/[escript]/branches/arrayview_from_1695_trunk/doc/user/Models.tex
ViewVC logotype

Contents of /branches/arrayview_from_1695_trunk/doc/user/Models.tex

Parent Directory Parent Directory | Revision Log Revision Log


Revision 1781 - (show annotations)
Thu Sep 11 05:03:14 2008 UTC (10 years, 11 months ago) by jfenwick
File MIME type: application/x-tex
File size: 3452 byte(s)
Branch commit

Merged changes from trunk version 1695 up to and including version 1779.


1 %
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 \rho c\hackscore{p} \left (\frac{\partial T}{\partial t} + \vec{v} \cdot \nabla T \right ) = k \nabla^{2}T
67 \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