[cig-commits] commit: Cleaned up notation in friction notes and lumped solver notes.
Mercurial
hg at geodynamics.org
Thu Apr 1 08:40:56 PDT 2010
changeset: 37:740a4c5d6641
tag: tip
user: Brad Aagaard <baagaard at usgs.gov>
date: Thu Apr 01 08:40:31 2010 -0700
files: friction.tex lumpedsolver.tex
description:
Cleaned up notation in friction notes and lumped solver notes.
diff -r b3db64b9d137 -r 740a4c5d6641 friction.tex
--- a/friction.tex Tue Mar 30 11:36:25 2010 -0700
+++ b/friction.tex Thu Apr 01 08:40:31 2010 -0700
@@ -39,7 +39,7 @@ We write the constraints on the fault sl
\matrix{C} \vec{u} = \vec{d} \text{ on } S_f,
\end{equation}
where $\vec{u}$ is the vector of displacements in the global
-coordinate system, $\matrix{C}$ is the matrix the transforms from the
+coordinate system, $\matrix{C}$ is the matrix that transforms from the
global coordinate system to the fault coordinate system and computes
the differential displacements across the fault, $\vec{d}$ is the vector of
fault slip in the fault coordinate system, and $S_f$ is the fault
@@ -48,12 +48,12 @@ surface. For Lagrange vertex $k$, conven
``positive'' side of the fault, the terms in the equation are
\begin{equation}
\left( \begin{array}{ccccccccc}
- \ldots & -C_{px} & -C_{py} & -C_{pz} & &
- +C_{px} & +C_{py} & +C_{pz} & \ldots \\
- \ldots & -C_{qx} & -C_{qy} & -C_{qz} & \ldots &
- +C_{qx} & +C_{qy} & +C_{qz} & \ldots \\
- \ldots & -C_{pr} & -C_{ry} & -C_{rz} & &
- +C_{pr} & +C_{ry} & +C_{rz} & \ldots \\
+ \ldots & -R_{px} & -R_{py} & -R_{pz} & &
+ +R_{px} & +R_{py} & +R_{pz} & \ldots \\
+ \ldots & -R_{qx} & -R_{qy} & -R_{qz} & \ldots &
+ +R_{qx} & +R_{qy} & +R_{qz} & \ldots \\
+ \ldots & -R_{rx} & -R_{ry} & -R_{rz} & &
+ +R_{rx} & +R_{ry} & +R_{rz} & \ldots \\
\end{array} \right)
\left( \begin{array}{c}
\vdots \\
@@ -72,7 +72,12 @@ The fault coordinate system is $(p,q,r)$
The fault coordinate system is $(p,q,r)$ and the global coordinate
system is $(x,y,z)$. In the fault coordinate system $p$ is the
horizontal along-strike direction, $q$ is the up-dip direction, and
-$r$ is the fault normal direction.
+$r$ is the fault normal direction. The components of $p$ in the global
+coordinate system are $(R_{px}, R_{py}, R_{pz})$. Similarly, the
+components of $q$ are $(R_{qx} R_{qy}, R_{qz})$ and the components of
+$r$ are $(R_{rx}, R_{ry}, R_{rz})$.
+
+\vdots
We start with the functional for the total potential energy, while
imposing the constraints associated with fault slip using Lagrange
@@ -85,8 +90,8 @@ Lagrange multipliers yields
Lagrange multipliers yields
\begin{gather}
\frac{\partial \Pi}{\partial u_k} = \ldots + \lambda_k^T
- \matrix{C_k}^T = 0, \\
- \frac{\partial \Pi}{\partial \lambda_k} = \matrix{C_k} \vec{u_k} -
+ \matrix{R_k}^T = 0, \\
+ \frac{\partial \Pi}{\partial \lambda_k} = \matrix{R_k} \vec{u_k} -
\vec{d_k} = 0.
\end{gather}
@@ -240,9 +245,9 @@ fault constitutive model and the one nee
We compute the sensitivity starting with equation ??,
\begin{gather}
- A u + C^T l = b \\
- A \partial u + C^T \partial l = 0 \\
- A \partial u = -C^T \partial l.
+ \matrix{A} \vec{u} + \matrix{C}^T \vec{l} = \vec{b} \\
+ \matrix{A} \partial \vec{u} + \matrix{C}^T \partial \vec{l} = 0 \\
+ \matrix{A} \partial \vec{u} = -\matrix{C}^T \partial \vec{l}.
\end{gather}
\subsubsection{Nondiagonal $A$}
@@ -270,15 +275,19 @@ contain entries coupling the two sides o
contain entries coupling the two sides of the fault. Using this
decomposition of $A$, we solve two linear problems,
\begin{gather}
- A_i \partial u_i = C^T \partial l \\
- A_j \partial u_j = -C^T \partial l,
+ \matrix{A_i} \partial \vec{u_i} = \matrix{C}^T \partial \vec{l} \\
+ \matrix{A_j} \partial \vec{u_j} = -\matrix{C}^T \partial \vec{l},
\end{gather}
and use the relation between displacement and slip
\begin{equation}
- \matrix{C} \vec{\partial u} = \vec{\partial d},
+ \matrix{C} \partial \vec{u} = \partial \vec{d},
\end{equation}
-to compute the change in slip associated with the change in the
-Lagrange multipliers.
+to estimate the change in slip associated with the change in the
+Lagrange multipliers. These two linear systems effectively assume that
+the deformation associated with the change in Lagrange multipliers is
+localized to the degrees of freedom adjacent to the fault (i.e., all
+vertices not on the fault are fixed). For most elastic problems, this
+approximation is very good.
These two linear problems involve only the degrees of freedom
associated with the cohesive cells. Furthermore, we associate the
@@ -295,11 +304,11 @@ In the case where $A$ is a diagonal matr
In the case where $A$ is a diagonal matrix, we can solve the
sensitivity equation directly,
\begin{equation}
- \partial u = -C A^{-1} C^T \partial l.
+ \partial \vec{u} = -\matrix{C} \matrix{A}^{-1} \matrix{C}^T \partial \vec{l}.
\end{equation}
Premultiplying by $\matrix{C}$ leads to
\begin{equation}
- \partial d = -C A^{-1} C^T \partial l.
+ \partial \vec{d} = -\matrix{C} \matrix{A}^{-1} \matrix{C}^T \partial \vec{l}.
\end{equation}
For Lagrange vertex $k$, conventional vertex $i$ on the
@@ -321,29 +330,29 @@ For Lagrange vertex $k$, conventional ve
\end{equation}
\begin{align}
S_{pp} &=
- C_{px}^2\left(\frac{1}{A_{ix}}+\frac{1}{A_{jx}}\right) +
- C_{py}^2\left(\frac{1}{A_{iy}}+\frac{1}{A_{jy}}\right) +
- C_{pz}^2\left(\frac{1}{A_{iz}}+\frac{1}{A_{jz}}\right), \\
+ R_{px}^2\left(\frac{1}{A_{ix}}+\frac{1}{A_{jx}}\right) +
+ R_{py}^2\left(\frac{1}{A_{iy}}+\frac{1}{A_{jy}}\right) +
+ R_{pz}^2\left(\frac{1}{A_{iz}}+\frac{1}{A_{jz}}\right), \\
S_{pq} &=
- C_{px}C_{qx}\left(\frac{1}{A_{ix}}+\frac{1}{A_{jx}}\right) +
- C_{py}C_{qy}\left(\frac{1}{A_{iy}}+\frac{1}{A_{jy}}\right) +
- C_{pz}C_{qz}\left(\frac{1}{A_{iz}}+\frac{1}{A_{jz}}\right), \\
+ R_{px}R_{qx}\left(\frac{1}{A_{ix}}+\frac{1}{A_{jx}}\right) +
+ R_{py}R_{qy}\left(\frac{1}{A_{iy}}+\frac{1}{A_{jy}}\right) +
+ R_{pz}R_{qz}\left(\frac{1}{A_{iz}}+\frac{1}{A_{jz}}\right), \\
S_{pr} &=
- C_{px}C_{rx}\left(\frac{1}{A_{ix}}+\frac{1}{A_{jx}}\right) +
- C_{py}C_{ry}\left(\frac{1}{A_{iy}}+\frac{1}{A_{jy}}\right) +
- C_{pz}C_{rz}\left(\frac{1}{A_{iz}}+\frac{1}{A_{jz}}\right), \\
+ R_{px}R_{rx}\left(\frac{1}{A_{ix}}+\frac{1}{A_{jx}}\right) +
+ R_{py}R_{ry}\left(\frac{1}{A_{iy}}+\frac{1}{A_{jy}}\right) +
+ R_{pz}R_{rz}\left(\frac{1}{A_{iz}}+\frac{1}{A_{jz}}\right), \\
S_{qq} &=
- C_{qx}^2\left(\frac{1}{A_{ix}}+\frac{1}{A_{jx}}\right) +
- C_{qy}^2\left(\frac{1}{A_{iy}}+\frac{1}{A_{jy}}\right) +
- C_{qz}^2\left(\frac{1}{A_{iz}}+\frac{1}{A_{jz}}\right), \\
+ R_{qx}^2\left(\frac{1}{A_{ix}}+\frac{1}{A_{jx}}\right) +
+ R_{qy}^2\left(\frac{1}{A_{iy}}+\frac{1}{A_{jy}}\right) +
+ R_{qz}^2\left(\frac{1}{A_{iz}}+\frac{1}{A_{jz}}\right), \\
S_{qr} &=
- C_{qx}C_{rx}\left(\frac{1}{A_{ix}}+\frac{1}{A_{jx}}\right) +
- C_{qy}C_{ry}\left(\frac{1}{A_{iy}}+\frac{1}{A_{jy}}\right) +
- C_{qz}C_{rz}\left(\frac{1}{A_{iz}}+\frac{1}{A_{jz}}\right), \\
+ R_{qx}R_{rx}\left(\frac{1}{A_{ix}}+\frac{1}{A_{jx}}\right) +
+ R_{qy}R_{ry}\left(\frac{1}{A_{iy}}+\frac{1}{A_{jy}}\right) +
+ R_{qz}R_{rz}\left(\frac{1}{A_{iz}}+\frac{1}{A_{jz}}\right), \\
S_{rr} &=
- C_{rx}^2\left(\frac{1}{A_{ix}}+\frac{1}{A_{jx}}\right) +
- C_{ry}^2\left(\frac{1}{A_{iy}}+\frac{1}{A_{jy}}\right) +
- C_{rz}^2\left(\frac{1}{A_{iz}}+\frac{1}{A_{jz}}\right).
+ R_{rx}^2\left(\frac{1}{A_{ix}}+\frac{1}{A_{jx}}\right) +
+ R_{ry}^2\left(\frac{1}{A_{iy}}+\frac{1}{A_{jy}}\right) +
+ R_{rz}^2\left(\frac{1}{A_{iz}}+\frac{1}{A_{jz}}\right).
\end{align}
In the special case where A contains only intertial terms (e.g., there
@@ -369,9 +378,9 @@ so that the sensitivity matrix, S, is di
The main disadvantage of using tractions rather than forces for the
Lagrange multipliers is that it introduces asymmetry into the Jacobian
-matrix. The term $\matrix{C_u}^T$ now includes terms associated with
+matrix. The term $\matrix{R_u}^T$ now includes terms associated with
the fault area in addition to the direction cosines, whereas the term
-$\matrix{C_l}$ only includes terms associated with the direction
+$\matrix{R_l}$ only includes terms associated with the direction
cosines. For low-order basis functions there may be no advantage in
associating tractions with the Lagrange multiplier DOF rather than
forces.
diff -r b3db64b9d137 -r 740a4c5d6641 lumpedsolver.tex
--- a/lumpedsolver.tex Tue Mar 30 11:36:25 2010 -0700
+++ b/lumpedsolver.tex Thu Apr 01 08:40:31 2010 -0700
@@ -73,10 +73,16 @@ so that we solve the system
\matrix{A}_\mathit{diag} \vec{du} = \vec{r}_o, \\
\vec{r}_0 = r(du=0,dl=0).
\end{gather}
-For convenience we will drop the distinction between $r_o$ and $r$ and simply refer to $r_0$ as $r$, because we form the residual in the first iteration at a time step assuming $du=0$ and $dl=0$. Consider Lagrange multiplier vertex $k$ associated with the slip between conventional vertex $i$ on the ``negative'' side of the fault and conventional vertex $j$ on the ``positive'' side of the fault. The system of equations is in the form \begin{gather}
+For convenience we will drop the distinction between $r_o$ and $r$ and
+simply refer to $r_0$ as $r$, because we form the residual in the
+first iteration at a time step assuming $du=0$ and $dl=0$. Consider
+Lagrange multiplier vertex $k$ associated with the slip between
+conventional vertex $i$ on the ``negative'' side of the fault and
+conventional vertex $j$ on the ``positive'' side of the fault. The
+system of equations is in the form \begin{gather}
A_i du_i - C^T_{ki} dl_k = r_i, \\
A_j du_j + C^T_{kj} dl_k = r_j, \\
- -C_{ki} du_i + C_{kj} du_j = d_k + C_{ki} u_i - C_{kj} u_j, \\
+ -R_{ki} du_i + C_{kj} du_j = d_k + C_{ki} u_i - C_{kj} u_j, \\
C_{ki} = C_{kj},
\end{gather}
where $A_i$ is a diagonal matrix (3x3 in 3-D) associated with Jacobian
@@ -125,29 +131,29 @@ so that the sensitivity matrix, S, is fu
\end{equation}
\begin{align}
S_{pp} &=
- C_{px}^2\left(\frac{1}{A_{ix}}+\frac{1}{A_{jx}}\right) +
- C_{py}^2\left(\frac{1}{A_{iy}}+\frac{1}{A_{jy}}\right) +
- C_{pz}^2\left(\frac{1}{A_{iz}}+\frac{1}{A_{jz}}\right), \\
+ R_{px}^2\left(\frac{1}{A_{ix}}+\frac{1}{A_{jx}}\right) +
+ R_{py}^2\left(\frac{1}{A_{iy}}+\frac{1}{A_{jy}}\right) +
+ R_{pz}^2\left(\frac{1}{A_{iz}}+\frac{1}{A_{jz}}\right), \\
S_{pq} &=
- C_{px}C_{qx}\left(\frac{1}{A_{ix}}+\frac{1}{A_{jx}}\right) +
- C_{py}C_{qy}\left(\frac{1}{A_{iy}}+\frac{1}{A_{jy}}\right) +
- C_{pz}C_{qz}\left(\frac{1}{A_{iz}}+\frac{1}{A_{jz}}\right), \\
+ R_{px}R_{qx}\left(\frac{1}{A_{ix}}+\frac{1}{A_{jx}}\right) +
+ R_{py}R_{qy}\left(\frac{1}{A_{iy}}+\frac{1}{A_{jy}}\right) +
+ R_{pz}R_{qz}\left(\frac{1}{A_{iz}}+\frac{1}{A_{jz}}\right), \\
S_{pr} &=
- C_{px}C_{rx}\left(\frac{1}{A_{ix}}+\frac{1}{A_{jx}}\right) +
- C_{py}C_{ry}\left(\frac{1}{A_{iy}}+\frac{1}{A_{jy}}\right) +
- C_{pz}C_{rz}\left(\frac{1}{A_{iz}}+\frac{1}{A_{jz}}\right), \\
+ R_{px}R_{rx}\left(\frac{1}{A_{ix}}+\frac{1}{A_{jx}}\right) +
+ R_{py}R_{ry}\left(\frac{1}{A_{iy}}+\frac{1}{A_{jy}}\right) +
+ R_{pz}R_{rz}\left(\frac{1}{A_{iz}}+\frac{1}{A_{jz}}\right), \\
S_{qq} &=
- C_{qx}^2\left(\frac{1}{A_{ix}}+\frac{1}{A_{jx}}\right) +
- C_{qy}^2\left(\frac{1}{A_{iy}}+\frac{1}{A_{jy}}\right) +
- C_{qz}^2\left(\frac{1}{A_{iz}}+\frac{1}{A_{jz}}\right), \\
+ R_{qx}^2\left(\frac{1}{A_{ix}}+\frac{1}{A_{jx}}\right) +
+ R_{qy}^2\left(\frac{1}{A_{iy}}+\frac{1}{A_{jy}}\right) +
+ R_{qz}^2\left(\frac{1}{A_{iz}}+\frac{1}{A_{jz}}\right), \\
S_{qr} &=
- C_{qx}C_{rx}\left(\frac{1}{A_{ix}}+\frac{1}{A_{jx}}\right) +
- C_{qy}C_{ry}\left(\frac{1}{A_{iy}}+\frac{1}{A_{jy}}\right) +
- C_{qz}C_{rz}\left(\frac{1}{A_{iz}}+\frac{1}{A_{jz}}\right), \\
+ R_{qx}R_{rx}\left(\frac{1}{A_{ix}}+\frac{1}{A_{jx}}\right) +
+ R_{qy}R_{ry}\left(\frac{1}{A_{iy}}+\frac{1}{A_{jy}}\right) +
+ R_{qz}R_{rz}\left(\frac{1}{A_{iz}}+\frac{1}{A_{jz}}\right), \\
S_{rr} &=
- C_{rx}^2\left(\frac{1}{A_{ix}}+\frac{1}{A_{jx}}\right) +
- C_{ry}^2\left(\frac{1}{A_{iy}}+\frac{1}{A_{jy}}\right) +
- C_{rz}^2\left(\frac{1}{A_{iz}}+\frac{1}{A_{jz}}\right).
+ R_{rx}^2\left(\frac{1}{A_{ix}}+\frac{1}{A_{jx}}\right) +
+ R_{ry}^2\left(\frac{1}{A_{iy}}+\frac{1}{A_{jy}}\right) +
+ R_{rz}^2\left(\frac{1}{A_{iz}}+\frac{1}{A_{jz}}\right).
\end{align}
\begin{gather}
%
@@ -169,9 +175,9 @@ so that the sensitivity matrix, S, is fu
\begin{equation}
C_{ki} = \left( \begin{array}{ccc}
- C_{px} & C_{py} & C_{pz} \\
- C_{qx} & C_{qy} & C_{qz} \\
- C_{rx} & C_{ry} & C_{rz}
+ R_{px} & R_{py} & R_{pz} \\
+ R_{qx} & R_{qy} & R_{qz} \\
+ R_{rx} & R_{ry} & R_{rz}
\end{array} \right),
\end{equation}
where $pqr$ is the fault coordinate system and $xyz$ is the global
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