[cig-commits] commit: Worked on friction notes.

[Mercurial]

changeset:   35:a6f656b56869
tag:         tip
user:        Brad Aagaard <baagaard at usgs.gov>
date:        Mon Mar 29 17:35:40 2010 -0700
files:       friction.tex
description:
Worked on friction notes.


diff -r 23bf193950cd -r a6f656b56869 friction.tex
--- a/friction.tex	Mon Mar 15 11:08:28 2010 -0700
+++ b/friction.tex	Mon Mar 29 17:35:40 2010 -0700
@@ -240,23 +240,69 @@ We compute the sensitivity starting with
 \begin{gather}
   A u + C^T l = b \\
   A \partial u + C^T \partial l = 0 \\
-  \partial u = -A^{-1} C^T \partial l.
+  A \partial u = -C^T \partial l.
 \end{gather}
+
+\subsubsection{Nondiagonal $A$}
+
+In the case where $A$ is sparse matrix with off-diagonal terms, we solve
+two linear problems to compute the change in the slip field resulting
+from the change in Lagrange multiplier values associated with the
+friction criterion. We decompose $A$ into
+
+\begin{equation}
+  A = 
+  \left( \begin{array}{ccc} 
+      A_0 & A_1 & A_2  \\
+      A_3 & A_i & 0  \\
+      A_4 & 0 & A_j  \\
+    \end{array} \right),
+\end{equation}
+where $A_i$ is associated with pairs of degrees of freedom on the
+fault with both degrees
+of freedom on the ``negative'' side of the fault, $A_j$ is associated
+with pairs of degrees of freedom on the fault with both degrees of freedom on the
+``positive'' side of the fault, and $A_0$, $A_1$, $A_2$, $A_3$, and
+$A_4$ are associated with pairs of degrees of freedom where at least
+one degree of freedom is not on the fault. By design $A$ does not
+contain entries coupling the two sides of the fault. Using this
+decomposition of $A$, we solve two linear problems,
+\begin{gather}
+  A_i \partial u_i = C^T \partial l \\
+  A_j \partial u_j = -C^T \partial l,
+\end{gather}
+and use the relation between displacement and slip
+\begin{equation}
+  \matrix{C} \vec{\partial u} = \vec{\partial d},
+\end{equation}
+to compute the change in slip associated with the change in the
+Lagrange multipliers.
+
+These two linear problems involve only the degrees of freedom
+associated with the cohesive cells. Furthermore, we associate the
+entries of $A_i$ and $A_j$ with the degrees of freedom corresponding
+to the Lagrange multplier vertices rather than the degrees of freedom
+corresponding to the vertices on the ``positive'' and ``negative''
+sides of the fault. This means the structure (i.e., nonzero pattern)
+of the matrices is defined by the fault mesh, and we can use a single
+sparse matrix and extract the values from $A$ for either $A_i$ or
+$A_j$ as needed.
+
+\subsubsection{Diagonal $A$}
+
+In the case where $A$ is a diagonal matrix, we can solve the
+sensitivity equation directly,
+\begin{equation}
+  \partial u = -C A^{-1} C^T \partial l.
+\end{equation}
 Premultiplying by $\matrix{C}$ leads to
 \begin{equation}
   \partial d = -C A^{-1} C^T \partial l.
 \end{equation}
-Because we do not want to compute $A^{-1}$, we use the diagonal terms
-of $A$ to estimate $A^{-1}$,
-\begin{equation}
-  \partial d \approx -C A_\mathit{diagonal}^{-1} C^T \partial l.
-\end{equation}
-Note that this is not an approximation when $A$ is
-diagonal.
 
 For Lagrange vertex $k$, conventional vertex $i$ on the
-``positive'' side of the fault, and conventional vertex $j$ on the
-``negative'' side of the fault, we have
+``negative side of the fault, and conventional vertex $j$ on the
+``positive side of the fault, we have
 \begin{equation}
   \left( \begin{array}{c} 
       \partial d_{kp} \\ \partial d_{kq} \\ \partial d_{kr}