[cig-commits] commit: Minor cleanup of solver section for dynamic simulations.

[Mercurial]

changeset:   60:23dce361d7dd
tag:         tip
user:        Brad Aagaard <baagaard at usgs.gov>
date:        Wed Aug 31 15:40:07 2011 -0700
files:       faultRup.tex
description:
Minor cleanup of solver section for dynamic simulations.


diff -r d61aa0f9edd9 -r 23dce361d7dd faultRup.tex
--- a/faultRup.tex	Wed Aug 31 15:20:52 2011 -0700
+++ b/faultRup.tex	Wed Aug 31 15:40:07 2011 -0700
@@ -600,8 +600,6 @@ for details).
 
 \subsection{Dynamic Simulations}\brad{Rough draft}
 
-% Lumped Jacobian
-
 In the dynamic simulations the the Courant-Friderichs-Lewy condition
 controls the stability of the time integration. In most dynamic
 problems this dictates a relatively small time step so that a typical
@@ -609,41 +607,55 @@ very efficient solver in order to run a 
 very efficient solver in order to run a simulation in a reasonable
 amount of time. 
 
+% Lumped Jacobian
+
 The Jacobian for our system of equations involves two terms: the
 inertial term given by equation~(\ref{eqn:jacobian:explicit:inertia})
 and the fault slip constraint term given by
 equation~(\ref{eqn:jacobian:constraint}). Using conventional
 finite-element basis functions in these integrations results in a
 sparse matrix with off-diagonal terms. Although we can use the same
-solvers as we do for quasi-static simulations, sliminating the
+solvers as we do for quasi-static simulations, eliminating the
 off-diagonal terms so that the Jacobian is diagonal, results in a much
 faster solver. Not only is the number of operations proportional to
 the number of degrees of freedom, but the memory requirements are
 greatly reduced by storing the diagonal of the matrix using a vector
-rather than a sparse matrix.
+rather than a sparse matrix. However, the block structure of our
+Jacobian matrix, with the fault slip constraints occupying
+off-diagonal blocks requires a two step approach to being able to
+solve the linear system of equations without forming a sparse
+matrix.
 
-The current best available option for eliminating the off-diagonal
-terms focuses on choosing a set of orthogonal basis functions, such as
-the Legendre polynomials with Gauss-Lobatto-Lgendre quadrature points
-\cite{Komatitsch:Vilotte:1998}, which naturally eliminates the
-off-diagonal terms in the Jacobian for both the the interial term and
-the fault slip constraint term without introducing any additional
-approximations. In contrast, the more traditional finite-element
-approach does introduce additional approximations by constructing a
-diagonal approximation of the sparse matrix. In PyLith we employ one
-of these more traditional approaches because it produces reasonable
-approximations for many different choices of basis functions and
-quadrature points. We construct a diagonal approximation such that the
-action on rigid body motion is the same for the diagonal approximation
-as it is for the original Jacobian. The errors associated with
-this approximation are small as long as the deformation occurs at
-length scales significantly larger than the discretization size,
-which is consistent with resolving the wave propagation.
+First, we eliminate the off-diagonal entries in each of the blocks of
+the matrix.  The current best available option for eliminating the
+off-diagonal terms formed during the integration of the inertial term
+focuses on choosing a set of orthogonal basis functions, such as the
+Legendre polynomials with Gauss-Lobatto-Lgendre quadrature points
+\cite{Komatitsch:Vilotte:1998}, which naturally produces a diagonal
+matrix for each finite-element cell without introducing any additional
+approximations. Because the fault slip constraint term also involves
+integration of the products of the basis functions over
+lower-dimension cells, orthogonal basis functions also produces a
+diagonal matrix for this integration. 
+
+In contrast, the more traditional finite-element approach does
+introduce additional approximations by constructing a diagonal
+approximation of the sparse matrix. In PyLith we employ one of these
+more traditional approaches, because it produces good approximations
+for many different choices of basis functions and quadrature
+points. For each finite-element cell, we construct a diagonal
+approximation of the integral such that the action on rigid body
+motion is the same for the diagonal approximation of the integral as
+it is for the original integral. The errors associated with this
+approximation are small as long as the deformation occurs at length
+scales significantly larger than the discretization size, which is
+consistent with resolving the wave propagation.
 
 \brad{TODO: Add equations for intertial term and fault slip constraint
   for lumped Jacobian}
 
 % Schur complement
+