[cig-commits] commit: Fixed signs in orientation. Added special case for solution.

[Mercurial]

changeset:   33:ccfecef493da
tag:         tip
user:        Brad Aagaard <baagaard at usgs.gov>
date:        Sun Jan 31 10:26:51 2010 -0800
files:       lumpedsolver.tex
description:
Fixed signs in orientation. Added special case for solution.


diff -r a8b084dbb7e1 -r ccfecef493da lumpedsolver.tex
--- a/lumpedsolver.tex	Sun Jan 31 10:26:19 2010 -0800
+++ b/lumpedsolver.tex	Sun Jan 31 10:26:51 2010 -0800
@@ -74,9 +74,9 @@ so that we solve the system
   \vec{r}_0 = r(du=0,dl=0).
 \end{gather}
 For convenience we will drop the distinction between $r_o$ and $r$ and simply refer to $r_0$ as $r$, because we form the residual in the first iteration at a time step assuming $du=0$ and $dl=0$. Consider Lagrange multiplier vertex $k$ associated with the slip between conventional vertex $i$ on the ``positive'' side of the fault and conventional vertex $j$ on the negative side of the fault. The system of equations is in the form \begin{gather}
-  A_i du_i + C^T_{ki} dl_k = r_i, \\
-  A_j du_j - C^T_{kj} dl_k = r_j, \\
-  C_{ki} du_i - C_{kj} du_j = d_k - C_{ki} u_i + C_{kj} u_j, \\
+  A_i du_i - C^T_{ki} dl_k = r_i, \\
+  A_j du_j + C^T_{kj} dl_k = r_j, \\
+  -C_{ki} du_i + C_{kj} du_j = d_k + C_{ki} u_i - C_{kj} u_j, \\
   C_{ki} = C_{kj},
 \end{gather}
 where $A_i$ is a diagonal matrix (3x3 in 3-D) associated with Jacobian
@@ -86,13 +86,13 @@ Solving the first two equations for $du_
 Solving the first two equations for $du_i$ and $du_j$, respectively,
 and substituting them into the third equation, yields
 \begin{equation}
-  C_{ki}A_i^{-1}(r_i-C^T_{ki} dl_k)
-  - C_{kj}A_j^{-1}(r_j + C_{kj}^T dl_k) = d_k - C_{ki} u_i + C_{kj} u_j.
+  -C_{ki}A_i^{-1}(r_i+C^T_{ki} dl_k)
+  + C_{kj}A_j^{-1}(r_j + C_{kj}^T dl_k) = d_k + C_{ki} u_i - C_{kj} u_j.
 \end{equation}
 Isolating the term with $dl_k$, we have
 \begin{equation}
   \left( C_{ki} (A_i^{-1}+A_j^{-1}) C_{ki}^T \right) dl_k =
-  C_{ki} A_i^{-1} r_i - C_{kj} A_j^{-1} r_j - d_k + C_{ki} u_i - C_{kj} u_j
+  -C_{ki} A_i^{-1} r_i + C_{kj} A_j^{-1} r_j - d_k - C_{ki} u_i + C_{kj} u_j
 \end{equation}
 Letting
 \begin{equation}
@@ -100,16 +100,22 @@ Letting
 \end{equation}
 and taking advantage of the fact that $C_{ki} = C_{kj}$, we have
 \begin{equation}
-  dl_k = S^{-1} \left( C_{ki} (A_i^{-1} r_i - A_j^{-1} r_j + u_i-u_j) - d_k \right)
+  dl_k = S^{-1} \left( -C_{ki} (A_i^{-1} r_i - A_j^{-1} r_j + u_i-u_j) - d_k \right)
 \end{equation}
 The adjustment to the displacement increment (solution) to account for
 the Lagrange multipliers is
  \begin{gather}
-  \Delta du_i = -A_i^{-1} C_{ki}^T dl_k \\
-  \Delta du_j = +A_j^{-1} C_{kj}^T dl_k.
+  \Delta du_i = +A_i^{-1} C_{ki}^T dl_k \\
+  \Delta du_j = -A_j^{-1} C_{kj}^T dl_k.
 \end{gather}
 
-
+In the most general circumstances where A includes both intertial and
+damping terms,
+\begin{gather}
+  A_{ix} \neq A_{iy} \neq A_{iz} \\
+  A_{jx} \neq A_{jy} \neq A_{jz},
+\end{gather}
+so that the sensitivity matrix, S, is fully populated.
 \begin{equation}
   S = \left( \begin{array}{ccc}
       S_{pp} & S_{pq} & S_{pr} \\
@@ -171,6 +177,22 @@ where $pqr$ is the fault coordinate syst
 where $pqr$ is the fault coordinate system and $xyz$ is the global
 coordinate system.
 
+In the special case where A contains only intertial terms (e.g., there
+is no overlap between fault and absorbing dampers boundary
+conditions),
+\begin{gather}
+  A_{ix} = A_{iy} = A_{iz} \\
+  A_{jx} = A_{jy} = A_{jz},
+\end{gather}
+so that the sensitivity matrix, S, is diagonal with equal terms.
+\begin{gather}
+  S = \left( \begin{array}{ccc}
+      S_{pp} & 0 & 0 \\
+      0 & S_{qq} & 0 \\
+      0 & 0 & S_{rr} 
+    \end{array} \right), \\
+  S_{pp} = S_{qq} = S_{rr} = \frac{A_{ix} A_{jx}}{A_{ix} + A_{jx}}
+\end{gather}
 
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