[cig-commits] commit: Small edits.

[Mercurial]

changeset:   168:443f4b2a5a50
tag:         tip
user:        Brad Aagaard <baagaard at usgs.gov>
date:        Thu Feb 07 09:12:09 2013 -0800
files:       faultRup.tex
description:
Small edits.


diff -r f66e7a77e6a8 -r 443f4b2a5a50 faultRup.tex
--- a/faultRup.tex	Thu Feb 07 09:00:04 2013 -0800
+++ b/faultRup.tex	Thu Feb 07 09:12:09 2013 -0800
@@ -351,7 +351,7 @@ this effective plastic strain and the el
 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
+approach and the domain decomposition approach in which the Lagrange
 multipliers and the constraint 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
@@ -413,7 +413,7 @@ We want to solve these equations for the
 We want to solve these equations for the coefficients $\bm{u}_n$
 and $\bm{l}_p$ subject to $\bm{u} = \bm{u}_0 \text{ on
 }S_u$. When we prescribe the slip, we specify $\bm{d}$ on $S_f$,
-and when we use a fault constitutive model, we specify how the
+and when we use a fault constitutive model we specify how the
 Lagrange multipliers $\bm{l}$ depend on the fault slip, slip rate,
 and state variables.
 
@@ -445,9 +445,9 @@ time-dependence only enters through the 
 time-dependence only enters through the constitutive models and the
 loading conditions. As a result, the quasi-static simulations are a
 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 $t+\Delta t$,
+properties and boundary conditions. The temporal accuracy 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}
@@ -479,7 +479,7 @@ with parallel processing
 with parallel processing
 \citep{PETSC:efficient,PETSc:manual}. In solving the
 system, we compute the residual (i.e., $\bm{r} = \bm{b} -
-\bm{A} \cdot \bm{u}$ and the Jacobian of the system
+\bm{A} \cdot \bm{u}$) and the Jacobian of the system
 ($\bm{A}$). In our case the solution is $\bm{u} =
 \left( \begin{smallmatrix} \bm{u}_n \\
     \bm{l}_n \end{smallmatrix} \right)$, and the residual is
@@ -515,7 +515,7 @@ This matches the tangent stiffness matri
 This matches the tangent stiffness matrix in conventional solid
 mechanics finite-element formulations. In computing the residual, we
 use the expression given in equation~(\ref{eqn:residual:elasticity})
-with an implementation for infinitesimal strain and an implementation
+with one implementation for infinitesimal strain and another implementation
 for small strain and rigid body motion. Following a similar procedure,
 we find the portion of the Jacobian associated with the constraints,
 equation~(\ref{eqn:quasi-static:residual:fault}), is
@@ -556,14 +556,13 @@ associated with the Lagrange multipliers
 
 The matrix $\bm{L}$ defined in
 equation~(\ref{eqn:jacobian:constraint}) is spectrally equivalent to
-the identity, because it just involves integration of products of the
-basis functions. This makes the traditional LBB stability criterion
+the identity, because it involves integration of products of the basis
+functions. This makes the traditional LBB stability criterion
 \citep{Brenner:Scott:2008} trivial to satisfy by choosing the space of
 Lagrange multipliers to be exactly the space of displacements,
-restricted to the fault. In simple terms to specify the problem we
-need to know the distance between any pair of vertices spanning the
-fault, which can be expressed as a relative displacement (i.e., fault
-slip).
+restricted to the fault. This means we simly need to know the distance
+between any pair of vertices spanning the fault, which can be
+expressed as a relative displacement, i.e., fault slip.
 
 % ------------------------------------------------------------------
 \subsection{Dynamic Simulations}