[cig-commits] commit: Switch from \mathbf and \mathsf to \bm (works for Greek letters).
Mercurial
hg at geodynamics.org
Mon Aug 13 14:56:19 PDT 2012
changeset: 130:9268656f1d00
user: Brad Aagaard <baagaard at usgs.gov>
date: Mon Aug 13 09:33:09 2012 -0700
files: faultRup.tex
description:
Switch from \mathbf and \mathsf to \bm (works for Greek letters).
diff -r d25491644e1e -r 9268656f1d00 faultRup.tex
--- a/faultRup.tex Wed Jul 18 22:09:18 2012 -0500
+++ b/faultRup.tex Mon Aug 13 09:33:09 2012 -0700
@@ -6,6 +6,7 @@
\usepackage{array}
\usepackage{rotating}
\usepackage[centertags]{amsmath}
+\usepackage{bm}
\usepackage{url}
% :SUBMIT: if draft, comment out this line
@@ -237,19 +238,19 @@ a step-by-step description of the formul
We solve the elasticity equation including inertial terms,
\begin{gather}
- \rho \frac{\partial^2\mathbf{u}}{\partial t^2} - \mathbf{f}
- - \mathsf{\nabla} \cdot \mathsf{\sigma} = \mathbf{0} \text{ in }V, \\
- \mathsf{\sigma} \cdot \mathbf{n} = \mathbf{T} \text{ on }S_T, \\
- \mathbf{u} = \mathbf{u}_0 \text{ on }S_u, \\
- \mathbf{d} - (\mathbf{u}_{+} - \mathbf{u}_{-}) = \mathbf{0}
+ \rho \frac{\partial^2\bm{u}}{\partial t^2} - \bm{f}
+ - \bm{\nabla} \cdot \bm{\sigma} = \bm{0} \text{ in }V, \\
+ \bm{\sigma} \cdot \bm{n} = \bm{T} \text{ on }S_T, \\
+ \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}
-where $\mathbf{u}$ is the displacement vector, $\rho$ is the mass
-density, $\mathbf{f}$ is the body force vector, $\mathsf{\sigma}$ is the
-Cauchy stress tensor, and $t$ is time. We specify tractions $\mathbf{T}$
-on surface $S_T$, displacements $\mathbf{u_0}$ on surface $S_u$, and slip
-$\mathbf{d}$ on fault surface $S_f$, where the tractions and fault slip
-are in global coordinates. Because both $\mathbf{T}$ and $\mathbf{u}$ are vector
+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.
@@ -259,20 +260,20 @@ dot product of the governing equation wi
dot product of the governing equation with a weighting function and
setting the integral over the domain equal to zero,
\begin{equation}
- \int_{V} \mathbf{\phi} \cdot
- \left( \mathsf{\nabla} \cdot \mathsf{\sigma} + \mathbf{f} -
- \rho\frac{\partial^{2}\mathbf{u}}{\partial t^{2}} \right)
+ \int_{V} \bm{\phi} \cdot
+ \left( \bm{\nabla} \cdot \bm{\sigma} + \bm{f} -
+ \rho\frac{\partial^{2}\bm{u}}{\partial t^{2}} \right)
\, dV=0.
\end{equation}
-The weighting function $\mathbf{\phi}$ is a piecewise differential vector
-field with $\mathbf{\phi} = \mathbf{0}$ on $S_u$. After some algebra and
+The weighting function $\bm{\phi}$ is a piecewise differential vector
+field with $\bm{\phi} = \bm{0}$ on $S_u$. After some algebra and
use of the boundary conditions, we have
\begin{equation}
\begin{split}
- - \int_{V} \nabla \mathbf{\phi} : \mathsf{\sigma} \, dV
- + \int_{S_T} \mathbf{\phi} \cdot \mathbf{T} \, dS
- + \int_{V} \mathbf{\phi} \cdot \mathbf{f} \, dV \\
- - \int_{V} \mathbf{\phi} \cdot \rho \frac{\partial^{2}\mathbf{u}}{\partial t^{2}} \, dV
+ - \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.
\end{split}
\end{equation}
@@ -286,8 +287,8 @@ the fault. Slip on the fault is the disp
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 $\mathbf{l}_{+}
-- \mathbf{l}_{-} = 0$.
+fault, which we impose using Lagrange multipliers with $\bm{l}_{+}
+- \bm{l}_{-} = 0$.
Recognizing that the tractions on the fault surface are analogous to
@@ -295,12 +296,12 @@ the Lagrange multipliers (fault traction
the Lagrange multipliers (fault tractions) over the fault surface,
\begin{equation}
\begin{split}
- - \int_{V} \nabla\mathbf{\phi} : \mathsf{\sigma} \, dV
- + \int_{S_T} \mathbf{\phi} \cdot \mathbf{T} \, dS
- - \int_{S_{f^+}} \mathbf{\phi} \cdot \mathbf{l} \, dS
- + \int_{S_{f^-}} \mathbf{\phi} \cdot \mathbf{l} \, dS \\
- + \int_{V} \mathbf{\phi} \cdot \mathbf{f} \, dV
- - \int_{V} \mathbf{\phi} \cdot \rho \frac{\partial^{2}\mathbf{u}}{\partial t^{2}} \, dV
+ - \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
=0.
\end{split}
\end{equation}
@@ -311,60 +312,60 @@ 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{equation}
- \int_{S_f} \mathbf{\phi} \cdot
- \left(\mathbf{d} - \mathbf{u}_{+} + \mathbf{u}_{-} \right) \, dS = 0.
+ \int_{S_f} \bm{\phi} \cdot
+ \left(\bm{d} - \bm{u}_{+} + \bm{u}_{-} \right) \, dS = 0.
\end{equation}
-We express the weighting function $\mathbf{\phi}$, trial solution
-$\mathbf{u}$, Lagrange multipliers $\mathbf{l}$, and fault slip $\mathbf{d}$ as
+We express the weighting function $\bm{\phi}$, trial solution
+$\bm{u}$, Lagrange multipliers $\bm{l}$, and fault slip $\bm{d}$ as
linear combinations of basis functions,
\begin{gather}
-\mathbf{\phi} = \sum_{m} \mathbf{a}_m N_m, \\
-\mathbf{u} = \sum_{n} \mathbf{u}_n N_n, \\
-\mathbf{l} = \sum_{p} \mathbf{l}_p N_p, \\
-\mathbf{d} = \sum_{p} \mathbf{d}_p N_p.
+\bm{\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}
Because the weighting function is zero on $S_u$, the number of basis
-functions for the trial solution $\mathbf{u}$ is generally greater than
-the number of basis functions for the weighting function $\mathbf{\phi}$,
+functions for the trial solution $\bm{u}$ is generally greater than
+the number of basis functions for the weighting function $\bm{\phi}$,
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{gather}
-\mathbf{\phi} = \mathbf{N}_m \cdot \mathbf{a}_m, \\
-\mathbf{u} = \mathbf{N}_n \cdot \mathbf{u}_n, \\
-\mathbf{l} = \mathbf{N}_p \cdot \mathbf{l}_p, \\
-\mathbf{d} = \mathbf{N}_p \cdot \mathbf{d}_p.
+\bm{\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.
\end{gather}
The weighting function is arbitrary, so the integrands must be zero
-for all $\mathbf{a}_m$, which leads to
+for all $\bm{a}_m$, which leads to
\begin{gather}
\begin{split}
-- \int_{V} \nabla \mathbf{N}_m^T \cdot \mathsf{\sigma} \, dV
-+ \int_{S_T} \mathbf{N}_m^T \cdot \mathbf{T} \, dS
-- \int_{S_{f^+}} \mathbf{N}_m^T \cdot \mathbf{N}_p \cdot \mathbf{l}_p \, dS \\
-+ \int_{S_{f^-}} \mathbf{N}_m^T \cdot \mathbf{N}_p \cdot \mathbf{l}_p \, dS
-+ \int_{V} \mathbf{N}_m^T \cdot \mathbf{f} \, dV \\
-- \int_{V} \rho \mathbf{N}_m^T \cdot \mathbf{N}_n \cdot \frac{\partial^2 \mathbf{u}_n}{\partial
+- \int_{V} \nabla \bm{N}_m^T \cdot \bm{\sigma} \, dV
++ \int_{S_T} \bm{N}_m^T \cdot \bm{T} \, dS
+- \int_{S_{f^+}} \bm{N}_m^T \cdot \bm{N}_p \cdot \bm{l}_p \, dS \\
++ \int_{S_{f^-}} \bm{N}_m^T \cdot \bm{N}_p \cdot \bm{l}_p \, dS
++ \int_{V} \bm{N}_m^T \cdot \bm{f} \, dV \\
+- \int_{V} \rho \bm{N}_m^T \cdot \bm{N}_n \cdot \frac{\partial^2 \bm{u}_n}{\partial
t^2} \, dV
-=\mathbf{0},
+=\bm{0},
\end{split}
\\
%
- \int_{S_f} \mathbf{N}_p^T \cdot
- \left( \mathbf{N}_p \cdot \mathbf{d}_p
- - \mathbf{N}_{n^+} \cdot \mathbf{u}_{n^+}
- + \mathbf{N}_{n^-} \cdot \mathbf{u}_{n^-}
- \right) \, dS = \mathbf{0}.
+ \int_{S_f} \bm{N}_p^T \cdot
+ \left( \bm{N}_p \cdot \bm{d}_p
+ - \bm{N}_{n^+} \cdot \bm{u}_{n^+}
+ + \bm{N}_{n^-} \cdot \bm{u}_{n^-}
+ \right) \, dS = \bm{0}.
\end{gather}
-We want to solve these equations for the coefficients $\mathbf{u}_n$
-and $\mathbf{l}_p$ subject to $\mathbf{u} = \mathbf{u}_0 \text{ on
-}S_u$. When we prescribed the slip, we specify $\mathbf{d}$ on $S_f$,
+We want to solve these equations for the coefficients $\bm{u}_n$
+and $\bm{l}_p$ subject to $\bm{u} = \bm{u}_0 \text{ on
+}S_u$. When we prescribed the slip, we specify $\bm{d}$ on $S_f$,
and when we use a fault constitutive model, we specify how the
-Lagrange multipliers $\mathbf{l}$ depend on the fault slip, slip rate,
+Lagrange multipliers $\bm{l}$ depend on the fault slip, slip rate,
and state variables.
For nonlinear bulk rheologies it is convenient to work with the
@@ -372,10 +373,10 @@ equations in terms of the increment in t
equations in terms of the increment in the solution from time $t$ to
$t+\Delta t$ rather than the solution at time $t+\Delta t$.
Consequently, rather than constructing a system with the form
-$\mathbf{A} \cdot \mathbf{u}(t+\Delta t) = \mathbf{b}(t+\Delta t)$, we
-construct a system with the form $\mathbf{A} \cdot \mathbf{du} =
-\mathbf{b}(t+\Delta t) - \mathbf{A} \cdot \mathbf{u}(t)$, where
-$\mathbf{u}(t+\Delta t) = \mathbf{u}(t) + \mathbf{du}$.
+$\bm{A} \cdot \bm{u}(t+\Delta t) = \bm{b}(t+\Delta t)$, we
+construct a system with the form $\bm{A} \cdot \bm{du} =
+\bm{b}(t+\Delta t) - \bm{A} \cdot \bm{u}(t)$, where
+$\bm{u}(t+\Delta t) = \bm{u}(t) + \bm{du}$.
% ------------------------------------------------------------------
\subsection{Quasi-static Simulations}
@@ -387,24 +388,24 @@ and the loading conditions. Considering
\begin{gather}
\label{eqn:quasi-static:residual:elasticity}
\begin{split}
- - \int_{V} \nabla \mathbf{N}_m^T \cdot \mathsf{\sigma}(t+\Delta t) \, dV
- + \int_{S_T} \mathbf{N}_m^T \cdot \mathbf{T}(t+\Delta t) \, dS \\
- - \int_{S_{f^+}} \mathbf{N}_m^T \cdot \mathbf{N}_p \cdot \mathbf{l}_p(t+\Delta t) \, dS \\
- + \int_{S_{f^-}} \mathbf{N}_m^T \cdot \mathbf{N}_p \cdot \mathbf{l}_p(t+\Delta t) \, dS
+ - \int_{V} \nabla \bm{N}_m^T \cdot \bm{\sigma}(t+\Delta t) \, dV
+ + \int_{S_T} \bm{N}_m^T \cdot \bm{T}(t+\Delta t) \, dS \\
+ - \int_{S_{f^+}} \bm{N}_m^T \cdot \bm{N}_p \cdot \bm{l}_p(t+\Delta t) \, dS \\
+ + \int_{S_{f^-}} \bm{N}_m^T \cdot \bm{N}_p \cdot \bm{l}_p(t+\Delta t) \, dS
\\
- + \int_{V} \mathbf{N}_m^T \cdot \mathbf{f}(t+\Delta t) \, dV
- =\mathbf{0},
+ + \int_{V} \bm{N}_m^T \cdot \bm{f}(t+\Delta t) \, dV
+ =\bm{0},
\end{split}
\\
%
\label{eqn:quasi-static:residual:fault}
\begin{split}
- \int_{S_f} \mathbf{N}_p^T \cdot
- \left( \mathbf{N}_p \cdot \mathbf{d}_p(t+\Delta t)
- - \mathbf{N}_{n^+} \cdot \mathbf{u}_{n^+}(t+\Delta t)
- \right) \, dS \\ +\int_{S_f} \mathbf{N}_p^T \cdot \left(
- \mathbf{N}_{n^-} \cdot \mathbf{u}_{n^-}(t+\Delta t)
- \right) \, dS = \mathbf{0}.
+ \int_{S_f} \bm{N}_p^T \cdot
+ \left( \bm{N}_p \cdot \bm{d}_p(t+\Delta t)
+ - \bm{N}_{n^+} \cdot \bm{u}_{n^+}(t+\Delta t)
+ \right) \, dS \\ +\int_{S_f} \bm{N}_p^T \cdot \left(
+ \bm{N}_{n^-} \cdot \bm{u}_{n^-}(t+\Delta t)
+ \right) \, dS = \bm{0}.
\end{split}
\end{gather}
In order to march forward in time, we simply increment time, solve the
@@ -414,11 +415,11 @@ a suite of tools for solving linear syst
a suite of tools for solving linear systems of algebraic equations
with parallel processing
\citep{PETSc:manual,PETSC:efficient}. In solving the
-system, we compute the residual (i.e., $\mathbf{r} = \mathbf{b} -
-\mathbf{A} \cdot \mathbf{u}$ and the Jacobian of the system
-($\mathbf{A}$). In our case the solution is $\mathbf{u} =
-\left( \begin{smallmatrix} \mathbf{u}_n \\
- \mathbf{l}_n \end{smallmatrix} \right)$, and the residual is
+system, we compute the residual (i.e., $\bm{r} = \bm{b} -
+\bm{A} \cdot \bm{u}$ and the Jacobian of the system
+($\bm{A}$). In our case the solution is $\bm{u} =
+\left( \begin{smallmatrix} \bm{u}_n \\
+ \bm{l}_n \end{smallmatrix} \right)$, and the residual is
simply the left hand sides of
equations~(\ref{eqn:quasi-static:residual:elasticity})
and~(\ref{eqn:quasi-static:residual:fault}).
@@ -426,40 +427,40 @@ The Jacobian of the system is the action
The Jacobian of the system is the action (operation) on the increment
in the solution. To find the portion of the Jacobian associated with
equation~(\ref{eqn:quasi-static:residual:elasticity}), we let
-$\mathsf{\sigma}(t+\Delta t) = \mathsf{\sigma}(t) +
-\mathsf{d\sigma}(t)$. The action on the increment of the solution is
+$\bm{\sigma}(t+\Delta t) = \bm{\sigma}(t) +
+\bm{d\sigma}(t)$. The action on the increment of the solution is
associated with the increment in the stress tensor
-$\mathsf{d\sigma}(t)$. We approximate the increment in the stress
+$\bm{d\sigma}(t)$. We approximate the increment in the stress
tensor using linear elasticity and infinitesimal strains,
\begin{equation}
- \mathsf{d\sigma}(t) = \frac{1}{2} \mathsf{C}(t) \cdot (\nabla + \nabla^T)
- \mathbf{u}(t),
+ \bm{d\sigma}(t) = \frac{1}{2} \bm{C}(t) \cdot (\nabla + \nabla^T)
+ \bm{u}(t),
\end{equation}
-where $\mathsf{C}$ is the fourth order tensor of elastic
+where $\bm{C}$ is the fourth order tensor of elastic
constants. Substituting into the first term in
equation~(\ref{eqn:quasi-static:residual:elasticity}) and expressing the
displacement vector as a linear combination of basis functions, after
some algebra we find this portion of the Jacobian is
\begin{equation}
\label{eqn:jacobian:implicit:stiffness}
- \mathbf{K} = \frac{1}{4} \int_V
- (\nabla^T + \nabla) \mathbf{N}_m^T \cdot
- \mathsf{C} \cdot (\nabla + \nabla^T) \mathbf{N}_n \, dV.
+ \bm{K} = \frac{1}{4} \int_V
+ (\nabla^T + \nabla) \bm{N}_m^T \cdot
+ \bm{C} \cdot (\nabla + \nabla^T) \bm{N}_n \, dV.
\end{equation}
Following a similar procedure, we find the portion of the Jacobian
associated with the constraints, equation~(\ref{eqn:quasi-static:residual:fault}), is
\begin{equation}
\label{eqn:jacobian:constraint}
- \mathbf{L} = \int_{S_f} \mathbf{N}_p^T \cdot (\mathbf{N}_{n^+} - \mathbf{N}_{n^-}) \, dS.
+ \bm{L} = \int_{S_f} \bm{N}_p^T \cdot (\bm{N}_{n^+} - \bm{N}_{n^-}) \, dS.
\end{equation}
Thus, the Jacobian of the entire system has the form,
\begin{equation}\label{eqn:saddle:point}
- \mathbf{A} =
+ \bm{A} =
\left( \begin{array}{cc}
- \mathbf{K} & \mathbf{L}^T \\ \mathbf{L} & \mathbf{0}
+ \bm{K} & \bm{L}^T \\ \bm{L} & \bm{0}
\end{array} \right).
\end{equation}
-Note that the terms in $\mathbf{N_{n^+}}$ and $\mathbf{N_{n^-}}$ are
+Note that the terms in $\bm{N_{n^+}}$ and $\bm{N_{n^-}}$ are
identical, but they refer to degrees of freedom (DOF) on the positive and
negative sides of the fault, respectively. Consequently, in practice
we compute the terms for the positive side of the fault and assemble
@@ -467,16 +468,16 @@ the fault. Hence, we compute
the fault. Hence, we compute
\begin{equation}
\label{eqn:jacobian:constraint:code}
- \mathbf{L_p} = \int_{S_f} \mathbf{N}_p^T \cdot \mathbf{N}_{n^+} \, dS,
+ \bm{L_p} = \int_{S_f} \bm{N}_p^T \cdot \bm{N}_{n^+} \, dS,
\end{equation}
with the Jacobian of the entire system taking the form,
\begin{equation}\label{eqn:saddle:point:code}
- \mathbf{A} =
+ \bm{A} =
\left( \begin{array}{cccc}
- \mathbf{K}_{nn} & \mathbf{K}_{nn^+} & \mathbf{K}_{nn^-} & \mathbf{0} \\
- \mathbf{K}_{n^+n} & \mathbf{K}_{n^+n^+} & \mathbf{0} & \mathbf{L}_p^T \\
- \mathbf{K}_{n^-n} & \mathbf{0} & \mathbf{K}_{n^-n^-} & -\mathbf{L}_p^T \\
- \mathbf{0} & \mathbf{L}_p & -\mathbf{L}_p & \mathbf{0}
+ \bm{K}_{nn} & \bm{K}_{nn^+} & \bm{K}_{nn^-} & \bm{0} \\
+ \bm{K}_{n^+n} & \bm{K}_{n^+n^+} & \bm{0} & \bm{L}_p^T \\
+ \bm{K}_{n^-n} & \bm{0} & \bm{K}_{n^-n^-} & -\bm{L}_p^T \\
+ \bm{0} & \bm{L}_p & -\bm{L}_p & \bm{0}
\end{array} \right),
\end{equation}
where $n$ denotes DOF not associated with the fault,
@@ -491,21 +492,21 @@ In dynamic simulations we include the in
In dynamic simulations we include the inertial term in order to
resolve the propagation of seismic waves. The integral equation for
the fault slip constraint remains unchanged, so the corresponding
-portions of the Jacobian ($\mathbf{L}$) and residual ($\mathbf{r}_p$)
+portions of the Jacobian ($\bm{L}$) and residual ($\bm{r}_p$)
are exactly the same as in the quasi-static simulations. Including the
inertial term in equation~\ref{eqn:quasi-static:residual:elasticity} yields
\begin{equation}
\label{eqn:dynamic:residual:elasticity}
\begin{split}
- - \int_{V} \nabla \mathbf{N}_m^T \cdot \mathsf{\sigma}(t+\Delta t) \, dV
- + \int_{S_T} \mathbf{N}_m^T \cdot \mathbf{T}(t+\Delta t) \, dS \\
- - \int_{S_{f^+}} \mathbf{N}_m^T \cdot \mathbf{N}_p \cdot \mathbf{l}_p(t+\Delta t) \, dS \\
- + \int_{S_{f^-}} \mathbf{N}_m^T \cdot \mathbf{N}_p \cdot \mathbf{l}_p(t+\Delta t) \, dS
+ - \int_{V} \nabla \bm{N}_m^T \cdot \bm{\sigma}(t+\Delta t) \, dV
+ + \int_{S_T} \bm{N}_m^T \cdot \bm{T}(t+\Delta t) \, dS \\
+ - \int_{S_{f^+}} \bm{N}_m^T \cdot \bm{N}_p \cdot \bm{l}_p(t+\Delta t) \, dS \\
+ + \int_{S_{f^-}} \bm{N}_m^T \cdot \bm{N}_p \cdot \bm{l}_p(t+\Delta t) \, dS
\\
- + \int_{V} \mathbf{N}_m^T \cdot \mathbf{f}(t+\Delta t) \, dV \\
- - \int_{V} \rho \mathbf{N}_m^T \cdot \mathbf{N}_n \cdot
- \frac{\partial^2 \mathbf{u}_n}{\partial t^2} \, dV
- =\mathbf{0}.
+ + \int_{V} \bm{N}_m^T \cdot \bm{f}(t+\Delta t) \, dV \\
+ - \int_{V} \rho \bm{N}_m^T \cdot \bm{N}_n \cdot
+ \frac{\partial^2 \bm{u}_n}{\partial t^2} \, dV
+ =\bm{0}.
\end{split}
\end{equation}
We find the upper portion of the Jacobian of the system by considering
@@ -515,30 +516,30 @@ time using explicit time stepping via Ne
time using explicit time stepping via Newmark's method with a central
difference scheme wherein the acceleration and velocity are given by
\begin{gather}
- \frac{\partial^2 \mathbf{u}}{\partial t^2}(t) =
+ \frac{\partial^2 \bm{u}}{\partial t^2}(t) =
\frac{1}{\Delta t^2} \left(
- \mathbf{du} - \mathbf{u}(t) + \mathbf{u}(t-\Delta t)
+ \bm{du} - \bm{u}(t) + \bm{u}(t-\Delta t)
\right), \\
%
- \frac{\partial \mathbf{u}}{\partial t}(t) = \frac{1}{2\Delta t} \left(
- \mathbf{du} + \mathbf{u}(t) - \mathbf{u}(t-\Delta t)
+ \frac{\partial \bm{u}}{\partial t}(t) = \frac{1}{2\Delta t} \left(
+ \bm{du} + \bm{u}(t) - \bm{u}(t-\Delta t)
\right).
\end{gather}
Expanding the inertial term yields
\begin{equation}
\begin{split}
- - \int_{V} \rho \mathbf{N}_m^T \cdot \mathbf{N}_n \cdot \frac{\partial^2 \mathbf{u}_n}{\partial
+ - \int_{V} \rho \bm{N}_m^T \cdot \bm{N}_n \cdot \frac{\partial^2 \bm{u}_n}{\partial
t^2} \, dV = \\
- - \frac{1}{\Delta t^2} \int_{V} \rho \mathbf{N}_m^T \cdot
- \mathbf{N}_n \cdot
- \left( \mathbf{du}_n(t) - \mathbf{u}_n(t) + \mathbf{u}_n(t-\Delta t) \right) \, dV,
+ - \frac{1}{\Delta t^2} \int_{V} \rho \bm{N}_m^T \cdot
+ \bm{N}_n \cdot
+ \left( \bm{du}_n(t) - \bm{u}_n(t) + \bm{u}_n(t-\Delta t) \right) \, dV,
\end{split}
\end{equation}
so that the upper portion of the Jacobian is
\begin{equation}
\label{eqn:jacobian:explicit:inertia}
- \mathbf{K} =
- \frac{1}{\Delta t^2} \int_{V} \rho \mathbf{N}_m^T\ \cdot \mathbf{N}_n \, dV.
+ \bm{K} =
+ \frac{1}{\Delta t^2} \int_{V} \rho \bm{N}_m^T\ \cdot \bm{N}_n \, dV.
\end{equation}
Earthquake ruptures in which the slip has a short rise time tends to
@@ -550,12 +551,12 @@ artificial damping via Kelvin-Voigt visc
artificial damping via Kelvin-Voigt viscosity
\citep{Day:etal:2005,Kaneko:etal:2008} to the computation of the strain,
\begin{gather}
- \mathbf{\varepsilon} = \frac{1}{2} \left[ \nabla \mathbf{u} +
- (\nabla \mathbf{u})^T \right ], \\
- \mathbf{\varepsilon} \approx \frac{1}{2} \left[ \nabla \mathbf{u}_d +
- (\nabla \mathbf{u}_d)^T \right ], \\
- \mathbf{u_d} = \mathbf{u} + \eta^* \Delta t \frac{\partial
- \mathbf{u}}{\partial t},
+ \bm{\varepsilon} = \frac{1}{2} \left[ \nabla \bm{u} +
+ (\nabla \bm{u})^T \right ], \\
+ \bm{\varepsilon} \approx \frac{1}{2} \left[ \nabla \bm{u}_d +
+ (\nabla \bm{u}_d)^T \right ], \\
+ \bm{u_d} = \bm{u} + \eta^* \Delta t \frac{\partial
+ \bm{u}}{\partial t},
\end{gather}
where $\eta^*$ is a nomdimensional viscosity on the order of
0.1--1.0.
@@ -601,7 +602,7 @@ temporal scales.
\subsection{Prescribed Fault Rupture}
In a prescribed (kinematic) fault rupture we specify the slip-time
-history $\mathbf{d}(x,y,z,t)$ at every location on the fault surfaces.
+history $\bm{d}(x,y,z,t)$ at every location on the fault surfaces.
The slip-time history enters into the calculation of the residual as
do the Lagrange multipliers, which are available from the current
trial solution. In prescribing the slip-time history we do not specify
@@ -654,43 +655,43 @@ consider only the DOF associated with th
consider only the DOF associated with the fault
interface when computing how a perturbation in the Lagrange
multipliers corresponds to a change in fault slip. In terms of the
-general form of a linear system of equations ($\mathbf{A} \mathbf{u} =
-\mathbf{b}$), our subset of equations based on
+general form of a linear system of equations ($\bm{A} \bm{u} =
+\bm{b}$), our subset of equations based on
equation~(\ref{eqn:saddle:point:code}) has the form
\begin{equation}
\begin{pmatrix}
- \mathbf{K}_{n^+n^+} & 0 & \mathbf{L}_p^T \\
- 0 & \mathbf{K}_{n^-n^-} & -\mathbf{L}_p^T \\
- \mathbf{L}_p & -\mathbf{L}_p & 0
+ \bm{K}_{n^+n^+} & 0 & \bm{L}_p^T \\
+ 0 & \bm{K}_{n^-n^-} & -\bm{L}_p^T \\
+ \bm{L}_p & -\bm{L}_p & 0
\end{pmatrix}
\begin{pmatrix}
- \mathbf{u}_{n^+} \\
- \mathbf{u}_{n^-} \\
- \mathbf{l}_p \\
+ \bm{u}_{n^+} \\
+ \bm{u}_{n^-} \\
+ \bm{l}_p \\
\end{pmatrix}
=
\begin{pmatrix}
- \mathbf{b}_{n^+} \\
- \mathbf{b}_{n^-} \\
- \mathbf{b}_p \\
+ \bm{b}_{n^+} \\
+ \bm{b}_{n^-} \\
+ \bm{b}_p \\
\end{pmatrix},
\end{equation}
where $n^+$ and $n^-$ refer to the DOF associated with
the positive and negative sides of the fault,
-respectively. Furthermore, we can ignore the terms $\mathbf{b}_{n^+}$
-and $\mathbf{b}_{n^-}$ because they remain constant as we change the
+respectively. Furthermore, we can ignore the terms $\bm{b}_{n^+}$
+and $\bm{b}_{n^-}$ because they remain constant as we change the
Lagrange multipliers or fault slip. Our task reduces to solving the
following system of equations to estimate the change in fault slip
-$\partial \mathbf{d}$ associated with a perturbation in the Lagrange
-multipliers $\partial \mathbf{l}_p$:
+$\partial \bm{d}$ associated with a perturbation in the Lagrange
+multipliers $\partial \bm{l}_p$:
\begin{gather}
\label{eqn:spontaneous:rupture:update:lagrange}
- \mathbf{K}_{n^+n^+} \cdot \partial \mathbf{u}_{n^+} =
- - \mathbf{L}_p^T \cdot \partial \mathbf{l}_p, \\
- \mathbf{K}_{n^-n^-} \cdot \partial \mathbf{u}_{n^-} =
- \mathbf{L}_p^T \cdot \partial \mathbf{l}_p, \\
+ \bm{K}_{n^+n^+} \cdot \partial \bm{u}_{n^+} =
+ - \bm{L}_p^T \cdot \partial \bm{l}_p, \\
+ \bm{K}_{n^-n^-} \cdot \partial \bm{u}_{n^-} =
+ \bm{L}_p^T \cdot \partial \bm{l}_p, \\
\label{eqn:spontaneous:rupture:update:slip}
- \partial \mathbf{d}_p = \partial \mathbf{u}_{n^+} - \partial \mathbf{u}_{n^-}.
+ \partial \bm{d}_p = \partial \bm{u}_{n^+} - \partial \bm{u}_{n^-}.
\end{gather}
The efficiency of this iterative procedure depends on both the fault
@@ -710,10 +711,10 @@ model. Specifically, we search for $\alp
model. Specifically, we search for $\alpha$ using a bilinear search in
logarithmic space to minimize
\begin{equation}
- C = \| \mathbf{l}_p + \alpha \partial\mathbf{l}_p - f(\mathbf{d}_p +
- \alpha \partial\mathbf{d}_p) \|_2,
+ C = \| \bm{l}_p + \alpha \partial\bm{l}_p - f(\bm{d}_p +
+ \alpha \partial\bm{d}_p) \|_2,
\end{equation}
-where $f(\mathbf{d})$ corresponds to the fault constitutive model and
+where $f(\bm{d})$ corresponds to the fault constitutive model and
$\|x\|_2$ denotes the L$^2$-norm of $x$. Performing this search in
logarithmic space rather than linear space greatly accelerates the
convergence in rate-state fault constitutive models in which the
@@ -853,7 +854,7 @@ arising in our quasi-static simulations,
arising in our quasi-static simulations, we employ preconditioned
Krylov subspace methods~\citep{Saad03}. We create a sequence of
vectors by repeatedly applying the system matrix to the
-right-hand-side vector, $\mathbf{A^k} \cdot \mathbf{b}$, and they form
+right-hand-side vector, $\bm{A^k} \cdot \bm{b}$, and they form
a basis for a subspace, termed the Krylov space. We can efficiently
find an approximate solution in this subspace. Because sparse
matrix-vector multiplication is scalable via parallel processing, this
@@ -889,24 +890,24 @@ preconditioner, we exploit the structure
preconditioner, we exploit the structure of the sparse Jacobian
matrix. Our system Jacobian has the form
\begin{equation}
- \mathbf{A} = \left( \begin{array}{cc}
- \mathbf{K} & \mathbf{L}^T \\
- \mathbf{L} & \mathbf{0}
+ \bm{A} = \left( \begin{array}{cc}
+ \bm{K} & \bm{L}^T \\
+ \bm{L} & \bm{0}
\end{array} \right).
\end{equation}
-The Schur complement $\mathbf{S}$ of the submatrix $\mathbf{K}$ is given by,
+The Schur complement $\bm{S}$ of the submatrix $\bm{K}$ is given by,
\begin{equation}
- \mathbf{S} = -\mathbf{L} \mathbf{K}^{-1} \mathbf{L}^T
+ \bm{S} = -\bm{L} \bm{K}^{-1} \bm{L}^T
\end{equation}
-which leads to a simple block diagonal preconditioner for $\mathbf{A}$
+which leads to a simple block diagonal preconditioner for $\bm{A}$
\begin{equation}
- \mathbf{P} = \left( \begin{array}{cc}
- \mathbf{P}_\mathit{elasticity} & 0 \\
- 0 & \mathbf{P}_\mathit{fault}
+ \bm{P} = \left( \begin{array}{cc}
+ \bm{P}_\mathit{elasticity} & 0 \\
+ 0 & \bm{P}_\mathit{fault}
\end{array} \right)
= \left( \begin{array}{cc}
- \mathbf{K} & 0 \\
- 0 & -\mathbf{L} \mathbf{K}^{-1} \mathbf{L}^T
+ \bm{K} & 0 \\
+ 0 & -\bm{L} \bm{K}^{-1} \bm{L}^T
\end{array} \right).
\end{equation}
@@ -928,31 +929,31 @@ preconditioning matrix associated with t
preconditioning matrix associated with the Lagrange multipliers, since
stock PETSc preconditioners can handle the elastic portion as
discussed in the previous paragraph. In computing
-$\mathbf{P_\mathit{fault}}$ we approximate $\mathbf{K}^{-1}$ with
-the inverse of the diagonal portion of $\mathbf{K}$. $\mathbf{L}$, which
+$\bm{P_\mathit{fault}}$ we approximate $\bm{K}^{-1}$ with
+the inverse of the diagonal portion of $\bm{K}$. $\bm{L}$, which
consists of integrating the products of basis functions over the fault
faces. Its structure depends on the quadrature scheme and the choice
of basis functions. For conventional low order finite-elements and
-Gauss quadrature, $\mathbf{L}$ contains nonzero terms coupling the
+Gauss quadrature, $\bm{L}$ contains nonzero terms coupling the
degree of freedom for each coordinate axes of a vertex with the
corresponding degree of freedom of the other vertices in a
cell. However, if we collocate quadrature points at the cell vertices,
then only one basis function is nonzero at each quadrature point and
-$\mathbf{L}$ becomes block diagonal; this is also true for spectral
+$\bm{L}$ becomes block diagonal; this is also true for spectral
elements with Legendre polynomials and Gauss-Lobatto-Legendre
quadrature points. This leads to a diagonal matrix for the lower
portion of the conditioning matrix,
\begin{equation}
- \mathbf{P}_\mathit{fault} = -\mathbf{L}_p (\mathbf{K}_{n+n+} + \mathbf{K}_{n-n-}) \mathbf{L}_p^{T},
+ \bm{P}_\mathit{fault} = -\bm{L}_p (\bm{K}_{n+n+} + \bm{K}_{n-n-}) \bm{L}_p^{T},
\end{equation}
-where $\mathbf{L}_p$ is given in
-equation~(\ref{eqn:jacobian:constraint:code}) and $\mathbf{K}_{n+n+}$
-and $\mathbf{K}_{n-n-}$ are the diagonal terms from
+where $\bm{L}_p$ is given in
+equation~(\ref{eqn:jacobian:constraint:code}) and $\bm{K}_{n+n+}$
+and $\bm{K}_{n-n-}$ are the diagonal terms from
equation~(\ref{eqn:saddle:point:code}).
-% Matt conjectures that collocation, because it makes $\mathbf{L}$
+% Matt conjectures that collocation, because it makes $\bm{L}$
% block diagonal, is more tolerant of the diagonal approximation for
-% $\mathbf{K}$.}
+% $\bm{K}$.}
Our preferred setup uses the field splitting options in PETSc to
combine an AMG preconditioner for the elasticity submatrix with out
@@ -1014,14 +1015,14 @@ diagonal approximation of the integral a
diagonal approximation of the integral as it is for the original
integral,
\begin{equation}
- \mathbf{A} \cdot \mathbf{u}_\mathrm{rigid} =
- \mathbf{A}_\mathit{diagonal} \cdot \mathbf{u}_\mathrm{rigid}.
+ \bm{A} \cdot \bm{u}_\mathrm{rigid} =
+ \bm{A}_\mathit{diagonal} \cdot \bm{u}_\mathrm{rigid}.
\end{equation}
Expressing the diagonal block of the Jacobian matrix as a vector and
the matrix of basis functions as a vector we have,
\begin{equation}
- \mathbf{A} = \int_\Omega \mathbf{N}^T \cdot \mathbf{N} \, d\Omega \rightarrow
- \mathbf{A}_\mathit{diagonal} = \int_\Omega \mathbf{N} \sum_i N_i \, d\Omega,
+ \bm{A} = \int_\Omega \bm{N}^T \cdot \bm{N} \, d\Omega \rightarrow
+ \bm{A}_\mathit{diagonal} = \int_\Omega \bm{N} \sum_i N_i \, d\Omega,
\end{equation}
where $N_i$ is the scalar basis function for degree of freedom $i$ and
$\Omega$ may be the domain volume (as in the case of the inertial
@@ -1047,22 +1048,22 @@ associated with the fault slip constrain
associated with the fault slip constraint in solving the system of
equations via a Schur's complement algorithm. We compute an initial
residual assuming the increment in the solution is zero (i.e.,
-$\mathbf{du}_n = \mathbf{0}$ and $\mathbf{dl}_p = \mathbf{0}$,
+$\bm{du}_n = \bm{0}$ and $\bm{dl}_p = \bm{0}$,
\begin{equation}
- \mathbf{r}^* = \begin{pmatrix} \mathbf{r}_n^* \\ \mathbf{r}_p^* \end{pmatrix} =
- \begin{pmatrix} \mathbf{b}_n \\ \mathbf{b}_p \end{pmatrix}
+ \bm{r}^* = \begin{pmatrix} \bm{r}_n^* \\ \bm{r}_p^* \end{pmatrix} =
+ \begin{pmatrix} \bm{b}_n \\ \bm{b}_p \end{pmatrix}
- \begin{pmatrix}
- \mathbf{K} & \mathbf{L}^T \\ \mathbf{L} & 0
+ \bm{K} & \bm{L}^T \\ \bm{L} & 0
\end{pmatrix}
- \begin{pmatrix} \mathbf{u}_n \\ \mathbf{l}_p \end{pmatrix}.
+ \begin{pmatrix} \bm{u}_n \\ \bm{l}_p \end{pmatrix}.
\end{equation}
We compute a corresponding initial solution to the system of equations
-$\mathbf{du}_n^*$ ignoring the off-diagonal blocks in the Jacobian and
+$\bm{du}_n^*$ ignoring the off-diagonal blocks in the Jacobian and
the increment in the Lagrange multipliers.
\begin{equation}
-\mathbf{du}_n^* = \mathbf{K}^{-1} \cdot \mathbf{r}_n,
+\bm{du}_n^* = \bm{K}^{-1} \cdot \bm{r}_n,
\end{equation}
-taking advantage of the fact that we construct $\mathbf{K}$ so that it
+taking advantage of the fact that we construct $\bm{K}$ so that it
is diagonal.
We next compute the increment in the Lagrange multipliers in order to
@@ -1071,56 +1072,56 @@ residual is
residual is
\begin{equation}
\label{eqn:lumped:jacobian:residual}
- \mathbf{r} = \begin{pmatrix} \mathbf{r}_n \\ \mathbf{r}_p \end{pmatrix} =
- \begin{pmatrix} \mathbf{r}_n^* \\ \mathbf{r}_p^* \end{pmatrix}
+ \bm{r} = \begin{pmatrix} \bm{r}_n \\ \bm{r}_p \end{pmatrix} =
+ \begin{pmatrix} \bm{r}_n^* \\ \bm{r}_p^* \end{pmatrix}
- \begin{pmatrix}
- \mathbf{K} & \mathbf{L}^T \\ \mathbf{L} & 0
+ \bm{K} & \bm{L}^T \\ \bm{L} & 0
\end{pmatrix}
- \begin{pmatrix} \mathbf{du}_n \\ \mathbf{dl}_p \end{pmatrix}.
+ \begin{pmatrix} \bm{du}_n \\ \bm{dl}_p \end{pmatrix}.
\end{equation}
Solving the first row of equation~(\ref{eqn:lumped:jacobian:residual})
for the increment in the solution and accounting for the structure of
-$\mathbf{L}$ as we write the expressions for DOF on
+$\bm{L}$ as we write the expressions for DOF on
each side of the fault, we have
\begin{gather}
- \mathbf{du}_{n^+} =
- \mathbf{du}_{n^+}^* - \mathbf{K}_{n^+n^+}^{-1} \cdot \mathbf{L}_p^T \cdot \mathbf{dl}_p, \\
- \mathbf{du}_{n^-} =
- \mathbf{du}_{n^-}^* + \mathbf{K}_{n^-n^-}^{-1} \cdot \mathbf{L}_p^T \cdot \mathbf{dl}_p.
+ \bm{du}_{n^+} =
+ \bm{du}_{n^+}^* - \bm{K}_{n^+n^+}^{-1} \cdot \bm{L}_p^T \cdot \bm{dl}_p, \\
+ \bm{du}_{n^-} =
+ \bm{du}_{n^-}^* + \bm{K}_{n^-n^-}^{-1} \cdot \bm{L}_p^T \cdot \bm{dl}_p.
\end{gather}
Substituting into the second row of
equation~(\ref{eqn:lumped:jacobian:residual}) and isolating the term
with the increment in the Lagrange multipliers yields
\begin{equation}
- \mathbf{L}_p \cdot
- \left( \mathbf{K}_{n^+n^+}^{-1} + \mathbf{K}_{n^-n^-}^{-1} \right) \cdot
- \mathbf{L}^T_p \cdot \mathbf{dl}_p =
- -\mathbf{r}_p^* + \mathbf{L}_p \cdot
- \left( \mathbf{du}_{n^+}^* - \mathbf{du}_{n^-}^* \right).
+ \bm{L}_p \cdot
+ \left( \bm{K}_{n^+n^+}^{-1} + \bm{K}_{n^-n^-}^{-1} \right) \cdot
+ \bm{L}^T_p \cdot \bm{dl}_p =
+ -\bm{r}_p^* + \bm{L}_p \cdot
+ \left( \bm{du}_{n^+}^* - \bm{du}_{n^-}^* \right).
\end{equation}
Letting
\begin{equation}
- \mathbf{S}_p = \mathbf{L}_p \cdot
- \left( \mathbf{K}_{n^+n^+}^{-1} + \mathbf{K}_{n^-n^-}^{-1} \right) \cdot
- \mathbf{L}^T_p,
+ \bm{S}_p = \bm{L}_p \cdot
+ \left( \bm{K}_{n^+n^+}^{-1} + \bm{K}_{n^-n^-}^{-1} \right) \cdot
+ \bm{L}^T_p,
\end{equation}
-and recognizing that $\mathbf{S}_p$ is diagonal because $\mathbf{K}$
-and $\mathbf{L}_p$ are diagonal allows us to solve for the increment
+and recognizing that $\bm{S}_p$ is diagonal because $\bm{K}$
+and $\bm{L}_p$ are diagonal allows us to solve for the increment
in the Lagrange multipliers,
\begin{equation}
- \mathbf{dl}_p = \mathbf{S}_p^{-1} \cdot \left[
- -\mathbf{r}_p^* + \mathbf{L}_p \cdot
- \left( \mathbf{du}_{n^+}^* - \mathbf{du}_{n^-}^* \right)
+ \bm{dl}_p = \bm{S}_p^{-1} \cdot \left[
+ -\bm{r}_p^* + \bm{L}_p \cdot
+ \left( \bm{du}_{n^+}^* - \bm{du}_{n^-}^* \right)
\right].
\end{equation}
Now that we have the increment in the Lagrange multipliers, we can
-correct our initial solution $\mathbf{du}_n^*$ so that the true residual
+correct our initial solution $\bm{du}_n^*$ so that the true residual
is zero,
\begin{equation}
- \mathbf{du}_n =
- \mathbf{du}_n^* - \mathbf{K}^{-1} \cdot \mathbf{L}^T \cdot \mathbf{dl}_p.
+ \bm{du}_n =
+ \bm{du}_n^* - \bm{K}^{-1} \cdot \bm{L}^T \cdot \bm{dl}_p.
\end{equation}
-Because $\mathbf{K}$ and $\mathbf{L}$ are comprised of diagonal
+Because $\bm{K}$ and $\bm{L}$ are comprised of diagonal
blocks, this expression for the updates to the solution are local to
the DOF attached to the fault and the Lagrange
multipliers.
@@ -1129,14 +1130,14 @@ multipliers.
We also leverage the elimination of off-diagonal entries from the
blocks of the Jacobian in dynamic simulations when updating the slip
-in spontaneous rupture models. Because $\mathbf{K}$ is diagonal in
+in spontaneous rupture models. Because $\bm{K}$ is diagonal in
this case, the expression for the change in slip for a perturbation in
the Lagrange multipliers
(equations~(\ref{eqn:spontaneous:rupture:update:lagrange})--(\ref{eqn:spontaneous:rupture:update:slip}))
simplifies to
\begin{equation}
- \partial \mathbf{d}_p = - \left( \mathbf{K}_{n^+n^+}^{-1} + \mathbf{K}_{n^-n^-}^{-1} \right)
- \cdot \mathbf{L}_p^T \cdot \partial \mathbf{l}_p.
+ \partial \bm{d}_p = - \left( \bm{K}_{n^+n^+}^{-1} + \bm{K}_{n^-n^-}^{-1} \right)
+ \cdot \bm{L}_p^T \cdot \partial \bm{l}_p.
\end{equation}
Consequently, the increment in fault slip and Lagrange multipliers for
each vertex can be done independently. In dynamic simulations the time
@@ -1563,35 +1564,35 @@ simulations of earthquake rupture propag
% ----------------------------------------------------------------------
% Notation -- End each entry with a period.
\begin{notation}
- $\mathbf{A}$ & Matrix associated with Jacobian operator for the entire system of equations.\\
- $\mathsf{C}$ & Four order tensor of elastic constants.\\
- $\mathbf{d}$ & fault slip vector.\\
- $\mathbf{f}$ & body force vector.\\
- $\mathbf{l}$ & Lagrange multiplier vector corresponding to the fault traction vector.\\
- $\mathbf{L}$ & Matrix associated with Jacobian operator for constraint equation.\\
- $\mathbf{K}$ & Matrix associated with Jacobian operator for
+ $\bm{A}$ & Matrix associated with Jacobian operator for the entire system of equations.\\
+ $\bm{C}$ & Four order tensor of elastic constants.\\
+ $\bm{d}$ & fault slip vector.\\
+ $\bm{f}$ & body force vector.\\
+ $\bm{l}$ & Lagrange multiplier vector corresponding to the fault traction vector.\\
+ $\bm{L}$ & Matrix associated with Jacobian operator for constraint equation.\\
+ $\bm{K}$ & Matrix associated with Jacobian operator for
elasticity equation.\\
$\mu_f$ & coefficient of friction.\\
- $\mathbf{n}$ & normal vector.\\
- $\mathbf{P}$ & preconditioning matrix.\\
- $\mathbf{P}_\mathit{elastic}$ & preconditioning matrix associated with elasticity.\\
- $\mathbf{P}_\mathit{fault}$ & preconditioning matrix associated with fault slip constraints (Lagrange multipliers).\\
+ $\bm{n}$ & normal vector.\\
+ $\bm{P}$ & preconditioning matrix.\\
+ $\bm{P}_\mathit{elastic}$ & preconditioning matrix associated with elasticity.\\
+ $\bm{P}_\mathit{fault}$ & preconditioning matrix associated with fault slip constraints (Lagrange multipliers).\\
$S_f$ & fault surface.\\
$S_T$ & surface with Neumann boundary conditions.\\
$S_u$ & surface with Dirichlet boundary conditions.\\
$t$ & time.\\
- $\mathbf{T}$ & Traction vector.\\
+ $\bm{T}$ & Traction vector.\\
$T_c$ & scalar shear traction associated with cohesion.\\
$T_f$ & scalar shear traction associated with friction.\\
$T_n$ & scalar normal traction.\\
- $\mathbf{u}$ & displacement vector.\\
+ $\bm{u}$ & displacement vector.\\
$V$ & spatial domain of model.\\
$V_p$ & dilatational wave speed. \\
$V_s$ & shear wave speed.\\
$\Delta t$ & Time step.\\
- $\mathbf{\phi}$ & weighting function.\\
+ $\bm{\phi}$ & weighting function.\\
$\rho$ & mass density.\\
- $\mathsf{\sigma}$ & Cauchy stress tensor.
+ $\bm{\sigma}$ & Cauchy stress tensor.
\end{notation}
@@ -1818,7 +1819,7 @@ simulations of earthquake rupture propag
\label{tab:solver:options}
\centering
\begin{tabular}{ll}
- $\begin{pmatrix}\mathbf{K} & \mathbf{0} \\ \mathbf{0} & \mathbf{I}\end{pmatrix}$~\label{prec:add} & $\begin{pmatrix}\mathbf{K} & \mathbf{L}^T \\ \mathbf{0} & \mathbf{I}\end{pmatrix}$ \label{prec:mult}\\
+ $\begin{pmatrix}\bm{K} & \bm{0} \\ \bm{0} & \bm{I}\end{pmatrix}$~\label{prec:add} & $\begin{pmatrix}\bm{K} & \bm{L}^T \\ \bm{0} & \bm{I}\end{pmatrix}$ \label{prec:mult}\\
\texttt{[pylithapp.problem.formulation]} & \texttt{[pylithapp.problem.formulation]} \\
\texttt{split\_fields = True} & \texttt{split\_fields = True} \\
\texttt{matrix\_type = aij} & \texttt{matrix\_type = aij} \\
@@ -1831,7 +1832,7 @@ simulations of earthquake rupture propag
\texttt{fs\_fieldsplit\_1\_pc\_type = jacobi} & \texttt{fs\_fieldsplit\_1\_pc\_type = jacobi} \\
\texttt{fs\_fieldsplit\_1\_ksp\_type = gmres} & \texttt{fs\_fieldsplit\_1\_ksp\_type = gmres} \\
\smallskip \\
- $\begin{pmatrix}\mathbf{K} & \mathbf{L}^T \\ \mathbf{0} & \mathbf{S}\end{pmatrix}$ \label{prec:schurUpper} & $\begin{pmatrix}\mathbf{I} & \mathbf{0} \\ \mathbf{B}^T \mathbf{A}^{-1} & \mathbf{I}\end{pmatrix}\begin{pmatrix}\mathbf{A} & \mathbf{0} \\ \mathbf{0} & \mathbf{S}\end{pmatrix}\begin{pmatrix}\mathbf{I} & \mathbf{A}^{-1} \mathbf{B} \\ \mathbf{0} & \mathbf{I}\end{pmatrix}$ \label{prec:schurFull}\\
+ $\begin{pmatrix}\bm{K} & \bm{L}^T \\ \bm{0} & \bm{S}\end{pmatrix}$ \label{prec:schurUpper} & $\begin{pmatrix}\bm{I} & \bm{0} \\ \bm{B}^T \bm{A}^{-1} & \bm{I}\end{pmatrix}\begin{pmatrix}\bm{A} & \bm{0} \\ \bm{0} & \bm{S}\end{pmatrix}\begin{pmatrix}\bm{I} & \bm{A}^{-1} \bm{B} \\ \bm{0} & \bm{I}\end{pmatrix}$ \label{prec:schurFull}\\
\texttt{[pylithapp.problem.formulation]} & \texttt{[pylithapp.problem.formulation]} \\
\texttt{split\_fields = True} & \texttt{split\_fields = True} \\
\texttt{matrix\_type = aij} & \texttt{matrix\_type = aij} \\
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