# [cig-commits] commit: Started revising.

Mercurial hg at geodynamics.org
Mon Jan 14 13:48:31 PST 2013

changeset:   155:3041d51e7728
tag:         tip
user:        Brad Aagaard <baagaard at usgs.gov>
date:        Mon Jan 14 13:48:28 2013 -0800
files:       faultRup.tex
description:
Started revising.

diff -r a42afc877c82 -r 3041d51e7728 faultRup.tex
--- a/faultRup.tex	Tue Dec 18 08:02:44 2012 -0500
+++ b/faultRup.tex	Mon Jan 14 13:48:28 2013 -0800
@@ -222,44 +222,50 @@ We solve the elasticity equation includi
We solve the elasticity equation including inertial terms,
\begin{linenomath*}\begin{gather}
\rho \frac{\partial^2\bm{u}}{\partial t^2} - \bm{f}
-  - \bm{\nabla} \cdot \bm{\sigma} = \bm{0} \text{ in }V, \\
+  - \pmb{\nabla} \cdot \bm{\sigma} = \bm{0} \text{ in }V, \\
+  \label{eqn:bc:Neumann}
\bm{\sigma} \cdot \bm{n} = \bm{T} \text{ on }S_T, \\
+  \label{eqn:bc:Dirichlet}
\bm{u} = \bm{u}_0 \text{ on }S_u, \\
\bm{d} - (\bm{u}_{+} - \bm{u}_{-}) = \bm{0}
\text{ on }S_f, \label{eqn:fault:disp}
\end{gather}\end{linenomath*}
-where $\bm{u}$ is the displacement vector, $\rho$ is the mass
-density, $\bm{f}$ is the body force vector, $\bm{\sigma}$ is the
-Cauchy stress tensor, and $t$ is time. We specify tractions $\bm{T}$
-on surface $S_T$, displacements $\bm{u_0}$ on surface $S_u$, and slip
-$\bm{d}$ on fault surface $S_f$, where the tractions and fault slip
-are in global coordinates. Because both $\bm{T}$ and $\bm{u}$ are vector
-quantities, there can be some spatial overlap of the surfaces $S_T$
-and $S_u$; however, a degree of freedom at any location cannot be
-associated with both types of boundary conditions simultaneously.
+where $\bm{u}$ is the displacement vector, $\rho$ is the mass density,
+$\bm{f}$ is the body force vector, $\bm{\sigma}$ is the Cauchy stress
+tensor, and $t$ is time. We specify tractions $\bm{T}$ on surface
+$S_T$, displacements $\bm{u_0}$ on surface $S_u$, and slip $\bm{d}$ on
+fault surface $S_f$, where the tractions and fault slip are in global
+coordinates. Because both $\bm{T}$ and $\bm{u}$ are vector quantities,
+there can be some spatial overlap of the surfaces $S_T$ and $S_u$;
+however, a degree of freedom at any location cannot be associated with
+both prescribed displacements (Dirichlet) and traction (Neumann)
+boundary conditions simultaneously.

Following a conventional finite-element formulation (ignoring the
fault surface for a moment), we construct the weak form by taking the
dot product of the governing equation with a weighting function and
setting the integral over the domain equal to zero,
\begin{linenomath*}$$- \int_{V} \bm{\phi} \cdot - \left( \bm{\nabla} \cdot \bm{\sigma} + \bm{f} - + \int_{V} \pmb{\phi} \cdot + \left( \pmb{\nabla} \cdot \bm{\sigma} + \bm{f} - \rho\frac{\partial^{2}\bm{u}}{\partial t^{2}} \right) \, dV=0.$$\end{linenomath*}
-The weighting function $\bm{\phi}$ is a piecewise differentiable vector
-field with $\bm{\phi} = \bm{0}$ on $S_u$. After some algebra and
-use of the boundary conditions, we have
+The weighting function $\pmb{\phi}$ is a piecewise differentiable
+vector field with $\pmb{\phi} = \bm{0}$ on $S_u$. After some algebra
+and use of the boundary conditions (equations~(\ref{eqn:bc:Neumann})
+and~(\ref{eqn:bc:Dirichlet})), we have
\begin{linenomath*}$$\begin{split} - - \int_{V} \nabla \bm{\phi} : \bm{\sigma} \, dV - + \int_{S_T} \bm{\phi} \cdot \bm{T} \, dS - + \int_{V} \bm{\phi} \cdot \bm{f} \, dV \\ - - \int_{V} \bm{\phi} \cdot \rho \frac{\partial^{2}\bm{u}}{\partial t^{2}} \, dV - =0. + - \int_{V} \nabla \pmb{\phi} : \bm{\sigma} \, dV + + \int_{S_T} \pmb{\phi} \cdot \bm{T} \, dS + + \int_{V} \pmb{\phi} \cdot \bm{f} \, dV \\ + - \int_{V} \pmb{\phi} \cdot \rho \frac{\partial^{2}\bm{u}}{\partial t^{2}} \, dV + =0, \end{split}$$\end{linenomath*}
+where $\nabla \pmb{\phi} : \bm{\sigma}$ is the double inner product of
+the gradient of the weighting function and the stress tensor.

Using a domain decomposition approach, we consider the fault surface
as an interior boundary between two domains as shown in
@@ -269,9 +275,9 @@ the vector from the negative side of the
the vector from the negative side of the fault to the positive side of
the fault. Slip on the fault is the displacement of the positive side
relative to the negative side. Slip on the fault also corresponds to
-equal and opposite tractions on the positive and negative sides of the
-fault, which we impose using Lagrange multipliers with $\bm{l}_{+} -- \bm{l}_{-} = 0$.
+equal and opposite tractions on the positive ($\bm{l_{+}}$) and negative
+($\bm{l_{-}}$) sides of the fault, which we impose using Lagrange
+multipliers with $\bm{l}_{+} - \bm{l}_{-} = 0$.

Recognizing that the tractions on the fault surface are analogous to
@@ -279,12 +285,12 @@ the Lagrange multipliers (fault traction
the Lagrange multipliers (fault tractions) over the fault surface,
\begin{linenomath*}$$\begin{split} - - \int_{V} \nabla\bm{\phi} : \bm{\sigma} \, dV - + \int_{S_T} \bm{\phi} \cdot \bm{T} \, dS - - \int_{S_{f^+}} \bm{\phi} \cdot \bm{l} \, dS \\ - + \int_{S_{f^-}} \bm{\phi} \cdot \bm{l} \, dS - + \int_{V} \bm{\phi} \cdot \bm{f} \, dV - - \int_{V} \bm{\phi} \cdot \rho \frac{\partial^{2}\bm{u}}{\partial t^{2}} \, dV + - \int_{V} \nabla\pmb{\phi} : \bm{\sigma} \, dV + + \int_{S_T} \pmb{\phi} \cdot \bm{T} \, dS + - \int_{S_{f^+}} \pmb{\phi} \cdot \bm{l} \, dS \\ + + \int_{S_{f^-}} \pmb{\phi} \cdot \bm{l} \, dS + + \int_{V} \pmb{\phi} \cdot \bm{f} \, dV + - \int_{V} \pmb{\phi} \cdot \rho \frac{\partial^{2}\bm{u}}{\partial t^{2}} \, dV =0. \end{split}$$\end{linenomath*}
@@ -295,29 +301,29 @@ taking the dot product of the constraint
taking the dot product of the constraint equation with the weighting
function and setting the integral over the fault surface to zero,
\begin{linenomath*}$$- \int_{S_f} \bm{\phi} \cdot + \int_{S_f} \pmb{\phi} \cdot \left(\bm{d} - \bm{u}_{+} + \bm{u}_{-} \right) \, dS = 0.$$\end{linenomath*}

-We express the weighting function $\bm{\phi}$, trial solution
+We express the weighting function $\pmb{\phi}$, trial solution
$\bm{u}$, Lagrange multipliers $\bm{l}$, and fault slip $\bm{d}$ as
linear combinations of basis functions,
\begin{linenomath*}\begin{gather}
-\bm{\phi} = \sum_{m} \bm{a}_m N_m, \\
+\pmb{\phi} = \sum_{m} \bm{a}_m N_m, \\
\bm{u} = \sum_{n} \bm{u}_n N_n, \\
\bm{l} = \sum_{p} \bm{l}_p N_p, \\
\bm{d} = \sum_{p} \bm{d}_p N_p.
\end{gather}\end{linenomath*}
Because the weighting function is zero on $S_u$, the number of basis
functions for the trial solution $\bm{u}$ is generally greater than
-the number of basis functions for the weighting function $\bm{\phi}$,
+the number of basis functions for the weighting function $\pmb{\phi}$,
i.e., $n > m$. The basis functions for the Lagrange multipliers and
fault slip are associated with the fault surface, which is a lower
dimension than the domain, so $p \ll n$ in most cases. If we express
the linear combination of basis functions in terms of a matrix-vector
product, we have
\begin{linenomath*}\begin{gather}
-\bm{\phi} = \bm{N}_m \cdot \bm{a}_m, \\
+\pmb{\phi} = \bm{N}_m \cdot \bm{a}_m, \\
\bm{u} = \bm{N}_n \cdot \bm{u}_n, \\
\bm{l} = \bm{N}_p \cdot \bm{l}_p, \\
\bm{d} = \bm{N}_p \cdot \bm{d}_p.
@@ -790,7 +796,7 @@ We must also update all cells on the pos
We must also update all cells on the positive side of the fault that
touch the fault with only an edge or single vertex. We need to replace
the original vertices with the newly introduced vertices on the
-positive side of the fault. In cases where the the fault reaches the
+positive side of the fault. In cases where the fault reaches the
boundaries of the domain, it is relatively easy to identify these
cells because these vertices are shared with the cells that have faces
on the positive side of the fault. However, in the case of a fault
@@ -1598,7 +1604,7 @@ rupture propagation.
$V_p$ & dilatational wave speed. \\
$V_s$ & shear wave speed.\\
$\Delta t$ & Time step.\\
-  $\bm{\phi}$ & weighting function.\\
+  $\pmb{\phi}$ & weighting function.\\
$\rho$ & mass density.\\
$\bm{\sigma}$ & Cauchy stress tensor.
\end{notation}
@@ -1968,7 +1974,7 @@ rupture propagation.
multiplicative factorization and the custom fault block
preconditioner yield the best performance with only a
fraction of the iterates as the other preconditioners and a small
-  increase with problem size. Furthermore, the the field
+  increase with problem size. Furthermore, the field
split preconditioner with multiplicative factorization and the custom
fault block preconditioner provides the shortest runtime.}
\end{table}