[cig-commits] r16905 - in short/3D/PyLith/trunk: . doc/userguide/boundaryconditions doc/userguide/governingeqns doc/userguide/materials
brad at geodynamics.org
brad at geodynamics.org
Sat Jun 5 14:57:38 PDT 2010
Author: brad
Date: 2010-06-05 14:57:38 -0700 (Sat, 05 Jun 2010)
New Revision: 16905
Modified:
short/3D/PyLith/trunk/TODO
short/3D/PyLith/trunk/doc/userguide/boundaryconditions/boundaryconditions.lyx
short/3D/PyLith/trunk/doc/userguide/governingeqns/governingeqns.lyx
short/3D/PyLith/trunk/doc/userguide/materials/materials.lyx
Log:
Fixed some typos and formatting issues.
Modified: short/3D/PyLith/trunk/TODO
===================================================================
--- short/3D/PyLith/trunk/TODO 2010-06-05 19:42:09 UTC (rev 16904)
+++ short/3D/PyLith/trunk/TODO 2010-06-05 21:57:38 UTC (rev 16905)
@@ -16,6 +16,7 @@
MANUAL
adjustSolnLumped stuff
+ check indexing
Tutorials
3d/hex8 [Charles]
Modified: short/3D/PyLith/trunk/doc/userguide/boundaryconditions/boundaryconditions.lyx
===================================================================
--- short/3D/PyLith/trunk/doc/userguide/boundaryconditions/boundaryconditions.lyx 2010-06-05 19:42:09 UTC (rev 16904)
+++ short/3D/PyLith/trunk/doc/userguide/boundaryconditions/boundaryconditions.lyx 2010-06-05 21:57:38 UTC (rev 16905)
@@ -2390,17 +2390,17 @@
\begin_layout Standard
When we use a lumped system Jacobian matrix, we cannot lump the terms associated
with the Lagrange multipliers.
- 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.
- With the addition of the Lagrange multipliers the general expression for
- the residual at time
+ Instead, we formulate the Jacobian ignoring the contributions from the
+ Lagrange multipliers, and then adjust the solution after the solve to account
+ for their presence.
+ Including the Lagrange multipliers in the general expression for the residual
+ at time
\begin_inset Formula $t+\Delta t$
\end_inset
- is
+, we have
\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}
+r_{i}^{n}(t+\Delta t)=A_{ij}^{nm}(u_{j}^{m}(t)+du_{j}^{m}(t))+C_{ki}^{pn}(l_{k}^{p}(t)+dl_{k}^{p}(t)),\end{equation}
\end_inset
@@ -2422,7 +2422,8 @@
\end_inset
.
- When we formulate the residual, the values
+ When we solve the lumped system ignoring the Lagrange multipliers contributions
+ to the Jacobian, we formulate the residual assuming the values
\begin_inset Formula $du_{i}^{n}$
\end_inset
@@ -2430,19 +2431,20 @@
\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 $du_{i}^{n}$
+ are zero.
+ So our task is to determine the increment in the Lagrange multiplier,
+\begin_inset Formula $dl_{k}^{p}$
\end_inset
- and
-\begin_inset Formula $dl_{k}^{p}$
+, and the correction to the displacement increment,
+\begin_inset Formula $du_{i}^{n}$
\end_inset
- by setting the actual residual to be zero; thus, we have
+, and by setting the residual with all terms included to 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}
+A_{ij}^{nm}(u_{j}^{m}(t)+du_{j}^{m}(t))+C_{ki}^{pn}(l_{k}^{p}(t)+dl_{k}^{p}(t))=0\text{ subject to}\\
+C_{ij}^{pn}(u_{j}^{n}(t)+du_{j}^{n}(t))=d_{i}^{p}.\end{gather}
\end_inset
@@ -2454,39 +2456,43 @@
\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}
+r_{i}^{n}+A_{ij}^{nm}du_{j}^{m}+C_{ki}^{pn}dl_{k}^{p}=0\text{ subject to}\\
+C_{ij}^{pn}(u_{j}^{n}(t)+du_{j}^{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}
+r_{i}^{n-}+A_{ij}^{nm-}du_{j}^{m-}+R_{ki}^{pn}dl_{k}^{p}=0,\\
+r_{i}^{n+}+A_{ij}^{nm+}du_{j}^{m+}+R_{ki}^{pn}dl_{k}^{p}=0,\\
+R_{ij}^{pn}(u_{j}^{n+}+du_{j}^{n+}-u_{j}^{n-}-du_{j}^{n-})=d_{i}^{p}.\end{gather}
\end_inset
Solving the first two equations for
-\begin_inset Formula $du_{i}^{n-}$
+\begin_inset Formula $du_{j}^{m-}$
\end_inset
and
-\begin_inset Formula $du_{i}^{n+}$
+\begin_inset Formula $du_{j}^{m+}$
\end_inset
- and combining them with the third equation leads to
+ and combining them using 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}+(A_{ij}^{nm+})^{-1}\right)R_{ki}^{pn}dl_{k}^{p}=d_{i}^{p}-R_{ij}^{pn}(u_{j}^{n+}-u_{j}^{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.
+ so
+\begin_inset Formula $A_{ij}^{nm}$
+\end_inset
+
+ is the same for all components at a vertex.
As a result the matrix on the left hand side simplifies to
\begin_inset Formula \[
S_{ik}^{pn}=\delta_{ik}\left(\frac{1}{A^{nm+}}+\frac{1}{A^{nm-}}\right),\]
@@ -2495,20 +2501,31 @@
and
\begin_inset Formula \[
-dl_{k}^{p}=(S_{ik}^{pn})^{-1}\left(d_{i}^{p}-R_{ij}^{pn}(u_{i}^{n+}-u_{i}^{n-})+R_{ij}^{pn}\left((A_{ij}^{nm+})^{-1}r_{i}^{n+}-(A_{ij}^{nm-})^{-1}r_{i}^{n-}\right)\right).\]
+dl_{k}^{p}=(S_{ik}^{pn})^{-1}\left(d_{i}^{p}-R_{ij}^{pn}(u_{j}^{n+}-u_{j}^{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
+ time
+\begin_inset Formula $t$
+\end_inset
+
+ to time
+\begin_inset Formula $t+\Delta t$
+\end_inset
+
+, we can correct the value for the displacement increment from time
+\begin_inset Formula $t$
+\end_inset
+
+ to
+\begin_inset Formula $t+\Delta t$
+\end_inset
+
+ 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*}
+\Delta du_{j}^{n-}=(A_{ij}^{nm-})^{-1}C_{ki}^{pn}dl_{k}^{p}\text{ and}\\
+\Delta du_{j}^{n+}=-(A_{ij}^{nm+})^{-1}C_{ki}^{pn}dl_{k}^{p}.\end{gather*}
\end_inset
@@ -2611,7 +2628,7 @@
<features>
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
-<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="3.5in">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
Modified: short/3D/PyLith/trunk/doc/userguide/governingeqns/governingeqns.lyx
===================================================================
--- short/3D/PyLith/trunk/doc/userguide/governingeqns/governingeqns.lyx 2010-06-05 19:42:09 UTC (rev 16904)
+++ short/3D/PyLith/trunk/doc/userguide/governingeqns/governingeqns.lyx 2010-06-05 21:57:38 UTC (rev 16905)
@@ -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
@@ -1518,7 +1518,8 @@
of gravitational body forces and deformation on the overburden pressure.
In such cases we want to account for both rigid body motion and small strains.
The elasticity formulation in PyLith for small strains uses the Green-Lagrange
- strain tensor and the Second Piola-Kirchhoff stress tensor
+ strain tensor and the Second Piola-Kirchhoff stress tensor as is based
+ on the one presented by Bathe
\begin_inset CommandInset citation
LatexCommand cite
key "Bathe:1995"
Modified: short/3D/PyLith/trunk/doc/userguide/materials/materials.lyx
===================================================================
--- short/3D/PyLith/trunk/doc/userguide/materials/materials.lyx 2010-06-05 19:42:09 UTC (rev 16904)
+++ short/3D/PyLith/trunk/doc/userguide/materials/materials.lyx 2010-06-05 21:57:38 UTC (rev 16905)
@@ -4623,10 +4623,7 @@
status open
\begin_layout Plain Layout
-
-\end_layout
-
-\begin_layout Plain Layout
+\align center
\begin_inset Caption
\begin_layout Plain Layout
@@ -4829,10 +4826,6 @@
\end_layout
-\begin_layout Plain Layout
-
-\end_layout
-
\end_inset
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