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

Revision 3301 - (hide annotations)
Mon Oct 25 01:27:54 2010 UTC (8 years, 11 months ago) by jfenwick
File MIME type: application/x-tex
File size: 10385 byte(s)
Modified layout of descriptions.
Fixed some overfull hboxes

 1 gross 2960 \section{Darcy Flux} 2 \label{DARCY FLUX} 3 We want to calculate the velocity $u$ and pressure $p$ on a domain $\Omega$ solving 4 the Darcy flux problem \index{Darcy flux}\index{Darcy flow} 5 \begin{equation}\label{DARCY PROBLEM} 6 \begin{array}{rcl} 7 jfenwick 3295 u_{i} + \kappa_{ij} p_{,j} & = & g_{i} \\ 8 u_{k,k} & = & f 9 gross 2960 \end{array} 10 \end{equation} 11 with the boundary conditions 12 \begin{equation}\label{DARCY BOUNDARY} 13 \begin{array}{rcl} 14 jfenwick 3295 u_{i} \; n_{i} = u^{N}_{i} \; n_{i} & \mbox{ on } & \Gamma_{N} \\ 15 p = p^{D} & \mbox{ on } & \Gamma_{D} \\ 16 gross 2960 \end{array} 17 \end{equation} 18 jfenwick 3295 where $\Gamma_{N}$ and $\Gamma_{D}$ are a partition of the boundary of $\Omega$ with $\Gamma_{D}$ non empty, $n_{i}$ is the outer normal field of the boundary of $\Omega$, $u^{N}_{i}$ and $p^{D}$ are given functions on $\Omega$, $g_{i}$ and $f$ are given source terms and $\kappa_{ij}$ is the given permeability. We assume that $\kappa_{ij}$ is symmetric (which is not really required) and positive definite, i.e there are positive constants $\alpha_{0}$ and $\alpha_{1}$ which are independent from the location in $\Omega$ such that 19 gross 2960 \begin{equation} 20 jfenwick 3295 \alpha_{0} \; x_{i} x_{i} \le \kappa_{ij} x_{i} x_{j} \le \alpha_{1} \; x_{i} x_{i} 21 gross 2960 \end{equation} 22 jfenwick 3295 for all $x_{i}$. 23 gross 2960 24 \subsection{Solution Method \label{DARCY SOLVE}} 25 It is useful to write equation~\ref{DARCY PROBLEM} in operator form. For any pressure $p$ 26 we set 27 \begin{equation} 28 jfenwick 3295 (Gp)_{i} = p_{,j} 29 gross 2960 \end{equation} 30 and a velocity $v$ we set 31 \begin{equation} 32 jfenwick 3295 Dv = v_{k,k} 33 gross 2960 \end{equation} 34 jfenwick 3295 We $K=(\kappa_{ij})$ we can write the Darcy problem~\ref{DARCY PROBLEM} as 35 gross 2960 \begin{equation} 36 \begin{array}{rcl} 37 u + K \, Gp & = & g \\ 38 Du & = & f 39 \end{array} 40 \end{equation} 41 We solve this equation by minimising the functional 42 \begin{equation}\label{DARCY COST} 43 jfenwick 3295 J(u,p):=\|K^{-\frac{1}{2}}(u + K \, G p - g)\|^2_{0} + \|\lambda (Du-f) \|_{0}^2 44 gross 2960 \end{equation} 45 jfenwick 3295 over all suitable $u$ and $p$. In this equation we set $\|p\|^2_{0}=(p,p)_{0}$ with 46 gross 2960 \begin{equation} 47 jfenwick 3295 (p,q)_{0} = \int_{\Omega } p\cdot q \, dx 48 gross 2960 \end{equation} 49 The factor $\lambda>0$ is a weighting factor. 50 A simple calculation shows that 51 one has to solve 52 \begin{equation} 53 jfenwick 3295 ( K^{-1} (v + K \, Gq) , u +K \, G p - g)_{0} + (\lambda Dv,\lambda (Du-f) )_{0} =0 54 gross 2960 \end{equation} 55 for all velocities $v$ and pressure $q$ which fulfill the homogeneous boundary conditions~\ref{DARCY BOUNDARY}. 56 This so-called variational equation can be translated back into operator notation 57 \begin{equation} 58 \begin{array}{rcl} 59 (K^{-1}+ D^*\lambda^2 D)u + Gp & = & D^*\lambda f + K^{-1} g \\ 60 G^*u + G^*K \, G p & = & G^*g \\ 61 \end{array} 62 \end{equation} 63 jfenwick 3295 where $D^*$ and $Q^*$ denote the adjoint operators with respect to $(.,.)_{0}$. 64 gross 2960 In~\cite{LEASTSQUARESFEM1994} it has been shown that this problem is continuous and coercive and therefore has a unique solution. Also standard FEM methods can be used for discretization. It is also possible 65 jfenwick 3295 to solve the problem in coupled form, however this approach leads in some cases to a very ill-conditioned stiffness matrix in particular in the case of a very small or large permeability ($\alpha_{1} \ll 1$ or $\alpha_{0} \gg 1$) 66 gross 2960 67 The approach we are taking is to eliminate the velocity $v$ from the problem. Assuming that $p$ is known we have 68 \begin{equation}\label{DARCY V FORM} 69 v= (K^{-1}+ D^*\lambda^2 D)^{-1} ( D^*\lambda f + K^{-1} g - Gp) 70 \end{equation} 71 (notice that $K^{-1}+\lambda D^*D$ is coercive) which is inserted into the second equation 72 \begin{equation} 73 G^* (K^{-1}+\lambda D^*D)^{-1} (\lambda D^*f + K^{-1} g - Gp) + G^* KG p = G^*g 74 \end{equation} 75 which is 76 \begin{equation} 77 G^* ( K - (K^{-1}+ D^*\lambda^2 D)^{-1} ) G p = G^* (g-(K^{-1}+D^*\lambda^2 D)^{-1} ( D^*\lambda f + K^{-1} g) ) 78 \end{equation} 79 We use the PCG \index{linear solver!PCG}\index{PCG} method to solve this. 80 The residual $r$ is given as 81 \begin{equation} 82 \begin{array}{rcl} 83 r & =& G^* \left( g - K\, G p - v \right) 84 \end{array} 85 \end{equation} 86 for the current pressure approximation $p$ and current velocity $v$ defined by 87 equation~\ref{DARCY V FORM}. 88 So in a particular implementation we use $\hat{r}=g-K\, Gp-v$ to represent the residual. 89 The evaluation of the iteration operator for a given $p$ is then 90 returning $Qp+v$ where $v$ is the solution of 91 \begin{equation}\label{UPDATE W} 92 (K^{-1}+ D^*\lambda^2 D)v = Gp 93 \end{equation} 94 To derive a preconditioner we use the identity 95 \begin{equation} 96 \begin{array}{rcl} 97 98 G^* ( K - (K^{-1}+ D^*\lambda^2 D)^{-1} ) G & = & G^* (I - (I + K D^*\lambda^2 D)^{-1}) K G \\ 99 & \approx & G^* (K D^*\lambda^2 D) K G \\ 100 & = & G^* K ( D^* \lambda^2 D) K G \\ 101 & \approx & G^* \frac{\lambda^2}{dx^2} K^2 G 102 \end{array} 103 \end{equation} 104 where $dx$ is the local mesh size and we use the approximation 105 \begin{equation} 106 D^* \lambda^2 D \approx \frac{\lambda^2}{dx^2} 107 \end{equation} 108 Therefore we use $G^* \frac{\lambda^2}{dx^2} K^2 G$ as a preconditioner.To evalute the preconditioner 109 we need to solve the equation 110 \begin{equation}\label{UPDATE P} 111 G^* \frac{\lambda^2}{dx^2} K^2 G p = G^* \hat{r} 112 \end{equation} 113 It remains to answer the question how to choose $\lambda$. We need to balance the first and second 114 term in $J(u,p)$ in equation~\ref{DARCY COST}. We inspect $J$ for 115 $(\hat{u}, \hat{p})$ which is a perturbed exact solution $(u,p)$. 116 jfenwick 3295 Assuming $\hat{u}=u+u_{0}e^{ik^tx}$ 117 and $\hat{p}=p+p_{0}e^{ik^tx}$ and constant $K$ we get 118 gross 2960 \begin{equation} 119 jfenwick 3295 J(\hat{u},\hat{p}) = C \left[ ( \|K^{-1}\|_{2} |u_{0}|^2 + \|K\|_{2} \|k\|_{2}^2 |p_{0}|^2| ) 120 + \lambda^2 \|k\|_{2}^2 |u_{0}|^2 \right] 121 gross 2960 \end{equation} 122 with some constant $C>0$. The first two terms and the third term correspond to the first term and 123 jfenwick 3295 second term in the definition of $J(u,p)$ in equation~\ref{DARCY COST}. For small $\|k\|_{2}$ 124 (i.e. for a smooth perturbation) $J(\hat{u},\hat{p})$ is dominated by $\|K^{-1}\|_{2} |u_{0}|^2$. 125 gross 2960 To scale the second term which is corresponds to the incompressibility condition for the velocity 126 jfenwick 3295 we need to meet the condition $\|K^{-1}\|_{2} = \lambda^2 \|k\|_{2}^2$. 127 Taking the boundary conditions into consideration the smallest possible value for $\|k\|_{2} = \frac{\pi}{l}$ where $l$ is the longest edge of the domain. This leads to the 128 gross 2960 \begin{equation}\label{DARCY LAMBDA} 129 jfenwick 3295 \lambda = \|K^{-1}\|_{2}^{\frac{1}{2}} \frac{l}{\pi} 130 gross 2960 \end{equation} 131 Notice that with this setting the preconditioner $G^* \frac{\lambda^2}{dx^2} K^2 G$ becomes 132 equivalent to $G^* K G$ if $K$ is a diagonal matrix and the mesh has a constant mesh size. 133 134 The residual norm used in the PCG is given as 135 \begin{equation}\label{DARCY R NORM} 136 jfenwick 3295 \|r\|_{PCG}^2 = \int r (G^* \frac{\lambda^2}{dx^2} K^2 G)^{-1} r \; dx =\int \hat{r} G ( G^* \frac{\lambda^2}{dx^2} K^2 G)^{-1} G^* \hat{r} \; dx \approx 137 gross 2960 \int \hat{r} K^{-1} \hat{r} \; dx 138 \end{equation} 139 The iteration is terminated if 140 \begin{equation}\label{DARCY STOP} 141 jfenwick 3295 \|r\|_{PCG} \le \mbox{ATOL} 142 gross 2960 \end{equation} 143 where we set 144 \begin{equation}\label{DARCY ATOL DEF} 145 jfenwick 3295 \mbox{ATOL} = \mbox{atol} + \mbox{rtol} \cdot \left(\frac{1}{\|K^{-\frac{1}{2}}v\|_{0}} + \frac{1}{\|K^{\frac{1}{2}} G p\|_{0}} \right)^{-1} 146 gross 2960 \end{equation} 147 where rtol is a given relative tolerance and $\mbox{atol}$ is a given absolute tolerance (typically $=0$). 148 Notice that if $Gp$ and $v$ both are zero, the pair $(0,p)$ is a solution. 149 The problem is that ATOL is depending on the solution $p$ and $v$ calculated form~\ref{DARCY V FORM}. 150 In practice one use the initial guess for $p$ 151 to get a first value for ATOL. If the stopping criterion is met in the PCG iteration, a new $v$ is calculated from the current pressure approximation and ATOL is recalculated. If \ref{DARCY STOP} is still fulfilled the calculation is terminated and $(v,p)$ is returned. Otherwise PCG is restarted with a new ATOL. 152 153 \subsection{Functions} 154 \begin{classdesc}{DarcyFlow}{domain} 155 opens the Darcy flux problem\index{Darcy flux} on the \Domain domain. 156 \end{classdesc} 157 158 \begin{methoddesc}[DarcyFlow]{setValue}{\optional{f=None, \optional{g=None, \optional{location_of_fixed_pressure=None, \optional{location_of_fixed_flux=None, 159 \\\optional{permeability=None}}}}}} 160 assigns values to the model parameters. Values can be assigned using various calls - in particular 161 in a time dependent problem only values that change over time needs to be reset. The permeability can be defined as scalar (isotropic), a vector (orthotropic) or a matrix (anisotropic). 162 \var{f} and \var{g} are the corresponding parameters in~\ref{DARCY PROBLEM}. 163 The locations and components where the flux is prescribed are set by positive values in 164 \var{location_of_fixed_flux}. 165 The locations where the pressure is prescribed are set by 166 by positive values of \var{location_of_fixed_pressure}. 167 The values of the pressure and flux are defined by the initial guess. 168 Notice that at any point on the boundary of the domain the pressure or the normal component of 169 the flux must be defined. There must be at least one point where the pressure is prescribed. 170 The method will try to cast the given values to appropriate 171 \Data class objects. 172 \end{methoddesc} 173 174 \begin{methoddesc}[DarcyFlow]{setTolerance}{\optional{rtol=1e-4}} 175 sets the relative tolerance \mbox{rtol} in \ref{DARCY ATOL DEF}. 176 \end{methoddesc} 177 178 \begin{methoddesc}[DarcyFlow]{setAbsoluteTolerance}{\optional{atol=0.}} 179 sets the absolute tolerance \mbox{atol} in \ref{DARCY ATOL DEF}. 180 \end{methoddesc} 181 182 \begin{methoddesc}[DarcyFlow]{getSolverOptionsFlux}{} 183 jfenwick 3301 Returns the solver options used to solve the flux problems~(\ref{DARCY V FORM}) and~(\ref{UPDATE W}). Use this 184 \SolverOptions object to control the solution algorithms. 185 gross 2960 \end{methoddesc} 186 187 \begin{methoddesc}[DarcyFlow]{getSolverOptionsPressure}{} 188 Returns the solver options used to solve the pressure problems~(\ref{UPDATE P}) as a preconditioner. 189 jfenwick 3301 Use this \SolverOptions object to control the solution algorithms. 190 gross 2960 \end{methoddesc} 191 192 \begin{methoddesc}[DarcyFlow]{solve}{u0,p0, \optional{max_iter=100, \optional{verbose=False}}} 193 solves the problem. and returns approximations for the flux $v$ and the pressure $p$. 194 \var{u0} and \var{p0} define initial guess for flux and pressure. Values marked 195 by positive values \var{location_of_fixed_flux} and \var{location_of_fixed_pressure}, respectively, are kept unchanged. \var{max_iter} sets the maximum number of iterations steps allowed for solving the coupled problem. 196 \end{methoddesc} 197 198 199 \subsection{Example: Gravity Flow} 200 later