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

Revision 1709 - (show annotations)
Fri Aug 15 01:09:45 2008 UTC (13 years, 10 months ago) by lgraham
File MIME type: application/x-tex
File size: 3452 byte(s)
added separate section for Einstein notation


 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 10 % Primary Business: Queensland, Australia 11 % Licensed under the Open Software License version 3.0 12 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}