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

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A very quick edit of chapter 3 of the User Guide. More editing needed.


 1 % 2 % $Id$ 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{The Module \linearPDEs} 18 19 20 21 \section{Linear Partial Differential Equations} 22 \label{SEC LinearPDE} 23 24 The \LinearPDE class is used to define a general linear, steady, second order PDE 25 for an unknown function $u$ on a given $\Omega$ defined through a \Domain object. 26 In the following $\Gamma$ denotes the boundary of the domain $\Omega$. $n$ denotes 27 the outer normal field on $\Gamma$. 28 29 For a single PDE with a solution with a single component the linear PDE is defined in the 30 following form: 31 \begin{equation}\label{LINEARPDE.SINGLE.1} 32 -(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 \; . 33 \end{equation} 34 $u_{,j}$ 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. 35 The coefficients $A$, $B$, $C$, $D$, $X$ and $Y$ have to be specified through \Data objects in the 36 \Function on the PDE or objects that can be converted into such \Data objects. 37 $A$ is a \RankTwo, $B$, $C$ and $X$ are \RankOne and $D$ and $Y$ are scalar. 38 The following natural 39 boundary conditions are considered \index{boundary condition!natural} on $\Gamma$: 40 \begin{equation}\label{LINEARPDE.SINGLE.2} 41 n\hackscore{j}(A\hackscore{jl} u\hackscore{,l}+B\hackscore{j} u)+d u=n\hackscore{j}X\hackscore{j} + y \;. 42 \end{equation} 43 Notice that the coefficients $A$, $B$ and $X$ are defined in the PDE. The coefficients $d$ and $y$ are 44 each a \Scalar in the \FunctionOnBoundary. Constraints \index{constraint} for the solution prescribing the value of the 45 solution at certain locations in the domain. They have the form 46 \begin{equation}\label{LINEARPDE.SINGLE.3} 47 u=r \mbox{ where } q>0 48 \end{equation} 49 $r$ and $q$ are each \Scalar where $q$ is the characteristic function 50 \index{characteristic function} defining where the constraint is applied. 51 The constraints defined by \eqn{LINEARPDE.SINGLE.3} override any other condition set by \eqn{LINEARPDE.SINGLE.1} 52 or \eqn{LINEARPDE.SINGLE.2}. 53 54 For a system of PDEs and a solution with several components the PDE has the form 55 \begin{equation}\label{LINEARPDE.SYSTEM.1} 56 -(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} \; . 57 \end{equation} 58 $A$ is a \RankFour, $B$ and $C$ are each a \RankThree, $D$ and $X$ are each a \RankTwo and $Y$ is a \RankOne. 59 The natural boundary conditions \index{boundary condition!natural} take the form: 60 \begin{equation}\label{LINEARPDE.SYSTEM.2} 61 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} \;. 62 \end{equation} 63 The coefficient $d$ is a \RankTwo and $y$ is a 64 \RankOne both in the \FunctionOnBoundary. Constraints \index{constraint} take the form 65 \begin{equation}\label{LINEARPDE.SYSTEM.3} 66 u\hackscore{i}=r\hackscore{i} \mbox{ where } q\hackscore{i}>0 67 \end{equation} 68 $r$ and $q$ are each \RankOne. Notice that not necessarily all components must 69 have a constraint at all locations. 70 71 \LinearPDE also supports solution discontinuities \index{discontinuity} over contact region $\Gamma^{contact}$ 72 in the domain $\Omega$. To specify the conditions across the discontinuity we are using the 73 generalised flux $J$ which is in the case of a systems of PDEs and several components of the solution 74 defined as 75 \begin{equation}\label{LINEARPDE.SYSTEM.5} 76 J\hackscore{ij}=A\hackscore{ijkl}u\hackscore{k,l}+B\hackscore{ijk}u\hackscore{k}-X\hackscore{ij} 77 \end{equation} 78 For the case of single solution component and single PDE $J$ is defined 79 \begin{equation}\label{LINEARPDE.SINGLE.5} 80 J\hackscore{j}=A\hackscore{jl}u\hackscore{,l}+B\hackscore{j}u\hackscore{k}-X\hackscore{j} 81 \end{equation} 82 In the context of discontinuities \index{discontinuity} $n$ denotes the normal on the 83 discontinuity pointing from side 0 towards side 1. For a system of PDEs 84 the contact condition takes the form 85 \begin{equation}\label{LINEARPDE.SYSTEM.6} 86 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} \; . 87 \end{equation} 88 where $J^{0}$ and $J^{1}$ are the fluxes on side $0$ and side $1$ of the 89 discontinuity $\Gamma^{contact}$, respectively. $[u]$, which is the difference 90 of the solution at side 1 and at side 0, denotes the jump of $u$ across $\Gamma^{contact}$. 91 The coefficient $d^{contact}$ is a \RankTwo and $y^{contact}$ is a 92 \RankOne both in the \FunctionOnContactZero or \FunctionOnContactOne. 93 In case of a single PDE and a single component solution the contact condition takes the form 94 \begin{equation}\label{LINEARPDE.SINGLE.6} 95 n\hackscore{j} J^{0}\hackscore{j}=n\hackscore{j} J^{1}\hackscore{j}=y^{contact} - d^{contact}[u] 96 \end{equation} 97 In this case the the coefficient $d^{contact}$ and $y^{contact}$ are each \Scalar 98 both in the \FunctionOnContactZero or \FunctionOnContactOne. 99 100 The PDE is symmetrical \index{symmetrical} if 101 \begin{equation}\label{LINEARPDE.SINGLE.4} 102 A\hackscore{jl}=A\hackscore{lj} \mbox{ and } B\hackscore{j}=C\hackscore{j} 103 \end{equation} 104 The system of PDEs is symmetrical \index{symmetrical} if 105 \begin{eqnarray} 106 \label{LINEARPDE.SYSTEM.4} 107 A\hackscore{ijkl}=A\hackscore{klij} \\ 108 B\hackscore{ijk}=C\hackscore{kij} \\ 109 D\hackscore{ik}=D\hackscore{ki} \\ 110 d\hackscore{ik}=d\hackscore{ki} \\ 111 d^{contact}\hackscore{ik}=d^{contact}\hackscore{ki} 112 \end{eqnarray} 113 Note that different from the scalar case~\eqn{LINEARPDE.SINGLE.4} now the coefficients $D$, $d$ abd $d^{contact}$ 114 have to be inspected. 115 116 117 \subsection{Classes} 118 \declaremodule{extension}{esys.escript.linearPDEs} 119 \modulesynopsis{Linear partial differential equation handler} 120 The module \linearPDEs provides an interface to define and solve linear partial 121 differential equations within \escript. \linearPDEs does not provide any 122 solver capabilities in itself but hands the PDE over to 123 the PDE solver library defined through the \Domain of the PDE. 124 The general interface is provided through the \LinearPDE class. The 125 \AdvectivePDE which is derived from the \LinearPDE class 126 provides an interface to PDE dominated by its advective terms. The \Poisson 127 class which is also derived form the \LinearPDE class should be used 128 to define the Poisson equation \index{Poisson}. 129 130 \subsection{\LinearPDE class} 131 This is the general class to define a linear PDE in \escript. We list a selection of the most 132 important methods of the class only and refer to reference guide \ReferenceGuide for a complete list. 133 134 \begin{classdesc}{LinearPDE}{domain,numEquations=0,numSolutions=0} 135 opens a linear, steady, second order PDE on the \Domain \var{domain}. \var{numEquations} 136 and \var{numSolutions} gives the number of equations and the number of solution components. 137 If \var{numEquations} and \var{numSolutions} is non-positive, the number of equations 138 and the number solutions, respectively, stay undefined until a coefficient is 139 defined. 140 \end{classdesc} 141 142 \begin{methoddesc}[LinearPDE]{setValue}{ 143 \optional{A}\optional{, B}, 144 \optional{, C}\optional{, D} 145 \optional{, X}\optional{, Y} 146 \optional{, d}\optional{, y} 147 \optional{, d_contact}\optional{, y_contact} 148 \optional{, q}\optional{, r}} 149 assigns new values to coefficients. By default all values are assumed to be zero\footnote{ 150 In fact it is assumed they are not present by assigning the value \code{escript.Data()}. The 151 can by used by the solver library to reduce computational costs. 152 } 153 If the new coefficient value is not a \Data object, it is converted into a \Data object in the 154 appropriate \FunctionSpace. 155 \end{methoddesc} 156 157 \begin{methoddesc}[LinearPDE]{getCoefficient}{name} 158 return the value assigned to coefficient \var{name}. If \var{name} is not a valid name 159 an exception is raised. 160 \end{methoddesc} 161 162 \begin{methoddesc}[LinearPDE]{getShapeOfCoefficient}{name} 163 returns the shape of coefficient \var{name} even if no value has been assigned to it. 164 \end{methoddesc} 165 166 \begin{methoddesc}[LinearPDE]{getFunctionSpaceForCoefficient}{name} 167 returns the \FunctionSpace of coefficient \var{name} even if no value has been assigned to it. 168 \end{methoddesc} 169 170 \begin{methoddesc}[LinearPDE]{setDebugOn}{} 171 switches the debug mode to on. 172 \end{methoddesc} 173 174 \begin{methoddesc}[LinearPDE]{setDebugOff}{} 175 switches the debug mode to on. 176 \end{methoddesc} 177 178 \begin{methoddesc}[LinearPDE]{isUsingLumping}{} 179 returns \True if \LUMPING is set as the solver for the system of linear equations. 180 Otherwise \False is returned. 181 \end{methoddesc} 182 183 \begin{methoddesc}[LinearPDE]{setSolverMethod}{\optional{solver=LinearPDE.DEFAULT}\optional{, preconditioner=LinearPDE.DEFAULT}} 184 sets the solver method and preconditioner to be used. It is pointed out that a PDE solver library 185 may not know the specified solver method but may choose a similar method and preconditioner. 186 \end{methoddesc} 187 188 \begin{methoddesc}[LinearPDE]{getSolverMethodName}{} 189 returns the name of the solver method and preconditioner which is currently been used. 190 \end{methoddesc} 191 192 \begin{methoddesc}[LinearPDE]{getSolverMethod}{} 193 returns the solver method and preconditioner which is currently been used. 194 \end{methoddesc} 195 196 \begin{methoddesc}[LinearPDE]{setSolverPackage}{\optional{package=LinearPDE.DEFAULT}} 197 Set the solver package to be used by PDE library to solve the linear systems of equations. The 198 specified package may not be supported by the PDE solver library. In this case, depending on 199 the PDE solver, the default solver is used or an exception is thrown. 200 If \var{package} is not specified, the default package of the PDE solver library is used. 201 \end{methoddesc} 202 203 \begin{methoddesc}[LinearPDE]{getSolverPackage}{} 204 returns the linear solver package currently by the PDE solver library 205 \end{methoddesc} 206 207 208 \begin{methoddesc}[LinearPDE]{setTolerance}{\optional{tol=1.e-8}}: 209 resets the tolerance for solution. The actually meaning of tolerance is 210 depending on the underlying PDE library. In most cases, the tolerance 211 will only consider the error from solving the discrete problem but will 212 not consider any discretization error. 213 \end{methoddesc} 214 215 \begin{methoddesc}[LinearPDE]{getTolerance}{} 216 returns the current tolerance of the solution 217 \end{methoddesc} 218 219 \begin{methoddesc}[LinearPDE]{getDomain}{} 220 returns the \Domain of the PDE. 221 \end{methoddesc} 222 223 \begin{methoddesc}[LinearPDE]{getDim}{} 224 returns the spatial dimension of the PDE. 225 \end{methoddesc} 226 227 \begin{methoddesc}[LinearPDE]{getNumEquations}{} 228 returns the number of equations. 229 \end{methoddesc} 230 231 \begin{methoddesc}[LinearPDE]{getNumSolutions}{} 232 returns the number of components of the solution. 233 \end{methoddesc} 234 235 \begin{methoddesc}[LinearPDE]{checkSymmetry}{verbose=\False} 236 returns \True if the PDE is symmetric and \False otherwise. 237 The method is very computational expensive and should only be 238 called for testing purposes. The symmetry flag is not altered. 239 If \var{verbose}=\True information about where symmetry is violated 240 are printed. 241 \end{methoddesc} 242 243 \begin{methoddesc}[LinearPDE]{getFlux}{u} 244 returns the flux $J\hackscore{ij}$ \index{flux} for given solution \var{u} 245 defined by \eqn{LINEARPDE.SYSTEM.5} and \eqn{LINEARPDE.SINGLE.5}, respectively. 246 \end{methoddesc} 247 248 249 \begin{methoddesc}[LinearPDE]{isSymmetric}{} 250 returns \True if the PDE has been indicated to be symmetric. 251 Otherwise \False is returned. 252 \end{methoddesc} 253 254 \begin{methoddesc}[LinearPDE]{setSymmetryOn}{} 255 indicates that the PDE is symmetric. 256 \end{methoddesc} 257 258 \begin{methoddesc}[LinearPDE]{setSymmetryOff}{} 259 indicates that the PDE is not symmetric. 260 \end{methoddesc} 261 262 \begin{methoddesc}[LinearPDE]{setReducedOrderOn}{} 263 switches on the reduction of polynomial order for the solution and equation evaluation even if 264 a quadratic or higher interpolation order is defined in the \Domain. This feature may not 265 be supported by all PDE libraries. 266 \end{methoddesc} 267 268 \begin{methoddesc}[LinearPDE]{setReducedOrderOff}{} 269 switches off the reduction of polynomial order for the solution and 270 equation evaluation. 271 \end{methoddesc} 272 273 \begin{methoddesc}[LinearPDE]{getOperator}{} 274 returns the \Operator of the PDE. 275 \end{methoddesc} 276 277 \begin{methoddesc}[LinearPDE]{getRightHandSide}{} 278 returns the right hand side of the PDE as a \Data object. If 279 \var{ignoreConstraint}=\True the constraints are not considered 280 when building up the right hand side. 281 \end{methoddesc} 282 283 \begin{methoddesc}[LinearPDE]{getSystem}{} 284 returns the \Operator and right hand side of the PDE. 285 \end{methoddesc} 286 287 \begin{methoddesc}[LinearPDE]{getSolution}{ 288 \optional{verbose=False} 289 \optional{, reordering=LinearPDE.NO_REORDERING} 290 \optional{, iter_max=1000} 291 \optional{, drop_tolerance=0.01} 292 \optional{, drop_storage=1.20} 293 \optional{, truncation=-1} 294 \optional{, restart=-1} 295 } 296 returns (an approximation of) the solution of the PDE. If \code{verbose=\True} some information during the solution process printed. 297 \var{reordering} selects a reordering methods that is applied before or during the solution process 298 (=\NOREORDERING ,\MINIMUMFILLIN ,\NESTEDDESCTION). 299 \var{iter_max} specifies the maximum number of iteration steps that are allowed to reach the specified tolerance. 300 \var{drop_tolerance} specifies a relative tolerance for small elements to be dropped when building a preconditioner 301 (eg. in \ILUT). \var{drop_storage} limits the extra storage allowed when building a preconditioner 302 (eg. in \ILUT). The extra storage is given relative to the size of the stiffness matrix, eg. 303 \var{drop_storage=1.2} will allow the preconditioner to use the $1.2$ fold storage space than used 304 for the stiffness matrix. \var{truncation} defines the truncation. 305 \end{methoddesc} 306 307 \begin{memberdesc}[LinearPDE]{DEFAULT} 308 default method, preconditioner or package to be used to solve the PDE. An appropriate method should be 309 chosen by the used PDE solver library. 310 \end{memberdesc} 311 312 \begin{memberdesc}[LinearPDE]{SCSL} 313 the SCSL library by SGI,~\Ref{SCSL}\footnote{The SCSL library will only be available on SGI systems} 314 \end{memberdesc} 315 316 \begin{memberdesc}[LinearPDE]{MKL} 317 the MKL library by Intel,~\Ref{MKL}\footnote{The MKL library will only be available when the intel compilation environment is used.}. 318 \end{memberdesc} 319 320 \begin{memberdesc}[LinearPDE]{UMFPACK} 321 the UMFPACK,~\Ref{UMFPACK}. Remark: UMFPACK is not parallelized. 322 \end{memberdesc} 323 324 \begin{memberdesc}[LinearPDE]{PASO} 325 the solver library of \finley, see \Sec{CHAPTER ON FINLEY}. 326 \end{memberdesc} 327 328 \begin{memberdesc}[LinearPDE]{ITERATIVE} 329 the default iterative method and preconditioner. The actually used method depends on the 330 PDE solver library and the solver package been chosen. Typically, \PCG is used for symmetric PDEs 331 and \BiCGStab otherwise, both with \JACOBI preconditioner. 332 \end{memberdesc} 333 334 \begin{memberdesc}[LinearPDE]{DIRECT} 335 the default direct linear solver. 336 \end{memberdesc} 337 338 \begin{memberdesc}[LinearPDE]{CHOLEVSKY} 339 direct solver based on Cholevsky factorization (or similar), see~\Ref{Saad}. The solver will require a symmetric PDE. 340 \end{memberdesc} 341 342 \begin{memberdesc}[LinearPDE]{PCG} 343 preconditioned conjugate gradient method, see~\Ref{WEISS}\index{linear solver!PCG}\index{PCG}. The solver will require a symmetric PDE. 344 \end{memberdesc} 345 346 \begin{memberdesc}[LinearPDE]{GMRES} 347 the GMRES method, see~\Ref{WEISS}\index{linear solver!GMRES}\index{GMRES}. Truncation and restart are controlled by the parameters 348 \var{truncation} and \var{restart} of \method{getSolution}. 349 \end{memberdesc} 350 351 \begin{memberdesc}[LinearPDE]{LUMPING} 352 uses lumping to solve the system of linear equations~\index{linear solver!lumping}\index{lumping}. This solver technique 353 condenses the stiffness matrix to a diagonal matrix so the solution of the linear systems becomes very cheap. It can be used when 354 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 355 but is expect to converge to zero when the mesh gets finer. 356 Lumping does not use the linear system solver library. 357 \end{memberdesc} 358 359 \begin{memberdesc}[LinearPDE]{PRES20} 360 the GMRES method with truncation after five residuals and 361 restart after 20 steps, see~\Ref{WEISS}. 362 \end{memberdesc} 363 364 \begin{memberdesc}[LinearPDE]{CGS} 365 conjugate gradient squared method, see~\Ref{WEISS}. 366 \end{memberdesc} 367 368 \begin{memberdesc}[LinearPDE]{BICGSTAB} 369 stabilized bi-conjugate gradients methods, see~\Ref{WEISS}. 370 \end{memberdesc} 371 372 \begin{memberdesc}[LinearPDE]{SSOR} 373 symmetric successive over-relaxation method, see~\Ref{WEISS}. Typically used as preconditioner but some linear solver libraries support 374 this as a solver. 375 \end{memberdesc} 376 \begin{memberdesc}[LinearPDE]{ILU0} 377 the incomplete LU factorization preconditioner with no fill-in, see~\Ref{Saad}. 378 \end{memberdesc} 379 380 \begin{memberdesc}[LinearPDE]{ILUT} 381 the incomplete LU factorization preconditioner with fill-in, see~\Ref{Saad}. During the LU-factorization element with 382 relative size less then \var{drop_tolerance} are dropped. Moreover, the size of the LU-factorization is restricted to the 383 \var{drop_storage}-fold of the stiffness matrix. \var{drop_tolerance} and \var{drop_storage} are both set in the 384 \method{getSolution} call. 385 \end{memberdesc} 386 387 \begin{memberdesc}[LinearPDE]{JACOBI} 388 the Jacobi preconditioner, see~\Ref{Saad}. 389 \end{memberdesc} 390 391 \begin{memberdesc}[LinearPDE]{AMG} 392 the algebraic--multi grid method, see~\Ref{AMG}. This method can be used as linear solver method but is more robust when used 393 in a preconditioner. 394 \end{memberdesc} 395 396 \begin{memberdesc}[LinearPDE]{RILU} 397 recursive incomplete LU factorization preconditioner, see~\Ref{RILU}. This method is similar to \ILUT but uses smoothing 398 between levels. During the LU-factorization element with 399 relative size less then \var{drop_tolerance} are dropped. Moreover, the size of the LU-factorization is restricted to the 400 \var{drop_storage}-fold of the stiffness matrix. \var{drop_tolerance} and \var{drop_storage} are both set in the 401 \method{getSolution} call. 402 \end{memberdesc} 403 404 \begin{memberdesc}[LinearPDE]{NO_REORDERING} 405 no ordering is used during factorization. 406 \end{memberdesc} 407 408 \begin{memberdesc}[LinearPDE]{MINIMUM_FILL_IN} 409 applies reordering before factorization using a fill-in minimization strategy. You have to check with the particular solver library or 410 linear solver package if this is supported. In any case, it is advisable to apply reordering on the mesh to minimize fill-in. 411 \end{memberdesc} 412 413 \begin{memberdesc}[LinearPDE]{NESTED_DISSECTION} 414 applies reordering before factorization using a nested dissection strategy. You have to check with the particular solver library or 415 linear solver package if this is supported. In any case, it is advisable to apply reordering on the mesh to minimize fill-in. 416 \end{memberdesc} 417 418 \subsection{The \Poisson Class} 419 The \Poisson class provides an easy way to define and solve the Poisson 420 equation 421 \begin{equation}\label{POISSON.1} 422 -u\hackscore{,ii}=f\; . 423 \end{equation} 424 with homogeneous boundary conditions 425 \begin{equation}\label{POISSON.2} 426 n\hackscore{i}u\hackscore{,i}=0 427 \end{equation} 428 and homogeneous constraints 429 \begin{equation}\label{POISSON.3} 430 u=0 \mbox{ where } q>0 431 \end{equation} 432 $f$ has to be a \Scalar in the \Function and $q$ must be 433 a \Scalar in the \SolutionFS. 434 435 \begin{classdesc}{Poisson}{domain} 436 opens a Poisson equation on the \Domain domain. \Poisson is derived from \LinearPDE. 437 \end{classdesc} 438 \begin{methoddesc}[Poisson]{setValue}{f=escript.Data(),q=escript.Data()} 439 assigns new values to \var{f} and \var{q}. 440 \end{methoddesc} 441 442 \subsection{The \Helmholtz Class} 443 The \Helmholtz class defines the Helmholtz problem 444 \begin{equation}\label{HZ.1} 445 \omega \; u - (k\; u\hackscore{,j})\hackscore{,j} = f 446 \end{equation} 447 with natural boundary conditions 448 \begin{equation}\label{HZ.2} 449 k\; u\hackscore{,j} n\hackscore{,j} = g- \alpha \; u 450 \end{equation} 451 and constraints: 452 \begin{equation}\label{HZ.3} 453 u=r \mbox{ where } q>0 454 \end{equation} 455 $\omega$, $k$, $f$ have to be a \Scalar in the \Function, 456 $g$ and $\alpha$ must be a \Scalar in the \FunctionOnBoundary, 457 and $q$ and $r$ must be a \Scalar in the \SolutionFS or must be mapped or interpolated into the particular \FunctionSpace. 458 459 \begin{classdesc}{Helmholtz}{domain} 460 opens a Helmholtz equation on the \Domain domain. \Helmholtz is derived from \LinearPDE. 461 \end{classdesc} 462 \begin{methoddesc}[Helmholtz]{setValue}{ \optional{omega} \optional{, k} \optional{, f} \optional{, alpha} \optional{, g} \optional{, r} \optional{, q}} 463 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. 464 \end{methoddesc} 465 466 \subsection{The \Lame Class} 467 The \Lame class defines a Lame equation problem: 468 \begin{equation}\label{LE.1} 469 -\mu (u\hackscore{i,j}+u\hackscore{j,i})+\lambda u\hackscore{k,k})\hackscore{j} = F\hackscore{i}-\sigma\hackscore{ij,j} 470 \end{equation} 471 with natural boundary conditions: 472 \begin{equation}\label{LE.2} 473 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} 474 \end{equation} 475 and constraint 476 \begin{equation}\label{LE.3} 477 u\hackscore{i}=r\hackscore{i} \mbox{ where } q\hackscore{i}>0 478 \end{equation} 479 $\mu$, $\lambda$ have to be a \Scalar in the \Function, 480 $F$ has to be a \Vector in the \Function, 481 $\sigma$ has to be a \Tensor in the \Function, 482 $f$ must be a \Vector in the \FunctionOnBoundary, 483 and $q$ and $r$ must be a \Vector in the \SolutionFS or must be mapped or interpolated into the particular \FunctionSpace. 484 485 \begin{classdesc}{Lame}{domain} 486 opens a Lame equation on the \Domain domain. \Lame is derived from \LinearPDE. 487 \end{classdesc} 488 \begin{methoddesc}[Lame]{setValue}{ \optional{lame_lambda} \optional{, lame_mu} \optional{, F} \optional{, sigma} \optional{, f} \optional{, r} \optional{, q}} 489 assigns new values to 490 \var{lame_lambda}, 491 \var{lame_mu}, 492 \var{F}, 493 \var{sigma}, 494 \var{f}, 495 \var{r} and 496 \var{q} 497 By default all values are set to be zero. 498 \end{methoddesc} 499

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