[cig-commits] r12423 - in mc/2D/ConMan/trunk/doc: . images
sue at geodynamics.org
sue at geodynamics.org
Thu Jul 17 11:29:53 PDT 2008
Author: sue
Date: 2008-07-17 11:29:52 -0700 (Thu, 17 Jul 2008)
New Revision: 12423
Modified:
mc/2D/ConMan/trunk/doc/conman.lyx
mc/2D/ConMan/trunk/doc/images/conman_cover.pdf
mc/2D/ConMan/trunk/doc/images/conman_cover.psd
Log:
sjking changes, readded images, new cover figures, new pdf
Modified: mc/2D/ConMan/trunk/doc/conman.lyx
===================================================================
--- mc/2D/ConMan/trunk/doc/conman.lyx 2008-07-16 23:35:43 UTC (rev 12422)
+++ mc/2D/ConMan/trunk/doc/conman.lyx 2008-07-17 18:29:52 UTC (rev 12423)
@@ -4,13 +4,47 @@
\begin_header
\textclass book
\begin_preamble
+%% 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.
+
+
+
+\usepackage{geometry}
+
+\geometry{verbose,letterpaper,tmargin=1in,bmargin=1in,lmargin=1in,rmargin=1in}
+
+
+
+\usepackage{float}
+
+\usepackage{textcomp}
+
+\usepackage{url}
+
+
+
+\makeatletter
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands.
+%% Bold symbol macro for standard LaTeX users
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Textclass specific LaTeX commands.
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands.
\usepackage{hyperref}
+
\let\myUrl\url
\renewcommand{\url}[1]{(\myUrl{#1})}
+
+
+\makeatother
\end_preamble
\language english
-\inputencoding auto
+\inputencoding latin9
\font_roman default
\font_sans default
\font_typewriter default
@@ -23,21 +57,17 @@
\paperfontsize default
\spacing single
\papersize default
-\use_geometry true
-\use_amsmath 1
+\use_geometry false
+\use_amsmath 0
\use_esint 0
\cite_engine basic
\use_bibtopic false
\paperorientation portrait
-\leftmargin 1in
-\topmargin 1in
-\rightmargin 1in
-\bottommargin 1in
\secnumdepth 3
\tocdepth 3
\paragraph_separation indent
\defskip medskip
-\quotes_language swedish
+\quotes_language english
\papercolumns 1
\papersides 2
\paperpagestyle headings
@@ -52,8 +82,6 @@
\begin_layout Standard
\noindent
\align center
-
-\color none
\begin_inset ERT
status collapsed
@@ -74,14 +102,14 @@
status open
\begin_layout Standard
-\noindent
+
\begin_inset Graphics
- filename images/conman_cover.pdf
+ filename /Users/sue/conman/images/conman_cover.pdf
width 75page%
\end_inset
-
+
\end_layout
\end_inset
@@ -96,9 +124,9 @@
\begin_layout Author
© California Institute of Technology
\newline
-Scott King
+ Scott King
\newline
-Version 2.0
+ Version 2.0
\end_layout
\begin_layout Date
@@ -166,17 +194,32 @@
\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 written by Arthur Raefsky and Scott
- King while at Caltech (Scott King is now at Virginia Tech).
+ 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 and runs under Unix and Unicos
- operating systems.
+ ConMan is written in Standard Fortran 77 with cray pointers and runs on
+ most unix systems with many fortran comilers (see Section 3.4 [TODO -- 3.4
+ is placeholder section.
+ Is this correct? Need label to put cross ref here]).
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.
+ 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.
+ problems but make no promises regarding the timeliness of changes or fixes;
+ see Section
+\begin_inset LatexCommand ref
+reference "sec:Support"
+
+\end_inset
+
+ for instructions on how to report problems.
\end_layout
\begin_layout Section
@@ -197,7 +240,6 @@
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
@@ -206,8 +248,8 @@
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 the vectorization strategy and finally a descriptio
-n of all the input variables.
+ 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
@@ -217,27 +259,9 @@
\emph on
italicized
\emph default
- type, and important points or potential traps will be noted with
+ type.
\end_layout
-\begin_layout Quote
-\begin_inset Formula $\clubsuit$
-\end_inset
-
- This is an important point
-\end_layout
-
-\begin_layout Standard
-Questions and problems can be addressed to
-\begin_inset LatexCommand url
-name "Scott King"
-target "sdk at vt.edu"
-
-\end_inset
-
-.
-\end_layout
-
\begin_layout Section
Citation
\end_layout
@@ -251,24 +275,30 @@
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
+The code is based on the method described in
\end_layout
\begin_layout Itemize
-todo -- which item from BIBLIOGRAPHY should go here, if any???
+King, S.D., A.
+ Raefsky, and B.H.
+ Hager, ConMan: Vectorizing a finite element code for incompressible two-dimensi
+onal convection in the Earth's mantle,Phys.
+ Earth Planet.
+ Int., 59, 195-208, 1990.
+
\end_layout
\begin_layout Standard
-The code was originally developed by Scott King.
+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
-\begin_inset LatexCommand htmlurl
-name "CIG"
+ code and CIG
+\begin_inset LatexCommand url
target "www.geodynamics.org"
\end_inset
@@ -285,14 +315,12 @@
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.
-
\end_layout
\begin_layout Chapter
@@ -304,12 +332,34 @@
\end_layout
\begin_layout Standard
+I am going to follow a similar form to Hughes (The Finite Element Method)
+ Chapter\InsetSpace ~
+1 sections 1-4.
+ There are two forms we can write the equation, the strong and the weak
+ form.
+ You are used to seeing the strong form, but not used to seeing 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.
+ For viscous flow, there is also a variational form, but we will not go
+ into that.
+\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).
- For example, the strong form of this simple equation is stated as follows:
\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
@@ -361,10 +411,27 @@
\end_layout
\begin_layout Standard
-The weak form of the corresponding boundary value problem is stated:
+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
@@ -402,7 +469,7 @@
\begin_inset Formula $\nu$
\end_inset
- is the set of weighting functions defined by
+ is the set of weighting functions defined by
\end_layout
\begin_layout Standard
@@ -419,7 +486,7 @@
\begin_inset Formula $\mathcal{L}$
\end_inset
- is a set of trial solutions defined by
+ is a set of trial solutions defined by
\end_layout
\begin_layout Standard
@@ -439,7 +506,7 @@
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
+ The result is a large sparse matrix equation of the form
\end_layout
\begin_layout Standard
@@ -457,11 +524,770 @@
[K]
\emph default
is referred to as the element stiffness matrix.
- There will be more to say about the implementation in Section 4.
- TODO-ref for section
+ There will be more to say about the implementation in Section
+\begin_inset LatexCommand ref
+reference "cha:Input-Guide"
+
+\end_inset
+
+.
\end_layout
+\begin_layout Subsection
+Galerkin's Approximation
+\end_layout
+
\begin_layout Standard
+Now we have a start on the finite element method.
+ I want to continue to follow Hughes; however his notation becomes quite
+ difficult to keep up with.
+ Now, lets 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
+
+ where
+\begin_inset Formula $h$
+\end_inset
+
+ 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
+\begin_inset Formula $u$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+On a computer, we don't have a continuous solution.
+ We have a solution at descrete 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
+\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
+\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
+\begin_inset Formula \begin{equation}
+N_{n+1}(1)=1.\end{equation}
+
+\end_inset
+
+ Then,
+\begin_inset Formula $g^{h}$
+\end_inset
+
+ 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
+\begin_inset Formula $u^{h}$
+\end_inset
+
+ 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
+\begin_inset Formula $d_{A}$
+\end_inset
+
+'s are unknown constants to be solved for.
+\end_layout
+
+\begin_layout Standard
+In the next section we will make the shape functions more concrete.
+ I want you 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
+
+\series bold
+any
+\series default
+ 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.
+\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
+reference "eq:whapprox"
+
+\end_inset
+
+ satisfies the requirement that
+\begin_inset Formula $w^{h}(1)=0$
+\end_inset
+
+, as necessary.
+\end_layout
+
+\begin_layout Standard
+Using our definitions of the
+\begin_inset Formula $w^{h}$
+\end_inset
+
+'s and our approximation for
+\begin_inset Formula $u^{h}$
+\end_inset
+
+, we can get the messy expression for Equation\InsetSpace ~
+
+\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
+\begin_inset Formula $N_{A}\times N_{B}$
+\end_inset
+
+ is zero except when
+\begin_inset Formula $A=B$
+\end_inset
+
+.
+ 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
+\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
+reference "eq:feform"
+
+\end_inset
+
+ is known except the
+\begin_inset Formula $d_{B}$
+\end_inset
+
+'s.
+ This constitutes a system of
+\begin_inset Formula $n$
+\end_inset
+
+ equations and
+\begin_inset Formula $n$
+\end_inset
+
+ 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
+\begin_inset Formula \begin{equation}
+[k]=\left[\begin{array}{cccc}
+K_{11} & K_{12} & \cdots & K_{1n}\\
+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,
+\begin_inset Formula $\left[K\right]$
+\end_inset
+
+ is the stiffness matrix,
+\begin_inset Formula $\{f\}$
+\end_inset
+
+ is the force vector, and
+\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.
+\end_layout
+
+\begin_layout Subsection
+Shape Functions
+\end_layout
+
+\begin_layout Standard
+At this point, I will narrow the focus to deal with specifically the elements
+ in my code, ConMan.
+ It is possible to think very general shape functions, but in practice,
+ people use triangles or quadralaterals (in 2-D).
+ We will extend our finite element formulation to a 2-D equation next time.
+ In terms of the level of approximation, there are also a lot of possibilities.
+ We will stick to the simplest form, bi-linear elements, but you should
+ be aware that higher order elements (bi-quadratic or bi-cubic spline elements)
+ are also popular with some people.
+ I am condensing a lot of very useful material from Chapter 3 of Hughes'
+ book into one lecture.
+ 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
+Lets 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
+reference "eq:normalize"
+
+\end_inset
+
+ says that they are normalized, so that they sum to one (everywhere on X,Y).
+ Equations\InsetSpace ~
+
+\begin_inset LatexCommand ref
+reference "eq:interpx"
+
+\end_inset
+
+ and\InsetSpace ~
+
+\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,
+\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
+reference "eq:normalize"
+
+\end_inset
+
+ and Equations \InsetSpace ~
+
+\begin_inset LatexCommand ref
+reference "eq:interpx"
+
+\end_inset
+
+ and\InsetSpace ~
+
+\begin_inset LatexCommand ref
+reference "eq:interpy"
+
+\end_inset
+
+.
+ A good exercise would be to show this is true.
+\end_layout
+
+\begin_layout Quote
+
+\series bold
+Problem 7
+\series default
+ Show that the shape functions defined above satisfy Equation\InsetSpace ~
+
+\begin_inset LatexCommand ref
+reference "eq:normalize"
+
+\end_inset
+
+ and Equations\InsetSpace ~
+
+\begin_inset LatexCommand ref
+reference "eq:interpx"
+
+\end_inset
+
+ and\InsetSpace ~
+
+\begin_inset LatexCommand ref
+reference "eq:interpy"
+
+\end_inset
+
+.
+
+\end_layout
+
+\begin_layout Standard
+Notice that by convention, I start numbering my element nodes in the lower
+ left hand corner and work counter-clockwise.
+
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+
+\backslash
+em
+\end_layout
+
+\end_inset
+
+ This is a very important point.
+ It is followed through out all my finite element codes.
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+ There is no magic reason, you just have to choose a starting place.
+\end_layout
+
+\begin_layout Standard
+In ConMan, as in Hughes, we further choose to normalize this by setting
+
+\begin_inset Formula $a=1$
+\end_inset
+
+ and
+\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
+\begin_inset Formula $a=0.5$
+\end_inset
+
+ and
+\begin_inset Formula $b=0.5$
+\end_inset
+
+ so that the denominator goes to 1.
+\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
+
+
+\begin_inset Formula \begin{eqnarray}
+N_{1,y} & = & \frac{-(1-x)}{4}\\
+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
+\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 correspon
+ds to a stretching or a shrinking (notice we set
+\begin_inset Formula $a=b=1$
+\end_inset
+
+ above.
+ However, if we are thinking about a cylindrical geometry, for example,
+ we can use the Jacobian of the transformation between the geometries.
+ Lets 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
+\begin_inset Formula $\xi,\eta$
+\end_inset
+
+ for the
+\begin_inset Formula $X,Y$
+\end_inset
+
+ coordinate pair in the `parent element' So we could write
+\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}
+\left\{ \begin{array}{c}
+x\\
+y\end{array}\right\} =\left[\begin{array}{cc}
+0.05 & 0.0\\
+0.0 & 0.10\end{array}\right]\left\{ \begin{array}{c}
+\xi\\
+\eta\end{array}\right\} +\left\{ \begin{array}{c}
+0.1\\
+0.2\end{array}\right\} \end{equation}
+
+\end_inset
+
+ Where
+\begin_inset Formula $[J]$
+\end_inset
+
+ is the Jacobian of the transformation.
+ If the transformation were from an arbitratry shaped quadralateral to the
+ parent element, then the off diagonal terms 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
+
+ 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).
+\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:
+\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.
+ Lets begin with a 1-D example, f(x) = c
+\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, f(x) = a x + b
+\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
+\begin_inset Formula $\frac{-1}{\sqrt{3}},\frac{1}{\sqrt{3}}$
+\end_inset
+
+ are exact for a linear equation, but from what I showed, so would any
+\begin_inset Formula $-x,x$
+\end_inset
+
+ 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
+\begin_inset Formula $-\sqrt{\frac{3}{5}},0,\sqrt{\frac{3}{5}}$
+\end_inset
+
+.
+\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
+\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
+
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+-1,-1
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+ to
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+1,1
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+ domain.
+\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
+\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.
+\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 storage.
+ 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
@@ -482,7 +1308,6 @@
genshg
\series default
.
-
\end_layout
\begin_layout Standard
@@ -491,7 +1316,7 @@
\begin_inset Float figure
wide false
sideways false
-status open
+status collapsed
\begin_layout Standard
\begin_inset Caption
@@ -516,12 +1341,12 @@
\noindent
\align center
\begin_inset Graphics
- filename images/conman-fig1.png
+ filename /Users/sue/conman/images/conman-fig1.eps
scale 45
\end_inset
-
+
\end_layout
\end_inset
@@ -533,7 +1358,7 @@
For ConMan the choice was made to use bilinear quadrilaterals as the parent
elements (Figure
\begin_inset LatexCommand ref
-reference "fig:Figure-2"
+reference "fig:The-bilinear-shape"
\end_inset
@@ -557,6 +1382,11 @@
\begin_inset Caption
\begin_layout Standard
+\begin_inset LatexCommand label
+name "fig:The-bilinear-shape"
+
+\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).
@@ -572,12 +1402,12 @@
\noindent
\align center
\begin_inset Graphics
- filename images/conman-fig2.png
+ filename /Users/sue/conman/images/conman-fig2.png
scale 40
\end_inset
-
+
\end_layout
\end_inset
@@ -588,11 +1418,31 @@
\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.
-
\end_layout
\begin_layout Description
-id transforms global nodes to equation numbers (Figure
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+id
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+ transforms global nodes to equation numbers (Figure
\begin_inset LatexCommand ref
reference "fig:Example-relationship-between"
@@ -603,8 +1453,28 @@
\end_layout
\begin_layout Description
-ien transforms element local node numbers to global node numbers (Figure
-
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+ien
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+ transforms element local node numbers to global node numbers (Figure
\begin_inset LatexCommand ref
reference "fig:4-Example-relationship-between"
@@ -615,7 +1485,28 @@
\end_layout
\begin_layout Description
-lm transforms element local node numbers to global equation numbers.
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+lm
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+ transforms element local node numbers to global equation numbers.
\end_layout
@@ -651,12 +1542,12 @@
\noindent
\align center
\begin_inset Graphics
- filename images/conman-fig3.png
+ filename /Users/sue/conman/images/conman-fig3.eps
scale 40
\end_inset
-
+
\end_layout
\end_inset
@@ -677,7 +1568,6 @@
for computational efficiency.
These arrays spare the user from dealing with the transformations, while
making the code efficient.
-
\end_layout
\begin_layout Standard
@@ -712,12 +1602,12 @@
\noindent
\align center
\begin_inset Graphics
- filename images/conman-fig4.png
+ filename /Users/sue/conman/images/conman-fig4.eps
scale 40
\end_inset
-
+
\end_layout
\end_inset
@@ -726,435 +1616,539 @@
\end_layout
\begin_layout Standard
-In the code, the data structures for these two arrays are
+In the code, the data structures for these two arrays are
\end_layout
\begin_layout Description
+\begin_inset ERT
+status collapsed
-\family roman
-id ( degree-of-freedom , global-node-number ) = equation-number
+\begin_layout Standard
+
+{
\end_layout
-\begin_layout Description
+\end_inset
-\family roman
-ien ( local-node-number, element-number ) = global-node-number
-\end_layout
+id
+\begin_inset ERT
+status collapsed
-\begin_layout Description
+\begin_layout Standard
-\family roman
-lm ( degree-of-freedom, local-node-number, element-number ) = global-equation-nu
-mber
+}
\end_layout
-\begin_layout Subsection
-Equations
+\end_inset
+
+ ( degree-of-freedom , global-node-number ) = equation-number
\end_layout
+\begin_layout Description
+\begin_inset ERT
+status collapsed
+
\begin_layout Standard
-The equations for incompressible convection (in dimensionless form) are
- the equations of momentum
+
+{
\end_layout
+\end_inset
+
+ien
+\begin_inset ERT
+status collapsed
+
\begin_layout Standard
-\begin_inset Formula \begin{equation}
-\nabla^{2}u=-\nabla p+Ra\theta\hat{k}\label{eq:momentum}\end{equation}
+}
+\end_layout
+
\end_inset
-
+ ( local-node-number, element-number ) = global-node-number
\end_layout
+\begin_layout Description
+\begin_inset ERT
+status collapsed
+
\begin_layout Standard
-continuity
+
+{
\end_layout
+\end_inset
+
+lm
+\begin_inset ERT
+status collapsed
+
\begin_layout Standard
-\begin_inset Formula \begin{equation}
-\nabla\cdot u=0\label{eq:continuity}\end{equation}
+}
+\end_layout
+
\end_inset
+ ( 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
-and energy
+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 it's
+ value is the global node number.
+ It is easiest to look at some examples:
\end_layout
\begin_layout Standard
-\noindent
-\align center
-\begin_inset Formula \begin{equation}
-\frac{\partial\theta}{\partial t}=u\cdot\nabla\theta+\nabla^{2}\theta+H\label{eq:energy}\end{equation}
+Consider a 3 element by 3 element grid.
+ I will number my elements and node starting in the lower left hand corner,
+ and working up fastest.
+\end_layout
-\end_inset
+\begin_layout Standard
+\begin_inset ERT
+status collapsed
+\begin_layout Standard
+
+\backslash
+begin{verbatim}
\end_layout
-\begin_layout Standard
-where
-\emph on
-u
-\emph default
- is the dimensionless velocity,
-\begin_inset Formula $\theta$
\end_inset
- is the dimensionless temperature,
-\emph on
-p
-\emph default
- is the dimensionless pressure,
-\begin_inset Formula $\hat{k}$
+ For element 3: ien (3, 1) = 3 ien (3, 2) = 7 ien (3, 3) = 8 ien (3, 4)
+ = 4 For element 5: ien (5, 1) = 6 ien (5, 2) = 10 ien (5, 3) = 11 ien (5,
+ 4) = 7
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+
+\backslash
+end{verbatim}
+\end_layout
+
\end_inset
- is the unit vector in the vertical direction,
-\emph on
-H
-\emph default
- is the heat source term, and
-\emph on
-t
-\emph default
- is the dimensionless time.
- In this form, all the material properties are combined into one dimensionless
- parameter, the Rayleigh number, given by
+
\end_layout
\begin_layout Standard
-\begin_inset Formula \begin{equation}
-Ra=\frac{g\alpha\Delta Td^{3}}{\kappa\mu}\label{eq:Raleigh number}\end{equation}
+There are three kinds of opperations we might think of wanting.
+ The first is taking values of some function (coordinates, velocities, temperatu
+res, stresses, etc.) defined on the global grid and getting the values for
+ a single element, this is called a
+\begin_inset ERT
+status collapsed
+\begin_layout Standard
+
+{
+\end_layout
+
\end_inset
-\end_layout
+\begin_inset ERT
+status collapsed
\begin_layout Standard
-where
-\emph on
-g
-\emph default
- is the acceleration due to gravity,
-\begin_inset Formula $\alpha$
-\end_inset
- is the coefficient of thermal expansion,
-\begin_inset Formula $\Delta T$
-\end_inset
- is the temperature drop across the box,
-\emph on
-d
-\emph default
- is the depth of the box,
-\begin_inset Formula $\kappa$
+\backslash
+em
+\end_layout
+
\end_inset
- is the thermal diffusivity, and
-\emph on
-µ
-\emph default
-is the dynamic viscosity.
-\end_layout
+ gather
+\begin_inset ERT
+status collapsed
\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
-\begin_inset Formula $\Delta t\rightarrow0$
-\end_inset
-.
-
+}
\end_layout
+\end_inset
+
+ operation.
+ The next is taking values in an element and spreading them out to the global
+ array, this is called a
+\begin_inset ERT
+status collapsed
+
\begin_layout Standard
-The momentum equation is solved using the penalty method to enforce incompressib
-ility.
- The formal statement of the problem is as follows:
+
+{
\end_layout
+\end_inset
+
+
+\begin_inset ERT
+status collapsed
+
\begin_layout Standard
-Given:
+
+
+\backslash
+em
\end_layout
+\end_inset
+
+ scatter
+\begin_inset ERT
+status collapsed
+
\begin_layout Standard
-\begin_inset Formula \begin{equation}
-f:\Omega\rightarrow\Re^{n}\,\,\,\,\,\, body\, force\, vector\label{eq:body force vector}\end{equation}
+}
+\end_layout
+
\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
+\begin_inset ERT
+status collapsed
+\begin_layout Standard
+
+{
\end_layout
+\end_inset
+
+
+\begin_inset ERT
+status collapsed
+
\begin_layout Standard
-\begin_inset Formula \begin{equation}
-g:\Gamma_{g}\rightarrow\Re^{n}\,\,\,\,\,\, imposed\, velocity\, vector\label{eq:imposed velocity vector}\end{equation}
+
+\backslash
+em
+\end_layout
+
\end_inset
+ assembly
+\begin_inset ERT
+status collapsed
+\begin_layout Standard
+
+}
\end_layout
-\begin_layout Standard
-\begin_inset Formula \begin{equation}
-h:\Gamma_{h}\rightarrow\Re^{n}\,\,\,\,\,\, imposed\, traction\, vector\label{eq:imposed traction vector}\end{equation}
-
\end_inset
+ step.
+\end_layout
+\begin_layout Standard
+All three operations, gather, scatter, and assemble are done by the routine
+ local.
+ Because it is called by the genshp routine above, it is a good example
+ to look at.
\end_layout
+\begin_layout Subsection
+Equations
+\end_layout
+
\begin_layout Standard
-where
-\begin_inset Formula $\Gamma$
+\begin_inset Formula \begin{eqnarray}
+\tau_{ij,j}+f_{i} & = & 0\label{eq:motion}\\
+u_{i,i} & = & 0\end{eqnarray}
+
\end_inset
- is the boundary of the domain
-\begin_inset Formula $\Omega$
+ where
+\begin_inset Formula \begin{equation}
+\tau_{ij}=-p\delta_{ij}+2\mu u_{(i,j)}\label{eq:constit}\end{equation}
+
\end_inset
-.
-
-\begin_inset Formula $\Gamma_{g}$
+ where
+\begin_inset Formula \begin{equation}
+u_{(i,j)}=(u_{i,j}+u_{j,i})/2\end{equation}
+
\end_inset
- and
-\begin_inset Formula $\Gamma_{h}$
+ We replace equation\InsetSpace ~
+
+\begin_inset LatexCommand ref
+reference "eq:constit"
+
\end_inset
- are the parts of the boundary where velocities and tractions are specified.
-
-\end_layout
+ with the following relationships
+\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}
-\begin_layout Standard
-Find
-\begin_inset Formula $u:\Omega\rightarrow\Re^{n}$
\end_inset
- and
-\begin_inset Formula $p:\Omega\rightarrow\Re$
+ As
+\begin_inset Formula $\lambda$
\end_inset
+ approaches infinity, these relations approach the incompressible solution.
+ Also, as
+\begin_inset Formula $\lambda$
+\end_inset
-\end_layout
+ approaches infinity,
+\begin_inset Formula $p^{\lambda}$
+\end_inset
-\begin_layout Standard
-\begin_inset Formula \begin{equation}
-t_{ij,j}+f_{i}=0\,\,\,\,\,\, on\,\Omega\label{eq:}\end{equation}
-
+ 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 ~
-\end_layout
+\begin_inset LatexCommand ref
+reference "eq:constit-lam2"
-\begin_layout Standard
-\begin_inset Formula \begin{equation}
-u_{i,i}=0\,\,\,\,\,\, on\,\Omega\label{eq:}\end{equation}
-
\end_inset
+ into\InsetSpace ~
-\end_layout
+\begin_inset LatexCommand ref
+reference "eq:constit-lam"
-\begin_layout Standard
+\end_inset
+
+ we get
\begin_inset Formula \begin{equation}
-u_{i}=g_{i}\,\,\,\,\,\, on\,\,\Gamma_{g}\label{eq:}\end{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_layout
+\end_inset
-\begin_layout Standard
+ or
\begin_inset Formula \begin{equation}
-t_{ij}n_{j}=h_{i}\,\,\,\,\,\, on\,\,\Gamma_{h}\label{eq:}\end{equation}
+\tau_{ii}/3=-p=(\lambda+2/3\mu)u_{i,i}\end{equation}
\end_inset
-
-\end_layout
-
-\begin_layout Standard
-with the constitutive equation for a Newtonian fluid
-\end_layout
-
-\begin_layout Standard
+ but we also have
\begin_inset Formula \begin{equation}
-t_{ij}=-p\delta_{ij}+2\mu u_{\left(i,j\right)}\label{eq:}\end{equation}
+-p^{\lambda}=\lambda u_{i,i}\end{equation}
\end_inset
+ from Equation\InsetSpace ~
-\end_layout
+\begin_inset LatexCommand ref
+reference "eq:constit-lam2"
-\begin_layout Standard
-where
-\begin_inset Formula $t_{ij}$
\end_inset
- denotes the Cauchy stress tensor,
-\emph on
-p
-\emph default
- is the pressure,
-\begin_inset Formula $\delta_{ij}$
+.
+ Clearly in the incompressible limit
+\begin_inset Formula $\lambda\gg\mu$
\end_inset
- is the Kronecker delta and
-\begin_inset Formula $u_{(i,j)}=(u_{i,j}+u_{j,i})/2$
+ then
+\begin_inset Formula $\lambda+2/3\mu\rightarrow\lambda$
\end_inset
+ and
+\begin_inset Formula $p^{\lambda}\rightarrow p$
+\end_inset
+
.
-
+ Also note that the continuity equation is satisfied.
\end_layout
\begin_layout Standard
-In the penalty formulation, the above is replaced by
-\end_layout
+Now, substituting Equation\InsetSpace ~
-\begin_layout Standard
-\begin_inset Formula \begin{equation}
-t_{ij}^{\left(\lambda\right)}=-p^{\left(\lambda\right)}\delta_{ij}+2\mu u_{\left(i,j\right)}^{\left(\lambda\right)}\label{eq:}\end{equation}
+\begin_inset LatexCommand ref
+reference "eq:constit-lam-final"
\end_inset
+ into Equation\InsetSpace ~
-\end_layout
+\begin_inset LatexCommand ref
+reference "eq:motion"
-\begin_layout Standard
-where
-\end_layout
+\end_inset
-\begin_layout Standard
+ we have
\begin_inset Formula \begin{equation}
-p^{\left(\lambda\right)}=-\lambda u_{i,j}^{\left(\lambda\right)}\label{eq:}\end{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.
+ I will also specialize to 2-D:
+\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_layout
-
-\begin_layout Standard
-and
-\begin_inset Formula $\lambda$
\end_inset
- is the penalty parameter (repeated subscripts means summation over all
- indices).
+
\end_layout
\begin_layout Standard
-This formulation automatically enforces incompressibility since the solution
- converges to the incompressible stokes equation as
-\begin_inset Formula $\lambda$
+These are second order partial differential equations.
+ Simplifying, I 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
- approaches infinity (Temam, 1977).
- Also, the unknown pressure field is eliminated.
- This is quite useful not only because the amount of computational work
- is decreased because no pressure equation is solved, but also because it
- eliminates the need to create artificial boundary conditions for the pressure
- equation.
- There are no pressure boundary conditions in the formal specification of
- the problem.
- By examining the equation we see that the role of pressure is to balance
- the system, so physically the penalty formulation makes sense.
-
-\end_layout
-\begin_layout Standard
-The equation is cast in the weak form and the Galerkin formulation (i.e.,
- the weighting functions are the same as the basis functions) is used to
- solve the weak form of the equation.
-
\end_layout
\begin_layout Standard
-\begin_inset Formula \begin{equation}
-V=\left\{ w\epsilon H^{1}|w=0\,\,\,\,\,\, on\,\,\Gamma_{g}\right\} \label{eq:}\end{equation}
+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 in carefully by Hughes on pages 197-200, but he has left out
+ a number of intermediate steps.
+ Nothing about this step is hard, it is just tedious.
+ There is, however, a cleaver short cut.
+ 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.
+ If you are confused, or don't believe me, then you should take the equations
+ directly above this paragraph, multiply by a weighting function
+\begin_inset Formula $w$
+\end_inset
+ and integrate over the 2-D domain
+\begin_inset Formula $\Omega$
+\end_inset
+
+, 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}
-\emph on
-V
-\emph default
- is the set of all weighting functions
-\emph on
-
-\begin_inset Formula $w$
\end_inset
+ Note that we don't get something for nothing, this short cut 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}
-\emph default
- which vanish on the boundary.
- Similarly
-\begin_inset Formula $V^{h}$
\end_inset
- is a subset of
-\emph on
-V
-\emph default
- parameterized by
-\emph on
-h
-\emph default
-, the mesh parameter.
- Let
-\begin_inset Formula $g^{h}$
+ were in our case
+\begin_inset Formula $w$
\end_inset
- denote an approximation of
-\begin_inset Formula $g$
+ is the weighting function and
+\begin_inset Formula $v$
\end_inset
- which converges to
-\begin_inset Formula $g$
+ 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
+\begin_inset Formula \begin{eqnarray}
+\frac{\partial w}{\partial x}=N_{x}\\
+\frac{\partial w}{\partial z}=N_{z}\\
+\frac{\partial u}{\partial x}=u\, N_{x}\\
+\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
- as
-\begin_inset Formula $h\rightarrow0$
+ 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
-Find
-\begin_inset Formula $u^{h}=w^{h}+g^{h},w^{h}\epsilon V^{h},$
+At this point, it is useful to separate the equations into a
+\begin_inset Formula $\lambda$
\end_inset
- such that for all
-\begin_inset Formula $\bar{w}^{h}\epsilon V^{h}$
+ part and a
+\begin_inset Formula $\mu$
\end_inset
+ part.
+ We can also write them as a 2-D 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_layout
+\end_inset
-\begin_layout Standard
+ and
\begin_inset Formula \begin{equation}
-\int_{\Omega}\left(\lambda w_{j,j}\bar{w}_{i,i}+2\mu w_{i,j}\bar{w}_{i,j}\right)=\int_{\Omega}f_{i}\bar{w_{i}^{h}d\Omega}+\int_{\Gamma_{h}}h_{i}\bar{w_{i}^{h}d\Omega}\label{eq:}\end{equation}
+[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
@@ -1162,26 +2156,45 @@
\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:
\begin_inset Formula \begin{equation}
--\int_{\Omega}\left(\lambda g_{j,j}^{h}\bar{w}_{i,i}+2\mu g_{\left(i,j\right)}^{h}\bar{w}_{\left(i,j\right)}^{h}\right)d\Omega\label{eq:}\end{equation}
+[K_{\lambda}]+[K_{\mu}]=[B]^{T}[D][B]\end{equation}
\end_inset
+
+\begin_inset Formula \begin{equation}
+[D_{\lambda}]+[D_{\mu}]=[D]\end{equation}
-\end_layout
+\end_inset
-\begin_layout Standard
+ where
\begin_inset Formula \begin{equation}
-w_{i}^{h}=\sum N_{A}u_{iA}\label{eq:}\end{equation}
+[D_{\mu}]=\mu\left[\begin{array}{ccc}
+2 & 0 & 0\\
+0 & 2 & 0\\
+0 & 0 & 1\end{array}\right]\end{equation}
\end_inset
+ and
+\begin_inset Formula \begin{equation}
+[D_{\lambda}]=\lambda\left[\begin{array}{ccc}
+1 & 1 & 0\\
+1 & 1 & 0\\
+0 & 0 & 0\end{array}\right]\end{equation}
-\end_layout
+\end_inset
-\begin_layout Standard
+ and
\begin_inset Formula \begin{equation}
-u_{iA}=u_{i}\left(x_{A}\right)\label{eq:}\end{equation}
+[B]=\left[\begin{array}{cc}
+N_{x} & 0\\
+0 & N_{z}\\
+N_{z} & N_{x}\end{array}\right].\end{equation}
\end_inset
@@ -1189,12 +2202,23 @@
\end_layout
\begin_layout Standard
-where
-\begin_inset Formula $N_{A}$
+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
- is the shape function for node A for the element.
-
+.
\end_layout
\begin_layout Standard
@@ -1235,7 +2259,6 @@
matrix and save both storage and operations using Cholesky factorization.
More details of the method and a formal error analysis can be found in
Hughes, Liu and Brooks (1979).
-
\end_layout
\begin_layout Standard
@@ -1256,21 +2279,12 @@
\begin_inset Float figure
wide false
sideways false
-status collapsed
+status open
\begin_layout Standard
\begin_inset Caption
\begin_layout Standard
-
-\family roman
-\series medium
-\shape up
-\size normal
-\emph off
-\bar no
-\noun off
-\color none
\begin_inset LatexCommand label
name "fig:5-The-element-stiffness"
@@ -1299,20 +2313,69 @@
\noindent
\align center
\begin_inset Graphics
- filename images/conman-fig5.png
+ filename /Users/sue/conman/images/conman-fig5.eps
scale 40
\end_inset
+
+\end_layout
+\end_inset
+
+
\end_layout
+\begin_layout Standard
+I'm not sure if we need this too...
+\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 Standard
+\begin_inset LatexCommand label
+name "fig:8-The-storage-for"
+
\end_inset
+The storage for the stiffness matrix used in routine
+\series bold
+f_vstf
+\series default
+.
+\end_layout
+\end_inset
+
+
\end_layout
\begin_layout Standard
+\align center
+\begin_inset Graphics
+ filename /Users/sue/conman/images/conman-fig8.eps
+ scale 40
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
The energy equation is an advection-diffusion equation.
The formal statement is
\end_layout
@@ -1322,7 +2385,7 @@
\begin_inset Formula $T:\Omega\rightarrow R$
\end_inset
- such that
+ such that
\end_layout
\begin_layout Standard
@@ -1441,7 +2504,7 @@
\end_layout
\begin_layout Standard
-with
+with
\end_layout
\begin_layout Standard
@@ -1488,9 +2551,16 @@
\end_inset
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 form of discretization has no crosswind diffusion because the
+\begin_inset Quotes eld
+\end_inset
+
+artificial diffusion
+\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 (Hughes and Brooks, 1977).
@@ -1512,11 +2582,11 @@
to achieve stable solutions (Travis et al., 1989).
Time stepping in the energy equation is done using an explicit predictor-correc
tor algorithm.
- The form of the predictor-corrector algorithm is
+ The form of the predictor-corrector algorithm is
\end_layout
\begin_layout Standard
-Predict:
+Predict:
\end_layout
\begin_layout Standard
@@ -1538,7 +2608,7 @@
\end_layout
\begin_layout Standard
-Solve:
+Solve:
\end_layout
\begin_layout Standard
@@ -1569,7 +2639,7 @@
\end_layout
\begin_layout Standard
-Correct:
+Correct:
\end_layout
\begin_layout Standard
@@ -1633,7 +2703,6 @@
\end_inset
is diagonal.
-
\end_layout
\begin_layout Standard
@@ -1646,7 +2715,6 @@
= 0.5 and two iterations, the method is second order accurate (Hughes, 1986,
p.
562-566).
-
\end_layout
\begin_layout Standard
@@ -1686,462 +2754,163 @@
\end_layout
\begin_layout Standard
-ConMan was designed to take advantage of machines capable of vector instructions.
- Hence thoughout the code, operations which would be performed on an individual
- element on a scalar machine are grouped together so that they can be performed
- on a group of elements.
- This is transparent to the user and requires only a small modification
- to the program style and data structure.
- The only unfortunate side effect of this operation is that the arrays of
- element quantities (e.g.,
-\emph on
-ien, lm, evisc
-\emph default
-) are now shuffled to avoid vector recurrences.
- Consider Figure
-\begin_inset LatexCommand ref
-reference "fig:6-The-four-surrounding"
-
-\end_inset
-
-, a small rectangular grid with one degree of freedom per node.
- When assembling the equation for global degree of freedom 5 (marked N),
- there is a contribution from elements 1, 2, 3 and 4 (circled).
- This is updated as follows:
-\end_layout
-
-\begin_layout LyX-Code
-DO E=1,NUMEL
-\end_layout
-
-\begin_layout LyX-Code
+There are generally two phases to ConMan, input and time stepping.
+ The main program is found in
+\series bold
+ConMan.F
+\series default
.
-
-\end_layout
-
-\begin_layout LyX-Code
+ The input is read in the files
+\series bold
+input.F and elminp.F
+\series default
.
-
-\end_layout
-
-\begin_layout LyX-Code
-(localize data)
-\end_layout
-
-\begin_layout LyX-Code
+ Time stepping is doing in
+\series bold
+timdrv.F
+\series default
.
-
+ For legacy reasons, there is a rather complex structure of
+\series bold
+eglib
+\series default
+ calling
+\series bold
+eg2.F
+\series default
+ which calls the assembly and solve routines.
\end_layout
-\begin_layout LyX-Code
-.
-
-\end_layout
-
-\begin_layout LyX-Code
-(form local stiffness matrix in LOCAL)
-\end_layout
-
-\begin_layout LyX-Code
-.
-
-\end_layout
-
-\begin_layout LyX-Code
-.
-
-\end_layout
-
-\begin_layout LyX-Code
-DO N=1,4
-\end_layout
-
-\begin_layout LyX-Code
-GLOBAL( LM(E,N) ) = GLOBAL( LM(E,N) ) + LOCAL(N)
-\end_layout
-
-\begin_layout LyX-Code
-ENDDO
-\end_layout
-
-\begin_layout LyX-Code
-ENDDO
-\end_layout
-
\begin_layout Standard
-where
-\family typewriter
-E
-\family default
- is the element number,
-\family typewriter
-N
-\family default
- is the local node number,
-\family typewriter
-LOCAL(N)
-\family default
- is the value being assembled for element
-\family typewriter
-E
-\family default
- and local node
-\family typewriter
-N
-\family default
- and
-\family typewriter
-GLOBAL
-\family default
- is the equation for the global node.
- Now the task is to assemble m elements,
-\begin_inset Formula $e_{1},\ldots,e_{m}$
-\end_inset
-
-, at one time, making sure that no global node is updated twice.
- This can be accomplished if for all elements,
-\begin_inset Formula $e_{1},\ldots,e_{m}$
-\end_inset
-
-, lm
-\begin_inset Formula $\left(e_{a},\mathrm{n}\right)$
-\end_inset
-
- is not equal to lm
-\begin_inset Formula $\left(e_{b},\mathrm{n}\right)$
-\end_inset
-
- for
-\begin_inset Formula $e_{a}$
-\end_inset
-
- and
-\begin_inset Formula $e_{b}$
-\end_inset
-
- in
-\begin_inset Formula $e_{1},\ldots,e_{m}$
-\end_inset
-
-.
- Also since the innermost loops are the vectorizable loops, we unroll the
-
-\family typewriter
-N
-\family default
- loop over the local nodes.
+There are three significant differences that the user who has seen versions
+ of ConMan in the past 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 LyX-Code
-.
-
-\end_layout
-
-\begin_layout LyX-Code
-.
-\end_layout
-
-\begin_layout LyX-Code
-(loop over all blocks)
-\end_layout
-
-\begin_layout LyX-Code
-.
-\end_layout
-
-\begin_layout LyX-Code
-.
-\end_layout
-
-\begin_layout LyX-Code
-.
-
-\end_layout
-
-\begin_layout LyX-Code
-(localize a block of data)
-\end_layout
-
-\begin_layout LyX-Code
-.
-
-\end_layout
-
-\begin_layout LyX-Code
-.
-
-\end_layout
-
-\begin_layout LyX-Code
-(form a block of local stiffness matrices in LOCAL)
-\end_layout
-
-\begin_layout LyX-Code
-.
-
-\end_layout
-
-\begin_layout LyX-Code
-.
-
-\end_layout
-
-\begin_layout LyX-Code
-DO IV=1,NVEC
-\end_layout
-
-\begin_layout LyX-Code
-IVEL=IV+IEL-1
-\end_layout
-
-\begin_layout LyX-Code
-GLOBAL( LM(IVEL,1) ) = GLOBAL( LM(IVEL,1) ) + LOCAL(IV,1)
-\end_layout
-
-\begin_layout LyX-Code
-GLOBAL( LM(IVEL,2) ) = GLOBAL( LM(IVEL,2) ) + LOCAL(IV,2)
-\end_layout
-
-\begin_layout LyX-Code
-GLOBAL( LM(IVEL,3) ) = GLOBAL( LM(IVEL,3) ) + LOCAL(IV,3)
-\end_layout
-
-\begin_layout LyX-Code
-GLOBAL( LM(IVEL,4) ) = GLOBAL( LM(IVEL,4) ) + LOCAL(IV,4)
-\end_layout
-
-\begin_layout LyX-Code
-ENDDO
-\end_layout
-
\begin_layout Standard
-
-\family typewriter
-NVEC
-\family default
- elements at a time can be processed, and the array
-\family typewriter
-LOCAL
-\family default
- must be of length
-\family typewriter
-NVEC
-\family default
- by 4.
- Notice that the
-\family typewriter
-NUMEL
-\family default
- elements have been broken up into groups
-\family typewriter
-NVEC
-\family default
- long with
-\family typewriter
-IEL
-\family default
- marking the first element in the group (see Figure
-\begin_inset LatexCommand ref
-reference "fig:7-A-representation-of"
-
-\end_inset
-
-).
- The shuffling of the elements is done in routine
+With this version of ConMan, the reordering of elements into block 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 (
\series bold
-genshp
+f_vstf.F
\series default
-, and the variables in the code (
-\emph on
-iel, nvec, iv, ivel
-\emph default
-) are the same as described above.
-
+) and right-hand side, or residual, (
+\series bold
+f_vres.F
+\series default
+) 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's 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
-\noindent
-\align center
-\begin_inset Float figure
-wide false
-sideways false
-status collapsed
-
-\begin_layout Standard
-\begin_inset Caption
-
-\begin_layout Standard
-\begin_inset LatexCommand label
-name "fig:6-The-four-surrounding"
-
+The second major change is that we have implemented a Picard iteration algorithm
+ for steady-state problems (c.f.
+ van Keken - thesis).
+ 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
+\begin_inset Formula $\alpha$
\end_inset
-The four surrounding elements (1, 2, 3 and 4 circled) all contribute to
- the global equation 5.
+ 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 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 solution, the Picard results are not useful and
+ the explicit solver should be used.
\end_layout
-\end_inset
-
-
-\end_layout
-
\begin_layout Standard
-\noindent
-\align center
-\begin_inset Graphics
- filename images/conman-fig6.png
- scale 40
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float figure
-wide false
-sideways false
+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
+\series bold
+mmgetblk
+\series default
+.
+ The `mm' routines are a fancy wrapper around a c
+\begin_inset ERT
status collapsed
\begin_layout Standard
-\begin_inset Caption
-\begin_layout Standard
-\begin_inset LatexCommand label
-name "fig:7-A-representation-of"
-
-\end_inset
-
-A representation of the storage of array LM, which is NUMEL by 4 long.
- The NUMEL elements are broken up into groups of length NVEC for vector
- processing.
+{
\end_layout
\end_inset
-\end_layout
+\begin_inset ERT
+status collapsed
\begin_layout Standard
-\noindent
-\align center
-\begin_inset Graphics
- filename images/conman-fig7.png
- scale 40
-\end_inset
-
+\backslash
+em
\end_layout
\end_inset
+ malloc
+\begin_inset ERT
+status collapsed
-\end_layout
-
-\begin_layout Quote
-\begin_inset Formula $\clubsuit$
-\end_inset
-
- In
-\emph on
-common.h
-\emph default
- the parameter
-\emph on
-lvec
-\emph default
- sets the maximum length of the vectors.
- It is usually wise to set this to the length of the vector registers on
- the machine (64 for Cray, 128 for Convex).
-
-\end_layout
-
-\begin_layout Quote
-\begin_inset Formula $\clubsuit$
-\end_inset
-
- If a length greater than 128 is used, then the size of the commons
-\emph on
-temp1
-\emph default
- and
-\emph on
-tempx
-\emph default
- must be changed in routine flow.
-
-\end_layout
-
\begin_layout Standard
-The element stiffness matrix for the stokes equation is always upper triangular.
- Since
-\emph on
-NVEC
-\emph default
- matrices at a time are formed, to save storage only the upper triangular
- part is stored (36 entries instead of 64 for 8 by 8 matrix).
- The numbering scheme for the storage is shown in Figure
-\begin_inset LatexCommand ref
-reference "fig:8-The-storage-for"
-\end_inset
-
-.
+}
\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 Standard
-\begin_inset LatexCommand label
-name "fig:8-The-storage-for"
-
\end_inset
-The storage for the stiffness matrix used in routine
-\series bold
-f_vstf
-\series default
-.
+ call.
+ The reason for this change is fairly technical, and I will spare the users
+ with the details.
+ The memory manager source can be found in the directory mm.src.
+ Care has to be taken when compiling on 32-bit or 64-bit machines (change
+ the header file in mm.src from mm2000_32.h to mm2000_64.h) 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 consult CIG or
+ Scott King.
\end_layout
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\align center
-\begin_inset Graphics
- filename images/conman-fig8.png
- scale 40
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
\begin_layout Section
Material Properties
\end_layout
@@ -2187,7 +2956,6 @@
is the dynamic viscosity.
In ConMan, the input parameter is the buoyancy part of the Rayleigh number.
-
\end_layout
\begin_layout Standard
@@ -2235,7 +3003,6 @@
\end_inset
, are the same.
-
\end_layout
\begin_layout Standard
@@ -2246,12 +3013,12 @@
\series default
.
The user can easily modify the functional form for specific problems.
- The default functional form is
+ 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^{*}+V^{*}z}{T+T_{o}}\right\} -\exp\left\{ \frac{E^{*}+V^{*}z}{1+T_{o}}\right\} \right\} \label{eq:}\end{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
@@ -2284,20 +3051,57 @@
\end_inset
is the depth.
-
-\begin_inset Formula $\mu_{o},\, E^{*},\, V^{*},$
+ In the input files,
+\begin_inset Formula $\mu_{o}$
\end_inset
- and
+, is input on the viscosity card,
+\begin_inset Formula $E^{*}$
+\end_inset
+
+, is input as Tcon(1)
+\begin_inset Formula $V^{*}$
+\end_inset
+
+ is input as Tcon(2), and
\begin_inset Formula $T_{o}$
\end_inset
- are all input in the material section.
+ is hardwired in
+\series bold
+rheol.newt.F
+\series default
+ to be 273.
+ and
+\begin_inset Formula $\Delta T$
+\end_inset
+
+ is hardwired to be 2000.0.
\end_layout
\begin_layout Standard
+The scaling in
+\series bold
+rheol.newt.F
+\series default
+ is such that
+\begin_inset Formula $E^{*}$
+\end_inset
+
+ can be input in kJ/mole and
+\begin_inset Formula $V^{*}$
+\end_inset
+
+ can be input as cm
+\begin_inset Formula $^{3}$
+\end_inset
+
+.
+\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
+ If no bottom temperature is specified the Rayleigh number becomes
\end_layout
\begin_layout Standard
@@ -2345,33 +3149,6 @@
timer
\series default
.
- To run ConMan on a 32 bit machine, each routine needs to have the comment
- in front of the line with ``implicit double precision'' removed.
- In routine
-\series bold
-intemp
-\series default
- the functions
-\emph on
-cos
-\emph default
- and
-\emph on
-sin
-\emph default
- need to be changed to
-\emph on
-dcos
-\emph default
- and
-\emph on
-dsin
-\emph default
-.
- Also, in routine pickdt the functions abs and amin need to be changed to
- dabs and dmin.
- Some compilers have automatic options to force reals to be real*8, so check
- your manual.
\end_layout
\begin_layout Section
@@ -2385,31 +3162,31 @@
\begin_layout Standard
ConMan works on a variety of computational platforms and has been tested
- on workstations running
+ on workstations running
\end_layout
\begin_layout Itemize
-Mac OS X 10.4.6 (G4, G5, and Intel)
+Mac OS X 10.4.6 (G4, G5, and Intel)
\end_layout
\begin_layout Itemize
-Windows 2000 and XP SP2
+Windows 2000 and XP SP2
\end_layout
\begin_layout Itemize
-RedHat Fedora Core 5 (x86)
+RedHat Fedora Core 5 (x86)
\end_layout
\begin_layout Itemize
-OpenSuse 10.0 (x86)
+OpenSuse 10.0 (x86)
\end_layout
\begin_layout Itemize
-Gentoo (x86)
+Gentoo (x86)
\end_layout
\begin_layout Itemize
-Debian stable (x86 and AMD64), testing (x86), and unstable (x86)
+Debian stable (x86 and AMD64), testing (x86), and unstable (x86)
\end_layout
\begin_layout Standard
@@ -2422,18 +3199,9 @@
\end_layout
\begin_layout Standard
-In order to build ConMan, you must have the headers and development libraries
- for
+This version of ConMan is self-contained and requires no external libraries.
\end_layout
-\begin_layout Itemize
-MPI
-\end_layout
-
-\begin_layout Itemize
-???
-\end_layout
-
\begin_layout Subsection
\begin_inset LatexCommand label
name "sec:Downloading-the-Code"
@@ -2444,16 +3212,14 @@
\end_layout
\begin_layout Standard
-You can get the source for the latest release from the
-\begin_inset LatexCommand htmlurl
-name "ConMan web page"
+You can get the source for the latest release from the ConMan web page
+\begin_inset LatexCommand url
target "geodynamics.org/cig/software/packages/mc/conman/"
\end_inset
.
In that tarball is the file ???name???.
-
\end_layout
\begin_layout Subsubsection
@@ -2482,33 +3248,33 @@
\begin_layout Standard
Otherwise, you will need to download and install a Subversion client, available
- at the
-\begin_inset LatexCommand htmlurl
-name "Subversion Website"
+ at the Subversion Website
+\begin_inset LatexCommand url
target "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 LyX-Code
+\begin_layout Section
+\begin_inset LatexCommand label
+name "sec:Support"
-\end_layout
+\end_inset
-\begin_layout Section
Support
\end_layout
\begin_layout Standard
-The primary point of support for ConMan is the
+The primary point of support for ConMan is the CIG Mantle Convection Mailing
+ List
\begin_inset LatexCommand url
-name "CIG Mantle Convection Mailing List"
target "cig-mc at geodynamics.org"
\end_inset
@@ -2520,8 +3286,8 @@
\end_layout
\begin_layout LyX-Code
-\begin_inset LatexCommand htmlurl
-name "cig-mc Archives"
+cig-mc Archives
+\begin_inset LatexCommand url
target "geodynamics.org/pipermail/cig-mc/"
\end_inset
@@ -2534,8 +3300,8 @@
\end_layout
\begin_layout LyX-Code
-\begin_inset LatexCommand htmlurl
-name "Roundup"
+Roundup
+\begin_inset LatexCommand url
target "geodynamics.org/roundup"
\end_inset
@@ -2548,11 +3314,16 @@
\end_layout
\begin_layout Chapter
+\begin_inset LatexCommand label
+name "cha:Input-Guide"
+
+\end_inset
+
Input Guide
\end_layout
\begin_layout Standard
-To run ConMan a series of eight file names are needed, some for input and
+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
@@ -2565,7 +3336,7 @@
\series default
and are described in this section.
The third file is an output file showing all the input parameters in a
- nace [todo -- nace TYPO for nice?] fancy form.
+ 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.
@@ -2573,18 +3344,21 @@
\series bold
fluxke
\series default
-), the seventh file is the coordinates, velocities and temperatures and
- the eighth file is for mean value calculations (see routine
+), the seventh file is the coordinates, velocities and temperatures, the
+ eighth file is for stresses (see routine
\series bold
-mean
+stress.F
\series default
+) and the ninth file is for geoid and topography (see routine
+\series bold
+geoid.F
+\series default
).
These file names are read in routine
\series bold
-flow
+ConMan.F
\series default
.
-
\end_layout
\begin_layout Standard
@@ -2599,8 +3373,11 @@
igeom
\series default
, contains the coordinates, boundary values and connectivity information.
- For the UNIX and CRAY interfaces ConMan reads the file names to attach
- to these units from a file named ’runfile’:
+ 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
@@ -2609,8 +3386,11 @@
igeom
\series default
is attached to the file named on the second line (names must be ASCII with
- a length less than 13 characters long).
- The input deck was broken up so that an automatic grid generating routine
+ 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 interpolat
@@ -2622,10 +3402,24 @@
\series bold
iin
\series default
- 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.
+ unit (throughout this guide a
+\begin_inset Quotes eld
+\end_inset
+
+card
+\begin_inset Quotes erd
+\end_inset
+
+ will mean one line of an ASCII text file).
+ These constitute the parameter part of the input
+\begin_inset Quotes eld
+\end_inset
+
+deck
+\begin_inset Quotes erd
+\end_inset
+
+ for the program ConMan.
The format for this guide is a
\series bold
bold
@@ -2636,38 +3430,124 @@
\emph default
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
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
Title\InsetSpace ~
-Card
+Card
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+
\emph on
Any descriptive character string up to 80 characters long
+\emph default
+
\end_layout
\begin_layout Description
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
Global\InsetSpace ~
Constants\InsetSpace ~
-Card
+Card
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+
\emph on
numnp nsd ndof nelx nelz mprec iflow necho inrsts iorstr nodebn ntimvs ntseq
- numeg isky nwrap
+ numeg isky nwrap
+\emph default
+
\end_layout
\begin_deeper
\begin_layout Description
-numnp .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+numnp
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
+ .
total number of nodal points
\end_layout
\begin_layout Description
-nsd .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+nsd
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
@@ -2675,246 +3555,565 @@
.
.
.
- number of spatial dimensions
+ .
+ number of spatial dimensions (always 2)
\end_layout
\begin_layout Description
-ndof .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+ndof
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
- number of degrees of freedom
+ .
+ number of degrees of freedom (always 2)
\end_layout
\begin_layout Description
-nelx .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+nelx
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
+ .
number of elements in the x1 (horizontal) direction
\end_layout
\begin_layout Description
-nelz .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+nelz
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
+ .
number of elements in the x2 (vertical) direction
\end_layout
\begin_layout Description
-mprec .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+mprec
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
- precision flag
+ .
+ precision flag (always use double)
\end_layout
\begin_deeper
\begin_layout Standard
-1 - single
+1 - single
\end_layout
\begin_layout Standard
-2 - double
+2 - double
\end_layout
\end_deeper
\begin_layout Description
-iflow .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+iflow
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
- data check flag
+ .
+ data check flag
\end_layout
\begin_deeper
\begin_layout Standard
-0 - check data only
+0 - check data only
\end_layout
\begin_layout Standard
-1 - execute code
+1 - execute code
\end_layout
\end_deeper
\begin_layout Description
-necho .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+necho
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
- echo data flag
+ .
+ echo data flag
\end_layout
\begin_deeper
\begin_layout Standard
-0 - minimum data echo (terse)
+0 - minimum data echo (terse)
\end_layout
\begin_layout Standard
-1 - echo data to output file (verbose)
+1 - echo data to output file (verbose)
\end_layout
\end_deeper
\begin_layout Description
-inrstr .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+inrstr
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
- read restart file flag
+ .
+ read restart file flag
\end_layout
\begin_deeper
\begin_layout Standard
-0 - use default start (conductive)
+0 - use default start (conductive)
\end_layout
\begin_layout Standard
-1 - read restart file from unit 16
+1 - read restart file from unit 16
\end_layout
\end_deeper
\begin_layout Description
-iorstr .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+iorstr
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
- write restart file flag
+ .
+ write restart file flag
\end_layout
\begin_deeper
\begin_layout Standard
-0 - don’t write restart file
+0 - don
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+
+\backslash
+textquoteright
\end_layout
+\end_inset
+
+
+\begin_inset ERT
+status collapsed
+
\begin_layout Standard
-1 - write restart file to unit 17
+
+{}
\end_layout
+\end_inset
+
+t write restart file
+\end_layout
+
+\begin_layout Standard
+1 - write restart file to unit 17
+\end_layout
+
\end_deeper
\begin_layout Description
-nodebn .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+nodebn
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
+ .
number of edge nodes for nusselt smoother
\end_layout
\begin_layout Description
-ntimvs .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+ntimvs
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
- temperature dependent viscosity flag
+ .
+ temperature dependent viscosity flag
\end_layout
\begin_deeper
\begin_layout Standard
-0 - stiffness matrix factored once
+0 - stiffness matrix factored once
\end_layout
\begin_layout Standard
-1 - stiffness matrix factored every time step
+1 - stiffness matrix factored every time step
\end_layout
\end_deeper
\begin_layout Description
-ntseq .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+ntseq
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
- number of time sequences
+ .
+ number of time sequences (always 1)
\end_layout
\begin_deeper
\begin_layout Standard
-currently only one supported
+currently only one supported
\end_layout
\end_deeper
\begin_layout Description
-numeg .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+numeg
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
- number of element groups
+ .
+ number of element groups (always 1)
\end_layout
\begin_deeper
\begin_layout Standard
-currently only one supported
+currently only one supported
\end_layout
\end_deeper
\begin_layout Description
-isky .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+isky
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
- flag for skyline factor
+ .
+ flag for skyline factor
\end_layout
\begin_deeper
\begin_layout Standard
-0 - dmf solver (if supported)
+0 - regular skyline
\end_layout
\begin_layout Standard
-1 - regular skyline
+1 - vectorized skyline
\end_layout
\end_deeper
\begin_layout Description
-nwrap .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+nwrap
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\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
+equal to number of elements in vertical
\end_layout
\begin_layout Standard
-to use nodes must be numbered increasing
+to use nodes must be numbered increasing
\end_layout
\begin_layout Standard
@@ -2924,195 +4123,545 @@
\end_deeper
\end_deeper
\begin_layout Description
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
Time\InsetSpace ~
Sequence\InsetSpace ~
-Cards - ntseq cards
+Cards
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
\end_layout
+\end_inset
+
+ - ntseq cards
+\end_layout
+
\begin_deeper
\begin_layout Standard
\emph on
-nstep niter alpha delt epstol
+nstep niter alpha delt epstol
+\emph default
+
\end_layout
\begin_layout Description
-nstep .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+nstep
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
- number of time steps
+ .
+ number of time steps
\end_layout
\begin_layout Description
-niter .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+niter
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
- number of multicorrector iterations
+ .
+ number of multicorrector iterations
\end_layout
\begin_deeper
\begin_layout Standard
-2 for explicit 2nd order
+2 - second-order expicit
\end_layout
+\begin_layout Standard
+1 - picard
+\end_layout
+
\end_deeper
\begin_layout Description
-alpha .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+alpha
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
- multicorrector parameter
+ .
+ multicorrector parameter
\end_layout
\begin_deeper
\begin_layout Standard
-0.5 for explicit 2nd order
+0.5 for explicit 2nd order
\end_layout
+\begin_layout Standard
+1.0 for picard
+\end_layout
+
\end_deeper
\begin_layout Description
-delt .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+delt
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
+ .
time step (not used)
\end_layout
\begin_layout Description
-epstol .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+epstol
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
+ .
tolerance for hybrid method (not used)
\end_layout
\end_deeper
\begin_layout Description
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
Output\InsetSpace ~
Step\InsetSpace ~
-Card
+Card
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+
\emph on
-nsdprt nsvprt nstprt nsmprt
+nsdprt nsvprt nstprt nsmprt
+\emph default
+
\end_layout
\begin_deeper
\begin_layout Description
-nsdprt .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+nsdprt
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
+ .
steps between disk output
\end_layout
\begin_layout Description
-nsvprt .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+nsvprt
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
+ .
steps between velocity output (not used)
\end_layout
\begin_layout Description
-nstprt .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+nstprt
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
- steps between temperature output
+ .
+ steps between temperature, velocity & stress output
\end_layout
\begin_layout Description
-nsmprt .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+nsmprt
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
- steps between mean field output (not used)
+ .
+ steps between stress field output (not used)
\end_layout
\end_deeper
\begin_layout Description
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
Velocity\InsetSpace ~
Boundary\InsetSpace ~
Condition\InsetSpace ~
Flag\InsetSpace ~
-Cards
+Cards
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+
\emph on
-bnode enode incr (bcf(i), i=1,ndof)
+bnode enode incr (bcf(i), i=1,ndof)
+\emph default
+
\end_layout
\begin_deeper
\begin_layout Description
-bnode .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+bnode
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
+ .
beginning node
\end_layout
\begin_layout Description
-enode .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+enode
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
+ .
ending node
\end_layout
\begin_layout Description
-incr .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+incr
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
+ .
node increment
\end_layout
\begin_layout Description
-bcf(i) .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+bcf(i)
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\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
+0 - free slip
\end_layout
\begin_layout Standard
-1 - pinned degree of freedom
+1 - pinned degree of freedom
\end_layout
\end_deeper
@@ -3120,53 +4669,161 @@
\begin_layout Standard
\emph on
-0 0 0 0 0 to end VBCF cards
+0 0 0 0 0 to end VBCF cards
+\emph default
+
\end_layout
\begin_layout Description
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
Temperature\InsetSpace ~
Boundary\InsetSpace ~
Condition\InsetSpace ~
Flag\InsetSpace ~
-Cards
+Cards
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+
\emph on
-bnode enode incr bcf
+bnode enode incr bcf
+\emph default
+
\end_layout
\begin_deeper
\begin_layout Description
-bnode .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+bnode
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
+ .
beginning node
\end_layout
\begin_layout Description
-enode .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+enode
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
+ .
ending node
\end_layout
\begin_layout Description
-incr .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+incr
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
+ .
node increment
\end_layout
\begin_layout Description
-bcf .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+bcf
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
@@ -3174,23 +4831,36 @@
.
.
.
- boundary condition flag for temperature
+ .
+ boundary condition flag for temperature
\end_layout
\begin_deeper
\begin_layout Standard
-1- fixed temperature
+1- fixed temperature
\end_layout
\end_deeper
\begin_layout Standard
\emph on
-0 0 0 0 to end TBCF cards
+0 0 0 0 to end TBCF cards
+\emph default
+
\end_layout
\end_deeper
\begin_layout Description
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
Nusselt\InsetSpace ~
Number\InsetSpace ~
Boundary\InsetSpace ~
@@ -3199,41 +4869,116 @@
Cards\InsetSpace ~
-\InsetSpace ~
Edge\InsetSpace ~
-Nodes top and bottom rows of
- nodes
+Nodes
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+ top and bottom rows of nodes
\emph on
-bnode enode incr
+bnode enode incr
+\emph default
+
\end_layout
\begin_deeper
\begin_layout Description
-bnode .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+bnode
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
+ .
beginning node
\end_layout
\begin_layout Description
-enode .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+enode
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
+ .
ending node
\end_layout
\begin_layout Description
-incr .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+incr
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
+ .
node increment
\end_layout
@@ -3245,6 +4990,16 @@
\end_deeper
\begin_layout Description
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
Nusselt\InsetSpace ~
Number\InsetSpace ~
Boundary\InsetSpace ~
@@ -3254,41 +5009,116 @@
-\InsetSpace ~
Second\InsetSpace ~
Row\InsetSpace ~
-Nodes second from top and
- bottom rows of nodes
+Nodes
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+ second from top and bottom rows of nodes
\emph on
-bnode enode incr
+bnode enode incr
+\emph default
+
\end_layout
\begin_deeper
\begin_layout Description
-bnode .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+bnode
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
+ .
beginning node
\end_layout
\begin_layout Description
-enode .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+enode
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
+ .
ending node
\end_layout
\begin_layout Description
-incr .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+incr
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
+ .
node increment
\end_layout
@@ -3296,98 +5126,259 @@
\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
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
Initial\InsetSpace ~
Temperature\InsetSpace ~
-Card
+Card
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+
\emph on
pert xsize zsize
-\emph default
-
\end_layout
\begin_deeper
\begin_layout Description
-pert .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+pert
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
+ .
perturbation from conductive state
\end_layout
\begin_layout Description
-xsize .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+xsize
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
+ .
nondimensional length (x1 direction) of box
\end_layout
\begin_layout Description
-zsize .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+zsize
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
+ .
nondimensional height (x2 direction) of box
\end_layout
\end_deeper
\begin_layout Description
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
Element\InsetSpace ~
Parameter\InsetSpace ~
-Cards - numeg cards
+Cards
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+ - numeg cards
\emph on
-ntype numel nen nenl numat nedof numsuf nipt implv implt
+ntype numel nen nenl numat nedof numsuf nipt implv implt
+\emph default
+
\end_layout
\begin_deeper
\begin_layout Description
-ntype .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+ntype
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
- element type
+ .
+ element type
\end_layout
\begin_deeper
\begin_layout Standard
-2 - two dimensional elements
+2 - two dimensional elements
\end_layout
+\end_deeper
+\begin_layout Description
+\begin_inset ERT
+status collapsed
+
\begin_layout Standard
-3 - three dimensional elements
+
+{
\end_layout
-\end_deeper
-\begin_layout Description
-numel .
+\end_inset
+
+numel
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
+ .
total number of elements
\end_layout
\begin_layout Description
-nen .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+nen
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
@@ -3395,246 +5386,733 @@
.
.
.
- number of element nodes
+ .
+ number of element nodes (always 4)
\end_layout
\begin_layout Description
-nenl .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+nenl
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
- number of local element nodes
+ .
+ number of local element nodes (always 4)
\end_layout
\begin_layout Description
-numat .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+numat
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
+ .
number of material groups
\end_layout
\begin_layout Description
-nedof .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+nedof
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
- number of element degrees of freedom
+ .
+ number of element degrees of freedom (always 2)
\end_layout
\begin_layout Description
-numsuf .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+numsuf
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
- number of surface force flux cards
+ .
+ number of imposed stress/flux cards
\end_layout
\begin_layout Description
-nipt .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+nipt
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
- number of integration points per element
+ .
+ number of integration points per element (always 5)
\end_layout
\begin_layout Description
-implv .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+implv
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
+ .
currently unused
\end_layout
\begin_layout Description
-implt .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+implt
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
+ .
currently unused
\end_layout
\end_deeper
\begin_layout Description
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
Viscosity\InsetSpace ~
-Card
+Card
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+
\emph on
-visc(i), i=1,numat
+visc(i), i=1,numat
+\emph default
+
\end_layout
\begin_deeper
\begin_layout Description
-visc(i) .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+visc(i)
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
+ .
preexponential viscosity coefficient for ith element
\end_layout
\end_deeper
\begin_layout Description
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
Penalty\InsetSpace ~
-Card
+Card
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+
\emph on
-alam(i), i=1,numat
+alam(i), i=1,numat
+\emph default
+
\end_layout
\begin_deeper
\begin_layout Description
-alam(i) .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+alam(i)
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
+ .
penalty parameter for ith element
\end_layout
\end_deeper
\begin_layout Description
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
Diffusivity\InsetSpace ~
-Card
+Card
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+
\emph on
diff(i), i=1,numat
-\emph default
-
\end_layout
\begin_deeper
\begin_layout Description
-diff(i) .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+diff(i)
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
+ .
thermal diffusivity for ith element
\end_layout
\end_deeper
\begin_layout Description
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
Buoyancy\InsetSpace ~
Rayleigh\InsetSpace ~
Number\InsetSpace ~
-Card
+Card
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+
\emph on
-Ra(i), i=1,numat
+Ra(i), i=1,numat
+\emph default
+
\end_layout
\begin_deeper
\begin_layout Description
-Ra(i) .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+Ra(i)
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
+ .
bouyancy part of Rayleigh number for ith element
\end_layout
\end_deeper
\begin_layout Description
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
Internal\InsetSpace ~
Heating\InsetSpace ~
Parameter\InsetSpace ~
-Card
+Card
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+
\emph on
-dmhu(i), i=1,numat
+dmhu(i), i=1,numat
+\emph default
+
\end_layout
\begin_deeper
\begin_layout Description
-dmhu(i) .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+dmhu(i)
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\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
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
Activation\InsetSpace ~
Energy\InsetSpace ~
-Card
+Card
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+
\emph on
tcon(1,i), i=1,numat
-\emph default
-
\end_layout
\begin_deeper
\begin_layout Description
-tcon(1,i) .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+tcon(1,i)
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
+ .
nondimensional activation energy for ith material group for temperature
dependent viscosity
\end_layout
\end_deeper
\begin_layout Description
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
Temperature\InsetSpace ~
Dependent\InsetSpace ~
Viscosity\InsetSpace ~
Temperature\InsetSpace ~
Offset\InsetSpace ~
-Card
+Card
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+
\emph on
tcon(2,i), i=1,numat
-\emph default
-
\end_layout
\begin_deeper
\begin_layout Description
-tcon(2,i) .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+tcon(2,i)
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
+ .
nondimen temperature offset for ith material group for temperature dependent
viscosity
\end_layout
\end_deeper
\begin_layout Description
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
Surface\InsetSpace ~
Force/Flux\InsetSpace ~
-Cards - numsuf cards
+Cards
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+ - numsuf cards
\emph on
-nel side fnorm ftan flux
+nel side fnorm ftan flux
+\emph default
+
\end_layout
\begin_deeper
\begin_layout Description
-nel .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+nel
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
@@ -3642,62 +6120,146 @@
.
.
.
+ .
element number
\end_layout
\begin_layout Description
-side .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+side
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
- side to apply force and flux
+ .
+ side to apply force and flux
\end_layout
\begin_deeper
\begin_layout Standard
-1 - bottom
+1 - bottom
\end_layout
\begin_layout Standard
-2 - right side
+2 - right side
\end_layout
\begin_layout Standard
-3 - top
+3 - top
\end_layout
\begin_layout Standard
-4 - left side
+4 - left side
\end_layout
\end_deeper
\begin_layout Description
-fnorm .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+fnorm
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
+ .
normal surface force
\end_layout
\begin_layout Description
-ftan .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+ftan
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
+ .
tangential surface force
\end_layout
\begin_layout Description
-flux .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+flux
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
@@ -3705,7 +6267,8 @@
.
.
.
- heat flux
+ .
+ heat flux
\end_layout
\end_deeper
@@ -3715,8 +6278,15 @@
igeom
\series default
unit.
- These constitute the geometry part of the input ``deck'' for the program
-
+ These constitute the geometry part of the input
+\begin_inset Quotes eld
+\end_inset
+
+deck
+\begin_inset Quotes erd
+\end_inset
+
+ for the program
\family typewriter
conman
\family default
@@ -3729,27 +6299,91 @@
\end_layout
\begin_layout Description
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
Absolute\InsetSpace ~
Coordinate\InsetSpace ~
-Card
+Card
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+
\emph on
-node gp (x(i,node) i=1,nsd)
+node gp (x(i,node) i=1,nsd)
+\emph default
+
\end_layout
\begin_deeper
\begin_layout Description
-node .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+node
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
+ .
the node whose coordinates are to be specified
\end_layout
\begin_layout Description
-gp .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+gp
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
@@ -3757,7 +6391,8 @@
.
.
.
- generation parameter for automatic generation
+ .
+ generation parameter for automatic generation
\end_layout
\begin_deeper
@@ -3766,139 +6401,371 @@
\end_layout
\begin_layout Standard
-2 - generate a line using node as a starting point
+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
+4 - generate a box using node as the lower left corner
\end_layout
\end_deeper
\begin_layout Description
-x(i,node) .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+x(i,node)
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
+ .
coordinate value in the ith spatial dimension
\end_layout
\end_deeper
\begin_layout Description
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
Corner\InsetSpace ~
Generation\InsetSpace ~
-Cards - gp-1 cards
+Cards
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+ - gp-1 cards
\emph on
node mgen (x(i,node) i=1,nsd)
\end_layout
\begin_deeper
\begin_layout Description
-node .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+node
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
+ .
node number
\end_layout
\begin_layout Description
-mgen .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+mgen
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
- generation parameter
+ .
+ generation parameter
\end_layout
\begin_deeper
\begin_layout Standard
-0 - don’t use this as the start of a generation sequence
+0 - don
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+
+\backslash
+textquoteright
\end_layout
+\end_inset
+
+
+\begin_inset ERT
+status collapsed
+
\begin_layout Standard
-1 - use this as the start of a generation sequence
+
+{}
\end_layout
+\end_inset
+
+t use this as the start of a generation sequence
+\end_layout
+
+\begin_layout Standard
+1 - use this as the start of a generation sequence
+\end_layout
+
\end_deeper
\begin_layout Description
-x(i,node) .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+x(i,node)
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
+ .
coordinate value in the ith spatial dimension
\end_layout
\end_deeper
\begin_layout Description
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
Generation\InsetSpace ~
Increment\InsetSpace ~
-Card
+Card
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+
\emph on
ninc1 inc1 ninc2 inc2
-\emph default
-
\end_layout
\begin_deeper
\begin_layout Description
-ninc1 .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+ninc1
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
+ .
number of additional nodes to generate in x1 direction
\end_layout
\begin_layout Description
-inc1 .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+inc1
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
+ .
increment of nodes in x1 direction
\end_layout
\begin_layout Description
-ninc2 .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+ninc2
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\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
+0 - if gp equals 2
\end_layout
\end_deeper
\begin_layout Description
-inc2 .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+inc2
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\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
+0 - if gp equals 2
\end_layout
\end_deeper
@@ -3914,27 +6781,91 @@
\end_layout
\begin_layout Description
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
Absolute\InsetSpace ~
Velocity\InsetSpace ~
-Card
+Card
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+
\emph on
-node gp (v(i,node) i=1,nsd)
+node gp (v(i,node) i=1,nsd)
+\emph default
+
\end_layout
\begin_deeper
\begin_layout Description
-node .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+node
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
+ .
the node whose velocities are to be specified
\end_layout
\begin_layout Description
-gp .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+gp
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
@@ -3942,146 +6873,406 @@
.
.
.
- generation parameter for automatic generation
+ .
+ generation parameter for automatic generation
\end_layout
\begin_deeper
\begin_layout Standard
-0 - no autogeneration
+0 - no autogeneration
\end_layout
\begin_layout Standard
-2 - generate a line using node as a starting point
+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
+4 - generate a box using node as the lower left corner
\end_layout
\end_deeper
\begin_layout Description
-v(i,node) .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+v(i,node)
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
+ .
velocity value in the ith spatial dimension
\end_layout
\end_deeper
\begin_layout Description
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
Corner\InsetSpace ~
Generation\InsetSpace ~
-Cards - gp-1 cards
+Cards
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+ - gp-1 cards
\emph on
-node mgen (v(i,node) i=1,nsd)
+node mgen (v(i,node) i=1,nsd)
+\emph default
+
\end_layout
\begin_deeper
\begin_layout Description
-node .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+node
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
+ .
node number
\end_layout
\begin_layout Description
-mgen .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+mgen
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
- generation parameter
+ .
+ generation parameter
\end_layout
\begin_deeper
\begin_layout Standard
-0 - don’t use this as the start of a generation sequence
+0 - don
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+
+\backslash
+textquoteright
\end_layout
+\end_inset
+
+
+\begin_inset ERT
+status collapsed
+
\begin_layout Standard
-1 - use this as the start of a generation sequence
+
+{}
\end_layout
+\end_inset
+
+t use this as the start of a generation sequence
+\end_layout
+
+\begin_layout Standard
+1 - use this as the start of a generation sequence
+\end_layout
+
\end_deeper
\begin_layout Description
-v(i,node) .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+v(i,node)
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
+ .
velocity value in the ith spatial dimension
\end_layout
\end_deeper
\begin_layout Description
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
Generation\InsetSpace ~
Increment\InsetSpace ~
-Card
+Card
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+
\emph on
-ninc1 inc1 ninc2 inc2
+ninc1 inc1 ninc2 inc2
+\emph default
+
\end_layout
\begin_deeper
\begin_layout Description
-ninc1 .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+ninc1
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
+ .
number of additional nodes to generate in x1 direction
\end_layout
\begin_layout Description
-inc1 .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+inc1
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
+ .
increment of nodes in x1 direction
\end_layout
\begin_layout Description
-ninc2 .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+ninc2
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\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
-0 - if gp equals 2
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
\end_layout
+\end_inset
+
+0
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+ - if gp equals 2
+\end_layout
+
\end_deeper
\begin_layout Description
-inc2 .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+inc2
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\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
+0 - if gp equals 2
\end_layout
\end_deeper
@@ -4097,27 +7288,91 @@
\end_layout
\begin_layout Description
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
Absolute\InsetSpace ~
Temperature\InsetSpace ~
-Card
+Card
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+
\emph on
-node gp t(node)
+node gp t(node)
+\emph default
+
\end_layout
\begin_deeper
\begin_layout Description
-node .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+node
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
+ .
the node whose velocities are to be specified
\end_layout
\begin_layout Description
-gp .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+gp
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
@@ -4125,144 +7380,383 @@
.
.
.
- generation parameter for automatic generation
+ .
+ generation parameter for automatic generation
\end_layout
\begin_deeper
\begin_layout Standard
-0 - no autogeneration
+0 - no autogeneration
\end_layout
\begin_layout Standard
-2 - generate a line using node as a starting point
+2 - generate a line using node as a starting point
\end_layout
\end_deeper
\begin_layout Description
-t(node) .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+t(node)
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
+ .
temperature value
\end_layout
\end_deeper
\begin_layout Description
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
Corner\InsetSpace ~
Generation\InsetSpace ~
-Cards - gp-1 cards
+Cards
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+ - gp-1 cards
\emph on
-node mgen t(node)
+node mgen t(node)
+\emph default
+
\end_layout
\begin_deeper
\begin_layout Description
-node .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+node
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
+ .
node number
\end_layout
\begin_layout Description
-mgen .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+mgen
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
- generation parameter
+ .
+ generation parameter
\end_layout
\begin_deeper
\begin_layout Standard
-0 - don’t use this as the start of a generation sequence
+0 - don
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+
+\backslash
+textquoteright
\end_layout
+\end_inset
+
+
+\begin_inset ERT
+status collapsed
+
\begin_layout Standard
-1 - use this as the start of a generation sequence
+
+{}
\end_layout
+\end_inset
+
+t use this as the start of a generation sequence
+\end_layout
+
+\begin_layout Standard
+1 - use this as the start of a generation sequence
+\end_layout
+
\end_deeper
\begin_layout Description
-t(node) .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+t(node)
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
+ .
temperature value
\end_layout
\end_deeper
\begin_layout Description
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
Generation\InsetSpace ~
Increment\InsetSpace ~
-Card
+Card
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+
\emph on
-ninc1 inc1 ninc2 inc2
+ninc1 inc1 ninc2 inc2
+\emph default
+
\end_layout
\begin_deeper
\begin_layout Description
-ninc1 .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+ninc1
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
+ .
number of additional nodes to generate in x1 direction
\end_layout
\begin_layout Description
-inc1 .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+inc1
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
+ .
increment of nodes in x1 direction
\end_layout
\begin_layout Description
-ninc2 .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+ninc2
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\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
+0 - if gp equals 2
\end_layout
\end_deeper
\begin_layout Description
-inc2 .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+inc2
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\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
+0 - if gp equals 2
\end_layout
\end_deeper
@@ -4278,27 +7772,91 @@
\end_layout
\begin_layout Description
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
Absolution\InsetSpace ~
Element\InsetSpace ~
-Card
+Card
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+
\emph on
-elnu ng mat no (ien(elnu,i) i=1,nen)
+elnu ng mat no (ien(elnu,i) i=1,nen)
+\emph default
+
\end_layout
\begin_deeper
\begin_layout Description
-elnu .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+elnu
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
+ .
element number
\end_layout
\begin_layout Description
-ng .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+ng
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
@@ -4306,21 +7864,43 @@
.
.
.
- generation parameter
+ .
+ generation parameter
\end_layout
\begin_deeper
\begin_layout Standard
-0 - no generation
+0 - no generation
\end_layout
\begin_layout Standard
-1 - generate using increments from increment card
+1 - generate using increments from increment card
\end_layout
\end_deeper
\begin_layout Description
-mat no .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+mat
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+ no .
.
.
.
@@ -4330,87 +7910,257 @@
\end_layout
\begin_layout Description
-ien(elnu,i) .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+ien(elnu,i)
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
+ .
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
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
Increment\InsetSpace ~
-Card
+Card
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+
\emph on
-nel1 incel1 incn1 nel2 incel2 incn2
+nel1 incel1 incn1 nel2 incel2 incn2
+\emph default
+
\end_layout
\begin_deeper
\begin_layout Description
-nel1 .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+nel1
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
+ .
number of elements in x1 (horizontal) direction
\end_layout
\begin_layout Description
-incel1 .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+incel1
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
+ .
increment of elements in x1 (horizontal) direction
\end_layout
\begin_layout Description
-incn1 .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+incn1
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
+ .
increment of nodes in x1 (horizontal) direction
\end_layout
\begin_layout Description
-nel2 .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+nel2
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
+ .
number of elements in x2 (vertical) direction
\end_layout
\begin_layout Description
-incel2 .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+incel2
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
+ .
increment of elements in x2 (vertical) direction
\end_layout
\begin_layout Description
-incn2 .
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{
+\end_layout
+
+\end_inset
+
+incn2
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
.
.
.
.
.
.
+ .
increment of nodes in x2 (vertical) direction
\end_layout
@@ -4422,206 +8172,3140 @@
\end_deeper
\begin_layout Chapter
+\begin_inset LatexCommand label
+name "cha:Sample-Input-Files"
+
+\end_inset
+
Sample Input Files
\end_layout
\begin_layout Standard
-The lines below are a sample 32 element by 32 element input deck for a 1
- by 1 square.
-
+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
\begin_layout LyX-Code
-32 by 32 element input deck
+50 x 50 el.
+ plate problem from Blankenbach et al.
+ 1989 #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,DEL
+NXTLN 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
\end_layout
-\begin_layout LyX-Code
-1089 2 2 32 32 1 1 0 0 1 66 1 1 1 1 0
+\begin_layout Standard
+The lines below are a sample geometry file for the 50 by 50 element problem.
\end_layout
\begin_layout LyX-Code
-50 2 0.5 0.0005 0.000001
+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_layout
-\begin_layout LyX-Code
-50 50 50 50
+\begin_layout Chapter
+\begin_inset LatexCommand label
+name "cha:Output-Guide"
+
+\end_inset
+
+Output Guide
\end_layout
-\begin_layout LyX-Code
-1 1057 33 0 1
+\begin_layout Section
+The Output Files
\end_layout
-\begin_layout LyX-Code
-1057 1089 1 1 0
+\begin_layout Chapter
+\begin_inset LatexCommand label
+name "cha:The-Benchmark-Cases"
+
+\end_inset
+
+The Benchmark Cases
\end_layout
-\begin_layout LyX-Code
-1 33 1 1 0
+\begin_layout Section
+Constant Viscosity Benchmark for ConMan
\end_layout
-\begin_layout LyX-Code
-33 1089 33 0 1
+\begin_layout Standard
+Here we reproduce the results from Blankenbach et al.
+ (1989) 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
+ and
+\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
+\begin_inset Formula $10^{4}$
+\end_inset
+
+,
+\begin_inset Formula $10^{5}$
+\end_inset
+
+, and
+\begin_inset Formula $10^{6}$
+\end_inset
+
+ respectively and the dimensional parameters are listed in Table\InsetSpace ~
+7.1.
+ The time is the run time 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 LyX-Code
-1 1 1 1 1
+\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 ~
+7.2 (Rayleigh number
+\begin_inset Formula $10^{4}$
+\end_inset
+
+), Table\InsetSpace ~
+7.3 (Rayleigh number
+\begin_inset Formula $10^{5}$
+\end_inset
+
+), and Table\InsetSpace ~
+7.4 (Rayleigh number
+\begin_inset Formula $10^{6}$
+\end_inset
+
+).
+ For the globally 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
+\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.
\end_layout
-\begin_layout LyX-Code
-33 33 1 1 1
+\begin_layout Standard
+\begin_inset Float table
+placement htdp
+wide false
+sideways false
+status open
+
+\begin_layout Standard
+
\end_layout
-\begin_layout LyX-Code
-1057 1057 1 1 1
+\begin_layout Standard
+\align center
+\begin_inset Tabular
+<lyxtabular version="3" rows="11" columns="4">
+<features>
+<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
-\begin_layout LyX-Code
-1089 1089 1 1 1
+\end_inset
+</cell>
+<cell multicolumn="1" alignment="none" valignment="top" topline="true" usebox="none" special="c">
+\begin_inset Text
+
+\begin_layout Standard
+Symbol
\end_layout
-\begin_layout LyX-Code
-0 0 0 0 0
+\end_inset
+</cell>
+<cell multicolumn="1" alignment="none" valignment="top" topline="true" usebox="none" special="c">
+\begin_inset Text
+
+\begin_layout Standard
+Value
\end_layout
-\begin_layout LyX-Code
-1 1057 33 1
+\end_inset
+</cell>
+<cell alignment="none" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
\end_layout
-\begin_layout LyX-Code
-33 1089 33 1
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="left" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+depth of domain
\end_layout
-\begin_layout LyX-Code
-0 0 0 0
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $d$
+\end_inset
+
+
\end_layout
-\begin_layout LyX-Code
-1 1057 33
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $10^{6}$
+\end_inset
+
+\InsetSpace ~
+m
\end_layout
-\begin_layout LyX-Code
-33 1089 33
+\end_inset
+</cell>
+<cell alignment="none" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
\end_layout
-\begin_layout LyX-Code
-0 0 0
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="left" valignment="top" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+gravitational acceleration
\end_layout
-\begin_layout LyX-Code
-2 1058 33
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+g
\end_layout
-\begin_layout LyX-Code
-32 1088 33
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+10 m/s
+\begin_inset Formula $^{2}$
+\end_inset
+
+
\end_layout
-\begin_layout LyX-Code
-0 0 0
+\end_inset
+</cell>
+<cell alignment="none" valignment="top" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
\end_layout
-\begin_layout LyX-Code
-0.001 1.0 1.0
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="left" valignment="top" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+temperature difference
\end_layout
-\begin_layout LyX-Code
-2 1024 4 4 1 2 0 5 0 0
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $\Delta T$
+\end_inset
+
+
\end_layout
-\begin_layout LyX-Code
-1.0
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+1000 K
\end_layout
-\begin_layout LyX-Code
-1.0e07
+\end_inset
+</cell>
+<cell alignment="none" valignment="top" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
\end_layout
-\begin_layout LyX-Code
-1.0
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="left" valignment="top" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+density
\end_layout
-\begin_layout LyX-Code
-77927.0
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $\rho$
+\end_inset
+
+
\end_layout
-\begin_layout LyX-Code
-0.0
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+4000\InsetSpace ~
+kg m
+\begin_inset Formula $^{-3}$
+\end_inset
+
+
\end_layout
-\begin_layout LyX-Code
-0.0
+\end_inset
+</cell>
+<cell alignment="none" valignment="top" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
\end_layout
-\begin_layout LyX-Code
-0.0
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="left" valignment="top" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+thermal diffusivity
\end_layout
-\begin_layout LyX-Code
-0.0
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $\kappa$
+\end_inset
+
+
\end_layout
+\end_inset
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+
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+
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+
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+
+\end_inset
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+
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+
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+
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+
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+
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+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true" bottomline="true">
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+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+
+\backslash
+dag
+\end_layout
+
+\end_inset
+
+ Christensen
+\begin_inset Formula $^{\prime}$
+\end_inset
+
+s extrapolated values.
+\end_layout
+
+\end_inset
+</cell>
+<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
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+
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+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Caption
+
+\begin_layout Standard
+Blankenbach (1989) Benchmark 1b: Steady State, 2D, constant viscosity convection
+ in a 1 by 1 box with Rayleigh number
+\begin_inset Formula $10^{5}$
+\end_inset
+
+ using ConMan
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float table
+wide false
+sideways false
+status open
+
+\begin_layout Standard
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+
+\backslash
+centering
+\end_layout
+
+\end_inset
+
+
+\begin_inset Tabular
+<lyxtabular version="3" rows="6" columns="8">
+<features>
+<column alignment="right" valignment="top" leftline="true" width="0">
+<column alignment="center" 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">
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="0">
+<column alignment="right" valignment="top" rightline="true" width="0">
+<row topline="true">
+<cell multicolumn="1" alignment="none" valignment="top" topline="true" usebox="none" special="|c">
+\begin_inset Text
+
+\begin_layout Standard
+Grid
+\end_layout
+
+\end_inset
+</cell>
+<cell multicolumn="1" alignment="none" valignment="top" topline="true" usebox="none" special="c">
+\begin_inset Text
+
+\begin_layout Standard
+V
+\begin_inset Formula $_{rms}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell multicolumn="1" alignment="none" valignment="top" topline="true" usebox="none" special="c">
+\begin_inset Text
+
+\begin_layout Standard
+Nusselt No.
+\end_layout
+
+\end_inset
+</cell>
+<cell multicolumn="1" alignment="none" valignment="top" topline="true" usebox="none" special="c">
+\begin_inset Text
+
+\begin_layout Standard
+Topo
+\begin_inset Formula $_{L}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell multicolumn="1" alignment="none" valignment="top" topline="true" usebox="none" special="c">
+\begin_inset Text
+
+\begin_layout Standard
+Topo
+\begin_inset Formula $_{R}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell multicolumn="1" alignment="none" valignment="top" topline="true" usebox="none" special="c">
+\begin_inset Text
+
+\begin_layout Standard
+Geoid
+\begin_inset Formula $_{L}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell multicolumn="1" alignment="none" valignment="top" topline="true" usebox="none" special="c">
+\begin_inset Text
+
+\begin_layout Standard
+Geoid
+\begin_inset Formula $_{R}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell multicolumn="1" alignment="none" valignment="top" topline="true" usebox="none" special="c|">
+\begin_inset Text
+
+\begin_layout Standard
+Run Time (sec)
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="right" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+50
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+840.524
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
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+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+941.607
+\end_layout
+
+\end_inset
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+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+-1301.980
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+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+14.958
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
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+
+\begin_layout Standard
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+\end_inset
+</cell>
+<cell alignment="right" valignment="top" topline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+1.71
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="right" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+100
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
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+\end_inset
+</cell>
+<cell alignment="center" valignment="top" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+22.023
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+945.108
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+-1290.926
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+14.109
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+</cell>
+<cell alignment="center" valignment="top" usebox="none">
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+
+\begin_layout Standard
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+
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+<row>
+<cell alignment="right" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
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+
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+
+\begin_layout Standard
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+
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+<cell alignment="center" valignment="top" usebox="none">
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+
+\begin_layout Standard
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+
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+</cell>
+<cell alignment="center" valignment="top" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
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+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
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+
+\end_inset
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+<cell alignment="center" valignment="top" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+-15.204
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="right" valignment="top" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+184.27
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="right" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+
+\backslash
+dag
+\end_layout
+
+\end_inset
+
+ C
+\begin_inset Formula $_{ext}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+833.989
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+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+21.997
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+
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+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
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+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+-1283.813
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+13.452
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+-15.034
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="right" valignment="top" topline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+
+\backslash
+null
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true" bottomline="true">
+<cell multicolumn="1" alignment="none" valignment="top" topline="true" bottomline="true" usebox="none" special="|l|">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+
+\backslash
+dag
+\end_layout
+
+\end_inset
+
+ Christensen
+\begin_inset Formula $^{\prime}$
+\end_inset
+
+s extrapolated values.
+\end_layout
+
+\end_inset
+</cell>
+<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Caption
+
+\begin_layout Standard
+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
+
+ using ConMan.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+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
+\begin_inset Formula \begin{equation}
+\eta(T)=\eta_{o}\exp\left[{-\ln\{1000\}}\frac{{T}}{{\Delta T}}\right]\end{equation}
+
+\end_inset
+
+ where
+\begin_inset Formula $\eta_{o}=2.9\times10^{19}$
+\end_inset
+
+ and
+\begin_inset Formula $\Delta T=1000.0$
+\end_inset
+
+.
+ The other scaling parameters are the same as Table\InsetSpace ~
+7.1.
+ The results for a Rayleigh number of
+\begin_inset Formula $10^{4}$
+\end_inset
+
+ are presented in Table\InsetSpace ~
+7.5.
+ 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
+
+\begin_layout Standard
+\begin_inset Float table
+wide false
+sideways false
+status open
+
+\begin_layout Standard
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+
+\backslash
+centering
+\end_layout
+
+\end_inset
+
+
+\begin_inset Tabular
+<lyxtabular version="3" rows="7" columns="8">
+<features>
+<column alignment="right" valignment="top" leftline="true" width="0">
+<column alignment="center" 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">
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="0">
+<column alignment="right" valignment="top" rightline="true" width="0">
+<row topline="true">
+<cell multicolumn="1" alignment="none" valignment="top" topline="true" usebox="none" special="|c">
+\begin_inset Text
+
+\begin_layout Standard
+Grid
+\end_layout
+
+\end_inset
+</cell>
+<cell multicolumn="1" alignment="none" valignment="top" topline="true" usebox="none" special="c">
+\begin_inset Text
+
+\begin_layout Standard
+V
+\begin_inset Formula $_{rms}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell multicolumn="1" alignment="none" valignment="top" topline="true" usebox="none" special="c">
+\begin_inset Text
+
+\begin_layout Standard
+Nusselt No.
+\end_layout
+
+\end_inset
+</cell>
+<cell multicolumn="1" alignment="none" valignment="top" topline="true" usebox="none" special="c">
+\begin_inset Text
+
+\begin_layout Standard
+Topo
+\begin_inset Formula $_{L}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
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+\begin_inset Formula $^{\prime}$
+\end_inset
+
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+
+\end_inset
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+<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
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+<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
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+<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
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+<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
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+<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
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+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Caption
+
+\begin_layout Standard
+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
+
+ using ConMan.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\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
+\end_layout
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
\begin_layout Chapter
-\start_of_appendix
License
\end_layout
@@ -4634,9 +11318,7 @@
\series default
\newline
-
-\series medium
-Everyone is permitted to copy and distribute verbatim copies of this license
+ Everyone is permitted to copy and distribute verbatim copies of this license
document, but changing it is not allowed.
\end_layout
@@ -4690,6 +11372,7 @@
\begin_layout Enumerate
Offer you this license which gives you legal permission to copy, distribute
and/or modify the software.
+
\end_layout
\begin_layout Standard
@@ -4707,7 +11390,6 @@
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
@@ -4720,31 +11402,24 @@
AND MODIFICATION
\end_layout
-\begin_layout Standard
+\begin_layout Itemize
\begin_inset ERT
status collapsed
\begin_layout Standard
-
-\backslash
-begin{itemize}
+[
\end_layout
\end_inset
-
-\end_layout
-
-\begin_layout Standard
+0.
\begin_inset ERT
status collapsed
\begin_layout Standard
-
-\backslash
-item[0.]
+]
\end_layout
\end_inset
@@ -4759,34 +11434,17 @@
(Hereinafter, translation is included without limitation in the term "modificat
ion.") Each licensee is addressed as "you."
\newline
-
+
\newline
-Activities other than copying,
+ 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 Standard
-\begin_inset ERT
-status collapsed
-
-\begin_layout Standard
-
-
-\backslash
-end{itemize}
-\end_layout
-
-\end_inset
-
-
-\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 appropria
@@ -4794,14 +11452,12 @@
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.
-
\end_layout
\end_deeper
@@ -4809,7 +11465,7 @@
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 also meet all of these conditions:
\end_layout
\begin_deeper
@@ -4851,7 +11507,6 @@
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
@@ -4859,7 +11514,6 @@
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
@@ -4867,14 +11521,13 @@
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:
+ 2 above provided that you also do one of the following:
\end_layout
\begin_deeper
@@ -4919,7 +11572,6 @@
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
@@ -4989,13 +11641,11 @@
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.
-
\end_layout
\end_deeper
@@ -5015,7 +11665,6 @@
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
@@ -5046,31 +11695,24 @@
NO WARRANTY
\end_layout
-\begin_layout Standard
+\begin_layout Itemize
\begin_inset ERT
status collapsed
\begin_layout Standard
-
-\backslash
-begin{itemize}
+[
\end_layout
\end_inset
-
-\end_layout
-
-\begin_layout Standard
+11.
\begin_inset ERT
status collapsed
\begin_layout Standard
-
-\backslash
-item[11.]
+]
\end_layout
\end_inset
@@ -5085,18 +11727,26 @@
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
+
\begin_layout Standard
+
+[
+\end_layout
+
+\end_inset
+
+12.
\begin_inset ERT
status collapsed
\begin_layout Standard
-
-\backslash
-item[12.]
+]
\end_layout
\end_inset
@@ -5112,22 +11762,6 @@
OF SUCH DAMAGES.
\end_layout
-\begin_layout Standard
-\begin_inset ERT
-status collapsed
-
-\begin_layout Standard
-
-
-\backslash
-end{itemize}
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
\begin_layout Section*
END OF TERMS AND CONDITIONS
\end_layout
@@ -5140,7 +11774,6 @@
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
@@ -5156,8 +11789,8 @@
Copyright
\size footnotesize
© (
-\size default
-year) (name of author)
+\size normal
+year) (name of author)
\end_layout
\begin_layout Quote
@@ -5165,7 +11798,6 @@
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
@@ -5173,7 +11805,6 @@
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
@@ -5184,12 +11815,11 @@
\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:
+ it starts in an interactive mode:
\end_layout
\begin_layout Quote
@@ -5206,27 +11836,25 @@
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:
+ Here is a sample; alter the names:
\end_layout
\begin_layout Quote
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
+ 1 April 1989
\newline
-Ty Coon, President of Vice
+ Ty Coon, President of Vice
\end_layout
\begin_layout Standard
@@ -5249,7 +11877,6 @@
A Petrov-Galerkin finite-element formulation for convection dominated flows.
Ph.D.
Thesis, California Institute of Technology, Pasadena, CA.
-
\end_layout
\begin_layout Bibliography
@@ -5272,7 +11899,6 @@
and Eng., 32,
\emph default
199-259.
-
\end_layout
\begin_layout Bibliography
@@ -5288,7 +11914,7 @@
The finite element method.
\emph default
- Prentice-Hall, Inc., Englewood Cliffs, New Jersey
+ Prentice-Hall, Inc., Englewood Cliffs, New Jersey
\end_layout
\begin_layout Bibliography
@@ -5307,7 +11933,6 @@
ASME
\emph default
, New York, 34:19-35.
-
\end_layout
\begin_layout Bibliography
@@ -5326,7 +11951,6 @@
ASME,
\emph default
New York, 95:75-99.
-
\end_layout
\begin_layout Bibliography
@@ -5343,10 +11967,9 @@
\emph on
J.
Comput.
- Phys.,
+ Phys.,
\emph default
-30: 19-35.
-
+ 30: 19-35.
\end_layout
\begin_layout Bibliography
@@ -5369,7 +11992,6 @@
and Eng.,
\emph default
15, 63-81.
-
\end_layout
\begin_layout Bibliography
@@ -5383,7 +12005,6 @@
Navier-Stokes equations: therory and numerical analysis.
North-Holland.
Amsterdam.
-
\end_layout
\begin_layout Bibliography
@@ -5399,7 +12020,31 @@
Gable, B.H.
Hager, P.
Olson, R.J.
- O’Connell, A.
+ O
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+
+\backslash
+textquoteright
+\end_layout
+
+\end_inset
+
+
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+{}
+\end_layout
+
+\end_inset
+
+Connell, A.
Raefsky, and G.
Schubert, 1991.
A benchmark comparison of numerical methods for infinite Prandtl number
@@ -5410,6 +12055,7 @@
Fluid Dynamics,
\emph default
in press.
+
\end_layout
\end_body
Modified: mc/2D/ConMan/trunk/doc/images/conman_cover.pdf
===================================================================
(Binary files differ)
Modified: mc/2D/ConMan/trunk/doc/images/conman_cover.psd
===================================================================
(Binary files differ)
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