[cig-commits] commit: More work on dynamic solver section.

[Mercurial]

changeset:   65:736e9ef822bf
tag:         tip
user:        Brad Aagaard <baagaard at usgs.gov>
date:        Thu Sep 01 12:58:29 2011 -0700
files:       faultRup.tex
description:
More work on dynamic solver section.


diff -r 77811eb14719 -r 736e9ef822bf faultRup.tex
--- a/faultRup.tex	Thu Sep 01 10:57:29 2011 -0700
+++ b/faultRup.tex	Thu Sep 01 12:58:29 2011 -0700
@@ -652,7 +652,23 @@ finite-element cell, we construct a diag
 finite-element cell, we construct a diagonal approximation of the
 integral such that the action on rigid body motion is the same for the
 diagonal approximation of the integral as it is for the original
-integral. The errors associated with this approximation are small as
+integral,
+\begin{equation}
+  \tensor{A} \cdot \vec{u}_\mathrm{rigid} =
+  \tensor{A}_\mathit{diagonal} \vec{u}_\mathrm{rigid}.
+\end{equation}
+Expressing the diagonal block of the Jacobian matrix as a vector and
+the matrix of basis functions as a vector we have,
+\begin{equation}
+  \tensor{A}  = \int_\Omega \tensor{N}^T \tensor{N} \, d\Omega \rightarrow
+  \tensor{A}_\mathit{diagonal} = \int_\Omega \vec{N} \sum_i N_i \, d\Omega,
+\end{equation}
+where $N_i$ is the scalar basis function for degree of freedom $i$ and
+$\Omega$ may be the domain volume (as in the case of the inertial
+term) or a boundary (as in the case of the fault slip constraint
+term).
+
+The errors associated with this approximation are small as
 long as the deformation occurs at length scales significantly larger
 than the discretization size, which is consistent with resolving
 seismic wave propagation accurately. Furthermore, in contrast to other
@@ -664,8 +680,6 @@ equations, these approximations are also
 equations, these approximations are also applied to the inertial term
 and fault slip constraint term when computing the residual.
 
-\brad{TODO: Add equations for diagonal approximations of the inertial
-  term and fault slip constraint.}
 
 % Schur complement