[cig-commits] r16697 - in short/3D/PyLith/trunk: . doc/userguide/governingeqns
brad at geodynamics.org
brad at geodynamics.org
Wed May 12 21:33:40 PDT 2010
Author: brad
Date: 2010-05-12 21:33:40 -0700 (Wed, 12 May 2010)
New Revision: 16697
Modified:
short/3D/PyLith/trunk/TODO
short/3D/PyLith/trunk/doc/userguide/governingeqns/governingeqns.lyx
Log:
Updated TODO.
Modified: short/3D/PyLith/trunk/TODO
===================================================================
--- short/3D/PyLith/trunk/TODO 2010-05-13 03:31:31 UTC (rev 16696)
+++ short/3D/PyLith/trunk/TODO 2010-05-13 04:33:40 UTC (rev 16697)
@@ -10,12 +10,15 @@
Need to link isJacobianSymmetric() in Integrator with materials.
Need to pass hint when creating Jacobian.
+* Use node set names in MeshIOCubit.
+ ns_names
+
+* Friction template
+ viscous (velocity proportional friction)
+
* Output
MATT - Will try moving std::ostringstream outside of loop
-* Distribute
- MATT - needs to update calls to use new distribute
-
* Better preconditioning
+ Need settings for Schur complement
@@ -25,6 +28,8 @@
MATT is thinking about how to setup Schur complement settings
(recursive specification of field splits).
+ Need field split working for both SolverLinear and SolverNonlinear.
+
* Friction
+ Add stuff to manual
Modified: short/3D/PyLith/trunk/doc/userguide/governingeqns/governingeqns.lyx
===================================================================
--- short/3D/PyLith/trunk/doc/userguide/governingeqns/governingeqns.lyx 2010-05-13 03:31:31 UTC (rev 16696)
+++ short/3D/PyLith/trunk/doc/userguide/governingeqns/governingeqns.lyx 2010-05-13 04:33:40 UTC (rev 16697)
@@ -1,4 +1,4 @@
-#LyX 1.6.4.1 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
@@ -59,8 +59,8 @@
\begin_layout Standard
We present here a brief derivation of the equations for both quasi-static
and dynamic computations.
- Since the general equations are the same (except for the absence of acceleratio
-n terms in the quasi-static case), we first derive these equations.
+ Since the general equations are the same (except for the absence of inertial
+ terms in the quasi-static case), we first derive these equations.
We then present solution methods for each specific case.
In all of our derivations, we use the notation described in Table
\begin_inset CommandInset ref
@@ -610,7 +610,7 @@
\begin_inset Formula $S_{f}$
\end_inset
-.
+ (we will consider the case of fault constitutive models in a later section).
Note that since both
\begin_inset Formula $T_{i}$
\end_inset
@@ -767,7 +767,7 @@
\begin_layout Standard
We formulate a set of algebraic equations using Galerkin's method.
- We consider a trial solution,
+ We consider (1) a trial solution,
\begin_inset Formula $\vec{u}$
\end_inset
@@ -776,7 +776,7 @@
\begin_inset Formula $S_{u}$
\end_inset
-, and a weighting function,
+, and (2) a weighting function,
\begin_inset Formula $\vec{\phi}$
\end_inset
@@ -807,16 +807,6 @@
We construct the weak form by computing the dot product of the wave equation
and weighting function and setting the integral over the domain to zero:
-\begin_inset Note Greyedout
-status open
-
-\begin_layout Plain Layout
-Add fault constraint
-\end_layout
-
-\end_inset
-
-
\begin_inset Formula \begin{gather}
\int_{V}\left(\sigma_{ij,j}+f_{i}-\rho\ddot{u}_{i}\right)\phi_{i}\, dV=0\text{, or }\\
\int_{V}\sigma_{ij,j}\phi_{i}\: dV+\int_{V}f_{i}\phi_{i}\: dV-\int_{V}\rho\ddot{u}_{i}\phi_{i}\: dV=0.\end{gather}
@@ -842,16 +832,6 @@
\end_inset
Substituting into the weak form gives
-\begin_inset Note Greyedout
-status open
-
-\begin_layout Plain Layout
-Add fault constraint
-\end_layout
-
-\end_inset
-
-
\begin_inset Formula \begin{equation}
-\int_{V}\sigma_{ij}\phi_{i,j}\, dV+\int_{S}\sigma_{ij}\phi_{i}n_{i}\, dS+\int_{V}f_{i}\phi_{i}\, dV-\int_{V}\rho\ddot{u}_{i}\phi_{i}\, dV=0.\end{equation}
@@ -870,17 +850,11 @@
\begin_inset Formula $S_{u}$
\end_inset
-,
-\begin_inset Note Greyedout
-status open
-
-\begin_layout Plain Layout
-Add fault constraint
-\end_layout
-
+ (we will
+\begin_inset Formula $S_{f}$
\end_inset
-
+ in section ?? [TODO]),
\begin_inset Formula \begin{equation}
-\int_{V}\sigma_{ij}\phi_{i,j}\, dV+\int_{S_{T}}\sigma_{ij}\phi_{i}n_{i}\, dS+\int_{S_{u}}\sigma_{ij}\phi_{i}n_{i}\, dS+\int_{V}f_{i}\phi_{i}\, dV-\int_{V}\rho\ddot{u}_{i}\phi_{i}\, dV=0,\end{equation}
@@ -894,26 +868,48 @@
\end_inset
so that the equation reduces to
-\begin_inset Note Greyedout
-status open
+\begin_inset Formula \begin{equation}
+-\int_{V}\sigma_{ij}\phi_{i,j}\: dV+\int_{S_{T}}T_{i}\phi_{i}\, dS+\int_{V}f_{i}\phi_{i}\, dV-\int_{V}\rho\ddot{u}_{i}\phi_{i}\, dV=0.\label{eq:elasticity:integral}\end{equation}
-\begin_layout Plain Layout
-Add fault constraint
-\end_layout
+\end_inset
+Now,
+\begin_inset Formula $\sigma_{ij}\phi_{i,j}$
\end_inset
+ is a scalar, so it is symmetric,
+\begin_inset Formula \begin{equation}
+\sigma_{ij}\phi_{i,j}=\sigma_{ji}\phi_{j,i},\end{equation}
+\end_inset
+
+and we know that
+\begin_inset Formula $\sigma_{ij}$
+\end_inset
+
+ is symmetric, so
\begin_inset Formula \begin{equation}
--\int_{V}\sigma_{ij}\phi_{i,j}\: dV+\int_{S_{T}}T_{i}\phi_{i}\, dS+\int_{V}f_{i}\phi_{i}\, dV-\int_{V}\rho\ddot{u}_{i}\phi_{i}\, dV=0.\label{eq:elasticity:integral}\end{equation}
+\sigma_{ij}\phi_{i,j}=\sigma_{ij}\phi_{j,i},\end{equation}
\end_inset
+which means
+\begin_inset Formula \begin{equation}
+\phi_{i,j}=\phi_{j,i},\end{equation}
+
+\end_inset
+
+which we can write as
+\begin_inset Formula \begin{equation}
+\phi_{i,j}=\frac{1}{2}(\phi_{i,j}+\phi_{j,i}).\end{equation}
+
+\end_inset
+
We express the trial solution and weighting function as linear combinations
of basis functions,
\begin_inset Formula \begin{gather*}
-u=\sum_{m}a_{i}^{m}N^{m},\\
-\phi=\sum_{n}c_{i}^{n}N^{n}.\end{gather*}
+u_{i}=\sum_{m}a_{i}^{m}N^{m},\\
+\phi_{i}=\sum_{n}c_{i}^{n}N^{n}.\end{gather*}
\end_inset
@@ -934,8 +930,8 @@
Substituting in the expressions for the trial solution and weighting function
yields
\begin_inset Formula \begin{gather*}
--\int_{V}\sigma_{ij}\sum_{m}c_{i}^{m}N_{,j}^{m}\: dV+\int_{S_{T}}T_{i}\sum_{m}c_{i}^{m}N^{m}\, dS+\int_{V}f_{i}\sum_{m}c_{i}^{m}N^{m}\, dV-\int_{V}\rho\sum_{n}\ddot{a}_{i}^{n}N^{n}\sum_{m}c_{i}^{m}N^{m}\ dV=0,\text{ or}\\
-\sum_{m}c_{i}^{m}(-\int_{V}\sigma_{ij}N_{,j}^{m}\: dV+\int_{S_{T}}T_{i}N^{m}\, dS+\int_{V}f_{i}N^{m}\, dV-\int_{V}\rho\sum_{n}\ddot{a}_{i}^{n}N^{n}N^{m}\ dV)=0.\end{gather*}
+-\int_{V}\sigma_{ij}\sum_{n}c_{i}^{n}N_{,j}^{n}\: dV+\int_{S_{T}}T_{i}\sum_{n}c_{i}^{n}N^{n}\, dS+\int_{V}f_{i}\sum_{n}c_{i}^{n}N^{n}\, dV-\int_{V}\rho\sum_{m}\ddot{a}_{i}^{m}N^{m}\sum_{n}c_{i}^{n}N^{n}\ dV=0,\text{ or}\\
+\sum_{n}c_{i}^{n}(-\int_{V}\sigma_{ij}N_{,j}^{n}\: dV+\int_{S_{T}}T_{i}N^{n}\, dS+\int_{V}f_{i}N^{n}\, dV-\int_{V}\rho\sum_{m}\ddot{a}_{i}^{m}N^{m}N^{n}\ dV)=0.\end{gather*}
\end_inset
@@ -950,7 +946,7 @@
\begin_inset Formula \begin{equation}
--\int_{V}\sigma_{ij}N_{,j}^{n}\: dV+\int_{S_{T}}T_{i}N^{n}\, dS+\int_{V}f_{i}N^{n}\, dV-\int_{V}\rho\sum_{m}\ddot{a}_{i}^{m}N^{m}N^{n}\ dV=0.\label{eq:elasticity:integral-1}\end{equation}
+-\int_{V}\sigma_{ij}N_{,j}^{n}\: dV+\int_{S_{T}}T_{i}N^{n}\, dS+\int_{V}f_{i}N^{n}\, dV-\int_{V}\rho\sum_{m}\ddot{a}_{i}^{m}N^{m}N^{n}\ dV=\vec{0}.\label{eq:elasticity:integral-1}\end{equation}
\end_inset
@@ -958,10 +954,30 @@
\begin_inset Formula $a_{i}^{m}$
\end_inset
-.
-
+ subject to
\end_layout
+\begin_layout Standard
+
+\family roman
+\series medium
+\shape up
+\size normal
+\emph off
+\bar no
+\noun off
+\color none
+\begin_inset Formula \begin{gather*}
+\sigma_{ij,j}+f_{i}=\rho\ddot{u_{i}}\text{ in }V,\\
+\sigma_{ij}n_{j}=T_{i}\text{ on }S_{T},\\
+u_{i}=u_{i}^{o}\text{ on }S_{u},\text{ and}\\
+R_{ki}(u_{i}^{+}-u_{i}^{-})=d_{k}\text{ on }S_{f},\end{gather*}
+
+\end_inset
+
+
+\end_layout
+
\begin_layout Subsection
Old
\end_layout
@@ -1000,16 +1016,6 @@
\end_inset
Substituting into the first term gives
-\begin_inset Note Greyedout
-status open
-
-\begin_layout Plain Layout
-Add fault constraint
-\end_layout
-
-\end_inset
-
-
\begin_inset Formula \begin{equation}
-\int_{V}\frac{1}{2}\sigma_{ij}\left(\phi_{i,j}+\phi_{j,i}\right)\, dV+\int_{S}\sigma_{ij}\phi_{i}n_{i}\, dS+\int_{V}f_{i}\phi_{i}\, dV-\int_{V}\rho\ddot{u}_{i}\phi_{i}\, dV=0.\end{equation}
@@ -1044,7 +1050,7 @@
,
\begin_inset Formula \begin{equation}
--\int_{V}\sigma_{ij}(t+\Delta t)N_{,j}^{n}\: dV+\int_{S_{T}}T_{i}(t+\Delta t)N^{n}\, dS+\int_{V}f_{i}(t+\Delta t)N^{n}\, dV=\vec{0}.\label{eq:elasticity:integral:t+dt}\end{equation}
+-\int_{V}\sigma_{ij}(t+\Delta t)N_{,j}^{n}\: dV+\int_{S_{T}}T_{i}(t+\Delta t)N^{n}\, dS+\int_{V}f_{i}(t+\Delta t)N^{n}\, dV=\vec{0}.\label{eq:elasticity:integral:quasistatic}\end{equation}
\end_inset
@@ -1059,7 +1065,7 @@
We employ numerical quadrature in the finite-element discretization and
replace the integrals with sums over the cells and quadrature points,
\begin_inset Formula \[
-R_{i}^{n}=-\sum_{\text{vol cells}}\sum_{\text{quad pts}}\sigma_{ij}(x_{q},t+\Delta t)N^{n}(x_{q})\: w_{q}|J_{cell}(x_{q})|+\sum_{\text{vol cells}}\sum_{\text{quad pt}s}f_{i}(x_{q},t+\Delta t)N^{n}(x_{q})\, w_{q}|J_{cell}(x_{q})|+\sum_{\text{tract cells}}\sum_{\text{quad pts}}T_{i}(x_{q},t+\Delta t)N^{n}(x_{q})\, w_{q}|J_{cell}(x_{q})|,\]
+R_{i}^{n}=-\sum_{\text{vol cells}}\sum_{\text{quad pts}}\sigma_{ij}(x_{q},t+\Delta t)N_{,j}^{n}(x_{q})\: w_{q}|J_{cell}(x_{q})|+\sum_{\text{vol cells}}\sum_{\text{quad pt}s}f_{i}(x_{q},t+\Delta t)N^{n}(x_{q})\, w_{q}|J_{cell}(x_{q})|+\sum_{\text{tract cells}}\sum_{\text{quad pts}}T_{i}(x_{q},t+\Delta t)N^{n}(x_{q})\, w_{q}|J_{cell}(x_{q})|,\]
\end_inset
@@ -1091,7 +1097,7 @@
Substituting into equation
\begin_inset CommandInset ref
LatexCommand eqref
-reference "eq:elasticity:integral:dynamic:t"
+reference "eq:elasticity:integral:quasistatic"
\end_inset
@@ -1111,6 +1117,12 @@
\begin_inset Formula \[
+\int_{V}d\sigma_{ij}(t)N_{,j}^{n}\ dV=-\int_{V}\sigma_{ij}(t)N_{,j}^{n}\: dV+\int_{S_{T}}T_{i}(t+\Delta t)N^{n}\, dS+\int_{V}f_{i}(t+\Delta t)N^{n}\, dV.\]
+
+\end_inset
+
+
+\begin_inset Formula \[
\int_{V}\frac{1}{2}d\sigma_{ij}(t)(\phi_{i,j}+\phi_{j,i})\: dV=-\int_{V}\frac{1}{2}\sigma_{ij}(t)(\phi_{i,j}+\phi_{j,i})\: dV+\int_{S_{T}}T_{i}(t+\Delta t)\phi_{i}\, dS+\int_{V}f_{i}(t+\Delta t)\phi_{i}\, dV.\]
\end_inset
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