[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:
Added paragraph on numerical damping.
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.
\end{equation}
-\brad{Add a description of the numerical damping implementation}
+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}
}
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