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

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Fri Oct 3 03:57:52 2008 UTC (11 years, 10 months ago) by gross
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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 7 lgraham 1702 % 8 ksteube 1811 % Primary Business: Queensland, Australia 9 % Licensed under the Open Software License version 3.0 10 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}