doc/user/Models.tex

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% $Id: Models.tex 1316 2007-09-25 03:18:30Z ksteube $
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%           Copyright 2003-2007 by ACceSS MNRF
%       Copyright 2007 by University of Queensland
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%                http://esscc.uq.edu.au
%        Primary Business: Queensland, Australia
%  Licensed under the Open Software License version 3.0
%     http://www.opensource.org/licenses/osl-3.0.php
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\chapter{Models}

The following sections give a breif overview of the model classes and their corresponding methods.

\section{Stokes Cartesian (Saddle Point Problem)}

\subsection{Description}

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 $Q$:

\begin{equation}
\left[ \begin{array}{cc}
A     & B \\
b^{*} & 0 \\
\end{array} \right]
\left[ \begin{array}{c}
u \\
p \\
\end{array} \right]
=\left[ \begin{array}{c}
f \\
g \\
\end{array} \right]
\label{SADDLEPOINT}
\end{equation}

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}.

The Uzawa scheme scheme is used to solve the momentum equation with the secondary condition of incompressibility \cite{GROSS2006,AAMIRBERKYAN2008}.

\begin{classdesc}{StokesProblemCartesian}{domain,debug}
opens the stokes equations on the \Domain domain. Setting debug=True switches the debug mode to on.
\end{classdesc}

example usage:

solution=StokesProblemCartesian(mesh) \\
solution.setTolerance(TOL) \\
solution.initialize(fixed\_u\_mask=b\_c,eta=eta,f=Y) \\
velocity,pressure=solution.solve(velocity,pressure,max\_iter=max\_iter,solver=solver) \\

\subsection{Benchmark Problem}

Convection problem


\section{Temperature Cartesian}

\begin{equation}
\rho c\hackscore{p} \left (\frac{\partial T}{\partial t} + \vec{v} \cdot \nabla T \right ) = k \nabla^{2}T
\label{HEAT EQUATION}
\end{equation}

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.

\subsection{Description}

\subsection{Method}

\begin{classdesc}{TemperatureCartesian}{dom,theta=THETA,useSUPG=SUPG}
\end{classdesc}

\subsection{Benchmark Problem}


\section{Level Set Method}

\subsection{Description}

\subsection{Method}

Advection and Reinitialisation

\begin{classdesc}{LevelSet}{mesh, func\_new, reinit\_max, reinit\_each, tolerance, smooth}
\end{classdesc}

%example usage:

%levelset = LevelSet(mesh, func\_new, reinit\_max, reinit\_each, tolerance, smooth)

\begin{methoddesc}[LevelSet]{update\_parameter}{parameter}
Update the parameter.
\end{methoddesc}

\begin{methoddesc}[LevelSet]{update\_phi}{paramter}{velocity}{dt}{t\_step}
Update level set function; advection and reinitialization
\end{methoddesc}

\subsection{Benchmark Problem}

Rayleigh-Taylor instability problem


\section{Drucker Prager Model}

\section{Plate Mantel}
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}