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

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that does not look to bad now although more stuff could be added.

 1 % $Id$ 2 % 3 % Copyright © 2006 by ACcESS MNRF 4 % \url{http://www.access.edu.au 5 % Primary Business: Queensland, Australia. 6 % Licensed under the Open Software License version 3.0 7 8 % 9 10 11 \chapter{The module \linearPDEs} 12 13 \declaremodule{extension}{linearPDEs} \modulesynopsis{Linear partial pifferential equation handler} 14 The module \linearPDEs provides an interface to define and solve linear partial 15 differential equations within \escript. \linearPDEs does not provide any 16 solver capabilities in itself but hands the PDE over to 17 the PDE solver library defined through the \Domain of the PDE. 18 The general interface is provided through the \LinearPDE class. The 19 \AdvectivePDE which is derived from the \LinearPDE class 20 provides an interface to PDE dominated by its advective terms. The \Poisson 21 class which is also derived form the \LinearPDE class should be used 22 to define the Poisson equation \index{Poisson}. 23 24 \section{\LinearPDE Class} 25 \label{SEC LinearPDE} 26 27 The \LinearPDE class is used to define a general linear, steady, second order PDE 28 for an unknown function $u$ on a given $\Omega$ defined through a \Domain object. 29 In the following $\Gamma$ denotes the boundary of the domain $\Omega$. $n$ denotes 30 the outer normal field on $\Gamma$. 31 32 For a single PDE with a solution with a single component the linear PDE is defined in the 33 following form: 34 \begin{equation}\label{LINEARPDE.SINGLE.1} 35 -(A\hackscore{jl} u\hackscore{,l})\hackscore{,j}-(B\hackscore{j} u)\hackscore{,j}+C\hackscore{l} u\hackscore{,l}+D u =-X\hackscore{j,j}+Y \; . 36 \end{equation} 37 $u_{,j}$ denotes the derivative of $u$ with respect to the $j$-th spatial direction. Einstein's summation convention, ie. summation over indexes appearing twice in a term of a sum is performed, is used. 38 The coefficients $A$, $B$, $C$, $D$, $X$ and $Y$ have to be specified through \Data objects in the 39 \Function on the PDE or objects that can be converted into such \Data objects. 40 $A$ is a \RankTwo, $B$, $C$ and $X$ are \RankOne and $D$ and $Y$ are scalar. 41 The following natural 42 boundary conditions are considered \index{boundary condition!natural} on $\Gamma$: 43 \begin{equation}\label{LINEARPDE.SINGLE.2} 44 n\hackscore{j}(A\hackscore{jl} u\hackscore{,l}+B\hackscore{j} u)+d u=n\hackscore{j}X\hackscore{j} + y \;. 45 \end{equation} 46 Notice that the coefficients $A$, $B$ and $X$ are defined in the PDE. The coefficients $d$ and $y$ are 47 each a \Scalar in the \FunctionOnBoundary. Constraints \index{constraint} for the solution prescribing the value of the 48 solution at certain locations in the domain. They have the form 49 \begin{equation}\label{LINEARPDE.SINGLE.3} 50 u=r \mbox{ where } q>0 51 \end{equation} 52 $r$ and $q$ are each \Scalar where $q$ is the characteristic function 53 \index{characteristic function} defining where the constraint is applied. 54 The constraints defined by \eqn{LINEARPDE.SINGLE.3} override any other condition set by \eqn{LINEARPDE.SINGLE.1} 55 or \eqn{LINEARPDE.SINGLE.2}. 56 57 For a system of PDEs and a solution with several components the PDE has the form 58 \begin{equation}\label{LINEARPDE.SYSTEM.1} 59 -(A\hackscore{ijkl} u\hackscore{k,l})\hackscore{,j}-(B\hackscore{ijk} u\hackscore{k})\hackscore{,j}+C\hackscore{ikl} u\hackscore{k,l}+D\hackscore{ik} u\hackscore{k} =-X\hackscore{ij,j}+Y\hackscore{i} \; . 60 \end{equation} 61 $A$ is a \RankFour, $B$ and $C$ are each a \RankThree, $D$ and $X$ are each a \RankTwo and $Y$ is a \RankOne. 62 The natural boundary conditions \index{boundary condition!natural} take the form: 63 \begin{equation}\label{LINEARPDE.SYSTEM.2} 64 n\hackscore{j}(A\hackscore{ijkl} u\hackscore{k,l}+B\hackscore{ijk} u\hackscore{k})+d\hackscore{ik} u\hackscore{k}=n\hackscore{j}X\hackscore{ij}+y\hackscore{i} \;. 65 \end{equation} 66 The coefficient $d$ is a \RankTwo and $y$ is a 67 \RankOne both in the \FunctionOnBoundary. Constraints \index{constraint} take the form 68 \begin{equation}\label{LINEARPDE.SYSTEM.3} 69 u\hackscore{i}=r\hackscore{i} \mbox{ where } q\hackscore{i}>0 70 \end{equation} 71 $r$ and $q$ are each \RankOne. Notice that not necessarily all components must 72 have a constraint at all locations. 73 74 \LinearPDE also supports solution discontinuities \index{discontinuity} over contact region $\Gamma^{contact}$ 75 in the domain $\Omega$. To specify the conditions across the discontinuity we are using the 76 generalised flux $J$ which is in the case of a systems of PDEs and several components of the solution 77 defined as 78 \begin{equation}\label{LINEARPDE.SYSTEM.5} 79 J\hackscore{ij}=A\hackscore{ijkl}u\hackscore{k,l}+B\hackscore{ijk}u\hackscore{k}-X\hackscore{ij} 80 \end{equation} 81 For the case of single solution component and single PDE $J$ is defined 82 \begin{equation}\label{LINEARPDE.SINGLE.5} 83 J\hackscore{j}=A\hackscore{jl}u\hackscore{,l}+B\hackscore{j}u\hackscore{k}-X\hackscore{j} 84 \end{equation} 85 In the context of discontinuities \index{discontinuity} $n$ denotes the normal on the 86 discontinuity pointing from side 0 towards side 1. For a system of PDEs 87 the contact condition takes the form 88 \begin{equation}\label{LINEARPDE.SYSTEM.6} 89 n\hackscore{j} J^{0}\hackscore{ij}=n\hackscore{j} J^{1}\hackscore{ij}=y^{contact}\hackscore{i} - d^{contact}\hackscore{ik} [u]\hackscore{k} \; . 90 \end{equation} 91 where $J^{0}$ and $J^{1}$ are the fluxes on side $0$ and side $1$ of the 92 discontinuity $\Gamma^{contact}$, respectively. $[u]$, which is the difference 93 of the solution at side 1 and at side 0, denotes the jump of $u$ across $\Gamma^{contact}$. 94 The coefficient $d^{contact}$ is a \RankTwo and $y^{contact}$ is a 95 \RankOne both in the \FunctionOnContactZero or \FunctionOnContactOne. 96 In case of a single PDE and a single component solution the contact condition takes the form 97 \begin{equation}\label{LINEARPDE.SINGLE.6} 98 n\hackscore{j} J^{0}\hackscore{j}=n\hackscore{j} J^{1}\hackscore{j}=y^{contact} - d^{contact}[u] 99 \end{equation} 100 In this case the the coefficient $d^{contact}$ and $y^{contact}$ are eaach \Scalar 101 both in the \FunctionOnContactZero or \FunctionOnContactOne. 102 103 The PDE is symmetrical \index{symmetrical} if 104 \begin{equation}\label{LINEARPDE.SINGLE.4} 105 A\hackscore{jl}=A\hackscore{lj} \mbox{ and } B\hackscore{j}=C\hackscore{j} 106 \end{equation} 107 The system of PDEs is symmetrical \index{symmetrical} if 108 \begin{eqnarray} 109 \label{LINEARPDE.SYSTEM.4} 110 A\hackscore{ijkl}=A\hackscore{klij} \\ 111 B\hackscore{ijk}=C\hackscore{kij} \\ 112 D\hackscore{ik}=D\hackscore{ki} \\ 113 d\hackscore{ik}=d\hackscore{ki} \\ 114 d^{contact}\hackscore{ik}=d^{contact}\hackscore{ki} 115 \end{eqnarray} 116 Note that different from the scalar case~\eqn{LINEARPDE.SINGLE.4} now the coefficients $D$, $d$ abd $d^{contact}$ 117 have to be inspected. 118 119 \section{\LinearPDE class} 120 This is the general class to define a linear PDE in \escript. We list a selction of the most 121 important methods of the class only and refer to reference guide \ReferenceGuide for a complete list. 122 123 \begin{classdesc}{LinearPDE}{domain,numEquations=0,numSolutions=0} 124 opens a linear, steady, second order PDE on the \Domain \var{domain}. \var{numEquations} 125 and \var{numSolutions} gives the number of equations and the number of solutiopn components. 126 If \var{numEquations} and \var{numSolutions} is non-positive, the number of equations 127 and the number solutions, respctively, stay undefined until a coefficient is 128 defined. 129 \end{classdesc} 130 131 \begin{methoddesc}[LinearPDE]{setValue}{ 132 \optional{A}\optional{, B}, 133 \optional{, C}\optional{, D} 134 \optional{, X}\optional{, Y} 135 \optional{, d}\optional{, y} 136 \optional{, d_contact}\optional{, y_contact} 137 \optional{, q}\optional{, r}} 138 assigns new values to coefficients. By dafault all values are assumed to be zero\footnote{ 139 In fact it is assumed they are not present by assigning the value \code{escript.Data()}. The 140 can by used by the solver library to reduce computational costs. 141 } 142 If the new coefficient value is not a \Data object, it is converted into a \Data object in the 143 appropriate \FunctionSpace. 144 \end{methoddesc} 145 146 \begin{methoddesc}[LinearPDE]{getCoefficient}{name} 147 return the value assigned to coefficient \var{name}. If \var{name} is not a valid name 148 an exception is raised. 149 \end{methoddesc} 150 151 \begin{methoddesc}[LinearPDE]{getShapeOfCoefficient}{name} 152 returns the shape of coefficient \var{name} even if no value has been assigned to it. 153 \end{methoddesc} 154 155 \begin{methoddesc}[LinearPDE]{getFunctionSpaceForCoefficient}{name} 156 returns the \FunctionSpace of coefficient \var{name} even if no value has been assigned to it. 157 \end{methoddesc} 158 159 \begin{methoddesc}[LinearPDE]{setDebugOn}{} 160 switches the debug mode to on. 161 \end{methoddesc} 162 163 \begin{methoddesc}[LinearPDE]{setDebugOff}{} 164 switches the debug mode to on. 165 \end{methoddesc} 166 167 \begin{methoddesc}[LinearPDE]{isUsingLumping}{} 168 returns \True if \LUMPING is set as the solver for the system of lienar equations. 169 Otherwise \False is returned. 170 \end{methoddesc} 171 172 \begin{methoddesc}[LinearPDE]{setSolverMethod}{\optional{solver=LinearPDE.DEFAULT}\optional{, preconditioner=LinearPDE.DEFAULT}} 173 sets the solver method and preconditioner to be used. It is pointed out that a PDE solver library 174 may not know the specified solver method but may choose a similar method and preconditioner. 175 \end{methoddesc} 176 177 \begin{methoddesc}[LinearPDE]{getSolverMethodName}{} 178 returns the name of the solver method and preconditioner which is currently been used. 179 \end{methoddesc} 180 181 \begin{methoddesc}[LinearPDE]{getSolverMethod}{} 182 returns the solver method and preconditioner which is currently been used. 183 \end{methoddesc} 184 185 \begin{methoddesc}[LinearPDE]{setSolverPackage}{\optional{package=LinearPDE.DEFAULT}} 186 Set the solver package to be used by PDE library to solve the linear systems of equations. The 187 specified package may not be supported by the PDE solver library. In this case, dependng on 188 the PDE solver, the default solver is used or an exeption is thrown. 189 If \var{package} is not specified, the default package of the PDE solver library is used. 190 \end{methoddesc} 191 192 \begin{methoddesc}[LinearPDE]{getSolverPackage}{} 193 returns the linear solver package currently by the PDE solver library 194 \end{methoddesc} 195 196 197 \begin{methoddesc}[LinearPDE]{setTolerance}{\optional{tol=1.e-8}}: 198 resets the tolerance for solution. The actually meaning of tolerance is 199 depending on the underlying PDE library. In most cases, the tolerance 200 will only consider the error from solving the discerete problem but will 201 not consider any discretization error. 202 \end{methoddesc} 203 204 \begin{methoddesc}[LinearPDE]{getTolerance}{} 205 returns the current tolerance of the solution 206 \end{methoddesc} 207 208 \begin{methoddesc}[LinearPDE]{getDomain}{} 209 returns the \Domain of the PDE. 210 \end{methoddesc} 211 212 \begin{methoddesc}[LinearPDE]{getDim}{} 213 returns the spatial dimension of the PDE. 214 \end{methoddesc} 215 216 \begin{methoddesc}[LinearPDE]{getNumEquations}{} 217 returns the number of equations. 218 \end{methoddesc} 219 220 \begin{methoddesc}[LinearPDE]{getNumSolutions}{} 221 returns the number of components of the solution. 222 \end{methoddesc} 223 224 \begin{methoddesc}[LinearPDE]{checkSymmetry}{verbose=\False} 225 returns \True if the PDE is symmetric and \False otherwise. 226 The method is very computational expensive and should only be 227 called for testing purposes. The symmetry flag is not altered. 228 If \var{verbose}=\True information about where symmetry is violated 229 are printed. 230 \end{methoddesc} 231 232 \begin{methoddesc}[LinearPDE]{getFlux}{u} 233 returns the flux $J\hackscore{ij}$ \index{flux} for given solution \var{u} 234 defined by \eqn{LINEARPDE.SYSTEM.5} and \eqn{LINEARPDE.SINGLE.5}, respectively. 235 \end{methoddesc} 236 237 238 \begin{methoddesc}[LinearPDE]{isSymmetric}{} 239 returns \True if the PDE has been indicated to be symmetric. 240 Otherwise \False is returned. 241 \end{methoddesc} 242 243 \begin{methoddesc}[LinearPDE]{setSymmetryOn}{} 244 indicates that the PDE is symmetric. 245 \end{methoddesc} 246 247 \begin{methoddesc}[LinearPDE]{setSymmetryOff}{} 248 indicates that the PDE is not symmetric. 249 \end{methoddesc} 250 251 \begin{methoddesc}[LinearPDE]{setReducedOrderOn}{} 252 switches on the reduction of polynomial order for the solution and equation evaluation even if 253 a quadratic or higher interpolation order is defined in the \Domain. This feature may not 254 be supported by all PDE libraries. 255 \end{methoddesc} 256 257 \begin{methoddesc}[LinearPDE]{setReducedOrderOff}{} 258 switches off the reduction of polynomial order for the solution and 259 equation evaluation. 260 \end{methoddesc} 261 262 \begin{methoddesc}[LinearPDE]{getOperator}{} 263 returns the \Operator of the PDE. 264 \end{methoddesc} 265 266 \begin{methoddesc}[LinearPDE]{getRightHandSide}{} 267 returns the right hand side of the PDE as a \Data object. If 268 \var{ignoreConstraint}=\True the constraints are not considered 269 when building up the right hand side. 270 \end{methoddesc} 271 272 \begin{methoddesc}[LinearPDE]{getSystem}{} 273 returns the \Operator and right hand side of the PDE. 274 \end{methoddesc} 275 276 \begin{methoddesc}[LinearPDE]{getSolution}{ 277 \optional{verbose=False} 278 \optional{, reordering=LinearPDE.NO_REORDERING} 279 \optional{, iter_max=1000} 280 \optional{, drop_tolerance=0.01} 281 \optional{, drop_storage=1.20} 282 \optional{, truncation=-1} 283 \optional{, restart=-1} 284 } 285 returns (an approximation of) the solution of the PDE. If \code{verbose=\True} some information during the solution process printed. 286 \var{reordering} selects a reordering methods that is applied before or during the solution process 287 (=\NOREORDERING ,\MINIMUMFILLIN ,\NESTEDDESCTION). 288 \var{iter_max} specifies the maximum number of iteration steps that are allowed to reach the specified tolerance. 289 \var{drop_tolerance} specifies a relative tolerance for small elements to be dropped when building a preconditioner 290 (eg. in \ILUT). \var{drop_storage} limits the extra storage allowed when building a preconditioner 291 (eg. in \ILUT). The extra storage is given relative to the size of the stiffness matrix, eg. 292 \var{drop_storage=1.2} will allow the preconditioner to use the $1.2$ fold storage space than used 293 for the stiffness matrix. \var{truncation} defines the truncation. 294 \end{methoddesc} 295 296 \begin{memberdesc}[LinearPDE]{DEFAULT} 297 default method, preconditioner or package to be used to solve the PDE. An appropriate method should be 298 chosen by the used PDE solver library. 299 \end{memberdesc} 300 301 \begin{memberdesc}[LinearPDE]{SCSL} 302 the SCSL library by SGI,~\Ref{SCSL}\footnote{The SCSL library will only be available on SGI systems} 303 \end{memberdesc} 304 305 \begin{memberdesc}[LinearPDE]{MKL} 306 the MKL library by Intel,~\Ref{MKL}\footnote{The MKL library will only be available when the intel compilation environment is used.}. 307 \end{memberdesc} 308 309 \begin{memberdesc}[LinearPDE]{UMFPACK} 310 the UMFPACK,~\Ref{UMFPACK}. Remark: UMFPACK is not parallelized. 311 \end{memberdesc} 312 313 \begin{memberdesc}[LinearPDE]{PASO} 314 the solver library of \finley, see \Sec{CHAPTER ON FINLEY}. 315 \end{memberdesc} 316 317 \begin{memberdesc}[LinearPDE]{ITERATIVE} 318 the default iterative method and preconditioner. The actually used method depends on the 319 PDE solver library and the solver package been choosen. Typically, \PCG is used for symmetric PDEs 320 and \BiCGStab otherwise, both with \JACOBI preconditioner. 321 \end{memberdesc} 322 323 \begin{memberdesc}[LinearPDE]{DIRECT} 324 the default direct linear solver. 325 \end{memberdesc} 326 327 \begin{memberdesc}[LinearPDE]{CHOLEVSKY} 328 direct solver based on Cholevsky factorization (or similar), see~\Ref{Saad}. The solver will require a symmetric PDE. 329 \end{memberdesc} 330 331 \begin{memberdesc}[LinearPDE]{PCG} 332 preconditioned conjugate gradient method, see~\Ref{WEISS}\index{linear solver!PCG}\index{PCG}. The solver will require a symmetric PDE. 333 \end{memberdesc} 334 335 \begin{memberdesc}[LinearPDE]{GMRES} 336 the GMRES method, see~\Ref{WEISS}\index{linear solver!GMRES}\index{GMRES}. Truncation and restart are controlled by the parameters 337 \var{truncation} and \var{restart} of \method{getSolution}. 338 \end{memberdesc} 339 340 \begin{memberdesc}[LinearPDE]{LUMPING} 341 uses lumping to solve the system of linear equations~\index{linear solver!lumping}\index{lumping}. This solver technique 342 condenses the stiffness matrix to a diagonal matrix so the solution of the linear systems becomes very cheap. It can be used when 343 only \var{D} is present but in any case has to applied with care. The difference in the solutions with and without lumping can be significant 344 but is expect to converge to zero when the mesh gets finer. 345 Lumping does not use the linear system solver library. 346 \end{memberdesc} 347 348 \begin{memberdesc}[LinearPDE]{PRES20} 349 the GMRES method with truncation after five residuals and 350 restart after 20 steps, see~\Ref{WEISS}. 351 \end{memberdesc}[LinearPDE]{CR} 352 353 \begin{memberdesc}[LinearPDE]{CGS} 354 conjugate gradient squared method, see~\Ref{WEISS}. 355 \end{memberdesc} 356 357 \begin{memberdesc}[LinearPDE]{BICGSTAB} 358 stabilized bi-conjugate gradients methods, see~\Ref{WEISS}. 359 \end{memberdesc} 360 361 \begin{memberdesc}[LinearPDE]{SSOR} 362 symmetric successive over-relaxation method, see~\Ref{WEISS}. Typically used as preconditioner but some linear solver libraries support 363 this as a solver. 364 \end{memberdesc} 365 \begin{memberdesc}[LinearPDE]{ILU0} 366 the incomplete LU factorization preconditioner with no fill-in, see~\Ref{Saad}. 367 \end{memberdesc} 368 369 \begin{memberdesc}[LinearPDE]{ILUT} 370 the incomplete LU factorization preconditioner with fill-in, see~\Ref{Saad}. During the LU-factorization element with 371 relative size less then \var{drop_tolerance} are dropped. Moreover, the size of the LU-factorization is restricted to the 372 \var{drop_storage}-fold of the stiffness matrix. \var{drop_tolerance} and \var{drop_storage} are both set in the 373 \method{getSolution} call. 374 \end{memberdesc} 375 376 \begin{memberdesc}[LinearPDE]{JACOBI} 377 the Jacobi preconditioner, see~\Ref{Saad}. 378 \end{memberdesc} 379 380 \begin{memberdesc}[LinearPDE]{AMG} 381 the algebraic--multi grid method, see~\Ref{AMG}. This method can be used as linear solver method but is more robust when used 382 in a preconditioner. 383 \end{memberdesc} 384 385 \begin{memberdesc}[LinearPDE]{RILU} 386 recursive incomplete LU factorization preconditioner, see~\Ref{RILU}. This method is similar to \ILUT but uses smoothing 387 between levels. During the LU-factorization element with 388 relative size less then \var{drop_tolerance} are dropped. Moreover, the size of the LU-factorization is restricted to the 389 \var{drop_storage}-fold of the stiffness matrix. \var{drop_tolerance} and \var{drop_storage} are both set in the 390 \method{getSolution} call. 391 \end{memberdesc} 392 393 \begin{memberdesc}[LinearPDE]{NO_REORDERING} 394 no ordering is used during factorization. 395 \end{memberdesc} 396 397 \begin{memberdesc}[LinearPDE]{MINIMUM_FILL_IN} 398 applies reordering before factorization using a fill-in minimization strategy. You have to check with the particular solver library or 399 linear solver package if this is supported. In any case, it is advisable to apply reordering on the mesh to minimize fill-in. 400 \end{memberdesc} 401 402 \begin{memberdesc}[LinearPDE]{NESTED_DISSECTION} 403 applies reordering before factorization using a nested dissection strategy. You have to check with the particular solver library or 404 linear solver package if this is supported. In any case, it is advisable to apply reordering on the mesh to minimize fill-in. 405 \end{memberdesc} 406 407 \section{The \Poisson Class} 408 The \Poisson class provides an easy way to define and solve the Poisson 409 equation 410 \begin{equation}\label{POISSON.1} 411 -u\hackscore{,ii}=f\; . 412 \end{equation} 413 with homogeneous boundary conditions 414 \begin{equation}\label{POISSON.2} 415 n\hackscore{i}u\hackscore{,i}=0 416 \end{equation} 417 and homogeneous constraints 418 \begin{equation}\label{POISSON.3} 419 u=0 \mbox{ where } q>0 420 \end{equation} 421 $f$ has to be a \Scalar in the \Function and $q$ must be 422 a \Scalar in the \SolutionFS. 423 424 \begin{classdesc}{Poisson}{domain} 425 opens a Poisson equation on the \Domain domain. \Poisson is derived from \LinearPDE. 426 \end{classdesc} 427 \begin{methoddesc}[Poisson]{setValue}{f=escript.Data(),q=escript.Data()} 428 assigns new values to \var{f} and \var{q}. 429 \end{methoddesc} 430 431 \section{The \Helmholtz Class} 432 The \Helmholtz class defines the Helmholtz problem 433 \begin{equation}\label{HZ.1} 434 \omega \; u - (k\; u\hackscore{,j})\hackscore{,j} = f 435 \end{equation} 436 with natural boundary conditons 437 \begin{equation}\label{HZ.2} 438 k\; u\hackscore{,j} n\hackscore{,j} = g- \alpha \; u 439 \end{equation} 440 and constraints: 441 \begin{equation}\label{HZ.3} 442 u=r \mbox{ where } q>0 443 \end{equation} 444 $\omega$, $k$, $f$ have to be a \Scalar in the \Function, 445 $g$ and $\alpha$ must be a \Scalar in the \FunctionOnBoundary, 446 and $q$ and $r$ must be a \Scalar in the \SolutionFS or must be mapped or interpolated into the particular \FunctionSpace. 447 448 \begin{classdesc}{Helmholtz}{domain} 449 opens a Helmholtz equation on the \Domain domain. \Helmholtz is derived from \LinearPDE. 450 \end{classdesc} 451 \begin{methoddesc}[Helmholtz]{setValue}{ \optional{omega} \optional{, k} \optional{, f} \optional{, alpha} \optional{, g} \optional{, r} \optional{, q}} 452 assigns new values to \var{omega}, \var{k}, \var{f}, \var{alpha}, \var{g}, \var{r}, \var{q}. By default all values are set to be zero. 453 \end{methoddesc} 454 455 \section{The \Lame Class} 456 The \Lame class defines a Lame equation problem: 457 \begin{equation}\label{LE.1} 458 -\mu (u\hackscore{i,j}+u\hackscore{j,i})+\lambda u\hackscore{k,k})\hackscore{j} = F\hackscore{i}-\sigma\hackscore{ij,j} 459 \end{equation} 460 with natural boundary conditons: 461 \begin{equation}\label{LE.2} 462 n\hackscore{j}(\mu \; (u\hackscore{i,j}+u\hackscore{j,i})+\lambda*u\hackscore{k,k}) = f\hackscore{i}+n\hackscore{j}\sigma\hackscore{ij} 463 \end{equation} 464 and constraint 465 \begin{equation}\label{LE.3} 466 u\hackscore{i}=r\hackscore{i} \mbox{ where } q\hackscore{i}>0 467 \end{equation} 468 $\mu$, $\lambda$ have to be a \Scalar in the \Function, 469 $F$ has to be a \Vector in the \Function, 470 $\sigma$ has to be a \Tensor in the \Function, 471 $f$ must be a \Vector in the \FunctionOnBoundary, 472 and $q$ and $r$ must be a \Vector in the \SolutionFS or must be mapped or interpolated into the particular \FunctionSpace. 473 474 \begin{classdesc}{Lame}{domain} 475 opens a Lame equation on the \Domain domain. \Lame is derived from \LinearPDE. 476 \end{classdesc} 477 \begin{methoddesc}[Lame]{setValue}{ \optional{lame_lambda} \optional{, lame_mu} \optional{, F} \optional{, sigma} \optional{, f} \optional{, r} \optional{, q}} 478 assigns new values to 479 \var{lame_lambda}, 480 \var{lame_mu}, 481 \var{F}, 482 \var{sigma}, 483 \var{f}, 484 \var{r} and 485 \var{q} 486 By default all values are set to be zero. 487 \end{methoddesc} 488

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