# [cig-commits] commit: More revising (double couple section and continuing from there).

Mercurial hg at geodynamics.org
Tue Jan 15 14:33:01 PST 2013

changeset:   157:4b5b01bbf44d
tag:         tip
user:        Brad Aagaard <baagaard at usgs.gov>
date:        Tue Jan 15 14:32:56 2013 -0800
files:       faultRup.tex references.bib
description:
More revising (double couple section and continuing from there).

diff -r 71f4a93fa393 -r 4b5b01bbf44d faultRup.tex
--- a/faultRup.tex	Mon Jan 14 19:41:22 2013 -0800
+++ b/faultRup.tex	Tue Jan 15 14:32:56 2013 -0800
@@ -311,30 +311,38 @@ The domain decomposition approach for im
The domain decomposition approach for imposing fault slip or tractions
on a fault is similar to the traction at split nodes'' (TSN)
technique used in a number of finite-difference and finite-element
-codes \cite{ADD_CITATIONS_HERE}, but differs from imposing fault slip
-via double-couple point sources. The domain decomposition approach
-treats the fault surface as a frictional contact interface, and the
-tractions correspond directly to the Lagrange multipliers needed to
-satisfy the constraint equation involving the jump in the displacement
-field across the fault and the fault slip. As a result, the fault
-tractions are equal and opposite on the two sides of the fault and
-satisfy equilibrium.  The TSN technique ... ADD STUFF HERE.
+codes
+\citep{Andrews:1999,Bizzarri:Cocco:2005,Day:etal:2005,Duan:Oglesby:2005,Dalguer:Day:2007,Moczo:etal:2007},
+but differs from imposing fault slip via double-couple point
+sources. The domain decomposition approach treats the fault surface as
+a frictional contact interface, and the tractions correspond directly
+to the Lagrange multipliers needed to satisfy the constraint equation
+involving the jump in the displacement field across the fault and the
+fault slip. As a result, the fault tractions are equal and opposite on
+the two sides of the fault and satisfy equilibrium.  The TSN technique
+is often applied in dynamic spontaneous rupture models with explicit
+time stepping and a diagonal system Jacobian, so that the fault
+tractions are explicitly computed as part of the solution of the
+uncoupled equations. In this way, the TSN technique as described by
+\citet{Andrews:1999} could be considered an optimization of the domain
+decomposition technique for the special case of dynamic spontaneous
+rupture with a fault consitutitive model and explicit time stepping.

Imposing fault slip via double couple point sources involves imposing
body forces consistent with an effective plastic strain associated
-with fault slip. These body forces resolved onto the fault surface do
-not directly correspond to the tractions on the fault
-surface. Instead, the fault tractions come from resolving the
-superposition of the stress from the elasticity equation and the body
-forces onto the fault surface.  Because the strain is imposed in a
-continuous medium, the body forces depend on the elastic modulii, so
-in the case of a contast in the elastic modulii across the fault, the
-body forces on the two sides of the fault differ. This introduces some
-greater complexity into imposing a desired fault slip and illustates
-the simplicity of the domain decomposition approach that directly
-relates the fault tractions to the enforcement of the constraint that
-the fault slip matches the jump in the displacement field across the
-fault.
+with fault slip (sometimes called the stress-free strain'',
+\citet{Aki:Richards:2002}). The total strain is the superposition of
+this effective plastic strain and the elastic strain. The fault
+tractions are associated with the total strain, not the effective
+plastic strain. This illustrates a key difference between this
+approach and the domain decomposition approach, in which the Lagrange
+multipliers and the constrain equation directly relate the fault slip
+to the fault tractions (Lagrange multipliers). One implication of this
+difference is that when using double couple point forces, the body
+forces driving slip depend on the elastic modulii and will differ
+across a fault surface with a contrast in the elastic modulii, whereas
+the fault tractions (Lagrange multipliers) in the domain decomposition
+approach will be equal in magnitude across the fault.

We express the weighting function $\pmb{\phi}$, trial solution
$\bm{u}$, Lagrange multipliers $\bm{l}$, and fault slip $\bm{d}$ as
@@ -359,10 +367,14 @@ product, we have
\bm{l} = \bm{N}_p \cdot \bm{l}_p, \\
\bm{d} = \bm{N}_p \cdot \bm{d}_p.
\end{gather}\end{linenomath*}
+The first term on the right hand side of these equations is a matrix
+of the basis functions. For example, in three dimensions $\bm{N}_m$ is
+a $3 \times 3m$ matrix, where $m$ is the number of basis functions.

The weighting function is arbitrary, so the integrands must be zero
for all $\bm{a}_m$, which leads to
\begin{linenomath*}\begin{gather}
+  \label{eqn:residual:elasticity}
\begin{split}
- \int_{V} \nabla \bm{N}_m^T \cdot \bm{\sigma} \, dV
+ \int_{S_T} \bm{N}_m^T \cdot \bm{T} \, dS
@@ -374,6 +386,7 @@ for all $\bm{a}_m$, which leads to
=\bm{0},
\end{split}
\\
+  \label{eqn:residual:constraint}
%
\int_{S_f} \bm{N}_p^T \cdot
\left( \bm{N}_p \cdot \bm{d}_p
@@ -388,23 +401,37 @@ Lagrange multipliers $\bm{l}$ depend on
Lagrange multipliers $\bm{l}$ depend on the fault slip, slip rate,
and state variables.

-For nonlinear bulk rheologies it is convenient to work with the
-increment in stress and strain, so we formulate the solution of the
-equations in terms of the increment in the solution from time $t$ to
-$t+\Delta t$ rather than the solution at time $t+\Delta t$.
-Consequently, rather than constructing a system with the form
-$\bm{A} \cdot \bm{u}(t+\Delta t) = \bm{b}(t+\Delta t)$, we
-construct a system with the form $\bm{A} \cdot \bm{du} = -\bm{b}(t+\Delta t) - \bm{A} \cdot \bm{u}(t)$, where
-$\bm{u}(t+\Delta t) = \bm{u}(t) + \bm{du}$.
+We evaluate the integrals in equations~(\ref{eqn:residual:elasticity})
+\citep{Zienkiewicz:Taylor:2005}. This involves evaluating the
+integrands at the quadrature points, multiplying by the corresponding
+weighting function, and summing over the quadrature points. With an
+appropriate choice for the quadrature scheme the finite-element method
+allows inclusion of spatial variations of boundary tractions, density,
+body forces, and physical properties within the cells.
+
+To solve equations~(\ref{eqn:residual:elasticity})
+and~(\ref{eqn:residual:constraint}), we construct a linear system of
+equations.  For nonlinear bulk rheologies it is convenient to work
+with the increment in stress and strain, so we formulate the solution
+of the equations in terms of the increment in the solution from time
+$t$ to $t+\Delta t$ rather than the solution at time $t+\Delta t$.
+Consequently, rather than constructing a system with the form $\bm{A} +\cdot \bm{u}(t+\Delta t) = \bm{b}(t+\Delta t)$, we construct a system
+with the form $\bm{A} \cdot \bm{du} = \bm{b}(t+\Delta t) - \bm{A} +\cdot \bm{u}(t)$, where $\bm{u}(t+\Delta t) = \bm{u}(t) + \bm{du}$. We
+use an initial guess of zero for the increment in the solution.

% ------------------------------------------------------------------
\subsection{Quasi-static Simulations}

For quasi-static simulations we ignore the inertial term and
-time-dependence only enters through the constitutive models
-$t+\Delta t$,
+time-dependence only enters through the constitutive models and the
+series of static problems with potentially time-varying physical
+properties and boundary conditions. The stability of the solution is
+limited to resolving these temporal variations. Considering the
+deformation at time $t+\Delta t$,
\begin{linenomath*}\begin{gather}
\label{eqn:quasi-static:residual:elasticity}
\begin{split}
diff -r 71f4a93fa393 -r 4b5b01bbf44d references.bib
--- a/references.bib	Mon Jan 14 19:41:22 2013 -0800
+++ b/references.bib	Tue Jan 15 14:32:56 2013 -0800
@@ -190,7 +190,7 @@
}

@article{Duan:Oglesby:2005,
-  author =	 {Duan, Benchun and Oglesby, David~D.},
+  author =	 {Duan, B. and Oglesby, D.~D.},
title =	 {Multicycle dynamics of nonplanar strike-slip faults},
journal =	 JGR,
volume =	 {110},
@@ -1519,6 +1519,78 @@
note = {English translation of the original 1928 paper published in {\it Mathematische Annalen}}
}

+ at incollection{Moczo:etal:2007,
+  author = {Moczo, P. and Robertsson, O.~A. and Eisner, L.},
+  title = {The finite-difference time-domain method for modeling of
+                  seismic wave propagation},
+  booktitle = {Advances in Wave Propagation in Heterogenous Earth},
+  publisher = {Elsevier},
+  year = 2007,
+  volume = 48,
+  pages = {421--516},
+  series = {Advances in Geophysics},
+  doi = {10.1016/S0065-2687(06)48008-0},
+}
+
+ at article{Andrews:1999,
+  author =	 {Andrews, D.~J.},
+  title =	 {Test of two methods for faulting in finite-
+                  difference calculations},
+  journal =	 BSSA,
+  volume =	 {89},
+  number =	 {4},
+  year =	 {1999},
+  month =	 aug,
+  pages =	 {931--937},
+  abstract =	 {Tests of two fault boundary conditions show that
+                  each converges with second order accuracy as the
+                  finite-difference grid is refined. The first method
+                  uses split nodes so that there are disjoint grids
+                  that interact via surface traction. The 3D version
+                  described here is a generalization of a method I
+                  have used extensively in 2D; it is as accurate as
+                  the 2D version. The second method represents fault
+                  slip as inelastic strain in a fault zone. Offset of
+                  stress from its elastic value is seismic moment
+                  density. Implementation of this method is quite
+                  simple in a finite-difference scheme using velocity
+                  and stress as dependent variables.},
+}
+
+ at article{Dalguer:Day:2007,
+  author =	 {Dalguer, L.~A. and Day, S.~M.},
+  title =	 {Staggered-grid split-node method for spontaneous
+                  rupture simulation},
+  journal =	 JGR-SE,
+  volume =	 {112},
+  number =	 {82},
+  year =	 {2007},
+  pages =	 {art. no.--B02302},
+  doi =	 {10.1029/2006JB004467},
+  abstract =	 {},
+}
+
+ at article{Bizzarri:Cocco:2005,
+  author = {Bizzarri, A. and Cocco, M.},
+  title = {3D dynamic simulations of spontaneous rupture propagation
+                  governed by different constitutive laws with rake
+                  rotation allowed},
+  journal = {Annals of Geophysics},
+  volume = {48},
+  number = {2},
+  year = {2005},
+  doi =	 {10.4401/ag-3201},
+  abstract =	 {},
+}
+
+ at book{Aki:Richards:2002,
+  author =	 {Aki, K. and Richards, P.~G.},
+  title =	 {Quantitative {Seismology}},
+  year =	 2002,
+  publisher =	 {University Science Books},