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

[Mercurial]

changeset:   61:38890ae93959
tag:         tip
user:        Brad Aagaard <baagaard at usgs.gov>
date:        Wed Aug 31 16:16:36 2011 -0700
files:       faultRup.tex references.bib
description:
More work on solver section for dynamic simulations.


diff -r 23dce361d7dd -r 38890ae93959 faultRup.tex
--- a/faultRup.tex	Wed Aug 31 15:40:07 2011 -0700
+++ b/faultRup.tex	Wed Aug 31 16:16:36 2011 -0700
@@ -627,16 +627,17 @@ matrix.
 matrix.
 
 First, we eliminate the off-diagonal entries in each of the blocks of
-the matrix.  The current best available option for eliminating the
-off-diagonal terms formed during the integration of the inertial term
-focuses on choosing a set of orthogonal basis functions, such as the
-Legendre polynomials with Gauss-Lobatto-Lgendre quadrature points
+the matrix during the finite-element integrations.  The current best
+available option for eliminating the off-diagonal terms formed during
+the integration of the inertial term focuses on choosing a set of
+orthogonal basis functions, such as the Legendre polynomials with
+Gauss-Lobatto-Lgendre quadrature points
 \cite{Komatitsch:Vilotte:1998}, which naturally produces a diagonal
 matrix for each finite-element cell without introducing any additional
 approximations. Because the fault slip constraint term also involves
 integration of the products of the basis functions over
 lower-dimension cells, orthogonal basis functions also produces a
-diagonal matrix for this integration. 
+diagonal matrix for this integration.
 
 In contrast, the more traditional finite-element approach does
 introduce additional approximations by constructing a diagonal
@@ -649,14 +650,40 @@ it is for the original integral. The err
 it is for the original integral. 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 the wave propagation.
+consistent with resolving the wave propagation. Furthermore, in
+contrast to other approaches that choose basis functions or quadrature
+points that affect the accuracy of the elasticity terms as well, this
+approach only affects the accuracy of the terms involved in the
+Jacobian. For consistency in the formulation of the system of
+equations, these approximations are also applied to the inertial term
+and fault slip constraint term when computing the residual.
 
-\brad{TODO: Add equations for intertial term and fault slip constraint
-  for lumped Jacobian}
+\brad{TODO: Add equations for diagonal approximations of the intertial
+  term and fault slip constraint.}
 
 % Schur complement
 
-
+Second, we leverage the block structure of the off-diagonal blocks
+associated with the fault slip constraint in solving the system of
+equations. We compute an initial residual assuming the increment in
+the solution is zero (i.e., $d\vec{u}_m = \vec{0}$ and $d\vec{l}_p =
+\vec{0}$,
+\begin{equation}
+  \vec{r}^* = \begin{matrix} \vec{r}_m^* \\ \vec{r}_p^* \end{matrix} =
+  \begin{matrix} \vec{b}_m \\ \vec{b}_p \end{matrix}
+  - \begin{matrix}
+    \tensor{K} & \tensor{L}^T \\ \tensor{L} & 0
+  \end{matrix}
+  \begin{matrix} \vec{u}_m \\ \vec{l}_m \end{matrix}.
+\end{equation}
+ We compute an initial solution to the system of equations
+$\vec{u}_m^*}$ ignoring the off-diagonal blocks in the Jacobian and
+the increment in the Lagrange multipliers.
+\begin{equation}
+d\vec{u}_m^* = \tensor{K}^{-1} \vec{r}_m,
+\end{equation}
+taking advantage of the fact that we constructed $\tensor{K}$ so that
+it is diagonal.
 
 
 % Spontaneous rupture
diff -r 23dce361d7dd -r 38890ae93959 references.bib
--- a/references.bib	Wed Aug 31 15:40:07 2011 -0700
+++ b/references.bib	Wed Aug 31 16:16:36 2011 -0700
@@ -600,7 +600,7 @@
                   ± 10\%. Events M ≥ 6 have as much as a 60\%
                   probability of recurrence within 5 years due to the
                   clustering of small earthquakes in foreshocks and
-                  aftershocks. This probability drops to less than 15%
+                  aftershocks. This probability drops to less than 15\%
                   for M ≥ 7 events. Increasing gap time generally
                   increases conditional probability of earthquake
                   occurrence, but the effect is weak. For the MAT, the