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modification on LinearPDE class and a first version of Transport class
1 ksteube 1811
2     %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
3 lgraham 1702 %
4 ksteube 1811 % Copyright (c) 2003-2008 by University of Queensland
5     % Earth Systems Science Computational Center (ESSCC)
6     % http://www.uq.edu.au/esscc
7 lgraham 1702 %
8 ksteube 1811 % Primary Business: Queensland, Australia
9     % Licensed under the Open Software License version 3.0
10     % http://www.opensource.org/licenses/osl-3.0.php
11 lgraham 1702 %
12 ksteube 1811 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
13 lgraham 1702
14 ksteube 1811
15 lgraham 1702 \chapter{Models}
16    
17     The following sections give a breif overview of the model classes and their corresponding methods.
18    
19     \section{Stokes Cartesian (Saddle Point Problem)}
20    
21     \subsection{Description}
22    
23     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 $Q$:
24    
25     \begin{equation}
26     \left[ \begin{array}{cc}
27     A & B \\
28     b^{*} & 0 \\
29     \end{array} \right]
30     \left[ \begin{array}{c}
31     u \\
32     p \\
33     \end{array} \right]
34     =\left[ \begin{array}{c}
35     f \\
36     g \\
37     \end{array} \right]
38     \label{SADDLEPOINT}
39     \end{equation}
40    
41     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}.
42    
43     The Uzawa scheme scheme is used to solve the momentum equation with the secondary condition of incompressibility \cite{GROSS2006,AAMIRBERKYAN2008}.
44    
45     \begin{classdesc}{StokesProblemCartesian}{domain,debug}
46     opens the stokes equations on the \Domain domain. Setting debug=True switches the debug mode to on.
47     \end{classdesc}
48    
49     example usage:
50    
51     solution=StokesProblemCartesian(mesh) \\
52     solution.setTolerance(TOL) \\
53     solution.initialize(fixed\_u\_mask=b\_c,eta=eta,f=Y) \\
54     velocity,pressure=solution.solve(velocity,pressure,max\_iter=max\_iter,solver=solver) \\
55    
56     \subsection{Benchmark Problem}
57    
58     Convection problem
59    
60    
61     \section{Temperature Cartesian}
62    
63     \begin{equation}
64 lgraham 1709 \rho c\hackscore{p} \left (\frac{\partial T}{\partial t} + \vec{v} \cdot \nabla T \right ) = k \nabla^{2}T
65 lgraham 1702 \label{HEAT EQUATION}
66     \end{equation}
67    
68     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.
69    
70     \subsection{Description}
71    
72     \subsection{Method}
73    
74     \begin{classdesc}{TemperatureCartesian}{dom,theta=THETA,useSUPG=SUPG}
75     \end{classdesc}
76    
77     \subsection{Benchmark Problem}
78    
79    
80     \section{Level Set Method}
81    
82     \subsection{Description}
83    
84     \subsection{Method}
85    
86     Advection and Reinitialisation
87    
88     \begin{classdesc}{LevelSet}{mesh, func\_new, reinit\_max, reinit\_each, tolerance, smooth}
89     \end{classdesc}
90    
91     %example usage:
92    
93     %levelset = LevelSet(mesh, func\_new, reinit\_max, reinit\_each, tolerance, smooth)
94    
95     \begin{methoddesc}[LevelSet]{update\_parameter}{parameter}
96     Update the parameter.
97     \end{methoddesc}
98    
99     \begin{methoddesc}[LevelSet]{update\_phi}{paramter}{velocity}{dt}{t\_step}
100     Update level set function; advection and reinitialization
101     \end{methoddesc}
102    
103     \subsection{Benchmark Problem}
104    
105     Rayleigh-Taylor instability problem
106    
107    
108     \section{Drucker Prager Model}
109    
110 gross 1841 \section{Isotropic Kelvin Material \label{IKM}}
111    
112    
113    
114     \begin{equation}\label{IKM-EQU-2}
115     D_{ij}=D_{ij}^{el}+D_{ij}^{vp}
116     \end{equation}
117     with the elastic stretching
118     \begin{equation}\label{IKM-EQU-3}
119     D_{ij}^{el}=\frac{2 \mu} \sigma'_{ij}
120     \end{equation}
121     \begin{equation}\label{IKM-EQU-4}
122     D_{ij}^{vp}=\sum_{q} D_{ij}^{q}
123     \end{equation}
124     \begin{equation}\label{IKM-EQU-5}
125     D_{ij}^{q}=\frac{1}{2 \eta^{q}} \sigma'_{ij} \mbox{ with } \eta^{q}=\eta^{q}_N \left(\frac{\tau}{\tau_t^q}\right){\frac{1}{n^{q}}-1}
126     \end{equation}
127     After inserting equation~\ref{IKM-EQU-5} into equation \ref{IKM-EQU-4} one gets:
128     \begin{equation}\label{IKM-EQU-4}
129     D_{ij}^{vp}=\frac{1}{2 \eta^{vp}} \sigma'_{ij}
130     \end{equation}
131    
132    
133     \begin{equation}\label{IKM-EQU-1}
134     -\sigma'_{ij,j}+p_j=F_j
135     \end{equation}
136    
137     \begin{equation}\label{IKM-EQU-2}
138     -v_{i,i}=0
139     \end{equation}
140    
141     \begin{equation}\label{IKM-EQU-3}
142     \sigma_{ij}=\sigma'_{ij,j}-\frac{1}{d} p \delta_{ij}
143     \end{equation}

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