# [cig-commits] commit: Added paragraph on numerical damping.

Mercurial hg at geodynamics.org
Thu Apr 26 11:03:01 PDT 2012

changeset:   112:4ea828744ec6
tag:         tip
user:        Brad Aagaard <baagaard at usgs.gov>
date:        Thu Apr 26 11:02:53 2012 -0700
files:       faultRup.tex references.bib
description:

diff -r 62b020c5b3d0 -r 4ea828744ec6 faultRup.tex
--- a/faultRup.tex	Wed Apr 25 15:34:05 2012 -0700
+++ b/faultRup.tex	Thu Apr 26 11:02:53 2012 -0700
@@ -539,7 +539,24 @@ 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.

+Earthquake ruptures in which the slip has a short rise time tends to
+introduce deformation at short length scales (high frequencies) that
+the numerical models can resolve accurately. This is especially true
+in spontaneous rupture simulations, because the rise time is sensitive
+to the evolution of the fault rupture. In order to reduce the
+introduction of deformation at such short length scales we add
+artificial damping via Kelvin-Voigt viscosity
+\citep{Day:etal:2005,Kaneko:etal:2008} to the computation of the strain,
+\begin{gather}
+  \mathbf{\varepsilon} = \frac{1}{2} \left[ \nabla \mathbf{u} +
+    (\nabla \mathbf{u})^T \right ], \\
+  \mathbf{\varepsilon} \approx \frac{1}{2} \left[ \nabla \mathbf{u}_d +
+    (\nabla \mathbf{u}_d)^T \right ], \\
+  \mathbf{u_d} = \mathbf{u} + \eta^* \Delta t \frac{\partial
+    \mathbf{u}}{\partial t},
+\end{gather}
+where $\eta^*$ is a nomdimensional viscosity on the order of
+0.1--1.0.

% ------------------------------------------------------------------
\subsection{Leveraging Common Components}
diff -r 62b020c5b3d0 -r 4ea828744ec6 references.bib
--- a/references.bib	Wed Apr 25 15:34:05 2012 -0700
+++ b/references.bib	Thu Apr 26 11:02:53 2012 -0700
@@ -342,6 +342,66 @@
doi =          {10.1785/0120100075}
}

+ at Article{Day:etal:2005,
+  author = 	 {Day, S.~M. and Dalguer, L.~A. and Lapusta, N. and Liu, Y.},
+  title = 	 {Comparison of finite difference and boundary
+                  integral solutions to three-dimensional spontaneous
+                  rupture},
+  journal = 	 JGR,
+  year = 	 {2005},
+  volume = 	 {110},
+  number =       {B09317},
+  doi =          {10.1029/2007JB005553},
+  abstract =     {The spontaneously propagating shear crack on a
+                  frictional interface has proven to be a useful
+                  idealization of a natural earthquake. The
+                  corresponding boundary value problems are nonlinear
+                  and usually require computationally intensive
+                  numerical methods for their solution. Assessing the
+                  convergence and accuracy of the numerical methods is
+                  challenging, as we lack appropriate analytical
+                  solutions for comparison. As a complement to other
+                  methods of assessment, we compare solutions obtained
+                  by two independent numerical methods, a finite
+                  difference method and a boundary integral (BI)
+                  method. The finite difference implementation, called
+                  DFM, uses a traction-at-split-node formulation of
+                  the fault discontinuity. The BI implementation
+                  employs spectral representation of the stress
+                  transfer functional. The three-dimensional (3-D)
+                  test problem involves spontaneous rupture spreading
+                  on a planar interface governed by linear
+                  slip-weakening friction that essentially defines a
+                  cohesive law. To get a priori understanding of the
+                  spatial resolution that would be required in this
+                  and similar problems, we review and combine some
+                  simple estimates of the cohesive zone sizes which
+                  correspond quite well to the sizes observed in
+                  simulations. We have assessed agreement between the
+                  methods in terms of the RMS differences in rupture
+                  time, final slip, and peak slip rate and related
+                  these to median and minimum measures of the cohesive
+                  zone resolution observed in the numerical solutions.
+                  The BI and DFM methods give virtually
+                  indistinguishable solutions to the 3-D spontaneous
+                  rupture test problem when their grid spacing Dx is
+                  small enough so that the solutions adequately
+                  resolve the cohesive zone, with at least three
+                  points for BI and at least five node points for
+                  DFM. Furthermore, grid-dependent differences in the
+                  results, for each of the two methods taken
+                  separately, decay as a power law in Dx, with the
+                  same convergence rate for each method, the
+                  calculations apparently converging to a common, grid
+                  interval invariant solution. This result provides
+                  strong evidence for the accuracy of both methods. In
+                  addition, the specific solution presented here, by
+                  virtue of being demonstrably grid-independent and
+                  consistent between two very different numerical
+                  methods, may prove useful for testing new numerical
+                  methods for spontaneous rupture problems.},
+}
+
@Article{Kaneko:etal:2008,
author = 	 {Kaneko, Y. and Lapusta, N. and Ampuero, J.-P.},
title = 	 {Spectral element modeling of spontaneous earthquake
@@ -350,7 +410,7 @@
journal = 	 JGR,
year = 	 {2008},
volume = 	 {113},
-  pages =        {B09317},
+  number =        {B09317},
doi =          {10.1029/2007JB005553}
}