[cig-commits] r12739 - in mc/2D/ConMan/trunk/doc: . images
sue at geodynamics.org
sue at geodynamics.org
Wed Aug 27 15:24:31 PDT 2008
Author: sue
Date: 2008-08-27 15:24:30 -0700 (Wed, 27 Aug 2008)
New Revision: 12739
Modified:
mc/2D/ConMan/trunk/doc/conman.lyx
mc/2D/ConMan/trunk/doc/conman.pdf
mc/2D/ConMan/trunk/doc/conman.tex
mc/2D/ConMan/trunk/doc/images/conman-fig1.eps
mc/2D/ConMan/trunk/doc/images/conman-fig2.eps
mc/2D/ConMan/trunk/doc/images/conman-fig3.eps
mc/2D/ConMan/trunk/doc/images/conman-fig4.eps
mc/2D/ConMan/trunk/doc/images/conman-fig5.eps
mc/2D/ConMan/trunk/doc/images/conman-fig6.eps
mc/2D/ConMan/trunk/doc/images/conman-fig7.eps
mc/2D/ConMan/trunk/doc/images/conman-fig8.eps
Log:
fixed verbatum sections to lyx code
Modified: mc/2D/ConMan/trunk/doc/conman.lyx
===================================================================
--- mc/2D/ConMan/trunk/doc/conman.lyx 2008-08-27 22:15:35 UTC (rev 12738)
+++ mc/2D/ConMan/trunk/doc/conman.lyx 2008-08-27 22:24:30 UTC (rev 12739)
@@ -1,5 +1,5 @@
-#LyX file created by tex2lyx 0.1.2
-\lyxformat 245
+#LyX 1.5.1 created this file. For more info see http://www.lyx.org/
+\lyxformat 276
\begin_document
\begin_header
\textclass book
@@ -122,18 +122,24 @@
\makeatother
-
-
\end_preamble
\language english
\inputencoding latin9
-\fontscheme default
+\font_roman default
+\font_sans default
+\font_typewriter default
+\font_default_family default
+\font_sc false
+\font_osf false
+\font_sf_scale 100
+\font_tt_scale 100
\graphics default
\paperfontsize default
\spacing single
\papersize default
\use_geometry false
\use_amsmath 0
+\use_esint 0
\cite_engine basic
\use_bibtopic false
\paperorientation portrait
@@ -147,6 +153,7 @@
\paperpagestyle headings
\tracking_changes false
\output_changes false
+\author ""
\end_header
\begin_body
@@ -154,8 +161,6 @@
\begin_layout Standard
\noindent
\align center
-
-
\begin_inset ERT
status collapsed
@@ -175,11 +180,9 @@
sideways false
status open
-
\begin_layout Standard
-
-\begin_inset Graphics
+\begin_inset Graphics
filename images/conman_cover.pdf
width 75page%
@@ -194,13 +197,11 @@
\end_layout
\begin_layout Title
-
ConMan
\end_layout
\begin_layout Author
-
-© California Institute of Technology
+© California Institute of Technology
\newline
Scott King
\newline
@@ -208,8 +209,6 @@
\end_layout
\begin_layout Date
-
-
\begin_inset ERT
status collapsed
@@ -226,18 +225,14 @@
\end_layout
\begin_layout Standard
+\begin_inset LatexCommand tableofcontents
-
-\begin_inset LatexCommand \tableofcontents
-
\end_inset
\end_layout
\begin_layout Standard
-
-
\begin_inset FloatList figure
\end_inset
@@ -246,8 +241,6 @@
\end_layout
\begin_layout Standard
-
-
\begin_inset ERT
status collapsed
@@ -265,34 +258,46 @@
\begin_layout Standard
-
\newpage
\end_layout
\begin_layout Chapter
-
Preface
\end_layout
\begin_layout Section
-
Abstract
\end_layout
\begin_layout Standard
-
-This manual serves as a user guide for ConMan, a vectorized finite element program for the solution of the equations of incompressible, infinite-Prandtl number convection in two dimensions originally written by Arthur Raefsky, Scott King, and Brad Hager. ConMan is a public domain program and is distributed free of charge to anyone who wishes to use it and may be freely copied and modified. ConMan is written in Standard Fortran 77 with cray pointers and runs on most Unix systems with many Fortran compilers (see Section 3.4
+This manual serves as a user guide for ConMan, a vectorized finite element
+ program for the solution of the equations of incompressible, infinite-Prandtl
+ number convection in two dimensions originally written by Arthur Raefsky,
+ Scott King, and Brad Hager.
+ ConMan is a public domain program and is distributed free of charge to
+ anyone who wishes to use it and may be freely copied and modified.
+ ConMan is written in Standard Fortran 77 with cray pointers and runs on
+ most Unix systems with many Fortran compilers (see Section 3.4
\series bold
-[TODO -- 3.4 is placeholder section. Is this correct? Need label to put cross ref here]
+[TODO -- 3.4 is placeholder section.
+ Is this correct? Need label to put cross ref here]
\series default
-). Porting it to other systems should be straightforward. As with anything free it comes with no guarantees, but it has been benchmarked against other existing codes (see Chapter
-\begin_inset LatexCommand \ref{cha:The-Benchmark-Cases}
+).
+ Porting it to other systems should be straightforward.
+ As with anything free it comes with no guarantees, but it has been benchmarked
+ against other existing codes (see Chapter
+\begin_inset LatexCommand ref
+reference "cha:The-Benchmark-Cases"
\end_inset
-). The authors would appreciate any information regarding bugs or potential problems but make no promises regarding the timeliness of changes or fixes; see Section
-\begin_inset LatexCommand \ref{sec:Support}
+).
+ The authors would appreciate any information regarding bugs or potential
+ problems but make no promises regarding the timeliness of changes or fixes;
+ see Section
+\begin_inset LatexCommand ref
+reference "sec:Support"
\end_inset
@@ -300,22 +305,35 @@
\end_layout
\begin_layout Section
-
Introduction
\end_layout
\begin_layout Standard
-
-This manual contains all of the necessary information for setting up input and running ConMan. It assumes some familiarity with the finite element method and Fortran. An excellent reference book for more detail on the finite element method is
+This manual contains all of the necessary information for setting up input
+ and running ConMan.
+ It assumes some familiarity with the finite element method and Fortran.
+ An excellent reference book for more detail on the finite element method
+ is
\emph on
The Finite Element Method
\emph default
- by T.J.R. Hughes. All of the data structures and bookkeeping arrays in ConMan follow the conventions in Hughes so for the person who wishes to make extensive use of ConMan, this book is a worthwhile investment.
+ by T.J.R.
+ Hughes.
+ All of the data structures and bookkeeping arrays in ConMan follow the
+ conventions in Hughes so for the person who wishes to make extensive use
+ of ConMan, this book is a worthwhile investment.
\end_layout
\begin_layout Standard
-
-This manual is broken up into several parts: it begins with a brief introduction to the finite element method and the notation that is used throughout the manual and ConMan. There is a discussion of the equations solved and the material properties including how and where to modify the code. There is also discussion of some key points concerning the implementation and finally a description of all the input variables. Within this document the following convention will be followed: subroutine names from ConMan will be given in
+This manual is broken up into several parts: it begins with a brief introduction
+ to the finite element method and the notation that is used throughout the
+ manual and ConMan.
+ There is a discussion of the equations solved and the material properties
+ including how and where to modify the code.
+ There is also discussion of some key points concerning the implementation
+ and finally a description of all the input variables.
+ Within this document the following convention will be followed: subroutine
+ names from ConMan will be given in
\series bold
bold
\series default
@@ -327,91 +345,109 @@
\end_layout
\begin_layout Section
-
Citation
\end_layout
\begin_layout Standard
-
-Computational Infrastructure for Geodynamics (CIG) is making this source code available to you in the hope that the software will enhance your research in geophysics. The ConMan code was donated to CIG in June 2008. A number of individuals have contributed a significant portion of their careers toward the development of ConMan. It is essential that you recognize these individuals in the normal scientific practice by making appropriate acknowledgements.
+Computational Infrastructure for Geodynamics (CIG) is making this source
+ code available to you in the hope that the software will enhance your research
+ in geophysics.
+ The ConMan code was donated to CIG in June 2008.
+ A number of individuals have contributed a significant portion of their
+ careers toward the development of ConMan.
+ It is essential that you recognize these individuals in the normal scientific
+ practice by making appropriate acknowledgements.
\end_layout
\begin_layout Standard
-
The code is based on the method described in
\end_layout
\begin_layout Itemize
-
-King, S.D., A. Raefsky, and B.H. Hager (1990), ConMan: Vectorizing a finite element code for incompressible two-dimensional convection in the Earth's mantle,
+King, S.D., A.
+ Raefsky, and B.H.
+ Hager (1990), ConMan: Vectorizing a finite element code for incompressible
+ two-dimensional convection in the Earth's mantle,
\emph on
-Phys. Earth Planet. Int., 59,
+Phys.
+ Earth Planet.
+ Int., 59,
\emph default
-195-208.
+ 195-208.
+
\end_layout
\begin_layout Standard
+The code was originally developed by Scott King, Arthur Raefsky and Brad
+ Hager, although many people have contributed improvements to ConMan over
+ the past 15 years.
+ The ConMan team requests that in your oral presentations and in your papers
+ that you indicate your use of this code and acknowledge the author of the
+ code and CIG
+\begin_inset LatexCommand url
+target "geodynamics.org"
-The code was originally developed by Scott King, Arthur Raefsky and Brad Hager, although many people have contributed improvements to ConMan over the past 15 years. The ConMan team requests that in your oral presentations and in your papers that you indicate your use of this code and acknowledge the author of the code and CIG
-\begin_inset LatexCommand \url{geodynamics.org}
-
\end_inset
.
\end_layout
\begin_layout Section
-
Support
\end_layout
\begin_layout Standard
-
-ConMan maintenance is supported by a grant from the National Science Foundation to CIG, managed by the California Institute of Technology, under Grant No. EAR-0406751.
+ConMan maintenance is supported by a grant from the National Science Foundation
+ to CIG, managed by the California Institute of Technology, under Grant
+ No.
+ EAR-0406751.
\end_layout
\begin_layout Standard
-
-Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.
+Any opinions, findings, and conclusions or recommendations expressed in
+ this material are those of the author and do not necessarily reflect the
+ views of the National Science Foundation.
\end_layout
\begin_layout Chapter
-
Computational Approach and Governing Equations
\end_layout
\begin_layout Section
-
The Finite Element Method
\end_layout
\begin_layout Standard
-
This section closely follows Hughes (
\emph on
The Finite Element Method
\emph default
) Chapter\InsetSpace ~
-1, sections 1-4. There are two ways we can write the equation, the strong and the weak form. More readers are probably more familiar with the strong form, and less familiar with the weak form. The finite element method is cast in the weak form. In elasticity, for example, the weak form comes from a variational principal, such as the principal of virtual displacements in elasticity. For viscous flow, there is also a variational form, but we will not discuss that here.
+1, sections 1-4.
+ There are two ways we can write the equation, the strong and the weak form.
+ More readers are probably more familiar with the strong form, and less
+ familiar with the weak form.
+ The finite element method is cast in the weak form.
+ In elasticity, for example, the weak form comes from a variational principal,
+ such as the principal of virtual displacements in elasticity.
+ For viscous flow, there is also a variational form, but we will not discuss
+ that here.
\end_layout
\begin_layout Standard
-
-In general, the finite element method takes a differential equation (strong form) and transforms it into an integral equation (weak form).
+In general, the finite element method takes a differential equation (strong
+ form) and transforms it into an integral equation (weak form).
\end_layout
\begin_layout Subsection
-
The Strong Form
\end_layout
\begin_layout Standard
-
For example, the strong form of this simple equation is stated as follows:
\end_layout
\begin_layout Standard
-
Given
\begin_inset Formula $f\left(x\right):\left[0,1\right]$
\end_inset
@@ -436,30 +472,27 @@
\end_layout
\begin_layout Standard
-
-
\begin_inset Formula \begin{equation}
u,_{xx}\left(x\right)+f\left(x\right)=0\label{eq:}\end{equation}
+
\end_inset
\end_layout
\begin_layout Standard
-
-
\begin_inset Formula \begin{equation}
u\left(1\right)=g\label{eq:}\end{equation}
+
\end_inset
\end_layout
\begin_layout Standard
-
-
\begin_inset Formula \begin{equation}
-u,_{x}\left(0\right)=h\label{eq:}\end{equation}
+
\end_inset
@@ -467,27 +500,27 @@
\begin_layout Standard
\noindent
-
-This choice of initial conditions allows us to examine both kinds of boundary conditions. The solution is trivial, but that does not matter. For completeness, it is
+This choice of initial conditions allows us to examine both kinds of boundary
+ conditions.
+ The solution is trivial, but that does not matter.
+ For completeness, it is
\begin_inset Formula \begin{equation}
u(x)=g+(1-x)h+\int_{x}^{1}\left(\int_{0}^{y}f(z)dz\right)dy\end{equation}
+
\end_inset
\end_layout
\begin_layout Subsection
-
The Weak Form
\end_layout
\begin_layout Standard
-
The weak form of the corresponding boundary value problem is stated:
\end_layout
\begin_layout Standard
-
Given
\emph on
f
@@ -500,7 +533,8 @@
\emph on
h
\emph default
-, as before. Find
+, as before.
+ Find
\begin_inset Formula $u\left(x\right)\epsilon\mathcal{L}$
\end_inset
@@ -512,10 +546,9 @@
\end_layout
\begin_layout Standard
-
-
\begin_inset Formula \begin{equation}
\intop_{0}^{1}w,_{x}\left(x\right)u,_{x}\left(x\right)=\intop_{0}^{1}w\left(x\right)f\left(x\right)dx+w\left(0\right)h\label{eq:}\end{equation}
+
\end_inset
@@ -523,8 +556,6 @@
\begin_layout Standard
\noindent
-
-
\begin_inset Formula $\nu$
\end_inset
@@ -532,10 +563,9 @@
\end_layout
\begin_layout Standard
-
-
\begin_inset Formula \begin{equation}
\nu=\left\{ w\left(x\right)|w\left(x\right)\epsilon H^{1},\, w\left(1\right)=0\right\} \label{eq:}\end{equation}
+
\end_inset
@@ -543,7 +573,6 @@
\begin_layout Standard
\noindent
-
and
\begin_inset Formula $\mathcal{L}$
\end_inset
@@ -552,10 +581,9 @@
\end_layout
\begin_layout Standard
-
-
\begin_inset Formula \begin{equation}
\mathcal{L=\left\{ \mathrm{\mathit{u\left(x\right)|u\left(x\right)}\epsilon H^{1},\,\mathit{u}\left(1\right)=\mathit{g}}\right\} }\label{eq:}\end{equation}
+
\end_inset
@@ -563,19 +591,20 @@
\begin_layout Standard
\noindent
-
-
\begin_inset Formula $H^{1}$
\end_inset
- is the set of all functions whose first derivatives are square integrable on [0, 1]. The integral equation is then solved by integrating over each element in the domain and adding the result. The result is a large sparse matrix equation of the form
+ is the set of all functions whose first derivatives are square integrable
+ on [0, 1].
+ The integral equation is then solved by integrating over each element in
+ the domain and adding the result.
+ The result is a large sparse matrix equation of the form
\end_layout
\begin_layout Standard
-
-
\begin_inset Formula \begin{equation}
\left[K\right]x=b\label{eq:}\end{equation}
+
\end_inset
@@ -583,13 +612,14 @@
\begin_layout Standard
\noindent
-
where
\emph on
[K]
\emph default
- is referred to as the element stiffness matrix. There will be more to say about the implementation in Section
-\begin_inset LatexCommand \ref{cha:Input-Guide}
+ is referred to as the element stiffness matrix.
+ There will be more to say about the implementation in Section
+\begin_inset LatexCommand ref
+reference "cha:Input-Guide"
\end_inset
@@ -597,13 +627,18 @@
\end_layout
\begin_layout Subsection
-
Galerkin's Approximation
\end_layout
\begin_layout Standard
-
-Now we have a start on the finite element method. We continue to follow Hughes; however, his notation becomes quite difficult to keep up with. Now, let's begin to think about putting a solution on the computer. Because we will have a finite approximation, related to how fine we space our grid, our solution will only approximate the real solution. Following Hughes' notation, the solution on the grid will be denoted as
+Now we have a start on the finite element method.
+ We continue to follow Hughes; however, his notation becomes quite difficult
+ to keep up with.
+ Now, let's begin to think about putting a solution on the computer.
+ Because we will have a finite approximation, related to how fine we space
+ our grid, our solution will only approximate the real solution.
+ Following Hughes' notation, the solution on the grid will be denoted as
+
\begin_inset Formula $u^{h}$
\end_inset
@@ -611,9 +646,11 @@
\begin_inset Formula $h$
\end_inset
- is some measure of the spacing at the grid. Then,
+ is some measure of the spacing at the grid.
+ Then,
\begin_inset Formula \begin{equation}
\int_{0}^{1}w^{h},_{x}u^{h},_{x}dx~=~\int_{0}^{1}w^{h}\, f^{h}\, dx+w^{h}(0)h.\label{eq:weak}\end{equation}
+
\end_inset
approximates our exact solution
@@ -624,22 +661,31 @@
\end_layout
\begin_layout Standard
-
-On a computer, we don't have a continuous solution. We have a solution at discrete points. We need to approximate the solution between the points (in order to integrate over the function). We will do this with
+On a computer, we don't have a continuous solution.
+ We have a solution at discrete points.
+ We need to approximate the solution between the points (in order to integrate
+ over the function).
+ We will do this with
\series bold
shape functions
\series default
-, as they are usually called in the finite element language. Hughes uses
+, as they are usually called in the finite element language.
+ Hughes uses
\begin_inset Formula $N_{A}A=1,2,\cdots,n$
\end_inset
- to denote the shape functions. You can also think of these as basis functions or interpolation functions. We require
+ to denote the shape functions.
+ You can also think of these as basis functions or interpolation functions.
+ We require
\begin_inset Formula $N_{A}(1)=0,A=1,2,\cdots,n$
\end_inset
-. In order to specify our boundary condition, we need another shape function which has the property
+.
+ In order to specify our boundary condition, we need another shape function
+ which has the property
\begin_inset Formula \begin{equation}
N_{n+1}(1)=1.\end{equation}
+
\end_inset
Then,
@@ -649,11 +695,13 @@
is given by,
\begin_inset Formula \begin{equation}
g^{h}=gN_{n+1}\end{equation}
+
\end_inset
and thus,
\begin_inset Formula \begin{equation}
g^{h}(1)=g.\end{equation}
+
\end_inset
With these definitions, we can write our solution
@@ -663,6 +711,7 @@
as
\begin_inset Formula \begin{equation}
u^{h}=\sum_{A=1}^{n}\, d_{A}\, N_{A}+gN_{n+1}\end{equation}
+
\end_inset
where the
@@ -673,41 +722,49 @@
\end_layout
\begin_layout Standard
-
-In the next section we will make the shape functions more concrete. It is useful to see how general this is, because in principle there is a great deal of flexibility in how we choose the shape functions.
+In the next section we will make the shape functions more concrete.
+ It is useful to see how general this is, because in principle there is
+ a great deal of flexibility in how we choose the shape functions.
\end_layout
\begin_layout Standard
-
We have not said anything more about this function
\begin_inset Formula $w^{h}$
\end_inset
- and how we are going to choose it. If our shape functions form a basis set for the grid, then we can represent
+ and how we are going to choose it.
+ If our shape functions form a basis set for the grid, then we can represent
+
\series bold
any
\series default
- function as a sum of the basis functions times some arbitrary coefficients
+ function as a sum of the basis functions times some arbitrary coefficients
+
\begin_inset Formula $c_{i}$
\end_inset
,
\begin_inset Formula \begin{equation}
w^{h}=\sum_{A=1}^{n}\, c_{A}\, N_{A}=c_{1}\, N_{1}~+~c_{2}\, N_{2}~+~\cdots~+~c_{n}\, N_{n}\label{eq:whapprox}\end{equation}
+
\end_inset
- If you don't remember this part of your mathematics background think of Fourier series. Any function one-dimensional function can be represented as an infinite series of sines and cosines times some unique set of coefficients. The shape functions form a similar kind of basis set.
+ If you don't remember this part of your mathematics background think of
+ Fourier series.
+ Any function one-dimensional function can be represented as an infinite
+ series of sines and cosines times some unique set of coefficients.
+ The shape functions form a similar kind of basis set.
\end_layout
\begin_layout Standard
-
Notice that because we required that
\begin_inset Formula $N_{A}(1)=0,A=1,2,\cdots,n$
\end_inset
, Equation\InsetSpace ~
-\begin_inset LatexCommand \ref{eq:whapprox}
+\begin_inset LatexCommand ref
+reference "eq:whapprox"
\end_inset
@@ -719,7 +776,6 @@
\end_layout
\begin_layout Standard
-
Using our definitions of the
\begin_inset Formula $w^{h}$
\end_inset
@@ -730,33 +786,39 @@
, we can get the messy expression for Equation\InsetSpace ~
-\begin_inset LatexCommand \ref{eq:weak}
+\begin_inset LatexCommand ref
+reference "eq:weak"
\end_inset
\begin_inset Formula \[
\int_{0}^{1}\left(\frac{\partial}{\partial x}\left(\sum_{A=1}^{n}\, c_{A}\, N_{A}\right)\frac{\partial}{\partial x}\left(\sum_{B=1}^{n}\, d_{B}\, N_{B}+gN_{n+1}\right)\right)dx~=~\]
+
\end_inset
\begin_inset Formula \begin{equation}
\int_{0}^{1}\sum_{A=1}^{n}\, c_{A}\, N_{A}\, f^{h}\, dx+\sum_{A=1}^{n}\, c_{A}N_{A}(0)h.\end{equation}
+
\end_inset
By rearranging, we can write
\begin_inset Formula \begin{equation}
\sum_{A=1}^{n}G_{A}c_{A}=0\end{equation}
+
\end_inset
where
\begin_inset Formula \[
G_{A}~=~\int_{0}^{1}\left(\frac{\partial N_{A}}{\partial x}\right)\left(\sum_{B=1}^{n}\, d_{B}\,\frac{\partial N_{B}}{\partial x}\right)dx\]
+
\end_inset
\begin_inset Formula \begin{equation}
-\int_{0}^{1}N_{A}\, f^{h}\, dx-N_{A}(0)h+\int_{0}^{1}\frac{\partial N_{A}}{\partial x}\frac{\partial N_{n+1}}{\partial x}\, g\, dx\end{equation}
+
\end_inset
Now I use the fact that the shape functions are basis functions, so
@@ -767,27 +829,32 @@
\begin_inset Formula $A=B$
\end_inset
-. We could equally well use the fact that the
+.
+ We could equally well use the fact that the
\begin_inset Formula $c_{A}$
\end_inset
-'s are arbitrary. Both of these force us to conclude that each
+'s are arbitrary.
+ Both of these force us to conclude that each
\begin_inset Formula $G_{A}$
\end_inset
must be identically zero and we get
\begin_inset Formula \[
\sum_{B=1}^{n}\,\left(\int_{0}^{1}\frac{\partial N_{A}}{\partial x}\frac{\partial N_{B}}{\partial x}\, dx\right)\, d_{B}=\]
+
\end_inset
\begin_inset Formula \begin{equation}
\int_{0}^{1}N_{A}\, f^{h}\, dx+N_{A}(0)h-g\,\int_{0}^{1}\frac{\partial N_{A}}{\partial x}\frac{\partial N_{n+1}}{\partial x}\, dx\label{eq:feform}\end{equation}
+
\end_inset
Everything in Equation\InsetSpace ~
-\begin_inset LatexCommand \ref{eq:feform}
+\begin_inset LatexCommand ref
+reference "eq:feform"
\end_inset
@@ -795,7 +862,8 @@
\begin_inset Formula $d_{B}$
\end_inset
-'s. This constitutes a system of
+'s.
+ This constitutes a system of
\begin_inset Formula $n$
\end_inset
@@ -803,23 +871,27 @@
\begin_inset Formula $n$
\end_inset
- unknowns. We can think of the left-hand side as a matrix,
+ unknowns.
+ We can think of the left-hand side as a matrix,
\begin_inset Formula $K_{AB}$
\end_inset
whose entries are
\begin_inset Formula \begin{equation}
\int_{0}^{1}\frac{\partial N_{A}}{\partial x}\frac{\partial N_{B}}{\partial x}\, dx\label{eq:Kelements}\end{equation}
+
\end_inset
We can write
\begin_inset Formula \begin{equation}
\sum_{B=1}^{n}\, K_{AB}d_{B}=F_{A},~~~~~A=1,2,\cdots,n\end{equation}
+
\end_inset
or as a matrix equation
\begin_inset Formula \begin{equation}
\left[K\right]\,\{d\}=\{f\}\end{equation}
+
\end_inset
where
@@ -829,6 +901,7 @@
K_{21} & K_{22} & \cdots & K_{2n}\\
\vdots & \vdots & & \vdots\\
K_{n1} & K_{n2} & \cdots & K_{nn}\end{array}\right]\label{Kmatrix}\end{equation}
+
\end_inset
By tradition,
@@ -843,74 +916,102 @@
\begin_inset Formula $\{d\}$
\end_inset
- is the displacement vector. When the problem under consideration pertains to a mechanical system, this makes the most sense, but even in heat conduction problems, or fluid flow problems, the terminology is still (often) retained.
+ is the displacement vector.
+ When the problem under consideration pertains to a mechanical system, this
+ makes the most sense, but even in heat conduction problems, or fluid flow
+ problems, the terminology is still (often) retained.
\end_layout
\begin_layout Subsection
-
Shape Functions
\end_layout
\begin_layout Standard
-
-At this point, we narrow the focus to deal with specifically the elements in ConMan. It is possible to think very general shape functions, but in practice, people use triangles or quadralaterals (in 2D). In terms of the level of approximation, there are also a lot of possibilities. We will stick to the simplest form, bilinear elements, but you should be aware that higher order elements (biquadratic or bicubic spline elements) are also popular with some people. This section is condensing a lot of very useful material from Chapter 3 of Hughes' book into a short overview. If you want to see more complete derivations, proofs of convergence, etc., of how to go about using higher order elements, look at Hughes book, Chapter 3.
+At this point, we narrow the focus to deal with specifically the elements
+ in ConMan.
+ It is possible to think very general shape functions, but in practice,
+ people use triangles or quadralaterals (in 2D).
+ In terms of the level of approximation, there are also a lot of possibilities.
+ We will stick to the simplest form, bilinear elements, but you should be
+ aware that higher order elements (biquadratic or bicubic spline elements)
+ are also popular with some people.
+ This section is condensing a lot of very useful material from Chapter 3
+ of Hughes' book into a short overview.
+ If you want to see more complete derivations, proofs of convergence, etc.,
+ of how to go about using higher order elements, look at Hughes book, Chapter
+ 3.
\end_layout
\begin_layout Standard
-
-Let's start by thinking of a rectangle that is 2a by 2b in length centered at (0,0). There are two properties we would like the shape functions to have
+Let's start by thinking of a rectangle that is 2a by 2b in length centered
+ at (0,0).
+ There are two properties we would like the shape functions to have
\begin_inset Formula \begin{eqnarray}
\sum_{A=1}^{4}N_{A}(X,Y) & = & 1\label{eq:normalize}\\
\sum_{A=1}^{4}N_{A}(X,Y)\, X_{A} & = & X\label{eq:interpx}\\
\sum_{A=1}^{4}N_{A}(X,Y)\, Y_{A} & = & Y\label{eq:interpy}\end{eqnarray}
+
\end_inset
Equation\InsetSpace ~
-\begin_inset LatexCommand \ref{eq:normalize}
+\begin_inset LatexCommand ref
+reference "eq:normalize"
\end_inset
- says that they are normalized, so that they sum to one (everywhere on X,Y). Equations\InsetSpace ~
+ says that they are normalized, so that they sum to one (everywhere on X,Y).
+ Equations\InsetSpace ~
-\begin_inset LatexCommand \ref{eq:interpx}
+\begin_inset LatexCommand ref
+reference "eq:interpx"
\end_inset
and\InsetSpace ~
-\begin_inset LatexCommand \ref{eq:interpy}
+\begin_inset LatexCommand ref
+reference "eq:interpy"
\end_inset
- state that the shape functions are also interpolation functions. Without doing a lot of derivation, I will claim that for the rectangle described above,
+ state that the shape functions are also interpolation functions.
+ Without doing a lot of derivation, I will claim that for the rectangle
+ described above,
\begin_inset Formula \begin{eqnarray}
N_{1} & = & \frac{(a-x)(b-y)}{4ab}\\
N_{2} & = & \frac{(a+x)(b-y)}{4ab}\\
N_{3} & = & \frac{(a+x)(b+y)}{4ab}\\
N_{4} & = & \frac{(a-x)(b+y)}{4ab}\end{eqnarray}
+
\end_inset
these shape functions satisfy the conditions in Equation\InsetSpace ~
-\begin_inset LatexCommand \ref{eq:normalize}
+\begin_inset LatexCommand ref
+reference "eq:normalize"
\end_inset
and Equations \InsetSpace ~
-\begin_inset LatexCommand \ref{eq:interpx}
+\begin_inset LatexCommand ref
+reference "eq:interpx"
\end_inset
and\InsetSpace ~
-\begin_inset LatexCommand \ref{eq:interpy}
+\begin_inset LatexCommand ref
+reference "eq:interpy"
\end_inset
-. A good exercise would be to show this is true. A visual representation of these shape functions is shown in Figure
-\begin_inset LatexCommand \ref{fig:The-bilinear-shape}
+.
+ A good exercise would be to show this is true.
+ A visual representation of these shape functions is shown in Figure
+\begin_inset LatexCommand ref
+reference "fig:The-bilinear-shape"
\end_inset
@@ -920,31 +1021,36 @@
\begin_layout Standard
\noindent
\align center
-
-
\begin_inset Float figure
wide false
sideways false
status open
+\begin_layout Standard
+\begin_inset Caption
-\begin_layout Caption
+\begin_layout Standard
+\begin_inset LatexCommand label
+name "fig:The-bilinear-shape"
+\end_inset
-\begin_inset LatexCommand \label{fig:The-bilinear-shape}
+The bilinear shape function for a single element (top) and the four elements
+ whose shape functions combine to form the global shape function for node
+ A (bottom).
+ Figure taken from Hughes, Section 3.2.
+\end_layout
\end_inset
-The bilinear shape function for a single element (top) and the four elements whose shape functions combine to form the global shape function for node A (bottom). Figure taken from Hughes, Section 3.2.
+
\end_layout
\begin_layout Standard
\noindent
\align center
-
-
-\begin_inset Graphics
- filename images/conman-fig2.eps
+\begin_inset Graphics
+ filename images/conman-fig2.png
scale 40
\end_inset
@@ -958,7 +1064,6 @@
\end_layout
\begin_layout Standard
-
In ConMan we further choose to normalize this by setting
\begin_inset Formula $a=1$
\end_inset
@@ -967,7 +1072,11 @@
\begin_inset Formula $b=1$
\end_inset
-. This choice gives us an element whose area is 1, which is a convenient way to think about things. (This is because we left the factor of 1/4 in the denominator). In my code, to make one less set of computations, I in effect set
+.
+ This choice gives us an element whose area is 1, which is a convenient
+ way to think about things.
+ (This is because we left the factor of 1/4 in the denominator).
+ In my code, to make one less set of computations, I in effect set
\begin_inset Formula $a=0.5$
\end_inset
@@ -979,13 +1088,13 @@
\end_layout
\begin_layout Standard
-
Notice it is pretty easy to take derivatives of these shape functions
\begin_inset Formula \begin{eqnarray}
N_{1,x} & = & \frac{-(1-y)}{4}\\
N_{2,x} & = & \frac{(1-y)}{4}\\
N_{3,x} & = & \frac{(1+y)}{4}\\
N_{4,x} & = & \frac{-(1+y)}{4}\end{eqnarray}
+
\end_inset
@@ -994,23 +1103,31 @@
N_{2,y} & = & \frac{-(1+x)}{4}\\
N_{3,y} & = & \frac{(1+x)}{4}\\
N_{4,y} & = & \frac{(1-x)}{4}\end{eqnarray}
+
\end_inset
- What do we do if we want to solve a problem on a domain that is not convenient to split into a grid of 1 by 1 unit elements? We use an important principle of mathematics, the Jacobian of the transformation
+ What do we do if we want to solve a problem on a domain that is not convenient
+ to split into a grid of 1 by 1 unit elements? We use an important principle
+ of mathematics, the Jacobian of the transformation
\begin_inset Formula \begin{equation}
K_{11}~=~\int_{A}^{B}N_{1,x}N_{1,x}\, dx~=~\int_{0}^{1}N_{1,x}N_{1,x}\, Jdx\end{equation}
+
\end_inset
where
\begin_inset Formula $J$
\end_inset
- is the Jacobian of the transformation. This is a very powerful point. When we are thinking of solving a regular Cartesian domain, this just corresponds to a stretching or a shrinking (notice we set
+ is the Jacobian of the transformation.
+ This is a very powerful point.
+ When we are thinking of solving a regular Cartesian domain, this just correspon
+ds to a stretching or a shrinking (notice we set
\begin_inset Formula $a=b=1$
\end_inset
above (see Figure
-\begin_inset LatexCommand \ref{fig:Figure-1}
+\begin_inset LatexCommand ref
+reference "fig:Figure-1"
\end_inset
@@ -1020,31 +1137,35 @@
\begin_layout Standard
\noindent
\align center
-
-
\begin_inset Float figure
wide false
sideways false
status open
+\begin_layout Standard
+\begin_inset Caption
-\begin_layout Caption
+\begin_layout Standard
+\begin_inset LatexCommand label
+name "fig:Figure-1"
+\end_inset
-\begin_inset LatexCommand \label{fig:Figure-1}
+The mapping between the global domain (right) and the parent element domain
+ (left) using the shape functions.
+ Figure taken from Hughes, Section 3.2.
+\end_layout
\end_inset
-The mapping between the global domain (right) and the parent element domain (left) using the shape functions. Figure taken from Hughes, Section 3.2.
+
\end_layout
\begin_layout Standard
\noindent
\align center
-
-
-\begin_inset Graphics
- filename images/conman-fig1.eps
+\begin_inset Graphics
+ filename images/conman-fig1.png
scale 45
\end_inset
@@ -1058,13 +1179,15 @@
\end_layout
\begin_layout Standard
-
-However, if we are thinking about a cylindrical geometry, for example, we can use the Jacobian of the transformation between the geometries. Let's look at two examples:
+However, if we are thinking about a cylindrical geometry, for example, we
+ can use the Jacobian of the transformation between the geometries.
+ Let's look at two examples:
\end_layout
\begin_layout Standard
-
-Converting an element 0.05 by 0.10 centered at (0.1,0.2) to the `parent element' centered at (0,0). Hughes also uses
+Converting an element 0.05 by 0.10 centered at (0.1,0.2) to the `parent element'
+ centered at (0,0).
+ Hughes also uses
\begin_inset Formula $\xi,\eta$
\end_inset
@@ -1076,10 +1199,11 @@
\begin_inset Formula \begin{eqnarray}
x & = & 0.1+{0.05}\xi+0.0\eta\\
y & = & 0.2+{0.0}\xi+{0.10}\eta\end{eqnarray}
+
\end_inset
or in matrix form we could write
-\begin_inset Formula \begin{equation}\label{eq:transform}
+\begin_inset Formula \begin{equation}
\left\{ \begin{array}{c}
x\\
y\end{array}\right\} =\left[\begin{array}{cc}
@@ -1088,75 +1212,88 @@
\xi\\
\eta\end{array}\right\} +\left\{ \begin{array}{c}
0.1\\
-0.2\end{array}\right\} \end{equation}
+0.2\end{array}\right\} \label{eq:transform}\end{equation}
+
\end_inset
- If the transformation was from an arbitratry shaped quadrilateral to the parent element, then the off diagonal terms in the mateix in equation\InsetSpace ~
+ If the transformation was from an arbitratry shaped quadrilateral to the
+ parent element, then the off diagonal terms in the matrix in equation\InsetSpace ~
-\begin_inset LatexCommand \ref{eq:transform}
+\begin_inset LatexCommand ref
+reference "eq:transform"
\end_inset
- will not be zero. It is easy enough to show that
+ will not be zero.
+ It is easy enough to show that
\begin_inset Formula \begin{equation}
\int_{x_{1}}^{x_{2}}\int_{y_{1}}^{y_{2}}f(x,y)\, dx\, dy=\int_{-1}^{1}\int_{-1}^{1}f(\xi,\eta)\,{\rm det}[J]\, d\xi\, d\eta\end{equation}
+
\end_inset
where
\begin_inset Formula $[J]$
\end_inset
- is the Jacobian of the transformation. It turns out, and it is also easy to show, that
+ is the Jacobian of the transformation.
+ It turns out, and it is also easy to show, that
\begin_inset Formula ${\rm det}[J]$
\end_inset
- is the ratio of the areas when going from one rectangle to another (in fact any Cartesian to Cartesian transformation).
+ is the ratio of the areas when going from one rectangle to another (in
+ fact any Cartesian to Cartesian transformation).
\end_layout
\begin_layout Standard
-
\series bold
Advanced Topic:
\series default
- Now suppose we want to map a cylindrical domain to our `parent element.' We can use the same principle in this case:
+ Now suppose we want to map a cylindrical domain to our `parent element.'
+ We can use the same principle in this case:
\begin_inset Formula \begin{eqnarray}
x & = & r\cos\theta=\cos\theta\,\xi-r\sin\theta\,\eta\\
y & = & r\sin\theta=\sin\theta\,\xi+r\cos\theta\,\eta\end{eqnarray}
+
\end_inset
so
\begin_inset Formula \begin{equation}
{\rm det}[J_{geometry}]~=~r\cos^{2}\theta+r\sin^{2}\theta=r.\end{equation}
+
\end_inset
of we would get
\begin_inset Formula \begin{equation}
\int_{r_{1}}^{r_{2}}\int_{\theta_{1}}^{\theta_{2}}f(r\cos\theta,r\sin\theta)\, r\, dr\, d\theta=\int_{-1}^{1}\int_{-1}^{1}f(\xi,\eta)\,{\rm det}[J_{area}]\, d\xi\, d\eta\end{equation}
+
\end_inset
\end_layout
\begin_layout Subsubsection
-
Gauss Quadrature
\end_layout
\begin_layout Standard
-
-An amazing fact, that makes the idea of finite elements easy and powerful, is Gauss Quadrature. Gauss Quadrature is a way to turn an integral into a summation. Let's begin with a 1D example,
+An amazing fact, that makes the idea of finite elements easy and powerful,
+ is Gauss Quadrature.
+ Gauss Quadrature is a way to turn an integral into a summation.
+ Let's begin with a 1D example,
\begin_inset Formula $f(x)=c$
\end_inset
\begin_inset Formula \begin{equation}
\int_{-1}^{1}c\, dx=cx|_{-1}^{1}=2c\end{equation}
+
\end_inset
Gauss noted that for any linear function
\begin_inset Formula \begin{equation}
\int_{-1}^{1}f(x)\, dx=2.0\times f(0)=2c\end{equation}
+
\end_inset
For a linear function,
@@ -1166,16 +1303,19 @@
\begin_inset Formula \begin{equation}
\int_{-1}^{1}f(x)\, dx=f(\frac{-1}{\sqrt{3}})+f(\frac{-1}{\sqrt{3}})\end{equation}
+
\end_inset
The direct way,
\begin_inset Formula \begin{equation}
\int_{-1}^{1}(ax+b)\, dx=(\frac{ax^{2}}{2}+bx)|_{-1}^{1}=\frac{a}{2}+b-(\frac{a}{2}-b)=2b\end{equation}
+
\end_inset
Gauss' way
\begin_inset Formula \begin{equation}
\int_{-1}^{1}(ax+b)\, dx=a\frac{-1}{\sqrt{3}}+b+a\frac{1}{\sqrt{3}}+b=2b\end{equation}
+
\end_inset
It turns out that
@@ -1186,11 +1326,13 @@
\begin_inset Formula $-x,x$
\end_inset
- combination, but what Gauss showed was more powerful, that if the function is of higher order, the
+ combination, but what Gauss showed was more powerful, that if the function
+ is of higher order, the
\begin_inset Formula $\frac{-1}{\sqrt{3}},\frac{1}{\sqrt{3}}$
\end_inset
- choice is the best approximation you can make with only two terms. If we go to three terms, the choice would be
+ choice is the best approximation you can make with only two terms.
+ If we go to three terms, the choice would be
\begin_inset Formula $-\sqrt{\frac{3}{5}},0,\sqrt{\frac{3}{5}}$
\end_inset
@@ -1198,16 +1340,18 @@
\end_layout
\begin_layout Standard
-
-To integrate a 2D Cartesian region, like our parent element, it turns out that 2 by 2 quadrature, or the four points
+To integrate a 2D Cartesian region, like our parent element, it turns out
+ that 2 by 2 quadrature, or the four points
\begin_inset Formula \begin{eqnarray}
\xi=\frac{-1}{\sqrt{3}} & ~~~ & \eta=\frac{-1}{\sqrt{3}}\\
\xi=\frac{1}{\sqrt{3}} & ~~~ & \eta=\frac{-1}{\sqrt{3}}\\
\xi=\frac{1}{\sqrt{3}} & ~~~ & \eta=\frac{1}{\sqrt{3}}\\
\xi=\frac{-1}{\sqrt{3}} & ~~~ & \eta=\frac{1}{\sqrt{3}}\end{eqnarray}
+
\end_inset
- are sufficient to exactly integrate our bilinear shape functions over the
+ are sufficient to exactly integrate our bilinear shape functions over the
+
\begin_inset ERT
status collapsed
@@ -1255,31 +1399,50 @@
\end_layout
\begin_layout Standard
-
-At this point, it would be worth talking about the code ConMan for a minute. The shape functions are generated in ConMan in the routine
+At this point, it would be worth talking about the code ConMan for a minute.
+ The shape functions are generated in ConMan in the routine
\series bold
genshp
\series default
- for GENerate SHape functions Parent domain. If you look at the routine you will find the first part of it is pretty easy to follow from the discussion above. Some of the second part is a little tricky in the details, but generally it is also pretty easy to follow.
+ for GENerate SHape functions Parent domain.
+ If you look at the routine you will find the first part of it is pretty
+ easy to follow from the discussion above.
+ Some of the second part is a little tricky in the details, but generally
+ it is also pretty easy to follow.
\end_layout
\begin_layout Standard
-
The subroutine
\series bold
genshg
\series default
- deals with the global shape functions (i.e., deals with the geometry and size). Originally, we called this routine once and stored all the shape functions. As problem sizes have grown, this took a lot of storage, so now we call it on the fly for each element when needed. It is computationally more expensive but cuts the storage requirement. This strategy will also be necessary for a Lagrangian formulation or adaptive gridding.
+ deals with the global shape functions (i.e., deals with the geometry and
+ size).
+ Originally, we called this routine once and stored all the shape functions.
+ As problem sizes have grown, this took a lot of storage, so now we call
+ it on the fly for each element when needed.
+ It is computationally more expensive but cuts the storage requirement.
+ This strategy will also be necessary for a Lagrangian formulation or adaptive
+ gridding.
\end_layout
\begin_layout Standard
+There are two domains to keep in mind when thinking about the finite element
+ method: the global domain and the parent element domain (Figure
+\begin_inset LatexCommand ref
+reference "fig:Figure-1"
-There are two domains to keep in mind when thinking about the finite element method: the global domain and the parent element domain (Figure
-\begin_inset LatexCommand \ref{fig:Figure-1}
-
\end_inset
-). All calculations are done in the parent element domain and the results are assembled into the global equations. This means all calculations can be done for a single parent element. Elements of different sizes or shapes filling an irregular global domain geometry (i.e., non-rectangular) can be solved by the same program. The only difference between these elements is the Jacobian of the transformation between the input domain and the parent element domain, which is calculated in routine
+).
+ All calculations are done in the parent element domain and the results
+ are assembled into the global equations.
+ This means all calculations can be done for a single parent element.
+ Elements of different sizes or shapes filling an irregular global domain
+ geometry (i.e., non-rectangular) can be solved by the same program.
+ The only difference between these elements is the Jacobian of the transformatio
+n between the input domain and the parent element domain, which is calculated
+ in routine
\series bold
genshg
\series default
@@ -1287,23 +1450,30 @@
\end_layout
\begin_layout Standard
+For ConMan the choice was made to use bilinear quadrilaterals as the parent
+ elements (Figure
+\begin_inset LatexCommand ref
+reference "fig:The-bilinear-shape"
-For ConMan the choice was made to use bilinear quadrilaterals as the parent elements (Figure
-\begin_inset LatexCommand \ref{fig:The-bilinear-shape}
-
\end_inset
-). Higher order elements (i.e., biquadratic or bicubic-spline) require more computational work per element. It has been our experience that using grid refinement, rather than using high-order elements, is the best strategy for an efficient, accurate code for incompressible, advection-diffusion problems.
+).
+ Higher order elements (i.e., biquadratic or bicubic-spline) require more
+ computational work per element.
+ It has been our experience that using grid refinement, rather than using
+ high-order elements, is the best strategy for an efficient, accurate code
+ for incompressible, advection-diffusion problems.
\end_layout
\begin_layout Standard
-
-Because of the changing between domains, it is necessary to define several bookkeeping arrays to identify nodes and elements in each of the domains. It is worth noting that the numbering of local elements always begins with 1 in the lower left-hand corner. There is no special reason; you just have to choose a convention.
+Because of the changing between domains, it is necessary to define several
+ bookkeeping arrays to identify nodes and elements in each of the domains.
+ It is worth noting that the numbering of local elements always begins with
+ 1 in the lower left-hand corner.
+ There is no special reason; you just have to choose a convention.
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -1370,16 +1540,16 @@
\end_inset
transforms global nodes to equation numbers (Figure
-\begin_inset LatexCommand \ref{fig:Example-relationship-between}
+\begin_inset LatexCommand ref
+reference "fig:Example-relationship-between"
\end_inset
-).
+).
+
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -1446,16 +1616,16 @@
\end_inset
transforms element local node numbers to global node numbers (Figure
-\begin_inset LatexCommand \ref{fig:4-Example-relationship-between}
+\begin_inset LatexCommand ref
+reference "fig:4-Example-relationship-between"
\end_inset
-).
+).
+
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -1521,37 +1691,43 @@
\end_inset
- transforms element local node numbers to global equation numbers.
+ transforms element local node numbers to global equation numbers.
+
\end_layout
\begin_layout Standard
\noindent
\align center
-
-
\begin_inset Float figure
wide false
sideways false
status open
+\begin_layout Standard
+\begin_inset Caption
-\begin_layout Caption
+\begin_layout Standard
+\begin_inset LatexCommand label
+name "fig:Example-relationship-between"
+\end_inset
-\begin_inset LatexCommand \label{fig:Example-relationship-between}
+ Example relationship between global nodes and equation numbers for a 2
+ degree of freedom problem using the id array.
+ An equation number of zero denotes a boundary condition.
+ Figure taken from Hughes, Section 3.2.
+\end_layout
\end_inset
- Example relationship between global nodes and equation numbers for a 2 degree of freedom problem using the id array. An equation number of zero denotes a boundary condition. Figure taken from Hughes, Section 3.2.
+
\end_layout
\begin_layout Standard
\noindent
\align center
-
-
-\begin_inset Graphics
- filename images/conman-fig3.eps
+\begin_inset Graphics
+ filename images/conman-fig3.png
scale 40
\end_inset
@@ -1565,38 +1741,53 @@
\end_layout
\begin_layout Standard
-
-With these, the code is able to go back and forth between the parent element domain and the global domain. Global node numbering is specified by the user, and equation numbers are assigned by the code to denote the row in the stiffness matrix corresponding to the degree(s) of freedom for that node. One global node may have more than one equation number (since there may be more than one degree of freedom per node). Boundary conditions are specified with a zero equation number. Since it is a sparse matrix, it is desirable to permute the stiffness matrix for computational efficiency. These arrays spare the user from dealing with the transformations, while making the code efficient.
+With these, the code is able to go back and forth between the parent element
+ domain and the global domain.
+ Global node numbering is specified by the user, and equation numbers are
+ assigned by the code to denote the row in the stiffness matrix corresponding
+ to the degree(s) of freedom for that node.
+ One global node may have more than one equation number (since there may
+ be more than one degree of freedom per node).
+ Boundary conditions are specified with a zero equation number.
+ Since it is a sparse matrix, it is desirable to permute the stiffness matrix
+ for computational efficiency.
+ These arrays spare the user from dealing with the transformations, while
+ making the code efficient.
\end_layout
\begin_layout Standard
\noindent
\align center
-
-
\begin_inset Float figure
wide false
sideways false
status open
+\begin_layout Standard
+\begin_inset Caption
-\begin_layout Caption
+\begin_layout Standard
+\begin_inset LatexCommand label
+name "fig:4-Example-relationship-between"
+\end_inset
-\begin_inset LatexCommand \label{fig:4-Example-relationship-between}
+Example relationship between global node numbers and local element numbers
+ using the ien array.
+ Local nodes are numbered counterclockwise from the bottom left-hand corner.
+ Figure taken from Hughes, Section 3.2.
+\end_layout
\end_inset
-Example relationship between global node numbers and local element numbers using the ien array. Local nodes are numbered counterclockwise from the bottom left-hand corner. Figure taken from Hughes, Section 3.2.
+
\end_layout
\begin_layout Standard
\noindent
\align center
-
-
-\begin_inset Graphics
- filename images/conman-fig4.eps
+\begin_inset Graphics
+ filename images/conman-fig4.png
scale 40
\end_inset
@@ -1610,13 +1801,10 @@
\end_layout
\begin_layout Standard
-
In the code, the data structures for these two arrays are
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -1686,8 +1874,6 @@
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -1757,8 +1943,6 @@
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -1824,27 +2008,30 @@
\end_inset
- ( degree-of-freedom, local-node-number, element-number ) = global-equation-number
+ ( degree-of-freedom, local-node-number, element-number ) = global-equation-numb
+er
\end_layout
\begin_layout Subsection
-
The Element Point of View
\end_layout
\begin_layout Standard
-
-Transforming between the element and global points of view is done with the data structure called the IEN array for Element to Node transformation. The IEN array takes an element number and a local node number, and its value is the global node number. It is easiest to look at some examples:
+Transforming between the element and global points of view is done with
+ the data structure called the IEN array for Element to Node transformation.
+ The IEN array takes an element number and a local node number, and its
+ value is the global node number.
+ It is easiest to look at some examples:
\end_layout
\begin_layout Standard
-
-Consider a 3-element by 3-element grid. I will number my elements and nodes starting in the lower left hand corner, and the node and element numbers will increase linearly in the vertical direction.
+Consider a 3-element by 3-element grid.
+ I will number my elements and nodes starting in the lower left hand corner,
+ and the node and element numbers will increase linearly in the vertical
+ direction.
\end_layout
\begin_layout Standard
-
-
\begin_inset ERT
status collapsed
@@ -1857,12 +2044,13 @@
\end_inset
- For element 3: ien (3, 1) = 3 ien (3, 2) = 7 ien (3, 3) = 8 ien (3, 4) = 4
+ For element 3: ien (3, 1) = 3 ien (3, 2) = 7 ien (3, 3) = 8 ien (3, 4)
+ = 4
\end_layout
\begin_layout Standard
-
-For element 5: ien (5, 1) = 6 ien (5, 2) = 10 ien (5, 3) = 11 ien (5, 4) = 7
+For element 5: ien (5, 1) = 6 ien (5, 2) = 10 ien (5, 3) = 11 ien (5, 4)
+ = 7
\begin_inset ERT
status collapsed
@@ -1879,8 +2067,10 @@
\end_layout
\begin_layout Standard
-
-There are three kinds of operations we might want to use. The first is taking the values of some function (coordinates, velocities, temperatures, stresses, etc.) defined on the global grid and getting the values for a single element, which is called a
+There are three kinds of operations we might want to use.
+ The first is taking the values of some function (coordinates, velocities,
+ temperatures, stresses, etc.) defined on the global grid and getting the
+ values for a single element, which is called a
\begin_inset ERT
status collapsed
@@ -1915,7 +2105,9 @@
\end_inset
- operation. The next is taking values in an element and spreading them out to the global array, which is called a
+ operation.
+ The next is taking values in an element and spreading them out to the global
+ array, which is called a
\begin_inset ERT
status collapsed
@@ -1950,7 +2142,9 @@
\end_inset
- operation. The third operation is to take the value at a local node and add it to the global value for that node, an
+ operation.
+ The third operation is to take the value at a local node and add it to
+ the global value for that node, an
\begin_inset ERT
status collapsed
@@ -1989,12 +2183,13 @@
\end_layout
\begin_layout Standard
-
-All three operations -- gather, scatter, and assemble -- are done by the routine
+All three operations -- gather, scatter, and assemble -- are done by the
+ routine
\series bold
local
\series default
-. Because it is called by the
+.
+ Because it is called by the
\series bold
genshp
\series default
@@ -2002,31 +2197,32 @@
\end_layout
\begin_layout Subsection
-
Equations
\end_layout
\begin_layout Standard
-
-
\begin_inset Formula \begin{eqnarray}
\tau_{ij,j}+f_{i} & = & 0\label{eq:motion}\\
u_{i,i} & = & 0\end{eqnarray}
+
\end_inset
where
\begin_inset Formula \begin{equation}
\tau_{ij}=-p\delta_{ij}+2\mu u_{(i,j)}\label{eq:constit}\end{equation}
+
\end_inset
where
\begin_inset Formula \begin{equation}
u_{(i,j)}=(u_{i,j}+u_{j,i})/2\end{equation}
+
\end_inset
We replace Equation\InsetSpace ~
-\begin_inset LatexCommand \ref{eq:constit}
+\begin_inset LatexCommand ref
+reference "eq:constit"
\end_inset
@@ -2034,13 +2230,15 @@
\begin_inset Formula \begin{eqnarray}
\tau_{ij}=-p^{\lambda}\delta_{ij}+2\mu u_{(i,j)}\label{eq:constit-lam}\\
0=u_{i,i}+p^{\lambda}/\lambda.\label{eq:constit-lam2}\end{eqnarray}
+
\end_inset
As
\begin_inset Formula $\lambda$
\end_inset
- approaches infinity, these relations approach the incompressible solution. Also, as
+ approaches infinity, these relations approach the incompressible solution.
+ Also, as
\begin_inset Formula $\lambda$
\end_inset
@@ -2048,49 +2246,59 @@
\begin_inset Formula $p^{\lambda}$
\end_inset
- approaches the hydrostatic pressure in the incompressible case. In general, the hydrostatic pressure is
+ approaches the hydrostatic pressure in the incompressible case.
+ In general, the hydrostatic pressure is
\begin_inset Formula $-\tau_{ii}/3$
\end_inset
-. Substituting Equation\InsetSpace ~
+.
+ Substituting Equation\InsetSpace ~
-\begin_inset LatexCommand \ref{eq:constit-lam2}
+\begin_inset LatexCommand ref
+reference "eq:constit-lam2"
\end_inset
into\InsetSpace ~
-\begin_inset LatexCommand \ref{eq:constit-lam}
+\begin_inset LatexCommand ref
+reference "eq:constit-lam"
\end_inset
we get
\begin_inset Formula \begin{equation}
\tau_{ij}=\lambda u_{i,i}\delta_{ij}+2\mu u_{(i,j)}\label{eq:constit-lam-final}\end{equation}
+
\end_inset
or
\begin_inset Formula \begin{equation}
\tau_{ii}=3\lambda u_{i,i}+2\mu u_{i,i}\end{equation}
+
\end_inset
or
\begin_inset Formula \begin{equation}
\tau_{ii}/3=-p=(\lambda+2/3\mu)u_{i,i}\end{equation}
+
\end_inset
but we also have
\begin_inset Formula \begin{equation}
-p^{\lambda}=\lambda u_{i,i}\end{equation}
+
\end_inset
from Equation\InsetSpace ~
-\begin_inset LatexCommand \ref{eq:constit-lam2}
+\begin_inset LatexCommand ref
+reference "eq:constit-lam2"
\end_inset
-. Clearly in the incompressible limit
+.
+ Clearly in the incompressible limit
\begin_inset Formula $\lambda\gg\mu$
\end_inset
@@ -2102,63 +2310,82 @@
\begin_inset Formula $p^{\lambda}\rightarrow p$
\end_inset
-. Also note that the continuity equation is satisfied.
+.
+ Also note that the continuity equation is satisfied.
\end_layout
\begin_layout Standard
-
Now, substituting Equation\InsetSpace ~
-\begin_inset LatexCommand \ref{eq:constit-lam-final}
+\begin_inset LatexCommand ref
+reference "eq:constit-lam-final"
\end_inset
into Equation\InsetSpace ~
-\begin_inset LatexCommand \ref{eq:motion}
+\begin_inset LatexCommand ref
+reference "eq:motion"
\end_inset
we have
\begin_inset Formula \begin{equation}
\{\lambda u_{i,i}\delta_{ij}+2\mu u_{(i,j)}\},j+f_{i}=0\end{equation}
+
\end_inset
- At this point, it is probably easier to switch to differential notation. These will also specialize to 2D:
+ At this point, it is probably easier to switch to differential notation.
+ These will also specialize to 2D:
\begin_inset Formula \begin{eqnarray}
\frac{\partial}{\partial x}\{\lambda(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial z})+2\mu(\frac{\partial u}{\partial x}+\frac{\partial u}{\partial x})/2\}+\frac{\partial}{\partial z}\{2\mu(\frac{\partial v}{\partial x}+\frac{\partial u}{\partial z})/2\}+f_{x}=0\\
\frac{\partial}{\partial x}\{2\mu(\frac{\partial u}{\partial z}+\frac{\partial v}{\partial x})/2\}+\frac{\partial}{\partial z}\{\lambda(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial z})+2\mu(\frac{\partial v}{\partial z}+\frac{\partial v}{\partial z})/2\}+f_{z}=0\end{eqnarray}
+
\end_inset
\end_layout
\begin_layout Standard
-
-These are second order partial differential equations. Simplifying, we get
+These are second order partial differential equations.
+ Simplifying, we get
\begin_inset Formula \begin{eqnarray}
\lambda(\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}v}{\partial x\partial z})+2\mu\frac{\partial^{2}u}{\partial x^{2}}+\mu(\frac{\partial^{2}u}{\partial z^{2}}+\frac{\partial^{2}v}{\partial z\partial x})+f_{x}=0\\
\lambda(\frac{\partial^{2}u}{\partial z\partial x}+\frac{\partial^{2}v}{\partial z^{2}})+\mu(\frac{\partial^{2}u}{\partial x\partial z}+\frac{\partial^{2}v}{\partial x^{2}})+2\mu\frac{\partial^{2}v}{\partial z^{2}}+f_{z}=0\end{eqnarray}
+
\end_inset
\end_layout
\begin_layout Standard
-
-Now we use the same technique (approach) as we used in Possion's equation to turn the differential form into an integral form. You can either look at it as we find the variational form of the Stokes equation (which is what we are doing) or you can think of it as multiplying by a weighting function
+Now we use the same technique (approach) as we used in Possion's equation
+ to turn the differential form into an integral form.
+ You can either look at it as we find the variational form of the Stokes
+ equation (which is what we are doing) or you can think of it as multiplying
+ by a weighting function
\begin_inset Formula $w$
\end_inset
- and integrating over the domain. Then using integration by parts to convert the second derivatives to first derivatives. This is done carefully by Hughes in
+ and integrating over the domain.
+ Then using integration by parts to convert the second derivatives to first
+ derivatives.
+ This is done carefully by Hughes in
\emph on
The Finite Element Method
\emph default
- on pages 197-200, but he has left out a number of intermediate steps. Nothing about this step is hard, just tedious. There is, however, a clever shortcut. If we return to the messy equations at the top of the page, multiply them by the weighting function
+ on pages 197-200, but he has left out a number of intermediate steps.
+ Nothing about this step is hard, just tedious.
+ There is, however, a clever shortcut.
+ If we return to the messy equations at the top of the page, multiply them
+ by the weighting function
\begin_inset Formula $w$
\end_inset
- and integrate over the domain, then we do not have to use integration by parts. To see this for yourself, simply take the equations directly above this paragraph, multiply by a weighting function
+ and integrate over the domain, then we do not have to use integration by
+ parts.
+ To see this for yourself, simply take the equations directly above this
+ paragraph, multiply by a weighting function
\begin_inset Formula $w$
\end_inset
@@ -2166,22 +2393,26 @@
\begin_inset Formula $\Omega$
\end_inset
-, then use integration by parts. You will find (after a little algebra)
+, then use integration by parts.
+ You will find (after a little algebra)
\end_layout
\begin_layout Standard
-
-
\begin_inset Formula \begin{eqnarray}
\int\int_{\Omega}\frac{\partial w}{\partial x}\{\lambda(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial z})+2\mu\frac{\partial u}{\partial x}\}+\frac{\partial w}{\partial z}\{2\mu(\frac{\partial v}{\partial x}+\frac{\partial u}{\partial z})/2\}\, d\Omega+\nonumber \\
\int\int_{\Omega}f_{x}\, w\, d\Omega=b.c.~terms\\
\int\int_{\Omega}\frac{\partial w}{\partial x}\{2\mu(\frac{\partial u}{\partial z}+\frac{\partial v}{\partial x})/2\}+\frac{\partial w}{\partial z}\{\lambda(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial z})+2\mu\frac{\partial v}{\partial z}\}\, d\Omega+\nonumber \\
\int\int_{\Omega}f_{z}\, w\, d\Omega=b.c.~terms\end{eqnarray}
+
\end_inset
- Note that we don't get something for nothing; this shortcut does not give us the boundary condition terms (velocity or flux). These would fall out of the integration by parts. Recall,
+ Note that we don't get something for nothing; this shortcut does not give
+ us the boundary condition terms (velocity or flux).
+ These would fall out of the integration by parts.
+ Recall,
\begin_inset Formula \begin{equation}
\int_{a}^{b}w\, dv=w\, v|_{a}^{b}-\int_{a}^{b}v\, dw\end{equation}
+
\end_inset
where in our case
@@ -2192,12 +2423,17 @@
\begin_inset Formula $v$
\end_inset
- is the second derivative term. The first term gives us the flux (first derivative) boundary conditions. In the case of the momentum equations, that is the applied tractions (or stress boundary conditions).
+ is the second derivative term.
+ The first term gives us the flux (first derivative) boundary conditions.
+ In the case of the momentum equations, that is the applied tractions (or
+ stress boundary conditions).
\end_layout
\begin_layout Standard
-
-Now we make use of Galerkin's approximation, or more simply, we use the same weighting functions as we use for interpolation function, i.e., the shape functions, N. So we substitute
+Now we make use of Galerkin's approximation, or more simply, we use the
+ same weighting functions as we use for interpolation function, i.e., the
+ shape functions, N.
+ So we substitute
\begin_inset Formula \begin{eqnarray}
\frac{\partial w}{\partial x}=N_{x}\\
\frac{\partial w}{\partial z}=N_{z}\\
@@ -2205,21 +2441,24 @@
\frac{\partial u}{\partial z}=u\, N_{z}\\
\frac{\partial v}{\partial x}=v\, N_{x}\\
\frac{\partial v}{\partial z}=v\, N_{z}\end{eqnarray}
+
\end_inset
- into our weak form equations. Although messy, that is straight-forward.
+ into our weak form equations.
+ Although messy, that is straight-forward.
+
\begin_inset Formula \begin{eqnarray}
\int\int_{\Omega}N_{x}\{\lambda(u\, N_{x}+v\, N_{z})+2\mu u\, N_{x}\}+N_{z}\{\mu(v\, N_{x}+u\, N_{z})\}\, d\Omega+\nonumber \\
\int\int_{\Omega}f_{x}\, w\, d\Omega=b.c.~terms\\
\int\int_{\Omega}N_{x}\{\mu(u\, N_{z}+v\, N_{x})\}+N_{z}\{\lambda(u\, N_{x}+v\, N_{z})+2\mu v\, N_{z}\}\, d\Omega+\nonumber \\
\int\int_{\Omega}f_{z}\, w\, d\Omega=b.c.~terms\end{eqnarray}
+
\end_inset
\end_layout
\begin_layout Standard
-
At this point, it is useful to separate the equations into a
\begin_inset Formula $\lambda$
\end_inset
@@ -2228,11 +2467,13 @@
\begin_inset Formula $\mu$
\end_inset
- part. We can also write them as a 2D matrix equation
+ part.
+ We can also write them as a 2D matrix equation
\begin_inset Formula \begin{equation}
[K_{\lambda}]=\left[\begin{array}{cc}
N_{x}\lambda N_{x} & N_{x}\lambda N_{z}\\
N_{z}\lambda N_{x} & N_{z}\lambda N_{z}\end{array}\right]\label{eq:Klambda}\end{equation}
+
\end_inset
and
@@ -2240,21 +2481,26 @@
[K_{\mu}]=\left[\begin{array}{cc}
N_{x}2\mu N_{x}+N_{z}\mu N_{z} & N_{z}\mu N_{x}\\
N_{x}\mu N_{z} & N_{z}2\mu N_{z}+N_{x}\mu N_{x}\end{array}\right].\label{eq:Kmu}\end{equation}
+
\end_inset
\end_layout
\begin_layout Standard
-
-Hughes makes use of an interesting and important observation. This observation will greatly simplify constructing the stiffness matrix for arbitrary coordinate systems. We can rewrite the stiffness matrices above in the following form:
+Hughes makes use of an interesting and important observation.
+ This observation will greatly simplify constructing the stiffness matrix
+ for arbitrary coordinate systems.
+ We can rewrite the stiffness matrices above in the following form:
\begin_inset Formula \begin{equation}
[K_{\lambda}]+[K_{\mu}]=[B]^{T}[D][B]\label{eq:84}\end{equation}
+
\end_inset
\begin_inset Formula \begin{equation}
[D_{\lambda}]+[D_{\mu}]=[D]\end{equation}
+
\end_inset
where
@@ -2263,6 +2509,7 @@
2 & 0 & 0\\
0 & 2 & 0\\
0 & 0 & 1\end{array}\right]\end{equation}
+
\end_inset
and
@@ -2271,6 +2518,7 @@
1 & 1 & 0\\
1 & 1 & 0\\
0 & 0 & 0\end{array}\right]\end{equation}
+
\end_inset
and
@@ -2279,14 +2527,26 @@
N_{x} & 0\\
0 & N_{z}\\
N_{z} & N_{x}\end{array}\right].\end{equation}
+
\end_inset
\end_layout
\begin_layout Standard
-
-The momentum and energy equations form a simple coupled system of differential equations. We treat the incompressibility equation as a constraint on the momentum equation and enforce incompressibility in the solution of the momentum equation using a penalty formulation described below. Since the temperatures provide the buoyancy (body force) to drive the momentum equation and since there is no time-dependence in the momentum equation, the algorithm to solve the system is a simple one: Given an initial temperature field, calculate the resulting velocity field. Use the velocities to advect the temperatures for the next time step and solve for a new temperature field. If the time stepping for the temperature equation is stable, then this method is stable and converges as
+The momentum and energy equations form a simple coupled system of differential
+ equations.
+ We treat the incompressibility equation as a constraint on the momentum
+ equation and enforce incompressibility in the solution of the momentum
+ equation using a penalty formulation described below.
+ Since the temperatures provide the buoyancy (body force) to drive the momentum
+ equation and since there is no time-dependence in the momentum equation,
+ the algorithm to solve the system is a simple one: Given an initial temperature
+ field, calculate the resulting velocity field.
+ Use the velocities to advect the temperatures for the next time step and
+ solve for a new temperature field.
+ If the time stepping for the temperature equation is stable, then this
+ method is stable and converges as
\begin_inset Formula $\Delta t\rightarrow0$
\end_inset
@@ -2294,29 +2554,41 @@
\end_layout
\begin_layout Standard
-
The element stiffness matrix (Equation\InsetSpace ~
-\begin_inset LatexCommand \ref{eq:84}
+\begin_inset LatexCommand ref
+reference "eq:84"
\end_inset
-) is made up of the two terms from the left hand side of the integral equation. The full element stiffness matrix for the quadralilateral element is an 8 by 8 matrix made up of 16 of the 2 by 2 matrices as shown in Figure\InsetSpace ~
+) is made up of the two terms from the left hand side of the integral equation.
+ The full element stiffness matrix for the quadralilateral element is an
+ 8 by 8 matrix made up of 16 of the 2 by 2 matrices as shown in Figure\InsetSpace ~
-\begin_inset LatexCommand \ref{fig:8-The-storage-for}
+\begin_inset LatexCommand ref
+reference "fig:8-The-storage-for"
\end_inset
-. Because of symmetry we only need to form and store the upper triangular part of the matrix. The integration is done using two by two gauss quadrature, which is exact when the elements are rectangular and bilinear shape functions are used. The
+.
+ Because of symmetry we only need to form and store the upper triangular
+ part of the matrix.
+ The integration is done using two by two gauss quadrature, which is exact
+ when the elements are rectangular and bilinear shape functions are used.
+ The
\begin_inset Formula $\lambda$
\end_inset
- term is under-integrated (one point rule) to keep the large penalty value from effectively locking the element
-\begin_inset LatexCommand \cite{Malkus and Hughes 1978}
+ term is under-integrated (one point rule) to keep the large penalty value
+ from effectively locking the element
+\begin_inset LatexCommand cite
+key "Malkus and Hughes 1978"
\end_inset
-. The right-hand side is made up of three known parts, the body force term (
+.
+ The right-hand side is made up of three known parts, the body force term
+ (
\begin_inset Formula $f_{i}$
\end_inset
@@ -2328,13 +2600,22 @@
\begin_inset Formula $g_{i}$
\end_inset
-). The momentum equation is equivalent to an incompressible elastic problem, and the resulting stiffness matrix will always be positive definite
-\begin_inset LatexCommand \cite[p. 84-89]{Hughes 1987}
+).
+ The momentum equation is equivalent to an incompressible elastic problem,
+ and the resulting stiffness matrix will always be positive definite
+\begin_inset LatexCommand cite
+after "p. 84-89"
+key "Hughes 1987"
\end_inset
-. This allows us to consider only the upper triangular part of the stiffness matrix and save both storage and operations using Cholesky factorization. More details of the method and a formal error analysis can be found in
-\begin_inset LatexCommand \cite{Hughes et al 1979}
+.
+ This allows us to consider only the upper triangular part of the stiffness
+ matrix and save both storage and operations using Cholesky factorization.
+ More details of the method and a formal error analysis can be found in
+
+\begin_inset LatexCommand cite
+key "Hughes et al 1979"
\end_inset
@@ -2342,7 +2623,6 @@
\end_layout
\begin_layout Standard
-
The stiffness matrix is formed in routine
\series bold
f_vstf
@@ -2355,8 +2635,6 @@
\end_layout
\begin_layout Standard
-
-
\begin_inset ERT
status collapsed
@@ -2371,7 +2649,6 @@
\begin_layout Standard
-
\end_layout
\end_inset
@@ -2389,7 +2666,6 @@
\begin_layout Standard
-
\end_layout
\end_inset
@@ -2409,7 +2685,6 @@
\begin_layout Standard
-
\end_layout
\end_inset
@@ -2420,12 +2695,12 @@
\begin_layout Standard
-%the 2D Cartesian stokes equation. The 8 by 8 matrix is make up of
+%the 2D Cartesian stokes equation.
+ The 8 by 8 matrix is make up of
\end_layout
\begin_layout Standard
-
\end_layout
\end_inset
@@ -2436,14 +2711,14 @@
\begin_layout Standard
-%16 2 by 2 submatrices of the form shown below. The $
+%16 2 by 2 submatrices of the form shown below.
+ The $
\backslash
lambda$ and
\end_layout
\begin_layout Standard
-
\end_layout
\end_inset
@@ -2461,7 +2736,6 @@
\begin_layout Standard
-
\end_layout
\end_inset
@@ -2481,7 +2755,6 @@
\begin_layout Standard
-
\end_layout
\end_inset
@@ -2494,12 +2767,11 @@
%
\backslash
-includegraphics[scale=0.4]{images/conman-fig5.eps}
+includegraphics[scale=0.4]{images/conman-fig5.png}
\end_layout
\begin_layout Standard
-
\end_layout
\end_inset
@@ -2519,7 +2791,6 @@
\begin_layout Standard
-
\end_layout
\end_inset
@@ -2537,7 +2808,6 @@
\begin_layout Standard
-
\end_layout
\end_inset
@@ -2557,7 +2827,6 @@
\begin_layout Standard
-
\end_layout
\end_inset
@@ -2573,7 +2842,6 @@
\begin_layout Standard
-
\end_layout
\end_inset
@@ -2584,19 +2852,18 @@
\begin_layout Standard
\noindent
\align center
-
-
\begin_inset Float figure
wide false
sideways false
status open
+\begin_layout Standard
+\begin_inset Caption
-\begin_layout Caption
+\begin_layout Standard
+\begin_inset LatexCommand label
+name "fig:8-The-storage-for"
-
-\begin_inset LatexCommand \label{fig:8-The-storage-for}
-
\end_inset
The storage for the stiffness matrix used in routine
@@ -2606,12 +2873,15 @@
.
\end_layout
+\end_inset
+
+
+\end_layout
+
\begin_layout Standard
\align center
-
-
-\begin_inset Graphics
- filename images/conman-fig8.eps
+\begin_inset Graphics
+ filename images/conman-fig8.png
scale 40
\end_inset
@@ -2625,12 +2895,11 @@
\end_layout
\begin_layout Standard
-
-The energy equation is an advection-diffusion equation. The formal statement is
+The energy equation is an advection-diffusion equation.
+ The formal statement is
\end_layout
\begin_layout Standard
-
Find
\begin_inset Formula $T:\Omega\rightarrow R$
\end_inset
@@ -2639,30 +2908,27 @@
\end_layout
\begin_layout Standard
-
-
\begin_inset Formula \begin{equation}
\dot{T}+u_{i}T_{,i}=\kappa T_{,ii}+H\,\,\,\,\,\, on\,\,\Omega\label{eq:}\end{equation}
+
\end_inset
\end_layout
\begin_layout Standard
-
-
\begin_inset Formula \begin{equation}
T=b\,\,\,\,\,\, on\,\,\Gamma_{b}\label{eq:}\end{equation}
+
\end_inset
\end_layout
\begin_layout Standard
-
-
\begin_inset Formula \begin{equation}
T_{,j}n_{j}=q\,\,\,\,\,\, on\,\,\Gamma_{q}\label{eq:}\end{equation}
+
\end_inset
@@ -2670,7 +2936,6 @@
\begin_layout Standard
\noindent
-
where
\begin_inset Formula $T$
\end_inset
@@ -2687,24 +2952,23 @@
\begin_inset Formula $H$
\end_inset
- is the internal heat source. The weak form of the energy equation is given by
+ is the internal heat source.
+ The weak form of the energy equation is given by
\end_layout
\begin_layout Standard
-
-
\begin_inset Formula \begin{equation}
\int_{\Omega}\left(w+p\right)\dot{T}d\Omega=-\int_{\Omega}\left(w+p\right)\left(u_{i}T_{,i}\right)d\Omega\label{eq:}\end{equation}
+
\end_inset
\end_layout
\begin_layout Standard
-
-
\begin_inset Formula \begin{equation}
-\kappa\int_{\Omega}w_{,i}T_{,i}d\Omega+\int_{\Gamma_{q}}wT_{,j}n_{j}d\Gamma_{q}\label{eq:}\end{equation}
+
\end_inset
@@ -2712,7 +2976,6 @@
\begin_layout Standard
\noindent
-
where
\begin_inset Formula $\dot{T}$
\end_inset
@@ -2729,34 +2992,39 @@
\begin_inset Formula $\left(w+p\right)$
\end_inset
- is the Petrov-Galerkin weighting function with p, the discontinuous streamline upwind part of the Petrov-Galerkin weighting function, given by
+ is the Petrov-Galerkin weighting function with p, the discontinuous streamline
+ upwind part of the Petrov-Galerkin weighting function, given by
\end_layout
\begin_layout Standard
-
-
\begin_inset Formula \begin{equation}
p=\tau u\nabla T=\tilde{k}\frac{u_{i}w_{,i}}{||u||^{2}}\label{eq:}\end{equation}
+
\end_inset
\end_layout
\begin_layout Standard
+The energy equation is solved using Petrov-Galerkin weighting functions
+ on the internal heat source and advective terms to correct for the under-diffus
+ion and remove the oscillations which would result from the standard Galerkin
+ method for an advection dominated problem
+\begin_inset LatexCommand cite
+key "Hughes and Brooks 1979"
-The energy equation is solved using Petrov-Galerkin weighting functions on the internal heat source and advective terms to correct for the under-diffusion and remove the oscillations which would result from the standard Galerkin method for an advection dominated problem
-\begin_inset LatexCommand \cite{Hughes and Brooks 1979}
-
\end_inset
-. The Petrov-Galerkin function can be thought of as a standard Galerkin method in which we counterbalance the numerical underdiffusion by adding an artificial diffusivity of the form
+.
+ The Petrov-Galerkin function can be thought of as a standard Galerkin method
+ in which we counterbalance the numerical underdiffusion by adding an artificial
+ diffusivity of the form
\end_layout
\begin_layout Standard
-
-
\begin_inset Formula \begin{equation}
\left(\xi u_{\xi}h_{\xi}+\eta u_{\eta}h_{\eta}\right)/2\label{eq:}\end{equation}
+
\end_inset
@@ -2764,25 +3032,22 @@
\begin_layout Standard
\noindent
-
with
\end_layout
\begin_layout Standard
-
-
\begin_inset Formula \begin{equation}
\xi=1-\frac{2\kappa}{u_{\xi}h_{\xi}}\label{eq:}\end{equation}
+
\end_inset
\end_layout
\begin_layout Standard
-
-
\begin_inset Formula \begin{equation}
\eta=1-\frac{2\kappa}{u_{\eta}h_{\eta}}\label{eq:}\end{equation}
+
\end_inset
@@ -2790,7 +3055,6 @@
\begin_layout Standard
\noindent
-
where
\begin_inset Formula $h_{\xi}$
\end_inset
@@ -2815,7 +3079,8 @@
\begin_inset Formula $\eta$
\end_inset
- system) evaluated at the element center. This form of discretization has no crosswind diffusion because the
+ system) evaluated at the element center.
+ This form of discretization has no crosswind diffusion because the
\begin_inset Quotes eld
\end_inset
@@ -2823,13 +3088,22 @@
\begin_inset Quotes erd
\end_inset
- acts only in the direction of the flow (i.e., it follows the streamline), hence the name Streamline Upwind Petrov-Galerkin (SUPG). This makes it a better approximation than straight upwinding, and it has been demonstrated to be more accurate than Galerkin or straight upwinding in advection dominated problems
-\begin_inset LatexCommand \cite{Hughes and Brooks 1979}
+ acts only in the direction of the flow (i.e., it follows the streamline),
+ hence the name Streamline Upwind Petrov-Galerkin (SUPG).
+ This makes it a better approximation than straight upwinding, and it has
+ been demonstrated to be more accurate than Galerkin or straight upwinding
+ in advection dominated problems
+\begin_inset LatexCommand cite
+key "Hughes and Brooks 1979"
\end_inset
-. It has recently been shown that the SUPG method is one of a broader class of methods for advection-diffusion equations referred to as Galerkin/Least-Squares methods
-\begin_inset LatexCommand \cite{Hughes et al 1988}
+.
+ It has recently been shown that the SUPG method is one of a broader class
+ of methods for advection-diffusion equations referred to as Galerkin/Least-Squa
+res methods
+\begin_inset LatexCommand cite
+key "Hughes et al 1988"
\end_inset
@@ -2837,95 +3111,97 @@
\end_layout
\begin_layout Standard
+The resulting matrix equation is not symmetric, but since the energy equation
+ only has one degree of freedom per node, while the momentum equation has
+ two or three, the storage for the energy equation is small compared to
+ the momentum equation.
+ Since we use an explicit time stepping method, the energy equation is not
+ implemented in matrix form.
+ The added cost of calculating the Petrov-Galerkin weighting functions is
+ much less than the cost of using a refined grid with the Galerkin method.
+ The Galerkin method requires a finer grid then the Petrov-Galerkin method
+ to achieve stable solutions
+\begin_inset LatexCommand cite
+key "Travis et al 1990"
-The resulting matrix equation is not symmetric, but since the energy equation only has one degree of freedom per node, while the momentum equation has two or three, the storage for the energy equation is small compared to the momentum equation. Since we use an explicit time stepping method, the energy equation is not implemented in matrix form. The added cost of calculating the Petrov-Galerkin weighting functions is much less than the cost of using a refined grid with the Galerkin method. The Galerkin method requires a finer grid then the Petrov-Galerkin method to achieve stable solutions
-\begin_inset LatexCommand \cite{Travis et al 1990}
-
\end_inset
-. Time stepping in the energy equation is done using an explicit predictor-corrector algorithm. The form of the predictor-corrector algorithm is
+.
+ Time stepping in the energy equation is done using an explicit predictor-correc
+tor algorithm.
+ The form of the predictor-corrector algorithm is
\end_layout
\begin_layout Standard
-
Predict:
\end_layout
\begin_layout Standard
-
-
\begin_inset Formula \begin{equation}
T_{n+1}^{\left(0\right)}=T_{n}+\Delta t\left(1-\alpha\right)\dot{T}_{n}\label{eq:}\end{equation}
+
\end_inset
\end_layout
\begin_layout Standard
-
-
\begin_inset Formula \begin{equation}
\dot{T}_{n=1}^{\left(0\right)}=0\label{eq:}\end{equation}
+
\end_inset
\end_layout
\begin_layout Standard
-
Solve:
\end_layout
\begin_layout Standard
-
-
\begin_inset Formula \begin{equation}
M^{*}\Delta\dot{T}_{n+1}^{\left(i\right)}=R_{n+1}^{\left(i\right)}\label{eq:}\end{equation}
+
\end_inset
\end_layout
\begin_layout Standard
-
-
\begin_inset Formula \begin{equation}
R_{n+1}^{\left(i\right)}=-\left[\dot{T}_{n+1}^{\left(i\right)}+u\cdot\left(T_{n+1}^{\left(i\right)}\right),x\right]\left(w+p\right)-\label{eq:}\end{equation}
+
\end_inset
\end_layout
\begin_layout Standard
-
-
\begin_inset Formula \begin{equation}
\tilde{k}w_{,x}\left(T_{n+1}^{\left(i\right)}\right),x+\,\,\,\left(boundary\, condition\, terms\right)\label{eq:}\end{equation}
+
\end_inset
\end_layout
\begin_layout Standard
-
Correct:
\end_layout
\begin_layout Standard
-
-
\begin_inset Formula \begin{equation}
T_{n+1}^{\left(i+1\right)}=T_{n+1}^{\left(i\right)}+\Delta t\alpha\dot{T}_{n+1}^{\left(i\right)}\label{eq:}\end{equation}
+
\end_inset
\end_layout
\begin_layout Standard
-
-
\begin_inset Formula \begin{equation}
\dot{T}_{n+1}^{\left(i+1\right)}=\dot{T}_{n+1}^{\left(i\right)}+\Delta\dot{T}_{n+1}^{\left(i\right)}\label{eq:}\end{equation}
+
\end_inset
@@ -2933,7 +3209,6 @@
\begin_layout Standard
\noindent
-
where
\begin_inset Formula $i$
\end_inset
@@ -2970,7 +3245,8 @@
\begin_inset Formula $\alpha$
\end_inset
- is a convergence parameter. Note that in the explicit formulation
+ is a convergence parameter.
+ Note that in the explicit formulation
\begin_inset Formula $M^{*}$
\end_inset
@@ -2978,13 +3254,16 @@
\end_layout
\begin_layout Standard
-
-The time step is dynamically chosen, and corresponds to the Courant time step (the largest step that can be taken explicitly and maintain stability). With the appropriate choice of variables,
+The time step is dynamically chosen, and corresponds to the Courant time
+ step (the largest step that can be taken explicitly and maintain stability).
+ With the appropriate choice of variables,
\begin_inset Formula $\alpha$
\end_inset
= 0.5 and two iterations, the method is second order accurate
-\begin_inset LatexCommand \cite[p. 562-566]{Hughes 1987}
+\begin_inset LatexCommand cite
+after "p. 562-566"
+key "Hughes 1987"
\end_inset
@@ -2992,7 +3271,6 @@
\end_layout
\begin_layout Standard
-
The predict step is done in routine
\series bold
timdrv
@@ -3021,30 +3299,31 @@
\end_layout
\begin_layout Chapter
-
Implementation
\end_layout
\begin_layout Section
-
Introduction
\end_layout
\begin_layout Standard
-
-There are generally two phases to ConMan, input and time stepping. The main program is found in
+There are generally two phases to ConMan, input and time stepping.
+ The main program is found in
\series bold
ConMan.F
\series default
-. The input is read in the files
+.
+ The input is read in the files
\series bold
input.F and elminp.F
\series default
-. Time stepping is doing in
+.
+ Time stepping is doing in
\series bold
timdrv.F
\series default
-. For legacy reasons, there is a rather complex structure of
+.
+ For legacy reasons, there is a rather complex structure of
\series bold
eglib
\series default
@@ -3056,13 +3335,23 @@
\end_layout
\begin_layout Standard
-
-There are three significant differences that the user familiar with past versions of ConMan will find in this version. The original version of ConMan, distributed by King and/or Hager, was designed to take advantage of machines with vector registers, such as the Cray X-MP or Y-MP. Hence thoughout the code, operations that would be performed on an individual element on a scalar machine were grouped together so that they could be performed on a group of elements.
+There are three significant differences that the user familiar with past
+ versions of ConMan will find in this version.
+ The original version of ConMan, distributed by King and/or Hager, was designed
+ to take advantage of machines with vector registers, such as the Cray X-MP
+ or Y-MP.
+ Hence thoughout the code, operations that would be performed on an individual
+ element on a scalar machine were grouped together so that they could be
+ performed on a group of elements.
\end_layout
\begin_layout Standard
-
-In this version of ConMan, the reordering of elements into blocks with independent degrees of freedom and the reorganization of routines into loops over element groups with inner do-loops having lengths equal to the length of vector registers has been removed. This means that the structure of the routines that form the element stiffness matrix (
+In this version of ConMan, the reordering of elements into blocks with independe
+nt degrees of freedom and the reorganization of routines into loops over
+ element groups with inner do-loops having lengths equal to the length of
+ vector registers has been removed.
+ This means that the structure of the routines that form the element stiffness
+ matrix (
\series bold
f_vstf.F
\series default
@@ -3070,28 +3359,47 @@
\series bold
f_vres.F
\series default
-) for the Stokes equation, and the routines for the calculation of the right-hand side of the energy eqation (
+) for the Stokes equation, and the routines for the calculation of the right-han
+d side of the energy eqation (
\series bold
f_tmres.F
\series default
-) are now loops over elements with short inner loops over local element nodes and/or integration points. One could argue that because modern CPU is relying on fast cache to keep the arithmatic units busy, the kind of grouping we did to take advantage of vector registers is still useful for modern processors. However, the element reordering made algorithms such as particle tracking (not implemented in this version) more challenging and was difficult for users new to the finite element method to understand. Therefore, we have removed the element reordering and block vectorization.
+) are now loops over elements with short inner loops over local element
+ nodes and/or integration points.
+ One could argue that because modern CPU is relying on fast cache to keep
+ the arithmatic units busy, the kind of grouping we did to take advantage
+ of vector registers is still useful for modern processors.
+ However, the element reordering made algorithms such as particle tracking
+ (not implemented in this version) more challenging and was difficult for
+ users new to the finite element method to understand.
+ Therefore, we have removed the element reordering and block vectorization.
\end_layout
\begin_layout Standard
-
-The second major change is that we have implemented a Picard iteration algorithm for steady-state problems (c.f. van Keken - thesis
+The second major change is that we have implemented a Picard iteration algorithm
+ for steady-state problems (c.f.
+ van Keken - thesis
\series bold
-[TODO: This sounds different from the van Keken bib entry we have now. What thesis?]
+[TODO: This sounds different from the van Keken bib entry we have now.
+ What thesis?]
\series default
-). Currently, this is implemented by changing compiler flags in the Makefile. Picard iteration is an implicit solution to the energy equation (as opposed to the predictor-corrector method described above); hence, non-symmetric factor and back-solve routines have been added as well as a routine to calculate the energy equation matrix and right-hand side vectors (
+).
+ Currently, this is implemented by changing compiler flags in the Makefile.
+ Picard iteration is an implicit solution to the energy equation (as opposed
+ to the predictor-corrector method described above); hence, non-symmetric
+ factor and back-solve routines have been added as well as a routine to
+ calculate the energy equation matrix and right-hand side vectors (
\series bold
f_trhsimp.F
\series default
-). To use Picard iteration you change the variable
+).
+ To use Picard iteration you change the variable
\begin_inset Formula $\alpha$
\end_inset
- in the Time Sequence card (second line of the input file) from 0.5 to 1.0 and change the iterations from 2 to 1 (same line). You need to
+ in the Time Sequence card (second line of the input file) from 0.5 to 1.0
+ and change the iterations from 2 to 1 (same line).
+ You need to
\begin_inset Quotes eld
\end_inset
@@ -3099,20 +3407,27 @@
\begin_inset Quotes erd
\end_inset
- and remake with the target Picard, to generate a new executable (conman.pic). A future revision may merge these energy equation solvers into a single code and hide the need to change the iteration steps from the user. When you have a steady-state solution, Picard iteration usually converges in 10-100 steps, as opposed to several thousand steps. When there is no steady-state solution, the Picard results are not useful and the explicit solver should be used.
+ and remake with the target Picard, to generate a new executable (conman.pic).
+ A future revision may merge these energy equation solvers into a single
+ code and hide the need to change the iteration steps from the user.
+ When you have a steady-state solution, Picard iteration usually converges
+ in 10-100 steps, as opposed to several thousand steps.
+ When there is no steady-state solution, the Picard results are not useful
+ and the explicit solver should be used.
\end_layout
\begin_layout Standard
-
The third significant change is the replacement of the
\series bold
mpoint
\series default
- function which allocated memory in the blank common array `a' with the call
+ function which allocated memory in the blank common array `a' with the
+ call
\series bold
mmgetblk
\series default
-. The `mm' routines are a fancy wrapper around a c
+.
+ The `mm' routines are a fancy wrapper around a c
\begin_inset ERT
status collapsed
@@ -3147,11 +3462,16 @@
\end_inset
- call. The reason for this change is fairly technical, and I will spare the user the details. The memory manager source can be found in the directory
+ call.
+ The reason for this change is fairly technical, and I will spare the user
+ the details.
+ The memory manager source can be found in the directory
\family typewriter
mm.src
\family default
-. Care has to be taken when compiling on 32-bit or 64-bit machines (change the header file in
+.
+ Care has to be taken when compiling on 32-bit or 64-bit machines (change
+ the header file in
\family typewriter
mm.src
\family default
@@ -3163,8 +3483,14 @@
\family typewriter
mm2000_64.h
\family default
-) and change the compile flags in the Makefile. It is now possible to use Fortran90's allocate routine for dynamic memory allocation, and the memory manager should be removed in a future revision. Making changes in the memory manager is tricky and this package is fragile. Users who have a problem with the memory manager should request assistance by contacting the CIG Mantle Convection Mailing List
-\begin_inset LatexCommand \url{cig-mc at geodynamics.org}
+) and change the compile flags in the Makefile.
+ It is now possible to use Fortran90's allocate routine for dynamic memory
+ allocation, and the memory manager should be removed in a future revision.
+ Making changes in the memory manager is tricky and this package is fragile.
+ Users who have a problem with the memory manager should request assistance
+ by contacting the CIG Mantle Convection Mailing List
+\begin_inset LatexCommand url
+target "cig-mc at geodynamics.org"
\end_inset
@@ -3172,27 +3498,24 @@
\end_layout
\begin_layout Section
-
Material Properties
\end_layout
\begin_layout Standard
-
-As discussed above, the equations in dimensionless form have one dimensionless parameter, the Rayleigh number.
+As discussed above, the equations in dimensionless form have one dimensionless
+ parameter, the Rayleigh number.
\end_layout
\begin_layout Standard
-
-
\begin_inset Formula \begin{equation}
Ra=\frac{g\alpha\Delta Td^{3}}{\kappa\mu}\label{eq:}\end{equation}
+
\end_inset
\end_layout
\begin_layout Standard
-
where
\begin_inset Formula $g$
\end_inset
@@ -3217,21 +3540,20 @@
\begin_inset Formula $\mu$
\end_inset
- is the dynamic viscosity. In ConMan, the input parameter is the buoyancy part of the Rayleigh number.
+ is the dynamic viscosity.
+ In ConMan, the input parameter is the buoyancy part of the Rayleigh number.
\end_layout
\begin_layout Standard
-
-
\begin_inset Formula \begin{equation}
Ra_{buoy}=g\alpha\label{eq:}\end{equation}
+
\end_inset
\end_layout
\begin_layout Standard
-
The depth,
\begin_inset Formula $d$
\end_inset
@@ -3240,7 +3562,8 @@
\begin_inset Formula $\Delta T$
\end_inset
- are specified from the grid and the temperature boundary conditions.
+ are specified from the grid and the temperature boundary conditions.
+
\begin_inset Formula $\kappa$
\end_inset
@@ -3248,7 +3571,8 @@
\begin_inset Formula $\mu$
\end_inset
- are separate input parameters. If the depth, temperature difference,
+ are separate input parameters.
+ If the depth, temperature difference,
\begin_inset Formula $\kappa$
\end_inset
@@ -3268,26 +3592,26 @@
\end_layout
\begin_layout Standard
-
-The viscosity can be a function of temperature and/or depth. This is done in routine
+The viscosity can be a function of temperature and/or depth.
+ This is done in routine
\series bold
rheol
\series default
-. The user can easily modify the functional form for specific problems. The default functional form is
+.
+ The user can easily modify the functional form for specific problems.
+ The default functional form is
\end_layout
\begin_layout Standard
-
-
\begin_inset Formula \begin{equation}
\mu\left(T,Z\right)=\mu_{o}\left\{ \exp\left\{ \frac{E^{*}*1.0e3+V^{*}z}{R*(T+T_{o})}\right\} -\exp\left\{ \frac{E^{*}*1.0e3+V^{*}z}{R*(1+T_{o})}\right\} \right\} \label{eq:}\end{equation}
+
\end_inset
\end_layout
\begin_layout Standard
-
where
\begin_inset Formula $\mu_{o}$
\end_inset
@@ -3312,7 +3636,8 @@
\begin_inset Formula $z$
\end_inset
- is the depth. In the input files,
+ is the depth.
+ In the input files,
\begin_inset Formula $\mu_{o}$
\end_inset
@@ -3332,7 +3657,8 @@
\series bold
rheol.newt.F
\series default
- to be 273.
+ to be 273.
+
\begin_inset Formula $\Delta T$
\end_inset
@@ -3340,7 +3666,6 @@
\end_layout
\begin_layout Standard
-
The scaling in
\series bold
rheol.newt.F
@@ -3361,22 +3686,20 @@
\end_layout
\begin_layout Standard
-
-Internal heating can be specified through the internal heating parameter. If no bottom temperature is specified, the Rayleigh number becomes
+Internal heating can be specified through the internal heating parameter.
+ If no bottom temperature is specified, the Rayleigh number becomes
\end_layout
\begin_layout Standard
-
-
\begin_inset Formula \begin{equation}
Ra=\frac{g\alpha Hd^{5}}{k\kappa\mu}\label{eq:}\end{equation}
+
\end_inset
\end_layout
\begin_layout Standard
-
where
\begin_inset Formula $H$
\end_inset
@@ -3385,21 +3708,25 @@
\begin_inset Formula $k$
\end_inset
- is the thermal conductivity. The grid can have multiple material groups, each with its own set of material properties.
+ is the thermal conductivity.
+ The grid can have multiple material groups, each with its own set of material
+ properties.
\end_layout
\begin_layout Section
-
Installation
\end_layout
\begin_layout Standard
-
-ConMan comes ready to run with a Unicos makefile. The file
+ConMan comes ready to run with a Unicos makefile.
+ The file
\series bold
Makefile
\series default
- contains the system calls for the compiler and the loader, FC and LD respectively. These need to be changed for your machine. Also the calls to
+ contains the system calls for the compiler and the loader, FC and LD respective
+ly.
+ These need to be changed for your machine.
+ Also the calls to
\series bold
second
\series default
@@ -3411,149 +3738,143 @@
\end_layout
\begin_layout Section
-
-Building from Source [TODO WEI -- thought you might need this section. Or else delete]
+Building from Source [TODO WEI -- thought you might need this section.
+ Or else delete]
\end_layout
\begin_layout Subsection
-
System Requirements
\end_layout
\begin_layout Standard
-
-ConMan works on a variety of computational platforms and has been tested on workstations running
+ConMan works on a variety of computational platforms and has been tested
+ on workstations running
\end_layout
\begin_layout Itemize
-
Mac OS X 10.4.6 (G4, G5, and Intel)
\end_layout
\begin_layout Itemize
-
Windows 2000 and XP SP2
\end_layout
\begin_layout Itemize
-
RedHat Fedora Core 5 (x86)
\end_layout
\begin_layout Itemize
-
OpenSuse 10.0 (x86)
\end_layout
\begin_layout Itemize
-
Gentoo (x86)
\end_layout
\begin_layout Itemize
-
Debian stable (x86 and AMD64), testing (x86), and unstable (x86)
\end_layout
\begin_layout Standard
-
-ConMan has also been tested on clusters running Redhat 7.2 (x86) and RedHat Enterprise Linux 3 (EM64T).
+ConMan has also been tested on clusters running Redhat 7.2 (x86) and RedHat
+ Enterprise Linux 3 (EM64T).
\end_layout
\begin_layout Subsection
-
Dependencies
\end_layout
\begin_layout Standard
-
This version of ConMan is self-contained and requires no external libraries.
\end_layout
\begin_layout Subsection
+\begin_inset LatexCommand label
+name "sec:Downloading-the-Code"
-
-\begin_inset LatexCommand \label{sec:Downloading-the-Code}
-
\end_inset
Downloading the Code
\end_layout
\begin_layout Standard
-
You can get the source for the latest release from the ConMan web page
-\begin_inset LatexCommand \url{geodynamics.org/cig/software/packages/mc/conman/}
+\begin_inset LatexCommand url
+target "geodynamics.org/cig/software/packages/mc/conman/"
\end_inset
-. In that tarball is the file ???name???.
+.
+ In that tarball is the file ???name???.
\end_layout
\begin_layout Subsubsection
-
Source Code Repository (Experts Only)
\end_layout
\begin_layout Standard
-
-Advanced users and software developers may be interested in downloading the latest ConMan source code directly from the CIG source code repository, instead of using the prepared source package. To check whether you have a subversion client installed on your machine, type:
+Advanced users and software developers may be interested in downloading
+ the latest ConMan source code directly from the CIG source code repository,
+ instead of using the prepared source package.
+ To check whether you have a subversion client installed on your machine,
+ type:
\end_layout
\begin_layout LyX-Code
-
svn
\end_layout
\begin_layout Standard
-
You should get a response that looks something like this:
\end_layout
\begin_layout LyX-Code
-
Type `svn help' for usage.
\end_layout
\begin_layout Standard
+Otherwise, you will need to download and install a Subversion client, available
+ at the Subversion Website
+\begin_inset LatexCommand url
+target "subversion.tigris.org/project_packages.html"
-Otherwise, you will need to download and install a Subversion client, available at the Subversion Website
-\begin_inset LatexCommand \url{subversion.tigris.org/project_packages.html}
-
\end_inset
-. Then the code can be checked out with the following command:
+.
+ Then the code can be checked out with the following command:
\end_layout
\begin_layout LyX-Code
-
svn checkout http://geodynamics.org/svn/cig/mc/3D/ConMan/trunk ConMan
\end_layout
\begin_layout Section
+\begin_inset LatexCommand label
+name "sec:Support"
-
-\begin_inset LatexCommand \label{sec:Support}
-
\end_inset
Support
\end_layout
\begin_layout Standard
+The primary point of support for ConMan is the CIG Mantle Convection Mailing
+ List
+\begin_inset LatexCommand url
+target "cig-mc at geodynamics.org"
-The primary point of support for ConMan is the CIG Mantle Convection Mailing List
-\begin_inset LatexCommand \url{cig-mc at geodynamics.org}
-
\end_inset
-. Feel free to send questions, comments, feature requests, and bugs to the list. The mailing list is archived at
+.
+ Feel free to send questions, comments, feature requests, and bugs to the
+ list.
+ The mailing list is archived at
\end_layout
\begin_layout LyX-Code
-
cig-mc Archives
-\begin_inset LatexCommand \url{geodynamics.org/pipermail/cig-mc/}
+\begin_inset LatexCommand url
+target "geodynamics.org/pipermail/cig-mc/"
\end_inset
@@ -3561,14 +3882,13 @@
\end_layout
\begin_layout Standard
-
You may also use the bug tracker
\end_layout
\begin_layout LyX-Code
-
Roundup
-\begin_inset LatexCommand \url{geodynamics.org/roundup}
+\begin_inset LatexCommand url
+target "geodynamics.org/roundup"
\end_inset
@@ -3576,23 +3896,23 @@
\end_layout
\begin_layout Standard
-
to submit bugs and requests for new features.
\end_layout
\begin_layout Chapter
+\begin_inset LatexCommand label
+name "cha:Input-Guide"
-
-\begin_inset LatexCommand \label{cha:Input-Guide}
-
\end_inset
Input Guide
\end_layout
\begin_layout Standard
-
-To run ConMan a series of nine file names are needed, some for input and some for output. Usually these are read from a runfile. The first two files are input files
+To run ConMan a series of nine file names are needed, some for input and
+ some for output.
+ Usually these are read from a runfile.
+ The first two files are input files
\series bold
input
\series default
@@ -3600,11 +3920,18 @@
\series bold
geom
\series default
- and are described in this section. The third file is an output file showing all the input parameters in a verbose form. The fourth and fifth files are an input temperature file (optional) and an output temperature file. These are for starting a new run from a previous run. The sixth file is a time series file (see routine
+ and are described in this section.
+ The third file is an output file showing all the input parameters in a
+ verbose form.
+ The fourth and fifth files are an input temperature file (optional) and
+ an output temperature file.
+ These are for starting a new run from a previous run.
+ The sixth file is a time series file (see routine
\series bold
fluxke
\series default
-), the seventh file is the coordinates, velocities and temperatures, the eighth file is for stresses (see routine
+), the seventh file is the coordinates, velocities and temperatures, the
+ eighth file is for stresses (see routine
\series bold
stress.F
\series default
@@ -3612,7 +3939,8 @@
\series bold
geoid.F
\series default
-). These file names are read in routine
+).
+ These file names are read in routine
\series bold
ConMan.F
\series default
@@ -3620,16 +3948,22 @@
\end_layout
\begin_layout Standard
-
-The input for ConMan is read from two different FORTRAN units. The first unit,
+The input for ConMan is read from two different FORTRAN units.
+ The first unit,
\series bold
iin
\series default
-, contains the time stepping, output, and material parameters as well as element type information while the second unit,
+, contains the time stepping, output, and material parameters as well as
+ element type information while the second unit,
\series bold
igeom
\series default
-, contains the coordinates, boundary values and connectivity information. ConMan reads the file names to attach to these units from standard input. The typical way to run ConMan is to create a file with nine lines, one file name per line, and redirect this into the executable (i.e., % conman.pic < runfile & ).
+, contains the coordinates, boundary values and connectivity information.
+ ConMan reads the file names to attach to these units from standard input.
+ The typical way to run ConMan is to create a file with nine lines, one
+ file name per line, and redirect this into the executable (i.e., % conman.pic
+ < runfile & ).
+
\series bold
iin
\series default
@@ -3637,16 +3971,19 @@
\series bold
igeom
\series default
- is attached to the file named on the second line (names must be ASCII with a length less than 80 characters long).
+ is attached to the file named on the second line (names must be ASCII with
+ a length less than 80 characters long).
\end_layout
\begin_layout Standard
-
-The input deck was broken up so that an automatic grid generating routine could be used to generate coordinates, boundary conditions and element connectivities separate from ConMan. The only automatic grid generation ConMan does is linear or bilinear interpolation which is described in the appropriate sections of this guide.
+The input deck was broken up so that an automatic grid generating routine
+ could be used to generate coordinates, boundary conditions and element
+ connectivities separate from ConMan.
+ The only automatic grid generation ConMan does is linear or bilinear interpolat
+ion which is described in the appropriate sections of this guide.
\end_layout
\begin_layout Standard
-
The following sixteen cards or groups of cards are read from the
\series bold
iin
@@ -3659,7 +3996,8 @@
\begin_inset Quotes erd
\end_inset
- will mean one line of an ASCII text file). These constitute the parameter part of the input
+ will mean one line of an ASCII text file).
+ These constitute the parameter part of the input
\begin_inset Quotes eld
\end_inset
@@ -3667,7 +4005,8 @@
\begin_inset Quotes erd
\end_inset
- for the program ConMan. The format for this guide is a
+ for the program ConMan.
+ The format for this guide is a
\series bold
bold
\series default
@@ -3675,12 +4014,11 @@
\emph on
italicized
\emph default
- line showing the order of the parameters and a listing of the parameters (with a brief explanation).
+ line showing the order of the parameters and a listing of the parameters
+ (with a brief explanation).
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -3755,8 +4093,6 @@
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -3826,15 +4162,12 @@
\emph on
-numnp nsd ndof nelx nelz mprec iflow necho inrsts iorstr nodebn ntimvs ntseq numeg isky nwrap
-\emph default
-
+numnp nsd ndof nelx nelz mprec iflow necho inrsts iorstr nodebn ntimvs ntseq
+ numeg isky nwrap
\end_layout
\begin_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -3900,12 +4233,16 @@
\end_inset
- . . . . . . total number of nodal points
+ .
+ .
+ .
+ .
+ .
+ .
+ total number of nodal points
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -3971,12 +4308,18 @@
\end_inset
- . . . . . . . . number of spatial dimensions (always 2)
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ number of spatial dimensions (always 2)
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -4042,12 +4385,17 @@
\end_inset
- . . . . . . . number of degrees of freedom (always 2)
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ number of degrees of freedom (always 2)
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -4113,12 +4461,17 @@
\end_inset
- . . . . . . . number of elements in the x1 (horizontal) direction
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ number of elements in the x1 (horizontal) direction
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -4184,12 +4537,17 @@
\end_inset
- . . . . . . . number of elements in the x2 (vertical) direction
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ number of elements in the x2 (vertical) direction
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -4255,24 +4613,26 @@
\end_inset
- . . . . . . precision flag (always use double)
+ .
+ .
+ .
+ .
+ .
+ .
+ precision flag (always use double)
\end_layout
\begin_deeper
\begin_layout Standard
-
1 - single
\end_layout
\begin_layout Standard
-
2 - double
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -4338,24 +4698,27 @@
\end_inset
- . . . . . . . data check flag
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ data check flag
\end_layout
\begin_deeper
\begin_layout Standard
-
0 - check data only
\end_layout
\begin_layout Standard
-
1 - execute code
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -4421,24 +4784,26 @@
\end_inset
- . . . . . . echo data flag
+ .
+ .
+ .
+ .
+ .
+ .
+ echo data flag
\end_layout
\begin_deeper
\begin_layout Standard
-
0 - minimum data echo (terse)
\end_layout
\begin_layout Standard
-
1 - echo data to output file (verbose)
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -4504,24 +4869,27 @@
\end_inset
- . . . . . . . read restart file flag
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ read restart file flag
\end_layout
\begin_deeper
\begin_layout Standard
-
0 - use default start (conductive)
\end_layout
\begin_layout Standard
-
1 - read restart file from unit 16
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -4587,12 +4955,18 @@
\end_inset
- . . . . . . . write restart file flag
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ write restart file flag
\end_layout
\begin_deeper
\begin_layout Standard
-
0 - don
\begin_inset ERT
status collapsed
@@ -4621,14 +4995,11 @@
\end_layout
\begin_layout Standard
-
1 - write restart file to unit 17
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -4694,12 +5065,16 @@
\end_inset
- . . . . . . number of edge nodes for nusselt smoother
+ .
+ .
+ .
+ .
+ .
+ .
+ number of edge nodes for nusselt smoother
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -4765,24 +5140,26 @@
\end_inset
- . . . . . . temperature dependent viscosity flag
+ .
+ .
+ .
+ .
+ .
+ .
+ temperature dependent viscosity flag
\end_layout
\begin_deeper
\begin_layout Standard
-
0 - stiffness matrix factored once
\end_layout
\begin_layout Standard
-
1 - stiffness matrix factored every time step
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -4848,19 +5225,23 @@
\end_inset
- . . . . . . . number of time sequences (always 1)
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ number of time sequences (always 1)
\end_layout
\begin_deeper
\begin_layout Standard
-
currently only one supported
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -4926,19 +5307,22 @@
\end_inset
- . . . . . . number of element groups (always 1)
+ .
+ .
+ .
+ .
+ .
+ .
+ number of element groups (always 1)
\end_layout
\begin_deeper
\begin_layout Standard
-
currently only one supported
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -5004,24 +5388,27 @@
\end_inset
- . . . . . . . flag for skyline factor
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ flag for skyline factor
\end_layout
\begin_deeper
\begin_layout Standard
-
0 - regular skyline
\end_layout
\begin_layout Standard
-
1 - vectorized skyline
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -5087,30 +5474,31 @@
\end_inset
- . . . . . . number of nodes to wrap
+ .
+ .
+ .
+ .
+ .
+ .
+ number of nodes to wrap
\end_layout
\begin_deeper
\begin_layout Standard
-
equal to number of elements in vertical
\end_layout
\begin_layout Standard
-
to use nodes must be numbered increasing
\end_layout
\begin_layout Standard
-
fastest in vertical direction
\end_layout
\end_deeper
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -5184,16 +5572,11 @@
\begin_deeper
\begin_layout Standard
-
\emph on
nstep niter alpha delt epstol
-\emph default
-
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -5259,12 +5642,17 @@
\end_inset
- . . . . . . . number of time steps
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ number of time steps
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -5330,24 +5718,27 @@
\end_inset
- . . . . . . . number of multicorrector iterations
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ number of multicorrector iterations
\end_layout
\begin_deeper
\begin_layout Standard
-
2 - second-order expicit
\end_layout
\begin_layout Standard
-
1 - picard
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -5413,24 +5804,27 @@
\end_inset
- . . . . . . . multicorrector parameter
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ multicorrector parameter
\end_layout
\begin_deeper
\begin_layout Standard
-
0.5 for explicit 2nd order
\end_layout
\begin_layout Standard
-
1.0 for picard
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -5496,12 +5890,17 @@
\end_inset
- . . . . . . . time step (not used)
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ time step (not used)
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -5567,13 +5966,17 @@
\end_inset
- . . . . . . tolerance for hybrid method (not used)
+ .
+ .
+ .
+ .
+ .
+ .
+ tolerance for hybrid method (not used)
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -5644,14 +6047,10 @@
\emph on
nsdprt nsvprt nstprt nsmprt
-\emph default
-
\end_layout
\begin_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -5717,12 +6116,16 @@
\end_inset
- . . . . . . steps between disk output
+ .
+ .
+ .
+ .
+ .
+ .
+ steps between disk output
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -5788,12 +6191,16 @@
\end_inset
- . . . . . . steps between velocity output (not used)
+ .
+ .
+ .
+ .
+ .
+ .
+ steps between velocity output (not used)
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -5859,12 +6266,16 @@
\end_inset
- . . . . . . steps between temperature, velocity & stress output
+ .
+ .
+ .
+ .
+ .
+ .
+ steps between temperature, velocity & stress output
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -5930,13 +6341,17 @@
\end_inset
- . . . . . . steps between stress field output (not used)
+ .
+ .
+ .
+ .
+ .
+ .
+ steps between stress field output (not used)
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -6009,14 +6424,10 @@
\emph on
bnode enode incr (bcf(i), i=1,ndof)
-\emph default
-
\end_layout
\begin_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -6082,12 +6493,16 @@
\end_inset
- . . . . . . beginning node
+ .
+ .
+ .
+ .
+ .
+ .
+ beginning node
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -6153,12 +6568,16 @@
\end_inset
- . . . . . . ending node
+ .
+ .
+ .
+ .
+ .
+ .
+ ending node
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -6224,12 +6643,17 @@
\end_inset
- . . . . . . . node increment
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ node increment
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -6295,17 +6719,22 @@
\end_inset
- . . . . . . . boundary condition flag for ith degree of freedom
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ boundary condition flag for ith degree of freedom
\end_layout
\begin_deeper
\begin_layout Standard
-
0 - free slip
\end_layout
\begin_layout Standard
-
1 - pinned degree of freedom
\end_layout
@@ -6313,16 +6742,11 @@
\end_deeper
\begin_layout Standard
-
\emph on
0 0 0 0 0 to end VBCF cards
-\emph default
-
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -6395,14 +6819,10 @@
\emph on
bnode enode incr bcf
-\emph default
-
\end_layout
\begin_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -6468,12 +6888,16 @@
\end_inset
- . . . . . . beginning node
+ .
+ .
+ .
+ .
+ .
+ .
+ beginning node
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -6539,12 +6963,16 @@
\end_inset
- . . . . . . ending node
+ .
+ .
+ .
+ .
+ .
+ .
+ ending node
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -6610,12 +7038,17 @@
\end_inset
- . . . . . . . node increment
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ node increment
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -6681,31 +7114,31 @@
\end_inset
- . . . . . . . . boundary condition flag for temperature
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ boundary condition flag for temperature
\end_layout
\begin_deeper
\begin_layout Standard
-
1- fixed temperature
\end_layout
\end_deeper
-\end_deeper
-\begin_deeper
\begin_layout Standard
-
\emph on
0 0 0 0 to end TBCF cards
-\emph default
-
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -6782,14 +7215,10 @@
top and bottom rows of nodes
\emph on
bnode enode incr
-\emph default
-
\end_layout
\begin_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -6855,12 +7284,16 @@
\end_inset
- . . . . . . beginning node
+ .
+ .
+ .
+ .
+ .
+ .
+ beginning node
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -6926,12 +7359,16 @@
\end_inset
- . . . . . . ending node
+ .
+ .
+ .
+ .
+ .
+ .
+ ending node
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -6997,24 +7434,24 @@
\end_inset
- . . . . . . . node increment
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ node increment
\end_layout
-\end_deeper
-\begin_deeper
\begin_layout Standard
-
\emph on
0 0 0 to end NNBCF (type a) cards
-\emph default
-
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -7092,14 +7529,10 @@
second from top and bottom rows of nodes
\emph on
bnode enode incr
-\emph default
-
\end_layout
\begin_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -7165,12 +7598,16 @@
\end_inset
- . . . . . . beginning node
+ .
+ .
+ .
+ .
+ .
+ .
+ beginning node
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -7236,12 +7673,16 @@
\end_inset
- . . . . . . ending node
+ .
+ .
+ .
+ .
+ .
+ .
+ ending node
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -7307,24 +7748,24 @@
\end_inset
- . . . . . . . node increment
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ node increment
\end_layout
-\end_deeper
-\begin_deeper
\begin_layout Standard
-
\emph on
0 0 0 to end NNBCF (type b) cards
-\emph default
-
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -7395,14 +7836,10 @@
\emph on
pert xsize zsize
-\emph default
-
\end_layout
\begin_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -7468,12 +7905,17 @@
\end_inset
- . . . . . . . perturbation from conductive state
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ perturbation from conductive state
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -7539,12 +7981,17 @@
\end_inset
- . . . . . . . nondimensional length (x1 direction) of box
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ nondimensional length (x1 direction) of box
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -7610,13 +8057,18 @@
\end_inset
- . . . . . . . nondimensional height (x2 direction) of box
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ nondimensional height (x2 direction) of box
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -7687,14 +8139,10 @@
- numeg cards
\emph on
ntype numel nen nenl numat nedof numsuf nipt implv implt
-\emph default
-
\end_layout
\begin_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -7760,19 +8208,23 @@
\end_inset
- . . . . . . . element type
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ element type
\end_layout
\begin_deeper
\begin_layout Standard
-
2 - two dimensional elements
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -7838,12 +8290,16 @@
\end_inset
- . . . . . . total number of elements
+ .
+ .
+ .
+ .
+ .
+ .
+ total number of elements
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -7909,12 +8365,18 @@
\end_inset
- . . . . . . . . number of element nodes (always 4)
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ number of element nodes (always 4)
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -7980,12 +8442,17 @@
\end_inset
- . . . . . . . number of local element nodes (always 4)
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ number of local element nodes (always 4)
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -8051,12 +8518,16 @@
\end_inset
- . . . . . . number of material groups
+ .
+ .
+ .
+ .
+ .
+ .
+ number of material groups
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -8122,12 +8593,17 @@
\end_inset
- . . . . . . . number of element degrees of freedom (always 2)
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ number of element degrees of freedom (always 2)
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -8193,12 +8669,16 @@
\end_inset
- . . . . . . number of imposed stress/flux cards
+ .
+ .
+ .
+ .
+ .
+ .
+ number of imposed stress/flux cards
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -8264,12 +8744,17 @@
\end_inset
- . . . . . . . number of integration points per element (always 5)
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ number of integration points per element (always 5)
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -8335,12 +8820,17 @@
\end_inset
- . . . . . . . currently unused
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ currently unused
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -8406,13 +8896,18 @@
\end_inset
- . . . . . . . currently unused
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ currently unused
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -8482,14 +8977,10 @@
\emph on
visc(i), i=1,numat
-\emph default
-
\end_layout
\begin_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -8555,13 +9046,17 @@
\end_inset
- . . . . . . preexponential viscosity coefficient for ith element
+ .
+ .
+ .
+ .
+ .
+ .
+ preexponential viscosity coefficient for ith element
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -8631,14 +9126,10 @@
\emph on
alam(i), i=1,numat
-\emph default
-
\end_layout
\begin_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -8704,13 +9195,17 @@
\end_inset
- . . . . . . penalty parameter for ith element
+ .
+ .
+ .
+ .
+ .
+ .
+ penalty parameter for ith element
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -8780,14 +9275,10 @@
\emph on
diff(i), i=1,numat
-\emph default
-
\end_layout
\begin_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -8853,13 +9344,18 @@
\end_inset
- . . . . . . . thermal diffusivity for ith element
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ thermal diffusivity for ith element
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -8931,14 +9427,10 @@
\emph on
Ra(i), i=1,numat
-\emph default
-
\end_layout
\begin_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -9004,13 +9496,18 @@
\end_inset
- . . . . . . . bouyancy part of Rayleigh number for ith element
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ bouyancy part of Rayleigh number for ith element
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -9082,14 +9579,10 @@
\emph on
dmhu(i), i=1,numat
-\emph default
-
\end_layout
\begin_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -9155,13 +9648,17 @@
\end_inset
- . . . . . . internal heat source for ith material group
+ .
+ .
+ .
+ .
+ .
+ .
+ internal heat source for ith material group
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -9232,14 +9729,10 @@
\emph on
tcon(1,i), i=1,numat
-\emph default
-
\end_layout
\begin_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -9305,13 +9798,17 @@
\end_inset
- . . . . . activation energy for ith material group for temperature dependent viscosity (kJ/mole)
+ .
+ .
+ .
+ .
+ .
+ activation energy for ith material group for temperature dependent viscosity
+ (kJ/mole)
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -9382,14 +9879,10 @@
\emph on
tcon(2,i), i=1,numat
-\emph default
-
\end_layout
\begin_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -9455,7 +9948,13 @@
\end_inset
- . . . . . activation volume for ith material group for temperature dependent viscosity (cm
+ .
+ .
+ .
+ .
+ .
+ activation volume for ith material group for temperature dependent viscosity
+ (cm
\begin_inset Formula $^{3}$
\end_inset
@@ -9464,8 +9963,6 @@
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -9536,14 +10033,10 @@
\emph on
tcon(3,i), i=1,numat
-\emph default
-
\end_layout
\begin_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -9609,13 +10102,16 @@
\end_inset
- . . . . . maximum value of the viscosity for the ith material group
+ .
+ .
+ .
+ .
+ .
+ maximum value of the viscosity for the ith material group
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -9686,14 +10182,10 @@
- numsuf cards
\emph on
nel side fnorm ftan flux
-\emph default
-
\end_layout
\begin_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -9759,12 +10251,18 @@
\end_inset
- . . . . . . . . element number
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ element number
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -9830,34 +10328,35 @@
\end_inset
- . . . . . . . side to apply force and flux
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ side to apply force and flux
\end_layout
\begin_deeper
\begin_layout Standard
-
1 - bottom
\end_layout
\begin_layout Standard
-
2 - right side
\end_layout
\begin_layout Standard
-
3 - top
\end_layout
\begin_layout Standard
-
4 - left side
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -9923,12 +10422,17 @@
\end_inset
- . . . . . . . normal surface force
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ normal surface force
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -9994,12 +10498,17 @@
\end_inset
- . . . . . . . tangential surface force
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ tangential surface force
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -10065,17 +10574,25 @@
\end_inset
- . . . . . . . . heat flux
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ heat flux
\end_layout
\end_deeper
\begin_layout Standard
-
The following four groups of cards are read from the
\series bold
igeom
\series default
- unit. These constitute the geometry part of the input
+ unit.
+ These constitute the geometry part of the input
\begin_inset Quotes eld
\end_inset
@@ -10087,17 +10604,15 @@
\family typewriter
conman
\family default
-. The format of this section is the same as above.
+.
+ The format of this section is the same as above.
\end_layout
\begin_layout Section
-
Coordinate Group
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -10168,14 +10683,10 @@
\emph on
node gp (x(i,node) i=1,nsd)
-\emph default
-
\end_layout
\begin_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -10241,12 +10752,17 @@
\end_inset
- . . . . . . . the node whose coordinates are to be specified
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ the node whose coordinates are to be specified
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -10312,29 +10828,32 @@
\end_inset
- . . . . . . . . generation parameter for automatic generation
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ generation parameter for automatic generation
\end_layout
\begin_deeper
\begin_layout Standard
-
0 - no autogeneration
\end_layout
\begin_layout Standard
-
2 - generate a line using node as a starting point
\end_layout
\begin_layout Standard
-
4 - generate a box using node as the lower left corner
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -10400,13 +10919,16 @@
\end_inset
- . . . . . coordinate value in the ith spatial dimension
+ .
+ .
+ .
+ .
+ .
+ coordinate value in the ith spatial dimension
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -10477,14 +10999,10 @@
- gp-1 cards
\emph on
node mgen (x(i,node) i=1,nsd)
-\emph default
-
\end_layout
\begin_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -10550,12 +11068,17 @@
\end_inset
- . . . . . . . node number
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ node number
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -10621,12 +11144,18 @@
\end_inset
- . . . . . . . generation parameter
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ generation parameter
\end_layout
\begin_deeper
\begin_layout Standard
-
0 - don
\begin_inset ERT
status collapsed
@@ -10655,14 +11184,11 @@
\end_layout
\begin_layout Standard
-
1 - use this as the start of a generation sequence
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -10728,13 +11254,16 @@
\end_inset
- . . . . . coordinate value in the ith spatial dimension
+ .
+ .
+ .
+ .
+ .
+ coordinate value in the ith spatial dimension
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -10805,14 +11334,10 @@
\emph on
ninc1 inc1 ninc2 inc2
-\emph default
-
\end_layout
\begin_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -10878,12 +11403,17 @@
\end_inset
- . . . . . . . number of additional nodes to generate in x1 direction
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ number of additional nodes to generate in x1 direction
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -10949,12 +11479,17 @@
\end_inset
- . . . . . . . increment of nodes in x1 direction
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ increment of nodes in x1 direction
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -11020,19 +11555,23 @@
\end_inset
- . . . . . . . number of additional nodes to generate in x2 direction
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ number of additional nodes to generate in x2 direction
\end_layout
\begin_deeper
\begin_layout Standard
-
0 - if gp equals 2
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -11098,36 +11637,34 @@
\end_inset
- . . . . . . . increment of nodes in x2 direction
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ increment of nodes in x2 direction
\end_layout
\begin_deeper
\begin_layout Standard
-
0 - if gp equals 2
\end_layout
\end_deeper
-\end_deeper
-\begin_deeper
\begin_layout Standard
-
\emph on
0 0 0 0 to end coordinate group
-\emph default
-
\end_layout
\end_deeper
\begin_layout Section
-
Velocity Boundary Condition Group
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -11198,14 +11735,10 @@
\emph on
node gp (v(i,node) i=1,nsd)
-\emph default
-
\end_layout
\begin_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -11271,12 +11804,17 @@
\end_inset
- . . . . . . . the node whose velocities are to be specified
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ the node whose velocities are to be specified
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -11342,29 +11880,32 @@
\end_inset
- . . . . . . . . generation parameter for automatic generation
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ generation parameter for automatic generation
\end_layout
\begin_deeper
\begin_layout Standard
-
0 - no autogeneration
\end_layout
\begin_layout Standard
-
2 - generate a line using node as a starting point
\end_layout
\begin_layout Standard
-
4 - generate a box using node as the lower left corner
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -11430,13 +11971,16 @@
\end_inset
- . . . . . velocity value in the ith spatial dimension
+ .
+ .
+ .
+ .
+ .
+ velocity value in the ith spatial dimension
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -11507,14 +12051,10 @@
- gp-1 cards
\emph on
node mgen (v(i,node) i=1,nsd)
-\emph default
-
\end_layout
\begin_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -11580,12 +12120,17 @@
\end_inset
- . . . . . . . node number
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ node number
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -11651,12 +12196,18 @@
\end_inset
- . . . . . . . generation parameter
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ generation parameter
\end_layout
\begin_deeper
\begin_layout Standard
-
0 - don
\begin_inset ERT
status collapsed
@@ -11685,14 +12236,11 @@
\end_layout
\begin_layout Standard
-
1 - use this as the start of a generation sequence
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -11758,13 +12306,16 @@
\end_inset
- . . . . . velocity value in the ith spatial dimension
+ .
+ .
+ .
+ .
+ .
+ velocity value in the ith spatial dimension
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -11835,14 +12386,10 @@
\emph on
ninc1 inc1 ninc2 inc2
-\emph default
-
\end_layout
\begin_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -11908,12 +12455,17 @@
\end_inset
- . . . . . . . number of additional nodes to generate in x1 direction
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ number of additional nodes to generate in x1 direction
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -11979,12 +12531,17 @@
\end_inset
- . . . . . . . increment of nodes in x1 direction
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ increment of nodes in x1 direction
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -12050,13 +12607,18 @@
\end_inset
- . . . . . . . number of additional nodes to generate in x2 direction
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ number of additional nodes to generate in x2 direction
\end_layout
\begin_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -12127,8 +12689,6 @@
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -12194,36 +12754,34 @@
\end_inset
- . . . . . . . increment of nodes in x2 direction
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ increment of nodes in x2 direction
\end_layout
\begin_deeper
\begin_layout Standard
-
0 - if gp equals 2
\end_layout
\end_deeper
-\end_deeper
-\begin_deeper
\begin_layout Standard
-
\emph on
0 0 0 0 to end velocity group
-\emph default
-
\end_layout
\end_deeper
\begin_layout Section
-
Temperature Boundary Condition Group
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -12294,14 +12852,10 @@
\emph on
node gp t(node)
-\emph default
-
\end_layout
\begin_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -12367,12 +12921,17 @@
\end_inset
- . . . . . . . the node whose velocities are to be specified
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ the node whose velocities are to be specified
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -12438,24 +12997,28 @@
\end_inset
- . . . . . . . . generation parameter for automatic generation
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ generation parameter for automatic generation
\end_layout
\begin_deeper
\begin_layout Standard
-
0 - no autogeneration
\end_layout
\begin_layout Standard
-
2 - generate a line using node as a starting point
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -12521,13 +13084,17 @@
\end_inset
- . . . . . . temperature value
+ .
+ .
+ .
+ .
+ .
+ .
+ temperature value
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -12598,14 +13165,10 @@
- gp-1 cards
\emph on
node mgen t(node)
-\emph default
-
\end_layout
\begin_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -12671,12 +13234,17 @@
\end_inset
- . . . . . . . node number
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ node number
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -12742,12 +13310,18 @@
\end_inset
- . . . . . . . generation parameter
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ generation parameter
\end_layout
\begin_deeper
\begin_layout Standard
-
0 - don
\begin_inset ERT
status collapsed
@@ -12776,14 +13350,11 @@
\end_layout
\begin_layout Standard
-
1 - use this as the start of a generation sequence
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -12849,13 +13420,17 @@
\end_inset
- . . . . . . temperature value
+ .
+ .
+ .
+ .
+ .
+ .
+ temperature value
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -12926,14 +13501,10 @@
\emph on
ninc1 inc1 ninc2 inc2
-\emph default
-
\end_layout
\begin_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -12999,12 +13570,17 @@
\end_inset
- . . . . . . . number of additional nodes to generate in x1 direction
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ number of additional nodes to generate in x1 direction
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -13070,12 +13646,17 @@
\end_inset
- . . . . . . . increment of nodes in x1 direction
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ increment of nodes in x1 direction
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -13141,19 +13722,23 @@
\end_inset
- . . . . . . . number of additional nodes to generate in x2 direction
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ number of additional nodes to generate in x2 direction
\end_layout
\begin_deeper
\begin_layout Standard
-
0 - if gp equals 2
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -13219,36 +13804,34 @@
\end_inset
- . . . . . . . increment of nodes in x2 direction
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ increment of nodes in x2 direction
\end_layout
\begin_deeper
\begin_layout Standard
-
0 - if gp equals 2
\end_layout
\end_deeper
-\end_deeper
-\begin_deeper
\begin_layout Standard
-
\emph on
0 0 to end temperature group
-\emph default
-
\end_layout
\end_deeper
\begin_layout Section
-
Element Connectivity (ien) Generation Group
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -13319,14 +13902,10 @@
\emph on
elnu ng mat no (ien(elnu,i) i=1,nen)
-\emph default
-
\end_layout
\begin_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -13392,12 +13971,17 @@
\end_inset
- . . . . . . . element number
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ element number
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -13463,24 +14047,28 @@
\end_inset
- . . . . . . . . generation parameter
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ generation parameter
\end_layout
\begin_deeper
\begin_layout Standard
-
0 - no generation
\end_layout
\begin_layout Standard
-
1 - generate using increments from increment card
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -13546,12 +14134,16 @@
\end_inset
- no . . . . . . material number for this element
+ no .
+ .
+ .
+ .
+ .
+ .
+ material number for this element
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -13617,13 +14209,17 @@
\end_inset
- . . . . . global node number for the ith local node of element counterclockwise from lower left corner
+ .
+ .
+ .
+ .
+ .
+ global node number for the ith local node of element counterclockwise from
+ lower left corner
\end_layout
\end_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -13693,14 +14289,10 @@
\emph on
nel1 incel1 incn1 nel2 incel2 incn2
-\emph default
-
\end_layout
\begin_deeper
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -13766,12 +14358,17 @@
\end_inset
- . . . . . . . number of elements in x1 (horizontal) direction
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ number of elements in x1 (horizontal) direction
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -13837,12 +14434,17 @@
\end_inset
- . . . . . . . increment of elements in x1 (horizontal) direction
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ increment of elements in x1 (horizontal) direction
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -13908,12 +14510,17 @@
\end_inset
- . . . . . . . increment of nodes in x1 (horizontal) direction
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ increment of nodes in x1 (horizontal) direction
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -13979,12 +14586,17 @@
\end_inset
- . . . . . . . number of elements in x2 (vertical) direction
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ number of elements in x2 (vertical) direction
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -14050,12 +14662,17 @@
\end_inset
- . . . . . . . increment of elements in x2 (vertical) direction
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ increment of elements in x2 (vertical) direction
\end_layout
\begin_layout Description
-
-
\begin_inset ERT
status collapsed
@@ -14121,162 +14738,370 @@
\end_inset
- . . . . . . . increment of nodes in x2 (vertical) direction
+ .
+ .
+ .
+ .
+ .
+ .
+ .
+ increment of nodes in x2 (vertical) direction
\end_layout
-\end_deeper
-\begin_deeper
\begin_layout Standard
-
\emph on
0 0 0 0 0 0 to end element connectivity group
-\emph default
-
\end_layout
\end_deeper
\begin_layout Chapter
+\begin_inset LatexCommand label
+name "cha:Sample-Input-Files"
-
-\begin_inset LatexCommand \label{cha:Sample-Input-Files}
-
\end_inset
Sample Input Files
\end_layout
\begin_layout Standard
+The lines below are a sample 50 element by 50 element input deck for a 1
+ by 1 square, constant viscosity with the Picard method.
+ This is Blankenbach 1a:
+\end_layout
-The lines below are a sample 50 element by 50 element input deck for a 1 by 1 square, constant viscosity with the Picard method. This is Blankenbach 1a
+\begin_layout LyX-Code
+50 x 50 el.
+ plate problem from Blankenbach et al., 1989
\end_layout
-\begin_layout Standard
+\begin_layout LyX-Code
+ #Nds sdm dof X Z prc ck echo rrst wrst nus tdvf tseq nelg sky wr
+\end_layout
+\begin_layout LyX-Code
+ 2601 2 2 50 50 2 1 0 0 1 102 0 1 1 1 0
+\end_layout
-\begin_inset ERT
-status collapsed
+\begin_layout LyX-Code
+ time step information
+\end_layout
-\begin_layout Standard
+\begin_layout LyX-Code
+ 100 1 1.0 1.0 0.50000
+\end_layout
+\begin_layout LyX-Code
+ output information
+\end_layout
-\backslash
-begin{verbatim}
+\begin_layout LyX-Code
+ 100 100 100 100
\end_layout
-\end_inset
+\begin_layout LyX-Code
+ velocity boundary condition flags: IFCMT,DELNXTLN
+\end_layout
- 50 x 50 el. problem for Blankenbach Benchmark numnp nsd ndof nelx nelz prc ck iecho rrst wrst nus tdvf tseq nelg sky wr 2601 2 2 50 50 2 1 0 0 1 102 0 1 1 1 0 time step information 100 1 1.0 1.0 0.50000 output information 100 100 100 100 velocity boundary condition flags bnode enode incr bcf1 bcf2 1 2551 51 0 1 2551 2601 1 1 0 1 51 1 1 0 51 2601 51 0 1 1 1 1 1 1 51 51 1 1 1 2551 2551 1 1 1 2601 2601 1 1 1 0 0 0 0 0 temperature boundary condition flags 1 2551 51 1 51 2601 51 1 0 0 0 0 bndy info (top - bottom rows) 1 2551 51 51 2601 51 0 0 0 bndy info (2nd from top - 2nd from bottom rows) 2 2552 51 50 2600 51 0 0 0 initial condition information 0.1 1.0 1.0 1.0 element information 2 2500 4 4 1 2 0 5 0 0 viscosity 1.0e0 penalty number 0.1E+08 diffusivity (always one) 1.0 Rayleigh number 1.0e+04 internal heating parameter 0.0 activation energy 0.0 activation volume 0.0 viscosity cutoff 1.0e7
-\begin_inset ERT
-status collapsed
+\begin_layout LyX-Code
+ bnode enode incr bcf1 bcf2
+\end_layout
-\begin_layout Standard
+\begin_layout LyX-Code
+ 1 2551 51 0 1
+\end_layout
+\begin_layout LyX-Code
+ 2551 2601 1 1 0
+\end_layout
-\backslash
-end{verbatim}
+\begin_layout LyX-Code
+ 1 51 1 1 0
\end_layout
-\end_inset
+\begin_layout LyX-Code
+ 51 2601 51 0 1
+\end_layout
- The lines below are a sample geometry file for the 50 by 50 element problem.
+\begin_layout LyX-Code
+ 1 1 1 1 1
\end_layout
-\begin_layout Standard
+\begin_layout LyX-Code
+ 51 51 1 1 1
+\end_layout
+\begin_layout LyX-Code
+ 2551 2551 1 1 1
+\end_layout
-\begin_inset ERT
-status collapsed
+\begin_layout LyX-Code
+ 2601 2601 1 1 1
+\end_layout
-\begin_layout Standard
+\begin_layout LyX-Code
+ 0 0 0 0 0
+\end_layout
+\begin_layout LyX-Code
+ temperature boundary condition flags
+\end_layout
-\backslash
-begin{verbatim}
+\begin_layout LyX-Code
+ 1 2551 51 1
\end_layout
-\end_inset
+\begin_layout LyX-Code
+ 51 2601 51 1
+\end_layout
- coordinates 1 4 0.0 0.0 2551 1 1.0 0.0 2601 1 1.0 1.0 51 1 0.0 1.0 50 51 50 1 0 0 0.0 0.0 velocity boundary conditions (non-zero) 0 0 0.0 0.0 temperature boundary conditions (non-zero) 1 2 1.0 2551 0 1.0 50 51 0 0 0.0 element connectivity and material groups 1 1 1 1 52 53 2 50 50 51 50 1 1 0 0 0 0 0 0 0
-\begin_inset ERT
-status collapsed
+\begin_layout LyX-Code
+ 0 0 0 0
+\end_layout
+\begin_layout LyX-Code
+ bndy info (top - bottom rows)
+\end_layout
+
+\begin_layout LyX-Code
+ 1 2551 51
+\end_layout
+
+\begin_layout LyX-Code
+ 51 2601 51
+\end_layout
+
+\begin_layout LyX-Code
+ 0 0 0
+\end_layout
+
+\begin_layout LyX-Code
+ bndy info (2nd from top - 2nd from bottom rows)
+\end_layout
+
+\begin_layout LyX-Code
+ 2 2552 51
+\end_layout
+
+\begin_layout LyX-Code
+ 50 2600 51
+\end_layout
+
+\begin_layout LyX-Code
+ 0 0 0
+\end_layout
+
+\begin_layout LyX-Code
+ initial condition information
+\end_layout
+
+\begin_layout LyX-Code
+ 0.1 1.0 1.0 1.0
+\end_layout
+
+\begin_layout LyX-Code
+ element information
+\end_layout
+
+\begin_layout LyX-Code
+ 2 2500 4 4 1 2 0 5 0 0
+\end_layout
+
+\begin_layout LyX-Code
+ viscosity
+\end_layout
+
+\begin_layout LyX-Code
+ 1.0e0
+\end_layout
+
+\begin_layout LyX-Code
+ penalty number
+\end_layout
+
+\begin_layout LyX-Code
+ 0.1E+08
+\end_layout
+
+\begin_layout LyX-Code
+ diffusivity (always one)
+\end_layout
+
+\begin_layout LyX-Code
+ 1.0
+\end_layout
+
+\begin_layout LyX-Code
+ Rayleigh number
+\end_layout
+
+\begin_layout LyX-Code
+ 1.0e+04
+\end_layout
+
+\begin_layout LyX-Code
+ internal heating parameter
+\end_layout
+
+\begin_layout LyX-Code
+ 0.0
+\end_layout
+
+\begin_layout LyX-Code
+ 0.0
+\end_layout
+
+\begin_layout LyX-Code
+ 0.0
+\end_layout
+
+\begin_layout LyX-Code
+ 1.0e7
+\end_layout
+
\begin_layout Standard
+The lines below are a sample geometry file for the 50 by 50 element problem.
+\end_layout
+\begin_layout LyX-Code
+coordinates
+\end_layout
-\backslash
-end{verbatim}
+\begin_layout LyX-Code
+ 1 4 0.0 0.0
\end_layout
-\end_inset
+\begin_layout LyX-Code
+ 2551 1 1.0 0.0
+\end_layout
+\begin_layout LyX-Code
+ 2601 1 1.0 1.0
+\end_layout
+\begin_layout LyX-Code
+ 51 1 0.0 1.0
\end_layout
-\begin_layout Chapter
+\begin_layout LyX-Code
+ 50 51 50 1
+\end_layout
+\begin_layout LyX-Code
+ 0 0 0.0 0.0
+\end_layout
-\begin_inset LatexCommand \label{cha:Output-Guide}
+\begin_layout LyX-Code
+ velocity boundary conditions (non-zero)
+\end_layout
+\begin_layout LyX-Code
+ 0 0 0.0 0.0
+\end_layout
+
+\begin_layout LyX-Code
+ temperature boundary conditions (non-zero)
+\end_layout
+
+\begin_layout LyX-Code
+ 1 2 1.0
+\end_layout
+
+\begin_layout LyX-Code
+ 2551 0 1.0
+\end_layout
+
+\begin_layout LyX-Code
+ 50 51
+\end_layout
+
+\begin_layout LyX-Code
+ 0 0 0.0
+\end_layout
+
+\begin_layout LyX-Code
+ element connectivity and material groups
+\end_layout
+
+\begin_layout LyX-Code
+ 1 1 1 1 52 53 2
+\end_layout
+
+\begin_layout LyX-Code
+ 50 50 51 50 1 1
+\end_layout
+
+\begin_layout LyX-Code
+ 0 0 0 0 0 0 0
+\end_layout
+
+\begin_layout Chapter
+\begin_inset LatexCommand label
+name "cha:Output-Guide"
+
\end_inset
Output Guide
\end_layout
\begin_layout Section
-
The Output Files
\end_layout
\begin_layout Standard
-
-The output files names are taken from the names in the runfile. The execution of
+The output files names are taken from the names in the runfile.
+ The execution of
\family typewriter
conman
\family default
- proceeds by % conman.pic < runfile where the runfile has a list of filenames that can be up to 80 characters long.
+ proceeds by % conman.pic < runfile where the runfile has a list of filenames
+ that can be up to 80 characters long.
\end_layout
\begin_layout Standard
\noindent
-
The names in the runfile are attached to the following input or output files.
\end_layout
-\begin_layout Standard
+\begin_layout LyX-Code
+input
+\end_layout
+\begin_layout LyX-Code
+geometry
+\end_layout
-\begin_inset ERT
-status collapsed
+\begin_layout LyX-Code
+output
+\end_layout
-\begin_layout Standard
+\begin_layout LyX-Code
+restart input
+\end_layout
+\begin_layout LyX-Code
+restart output
+\end_layout
-\backslash
-begin{verbatim}
+\begin_layout LyX-Code
+time series output
\end_layout
-\end_inset
+\begin_layout LyX-Code
+temperature (and velocity) field output
+\end_layout
- input geometry output restart input restart output time series output temperature (and velocity) field output stress (and viscosity) field output geoid output
-\begin_inset ERT
-status collapsed
-
-\begin_layout Standard
-
-
-\backslash
-end{verbatim}
+\begin_layout LyX-Code
+stress (and viscosity) field output
\end_layout
-\end_inset
-
-
+\begin_layout LyX-Code
+geoid output
\end_layout
\begin_layout Standard
-
-All files are ASCII files. The
+All files are ASCII files.
+ The
\begin_inset ERT
status collapsed
@@ -14350,7 +15175,6 @@
\end_layout
\begin_layout Standard
-
The
\begin_inset ERT
status collapsed
@@ -14386,7 +15210,8 @@
\end_inset
- file is a formatted record of the input. If the variable
+ file is a formatted record of the input.
+ If the variable
\begin_inset ERT
status collapsed
@@ -14421,7 +15246,9 @@
\end_inset
- is set to one, then values of the coordinates and boundary conditions are output and the file can become large. Near the end of the
+ is set to one, then values of the coordinates and boundary conditions are
+ output and the file can become large.
+ Near the end of the
\begin_inset ERT
status collapsed
@@ -14460,7 +15287,6 @@
\end_layout
\begin_layout Standard
-
The
\begin_inset ERT
status collapsed
@@ -14531,11 +15357,15 @@
\end_inset
- is set to one. The file is ASCII but formatted and the format statement can be found in the file
+ is set to one.
+ The file is ASCII but formatted and the format statement can be found in
+ the file
\series bold
input.F
\series default
-. The first line contains the initial timestep and time, the second line is a header, and the third through
+.
+ The first line contains the initial timestep and time, the second line
+ is a header, and the third through
\begin_inset ERT
status collapsed
@@ -14574,7 +15404,6 @@
\end_layout
\begin_layout Standard
-
The
\begin_inset ERT
status collapsed
@@ -14645,7 +15474,9 @@
\end_inset
- is set to one. We recommend this always be set to run. This file is overwritten every
+ is set to one.
+ We recommend this always be set to run.
+ This file is overwritten every
\begin_inset ERT
status collapsed
@@ -14683,12 +15514,9 @@
steps and the output is written from the file
\series bold
timdrv.F
-\series default
-
\end_layout
\begin_layout Standard
-
The
\begin_inset ERT
status collapsed
@@ -14724,15 +15552,16 @@
\end_inset
- file is written every time step. The values are ASCII and are heat flux at the bottom, heat flux at the top, kinetic energy, and time. All values are dimensionless. This file is written from the routine
+ file is written every time step.
+ The values are ASCII and are heat flux at the bottom, heat flux at the
+ top, kinetic energy, and time.
+ All values are dimensionless.
+ This file is written from the routine
\series bold
fluxke.F
-\series default
-
\end_layout
\begin_layout Standard
-
The
\begin_inset ERT
status collapsed
@@ -14772,7 +15601,8 @@
\series bold
print.F
\series default
-. There are two header lines that contain
+.
+ There are two header lines that contain
\begin_inset ERT
status collapsed
@@ -14807,7 +15637,10 @@
\end_inset
-. The second line labels the output and the third through numnp lines list the node, x, z, vx, vz, and temperature values at each node. This file is output every
+.
+ The second line labels the output and the third through numnp lines list
+ the node, x, z, vx, vz, and temperature values at each node.
+ This file is output every
\begin_inset ERT
status collapsed
@@ -14842,7 +15675,8 @@
\end_inset
- steps and each successive set of values is appended to the end of the file. The unix command
+ steps and each successive set of values is appended to the end of the file.
+ The unix command
\begin_inset ERT
status collapsed
@@ -14938,7 +15772,6 @@
\end_layout
\begin_layout Standard
-
The
\begin_inset ERT
status collapsed
@@ -14982,7 +15815,9 @@
\series bold
stress.F
\series default
-. Like the temperature velocity file there are two header lines that contain
+.
+ Like the temperature velocity file there are two header lines that contain
+
\begin_inset ERT
status collapsed
@@ -15017,11 +15852,15 @@
\end_inset
-. The second line labels the output and the third through
+.
+ The second line labels the output and the third through
\family typewriter
numnp
\family default
- lines list the node, x, z, txx, tzz. txz. p and viscosity at each node. This file is output every
+ lines list the node, x, z, txx, tzz.
+ txz.
+ p and viscosity at each node.
+ This file is output every
\begin_inset ERT
status collapsed
@@ -15056,7 +15895,8 @@
\end_inset
- steps and each successive set of values is appended to the end of the file. The unix command
+ steps and each successive set of values is appended to the end of the file.
+ The unix command
\begin_inset ERT
status collapsed
@@ -15152,7 +15992,6 @@
\end_layout
\begin_layout Standard
-
The
\begin_inset ERT
status collapsed
@@ -15188,15 +16027,22 @@
\end_inset
- file contains the horizontal coordinate, the dynamic topography, the geoid contribution from the temperature only and the total geoid (from surface and cmb topography, and internal (i.e., temperature) density contrasts). This is written from the file
+ file contains the horizontal coordinate, the dynamic topography, the geoid
+ contribution from the temperature only and the total geoid (from surface
+ and cmb topography, and internal (i.e., temperature) density contrasts).
+ This is written from the file
\series bold
geoid.F
\series default
-. The dimensional values are hard wired into the code and can be found at the top of
+.
+ The dimensional values are hard wired into the code and can be found at
+ the top of
\series bold
geoid.F
\series default
-. This version of the geoid code has not been tested with flow-through boundary conditions (i.e,
+.
+ This version of the geoid code has not been tested with flow-through boundary
+ conditions (i.e,
\begin_inset ERT
status collapsed
@@ -15235,8 +16081,6 @@
\end_layout
\begin_layout Standard
-
-
\begin_inset ERT
status collapsed
@@ -15292,7 +16136,9 @@
\begin_inset Formula $10^{7}$
\end_inset
- and does not adjust the parameters as the Rayleigh number changes. This would be a trivial fix but still may not yield the results intended. It is best to carefully check the
+ and does not adjust the parameters as the Rayleigh number changes.
+ This would be a trivial fix but still may not yield the results intended.
+ It is best to carefully check the
\series bold
geoid.F
\series default
@@ -15311,18 +16157,17 @@
\end_layout
\begin_layout Chapter
+\begin_inset LatexCommand label
+name "cha:The-Benchmark-Cases"
-
-\begin_inset LatexCommand \label{cha:The-Benchmark-Cases}
-
\end_inset
The Benchmark Cases
\end_layout
\begin_layout Standard
-
-We provide input and geometry files for the benchmark cases described below in the directories
+We provide input and geometry files for the benchmark cases described below
+ in the directories
\family typewriter
cookbook1
\family default
@@ -15334,17 +16179,19 @@
\end_layout
\begin_layout Section
-
Cookbook 1: Constant Viscosity Benchmark for ConMan
\end_layout
\begin_layout Standard
-
-Here we reproduce the results from Blankenbach et al. (1989)
+Here we reproduce the results from Blankenbach et al.
+ (1989)
\series bold
[TODO need reference for bibliography]
\series default
- for constant viscosity in a unit-aspect ratio domain, with free-slip boundary conditions, heated from below and cooled from above. The user should note that to run this problem, you will need to modify the routines
+ for constant viscosity in a unit-aspect ratio domain, with free-slip boundary
+ conditions, heated from below and cooled from above.
+ The user should note that to run this problem, you will need to modify
+ the routines
\series bold
rheol.newt.F
\series default
@@ -15352,7 +16199,9 @@
\series bold
geoid.F
\series default
- following the comments in those routines. The constant viscosity calculations (1a, 1b, and 1c) use a Rayleigh number of
+ following the comments in those routines.
+ The constant viscosity calculations (1a, 1b, and 1c) use a Rayleigh number
+ of
\begin_inset Formula $10^{4}$
\end_inset
@@ -15366,20 +16215,32 @@
respectively and the dimensional parameters are listed in Table\InsetSpace ~
-\begin_inset LatexCommand \ref{tab:Mantle-parameters-for}
+\begin_inset LatexCommand ref
+reference "tab:Mantle-parameters-for"
\end_inset
-. The time is the runtime in seconds for the Picard version of the code using 100 iterations on an intel MacPro with two 3\InsetSpace ~
-GHz processors using the Intel fortran compiler with -O2 optimization. While the problems in Blankenbach et al. are specified dimensionally, the equations are solved non-dimensionally within ConMan for numerical stability and the scaling factors are applied to the results.
+.
+ The time is the runtime in seconds for the Picard version of the code using
+ 100 iterations on an intel MacPro with two 3\InsetSpace ~
+GHz processors using the Intel
+ fortran compiler with -O2 optimization.
+ While the problems in Blankenbach et al.
+ are specified dimensionally, the equations are solved non-dimensionally
+ within ConMan for numerical stability and the scaling factors are applied
+ to the results.
\end_layout
\begin_layout Standard
+The results are computed on uniformly spaced grids and the global properties
+ of Nusselt number and root-mean-square velocity, as well as the topography
+ and geoid at the left- and right-hand side of the domain are reported,
+ along with the extrapolated value from Christensen's results (see Blankenbach
+ et al., 1989 for discussion) in Table\InsetSpace ~
-The results are computed on uniformly spaced grids and the global properties of Nusselt number and root-mean-square velocity, as well as the topography and geoid at the left- and right-hand side of the domain are reported, along with the extrapolated value from Christensen's results (see Blankenbach et al., 1989 for discussion) in Table\InsetSpace ~
+\begin_inset LatexCommand ref
+reference "tab:Blankenbach-(1989)-Benchmark-1a"
-\begin_inset LatexCommand \ref{tab:Blankenbach-(1989)-Benchmark-1a}
-
\end_inset
(Rayleigh number
@@ -15388,7 +16249,8 @@
), Table\InsetSpace ~
-\begin_inset LatexCommand \ref{tab:Blankenbach-(1989)-Benchmark-1b}
+\begin_inset LatexCommand ref
+reference "tab:Blankenbach-(1989)-Benchmark-1b"
\end_inset
@@ -15398,7 +16260,8 @@
), and Table\InsetSpace ~
-\begin_inset LatexCommand \ref{tab:Blankenbach-(1989)-Benchmark-1c}
+\begin_inset LatexCommand ref
+reference "tab:Blankenbach-(1989)-Benchmark-1c"
\end_inset
@@ -15406,50 +16269,51 @@
\begin_inset Formula $10^{6}$
\end_inset
-). For the global properties of Nusselt number and root-mean-square velocity, the 50 by 50 element grid are within 1% of Christensen's extrapolated results even for the Rayleigh number
+).
+ For the global properties of Nusselt number and root-mean-square velocity,
+ the 50 by 50 element grid are within 1% of Christensen's extrapolated results
+ even for the Rayleigh number
\begin_inset Formula $10^{6}$
\end_inset
- calculations. The agreement for the point values of topography and geoid are also 1% error on the 50 by 50 grid for the topography and geoid for the 50 by 50 element grid for the Rayleigh number
+ calculations.
+ The agreement for the point values of topography and geoid are also 1%
+ error on the 50 by 50 grid for the topography and geoid for the 50 by 50
+ element grid for the Rayleigh number
\begin_inset Formula $10^{6}$
\end_inset
calculations (Table\InsetSpace ~
-7.4). For the 200 by 200 grid, all values converge to Christensen's extrapolated results.
+7.4).
+ For the 200 by 200 grid, all values converge to Christensen's extrapolated
+ results.
\end_layout
\begin_layout Standard
-
-
\begin_inset Float table
placement htdp
wide false
sideways false
status open
-
\begin_layout Standard
-
\end_layout
\begin_layout Standard
\align center
-
-
-\begin_inset Tabular
+\begin_inset Tabular
<lyxtabular version="3" rows="11" columns="4">
<features>
-<column alignment="left" valignment="top">
-<column alignment="center" valignment="top">
-<column alignment="center" valignment="top">
-<column alignment="center" valignment="top">
+<column alignment="left" valignment="top" width="0">
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="0">
<row topline="true">
<cell multicolumn="1" alignment="none" valignment="top" topline="true" usebox="none" special="c">
\begin_inset Text
\begin_layout Standard
-
Parameter
\end_layout
@@ -15459,7 +16323,6 @@
\begin_inset Text
\begin_layout Standard
-
Symbol
\end_layout
@@ -15469,7 +16332,6 @@
\begin_inset Text
\begin_layout Standard
-
Value
\end_layout
@@ -15478,6 +16340,10 @@
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text
+\begin_layout Standard
+
+\end_layout
+
\end_inset
</cell>
</row>
@@ -15486,7 +16352,6 @@
\begin_inset Text
\begin_layout Standard
-
depth of domain
\end_layout
@@ -15496,8 +16361,6 @@
\begin_inset Text
\begin_layout Standard
-
-
\begin_inset Formula $d$
\end_inset
@@ -15510,8 +16373,6 @@
\begin_inset Text
\begin_layout Standard
-
-
\begin_inset Formula $10^{6}$
\end_inset
@@ -15524,6 +16385,10 @@
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text
+\begin_layout Standard
+
+\end_layout
+
\end_inset
</cell>
</row>
@@ -15532,7 +16397,6 @@
\begin_inset Text
\begin_layout Standard
-
gravitational acceleration
\end_layout
@@ -15542,7 +16406,6 @@
\begin_inset Text
\begin_layout Standard
-
g
\end_layout
@@ -15552,7 +16415,6 @@
\begin_inset Text
\begin_layout Standard
-
10 m/s
\begin_inset Formula $^{2}$
\end_inset
@@ -15565,6 +16427,10 @@
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text
+\begin_layout Standard
+
+\end_layout
+
\end_inset
</cell>
</row>
@@ -15573,7 +16439,6 @@
\begin_inset Text
\begin_layout Standard
-
temperature difference
\end_layout
@@ -15583,8 +16448,6 @@
\begin_inset Text
\begin_layout Standard
-
-
\begin_inset Formula $\Delta T$
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Grid
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@@ -17224,7 +18056,6 @@
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V
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-
Nusselt No.
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Topo
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Topo
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Geoid
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Geoid
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@@ -17736,14 +18553,17 @@
\end_layout
-\begin_layout Caption
+\begin_layout Standard
+\begin_inset Caption
+\begin_layout Standard
+\begin_inset LatexCommand label
+name "tab:Blankenbach-(1989)-Benchmark-1c"
-\begin_inset LatexCommand \label{tab:Blankenbach-(1989)-Benchmark-1c}
-
\end_inset
-Blankenbach (1989) Benchmark 1c: Steady State, 2D, constant viscosity convection in a 1 by 1 box with Rayleigh number
+Blankenbach (1989) Benchmark 1c: Steady State, 2D, constant viscosity convection
+ in a 1 by 1 box with Rayleigh number
\begin_inset Formula $10^{6}$
\end_inset
@@ -17755,16 +18575,24 @@
\end_layout
+\end_inset
+
+
+\end_layout
+
\begin_layout Section
-
Cookbook 2: Temperature-Dependent Viscosity Benchmark for ConMan
\end_layout
\begin_layout Standard
-
-Here we reproduce the results from Blankenbach et al. (1989) for temperature-dependent viscosity in a unit-aspect ratio domain, with free-slip boundary conditions, heated from below and cooled from above (case 2a). For this problem, the temperature-dependence of viscosity is given by
+Here we reproduce the results from Blankenbach et al.
+ (1989) for temperature-dependent viscosity in a unit-aspect ratio domain,
+ with free-slip boundary conditions, heated from below and cooled from above
+ (case 2a).
+ For this problem, the temperature-dependence of viscosity is given by
\begin_inset Formula \begin{equation}
\eta(T)=\eta_{o}\exp\left[{-\ln\{1000\}}\frac{{T}}{{\Delta T}}\right]\end{equation}
+
\end_inset
where
@@ -17775,37 +18603,43 @@
\begin_inset Formula $\Delta T=1000.0$
\end_inset
-. The other scaling parameters are the same as Table\InsetSpace ~
+.
+ The other scaling parameters are the same as Table\InsetSpace ~
-\begin_inset LatexCommand \ref{tab:Mantle-parameters-for}
+\begin_inset LatexCommand ref
+reference "tab:Mantle-parameters-for"
\end_inset
-. The results for a Rayleigh number of
+.
+ The results for a Rayleigh number of
\begin_inset Formula $10^{4}$
\end_inset
are presented in Table\InsetSpace ~
-\begin_inset LatexCommand \ref{tab:Blankenbach-(1989)-Benchmark-2a}
+\begin_inset LatexCommand ref
+reference "tab:Blankenbach-(1989)-Benchmark-2a"
\end_inset
-. Here once again the global properties of Nusselt number and root-mean-square velocity for the 50 by 50 grid are within 1-2% of Christensen's extrapolated results; however, in contrast to the constant viscosity cases, the values of topography and geoid in the corners differ from Christensen's extrapolated results by as much as 3% for topography and 7% for geoid on the 50 element by 50 element grid. Again, by the 400x400 grid, the values are well within 0.5%.
+.
+ Here once again the global properties of Nusselt number and root-mean-square
+ velocity for the 50 by 50 grid are within 1-2% of Christensen's extrapolated
+ results; however, in contrast to the constant viscosity cases, the values
+ of topography and geoid in the corners differ from Christensen's extrapolated
+ results by as much as 3% for topography and 7% for geoid on the 50 element
+ by 50 element grid.
+ Again, by the 400x400 grid, the values are well within 0.5%.
\end_layout
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-
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wide false
sideways false
status open
-
\begin_layout Standard
-
-
\begin_inset ERT
status collapsed
@@ -17819,23 +18653,22 @@
\end_inset
-\begin_inset Tabular
+\begin_inset Tabular
<lyxtabular version="3" rows="7" columns="8">
<features>
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-<column alignment="center" valignment="top">
-<column alignment="center" valignment="top">
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-<column alignment="center" valignment="top">
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\begin_layout Standard
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Grid
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V
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Nusselt No.
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Topo
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Topo
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Geoid
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+
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+\begin_layout Standard
+
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+
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+
+\end_layout
+
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+
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+
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<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
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+\begin_layout Standard
+
+\end_layout
+
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<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
\begin_inset Text
+\begin_layout Standard
+
+\end_layout
+
\end_inset
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<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
\begin_inset Text
+\begin_layout Standard
+
+\end_layout
+
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</row>
@@ -18439,14 +19249,18 @@
\end_layout
-\begin_layout Caption
+\begin_layout Standard
+\begin_inset Caption
+\begin_layout Standard
+\begin_inset LatexCommand label
+name "tab:Blankenbach-(1989)-Benchmark-2a"
-\begin_inset LatexCommand \label{tab:Blankenbach-(1989)-Benchmark-2a}
-
\end_inset
-Blankenbach (1989) Benchmark 2a: Steady State, 2D, temperature-dependent viscosity convection (b=6.907755279) in a 1 by 1 box with Rayleigh number
+Blankenbach (1989) Benchmark 2a: Steady State, 2D, temperature-dependent
+ viscosity convection (b=6.907755279) in a 1 by 1 box with Rayleigh number
+
\begin_inset Formula $10^{4}$
\end_inset
@@ -18458,19 +19272,28 @@
\end_layout
+\end_inset
+
+
+\end_layout
+
\begin_layout Section
-
Cookbook 3: Constant Viscosity Driven Slab Problem for ConMan
\end_layout
\begin_layout Standard
-
This geometry is based on the subduction zone benchmark by
-\begin_inset LatexCommand \cite{van Keken et al 2008}
+\begin_inset LatexCommand cite
+key "van Keken et al 2008"
\end_inset
-. A thermal structure from this is presented on the cover. The purpose here is to illustrate how one can (with some struggle) implement a deformed and variable mesh with the fairly basic grid generator in ConMan. One can use a more advanced grid generator and input the coordinates of every node directly as well as the
+.
+ A thermal structure from this is presented on the cover.
+ The purpose here is to illustrate how one can (with some struggle) implement
+ a deformed and variable mesh with the fairly basic grid generator in ConMan.
+ One can use a more advanced grid generator and input the coordinates of
+ every node directly as well as the
\series bold
ien
\series default
@@ -18478,15 +19301,18 @@
\end_layout
\begin_layout Standard
-
This problem will not reproduce the exact result presented in
-\begin_inset LatexCommand \cite{van Keken et al 2008}
+\begin_inset LatexCommand cite
+key "van Keken et al 2008"
\end_inset
- because this version of ConMan does not have the discontinuous velocity field, which turns out to be critical to match the benchmark results. The resulting thermal structure should look like Figure\InsetSpace ~
+ because this version of ConMan does not have the discontinuous velocity
+ field, which turns out to be critical to match the benchmark results.
+ The resulting thermal structure should look like Figure\InsetSpace ~
-\begin_inset LatexCommand \ref{fig:wedge}
+\begin_inset LatexCommand ref
+reference "fig:wedge"
\end_inset
@@ -18496,29 +19322,31 @@
\begin_layout Standard
\noindent
\align center
-
-
\begin_inset Float figure
wide false
sideways false
status open
+\begin_layout Standard
+\begin_inset Caption
-\begin_layout Caption
+\begin_layout Standard
+\begin_inset LatexCommand label
+name "fig:wedge"
+\end_inset
-\begin_inset LatexCommand \label{fig:wedge}
+ Thermal structure for the wedge problem in Cookbook 3 example.
+\end_layout
\end_inset
- Thermal structure for the wedge problem in Cookbook 3 example.
+
\end_layout
\begin_layout Standard
\align center
-
-
-\begin_inset Graphics
+\begin_inset Graphics
filename images/wedge.pdf
scale 60
@@ -18534,19 +19362,17 @@
\begin_layout Standard
\start_of_appendix
-
-
\begin_inset ERT
status collapsed
\begin_layout Standard
-%dummy comment inserted by tex2lyx to ensure that this paragraph is not empty
+%dummy comment inserted by tex2lyx to ensure that this paragraph is not
+ empty
\end_layout
\begin_layout Standard
-
\end_layout
\end_inset
@@ -18557,12 +19383,12 @@
\begin_layout Standard
-%dummy comment inserted by tex2lyx to ensure that this paragraph is not empty
+%dummy comment inserted by tex2lyx to ensure that this paragraph is not
+ empty
\end_layout
\begin_layout Standard
-
\end_layout
\end_inset
@@ -18573,12 +19399,12 @@
\begin_layout Standard
-%dummy comment inserted by tex2lyx to ensure that this paragraph is not empty
+%dummy comment inserted by tex2lyx to ensure that this paragraph is not
+ empty
\end_layout
\begin_layout Standard
-
\end_layout
\end_inset
@@ -18587,84 +19413,103 @@
\end_layout
\begin_layout Chapter
-
License
\end_layout
\begin_layout Standard
-
\series bold
-GNU GENERAL PUBLIC LICENSE Version 2, June 1991. Copyright (C) 1989, 1991 Free Software Foundation, Inc. 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
+GNU GENERAL PUBLIC LICENSE Version 2, June 1991.
+ Copyright (C) 1989, 1991 Free Software Foundation, Inc.
+ 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
\series default
\newline
- Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed.
+ Everyone is permitted to copy and distribute verbatim copies of this license
+ document, but changing it is not allowed.
\end_layout
\begin_layout Section*
-
Preamble
\end_layout
\begin_layout Standard
-
-The licenses for most software are designed to take away your freedom to share and change it. By contrast, the GNU General Public License is intended to guarantee your freedom to share and change free software -- to make sure the software is free for all its users. This General Public License applies to most of the Free Software Foundation's software and to any other program whose authors commit to using it. (Some other Free Software Foundation software is covered by the GNU Library General Public License instead.) You can apply it to your programs, too.
+The licenses for most software are designed to take away your freedom to
+ share and change it.
+ By contrast, the GNU General Public License is intended to guarantee your
+ freedom to share and change free software -- to make sure the software
+ is free for all its users.
+ This General Public License applies to most of the Free Software Foundation's
+ software and to any other program whose authors commit to using it.
+ (Some other Free Software Foundation software is covered by the GNU Library
+ General Public License instead.) You can apply it to your programs, too.
\end_layout
\begin_layout Standard
-
-When we speak of free software, we are referring to freedom, not price. Our General Public Licenses are designed to make sure that you have the freedom to distribute copies of free software (and charge for this service if you wish), that you receive source code or can get it if you want it, that you can change the software or use pieces of it in new free programs; and that you know you can do these things.
+When we speak of free software, we are referring to freedom, not price.
+ Our General Public Licenses are designed to make sure that you have the
+ freedom to distribute copies of free software (and charge for this service
+ if you wish), that you receive source code or can get it if you want it,
+ that you can change the software or use pieces of it in new free programs;
+ and that you know you can do these things.
\end_layout
\begin_layout Standard
-
-To protect your rights, we need to make restrictions that forbid anyone to deny you these rights or to ask you to surrender the rights. These restrictions translate to certain responsibilities for you if you distribute copies of the software, or if you modify it.
+To protect your rights, we need to make restrictions that forbid anyone
+ to deny you these rights or to ask you to surrender the rights.
+ These restrictions translate to certain responsibilities for you if you
+ distribute copies of the software, or if you modify it.
\end_layout
\begin_layout Standard
-
-For example, if you distribute copies of such a program, whether gratis or for a fee, you must give the recipients all the rights that you have. You must make sure that they, too, receive or can get the source code. And you must show them these terms so they know their rights.
+For example, if you distribute copies of such a program, whether gratis
+ or for a fee, you must give the recipients all the rights that you have.
+ You must make sure that they, too, receive or can get the source code.
+ And you must show them these terms so they know their rights.
\end_layout
\begin_layout Standard
-
We protect your rights with two steps:
\end_layout
\begin_layout Enumerate
-
Copyright the software, and
\end_layout
\begin_layout Enumerate
-
-Offer you this license which gives you legal permission to copy, distribute and/or modify the software.
+Offer you this license which gives you legal permission to copy, distribute
+ and/or modify the software.
+
\end_layout
\begin_layout Standard
-
-Also, for each author's protection and ours, we want to make certain that everyone understands that there is no warranty for this free software. If the software is modified by someone else and passed on, we want its recipients to know that what they have is not the original, so that any problems introduced by others will not reflect on the original authors' reputations.
+Also, for each author's protection and ours, we want to make certain that
+ everyone understands that there is no warranty for this free software.
+ If the software is modified by someone else and passed on, we want its
+ recipients to know that what they have is not the original, so that any
+ problems introduced by others will not reflect on the original authors'
+ reputations.
\end_layout
\begin_layout Standard
-
-Finally, any free program is threatened constantly by software patents. We wish to avoid the danger that redistributors of a free program will individually obtain patent licenses, in effect making the program proprietary. To prevent this, we have made it clear that any patent must be licensed for everyone's free use or not licensed at all.
+Finally, any free program is threatened constantly by software patents.
+ We wish to avoid the danger that redistributors of a free program will
+ individually obtain patent licenses, in effect making the program proprietary.
+ To prevent this, we have made it clear that any patent must be licensed
+ for everyone's free use or not licensed at all.
\end_layout
\begin_layout Standard
-
-The precise terms and conditions for copying, distribution and modification follow.
+The precise terms and conditions for copying, distribution and modification
+ follow.
\end_layout
\begin_layout Section*
-
-GNU GENERAL PUBLIC LICENSE TERMS AND CONDITIONS FOR COPYING, DISTRIBUTION AND MODIFICATION
+GNU GENERAL PUBLIC LICENSE TERMS AND CONDITIONS FOR COPYING, DISTRIBUTION
+ AND MODIFICATION
\end_layout
\begin_layout Itemize
-
-
\begin_inset ERT
status collapsed
@@ -18686,165 +19531,279 @@
\end_inset
-This License applies to any program or other work which contains a notice placed by the copyright holder saying it may be distributed under the terms of this General Public License. The "Program" below refers to any such program or work, and a "work based on the Program" means either the Program or any derivative work under copyright law: that is to say, a work containing the Program or a portion of it, either verbatim or with modifications and/or translated into another language. (Hereinafter, translation is included without limitation in the term "modification.") Each licensee is addressed as "you."
+This License applies to any program or other work which contains a notice
+ placed by the copyright holder saying it may be distributed under the terms
+ of this General Public License.
+ The "Program" below refers to any such program or work, and a "work based
+ on the Program" means either the Program or any derivative work under copyright
+ law: that is to say, a work containing the Program or a portion of it,
+ either verbatim or with modifications and/or translated into another language.
+ (Hereinafter, translation is included without limitation in the term "modificat
+ion.") Each licensee is addressed as "you."
\newline
\newline
- Activities other than copying, distribution and modification are not covered by this License; they are outside its scope. The act of running the Program is not restricted, and the output from the Program is covered only if its contents constitute a work based on the Program (independent of having been made by running the Program). Whether that is true depends on what the Program does.
+ Activities other than copying,
+ distribution and modification are not covered by this License; they are
+ outside its scope.
+ The act of running the Program is not restricted, and the output from the
+ Program is covered only if its contents constitute a work based on the
+ Program (independent of having been made by running the Program).
+ Whether that is true depends on what the Program does.
+
\end_layout
\begin_layout Enumerate
-
-You may copy and distribute verbatim copies of the Program's source code as you receive it, in any medium, provided that you conspicuously and appropriately publish on each copy an appropriate copyright notice and disclaimer of warranty; keep intact all the notices that refer to this License and to the absence of any warranty; and give any other recipients of the Program a copy of this License along with the Program.
+You may copy and distribute verbatim copies of the Program's source code
+ as you receive it, in any medium, provided that you conspicuously and appropria
+tely publish on each copy an appropriate copyright notice and disclaimer
+ of warranty; keep intact all the notices that refer to this License and
+ to the absence of any warranty; and give any other recipients of the Program
+ a copy of this License along with the Program.
\end_layout
\begin_deeper
\begin_layout Standard
-
-You may charge a fee for the physical act of transferring a copy, and you may at your option offer warranty protection in exchange for a fee.
+You may charge a fee for the physical act of transferring a copy, and you
+ may at your option offer warranty protection in exchange for a fee.
\end_layout
\end_deeper
\begin_layout Enumerate
-
-You may modify your copy or copies of the Program or any portion of it, thus forming a work based on the Program, and copy and distribute such modifications or work under the terms of Section 1 above, provided that you also meet all of these conditions:
+You may modify your copy or copies of the Program or any portion of it,
+ thus forming a work based on the Program, and copy and distribute such
+ modifications or work under the terms of Section 1 above, provided that
+ you also meet all of these conditions:
\end_layout
\begin_deeper
\begin_layout Enumerate
-
-You must cause the modified files to carry prominent notices stating that you changed the files and the date of any change.
+You must cause the modified files to carry prominent notices stating that
+ you changed the files and the date of any change.
+
\end_layout
\begin_layout Enumerate
-
-You must cause any work that you distribute or publish, that in whole or in part contains or is derived from the Program or any part thereof, to be licensed as a whole at no charge to all third parties under the terms of this License.
+You must cause any work that you distribute or publish, that in whole or
+ in part contains or is derived from the Program or any part thereof, to
+ be licensed as a whole at no charge to all third parties under the terms
+ of this License.
+
\end_layout
\begin_layout Enumerate
-
-If the modified program normally reads commands interactively when run, you must cause it, when started running for such interactive use in the most ordinary way, to print or display an announcement including an appropriate copyright notice and a notice that there is no warranty (or else, saying that you provide a warranty) and that users may redistribute the program under these conditions, and telling the user how to view a copy of this License. (Exception: if the Program itself is interactive but does not normally print such an announcement, your work based on the Program is not required to print an announcement.)
+If the modified program normally reads commands interactively when run,
+ you must cause it, when started running for such interactive use in the
+ most ordinary way, to print or display an announcement including an appropriate
+ copyright notice and a notice that there is no warranty (or else, saying
+ that you provide a warranty) and that users may redistribute the program
+ under these conditions, and telling the user how to view a copy of this
+ License.
+ (Exception: if the Program itself is interactive but does not normally
+ print such an announcement, your work based on the Program is not required
+ to print an announcement.)
\end_layout
-\end_deeper
-\begin_deeper
\begin_layout Standard
-
-These requirements apply to the modified work as a whole. If identifiable sections of that work are not derived from the Program, and can be reasonably considered independent and separate works in themselves, then this License, and its terms, do not apply to those sections when you distribute them as separate works. But when you distribute the same sections as part of a whole which is a work based on the Program, the distribution of the whole must be on the terms of this License, whose permissions for other licensees extend to the entire whole, and thus to each and every part regardless of who wrote it.
+These requirements apply to the modified work as a whole.
+ If identifiable sections of that work are not derived from the Program,
+ and can be reasonably considered independent and separate works in themselves,
+ then this License, and its terms, do not apply to those sections when you
+ distribute them as separate works.
+ But when you distribute the same sections as part of a whole which is a
+ work based on the Program, the distribution of the whole must be on the
+ terms of this License, whose permissions for other licensees extend to
+ the entire whole, and thus to each and every part regardless of who wrote
+ it.
\end_layout
\begin_layout Standard
-
-Thus, it is not the intent of this section to claim rights or contest your rights to work written entirely by you; rather, the intent is to exercise the right to control the distribution of derivative or collective works based on the Program.
+Thus, it is not the intent of this section to claim rights or contest your
+ rights to work written entirely by you; rather, the intent is to exercise
+ the right to control the distribution of derivative or collective works
+ based on the Program.
\end_layout
\begin_layout Standard
-
-In addition, mere aggregation of another work not based on the Program with the Program (or with a work based on the Program) on a volume of a storage or distribution medium does not bring the other work under the scope of this License.
+In addition, mere aggregation of another work not based on the Program with
+ the Program (or with a work based on the Program) on a volume of a storage
+ or distribution medium does not bring the other work under the scope of
+ this License.
\end_layout
\end_deeper
\begin_layout Enumerate
-
-You may copy and distribute the Program (or a work based on it, under Section 2) in object code or executable form under the terms of Sections 1 and 2 above provided that you also do one of the following:
+You may copy and distribute the Program (or a work based on it, under Section
+ 2) in object code or executable form under the terms of Sections 1 and
+ 2 above provided that you also do one of the following:
\end_layout
\begin_deeper
\begin_layout Enumerate
-
-Accompany it with the complete corresponding machine-readable source code, which must be distributed under the terms of Sections 1 and 2 above on a medium customarily used for software interchange; or,
+Accompany it with the complete corresponding machine-readable source code,
+ which must be distributed under the terms of Sections 1 and 2 above on
+ a medium customarily used for software interchange; or,
\end_layout
\begin_layout Enumerate
-
-Accompany it with a written offer, valid for at least three years, to give any third party, for a charge no more than your cost of physically performing source distribution, a complete machine-readable copy of the corresponding source code, to be distributed under the terms of Sections 1 and 2 above on a medium customarily used for software interchange; or,
+Accompany it with a written offer, valid for at least three years, to give
+ any third party, for a charge no more than your cost of physically performing
+ source distribution, a complete machine-readable copy of the corresponding
+ source code, to be distributed under the terms of Sections 1 and 2 above
+ on a medium customarily used for software interchange; or,
\end_layout
\begin_layout Enumerate
-
-Accompany it with the information you received as to the offer to distribute corresponding source code. (This alternative is allowed only for noncommercial distribution and only if you received the program in object code or executable form with such an offer, in accord with Subsection b above.)
+Accompany it with the information you received as to the offer to distribute
+ corresponding source code.
+ (This alternative is allowed only for noncommercial distribution and only
+ if you received the program in object code or executable form with such
+ an offer, in accord with Subsection b above.)
\end_layout
-\end_deeper
-\begin_deeper
\begin_layout Standard
-
-The source code for a work means the preferred form of the work for making modifications to it. For an executable work, complete source code means all the source code for all modules it contains, plus any associated interface definition files, plus the scripts used to control compilation and installation of the executable. However, as a special exception, the source code distributed need not include anything that is normally distributed (in either source or binary form) with the major components (compiler, kernel, and so on) of the operating system on which the executable runs, unless that component itself accompanies the executable.
+The source code for a work means the preferred form of the work for making
+ modifications to it.
+ For an executable work, complete source code means all the source code
+ for all modules it contains, plus any associated interface definition files,
+ plus the scripts used to control compilation and installation of the executable.
+ However, as a special exception, the source code distributed need not include
+ anything that is normally distributed (in either source or binary form)
+ with the major components (compiler, kernel, and so on) of the operating
+ system on which the executable runs, unless that component itself accompanies
+ the executable.
\end_layout
\begin_layout Standard
-
-If distribution of executable or object code is made by offering access to copy from a designated place, then offering equivalent access to copy the source code from the same place counts as distribution of the source code, even though third parties are not compelled to copy the source along with the object code.
+If distribution of executable or object code is made by offering access
+ to copy from a designated place, then offering equivalent access to copy
+ the source code from the same place counts as distribution of the source
+ code, even though third parties are not compelled to copy the source along
+ with the object code.
\end_layout
\end_deeper
\begin_layout Enumerate
-
-You may not copy, modify, sublicense, or distribute the Program except as expressly provided under this License. Any attempt otherwise to copy, modify, sublicense or distribute the Program is void, and will automatically terminate your rights under this License. However, parties who have received copies, or rights, from you under this License will not have their licenses terminated so long as such parties remain in full compliance.
+You may not copy, modify, sublicense, or distribute the Program except as
+ expressly provided under this License.
+ Any attempt otherwise to copy, modify, sublicense or distribute the Program
+ is void, and will automatically terminate your rights under this License.
+ However, parties who have received copies, or rights, from you under this
+ License will not have their licenses terminated so long as such parties
+ remain in full compliance.
+
\end_layout
\begin_layout Enumerate
-
-You are not required to accept this License, since you have not signed it. However, nothing else grants you permission to modify or distribute the Program or its derivative works. These actions are prohibited by law if you do not accept this License. Therefore, by modifying or distributing the Program (or any work based on the Program), you indicate your acceptance of this License to do so, and all its terms and conditions for copying, distributing or modifying the Program or works based on it.
+You are not required to accept this License, since you have not signed it.
+ However, nothing else grants you permission to modify or distribute the
+ Program or its derivative works.
+ These actions are prohibited by law if you do not accept this License.
+ Therefore, by modifying or distributing the Program (or any work based
+ on the Program), you indicate your acceptance of this License to do so,
+ and all its terms and conditions for copying, distributing or modifying
+ the Program or works based on it.
+
\end_layout
\begin_layout Enumerate
-
-Each time you redistribute the Program (or any work based on the Program), the recipient automatically receives a license from the original licensor to copy, distribute or modify the Program subject to these terms and conditions. You may not impose any further restrictions on the recipients' exercise of the rights granted herein. You are not responsible for enforcing compliance by third parties to this License.
+Each time you redistribute the Program (or any work based on the Program),
+ the recipient automatically receives a license from the original licensor
+ to copy, distribute or modify the Program subject to these terms and conditions.
+ You may not impose any further restrictions on the recipients' exercise
+ of the rights granted herein.
+ You are not responsible for enforcing compliance by third parties to this
+ License.
+
\end_layout
\begin_layout Enumerate
-
-If, as a consequence of a court judgment or allegation of patent infringement or for any other reason (not limited to patent issues), conditions are imposed on you (whether by court order, agreement or otherwise) that contradict the conditions of this License, they do not excuse you from the conditions of this License. If you cannot distribute so as to satisfy simultaneously your obligations under this License and any other pertinent obligations, then as a consequence you may not distribute the Program at all. For example, if a patent license would not permit royalty-free redistribution of the Program by all those who receive copies directly or indirectly through you, then the only way you could satisfy both it and this License would be to refrain entirely from distribution of the Program.
+If, as a consequence of a court judgment or allegation of patent infringement
+ or for any other reason (not limited to patent issues), conditions are
+ imposed on you (whether by court order, agreement or otherwise) that contradict
+ the conditions of this License, they do not excuse you from the conditions
+ of this License.
+ If you cannot distribute so as to satisfy simultaneously your obligations
+ under this License and any other pertinent obligations, then as a consequence
+ you may not distribute the Program at all.
+ For example, if a patent license would not permit royalty-free redistribution
+ of the Program by all those who receive copies directly or indirectly through
+ you, then the only way you could satisfy both it and this License would
+ be to refrain entirely from distribution of the Program.
\end_layout
\begin_deeper
\begin_layout Standard
-
-If any portion of this section is held invalid or unenforceable under any particular circumstance, the balance of the section is intended to apply and the section as a whole is intended to apply in other circumstances.
+If any portion of this section is held invalid or unenforceable under any
+ particular circumstance, the balance of the section is intended to apply
+ and the section as a whole is intended to apply in other circumstances.
\end_layout
\begin_layout Standard
-
-It is not the purpose of this section to induce you to infringe any patents or other property right claims or to contest validity of any such claims; this section has the sole purpose of protecting the integrity of the free software distribution system, which is implemented by public license practices. Many people have made generous contributions to the wide range of software distributed through that system in reliance on consistent application of that system; it is up to the author/donor to decide if he or she is willing to distribute software through any other system and a licensee cannot impose that choice.
+It is not the purpose of this section to induce you to infringe any patents
+ or other property right claims or to contest validity of any such claims;
+ this section has the sole purpose of protecting the integrity of the free
+ software distribution system, which is implemented by public license practices.
+ Many people have made generous contributions to the wide range of software
+ distributed through that system in reliance on consistent application of
+ that system; it is up to the author/donor to decide if he or she is willing
+ to distribute software through any other system and a licensee cannot impose
+ that choice.
\end_layout
\begin_layout Standard
-
-This section is intended to make thoroughly clear what is believed to be a consequence of the rest of this License.
+This section is intended to make thoroughly clear what is believed to be
+ a consequence of the rest of this License.
\end_layout
\end_deeper
\begin_layout Enumerate
-
-If the distribution and/or use of the Program is restricted in certain countries either by patents or by copyrighted interfaces, the original copyright holder who places the Program under this License may add an explicit geographical distribution limitation excluding those countries, so that distribution is permitted only in or among countries not thus excluded. In such case, this License incorporates the limitation as if written in the body of this License.
+If the distribution and/or use of the Program is restricted in certain countries
+ either by patents or by copyrighted interfaces, the original copyright
+ holder who places the Program under this License may add an explicit geographic
+al distribution limitation excluding those countries, so that distribution
+ is permitted only in or among countries not thus excluded.
+ In such case, this License incorporates the limitation as if written in
+ the body of this License.
+
\end_layout
\begin_layout Enumerate
-
-The Free Software Foundation may publish revised and/or new versions of the General Public License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns.
+The Free Software Foundation may publish revised and/or new versions of
+ the General Public License from time to time.
+ Such new versions will be similar in spirit to the present version, but
+ may differ in detail to address new problems or concerns.
\end_layout
\begin_deeper
\begin_layout Standard
-
-Each version is given a distinguishing version number. If the Program specifies a version number of this License which applies to it and "any later version," you have the option of following the terms and conditions either of that version or of any later version published by the Free Software Foundation. If the Program does not specify a version number of this License, you may choose any version ever published by the Free Software Foundation.
+Each version is given a distinguishing version number.
+ If the Program specifies a version number of this License which applies
+ to it and "any later version," you have the option of following the terms
+ and conditions either of that version or of any later version published
+ by the Free Software Foundation.
+ If the Program does not specify a version number of this License, you may
+ choose any version ever published by the Free Software Foundation.
\end_layout
\end_deeper
\begin_layout Enumerate
-
-If you wish to incorporate parts of the Program into other free programs whose distribution conditions are different, write to the author to ask for permission. For software which is copyrighted by the Free Software Foundation, write to the Free Software Foundation; we sometimes make exceptions for this. Our decision will be guided by the two goals of preserving the free status of all derivatives of our free software and of promoting the sharing and reuse of software generally.
+If you wish to incorporate parts of the Program into other free programs
+ whose distribution conditions are different, write to the author to ask
+ for permission.
+ For software which is copyrighted by the Free Software Foundation, write
+ to the Free Software Foundation; we sometimes make exceptions for this.
+ Our decision will be guided by the two goals of preserving the free status
+ of all derivatives of our free software and of promoting the sharing and
+ reuse of software generally.
+
\end_layout
\begin_layout Subsection*
-
NO WARRANTY
\end_layout
\begin_layout Itemize
-
-
\begin_inset ERT
status collapsed
@@ -18866,12 +19825,20 @@
\end_inset
-BECAUSE THE PROGRAM IS LICENSED FREE OF CHARGE, THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY SERVICING, REPAIR OR CORRECTION.
+BECAUSE THE PROGRAM IS LICENSED FREE OF CHARGE, THERE IS NO WARRANTY FOR
+ THE PROGRAM, TO THE EXTENT PERMITTED BY APPLICABLE LAW.
+ EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND/OR OTHER
+ PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY OF ANY KIND, EITHER
+ EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
+ OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.
+ THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH
+ YOU.
+ SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY
+ SERVICING, REPAIR OR CORRECTION.
+
\end_layout
\begin_layout Itemize
-
-
\begin_inset ERT
status collapsed
@@ -18893,85 +19860,106 @@
\end_inset
-IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MAY MODIFY AND/OR REDISTRIBUTE THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS), EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES.
+IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING WILL
+ ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MAY MODIFY AND/OR REDISTRIBUTE
+ THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING
+ ANY GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF
+ THE USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS
+ OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR
+ THIRD PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS),
+ EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY
+ OF SUCH DAMAGES.
+
\end_layout
\begin_layout Section*
-
END OF TERMS AND CONDITIONS
\end_layout
\begin_layout Subsection*
-
How to Apply These Terms to Your New Programs
\end_layout
\begin_layout Standard
-
-If you develop a new program, and you want it to be of the greatest possible use to the public, the best way to achieve this is to make it free software which everyone can redistribute and change under these terms.
+If you develop a new program, and you want it to be of the greatest possible
+ use to the public, the best way to achieve this is to make it free software
+ which everyone can redistribute and change under these terms.
\end_layout
\begin_layout Standard
-
-To do so, attach the following notices to the program. It is safest to attach them to the start of each source file to most effectively convey the exclusion of warranty; and each file should have at least the "copyright" line and a pointer to where the full notice is found. For example:
+To do so, attach the following notices to the program.
+ It is safest to attach them to the start of each source file to most effectivel
+y convey the exclusion of warranty; and each file should have at least the
+ "copyright" line and a pointer to where the full notice is found.
+ For example:
\end_layout
\begin_layout Quote
-
-One line to give the program's name and a brief idea of what it does. Copyright
+One line to give the program's name and a brief idea of what it does.
+ Copyright
\size footnotesize
-© (
+© (
\size normal
year) (name of author)
\end_layout
\begin_layout Quote
-
-This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.
+This program is free software; you can redistribute it and/or modify it
+ under the terms of the GNU General Public License as published by the Free
+ Software Foundation; either version 2 of the License, or (at your option)
+ any later version.
\end_layout
\begin_layout Quote
-
-This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.
+This program is distributed in the hope that it will be useful, but WITHOUT
+ ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
+ FOR A PARTICULAR PURPOSE.
+ See the GNU General Public License for more details.
\end_layout
\begin_layout Quote
-
-You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
+You should have received a copy of the GNU General Public License along
+ with this program; if not, write to the Free Software Foundation, Inc.,
+ 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
\end_layout
\begin_layout Standard
-
Also add information on how to contact you by electronic and paper mail.
\end_layout
\begin_layout Standard
-
-If the program is interactive, make it output a short notice like this when it starts in an interactive mode:
+If the program is interactive, make it output a short notice like this when
+ it starts in an interactive mode:
\end_layout
\begin_layout Quote
-
-Gnomovision version 69, Copyright © year name of author Gnomovision comes with ABSOLUTELY NO WARRANTY; for details type `show w'. This is free software, and you are welcome to redistribute it under certain conditions; type `show c' for details.
+Gnomovision version 69, Copyright © year name of author Gnomovision comes
+ with ABSOLUTELY NO WARRANTY; for details type `show w'.
+ This is free software, and you are welcome to redistribute it under certain
+ conditions; type `show c' for details.
+
\end_layout
\begin_layout Standard
-
-The hypothetical commands `show w' and `show c' should show the appropriate parts of the General Public License. Of course, the commands you use may be called something other than `show w' and `show c'; they could even be mouse-clicks or menu items -- whatever suits your program.
+The hypothetical commands `show w' and `show c' should show the appropriate
+ parts of the General Public License.
+ Of course, the commands you use may be called something other than `show
+ w' and `show c'; they could even be mouse-clicks or menu items -- whatever
+ suits your program.
\end_layout
\begin_layout Standard
-
-You should also get your employer (if you work as a programmer) or your school, if any, to sign a "copyright disclaimer" for the program, if necessary. Here is a sample; alter the names:
+You should also get your employer (if you work as a programmer) or your
+ school, if any, to sign a "copyright disclaimer" for the program, if necessary.
+ Here is a sample; alter the names:
\end_layout
\begin_layout Quote
-
-Yoyodyne, Inc., hereby disclaims all copyright interest in the program `Gnomovision' (which makes passes at compilers) written by James Hacker.
+Yoyodyne, Inc., hereby disclaims all copyright interest in the program `Gnomovisi
+on' (which makes passes at compilers) written by James Hacker.
\end_layout
\begin_layout Quote
-
(signature of Ty Coon)
\newline
1 April 1989
@@ -18980,54 +19968,110 @@
\end_layout
\begin_layout Standard
-
-This General Public License does not permit incorporating your program into proprietary programs. If your program is a subroutine library, you may consider it more useful to permit linking proprietary applications with the library. If this is what you want to do, use the GNU Library General Public License instead of this License.
+This General Public License does not permit incorporating your program into
+ proprietary programs.
+ If your program is a subroutine library, you may consider it more useful
+ to permit linking proprietary applications with the library.
+ If this is what you want to do, use the GNU Library General Public License
+ instead of this License.
\end_layout
\begin_layout Bibliography
+\begin_inset LatexCommand bibitem
+label "1"
+key "Blankenbach et al, 1989"
-\bibitem [1]{Blankenbach et al, 1989}
-Blankenbach, B., F. Busse, U. Christensen, L. Cserepes, D. Gunkel, U. Hansen, H. Harder, G. Jarvis, M. Koch, G. Marquart, D. Moore, P. Olson, H. Schmeling and T. Schnaubelt (1989) A Benchmark comparison for mantle convection codes,
+\end_inset
+
+Blankenbach, B., F.
+ Busse, U.
+ Christensen, L.
+ Cserepes, D.
+ Gunkel, U.
+ Hansen, H.
+ Harder, G.
+ Jarvis, M.
+ Koch, G.
+ Marquart, D.
+ Moore, P.
+ Olson, H.
+ Schmeling and T.
+ Schnaubelt (1989) A Benchmark comparison for mantle convection codes,
\emph on
-Geophys. J. Int.
+Geophys.
+ J.
+ Int.
\emph default
, 98, 23-38.
\end_layout
\begin_layout Bibliography
+\begin_inset LatexCommand bibitem
+label "2"
+key "Brooks 1981"
-\bibitem [2]{Brooks 1981}
-Brooks, A.N.
+\end_inset
+
+Brooks, A.N.
+
\emph on
A Petrov-Galerkin finite-element formulation for convection dominated flows.
+
\emph default
- Unpublished doctoral thesis, California Institute of Technology, Pasadena, CA, 1981.
+ Unpublished doctoral thesis, California Institute of Technology, Pasadena,
+ CA, 1981.
\end_layout
\begin_layout Bibliography
+\begin_inset LatexCommand bibitem
+label "3"
+key "Brooks and Hughes 1990"
-\bibitem [3]{Brooks and Hughes 1990}
-Brooks, A.N. and Hughes, T.J.R. (1990), Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations.
+\end_inset
+
+Brooks, A.N.
+ and Hughes, T.J.R.
+ (1990), Streamline upwind/Petrov-Galerkin formulations for convection dominated
+ flows with particular emphasis on the incompressible Navier-Stokes equations.
+
\emph on
-Comp. Meth. in Appl. Mech. and Eng., 81(3),
+Comp.
+ Meth.
+ in Appl.
+ Mech.
+ and Eng., 81(3),
\emph default
199-259.
\end_layout
\begin_layout Bibliography
+\begin_inset LatexCommand bibitem
+label "4"
+key "Hughes 1987"
-\bibitem [4]{Hughes 1987}
-Hughes, T.J.R.
+\end_inset
+
+Hughes, T.J.R.
+
\emph on
The finite element method.
+
\emph default
Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1987.
\end_layout
\begin_layout Bibliography
+\begin_inset LatexCommand bibitem
+label "5"
+key "Hughes and Brooks 1979"
-\bibitem [5]{Hughes and Brooks 1979}
-Hughes, T.J.R. and Brooks, A.N. (1979), A multi-dimensional upwind scheme with no crosswind diffusion. In: Finite element methods for convection dominated flows (T.J.R. Hughes, ed.),
+\end_inset
+
+Hughes, T.J.R.
+ and Brooks, A.N.
+ (1979), A multi-dimensional upwind scheme with no crosswind diffusion.
+ In: Finite element methods for convection dominated flows (T.J.R.
+ Hughes, ed.),
\emph on
ASME,
\emph default
@@ -19039,13 +20083,20 @@
\end_layout
\begin_layout Bibliography
+\begin_inset LatexCommand bibitem
+label "6"
+key "Hughes et al 1988"
-\bibitem [6]{Hughes et al 1988}
-Hughes, T.J.R., Franca, L.P., Hulbert, G.M., Johan, Z., and Shakib, F. (1988), The Galerkin/least-squares method for advective-diffusive equations, in
+\end_inset
+
+Hughes, T.J.R., Franca, L.P., Hulbert, G.M., Johan, Z., and Shakib, F.
+ (1988), The Galerkin/least-squares method for advective-diffusive equations,
+ in
\emph on
Recent developments in computational fluid dynamics
\emph default
- (T.E. Tezduyar, ed.),
+ (T.E.
+ Tezduyar, ed.),
\emph on
ASME,
\emph default
@@ -19057,11 +20108,20 @@
\end_layout
\begin_layout Bibliography
+\begin_inset LatexCommand bibitem
+label "7"
+key "Hughes et al 1979"
-\bibitem [7]{Hughes et al 1979}
-Hughes, T.J.R., Liu, W.K., and Brooks, A.N. (1979), Finite element analysis of incompressible viscous flows by the penalty function formulation.
+\end_inset
+
+Hughes, T.J.R., Liu, W.K., and Brooks, A.N.
+ (1979), Finite element analysis of incompressible viscous flows by the
+ penalty function formulation.
+
\emph on
-J. Comput. Phys.,
+J.
+ Comput.
+ Phys.,
\emph default
\emph on
@@ -19071,11 +20131,23 @@
\end_layout
\begin_layout Bibliography
+\begin_inset LatexCommand bibitem
+label "8"
+key "Malkus and Hughes 1978"
-\bibitem [8]{Malkus and Hughes 1978}
-Malkus, D.S. and Hughes, T.J.R. (1978), Mixed finite element methods reduced and selective integration techniques: a unification of concepts.
+\end_inset
+
+Malkus, D.S.
+ and Hughes, T.J.R.
+ (1978), Mixed finite element methods reduced and selective integration
+ techniques: a unification of concepts.
+
\emph on
-Comp. Meth. in Appl. Mech. and Eng.,
+Comp.
+ Meth.
+ in Appl.
+ Mech.
+ and Eng.,
\emph default
\emph on
@@ -19085,21 +20157,38 @@
\end_layout
\begin_layout Bibliography
+\begin_inset LatexCommand bibitem
+label "9"
+key "Temam 1977"
-\bibitem [9]{Temam 1977}
-Temam, R.
+\end_inset
+
+Temam, R.
+
\emph on
Navier-Stokes equations: theory and numerical analysis
\emph default
-. North-Holland. Amsterdam, 1977.
+.
+ North-Holland.
+ Amsterdam, 1977.
\end_layout
\begin_layout Bibliography
+\begin_inset LatexCommand bibitem
+label "10"
+key "Travis et al 1990"
-\bibitem [10]{Travis et al 1990}
-Travis, B.J., Anderson, C., Baumgardner, J., Gable, C.W., Hager, B.H., O'Connell, R.J., Olson, P., Raefsky, A. and Schubert, G. (1990), A benchmark comparison of numerical methods for infinite Prandtl number convection in two-dimensional Cartesian geometry,
+\end_inset
+
+Travis, B.J., Anderson, C., Baumgardner, J., Gable, C.W., Hager, B.H., O'Connell,
+ R.J., Olson, P., Raefsky, A.
+ and Schubert, G.
+ (1990), A benchmark comparison of numerical methods for infinite Prandtl
+ number convection in two-dimensional Cartesian geometry,
\emph on
-Geophys. Astrophys. Fluid Dynamics,
+Geophys.
+ Astrophys.
+ Fluid Dynamics,
\emph default
\emph on
@@ -19109,19 +20198,36 @@
\end_layout
\begin_layout Bibliography
+\begin_inset LatexCommand bibitem
+label "11"
+key "van Keken 1993"
-\bibitem [11]{van Keken 1993}
-van Keken, P. E. (1993) Numerical modeling of thermochemically driven fluid flow with non-Newtonian rheology: Applied to the Earth's lithosphere and mantle, Ph.D. Thesis, Utrecht University.
+\end_inset
+
+van Keken, P.
+ E.
+ (1993) Numerical modeling of thermochemically driven fluid flow with non-Newton
+ian rheology: Applied to the Earth's lithosphere and mantle, Ph.D.
+ Thesis, Utrecht University.
\end_layout
\begin_layout Bibliography
+\begin_inset LatexCommand bibitem
+label "12"
+key "van Keken et al 2008"
-\bibitem [12]{van Keken et al 2008}
- van Keken, P.E., Currie, C., King, S.D., Behn, M.D., Canioncle, A., He, J., Katz, R.F., Lin, S.-C., Parmentier, E.M., Spiegelman, M., and Wang, K. (2008), A community benchmark for subduction zone modeling,
+\end_inset
+
+ van Keken, P.E., Currie, C., King, S.D., Behn, M.D., Canioncle, A., He, J., Katz,
+ R.F., Lin, S.-C., Parmentier, E.M., Spiegelman, M., and Wang, K.
+ (2008), A community benchmark for subduction zone modeling,
\emph on
-Phys. Earth Planet. Int.
+Phys.
+ Earth Planet.
+ Int.
\emph default
-, in press.
+, in press.
+
\end_layout
\end_body
Modified: mc/2D/ConMan/trunk/doc/conman.pdf
===================================================================
(Binary files differ)
Modified: mc/2D/ConMan/trunk/doc/conman.tex
===================================================================
--- mc/2D/ConMan/trunk/doc/conman.tex 2008-08-27 22:15:35 UTC (rev 12738)
+++ mc/2D/ConMan/trunk/doc/conman.tex 2008-08-27 22:24:30 UTC (rev 12739)
@@ -8,7 +8,6 @@
\setcounter{tocdepth}{3}
\usepackage{float}
\usepackage{textcomp}
-\usepackage{color}
\usepackage{graphicx}
\IfFileExists{url.sty}{\usepackage{url}}
{\newcommand{\url}{\texttt}}
@@ -35,6 +34,38 @@
{\end{list}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands.
+%% LyX 1.5.1 created this file. For more info, see http://www.lyx.org/.
+%% Do not edit unless you really know what you are doing.
+
+
+
+
+
+
+\usepackage{float}
+
+\usepackage{textcomp}
+
+\usepackage{color}
+
+
+\IfFileExists{url.sty}{\usepackage{url}
+}
+ {\newcommand{\url}{\texttt}}
+
+\makeatletter
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands.
+%% Bold symbol macro for standard LaTeX users
+
+
+%% Because html converters don't know tabularnewline
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Textclass specific LaTeX commands.
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands.
%% LyX 1.5.4 created this file. For more info, see http://www.lyx.org/.
%% Do not edit unless you really know what you are doing.
@@ -45,12 +76,15 @@
\usepackage{float}
+
\usepackage{textcomp}
+
\usepackage{url}
+
\makeatletter
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands.
@@ -72,6 +106,7 @@
\usepackage{geometry}
+
\geometry{verbose,letterpaper,tmargin=1in,bmargin=1in,lmargin=1in,rmargin=1in}
@@ -79,14 +114,17 @@
\usepackage{float}
+
\usepackage{textcomp}
+
\usepackage{url}
+
\makeatletter
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands.
@@ -102,6 +140,7 @@
+
\let\myUrl\url
\renewcommand{\url}[1]{(\myUrl{#1})}
@@ -111,6 +150,11 @@
\makeatother
+
+\makeatother
+
+
+
\usepackage{babel}
\makeatother
@@ -118,7 +162,7 @@
\noindent \begin{center}
\thispagestyle{empty}%
\begin{figure}[H]
- \includegraphics[width=0.75\paperwidth]{images/conman_cover.pdf}
+ \includegraphics[width=0.75\paperwidth]{images/conman_cover}
\end{figure}
\par\end{center}
@@ -205,7 +249,7 @@
\begin{itemize}
\item King, S.D., A. Raefsky, and B.H. Hager (1990), ConMan: Vectorizing
a finite element code for incompressible two-dimensional convection
-in the Earth's mantle, \emph{Phys. Earth Planet. Int., 59, }195-208.
+in the Earth's mantle, \emph{Phys. Earth Planet. Int., 59,} 195-208.
\end{itemize}
The code was originally developed by Scott King, Arthur Raefsky and
Brad Hager, although many people have contributed improvements to
@@ -378,7 +422,7 @@
\sum_{B=1}^{n}\,\left(\int_{0}^{1}\frac{\partial N_{A}}{\partial x}\frac{\partial N_{B}}{\partial x}\, dx\right)\, d_{B}=\]
\begin{equation}
\int_{0}^{1}N_{A}\, f^{h}\, dx+N_{A}(0)h-g\,\int_{0}^{1}\frac{\partial N_{A}}{\partial x}\frac{\partial N_{n+1}}{\partial x}\, dx\label{eq:feform}\end{equation}
-Everything in Equation~\ref{eq:feform} is known except the $d_{B}$'s.
+ Everything in Equation~\ref{eq:feform} is known except the $d_{B}$'s.
This constitutes a system of $n$ equations and $n$ unknowns. We
can think of the left-hand side as a matrix, $K_{AB}$ whose entries
are \begin{equation}
@@ -393,7 +437,7 @@
K_{21} & K_{22} & \cdots & K_{2n}\\
\vdots & \vdots & & \vdots\\
K_{n1} & K_{n2} & \cdots & K_{nn}\end{array}\right]\label{Kmatrix}\end{equation}
-By tradition, $\left[K\right]$ is the stiffness matrix, $\{f\}$
+ By tradition, $\left[K\right]$ is the stiffness matrix, $\{f\}$
is the force vector, and $\{d\}$ is the displacement vector. When
the problem under consideration pertains to a mechanical system, this
makes the most sense, but even in heat conduction problems, or fluid
@@ -420,7 +464,7 @@
\sum_{A=1}^{4}N_{A}(X,Y) & = & 1\label{eq:normalize}\\
\sum_{A=1}^{4}N_{A}(X,Y)\, X_{A} & = & X\label{eq:interpx}\\
\sum_{A=1}^{4}N_{A}(X,Y)\, Y_{A} & = & Y\label{eq:interpy}\end{eqnarray}
-Equation~\ref{eq:normalize} says that they are normalized, so that
+ Equation~\ref{eq:normalize} says that they are normalized, so that
they sum to one (everywhere on X,Y). Equations~\ref{eq:interpx}
and~\ref{eq:interpy} state that the shape functions are also interpolation
functions. Without doing a lot of derivation, I will claim that for
@@ -429,7 +473,7 @@
N_{2} & = & \frac{(a+x)(b-y)}{4ab}\\
N_{3} & = & \frac{(a+x)(b+y)}{4ab}\\
N_{4} & = & \frac{(a-x)(b+y)}{4ab}\end{eqnarray}
-these shape functions satisfy the conditions in Equation~\ref{eq:normalize}
+ these shape functions satisfy the conditions in Equation~\ref{eq:normalize}
and Equations ~\ref{eq:interpx} and~\ref{eq:interpy}. A good exercise
would be to show this is true. A visual representation of these shape
functions is shown in Figure\ref{fig:The-bilinear-shape}.
@@ -444,7 +488,7 @@
\noindent \begin{centering}
-\includegraphics[scale=0.4]{images/conman-fig2.png}
+\includegraphics[scale=0.4]{images/conman-fig2}
\par\end{centering}
\end{figure}
@@ -473,10 +517,10 @@
important principle of mathematics, the Jacobian of the transformation
\begin{equation}
K_{11}~=~\int_{A}^{B}N_{1,x}N_{1,x}\, dx~=~\int_{0}^{1}N_{1,x}N_{1,x}\, Jdx\end{equation}
-where $J$ is the Jacobian of the transformation. This is a very powerful
-point. When we are thinking of solving a regular Cartesian domain,
-this just corresponds to a stretching or a shrinking (notice we set
-$a=b=1$ above (see Figure \ref{fig:Figure-1}).
+ where $J$ is the Jacobian of the transformation. This is a very
+powerful point. When we are thinking of solving a regular Cartesian
+domain, this just corresponds to a stretching or a shrinking (notice
+we set $a=b=1$ above (see Figure \ref{fig:Figure-1}).
\noindent \begin{center}
%
@@ -487,7 +531,7 @@
\noindent \begin{centering}
-\includegraphics[scale=0.45]{images/conman-fig1.png}
+\includegraphics[scale=0.45]{images/conman-fig1}
\par\end{centering}
\end{figure}
@@ -502,7 +546,7 @@
coordinate pair in the `parent element.' So we could write \begin{eqnarray}
x & = & 0.1+{0.05}\xi+0.0\eta\\
y & = & 0.2+{0.0}\xi+{0.10}\eta\end{eqnarray}
- or in matrix form we could write \begin{equation}\label{eq:transform}
+ or in matrix form we could write \begin{equation}
\left\{ \begin{array}{c}
x\\
y\end{array}\right\} =\left[\begin{array}{cc}
@@ -511,15 +555,16 @@
\xi\\
\eta\end{array}\right\} +\left\{ \begin{array}{c}
0.1\\
-0.2\end{array}\right\} \end{equation}
-If the transformation was from an arbitratry shaped quadrilateral to the parent element,
-then the off diagonal terms in the matrix in equation~\ref{eq:transform} will not be zero. It is easy enough to
-show that \begin{equation}
+0.2\end{array}\right\} \label{eq:transform}\end{equation}
+ If the transformation was from an arbitratry shaped quadrilateral
+to the parent element, then the off diagonal terms in the matrix in
+equation~\ref{eq:transform} will not be zero. It is easy enough
+to show that \begin{equation}
\int_{x_{1}}^{x_{2}}\int_{y_{1}}^{y_{2}}f(x,y)\, dx\, dy=\int_{-1}^{1}\int_{-1}^{1}f(\xi,\eta)\,{\rm det}[J]\, d\xi\, d\eta\end{equation}
-where $[J]$ is the Jacobian of the transformation.
-It turns out, and it is also easy to show, that ${\rm det}[J]$ is
-the ratio of the areas when going from one rectangle to another (in
-fact any Cartesian to Cartesian transformation).
+ where $[J]$ is the Jacobian of the transformation. It turns out,
+and it is also easy to show, that ${\rm det}[J]$ is the ratio of
+the areas when going from one rectangle to another (in fact any Cartesian
+to Cartesian transformation).
\textbf{Advanced Topic:} Now suppose we want to map a cylindrical
domain to our `parent element.' We can use the same principle in this
@@ -547,7 +592,7 @@
\int_{-1}^{1}(ax+b)\, dx=(\frac{ax^{2}}{2}+bx)|_{-1}^{1}=\frac{a}{2}+b-(\frac{a}{2}-b)=2b\end{equation}
Gauss' way \begin{equation}
\int_{-1}^{1}(ax+b)\, dx=a\frac{-1}{\sqrt{3}}+b+a\frac{1}{\sqrt{3}}+b=2b\end{equation}
-It turns out that $\frac{-1}{\sqrt{3}},\frac{1}{\sqrt{3}}$ are exact
+ It turns out that $\frac{-1}{\sqrt{3}},\frac{1}{\sqrt{3}}$ are exact
for a linear equation, but from what I showed, so would any $-x,x$
combination, but what Gauss showed was more powerful, that if the
function is of higher order, the $\frac{-1}{\sqrt{3}},\frac{1}{\sqrt{3}}$
@@ -604,12 +649,12 @@
reason; you just have to choose a convention.
\begin{description}
-\item [{{{id}}}] transforms global nodes to equation numbers (Figure
+\item [{{{{id}}}}] transforms global nodes to equation numbers (Figure
\ref{fig:Example-relationship-between}).
-\item [{{{ien}}}] transforms element local node numbers to global node
-numbers (Figure \ref{fig:4-Example-relationship-between}).
-\item [{{{lm}}}] transforms element local node numbers to global equation
-numbers.
+\item [{{{{ien}}}}] transforms element local node numbers to global
+node numbers (Figure \ref{fig:4-Example-relationship-between}).
+\item [{{{{lm}}}}] transforms element local node numbers to global
+equation numbers.
\end{description}
\noindent \begin{center}
%
@@ -621,7 +666,7 @@
\noindent \begin{centering}
-\includegraphics[scale=0.4]{images/conman-fig3.png}
+\includegraphics[scale=0.4]{images/conman-fig3}
\par\end{centering}
\end{figure}
@@ -648,7 +693,7 @@
\noindent \begin{centering}
-\includegraphics[scale=0.4]{images/conman-fig4.png}
+\includegraphics[scale=0.4]{images/conman-fig4}
\par\end{centering}
\end{figure}
@@ -657,9 +702,9 @@
In the code, the data structures for these two arrays are
\begin{description}
-\item [{{{id}}}] ( degree-of-freedom , global-node-number ) = equation-number
-\item [{{{ien}}}] ( local-node-number, element-number ) = global-node-number
-\item [{{{lm}}}] ( degree-of-freedom, local-node-number, element-number
+\item [{{{{id}}}}] ( degree-of-freedom , global-node-number ) = equation-number
+\item [{{{{ien}}}}] ( local-node-number, element-number ) = global-node-number
+\item [{{{{lm}}}}] ( degree-of-freedom, local-node-number, element-number
) = global-equation-number
\end{description}
@@ -708,7 +753,7 @@
\begin{eqnarray}
\tau_{ij}=-p^{\lambda}\delta_{ij}+2\mu u_{(i,j)}\label{eq:constit-lam}\\
0=u_{i,i}+p^{\lambda}/\lambda.\label{eq:constit-lam2}\end{eqnarray}
-As $\lambda$ approaches infinity, these relations approach the incompressible
+ As $\lambda$ approaches infinity, these relations approach the incompressible
solution. Also, as $\lambda$ approaches infinity, $p^{\lambda}$
approaches the hydrostatic pressure in the incompressible case. In
general, the hydrostatic pressure is $-\tau_{ii}/3$. Substituting
@@ -721,7 +766,7 @@
\tau_{ii}/3=-p=(\lambda+2/3\mu)u_{i,i}\end{equation}
but we also have \begin{equation}
-p^{\lambda}=\lambda u_{i,i}\end{equation}
-from Equation~\ref{eq:constit-lam2}. Clearly in the incompressible
+ from Equation~\ref{eq:constit-lam2}. Clearly in the incompressible
limit $\lambda\gg\mu$ then $\lambda+2/3\mu\rightarrow\lambda$ and
$p^{\lambda}\rightarrow p$. Also note that the continuity equation
is satisfied.
@@ -763,12 +808,12 @@
\int\int_{\Omega}f_{x}\, w\, d\Omega=b.c.~terms\\
\int\int_{\Omega}\frac{\partial w}{\partial x}\{2\mu(\frac{\partial u}{\partial z}+\frac{\partial v}{\partial x})/2\}+\frac{\partial w}{\partial z}\{\lambda(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial z})+2\mu\frac{\partial v}{\partial z}\}\, d\Omega+\nonumber \\
\int\int_{\Omega}f_{z}\, w\, d\Omega=b.c.~terms\end{eqnarray}
-Note that we don't get something for nothing; this shortcut does not
-give us the boundary condition terms (velocity or flux). These would
-fall out of the integration by parts. Recall, \begin{equation}
+ Note that we don't get something for nothing; this shortcut does
+not give us the boundary condition terms (velocity or flux). These
+would fall out of the integration by parts. Recall, \begin{equation}
\int_{a}^{b}w\, dv=w\, v|_{a}^{b}-\int_{a}^{b}v\, dw\end{equation}
-where in our case $w$ is the weighting function and $v$ is the second
-derivative term. The first term gives us the flux (first derivative)
+ where in our case $w$ is the weighting function and $v$ is the
+second derivative term. The first term gives us the flux (first derivative)
boundary conditions. In the case of the momentum equations, that is
the applied tractions (or stress boundary conditions).
@@ -882,7 +927,7 @@
\begin{centering}
-\includegraphics[scale=0.4]{images/conman-fig8.png}
+\includegraphics[scale=0.4]{images/conman-fig8}
\par\end{centering}
\end{figure}
@@ -953,10 +998,10 @@
and $u_{\xi}$ and $u_{\eta}$ are the velocities in the local element
coordinate system ($\xi$ $\eta$ system) evaluated at the element
center. This form of discretization has no crosswind diffusion because
-the ``artificial diffusion'' acts only in the direction of the flow
-(i.e., it follows the streamline), hence the name Streamline Upwind
-Petrov-Galerkin (SUPG). This makes it a better approximation than
-straight upwinding, and it has been demonstrated to be more accurate
+the {}``artificial diffusion'' acts only in the direction of the
+flow (i.e., it follows the streamline), hence the name Streamline
+Upwind Petrov-Galerkin (SUPG). This makes it a better approximation
+than straight upwinding, and it has been demonstrated to be more accurate
than Galerkin or straight upwinding in advection dominated problems
\cite{Hughes and Brooks 1979}. It has recently been shown that the
SUPG method is one of a broader class of methods for advection-diffusion
@@ -1075,8 +1120,8 @@
side vectors (\textbf{f\_trhsimp.F}). To use Picard iteration you
change the variable $\alpha$ in the Time Sequence card (second line
of the input file) from 0.5 to 1.0 and change the iterations from
-2 to 1 (same line). You need to ``make clean'' and remake with the
-target Picard, to generate a new executable (conman.pic). A future
+2 to 1 (same line). You need to {}``make clean'' and remake with
+the target Picard, to generate a new executable (conman.pic). A future
revision may merge these energy equation solvers into a single code
and hide the need to change the iteration steps from the user. When
you have a steady-state solution, Picard iteration usually converges
@@ -1275,91 +1320,95 @@
is described in the appropriate sections of this guide.
The following sixteen cards or groups of cards are read from the \textbf{iin}
-unit (throughout this guide a ``card'' will mean one line of an ASCII
-text file). These constitute the parameter part of the input ``deck''
-for the program ConMan. The format for this guide is a \textbf{bold}
-title line giving the card title followed by an \emph{italicized}
+unit (throughout this guide a {}``card'' will mean one line of an
+ASCII text file). These constitute the parameter part of the input
+{}``deck'' for the program ConMan. The format for this guide is
+a \textbf{bold} title line giving the card title followed by an \emph{italicized}
line showing the order of the parameters and a listing of the parameters
(with a brief explanation).
\begin{description}
-\item [{{{Title~Card}}}] \emph{Any descriptive character string up
-to 80 characters long}
-\item [{{{Global~Constants~Card}}}] \emph{numnp nsd ndof nelx nelz
-mprec iflow necho inrsts iorstr nodebn ntimvs ntseq numeg isky nwrap}
+\item [{{{{Title~Card}}}}] \emph{Any descriptive character string
+up to 80 characters long}
+\item [{{{{Global~Constants~Card}}}}] \emph{numnp nsd ndof nelx
+nelz mprec iflow necho inrsts iorstr nodebn ntimvs ntseq numeg isky
+nwrap}
\begin{description}
-\item [{{{numnp}}}] . . . . . . total number of nodal points
-\item [{{{nsd}}}] . . . . . . . . number of spatial dimensions (always
+\item [{{{{numnp}}}}] . . . . . . total number of nodal points
+\item [{{{{nsd}}}}] . . . . . . . . number of spatial dimensions
+(always 2)
+\item [{{{{ndof}}}}] . . . . . . . number of degrees of freedom (always
2)
-\item [{{{ndof}}}] . . . . . . . number of degrees of freedom (always
-2)
-\item [{{{nelx}}}] . . . . . . . number of elements in the x1 (horizontal)
+\item [{{{{nelx}}}}] . . . . . . . number of elements in the x1 (horizontal)
direction
-\item [{{{nelz}}}] . . . . . . . number of elements in the x2 (vertical)
+\item [{{{{nelz}}}}] . . . . . . . number of elements in the x2 (vertical)
direction
-\item [{{{mprec}}}] . . . . . . precision flag (always use double)
+\item [{{{{mprec}}}}] . . . . . . precision flag (always use double)
1 - single
2 - double
-\item [{{{iflow}}}] . . . . . . . data check flag
+\item [{{{{iflow}}}}] . . . . . . . data check flag
0 - check data only
1 - execute code
-\item [{{{necho}}}] . . . . . . echo data flag
+\item [{{{{necho}}}}] . . . . . . echo data flag
0 - minimum data echo (terse)
1 - echo data to output file (verbose)
-\item [{{{inrstr}}}] . . . . . . . read restart file flag
+\item [{{{{inrstr}}}}] . . . . . . . read restart file flag
0 - use default start (conductive)
1 - read restart file from unit 16
-\item [{{{iorstr}}}] . . . . . . . write restart file flag
+\item [{{{{iorstr}}}}] . . . . . . . write restart file flag
0 - don\textquoteright{}t write restart file
1 - write restart file to unit 17
-\item [{{{nodebn}}}] . . . . . . number of edge nodes for nusselt smoother
-\item [{{{ntimvs}}}] . . . . . . temperature dependent viscosity flag
+\item [{{{{nodebn}}}}] . . . . . . number of edge nodes for nusselt
+smoother
+\item [{{{{ntimvs}}}}] . . . . . . temperature dependent viscosity
+flag
0 - stiffness matrix factored once
1 - stiffness matrix factored every time step
-\item [{{{ntseq}}}] . . . . . . . number of time sequences (always
+\item [{{{{ntseq}}}}] . . . . . . . number of time sequences (always
1)
currently only one supported
-\item [{{{numeg}}}] . . . . . . number of element groups (always 1)
+\item [{{{{numeg}}}}] . . . . . . number of element groups (always
+1)
currently only one supported
-\item [{{{isky}}}] . . . . . . . flag for skyline factor
+\item [{{{{isky}}}}] . . . . . . . flag for skyline factor
0 - regular skyline
1 - vectorized skyline
-\item [{{{nwrap}}}] . . . . . . number of nodes to wrap
+\item [{{{{nwrap}}}}] . . . . . . number of nodes to wrap
equal to number of elements in vertical
@@ -1369,50 +1418,51 @@
fastest in vertical direction
\end{description}
-\item [{{{Time~Sequence~Cards}}}] - ntseq cards
+\item [{{{{Time~Sequence~Cards}}}}] - ntseq cards
\emph{nstep niter alpha delt epstol}
\begin{description}
-\item [{{{nstep}}}] . . . . . . . number of time steps
-\item [{{{niter}}}] . . . . . . . number of multicorrector iterations
+\item [{{{{nstep}}}}] . . . . . . . number of time steps
+\item [{{{{niter}}}}] . . . . . . . number of multicorrector iterations
2 - second-order expicit
1 - picard
-\item [{{{alpha}}}] . . . . . . . multicorrector parameter
+\item [{{{{alpha}}}}] . . . . . . . multicorrector parameter
0.5 for explicit 2nd order
1.0 for picard
-\item [{{{delt}}}] . . . . . . . time step (not used)
-\item [{{{epstol}}}] . . . . . . tolerance for hybrid method (not used)
+\item [{{{{delt}}}}] . . . . . . . time step (not used)
+\item [{{{{epstol}}}}] . . . . . . tolerance for hybrid method (not
+used)
\end{description}
-\item [{{{Output~Step~Card}}}] \emph{nsdprt nsvprt nstprt nsmprt}
+\item [{{{{Output~Step~Card}}}}] \emph{nsdprt nsvprt nstprt nsmprt}
\begin{description}
-\item [{{{nsdprt}}}] . . . . . . steps between disk output
-\item [{{{nsvprt}}}] . . . . . . steps between velocity output (not
-used)
-\item [{{{nstprt}}}] . . . . . . steps between temperature, velocity
+\item [{{{{nsdprt}}}}] . . . . . . steps between disk output
+\item [{{{{nsvprt}}}}] . . . . . . steps between velocity output
+(not used)
+\item [{{{{nstprt}}}}] . . . . . . steps between temperature, velocity
\& stress output
-\item [{{{nsmprt}}}] . . . . . . steps between stress field output
+\item [{{{{nsmprt}}}}] . . . . . . steps between stress field output
(not used)
\end{description}
-\item [{{{Velocity~Boundary~Condition~Flag~Cards}}}] \emph{bnode
+\item [{{{{Velocity~Boundary~Condition~Flag~Cards}}}}] \emph{bnode
enode incr (bcf(i), i=1,ndof)}
\begin{description}
-\item [{{{bnode}}}] . . . . . . beginning node
-\item [{{{enode}}}] . . . . . . ending node
-\item [{{{incr}}}] . . . . . . . node increment
-\item [{{{bcf(i)}}}] . . . . . . . boundary condition flag for ith
-degree of freedom
+\item [{{{{bnode}}}}] . . . . . . beginning node
+\item [{{{{enode}}}}] . . . . . . ending node
+\item [{{{{incr}}}}] . . . . . . . node increment
+\item [{{{{bcf(i)}}}}] . . . . . . . boundary condition flag for
+ith degree of freedom
0 - free slip
@@ -1424,14 +1474,14 @@
\emph{0 0 0 0 0 to end VBCF cards}
\begin{description}
-\item [{{{Temperature~Boundary~Condition~Flag~Cards}}}] \emph{bnode
+\item [{{{{Temperature~Boundary~Condition~Flag~Cards}}}}] \emph{bnode
enode incr bcf}
\begin{description}
-\item [{{{bnode}}}] . . . . . . beginning node
-\item [{{{enode}}}] . . . . . . ending node
-\item [{{{incr}}}] . . . . . . . node increment
-\item [{{{bcf}}}] . . . . . . . . boundary condition flag for temperature
+\item [{{{{bnode}}}}] . . . . . . beginning node
+\item [{{{{enode}}}}] . . . . . . ending node
+\item [{{{{incr}}}}] . . . . . . . node increment
+\item [{{{{bcf}}}}] . . . . . . . . boundary condition flag for temperature
1- fixed temperature
@@ -1439,109 +1489,113 @@
\end{description}
\emph{0 0 0 0 to end TBCF cards}
-\item [{{{Nusselt~Number~Boundary~Condition~Flag~Cards~-~Edge~Nodes}}}] top
+\item [{{{{Nusselt~Number~Boundary~Condition~Flag~Cards~-~Edge~Nodes}}}}] top
and bottom rows of nodes \emph{bnode enode incr}
\begin{description}
-\item [{{{bnode}}}] . . . . . . beginning node
-\item [{{{enode}}}] . . . . . . ending node
-\item [{{{incr}}}] . . . . . . . node increment
+\item [{{{{bnode}}}}] . . . . . . beginning node
+\item [{{{{enode}}}}] . . . . . . ending node
+\item [{{{{incr}}}}] . . . . . . . node increment
\end{description}
\emph{0 0 0 to end NNBCF (type a) cards}
-\item [{{{Nusselt~Number~Boundary~Condition~Flag~Cards~-~Second~Row~Nodes}}}] second
+\item [{{{{Nusselt~Number~Boundary~Condition~Flag~Cards~-~Second~Row~Nodes}}}}] second
from top and bottom rows of nodes \emph{bnode enode incr}
\begin{description}
-\item [{{{bnode}}}] . . . . . . beginning node
-\item [{{{enode}}}] . . . . . . ending node
-\item [{{{incr}}}] . . . . . . . node increment
+\item [{{{{bnode}}}}] . . . . . . beginning node
+\item [{{{{enode}}}}] . . . . . . ending node
+\item [{{{{incr}}}}] . . . . . . . node increment
\end{description}
\emph{0 0 0 to end NNBCF (type b) cards}
-\item [{{{Initial~Temperature~Card}}}] \emph{pert xsize zsize}
+\item [{{{{Initial~Temperature~Card}}}}] \emph{pert xsize zsize}
\begin{description}
-\item [{{{pert}}}] . . . . . . . perturbation from conductive state
-\item [{{{xsize}}}] . . . . . . . nondimensional length (x1 direction)
+\item [{{{{pert}}}}] . . . . . . . perturbation from conductive state
+\item [{{{{xsize}}}}] . . . . . . . nondimensional length (x1 direction)
of box
-\item [{{{zsize}}}] . . . . . . . nondimensional height (x2 direction)
+\item [{{{{zsize}}}}] . . . . . . . nondimensional height (x2 direction)
of box
\end{description}
-\item [{{{Element~Parameter~Cards}}}] - numeg cards \emph{ntype numel
-nen nenl numat nedof numsuf nipt implv implt}
+\item [{{{{Element~Parameter~Cards}}}}] - numeg cards \emph{ntype
+numel nen nenl numat nedof numsuf nipt implv implt}
\begin{description}
-\item [{{{ntype}}}] . . . . . . . element type
+\item [{{{{ntype}}}}] . . . . . . . element type
2 - two dimensional elements
-\item [{{{numel}}}] . . . . . . total number of elements
-\item [{{{nen}}}] . . . . . . . . number of element nodes (always 4)
-\item [{{{nenl}}}] . . . . . . . number of local element nodes (always
+\item [{{{{numel}}}}] . . . . . . total number of elements
+\item [{{{{nen}}}}] . . . . . . . . number of element nodes (always
4)
-\item [{{{numat}}}] . . . . . . number of material groups
-\item [{{{nedof}}}] . . . . . . . number of element degrees of freedom
-(always 2)
-\item [{{{numsuf}}}] . . . . . . number of imposed stress/flux cards
-\item [{{{nipt}}}] . . . . . . . number of integration points per element
-(always 5)
-\item [{{{implv}}}] . . . . . . . currently unused
-\item [{{{implt}}}] . . . . . . . currently unused
+\item [{{{{nenl}}}}] . . . . . . . number of local element nodes
+(always 4)
+\item [{{{{numat}}}}] . . . . . . number of material groups
+\item [{{{{nedof}}}}] . . . . . . . number of element degrees of
+freedom (always 2)
+\item [{{{{numsuf}}}}] . . . . . . number of imposed stress/flux
+cards
+\item [{{{{nipt}}}}] . . . . . . . number of integration points per
+element (always 5)
+\item [{{{{implv}}}}] . . . . . . . currently unused
+\item [{{{{implt}}}}] . . . . . . . currently unused
\end{description}
-\item [{{{Viscosity~Card}}}] \emph{visc(i), i=1,numat}
+\item [{{{{Viscosity~Card}}}}] \emph{visc(i), i=1,numat}
\begin{description}
-\item [{{{visc(i)}}}] . . . . . . preexponential viscosity coefficient
+\item [{{{{visc(i)}}}}] . . . . . . preexponential viscosity coefficient
for ith element
\end{description}
-\item [{{{Penalty~Card}}}] \emph{alam(i), i=1,numat}
+\item [{{{{Penalty~Card}}}}] \emph{alam(i), i=1,numat}
\begin{description}
-\item [{{{alam(i)}}}] . . . . . . penalty parameter for ith element
+\item [{{{{alam(i)}}}}] . . . . . . penalty parameter for ith element
\end{description}
-\item [{{{Diffusivity~Card}}}] \emph{diff(i), i=1,numat}
+\item [{{{{Diffusivity~Card}}}}] \emph{diff(i), i=1,numat}
\begin{description}
-\item [{{{diff(i)}}}] . . . . . . . thermal diffusivity for ith element
+\item [{{{{diff(i)}}}}] . . . . . . . thermal diffusivity for ith
+element
\end{description}
-\item [{{{Buoyancy~Rayleigh~Number~Card}}}] \emph{Ra(i), i=1,numat}
+\item [{{{{Buoyancy~Rayleigh~Number~Card}}}}] \emph{Ra(i), i=1,numat}
\begin{description}
-\item [{{{Ra(i)}}}] . . . . . . . bouyancy part of Rayleigh number
+\item [{{{{Ra(i)}}}}] . . . . . . . bouyancy part of Rayleigh number
for ith element
\end{description}
-\item [{{{Internal~Heating~Parameter~Card}}}] \emph{dmhu(i), i=1,numat}
+\item [{{{{Internal~Heating~Parameter~Card}}}}] \emph{dmhu(i),
+i=1,numat}
\begin{description}
-\item [{{{dmhu(i)}}}] . . . . . . internal heat source for ith material
-group
+\item [{{{{dmhu(i)}}}}] . . . . . . internal heat source for ith
+material group
\end{description}
-\item [{{{Activation~Energy~Card}}}] \emph{tcon(1,i), i=1,numat}
+\item [{{{{Activation~Energy~Card}}}}] \emph{tcon(1,i), i=1,numat}
\begin{description}
-\item [{{{tcon(1,i)}}}] . . . . . activation energy for ith material
+\item [{{{{tcon(1,i)}}}}] . . . . . activation energy for ith material
group for temperature dependent viscosity (kJ/mole)
\end{description}
-\item [{{{Activation~Volume~Card}}}] \emph{tcon(2,i), i=1,numat}
+\item [{{{{Activation~Volume~Card}}}}] \emph{tcon(2,i), i=1,numat}
\begin{description}
-\item [{{{tcon(2,i)}}}] . . . . . activation volume for ith material
-group for temperature dependent viscosity (cm$^{3}$/mole)
+\item [{{{{tcon(2,i)}}}}] . . . . . activation volume for ith material
+group for temperature dependent viscosity (cm$^{3}$/mole)
\end{description}
-\item [{{{Viscosity~Cutoff~Card}}}] \emph{tcon(3,i), i=1,numat}
+\item [{{{{Viscosity~Cutoff~Card}}}}] \emph{tcon(3,i), i=1,numat}
\begin{description}
-\item [{{{tcon(3,i)}}}] . . . . . maximum value of the viscosity for
-the ith material group
+\item [{{{{tcon(3,i)}}}}] . . . . . maximum value of the viscosity
+for the ith material group
\end{description}
-\item [{{{Surface~Force/Flux~Cards}}}] - numsuf cards \emph{nel side
-fnorm ftan flux}
+\item [{{{{Surface~Force/Flux~Cards}}}}] - numsuf cards \emph{nel
+side fnorm ftan flux}
\begin{description}
-\item [{{{nel}}}] . . . . . . . . element number
-\item [{{{side}}}] . . . . . . . side to apply force and flux
+\item [{{{{nel}}}}] . . . . . . . . element number
+\item [{{{{side}}}}] . . . . . . . side to apply force and flux
1 - bottom
@@ -1552,26 +1606,27 @@
4 - left side
-\item [{{{fnorm}}}] . . . . . . . normal surface force
-\item [{{{ftan}}}] . . . . . . . tangential surface force
-\item [{{{flux}}}] . . . . . . . . heat flux
+\item [{{{{fnorm}}}}] . . . . . . . normal surface force
+\item [{{{{ftan}}}}] . . . . . . . tangential surface force
+\item [{{{{flux}}}}] . . . . . . . . heat flux
\end{description}
\end{description}
The following four groups of cards are read from the \textbf{igeom}
-unit. These constitute the geometry part of the input ``deck'' for
-the program \texttt{conman}. The format of this section is the same
-as above.
+unit. These constitute the geometry part of the input {}``deck''
+for the program \texttt{conman}. The format of this section is the
+same as above.
\section{Coordinate Group }
\begin{description}
-\item [{{{Absolute~Coordinate~Card}}}] \emph{node gp (x(i,node) i=1,nsd)}
+\item [{{{{Absolute~Coordinate~Card}}}}] \emph{node gp (x(i,node)
+i=1,nsd)}
\begin{description}
-\item [{{{node}}}] . . . . . . . the node whose coordinates are to
-be specified
-\item [{{{gp}}}] . . . . . . . . generation parameter for automatic
+\item [{{{{node}}}}] . . . . . . . the node whose coordinates are
+to be specified
+\item [{{{{gp}}}}] . . . . . . . . generation parameter for automatic
generation
@@ -1581,15 +1636,15 @@
4 - generate a box using node as the lower left corner
-\item [{{{x(i,node)}}}] . . . . . coordinate value in the ith spatial
+\item [{{{{x(i,node)}}}}] . . . . . coordinate value in the ith spatial
dimension
\end{description}
-\item [{{{Corner~Generation~Cards}}}] - gp-1 cards \emph{node mgen
-(x(i,node) i=1,nsd)}
+\item [{{{{Corner~Generation~Cards}}}}] - gp-1 cards \emph{node
+mgen (x(i,node) i=1,nsd)}
\begin{description}
-\item [{{{node}}}] . . . . . . . node number
-\item [{{{mgen}}}] . . . . . . . generation parameter
+\item [{{{{node}}}}] . . . . . . . node number
+\item [{{{{mgen}}}}] . . . . . . . generation parameter
0 - don\textquoteright{}t use this as the start of a generation
@@ -1597,22 +1652,23 @@
1 - use this as the start of a generation sequence
-\item [{{{x(i,node)}}}] . . . . . coordinate value in the ith spatial
+\item [{{{{x(i,node)}}}}] . . . . . coordinate value in the ith spatial
dimension
\end{description}
-\item [{{{Generation~Increment~Card}}}] \emph{ninc1 inc1 ninc2 inc2}
+\item [{{{{Generation~Increment~Card}}}}] \emph{ninc1 inc1 ninc2
+inc2}
\begin{description}
-\item [{{{ninc1}}}] . . . . . . . number of additional nodes to generate
-in x1 direction
-\item [{{{inc1}}}] . . . . . . . increment of nodes in x1 direction
-\item [{{{ninc2}}}] . . . . . . . number of additional nodes to generate
-in x2 direction
+\item [{{{{ninc1}}}}] . . . . . . . number of additional nodes to
+generate in x1 direction
+\item [{{{{inc1}}}}] . . . . . . . increment of nodes in x1 direction
+\item [{{{{ninc2}}}}] . . . . . . . number of additional nodes to
+generate in x2 direction
0 - if gp equals 2
-\item [{{{inc2}}}] . . . . . . . increment of nodes in x2 direction
+\item [{{{{inc2}}}}] . . . . . . . increment of nodes in x2 direction
0 - if gp equals 2
@@ -1625,12 +1681,13 @@
\section{Velocity Boundary Condition Group }
\begin{description}
-\item [{{{Absolute~Velocity~Card}}}] \emph{node gp (v(i,node) i=1,nsd)}
+\item [{{{{Absolute~Velocity~Card}}}}] \emph{node gp (v(i,node)
+i=1,nsd)}
\begin{description}
-\item [{{{node}}}] . . . . . . . the node whose velocities are to be
-specified
-\item [{{{gp}}}] . . . . . . . . generation parameter for automatic
+\item [{{{{node}}}}] . . . . . . . the node whose velocities are
+to be specified
+\item [{{{{gp}}}}] . . . . . . . . generation parameter for automatic
generation
@@ -1640,15 +1697,15 @@
4 - generate a box using node as the lower left corner
-\item [{{{v(i,node)}}}] . . . . . velocity value in the ith spatial
+\item [{{{{v(i,node)}}}}] . . . . . velocity value in the ith spatial
dimension
\end{description}
-\item [{{{Corner~Generation~Cards}}}] - gp-1 cards \emph{node mgen
-(v(i,node) i=1,nsd)}
+\item [{{{{Corner~Generation~Cards}}}}] - gp-1 cards \emph{node
+mgen (v(i,node) i=1,nsd)}
\begin{description}
-\item [{{{node}}}] . . . . . . . node number
-\item [{{{mgen}}}] . . . . . . . generation parameter
+\item [{{{{node}}}}] . . . . . . . node number
+\item [{{{{mgen}}}}] . . . . . . . generation parameter
0 - don\textquoteright{}t use this as the start of a generation
@@ -1656,22 +1713,23 @@
1 - use this as the start of a generation sequence
-\item [{{{v(i,node)}}}] . . . . . velocity value in the ith spatial
+\item [{{{{v(i,node)}}}}] . . . . . velocity value in the ith spatial
dimension
\end{description}
-\item [{{{Generation~Increment~Card}}}] \emph{ninc1 inc1 ninc2 inc2}
+\item [{{{{Generation~Increment~Card}}}}] \emph{ninc1 inc1 ninc2
+inc2}
\begin{description}
-\item [{{{ninc1}}}] . . . . . . . number of additional nodes to generate
-in x1 direction
-\item [{{{inc1}}}] . . . . . . . increment of nodes in x1 direction
-\item [{{{ninc2}}}] . . . . . . . number of additional nodes to generate
-in x2 direction
+\item [{{{{ninc1}}}}] . . . . . . . number of additional nodes to
+generate in x1 direction
+\item [{{{{inc1}}}}] . . . . . . . increment of nodes in x1 direction
+\item [{{{{ninc2}}}}] . . . . . . . number of additional nodes to
+generate in x2 direction
\begin{description}
-\item [{{{0}}}] - if gp equals 2
+\item [{{{{0}}}}] - if gp equals 2
\end{description}
-\item [{{{inc2}}}] . . . . . . . increment of nodes in x2 direction
+\item [{{{{inc2}}}}] . . . . . . . increment of nodes in x2 direction
0 - if gp equals 2
@@ -1684,12 +1742,12 @@
\section{Temperature Boundary Condition Group }
\begin{description}
-\item [{{{Absolute~Temperature~Card}}}] \emph{node gp t(node)}
+\item [{{{{Absolute~Temperature~Card}}}}] \emph{node gp t(node)}
\begin{description}
-\item [{{{node}}}] . . . . . . . the node whose velocities are to be
-specified
-\item [{{{gp}}}] . . . . . . . . generation parameter for automatic
+\item [{{{{node}}}}] . . . . . . . the node whose velocities are
+to be specified
+\item [{{{{gp}}}}] . . . . . . . . generation parameter for automatic
generation
@@ -1697,14 +1755,14 @@
2 - generate a line using node as a starting point
-\item [{{{t(node)}}}] . . . . . . temperature value
+\item [{{{{t(node)}}}}] . . . . . . temperature value
\end{description}
-\item [{{{Corner~Generation~Cards}}}] - gp-1 cards \emph{node mgen
-t(node)}
+\item [{{{{Corner~Generation~Cards}}}}] - gp-1 cards \emph{node
+mgen t(node)}
\begin{description}
-\item [{{{node}}}] . . . . . . . node number
-\item [{{{mgen}}}] . . . . . . . generation parameter
+\item [{{{{node}}}}] . . . . . . . node number
+\item [{{{{mgen}}}}] . . . . . . . generation parameter
0 - don\textquoteright{}t use this as the start of a generation
@@ -1712,21 +1770,22 @@
1 - use this as the start of a generation sequence
-\item [{{{t(node)}}}] . . . . . . temperature value
+\item [{{{{t(node)}}}}] . . . . . . temperature value
\end{description}
-\item [{{{Generation~Increment~Card}}}] \emph{ninc1 inc1 ninc2 inc2}
+\item [{{{{Generation~Increment~Card}}}}] \emph{ninc1 inc1 ninc2
+inc2}
\begin{description}
-\item [{{{ninc1}}}] . . . . . . . number of additional nodes to generate
-in x1 direction
-\item [{{{inc1}}}] . . . . . . . increment of nodes in x1 direction
-\item [{{{ninc2}}}] . . . . . . . number of additional nodes to generate
-in x2 direction
+\item [{{{{ninc1}}}}] . . . . . . . number of additional nodes to
+generate in x1 direction
+\item [{{{{inc1}}}}] . . . . . . . increment of nodes in x1 direction
+\item [{{{{ninc2}}}}] . . . . . . . number of additional nodes to
+generate in x2 direction
0 - if gp equals 2
-\item [{{{inc2}}}] . . . . . . . increment of nodes in x2 direction
+\item [{{{{inc2}}}}] . . . . . . . increment of nodes in x2 direction
0 - if gp equals 2
@@ -1739,37 +1798,38 @@
\section{Element Connectivity (ien) Generation Group }
\begin{description}
-\item [{{{Absolution~Element~Card}}}] \emph{elnu ng mat no (ien(elnu,i)
+\item [{{{{Absolution~Element~Card}}}}] \emph{elnu ng mat no (ien(elnu,i)
i=1,nen)}
\begin{description}
-\item [{{{elnu}}}] . . . . . . . element number
-\item [{{{ng}}}] . . . . . . . . generation parameter
+\item [{{{{elnu}}}}] . . . . . . . element number
+\item [{{{{ng}}}}] . . . . . . . . generation parameter
0 - no generation
1 - generate using increments from increment card
-\item [{{{mat}}}] no . . . . . . material number for this element
-\item [{{{ien(elnu,i)}}}] . . . . . global node number for the ith
-local node of element counterclockwise from lower left corner
+\item [{{{{mat}}}}] no . . . . . . material number for this element
+\item [{{{{ien(elnu,i)}}}}] . . . . . global node number for the
+ith local node of element counterclockwise from lower left corner
\end{description}
-\item [{{{Increment~Card}}}] \emph{nel1 incel1 incn1 nel2 incel2 incn2}
+\item [{{{{Increment~Card}}}}] \emph{nel1 incel1 incn1 nel2 incel2
+incn2}
\begin{description}
-\item [{{{nel1}}}] . . . . . . . number of elements in x1 (horizontal)
+\item [{{{{nel1}}}}] . . . . . . . number of elements in x1 (horizontal)
direction
-\item [{{{incel1}}}] . . . . . . . increment of elements in x1 (horizontal)
+\item [{{{{incel1}}}}] . . . . . . . increment of elements in x1
+(horizontal) direction
+\item [{{{{incn1}}}}] . . . . . . . increment of nodes in x1 (horizontal)
direction
-\item [{{{incn1}}}] . . . . . . . increment of nodes in x1 (horizontal)
+\item [{{{{nel2}}}}] . . . . . . . number of elements in x2 (vertical)
direction
-\item [{{{nel2}}}] . . . . . . . number of elements in x2 (vertical)
+\item [{{{{incel2}}}}] . . . . . . . increment of elements in x2
+(vertical) direction
+\item [{{{{incn2}}}}] . . . . . . . increment of nodes in x2 (vertical)
direction
-\item [{{{incel2}}}] . . . . . . . increment of elements in x2 (vertical)
-direction
-\item [{{{incn2}}}] . . . . . . . increment of nodes in x2 (vertical)
-direction
\end{description}
\emph{0 0 0 0 0 0 to end element connectivity group}
@@ -1779,31 +1839,144 @@
The lines below are a sample 50 element by 50 element input deck for
a 1 by 1 square, constant viscosity with the Picard method. This is
-Blankenbach 1a
+Blankenbach 1a:
-\begin{verbatim} 50 x 50 el. problem for Blankenbach Benchmark numnp
-nsd ndof nelx nelz prc ck iecho rrst wrst nus tdvf tseq nelg sky wr
-2601 2 2 50 50 2 1 0 0 1 102 0 1 1 1 0 time step information 100 1
-1.0 1.0 0.50000 output information 100 100 100 100 velocity boundary
-condition flags bnode enode incr bcf1 bcf2 1 2551 51 0 1 2551 2601
-1 1 0 1 51 1 1 0 51 2601 51 0 1 1 1 1 1 1 51 51 1 1 1 2551 2551 1
-1 1 2601 2601 1 1 1 0 0 0 0 0 temperature boundary condition flags
-1 2551 51 1 51 2601 51 1 0 0 0 0 bndy info (top - bottom rows) 1 2551
-51 51 2601 51 0 0 0 bndy info (2nd from top - 2nd from bottom rows)
-2 2552 51 50 2600 51 0 0 0 initial condition information 0.1 1.0 1.0
-1.0 element information 2 2500 4 4 1 2 0 5 0 0 viscosity 1.0e0 penalty
-number 0.1E+08 diffusivity (always one) 1.0 Rayleigh number 1.0e+04
-internal heating parameter 0.0 activation energy 0.0 activation volume
-0.0 viscosity cutoff 1.0e7 \end{verbatim} The lines below are a sample
-geometry file for the 50 by 50 element problem.
+\begin{lyxcode}
+50~x~50~el.~plate~problem~from~Blankenbach~et~al.,~1989
-\begin{verbatim} coordinates 1 4 0.0 0.0 2551 1 1.0 0.0 2601 1 1.0
-1.0 51 1 0.0 1.0 50 51 50 1 0 0 0.0 0.0 velocity boundary conditions
-(non-zero) 0 0 0.0 0.0 temperature boundary conditions (non-zero)
-1 2 1.0 2551 0 1.0 50 51 0 0 0.0 element connectivity and material
-groups 1 1 1 1 52 53 2 50 50 51 50 1 1 0 0 0 0 0 0 0 \end{verbatim}
+~\#Nds~~sdm~dof~~~X~~~~Z~prc~ck~echo~rrst~wrst~nus~tdvf~tseq~nelg~sky~wr
+~~2601~~2~~~2~50~~50~~~2~~1~~~~0~~~~0~~~~1~102~~~~0~~~~1~~~~1~~~1~~0
+~time~step~information
+
+~~~~100~~~1~~1.0~~1.0~~0.50000
+
+~output~information
+
+~~~~100~~~~100~~~~100~~~~100
+
+~velocity~boundary~condition~flags:~IFCMT,DELNXTLN
+
+~bnode~~~enode~~~incr~bcf1~bcf2
+
+~~~~~1~~2551~~~~~51~~~~0~~~~1
+
+~~2551~~2601~~~~~~1~~~~1~~~~0
+
+~~~~~~1~~~51~~~~~~1~~~~1~~~~0
+
+~~~~51~~2601~~~~~51~~~~0~~~~1
+
+~~~~~1~~~~~1~~~~~~1~~~~1~~~~1
+
+~~~~51~~~~51~~~~~~1~~~~1~~~~1
+
+~~2551~~2551~~~~~~1~~~~1~~~~1
+
+~~2601~~2601~~~~~~1~~~~1~~~~1
+
+~~~~~~0~~~~~~0~~~~~~0~~~~0~~~~0
+
+~temperature~boundary~condition~flags
+
+~~~~~1~~2551~~~~51~~~~1
+
+~~~~51~~2601~~~~51~~~~1
+
+~~~~~~0~~~~~~0~~~~~~0~~~~0
+
+~bndy~info~(top~-~bottom~rows)
+
+~~~~~1~~2551~~~~51
+
+~~~~51~~2601~~~~51
+
+~~~~~~0~~~~~~0~~~~~~0
+
+~bndy~info~(2nd~from~top~-~2nd~from~bottom~rows)
+
+~~~~~~2~~2552~~~~51
+
+~~~~~50~~2600~~~~51
+
+~~~~~~0~~~~~~0~~~~~~0
+
+~initial~condition~information
+
+~~~~0.1~~~~1.0~~~~1.0~~~~1.0
+
+~element~information
+
+~2~~2500~4~4~1~2~0~5~0~0
+
+~viscosity
+
+~~1.0e0~~
+
+~penalty~number
+
+~~0.1E+08~
+
+~diffusivity~(always~one)
+
+~~~~1.0~~~
+
+~Rayleigh~number
+
+~~1.0e+04~~
+
+~internal~heating~parameter
+
+~~~~0.0
+
+~~~~0.0
+
+~~~~0.0~~~
+
+~~~~1.0e7~~~
+\end{lyxcode}
+The lines below are a sample geometry file for the 50 by 50 element
+problem.
+
+\begin{lyxcode}
+coordinates
+
+~~~~~~1~~~~4~~0.0~~0.0
+
+~~~2551~~~~1~~1.0~~0.0
+
+~~~2601~~~~1~~1.0~~1.0
+
+~~~~~51~~~~1~~0.0~~1.0
+
+~~~~~50~~~51~~~50~~~~1
+
+~~~~~~0~~~~0~~0.0~~0.0
+
+~velocity~boundary~conditions~(non-zero)
+
+~~~~~~0~~~~0~~0.0~~0.0
+
+~temperature~boundary~conditions~(non-zero)
+
+~~~~~~1~~~~2~~1.0
+
+~~~2551~~~~0~~1.0
+
+~~~~~50~~~51
+
+~~~~~~0~~~~0~~0.0
+
+~element~connectivity~and~material~groups
+
+~~~~~~1~~~~~~1~~~~~~1~~~~~~1~~~~~52~~~~~53~~~~~~2
+
+~~~~~50~~~~~50~~~~~51~~~~~50~~~~~~1~~~~~~1
+
+~~~~~~0~~~~~~0~~~~~~0~~~~~~0~~~~~~0~~~~~~0~~~~~~0
+\end{lyxcode}
+
\chapter{\label{cha:Output-Guide}Output Guide}
@@ -1817,10 +1990,25 @@
\noindent The names in the runfile are attached to the following input
or output files.
-\begin{verbatim} input geometry output restart input restart output
-time series output temperature (and velocity) field output stress
-(and viscosity) field output geoid output \end{verbatim}
+\begin{lyxcode}
+input~
+geometry~
+
+output~
+
+restart~input~
+
+restart~output~
+
+time~series~output~
+
+temperature~(and~velocity)~
+
+field~output~stress~(and~viscosity)~
+
+field~output~geoid~output~
+\end{lyxcode}
All files are ASCII files. The {\em input} and {\em geometry}
files are as described in the previous section.
@@ -1926,6 +2114,8 @@
%
\begin{table}[htdp]
+
+
\begin{centering}
\begin{tabular}{lccc}
\hline
@@ -2030,7 +2220,7 @@
heated from below and cooled from above (case 2a). For this problem,
the temperature-dependence of viscosity is given by \begin{equation}
\eta(T)=\eta_{o}\exp\left[{-\ln\{1000\}}\frac{{T}}{{\Delta T}}\right]\end{equation}
-where $\eta_{o}=2.9\times10^{19}$ and $\Delta T=1000.0$. The other
+ where $\eta_{o}=2.9\times10^{19}$ and $\Delta T=1000.0$. The other
scaling parameters are the same as Table~\ref{tab:Mantle-parameters-for}.
The results for a Rayleigh number of $10^{4}$ are presented in Table~\ref{tab:Blankenbach-(1989)-Benchmark-2a}.
Here once again the global properties of Nusselt number and root-mean-square
@@ -2089,7 +2279,7 @@
\begin{centering}
-\includegraphics[scale=0.6]{images/wedge.pdf}
+\includegraphics[scale=0.6]{images/wedge}
\par\end{centering}
\end{figure}
@@ -2098,6 +2288,7 @@
\appendix
%dummy comment inserted by tex2lyx to ensure that this paragraph is not empty
%dummy comment inserted by tex2lyx to ensure that this paragraph is not empty
+%dummy comment inserted by tex2lyx to ensure that this paragraph is not empty
@@ -2451,11 +2642,13 @@
with the library. If this is what you want to do, use the GNU Library
General Public License instead of this License.
-\begin{thebibliography}{1}
+\begin{thebibliography}{10}
+\bibitem[1]{Blankenbach et al, 1989}Blankenbach, B., F. Busse, U.
+Christensen, L. Cserepes, D. Gunkel, U. Hansen, H. Harder, G. Jarvis,
+M. Koch, G. Marquart, D. Moore, P. Olson, H. Schmeling and T. Schnaubelt
+(1989) A Benchmark comparison for mantle convection codes, \emph{Geophys.
+J. Int.}, 98, 23-38.
-\bibitem[1]{Blankenbach et al, 1989}Blankenbach, B., F. Busse, U. Christensen, L. Cserepes, D. Gunkel, U. Hansen, H. Harder, G. Jarvis, M. Koch, G. Marquart, D. Moore, P. Olson, H. Schmeling and T. Schnaubelt (1989) A Benchmark comparison for mantle convection codes,
-\emph{Geophys. J. Int.}, 98, 23-38.
-
\bibitem[2]{Brooks 1981}Brooks, A.N. \emph{A Petrov-Galerkin finite-element
formulation for convection dominated flows.} Unpublished doctoral
thesis, California Institute of Technology, Pasadena, CA, 1981.
@@ -2500,13 +2693,14 @@
doi: 10.1080/03091929008204111
\bibitem[11]{van Keken 1993}van Keken, P. E. (1993) Numerical modeling
-of thermochemically driven fluid flow with non-Newtonian rheology: Applied
-to the Earth's lithosphere and mantle, Ph.D. Thesis, Utrecht University.
+of thermochemically driven fluid flow with non-Newtonian rheology:
+Applied to the Earth's lithosphere and mantle, Ph.D. Thesis, Utrecht
+University.
\bibitem[12]{van Keken et al 2008} van Keken, P.E., Currie, C., King,
S.D., Behn, M.D., Canioncle, A., He, J., Katz, R.F., Lin, S.-C., Parmentier,
E.M., Spiegelman, M., and Wang, K. (2008), A community benchmark for
-subduction zone modeling, \emph{Phys. Earth Planet. Int.}, in press.
+subduction zone modeling, \emph{Phys. Earth Planet. Int.}, in press.
\end{thebibliography}
\end{document}
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