/[escript]/trunk/doc/user/Models.tex
ViewVC logotype

Annotation of /trunk/doc/user/Models.tex

Parent Directory Parent Directory | Revision Log Revision Log


Revision 1709 - (hide annotations)
Fri Aug 15 01:09:45 2008 UTC (11 years, 6 months ago) by lgraham
File MIME type: application/x-tex
File size: 3452 byte(s)
added separate section for Einstein notation


1 lgraham 1702 %
2     % $Id: Models.tex 1316 2007-09-25 03:18:30Z ksteube $
3     %
4     %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
5     %
6     % Copyright 2003-2007 by ACceSS MNRF
7     % Copyright 2007 by University of Queensland
8     %
9     % http://esscc.uq.edu.au
10     % Primary Business: Queensland, Australia
11     % Licensed under the Open Software License version 3.0
12     % http://www.opensource.org/licenses/osl-3.0.php
13     %
14     %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
15     %
16    
17     \chapter{Models}
18    
19     The following sections give a breif overview of the model classes and their corresponding methods.
20    
21     \section{Stokes Cartesian (Saddle Point Problem)}
22    
23     \subsection{Description}
24    
25     Saddle point type problems emerge in a number of applications throughout physics and engineering. Finite element discretisation of the Navier-Stokes (momentum) equations for incompressible flow leads to equations of a saddle point type, which can be formulated as a solution of the following operator problem for $u \in V$ and $p \in Q$ with suitable Hilbert spaces $V$ and $Q$:
26    
27     \begin{equation}
28     \left[ \begin{array}{cc}
29     A & B \\
30     b^{*} & 0 \\
31     \end{array} \right]
32     \left[ \begin{array}{c}
33     u \\
34     p \\
35     \end{array} \right]
36     =\left[ \begin{array}{c}
37     f \\
38     g \\
39     \end{array} \right]
40     \label{SADDLEPOINT}
41     \end{equation}
42    
43     where $A$ is coercive, self-adjoint linear operator in $V$, $B$ is a linear operator from $Q$ into $V$ and $B^{*}$ is the adjoint operator of $B$. $f$ and $g$ are given elements from $V$ and $Q$ respectivitly. For more details on the mathematics see references \cite{AAMIRBERKYAN2008,MBENZI2005}.
44    
45     The Uzawa scheme scheme is used to solve the momentum equation with the secondary condition of incompressibility \cite{GROSS2006,AAMIRBERKYAN2008}.
46    
47     \begin{classdesc}{StokesProblemCartesian}{domain,debug}
48     opens the stokes equations on the \Domain domain. Setting debug=True switches the debug mode to on.
49     \end{classdesc}
50    
51     example usage:
52    
53     solution=StokesProblemCartesian(mesh) \\
54     solution.setTolerance(TOL) \\
55     solution.initialize(fixed\_u\_mask=b\_c,eta=eta,f=Y) \\
56     velocity,pressure=solution.solve(velocity,pressure,max\_iter=max\_iter,solver=solver) \\
57    
58     \subsection{Benchmark Problem}
59    
60     Convection problem
61    
62    
63     \section{Temperature Cartesian}
64    
65     \begin{equation}
66 lgraham 1709 \rho c\hackscore{p} \left (\frac{\partial T}{\partial t} + \vec{v} \cdot \nabla T \right ) = k \nabla^{2}T
67 lgraham 1702 \label{HEAT EQUATION}
68     \end{equation}
69    
70     where $\vec{v}$ is the velocity vector, $T$ is the temperature, $\rho$ is the density, $\eta$ is the viscosity, $c\hackscore{p}$ is the specific heat at constant pressure and $k$ is the thermal conductivity.
71    
72     \subsection{Description}
73    
74     \subsection{Method}
75    
76     \begin{classdesc}{TemperatureCartesian}{dom,theta=THETA,useSUPG=SUPG}
77     \end{classdesc}
78    
79     \subsection{Benchmark Problem}
80    
81    
82     \section{Level Set Method}
83    
84     \subsection{Description}
85    
86     \subsection{Method}
87    
88     Advection and Reinitialisation
89    
90     \begin{classdesc}{LevelSet}{mesh, func\_new, reinit\_max, reinit\_each, tolerance, smooth}
91     \end{classdesc}
92    
93     %example usage:
94    
95     %levelset = LevelSet(mesh, func\_new, reinit\_max, reinit\_each, tolerance, smooth)
96    
97     \begin{methoddesc}[LevelSet]{update\_parameter}{parameter}
98     Update the parameter.
99     \end{methoddesc}
100    
101     \begin{methoddesc}[LevelSet]{update\_phi}{paramter}{velocity}{dt}{t\_step}
102     Update level set function; advection and reinitialization
103     \end{methoddesc}
104    
105     \subsection{Benchmark Problem}
106    
107     Rayleigh-Taylor instability problem
108    
109    
110     \section{Drucker Prager Model}
111    
112     \section{Plate Mantel}

  ViewVC Help
Powered by ViewVC 1.1.26