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modification on LinearPDE class and a first version of Transport class
1
2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
3 %
4 % Copyright (c) 2003-2008 by University of Queensland
5 % Earth Systems Science Computational Center (ESSCC)
6 % http://www.uq.edu.au/esscc
7 %
8 % 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 %
12 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
13
14
15 \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 \rho c\hackscore{p} \left (\frac{\partial T}{\partial t} + \vec{v} \cdot \nabla T \right ) = k \nabla^{2}T
65 \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 \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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