[cig-commits] r13140 - doc/snac
sue at geodynamics.org
sue at geodynamics.org
Tue Oct 28 12:34:50 PDT 2008
Author: sue
Date: 2008-10-28 12:34:50 -0700 (Tue, 28 Oct 2008)
New Revision: 13140
Modified:
doc/snac/snac.lyx
Log:
fixed some formulas and citations which were preventing generating PDF. added some bib todos
Modified: doc/snac/snac.lyx
===================================================================
--- doc/snac/snac.lyx 2008-10-28 17:38:45 UTC (rev 13139)
+++ doc/snac/snac.lyx 2008-10-28 19:34:50 UTC (rev 13140)
@@ -1,4 +1,4 @@
-#LyX 1.5.3 created this file. For more info see http://www.lyx.org/
+#LyX 1.5.1 created this file. For more info see http://www.lyx.org/
\lyxformat 276
\begin_document
\begin_header
@@ -421,17 +421,10 @@
\end_inset
a).
- In this mesh hierarchy, called the mixed discretization
-\begin_inset ERT
-status collapsed
+ In this mesh hierarchy, called the mixed discretization
+\begin_inset LatexCommand cite
+key "MartCund1982"
-\begin_layout Standard
-
-
-\backslash
-citep{MartCund1982}
-\end_layout
-
\end_inset
, hexahedral elements are used only as an averaging unit for volumetric
@@ -442,17 +435,11 @@
response.
For a given loading, responses of one set of tetrahedra can be different
from those of the other set because of the differently orientated faces
- of tetrahedra in each set
-\begin_inset ERT
-status collapsed
+ of tetrahedra in each set, e.g.
+
+\begin_inset LatexCommand cite
+key "Zien_etal1995"
-\begin_layout Standard
-
-
-\backslash
-citep[e.g.,]{Zien_etal1995}
-\end_layout
-
\end_inset
.
@@ -465,18 +452,8 @@
status open
\begin_layout Standard
-\begin_inset ERT
-status collapsed
-
-\begin_layout Standard
-
-
-\backslash
-centering
-\end_layout
-
-\end_inset
-
+\noindent
+\align center
\begin_inset Graphics
filename figures/snac-mesh.pdf
@@ -491,17 +468,14 @@
\begin_inset Caption
\begin_layout Standard
-\begin_inset OptArg
-status open
+\begin_inset LatexCommand label
+name "fig:snac_mesh"
-\begin_layout Standard
-Configurations of tetrahedra and conventions for the notation
-\end_layout
-
\end_inset
-(a) Two configurations of five tetrahedra in a hexahedral element used in
- the mixed discretization.
+Configurations of tetrahedra and conventions for the notation.
+ (a) Two configurations of five tetrahedra in a hexahedral element used
+ in the mixed discretization.
Numbers next to apexes indicate the local node numbering.
(b) Conventions for the notation.
@@ -525,18 +499,9 @@
\end_layout
-\begin_layout Standard
-\begin_inset LatexCommand label
-name "fig:snac_mesh"
-
\end_inset
-
-\end_layout
-\end_inset
-
-
\end_layout
\begin_layout Subsection
@@ -545,17 +510,10 @@
\begin_layout Standard
The approximation of partial derivatives with respect to spatial variables
- follows the integral definitions
-\begin_inset ERT
-status collapsed
+ follows the integral definitions, e.g.,
+\begin_inset LatexCommand cite
+key "Wilkins1964"
-\begin_layout Standard
-
-
-\backslash
-citep[e.g.,]{Wilkins1964}
-\end_layout
-
\end_inset
:
@@ -622,22 +580,30 @@
\begin_inset Formula $f=1$
\end_inset
- in (
+ in Eq.
+
\begin_inset LatexCommand ref
reference "eq:def_fi"
\end_inset
-)),
+),
+\end_layout
+
+\begin_layout Standard
\begin_inset Formula \begin{equation}
-\begin{split}f_{,i} & =\frac{1}{V}\sum_{l=1}^{4}\bar{f}^{l}n_{i}^{l}A^{l}=\frac{1}{V}\sum_{l=1}^{4}\frac{1}{3}\sum_{m=1,\neq l}^{4}f^{m}n_{i}^{l}A^{l}\\
+\begin{array}{cc}
+f_{,i} & =\frac{1}{V}\sum_{l=1}^{4}\bar{f}^{l}n_{i}^{l}A^{l}=\frac{1}{V}\sum_{l=1}^{4}\frac{1}{3}\sum_{m=1,\neq l}^{4}f^{m}n_{i}^{l}A^{l}\\
& =\frac{1}{3V}\sum_{m=1}^{4}f^{m}\sum_{l=1,\neq m}^{4}n_{i}^{l}A^{l}\\
- & =-\frac{1}{3V}\sum_{m=1}^{4}f^{m}n_{i}^{m}A^{m},\end{split}
-\label{eq:formula_fi}\end{equation}
+ & =-\frac{1}{3V}\sum_{m=1}^{4}f^{m}n_{i}^{m}A^{m},\end{array}\label{eq:formula_fi}\end{equation}
\end_inset
- where
+
+\end_layout
+
+\begin_layout Standard
+where
\begin_inset Formula $l$
\end_inset
@@ -829,26 +795,26 @@
\begin_layout Standard
We seek static or quasi-static solutions through a dynamic relaxation method.
Instead of adding a usual velocity-dependent friction term, we adopt a
- local non-viscous damping scheme
-\begin_inset ERT
-status collapsed
+ local non-viscous damping scheme
+\begin_inset LatexCommand cite
+key "Cundall 1987"
-\begin_layout Standard
+\end_inset
-
-\backslash
-citep{Cundall1987}
+:
\end_layout
-\end_inset
-
-:
+\begin_layout Standard
\begin_inset Formula \begin{equation}
-F_{i}^{damped}=F_{i}-\alpha\text{ sgn}(v_{i})|F_{i}|,\label{eq:forcedamped}\end{equation}
+F_{i}^{damped}=F_{i}-\alpha\,\mathrm{sgn}\left(v_{i}\right)\mid F_{i}\mid\,,\label{eq:forcedamped}\end{equation}
\end_inset
- where
+
+\end_layout
+
+\begin_layout Standard
+where
\begin_inset Formula $F_{i}$
\end_inset
@@ -947,23 +913,10 @@
is the P wave velocity.
Instead, through an analogy of continuum to an infinite mass-spring system,
we use a criterion that does not explicitly include length scale and P
- wave velocity
-\begin_inset ERT
-status collapsed
+ wave velocity; see Chapter 9 in Bathe
+\begin_inset LatexCommand cite
+key "Bathe1996"
-\begin_layout Standard
-
-
-\backslash
-citep[see Ch.
- 9
-\end_layout
-
-\begin_layout Standard
-
-in]{Bathe1996}
-\end_layout
-
\end_inset
:
@@ -1027,13 +980,20 @@
\end_inset
:
+\end_layout
+
+\begin_layout Standard
\begin_inset Formula \begin{equation}
-\begin{split}\frac{1}{3}T_{i} & =-ku_{i}\Rightarrow\\
-\frac{1}{3}(\lambda+2\mu)(\dot{\epsilon}_{ii}dt)n_{i}S & =-k(v_{i}dt)\text{ (no sum)},\end{split}
-\label{eq:eqfork}\end{equation}
+\begin{array}{cc}
+\frac{1}{3}T_{i} & =-ku_{i}\Rightarrow\\
+\frac{1}{3}\left(\lambda+2\mu\right)\left(\bar{\epsilon}_{ii}dt\right)n_{i}S & =-k\left(v_{i}dt\right)\,\,(\mathrm{no\, sum}),\end{array}\label{eq:eqfork}\end{equation}
\end_inset
+
+\end_layout
+
+\begin_layout Standard
where only the volumetric contribution from internal forces is taken into
account.
By substituting the approximation for the partial derivative (
@@ -1052,6 +1012,10 @@
\end_inset
+
+\end_layout
+
+\begin_layout Standard
where
\begin_inset Formula $l$
\end_inset
@@ -1078,47 +1042,25 @@
\begin_layout Standard
SNAC uses a general elasto-visco-plastic rheological model to update the
- Cauchy stress tensor
-\begin_inset ERT
-status collapsed
+ Cauchy stress tensor, e.g., Albe, et al.
+\begin_inset LatexCommand cite
+key "Albe_etal2000"
-\begin_layout Standard
-
-
-\backslash
-citep[e.g.,]{Albe_etal2000}
-\end_layout
-
\end_inset
-.
First, the initial guess of stress is acquired by the Maxwell viscoelastic
- constitutive law
-\begin_inset ERT
-status collapsed
+ constitutive law
+\begin_inset LatexCommand cite
+key "Poli_etal1993a"
-\begin_layout Standard
-
-
-\backslash
-citep{Poli_etal1993a}
-\end_layout
-
\end_inset
.
If this initial guess exceeds a given yield stress, it is projected onto
- the yield surface using a return mapping method
-\begin_inset ERT
-status collapsed
+ the yield surface using a return mapping method
+\begin_inset LatexCommand cite
+key "SimoHugh2004"
-\begin_layout Standard
-
-
-\backslash
-citep{SimoHugh2004}
-\end_layout
-
\end_inset
; otherwise, the viscoelastic stress update is retained.
@@ -1163,19 +1105,13 @@
\begin_inset Formula $\Delta\sigma^{\Delta J}$
\end_inset
-)
-\begin_inset ERT
-status collapsed
+)
+\begin_inset LatexCommand cite
+key "RudnRice1975"
-\begin_layout Standard
-
-
-\backslash
-citep{RudnRice1975}
-\end_layout
-
\end_inset
+.
\begin_inset Formula \begin{equation}
\Delta\sigma^{\Delta J}=\frac{\partial(\Delta\sigma)}{\partial t}-W\cdot\Delta\sigma-\Delta\sigma\cdot W^{T},\label{eq:corotstress}\end{equation}
@@ -1480,8 +1416,13 @@
its closest neighbor tetrahedron of the old mesh, which is found based
on the distance between quadrature locations.
Such a transfer is proven to be variationally consistent and also preserves
- nonlinear internal constraints as in the case of plasticity [Ortiz and
- Quigley, 1991].
+ nonlinear internal constraints as in the case of plasticity
+\begin_inset LatexCommand cite
+key "Ortiz and Quigley, 1991"
+
+\end_inset
+
+.
\end_layout
@@ -1491,7 +1432,7 @@
\begin_inset Float figure
wide false
sideways false
-status collapsed
+status open
\begin_layout Standard
\noindent
@@ -1636,7 +1577,7 @@
\begin_inset Float figure
wide false
sideways false
-status collapsed
+status open
\begin_layout Standard
\begin_inset Graphics
@@ -1745,7 +1686,7 @@
\begin_inset Float figure
wide false
sideways false
-status collapsed
+status open
\begin_layout Standard
\noindent
@@ -1923,7 +1864,7 @@
\begin_inset Float figure
wide false
sideways false
-status collapsed
+status open
\begin_layout Standard
\noindent
@@ -2941,7 +2882,7 @@
\end_layout
\begin_layout Standard
-Nothing was under this section head.
+[TODO -- note: Nothing was under this section head.]
\end_layout
\begin_layout Section
@@ -8417,7 +8358,8 @@
\begin_layout Bibliography
\begin_inset LatexCommand bibitem
-key "key-13"
+label "14"
+key "MartCund1982"
\end_inset
@@ -8426,7 +8368,8 @@
\begin_layout Bibliography
\begin_inset LatexCommand bibitem
-key "key-14"
+label "15"
+key "Cundall 1989"
\end_inset
@@ -8435,7 +8378,8 @@
\begin_layout Bibliography
\begin_inset LatexCommand bibitem
-key "key-15"
+label "16"
+key "Wilkins1964"
\end_inset
@@ -8444,7 +8388,8 @@
\begin_layout Bibliography
\begin_inset LatexCommand bibitem
-key "key-16"
+label "17"
+key "Needleman 1988"
\end_inset
@@ -8453,7 +8398,8 @@
\begin_layout Bibliography
\begin_inset LatexCommand bibitem
-key "key-1"
+label "18"
+key "Ortiz and Simo, 1986"
\end_inset
@@ -8462,7 +8408,8 @@
\begin_layout Bibliography
\begin_inset LatexCommand bibitem
-key "key-1"
+label "19"
+key "Ortiz and Quigley, 1991"
\end_inset
@@ -8471,7 +8418,8 @@
\begin_layout Bibliography
\begin_inset LatexCommand bibitem
-key "key-2"
+label "20"
+key "Quenette, et al"
\end_inset
@@ -8481,7 +8429,8 @@
\begin_layout Bibliography
\begin_inset LatexCommand bibitem
-key "key-3"
+label "21"
+key "Tan, et al., 2004"
\end_inset
@@ -8490,7 +8439,8 @@
\begin_layout Bibliography
\begin_inset LatexCommand bibitem
-key "key-4"
+label "22"
+key "Jaeger 1969"
\end_inset
@@ -8501,12 +8451,84 @@
\begin_layout Bibliography
\begin_inset LatexCommand bibitem
-key "key-5"
+label "23"
+key "FLAC3D Manual"
\end_inset
FLAC3D Manual -- TODO
\end_layout
+\begin_layout Bibliography
+\begin_inset LatexCommand bibitem
+label "24"
+key "Zien_etal1995"
+
+\end_inset
+
+ Zien_etal1995 -- TODO
+\end_layout
+
+\begin_layout Bibliography
+\begin_inset LatexCommand bibitem
+label "25"
+key "Bathe1996"
+
+\end_inset
+
+Bathe1996 -- TODO
+\end_layout
+
+\begin_layout Bibliography
+\begin_inset LatexCommand bibitem
+label "26"
+key "Albe_etal2000"
+
+\end_inset
+
+Albe_etal2000 -- TODO
+\end_layout
+
+\begin_layout Bibliography
+\begin_inset LatexCommand bibitem
+label "27"
+key "Poli_etal1993a"
+
+\end_inset
+
+Poli_etal1993a -- TODO
+\end_layout
+
+\begin_layout Bibliography
+\begin_inset LatexCommand bibitem
+label "28"
+key "SimoHugh2004"
+
+\end_inset
+
+SimoHugh2004 -- TODO
+\end_layout
+
+\begin_layout Bibliography
+\begin_inset LatexCommand bibitem
+label "29"
+key "RudnRice1975"
+
+\end_inset
+
+RudnRice1975 -- TODO
+\end_layout
+
+\begin_layout Bibliography
+\begin_inset LatexCommand bibitem
+label "30"
+key "Cundall 1987"
+
+\end_inset
+
+Cundall 1987 (or is the date just wrong in the text? this was from your
+ ERT notation) -- TODO
+\end_layout
+
\end_body
\end_document
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