[cig-commits] commit: Started flushing out the governing equations section.

[Mercurial]

changeset:   52:b11cd13e7c17
tag:         tip
user:        Brad Aagaard <baagaard at usgs.gov>
date:        Fri Aug 26 17:31:57 2011 -0700
files:       faultRup.tex
description:
Started flushing out the governing equations section.


diff -r 377e0db916b4 -r b11cd13e7c17 faultRup.tex
--- a/faultRup.tex	Thu Aug 25 11:38:27 2011 -0700
+++ b/faultRup.tex	Fri Aug 26 17:31:57 2011 -0700
@@ -3,8 +3,8 @@
 
 \usepackage{array}
 \usepackage{epsfig}
+\usepackage{rotating}
 \usepackage[centertags]{amsmath}
-\usepackage{rotating}
 \usepackage{color}
 
 \newcommand\brad[1]{{\color{red}\bf [BRAD: #1]}}
@@ -12,6 +12,7 @@
 \newcommand\charles[1]{{\color{green}\bf [CHARLES: #1]}}
 
 \newcommand{\refeqn}[1]{~(\ref{#1})}
+\newcommand{\tensor}[1]{\underline{#1}}
 
 \widowpenalty=999999
 \clubpenalty=999999
@@ -162,23 +163,62 @@ and verification of its implementation u
 and verification of its implementation using a few benchmarks.
 
 % ------------------------------------------------------------------
-\section{Model}\brad{Rough draft}
-  \begin{itemize}
-    \item Lagrange multipliers
-    \item Slip functions
-    \item Fault constitutive models
-  \end{itemize}
+\section{Numerical Model of Fault Slip}\brad{Rough draft}
+
+We solve the elasticity equation including inertial terms,
+\begin{gather}
+  \rho \frac{\partial^2\vec{u}}{\partial t^2} - \vec{f} 
+  - \nabla \cdot \tensor{\sigma} = \vec{0} \text{ in }V, \\
+  \tensor{\sigma} \cdot \vec{n} = \vec{T} \text{ on }S_T, \\
+  \vec{u} = \vec{u}_0 \text{ on }S_u, \\
+  \tensor{R} \cdot \left( \vec{u}^{+} - \vec{u}^{-}\right) = \vec{d}
+  \text{ on }S_f,
+\end{gather}
+where $\vec{u}$ is the displacement vector, $\rho$ is the mass
+density, $f$ is the body force vector, $\tensor{\sigma}$ is the Cauchy
+stress tensor, and $t$ is time. We specify tractions $\vec{T}$ on
+surface $S_f$, displacements $\vec{u_0}$ on surface $S_u$, and slip
+$\vec{d}$ on fault surface $S_f$ (we also consider the case of a fault
+constitutive model later in this section). The rotation matrix
+$\tensor{R}$ transforms vectors from the global coordinate system to
+the fault coordinate system. Because both $\vec{T}$ and $\vec{u}$ are
+vector quantities, there can be some spatial overlap of the surfaces
+$S_T$ and $S_u$; however, a degree of freedom cannot simultaneously be
+associated with both types of boundary conditions.
+
+Construct the weak form ...
+
+We express the trial solution and weighting function as linear
+combinations of basis functions ...
+
+We want to solve this equation for the unknown coefficients ...
+subject to ...
+
+Quasi-static
+
+Dynamic
+
+Impose fault slip using Lagrange multipliers ...
+
+
+Kinematic rupture model
+
+Slip-time functions
+
+Dynamic (spontaneous) rupture model
+
+Fault constitutive models
 
 % ------------------------------------------------------------------
-\section{Geometry Processing}\matt{Rough draft}
+\subsection{Finite-Element Mesh Processing}\matt{Rough draft}
   adjusting topology
 
 % ------------------------------------------------------------------
-\section{Discretization}\brad{Rough draft}
+\subsection{Fault Discretization}\brad{Rough draft}
   cohesive cells
 
 % ------------------------------------------------------------------
-\section{Solver}\matt{Rough draft}
+\subsection{Solver Customization}\matt{Rough draft}
   \begin{itemize}
     \item saddle point
     \item custom preconditioner
@@ -187,11 +227,9 @@ and verification of its implementation u
 % ------------------------------------------------------------------
 \section{Code Verification Benchmarks}
 \begin{itemize}
-  \begin{itemize}
   \item Savage and Prescott\charles{Rough draft}
   \item Spontaneous rupture 2-D\brad{Rough draft}
   \item Spontaneous rupture 3-D\brad{Rough draft}
-  \end{itemize}
 \end{itemize}
 
 % ------------------------------------------------------------------