient
+cohesive zone \citep{Rice:1993}. The finer meshes provide sufficient
resolution of the cohesive zone so there is very little high-frequency
oscillation in the slip rate time histories. The triangular cells
result in less oscillation compared with quadrilateral cells.
-In this problem without an analytical solution, we rely on comparison
-with other spontaneous dynamic rupture modeling codes to verify the
-numerical implementation in
+In this problem without an analytical solution, as in all of the
+benchmarks in the SCEC spontaneous rupture benchmark suite, we rely on
+comparison with other spontaneous dynamic rupture modeling codes to
+verify the numerical implementation in
PyLith. Figure~\ref{fig:tpv13-2d:slip:rate}(b) compares the slip rate
time histories from PyLith with four other codes (see
-\citep{Harris:etal:SRL:2011} for a discussion of these other
-finite-element and finite-difference codes). The slip rate time
-histories agree very well, although some codes yield more oscillation
-than others. We attribute this to variations in the amount of
-numerical damping used across the various codes.
+\citep{Harris:etal:SRL:2011}, \citep{Andrews:etal:2007},
+\citep{Barall:2009}, and \citep{Dunham:etal:2011} for a discussion of
+these other finite-element and finite-difference codes). The slip
+rate time histories agree very well, although some codes yield more
+oscillation than others. We attribute this to variations in the amount
+of numerical damping used across the various codes.
+The results in for the 3-D version of the TPV13 benchmark yield
+similar results. Figure~\ref{fig:tpv13:rupture:time}(a) shows the same
+trends in rupture speed with discretization size that we observed in
+the 2-D version. In both cases models with insufficient resolution to
+resolve the cohesive zone propagate slightly slower than models with
+sufficient resolution. In this case the differences between the
+rupture times for the 200 m and 100 m resolution tetrahedral meshes
+are less than 0.1 seconds over the entire fault surface. Comparing the
+rupture times among the modeling codes in
+Figure~\ref{fig:tpv13:rupture:time}, we find that the four codes fall
+into two groups. In the mode-III (along-strike) direction, PyLith and
+the spectral element code by \citet{Kaneko:etal:2008} are essentially
+identical while the finite-element codes by \citet{Barall:2009} and
+\citet{Ma:Andrews:2010} are also essentially identical. In the mode-II
+(up-dip) direction all four codes agree very closely. As in the 2-D
+version, we attribute the differences among the codes not to the
+numerical implementation but the treatment of discontinuities in the
+spatial variation of the parameters. This explains why the
+higher-order spectral element code by \citet{Kaneko:etal:2008} agrees
+so closely with PyLith, a lower-order finite-element code.
+The slip rate and velocity time histories displayed in
+Figures~\ref{fig:tpv13:slip:rate} and~\ref{fig:tpv13:velocity} are
+consistent with the trends observed in the comparison of rupture
+times. Furthermore, the codes all produce consistent results
+throughout the entire time histories. The small differences in rupture
+time in the mode-II (along-strike) direction between the two groups of
+codes is evident in the slip rate time histories at a depth of 7.5 km
+and 12 km along strike
+(Figure~\ref{fig:tpv13:slip:rate}(f)). Nevertheless, this simply
+produces a small time shift in the time history.
From the 2-D and 3-D versions of the SCEC spontaneous rupture
benchmark TPV13, we conclude that the PyLith performs very similarly
to other finite-element and finite-difference spontaneous dynamic
-rupture modeling codes in this relatively complex problem involving a
-Drucker-Prager elastoplastic builk rheology, slip-weakening friction,
-and supershear rupture on a dipping normal fault.
-
-\begin{itemize}
-\item Spontaneous rupture benchmark: TPV13
- \begin{itemize}
- \item dipping fault, depth dependent stresses, super-shear rupture,
- Drucker-Prager elastoplastic model
- \item Not ideal due to discontinuities in spatial variation of parameters
- \item 2-D, compare quad4 and tri3
- \item 3-D, only tet4 (complex geometry)
- \item Figures
- \begin{itemize}
- \item 3-D geometry (points for comparison)
- \item 2-D: final slip profile
- \item 2-D: slip rate time histories (4 points) [TPV12, TPV13]
- \item 3-D: rupture contours
- \item 3-D: slip rate time histories (4 points) [TPV12, TPV13]
- \end{itemize}
- \end{itemize}
-\end{itemize}
-
+rupture modeling codes. In particular it is well-suited to problems
+with complex geometry as we are able to vary the discretization size
+while simulating a dipping normal fault. The code accurately captures
+supershear rupture and properly implements a Drucker-Prager
+elastoplastic bulk rheology and slip-weakening friction.
% ----------------------------------------------------------------------
% Notation -- End each entry with a period.
@@ -1382,7 +1417,7 @@ and supershear rupture on a dipping norm
$\mathbf{d}$ & fault slip vector.\\
$\mathbf{f}$ & body force vector.\\
$\mathbf{l}$ & Lagrange multiplier vector corresponding to the fault traction vector.\\
- $\mathbf{L}$ & Matrix associatd with Jacobian operator for constraint equation.\\
+ $\mathbf{L}$ & Matrix associated with Jacobian operator for constraint equation.\\
$\mathbf{K}$ & Matrix associated with Jacobian operator for
elasticity equation.\\
$\mu_f$ & coefficient of friction.\\
@@ -1491,8 +1526,8 @@ MGK acknowledges partial support from NS
fault in the Savage and Prescott benchmark during earthquake
cycles 3 and 10. The displacements values shown are
relative to the values at the beginning of the earthquake cycle to
- faciliate comparison between the analytical solution and the
- numerical models which require spinup to reach the steady-state
+ facilitate comparison between the analytical solution and the
+ numerical models which require spin-up to reach the steady-state
solution. Both the hexahedral (Hex8) and tetrahedral (Tet4)
discretizations resolve the viscoelastic deformation and display
excellent agreement with the steady-state solution by the tenth
@@ -1566,7 +1601,7 @@ MGK acknowledges partial support from NS
\noindent\includegraphics{figs/tpv13-2d_sliprate}
\caption{Slip rate time histories for SCEC spontaneous rupture benchmark
TPV13-2D. Locations correspond to the red dots along the
- centerline of the fault shown in
+ center-line of the fault shown in
Figure~\ref{fig:tpv13:geometry}. Panels (a)--(d) show convergence
of the solution for quadrilateral and triangular cells as a
function of discretization size, and panels (e)--(h) demonstrate
@@ -1608,8 +1643,8 @@ MGK acknowledges partial support from NS
SCEC spontaneous rupture benchmark TPV13. Panels (a)--(b) are
associated with a site that is on the hanging wall 3 km from the
fault trace and 12 km along strike, and panels (c)--(d) are
- assocaited with a site that is on the footwall 3 km from the fault
- trace along the fault centerline. As expected based on the close
+ associated with a site that is on the footwall 3 km from the fault
+ trace along the fault center-line. As expected based on the close
agreement in the rupture time contours and fault slip rates, the
velocity time histories from the difference dynamic rupture
modeling codes agree very closely.}
diff -r 01f064fce659 -r b8d8625c5901 references.bib
--- a/references.bib Tue Apr 24 17:15:57 2012 -0700
+++ b/references.bib Wed Apr 25 14:02:44 2012 -0700
@@ -231,20 +231,138 @@
compressive side of the bend.},
}
- at Article{Kaneko:Fialko:????,
+ at Article{Kaneko:Fialko:2011,
author = {Kaneko, Y. and Fialko, Y.},
title = {Shallow slip deficit due to large strike-slip
earthquakes in dynamic rupture simulations with
elasto-plastic off-fault response},
journal = GJI,
year = {2011},
- volume = {??},
- number = {??},
- pages = {??},
- month = {??},
- note = {in press},
+ volume = {186},
+ number = {3},
+ pages = {1389--1403},
doi = {10.1111/j.1365-246X.2011.05117.x},
- abstract = {},
+ abstract = {Slip inversions of geodetic data from several large
+ (magnitude ∼7) strike-slip earthquakes point to
+ coseismic slip deï¬cit at shallow depths (<3–4 km),
+ that is, coseismic slip appears to decrease towards
+ the Earth surface. While the inferred slip
+ distribution may be consistent with
+ laboratory-derived rate and state friction laws
+ suggesting that the uppermost brittle crust may be
+ velocity strengthening, there remains a question of
+ how the coseismic slip deï¬cit is accommodated
+ throughout the earthquake cycle. The consequence of
+ velocity-strengthening fault friction at shallow
+ depths is that the deï¬cit of coseismic slip is
+ relieved by post-seismic afterslip and interseismic
+ creep. However, many seismic events with inferred
+ shallow slip deï¬cit were not associated with either
+ resolvable shallow interseismic creep or robust
+ shallow afterslip. Hence, the origin of shallow
+ ‘slip deï¬cit’ remains uncertain. In this study, we
+ investigate whether inelastic failure in the shallow
+ crust due to dynamic earthquake rupture can explain
+ the inferred deï¬cit of shallow slip. Evidence for
+ such failure is emerging from geologic, seismic and
+ geodetic observations. We ï¬nd that the amount of
+ shallow slip deï¬cit is proportional to the amount of
+ inelastic deformation near the Earth surface. Such
+ deformation occurs under a wide range of parameters
+ that characterize rock strength in the upper crust.
+ However, the largest magnitude of slip deï¬cit in
+ models accounting for off-fault yielding is 2–4
+ times smaller than that inferred from kinematic
+ inversions of geodetic data. To explain this
+ discrepancy, we further explore to what extent
+ assumptions in the kinematic inversions may bias the
+ inferred slip distributions. Inelastic deformation
+ in the shallow crust reduces coseismic strain near
+ the fault, introducing an additional ‘artiï¬cial’
+ deï¬cit of up to 10 per cent of the maximum slip in
+ inversions of geodetic data that are based on purely
+ elastic models. The largest magnitude of slip deï¬cit
+ in our models combined with the bias in inversions
+ accounts for up to 25 per cent of shallow slip
+ deï¬cit, which is comparable, but still smaller than
+ 30– 60 per cent deï¬cit inferred from kinematic
+ inversions. We discuss potential mechanisms that may
+ account for the remaining discrepancy between slip
+ deï¬cit predicted by elasto-plastic rupture models
+ and that inferred from inversions of space geodetic
+ data},
+}
+
+ at Article{Andrews:etal:2007,
+ author = {Andrews, D.~J. and Hanks, T.~C. and Whitney, J.~W.},
+ title = {Physical limits on ground motion at Yucca Mountain},
+ journal = BSSA,
+ year = {2007},
+ volume = {97},
+ number = {6},
+ month = dec,
+ pages = {1771--1792},
+ doi = {10.1785/0120070014}
+}
+
+ at Article{Barall:2009,
+ author = {Barall, M.},
+ title = {A grid-doubling finite-element technique for
+ calculating dynamic three-dimensional spontaneous
+ rupture on an earthquake fault},
+ journal = GJI,
+ year = {2009},
+ volume = {178},
+ pages = {845--859},
+ doi = {10.1111/j.1365-246X.2009.04190.x}
+}
+
+ at Article{Duan:2009,
+ author = {Duan, B.},
+ title = {Effects of low-velocity fault zones on dynamic
+ ruptures with nonelastic off-fault response},
+ journal = GRL,
+ year = {2009},
+ volume = {35},
+ pages = {L04307},
+ doi = {10.1029/2008GL033171.}
+}
+
+ at Article{Dunham:etal:2011,
+ author = {Dunham, E.~M. and Belanger, D. and Cong, L. and
+ Kozdon, J.~E.},
+ title = {Earthquake ruptures with strongly rate-weakening
+ friction and off-fault plasticity: {Planar} faults},
+ journal = BSSA,
+ year = {2011},
+ volume = {101},
+ number = {5},
+ month = oct,
+ pages = {2308--2322},
+ doi = {10.1785/0120100075}
+}
+
+ at Article{Kaneko:etal:2008,
+ author = {Kaneko, Y. and Lapusta, N. and Ampuero, J.-P.},
+ title = {Spectral element modeling of spontaneous earthquake
+ rupture on rate and state faults: {Effect} of
+ velocity-strengthening friction at shallow depths},
+ journal = JGR,
+ year = {2008},
+ volume = {113},
+ pages = {B09317},
+ doi = {10.1029/2007JB005553}
+}
+
+ at Article{Ma:Andrews:2010,
+ author = {Ma, S. and Andrews, D.~J.},
+ title = {Inelastic off-fault response and three-dimensional
+ earthquake rupture dynamics on a strikeslip fault},
+ journal = JGR,
+ year = {2010},
+ volume = {115},
+ pages = {B04304},
+ doi = {10.1029/2009JB006382}
}
@Article{Harris:etal:SRL:2011,
@@ -283,20 +401,52 @@
doi = {10.1785/gssrl.80.1.119}
}
- at Article{Kaneko:etal:????,
+ at Article{Kaneko:etal:2011,
author = {Kaneko, Y. and Ampuero, J.-P. and Lapusta, N.},
title = {Spectral-element simulations of long-term fault
slip: {Effect} of low-rigidity layers on
earthquake-cycle dynamics},
journal = JGR,
year = {2011},
- volume = {??},
- number = {??},
- pages = {??},
- month = {??},
- note = {in press},
+ volume = {116},
+ number = {B10313},
+ pages = {18pp},
doi = {10.1029/2011JB008395},
- abstract = {},
+ abstract = {We develop a spectral element method for the
+ simulation of long-term histories of spontaneous
+ seismic and aseismic slip on faults subjected to
+ tectonic loading. Our approach reproduces all stages
+ of earthquake cycles: nucleation and propagation of
+ earthquake rupture, postseismic slip and
+ interseismic creep. We apply the developed
+ methodology to study the effects of low-rigidity
+ layers on the dynamics of the earthquake cycle in
+ 2-D. We consider two cases: small (M ∼ 1)
+ earthquakes on a fault surrounded by a damaged fault
+ zone and large (M ∼ 7) earthquakes on a vertical
+ strike-slip fault that cuts through shallow
+ low-rigidity layers. Our results indicate how the
+ source properties of repeating earthquakes are
+ affected by the presence of a damaged fault zone
+ with low rigidity. Compared to faults in homogeneous
+ media, we find (1) reduction in the earthquake
+ nucleation size, (2) amplification of slip rates
+ during dynamic rupture propagation, (3) larger
+ recurrence interval, and (4) smaller amount of
+ aseismic slip. Based on linear stability analysis,
+ we derive a theoretical estimate of the nucleation
+ size as a function of the width and rigidity
+ reduction of the fault zone layer, which is in good
+ agreement with simulated nucleation sizes. We
+ further examine the effects of vertically-stratified
+ layers (e.g., sedimentary basins) on the nature of
+ shallow coseismic slip deficit. Our results suggest
+ that low-rigidity shallow layers alone do not lead
+ to coseismic slip deficit. While the low-rigidity
+ layers result in lower interseismic stress
+ accumulation, they also cause dynamic amplification
+ of slip rates, with the net effect on slip being
+ nearly zero.},
}
@Article{Langbein:etal:2006,
@@ -513,6 +663,77 @@
crust with the upper mantle},
}
+ at Article{Rice:1993,
+ author = {Rice, J.~R.},
+ title = {Spatiotemporal complexity of slip on a fault},
+ journal = JGR,
+ year = 1993,
+ volume = 98,
+ number = {B6},
+ pages = {9885--9907},
+ month = jun # {~10},
+ abstract = {Three-dimensional analyses are reported of slip on a
+ long vertical strike-slip fault between steadily
+ driven elastic crustal blocks. A rate- and state-
+ dependent friction law governs motion on the fault;
+ the law includes a characteristic slip distance L
+ for evolution of surface state and slip
+ weakening. Because temperature and normal stress
+ vary with depth, frictional constitutive properties
+ (velocity weakening/strengthening) do also. Those
+ properties are taken either as uniform along-strike
+ at every depth or as perturbed modestly from
+ uniformity. The governing equations of quasi-static
+ elasticity and frictional slip are solved on a
+ computational grid of cells as a discrete numerical
+ system, and a viscous radiation damping term is
+ included to approximately represent inertial control
+ of slip rates during earthquake-like
+ instabilities. The numerical results show richly
+ complex slip, with a spectrum of event sizes, when
+ solved for a grid with oversized cells, that is,
+ with cell size h that is too large to validly
+ represent the underlying continuous system of
+ equations. However, in every case for which it has
+ been feasible to do the computations (moderately
+ large L only), that spatio- temporally complex slip
+ disappears in favor of simple limit cycles of
+ periodically repeated large earthquakes with
+ reduction of cell size h. Further study will be
+ necessary to determine whether a similar transition
+ occurs when the elastodynamics of rupture
+ propagation is treated more exactly, rather than in
+ the radiation damping approximation. The transition
+ from complex to ordered fault response occurs as h
+ is reduced below a theoretically derived nucleation
+ size h* which scales with L but is 2 x 10(4) to
+ 10(5) larger in cases considered. Cells larger than
+ h* can fail independently of one another, whereas
+ those much smaller than h* cannot slip unstably
+ alone and can do so only as part of a cooperating
+ group of cells. The results contradict an emergent
+ view that spatio-temporal complexity is a generic
+ feature of mechanical fault models. Such generic
+ complexity does apparently result from models which
+ are inherently discrete in the sense of having no
+ well-defined continuum limit as h diminishes. Those
+ models form a different class of dynamical systems
+ from models like the present one that do have a
+ continuum limit. Strongly oversized cells cause the
+ model developed here to mimic an inherently discrete
+ system. It is suggested that oversized cells,
+ capable of failing independently of one another, may
+ crudely represent geometrically disordered fault
+ zones, with quasi-independent fault segments that
+ join one another at kinks or jogs. Such geometric
+ disorder, at scales larger than h*, may force a
+ system with a well-defined continuum limit to mimic
+ an inherently discrete system and show
+ spatio-temporally complex slip at those larger
+ scales.},
+
+}
+
@Article{Reilinger:etal:2000,
author = {Reilinger, R.~E. and Ergintav, S. and Burgmann,
R. and McClusky, S. and Lenk, O. and Barka, A. and
@@ -648,6 +869,53 @@
predictability and causes earthquake recurrence to
be far more aperiodic than has been suggested. },
}
+
+ at article{Savage:Prescott:1978,
+ author = {Savage, J.~C. and Prescott, W.~H.},
+ title = {Asthenosphere Readjustment and the Earthquake Cycle},
+ journal = JGR,
+ year = 1978,
+ volume = 83,
+ number = {B7},
+ pages = {3369--3376},
+ doi = {10.1029/JB083iB07p03369},
+ abstract = { A simple two-dimensional model of the earthquake cycle
+ (preearthquake strain accumulation, coseismic strain
+ release, and postseismic readjustment) has been
+ constructed from the Nur-Mavko solution for a screw
+ dislocation in an elastic plate (lithosphere)
+ overlying a viscoelastic substrate
+ (asthenosphere). The deformation at the free surface
+ is calculated for an earthquake cycle imposed by
+ prescribed slip on a transform fault. This
+ deformation is compared to that produced by a
+ similar cycle in an elastic half space so that the
+ effects of viscoelastic relaxation in the
+ asthenosphere may be isolated. The following
+ conclusions are drawn: (1) The surface deformation
+ produced by viscoelastic relaxation in the
+ asthenosphere can be duplicated identically by a
+ reasonable distribution of slip at depth on a
+ vertical fault in an elastic half space. Thus
+ differentiation of two possible modes of
+ postearthquake readjustment will be difficult. (2)
+ The effect of asthenosphere relaxation is important
+ only if the depth of the seismic zone is comparable
+ to the thickness of the lithosphere. If the seismic
+ zone is 15 km deep and the lithosphere is 75 km
+ thick, as commonly estimated for the San Andreas
+ fault zone, asthenosphere relaxation is not
+ particularly significant in determining surface
+ deformation. (3) In a periodic sequence of
+ earthquakes the principal observable effects of
+ viscoelasticity in the asthenosphere are to produce
+ a rapid postearthquake deformation and to
+ concentrate strain accumulation and relaxation even
+ closer to the fault than in the elastic half-space
+ model.
+ },
+}
+
@Book{Smith:etal:1996,
author = {Smith, B.~F. and Bj{\o}rstad, P. and Gropp, W.~D.},
@@ -813,3 +1081,11 @@
}
+ at TechReport{ML:users:guide,
+ author = {Sala, M. and Hu, J.~J. and Tuminaro, R.~S.},
+ title = {{ML}3.1 {Smoothed} {Aggregation} {User}'s {Guide}},
+ institution = {Sandia National Laboratories},
+ number = {SAND2004-4821},
+ address = {Albuquerque, NM (USA)},
+ year = {2004}
+}
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