[cig-commits] r16903 - short/3D/PyLith/trunk/doc/userguide/boundaryconditions

[brad at geodynamics.org]

Author: brad
Date: 2010-06-04 21:12:33 -0700 (Fri, 04 Jun 2010)
New Revision: 16903

Modified:
   short/3D/PyLith/trunk/doc/userguide/boundaryconditions/boundaryconditions.lyx
Log:
Added equations for adjusting lumped solution.

Modified: short/3D/PyLith/trunk/doc/userguide/boundaryconditions/boundaryconditions.lyx
===================================================================
--- short/3D/PyLith/trunk/doc/userguide/boundaryconditions/boundaryconditions.lyx	2010-06-05 03:17:59 UTC (rev 16902)
+++ short/3D/PyLith/trunk/doc/userguide/boundaryconditions/boundaryconditions.lyx	2010-06-05 04:12:33 UTC (rev 16903)
@@ -1,4 +1,4 @@
-#LyX 1.6.4 created this file. For more info see http://www.lyx.org/
+#LyX 1.6.5 created this file. For more info see http://www.lyx.org/
 \lyxformat 345
 \begin_document
 \begin_header
@@ -2393,10 +2393,126 @@
  Instead, we ignore the formulate the Jacobian ignoring the contributions
  from the Lagrange multipliers, and then adjust the solution after the solve
  to account for their presence.
-\end_layout
+ With the addition of the Lagrange multipliers the general expression for
+ the residual at time 
+\begin_inset Formula $t+\Delta t$
+\end_inset
 
-\begin_layout Standard
-ADD STUFF HERE
+ is
+\begin_inset Formula \begin{equation}
+r_{i}^{n}(t+\Delta t)=A_{ij}^{nm}(u_{i}^{n}(t)+du_{i}^{n}(t))+C_{ki}^{pn}(l_{k}^{p}(t)+dl_{k}^{p}(t)),\end{equation}
+
+\end_inset
+
+where we have written the displacements and Lagrange multipliers at time
+ 
+\begin_inset Formula $t+\Delta t$
+\end_inset
+
+ in terms of the values at time 
+\begin_inset Formula $t$
+\end_inset
+
+ and the increment from time 
+\begin_inset Formula $t$
+\end_inset
+
+ to 
+\begin_inset Formula $t+\Delta t$
+\end_inset
+
+.
+ When we formulate the residual, the values 
+\begin_inset Formula $du_{i}^{n}$
+\end_inset
+
+(t) and 
+\begin_inset Formula $dl_{k}^{p}(t)$
+\end_inset
+
+ are unknown and assumed to be zero.
+ So our task is to determine 
+\begin_inset Formula $du_{i}^{n}$
+\end_inset
+
+ and 
+\begin_inset Formula $dl_{k}^{p}$
+\end_inset
+
+ by setting the actual residual to be zero; thus, we have
+\begin_inset Formula \begin{gather}
+A_{ij}^{nm}(u_{i}^{n}(t)+du_{i}^{n}(t))+C_{ki}^{pn}(l_{k}^{p}(t)+dl_{k}^{p}(t))=0\text{ subject to}\\
+C_{ij}^{pn}(u_{i}^{n}(t)+du_{i}^{n}(t))=d_{i}^{p}.\end{gather}
+
+\end_inset
+
+Making use of the residual computed with 
+\begin_inset Formula $du_{i}^{n}(t)=0$
+\end_inset
+
+ and 
+\begin_inset Formula $dl_{k}^{p}(t)=0$
+\end_inset
+
+, we have
+\begin_inset Formula \begin{gather}
+r_{i}^{n}+A_{ij}^{nm}du_{i}^{n}+C_{ki}^{pn}dl_{k}^{p}=0\text{ subject to}\\
+C_{ij}^{pn}(u_{i}^{n}(t)+du_{i}^{n}(t))=d_{i}^{p}.\end{gather}
+
+\end_inset
+
+Explicitly writing the equations for the vertices on the negative and positive
+ sides of the fault yields
+\begin_inset Formula \begin{gather}
+r_{i}^{n-}+A_{ij}^{nm-}du_{i}^{n-}+R_{ki}^{pn}dl_{k}^{p}=0,\\
+r_{i}^{n+}+A_{ij}^{nm+}du_{i}^{n+}+R_{ki}^{pn}dl_{k}^{p}=0,\\
+R_{ij}^{pn}(u_{i}^{n+}+du_{i}^{n+}-u_{i}^{n-}-du_{i}^{n-})=d_{i}^{p}.\end{gather}
+
+\end_inset
+
+Solving the first two equations for 
+\begin_inset Formula $du_{i}^{n-}$
+\end_inset
+
+ and 
+\begin_inset Formula $du_{i}^{n+}$
+\end_inset
+
+ and combining them with the third equation leads to
+\begin_inset Formula \begin{multline}
+R_{ij}^{pn}\left((A_{ij}^{nm+})^{-1}+(A_{ij}^{nm+})^{-1}\right)R_{ki}^{pn}dl_{k}^{p}=d_{i}^{p}-R_{ij}^{pn}(u_{i}^{n+}-u_{i}^{n-})\\
++R_{ij}^{pn}\left((A_{ij}^{nm+})^{-1}r_{i}^{n+}-(A_{ij}^{nm-})^{-1}r_{i}^{n-}\right).\end{multline}
+
+\end_inset
+
+We do not allow overlap between the fault interface and the absorbing boundary,
+ so A_{ij}^{nm} is the same for all components at a vertex.
+ As a result the matrix on the left hand side simplifies to
+\begin_inset Formula \[
+S_{ik}^{pn}=\delta_{ik}\left(\frac{1}{A^{nm+}}+\frac{1}{A^{nm-}}\right),\]
+
+\end_inset
+
+and
+\begin_inset Formula \[
+dl_{k}^{p}=(S_{ik}^{pn})^{-1}\left(d_{i}^{p}-R_{ij}^{pn}(u_{i}^{n+}-u_{i}^{n-})+R_{ij}^{pn}\left((A_{ij}^{nm+})^{-1}r_{i}^{n+}-(A_{ij}^{nm-})^{-1}r_{i}^{n-}\right)\right).\]
+
+\end_inset
+
+Now that we know the value of the increment in the Lagrange multiplier from
+ time t to time t+
+\backslash
+Delta t, we can correct the determine the correct value for the displacement
+ increment from time t to t+
+\backslash
+Delta t using
+\begin_inset Formula \begin{gather*}
+\Delta du_{i}^{n-}=-(A_{ij}^{nm-})^{-1}C_{ki}^{pn}dl_{k}^{p}\text{ and}\\
+\Delta du_{i}^{n+}=-(A_{ij}^{nm+})^{-1}C_{ki}^{pn}dl_{k}^{p}.\end{gather*}
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Subsubsection