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

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 1 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3 % 4 % Copyright (c) 2003-2010 by University of Queensland 5 % Earth Systems Science Computational Center (ESSCC) 6 7 % 8 % Primary Business: Queensland, Australia 9 % Licensed under the Open Software License version 3.0 10 11 % 12 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 13 14 \section{Darcy Flux} 15 \label{DARCY FLUX} 16 We want to calculate the flux $u$ and pressure $p$ on a domain $\Omega$ 17 solving the Darcy flux problem\index{Darcy flux}\index{Darcy flow} 18 \begin{equation}\label{DARCY PROBLEM} 19 \begin{array}{rcl} 20 u_{i} + \kappa_{ij} p_{,j} & = & g_{i} \\ 21 u_{k,k} & = & f 22 \end{array} 23 \end{equation} 24 with the boundary conditions 25 \begin{equation}\label{DARCY BOUNDARY} 26 \begin{array}{rcl} 27 u_{i} \; n_{i} = u^{N}_{i} \; n_{i} & \mbox{ on } & \Gamma_{N} \\ 28 p = p^{D} & \mbox{ on } & \Gamma_{D} \\ 29 \end{array} 30 \end{equation} 31 where $\Gamma_{N}$ and $\Gamma_{D}$ are a partition of the boundary of 32 $\Omega$ with $\Gamma_{D}$ non-empty, $n_{i}$ is the outer normal field of the 33 boundary of $\Omega$, $u^{N}_{i}$ and $p^{D}$ are given functions on $\Omega$, 34 $g_{i}$ and $f$ are given source terms and $\kappa_{ij}$ is the given 35 permeability. 36 We assume that $\kappa_{ij}$ is symmetric (which is not really required) and 37 positive definite, i.e. there are positive constants $\alpha_{0}$ and 38 $\alpha_{1}$ which are independent from the location in $\Omega$ such that 39 \begin{equation} 40 \alpha_{0} \; x_{i} x_{i} \le \kappa_{ij} x_{i} x_{j} \le \alpha_{1} \; x_{i} x_{i} 41 \end{equation} 42 for all $x_{i}$. 43 44 45 \subsection{Solution Method \label{DARCY SOLVE}} 46 Unfortunate equation~\ref{DARCY PROBLEM} can not solved directly in an easy way and requires mixed FEM. 47 We consider a few options to solve equation~\ref{DARCY PROBLEM} 48 \subsubsection{Evaluation}\label{SEC DARCY SIMPLE} 49 The first equation of equation~\ref{DARCY PROBLEM} is inserted into the second one: 50 \begin{equation}\label{DARCY PROBLEM SIMPLE} 51 - (\kappa_{ij} p_{,j})_{,i} = f - (g_{i})_{,i} 52 \end{equation} 53 54 with boundary conditions 55 \begin{equation}\label{DARCY BOUNDARY SIMPLE} 56 \begin{array}{rcl} 57 \kappa_{ij} p_{,j} \; n_{i} = ( g_{i} - u^{N}_{i} ) \; n_{i} & \mbox{ on } & \Gamma_{N} \\ 58 p = p^{D} & \mbox{ on } & \Gamma_{D} \\ 59 \end{array} 60 \end{equation} 61 Then the flux field is recovered by directly setting 62 \begin{equation}\label{DARCY PROBLEM SIMPLE FLUX} 63 u_{j} = g_j - \kappa_{ij} p_{,j} 64 \end{equation} 65 This simple recovery process will not ensure that the (numerically) calculated flux 66 meets the boundary conditions for flux or the incompressibility condition. 67 However this is a very fast way of calculating the flux. 68 69 70 \subsubsection{Global Postprocessing \label{SEC DARCY POST}} 71 An improved flux recovery can be achieved by solving a modified version of equation~\ref{DARCY PROBLEM SIMPLE FLUX} 72 adding the the gradient of the divergence of the flux: 73 \begin{equation}\label{DARCY PROBLEM POST FLUX} 74 \kappa^{-1}_{ij} u_{j} - 75 (\lambda \cdot u_{k,k} )_{,i}= 76 \kappa^{-1}_{ij} g_j- p_{,i} 77 - (\lambda \cdot f )_{,i} 78 \end{equation} 79 where 80 \begin{equation}\label{DARCY PROBLEM POST FLUX A} 81 \lambda = \omega \cdot |\kappa^{-1}| \cdot vol(\Omega)^{1/d} \cdot h 82 \end{equation} 83 with a non-negative factor $\omega$, $d$ is the spatial dimension and $h$ is the local element size. 84 \begin{equation}\label{DARCY PROBLEM POST FLUX BOUNDARY} 85 \begin{array}{rcl} 86 u_{i} \; n_{i} = u^{N}_{i} \; n_{i} & \mbox{ on } & \Gamma_{N} \\ 87 u_{k,k} = f & \mbox{ on } & \Gamma_{D} \\ 88 \end{array} 89 \end{equation} 90 Notice that the second condition is a natural boundary condition. 91 Global post-processing is more expense than direct pressure evaluation 92 however the flux is more accurate and asymptotic incompressibility 93 for mesh size towards zero can be shown, if $\omega>0$. 94 95 96 \subsection{Functions} 97 \begin{classdesc}{DarcyFlow}{domain, \optional{w=1., \optional{solver=\member{DarcyFlow.POST}, \optional{ 98 useReduced=\True, \optional{ verbose=\True} } }}} 99 opens the Darcy flux problem\index{Darcy flux} on the \Domain domain. 100 Reduced approximations for pressure and flux are used if \var{useReduced} is set. 101 Argument \var{solver} defines the solver method. 102 If \var{verbose} is set some information are printed. 103 \var{w} defines the weighting factor $\omega$ for global post-processing of the flux (see equation~\ref{DARCY PROBLEM POST FLUX A}.) 104 \end{classdesc} 105 106 \begin{memberdesc}[DarcyFlow]{EVAL} 107 flux is calculated directly from pressure evaluation, see section~\ref{SEC DARCY SIMPLE}. 108 \end{memberdesc} 109 110 \begin{memberdesc}[DarcyFlow]{SMOOTH} 111 solver using global post-processing of flux with weighting factor $\omega=0$, see section~\ref{SEC DARCY POST}. 112 \end{memberdesc} 113 114 \begin{memberdesc}[DarcyFlow]{POST} 115 solver using global post-processing of flux, see section~\ref{SEC DARCY POST}. 116 \end{memberdesc} 117 118 \begin{methoddesc}[DarcyFlow]{setValue}{\optional{f=None, \optional{g=None, \optional{location_of_fixed_pressure=None, \optional{location_of_fixed_flux=None, 119 \\\optional{permeability=None}}}}}} 120 assigns values to the model parameters. Values can be assigned using various 121 calls -- in particular in a time dependent problem only values that change 122 over time need to be reset. The permeability can be defined as a scalar 123 (isotropic), or a symmetric matrix (anisotropic). 124 \var{f} and \var{g} are the corresponding parameters in~\ref{DARCY PROBLEM}. 125 The locations and components where the flux is prescribed are set by positive 126 values in \var{location_of_fixed_flux}. 127 The locations where the pressure is prescribed are set by by positive values 128 of \var{location_of_fixed_pressure}. 129 The values of the pressure and flux are defined by the initial guess. 130 Notice that at any point on the boundary of the domain the pressure or the 131 normal component of the flux must be defined. There must be at least one point 132 where the pressure is prescribed. 133 The method will try to cast the given values to appropriate \Data class objects. 134 \end{methoddesc} 135 136 \begin{methoddesc}[DarcyFlow]{getSolverOptionsFlux}{} 137 returns the solver options used to solve the flux problems. 138 Use this \SolverOptions object to control the solution algorithms. 139 This option is only relevant if global postprocesing is used. 140 \end{methoddesc} 141 142 \begin{methoddesc}[DarcyFlow]{getSolverOptionsPressure}{} 143 returns a \SolverOptions object with the options used to solve the pressure 144 problems. 145 Use this object to control the solution algorithms. 146 \end{methoddesc} 147 148 \begin{methoddesc}[DarcyFlow]{solve}{u0,p0} 149 solves the problem and returns approximations for the flux $v$ and the pressure $p$. 150 \var{u0} and \var{p0} define initial guesses for flux and pressure. 151 Values marked by positive values \var{location_of_fixed_flux} and 152 \var{location_of_fixed_pressure}, respectively, are kept unchanged. 153 \end{methoddesc} 154 155 \begin{methoddesc}[DarcyFlow]{getFlux}{p, \optional{ u0 = None}} 156 returns the flux for a given pressure \var{p} where the flux is equal to \var{u0} 157 on locations where \var{location_of_fixed_flux} is positive, see \member{setValue}. 158 Notice that \var{g} and \var{f} are used. 159 \end{methoddesc} 160 161 \begin{figure} 162 \centerline{\includegraphics[width=\figwidth]{darcy_result}} 163 \caption{Flux and pressure field of the Dary flow example.} 164 \label{DIFFUSION FIG 1} 165 \end{figure} 166 167 \subsection{Example: Gravity Flow} 168 The following script \file{darcy.py}\index{scripts!\file{darcy.py.py}}\index{Darcy flow} 169 which is available in the \ExampleDirectory illustrates the usage of the 170 \class{DarcyFlow} class: 171 \begin{python} 172 from esys.escript import * 173 from esys.escript.models import DarcyFlow 174 from esys.finley import Rectangle 175 from esys.weipa import saveVTK 176 mydomain = Rectangle(l0=2.,l1=1.,n0=40, n1=20) 177 x = mydomain.getX() 178 p_BC=whereZero(x-1.)*wherePositive(x-1.) 179 u_BC=(whereZero(x)+whereZero(x-2.)) * [1.,0.] + \ 180 (whereZero(x) + whereZero(x-1.)*whereNonPositive(x-1.0)) * [0., 1.] 181 mypde = DarcyFlow(domain=mydomain) 182 mypde.setValue(g=[0., 2], 183 location_of_fixed_pressure=p_BC, 184 location_of_fixed_flux=u_BC, 185 permeability=100.) 186 187 u,p=mypde.solve(u0=x*[0., -1.], p0=0) 188 saveVTK("u.vtu",flux=u, pressure=p) 189 \end{python} 190 In the example the pressure is fixed to the initial presure \var{p0} on the right half of the top face. 191 The normal flux is set on all other faces. The corresponding values for the flux are set by the initial value 192 \var{u0}.