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

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Macro elements are implemented now. VTK writer for macro elements still needs testing.

 1 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3 % 4 % Copyright (c) 2003-2009 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 15 \chapter{ The Module \finley} 16 \label{CHAPTER ON FINLEY} 17 18 \begin{figure} 19 \centerline{\includegraphics[width=\figwidth]{figures/FinleyMesh}} 20 \caption{Subdivision of an Ellipse into triangles order 1 (\finleyelement{Tri3})} 21 \label{FINLEY FIG 0} 22 \end{figure} 23 24 \begin{figure} 25 \centerline{\includegraphics[width=\figwidth]{figures/FinleyContact}} 26 \caption{Mesh around a contact region (\finleyelement{Rec4})} 27 \label{FINLEY FIG 01} 28 \end{figure} 29 30 \declaremodule{extension}{finley} \modulesynopsis{Solving linear, steady partial differential equations using 31 finite elements} 32 33 {\it finley} is a library of C functions solving linear, steady partial differential equations 34 \index{partial differential equations} (PDEs) or systems of PDEs using isoparametrical finite 35 elements \index{FEM!isoparametrical}. 36 It supports unstructured, 1D, 2D and 3D meshes. The module \finley provides an access to the 37 library through the \LinearPDE class of \escript supporting its full functionality. {\it finley} 38 is parallelized using the OpenMP \index{OpenMP} paradigm. 39 40 \section{Formulation} 41 42 For a single PDE with a solution with a single component the linear PDE is defined in the 43 following form: 44 \begin{equation}\label{FINLEY.SINGLE.1} 45 \begin{array}{cl} & 46 \displaystyle{ 47 \int\hackscore{\Omega} 48 A\hackscore{jl} \cdot v\hackscore{,j}u\hackscore{,l}+ B\hackscore{j} \cdot v\hackscore{,j} u+ C\hackscore{l} \cdot v u\hackscore{,l}+D \cdot vu \; d\Omega } \\ 49 + & \displaystyle{\int\hackscore{\Gamma} d \cdot vu \; d{\Gamma} } 50 + \displaystyle{\int\hackscore{\Gamma^{contact}} d^{contact} \cdot [v][u] \; d{\Gamma} } \\ 51 = & \displaystyle{\int\hackscore{\Omega} X\hackscore{j} \cdot v\hackscore{,j}+ Y \cdot v \; d\Omega }\\ 52 + & \displaystyle{\int\hackscore{\Gamma} y \cdot v \; d{\Gamma}} + 53 \displaystyle{\int\hackscore{\Gamma^{contact}} y^{contact}\cdot [v] \; d{\Gamma}} \\ 54 \end{array} 55 \end{equation} 56 57 \section{Meshes} 58 To understand the usage of \finley one needs to have an understanding of how the finite element meshes 59 \index{FEM!mesh} are defined. \fig{FINLEY FIG 0} shows an example of the 60 subdivision of an ellipse into so called elements \index{FEM!elements} \index{element}. 61 In this case, triangles have been used but other forms of subdivisions 62 can be constructed, e.g. into quadrilaterals or, in the three dimensional case, into tetrahedrons 63 and hexahedrons. The idea of the finite element method is to approximate the solution by a function 64 which is a polynomial of a certain order and is continuous across it boundary to neighbor elements. 65 In the example of \fig{FINLEY FIG 0} a linear polynomial is used on each triangle. As one can see, the triangulation 66 is quite a poor approximation of the ellipse. It can be improved by introducing a midpoint on each element edge then 67 positioning those nodes located on an edge expected to describe the boundary, onto the boundary. 68 In this case the triangle gets a curved edge which requires a parametrization of the triangle using a 69 quadratic polynomial. For this case, the solution is also approximated by a piecewise quadratic polynomial 70 (which explains the name isoparametrical elements), see \Ref{Zienc,NumHand} for more details. 71 \finley supports macro elements\index{macro elements}. For these elements a piecewise linear approximation is used on an element which is further subdivided (in the case \finley halved). As such these elements do not provide more than a further mesh refinement but should be used in the case of incompressible flows, see \class{StokesProblemCartesian}. For these problems a linear approximation of the pressure across the element is used (use the \ReducedSolutionFS \FunctionSpace) while the refined element is used to approximate velocity. So a macro element provides a continuous pressure approximation together with a velocity approximation on a refined mesh. This approach is necessary to make sure that the incompressible flow has a unique solution. 72 73 The union of all elements defines the domain of the PDE. 74 Each element is defined by the nodes used to describe its shape. In \fig{FINLEY FIG 0} the element, 75 which has type \finleyelement{Tri3}, 76 with element reference number $19$ \index{element!reference number} is defined by the nodes 77 with reference numbers $9$, $11$ and $0$ \index{node!reference number}. Notice that the order is counterclockwise. 78 The coefficients of the PDE are evaluated at integration nodes with each individual element. 79 For quadrilateral elements a Gauss quadrature scheme is used. In the case of triangular elements a 80 modified form is applied. The boundary of the domain is also subdivided into elements. \index{element!face} In \fig{FINLEY FIG 0} 81 line elements with two nodes are used. The elements are also defined by their describing nodes, e.g. 82 the face element reference number $20$ which has type \finleyelement{Line2} is defined by the nodes 83 with the reference numbers $11$ and $0$. Again the order is crucial, if moving from the first 84 to second node the domain has to lie on the left hand side (in the case of a two dimension surface element 85 the domain has to lie on the left hand side when moving counterclockwise). If the gradient on the 86 surface of the domain is to be calculated rich face elements face to be used. Rich elements on a face 87 are identical to interior elements but with a modified order of nodes such that the 'first' face of the element aligns 88 with the surface of the domain. In \fig{FINLEY FIG 0} 89 elements of the type \finleyelement{Tri3Face} are used. 90 The face element reference number $20$ as a rich face element is defined by the nodes 91 with reference numbers $11$, $0$ and $9$. Notice that the face element $20$ is identical to the 92 interior element $19$ except that, in this case, the order of the node is different to align the first 93 edge of the triangle (which is the edge starting with the first node) with the boundary of the domain. 94 95 Be aware that face elements and elements in the interior of the domain must match, i.e. a face element must be the face 96 of an interior element or, in case of a rich face element, it must be identical to an interior element. 97 If no face elements are specified 98 \finley implicitly assumes homogeneous natural boundary conditions \index{natural boundary conditions!homogeneous}, 99 i.e. \var{d}=$0$ and \var{y}=$0$, on the entire boundary of the domain. For 100 inhomogeneous natural boundary conditions \index{natural boundary conditions!inhomogeneous}, 101 the boundary must be described by face elements. 102 103 If discontinuities of the PDE solution are considered contact elements 104 \index{element!contact}\index{contact conditions} are introduced to describe the contact region $\Gamma^{contact}$ 105 even if $d^{contact}$ and $y^{contact}$ are zero. \fig{FINLEY FIG 01} shows a simple example of a mesh 106 of rectangular elements around a contact region $\Gamma^{contact}$ \index{element!contact}. 107 The contact region is described by the 108 elements $4$, $3$ and $6$. Their element type is \finleyelement{Line2_Contact}. 109 The nodes $9$, $12$, $6$, $5$ define contact element $4$, where the coordinates of nodes $12$ and $5$ and 110 nodes $4$ and $6$ are identical with the idea that nodes $12$ and $9$ are located above and 111 nodes $5$ and $6$ below the contact region. 112 Again, the order of the nodes within an element is crucial. There is also the option of using rich elements 113 if the gradient is to be calculated on the contact region. Similarly to the rich face elements 114 these are constructed from two interior elements by reordering the nodes such that 115 the 'first' face of the element above and the 'first' face of the element below the 116 contact regions line up. The rich version of element 117 $4$ is of type \finleyelement{Rec4Face_Contact} and is defined by the nodes $9$, $12$, $16$, $18$, $6$, $5$, $0$ and 118 $2$. 119 120 121 122 \tab{FINLEY TAB 1} shows the interior element types and the corresponding element types to be used 123 on the face and contacts. \fig{FINLEY.FIG:1}, \fig{FINLEY.FIG:2} and \fig{FINLEY.FIG:4} show the ordering of 124 the nodes within an element. 125 126 \begin{table} 127 \begin{tablev}{l|llll}{textrm}{interior}{face}{rich face}{contact}{rich contact} 128 \linev{\finleyelement{Line2}}{\finleyelement{Point1}}{\finleyelement{Line2Face}}{\finleyelement{Point1_Contact}}{\finleyelement{Line2Face_Contact}} 129 \linev{\finleyelement{Line3}}{\finleyelement{Point1}}{\finleyelement{Line3Face}}{\finleyelement{Point1_Contact}}{\finleyelement{Line3Face_Contact}} 130 \linev{\finleyelement{Tri3}}{\finleyelement{Line2}}{\finleyelement{Tri3Face}}{\finleyelement{Line2_Contact}}{\finleyelement{Tri3Face_Contact}} 131 \linev{\finleyelement{Tri6}}{\finleyelement{Line3}}{\finleyelement{Tri6Face}}{\finleyelement{Line3_Contact}}{\finleyelement{Tri6Face_Contact}} 132 \linev{\finleyelement{Rec4}}{\finleyelement{Line2}}{\finleyelement{Rec4Face}}{\finleyelement{Line2_Contact}}{\finleyelement{Rec4Face_Contact}} 133 \linev{\finleyelement{Rec8}}{\finleyelement{Line3}}{\finleyelement{Rec8Face}}{\finleyelement{Line3_Contact}}{\finleyelement{Rec8Face_Contact}} 134 \linev{\finleyelement{Rec9}}{\finleyelement{Line3}}{\finleyelement{Rec9Face}}{\finleyelement{Line3_Contact}}{\finleyelement{Rec9Face_Contact}} 135 \linev{\finleyelement{Tet4}}{\finleyelement{Tri6}}{\finleyelement{Tet4Face}}{\finleyelement{Tri6_Contact}}{\finleyelement{Tet4Face_Contact}} 136 \linev{\finleyelement{Tet10}}{\finleyelement{Tri9}}{\finleyelement{Tet10Face}}{\finleyelement{Tri9_Contact}}{\finleyelement{Tet10Face_Contact}} 137 \linev{\finleyelement{Hex8}}{\finleyelement{Rec4}}{\finleyelement{Hex8Face}}{\finleyelement{Rec4_Contact}}{\finleyelement{Hex8Face_Contact}} 138 \linev{\finleyelement{Hex20}}{\finleyelement{Rec8}}{\finleyelement{Hex20Face}}{\finleyelement{Rec8_Contact}}{\finleyelement{Hex20Face_Contact}} 139 \linev{\finleyelement{Hex27}}{\finleyelement{Rec9}}{N\textbackslash A}{N\textbackslash A}{N\textbackslash A} 140 \linev{\finleyelement{Hex27Macro}}{\finleyelement{Rec9Macro}}{N\textbackslash A}{N\textbackslash A}{N\textbackslash A} 141 \linev{\finleyelement{Tet10Macro}}{\finleyelement{Tri6Macro}}{N\textbackslash A}{N\textbackslash A}{N\textbackslash A} 142 \linev{\finleyelement{Rec9Macro}}{\finleyelement{Line3Macro}}{N\textbackslash A}{N\textbackslash A}{N\textbackslash A} 143 \linev{\finleyelement{Tri6Macro}}{\finleyelement{Line3Macro}}{N\textbackslash A}{N\textbackslash A}{N\textbackslash A} 144 \end{tablev} 145 \caption{Finley elements and corresponding elements to be used on domain faces and contacts. 146 The rich types have to be used if the gradient of function is to be calculated on faces and contacts, respectively.} 147 \label{FINLEY TAB 1} 148 \end{table} 149 150 The native \finley file format is defined as follows. 151 Each node \var{i} has \var{dim} spatial coordinates \var{Node[i]}, a reference number 152 \var{Node_ref[i]}, a degree of freedom \var{Node_DOF[i]} and tag \var{Node_tag[i]}. 153 In most cases \var{Node_DOF[i]}=\var{Node_ref[i]} however, for periodic boundary conditions, 154 \var{Node_DOF[i]} is chosen differently, see example below. The tag can be used to mark nodes sharing 155 the same properties. Element \var{i} is defined by the \var{Element_numNodes} nodes \var{Element_Nodes[i]} 156 which is a list of node reference numbers. The order is crucial. 157 It has a reference number \var{Element_ref[i]} and a tag \var{Element_tag[i]}. The tag 158 can be used to mark elements sharing the same properties. For instance elements above 159 a contact region are marked with $2$ and elements below a contact region are marked with $1$. 160 \var{Element_Type} and \var{Element_Num} give the element type and the number of elements in the mesh. 161 Analogue notations are used for face and contact elements. The following Python script 162 prints the mesh definition in the \finley file format: 163 \begin{python} 164 print "%s\n"%mesh_name 165 # node coordinates: 166 print "%dD-nodes %d\n"%(dim,numNodes) 167 for i in range(numNodes): 168 print "%d %d %d"%(Node_ref[i],Node_DOF[i],Node_tag[i]) 169 for j in range(dim): print " %e"%Node[i][j] 170 print "\n" 171 # interior elements 172 print "%s %d\n"%(Element_Type,Element_Num) 173 for i in range(Element_Num): 174 print "%d %d"%(Element_ref[i],Element_tag[i]) 175 for j in range(Element_numNodes): print " %d"%Element_Nodes[i][j] 176 print "\n" 177 # face elements 178 print "%s %d\n"%(FaceElement_Type,FaceElement_Num) 179 for i in range(FaceElement_Num): 180 print "%d %d"%(FaceElement_ref[i],FaceElement_tag[i]) 181 for j in range(FaceElement_numNodes): print " %d"%FaceElement_Nodes[i][j] 182 print "\n" 183 # contact elements 184 print "%s %d\n"%(ContactElement_Type,ContactElement_Num) 185 for i in range(ContactElement_Num): 186 print "%d %d"%(ContactElement_ref[i],ContactElement_tag[i]) 187 for j in range(ContactElement_numNodes): print " %d"%ContactElement_Nodes[i][j] 188 print "\n" 189 # point sources (not supported yet) 190 write("Point1 0",face_element_type,numFaceElements) 191 \end{python} 192 193 The following example of a mesh file defines the mesh shown in \fig{FINLEY FIG 01}: 194 \begin{verbatim} 195 Example 1 196 2D Nodes 16 197 0 0 0 0. 0. 198 2 2 0 0.33 0. 199 3 3 0 0.66 0. 200 7 4 0 1. 0. 201 5 5 0 0. 0.5 202 6 6 0 0.33 0.5 203 8 8 0 0.66 0.5 204 10 10 0 1.0 0.5 205 12 12 0 0. 0.5 206 9 9 0 0.33 0.5 207 13 13 0 0.66 0.5 208 15 15 0 1.0 0.5 209 16 16 0 0. 1.0 210 18 18 0 0.33 1.0 211 19 19 0 0.66 1.0 212 20 20 0 1.0 1.0 213 Rec4 6 214 0 1 0 2 6 5 215 1 1 2 3 8 6 216 2 1 3 7 10 8 217 5 2 12 9 18 16 218 7 2 13 19 18 9 219 10 2 20 19 13 15 220 Line2 0 221 Line2_Contact 3 222 4 0 9 12 6 5 223 3 0 13 9 8 6 224 6 0 15 13 10 8 225 Point1 0 226 \end{verbatim} 227 Notice that the order in which the nodes and elements are given is arbitrary. 228 In the case that rich contact elements are used the contact element section gets 229 the form 230 \begin{verbatim} 231 Rec4Face_Contact 3 232 4 0 9 12 16 18 6 5 0 2 233 3 0 13 9 18 19 8 6 2 3 234 6 0 15 13 19 20 10 8 3 7 235 \end{verbatim} 236 Periodic boundary condition \index{boundary conditions!periodic} can be introduced by altering \var{Node_DOF}. 237 It allows identification of nodes even if they have different physical locations. For instance, to 238 enforce periodic boundary conditions at the face $x_0=0$ and $x_0=1$ one identifies 239 the degrees of freedom for nodes $0$, $5$, $12$ and $16$ with the degrees of freedom for 240 $7$, $10$, $15$ and $20$, respectively. The node section of the \finley mesh gets now the form: 241 \begin{verbatim} 242 2D Nodes 16 243 0 0 0 0. 0. 244 2 2 0 0.33 0. 245 3 3 0 0.66 0. 246 7 0 0 1. 0. 247 5 5 0 0. 0.5 248 6 6 0 0.33 0.5 249 8 8 0 0.66 0.5 250 10 5 0 1.0 0.5 251 12 12 0 0. 0.5 252 9 9 0 0.33 0.5 253 13 13 0 0.66 0.5 254 15 12 0 1.0 0.5 255 16 16 0 0. 1.0 256 18 18 0 0.33 1.0 257 19 19 0 0.66 1.0 258 20 16 0 1.0 1.0 259 \end{verbatim} 260 261 \clearpage 262 \input{finleyelements} 263 \clearpage 264 265 \section{Macro Elements} 266 \label{SEC FINLEY MACRO} 267 268 269 270 \begin{table} 271 {\scriptsize 272 \begin{tabular}{l||c|c|c|c|c|c|c|c} 273 \member{setSolverMethod} & \member{DIRECT}& \member{PCG} & \member{GMRES} & \member{TFQMR} & \member{MINRES} & \member{PRES20} & \member{BICGSTAB} & \member{LUMPING} \\ 274 \hline 275 \hline 276 \member{setReordering} & $\checkmark$ & & & & & &\\ 277 \hline \member{setRestart} & & & $\checkmark$ & & & $20$ & \\ 278 \hline\member{setTruncation} & & & $\checkmark$ & & & $5$ & \\ 279 \hline\member{setIterMax} & & $\checkmark$& $\checkmark$ & $\checkmark$& $\checkmark$& $\checkmark$ & $\checkmark$ \\ 280 \hline\member{setTolerance} & & $\checkmark$& $\checkmark$ & $\checkmark$& $\checkmark$& $\checkmark$ & $\checkmark$ \\ 281 \hline\member{setAbsoluteTolerance} & & $\checkmark$& $\checkmark$ & $\checkmark$& $\checkmark$& $\checkmark$ & $\checkmark$ \\ 282 \hline\member{setReordering} & $\checkmark$ & & & & & & & \\ 283 \end{tabular} 284 } 285 \caption{Solvers available for 286 \finley 287 and the \PASO package and the relevant options in \class{SolverOptions}. 288 \MKL supports 289 \MINIMUMFILLIN 290 and 291 \NESTEDDESCTION 292 reordering. 293 Currently the \UMFPACK interface does not support any reordering. 294 \label{TAB FINLEY SOLVER OPTIONS 1} } 295 \end{table} 296 297 \begin{table} 298 {\scriptsize 299 \begin{tabular}{l||c|c|c|c|c|c|c|c} 300 \member{setPreconditioner} & 301 \member{NO_PRECONDITIONER} & 302 \member{AMG} & 303 \member{JACOBI} & 304 \member{GAUSS_SEIDEL}& 305 \member{REC_ILU}& 306 \member{RILU} & 307 \member{ILU0} & 308 \member{DIRECT} \\ 309 \hline 310 status: & 311 later & 312 later & 313 $\checkmark$ & 314 $\checkmark$& 315 $\checkmark$ & 316 later & 317 $\checkmark$ & 318 later \\ 319 \hline 320 \hline 321 \member{setCoarsening}& 322 & 323 $\checkmark$ & 324 & 325 & 326 & 327 & 328 & 329 \\ 330 331 332 \hline\member{setLevelMax}& 333 & 334 $\checkmark$ & 335 & 336 & 337 & 338 & 339 & 340 \\ 341 342 \hline\member{setCoarseningThreshold}& 343 & 344 $\checkmark$ & 345 & 346 & 347 & 348 & 349 & 350 \\ 351 352 \hline\member{setMinCoarseMatrixSize} & 353 & 354 $\checkmark$ & 355 & 356 & 357 & 358 & 359 & 360 \\ 361 362 \hline\member{setNumSweeps} & 363 & 364 & 365 $\checkmark$ & 366 $\checkmark$ & 367 & 368 & 369 & 370 \\ 371 372 \hline\member{setNumPreSweeps}& 373 & 374 $\checkmark$ & 375 & 376 & 377 & 378 & 379 & 380 \\ 381 382 \hline\member{setNumPostSweeps} & 383 & 384 $\checkmark$ & 385 & 386 & 387 & 388 & 389 & 390 \\ 391 392 \hline\member{setInnerTolerance}& 393 & 394 & 395 & 396 & 397 & 398 & 399 & 400 \\ 401 402 \hline\member{setDropTolerance}& 403 & 404 & 405 & 406 & 407 & 408 & 409 & 410 \\ 411 412 \hline\member{setDropStorage}& 413 & 414 & 415 & 416 & 417 & 418 & 419 & 420 \\ 421 422 \hline\member{setRelaxationFactor}& 423 & 424 & 425 & 426 & 427 & 428 $\checkmark$ & 429 & 430 \\ 431 432 \hline\member{adaptInnerTolerance}& 433 & 434 & 435 & 436 & 437 & 438 & 439 & 440 \\ 441 442 \hline\member{setInnerIterMax}& 443 & 444 & 445 & 446 & 447 & 448 & 449 & 450 \\ 451 \end{tabular} 452 } 453 \caption{Preconditioners available for \finley and the \PASO package and the relevant options in \class{SolverOptions}. \label{TAB FINLEY SOLVER OPTIONS 2}} 454 \end{table} 455 456 \subsection{Linear Solvers in \SolverOptions} 457 Table~\ref{TAB FINLEY SOLVER OPTIONS 1} and 458 Table~\ref{TAB FINLEY SOLVER OPTIONS 2} show the solvers and preconditioners supported by 459 \finley through the \PASO library. Currently direct solvers are not supported under MPI. 460 By default, \finley is using the iterative solvers \PCG for symmetric and \BiCGStab for non-symmetric problems. 461 If the direct solver is selected which can be useful when solving very ill-posedequations 462 \finley uses the \MKL \footnote{If the stiffness matrix is non-regular \MKL may return without 463 returning a proper error code. If you observe suspicious solutions when using MKL, this may cause by a non-invertible operator. } solver package. If \MKL is not available \UMFPACK is used. If \UMFPACK is not available 464 a suitable iterative solver from the \PASO is used. 465 466 \subsection{Functions} 467 \begin{funcdesc}{ReadMesh}{fileName \optional{, \optional{integrationOrder=-1}, optimize=True}} 468 creates a \Domain object form the FEM mesh defined in 469 file \var{fileName}. The file must be given the \finley file format. 470 If \var{integrationOrder} is positive, a numerical integration scheme 471 chosen which is accurate on each element up to a polynomial of 472 degree \var{integrationOrder} \index{integration order}. Otherwise 473 an appropriate integration order is chosen independently. 474 By default the labeling of mesh nodes and element distribution is 475 optimized. Set \var{optimize=False} to switch off relabeling and redistribution. 476 \end{funcdesc} 477 478 \begin{funcdesc}{ReadGmsh}{fileName \optional{, \optional{integrationOrder=-1}, optimize=True\optional{, useMacroElements=False}}} 479 creates a \Domain object form the FEM mesh defined in 480 file \var{fileName}. The file must be given the \gmshextern file format. 481 If \var{integrationOrder} is positive, a numerical integration scheme 482 chosen which is accurate on each element up to a polynomial of 483 degree \var{integrationOrder} \index{integration order}. Otherwise 484 an appropriate integration order is chosen independently. 485 By default the labeling of mesh nodes and element distribution is 486 optimized. Set \var{optimize=False} to switch off relabeling and redistribution. 487 If \var{useMacroElements} is set, second order elements are interpreated as macro elements~\index{macro elements}. 488 Currently \function{ReadGmsh} does not support MPI. 489 \end{funcdesc} 490 491 \begin{funcdesc}{MakeDomain}{design\optional{, integrationOrder=-1\optional{, optimizeLabeling=True\optional{, useMacroElements=False}}}} 492 Creates a Finley \Domain from a \class{Design} object using \gmshextern. 493 The \class{Design} \var{design} defines the geometry. 494 If \var{integrationOrder} is positive, a numerical integration scheme 495 chosen which is accurate on each element up to a polynomial of 496 degree \var{integrationOrder} \index{integration order}. Otherwise 497 an appropriate integration order is chosen independently. 498 Set \var{optimizeLabeling=False} to switch off relabeling and redistribution (not recommended). 499 If \var{useMacroElements} is set, macro elements~\index{macro elements} are used. 500 Currently \function{MakeDomain} does not support MPI. 501 \end{funcdesc} 502 503 504 \begin{funcdesc}{load}{fileName} 505 recovers a \Domain object from a dump file created by the \ 506 eateseates a \Domain object form the FEM mesh defined in 507 file \var{fileName}. The file must be given the \finley file format. 508 If \var{integrationOrder} is positive, a numerical integration scheme 509 chosen which is accurate on each element up to a polynomial of 510 degree \var{integrationOrder} \index{integration order}. Otherwise 511 an appropriate integration order is chosen independently. 512 \end{funcdesc} 513 514 515 \begin{funcdesc}{Rectangle}{n0,n1,order=1,l0=1.,l1=1., integrationOrder=-1, \\ 516 periodic0=\False, periodic1=\False, useElementsOnFace=\False, useMacroElements=\False, optimize=\False} 517 Generates a \Domain object representing a two dimensional rectangle between 518 $(0,0)$ and $(l0,l1)$ with orthogonal edges. The rectangle is filled with 519 \var{n0} elements along the $x_0$-axis and 520 \var{n1} elements along the $x_1$-axis. 521 For \var{order}=1 and \var{order}=2 522 \finleyelement{Rec4} and 523 \finleyelement{Rec8} are used, respectively. 524 In the case of \var{useElementsOnFace}=\False, 525 \finleyelement{Line2} and 526 \finleyelement{Line3} are used to subdivide the edges of the rectangle, respectively. 527 If \var{order}=-1, \finleyelement{Rec8Macro} and \finleyelement{Line3Macro}~\index{macro elements}. This option should be used when solving incompressible fluid flow problem, e.g. \class{StokesProblemCartesian}. 528 In the case of \var{useElementsOnFace}=\True (this option should be used if gradients 529 are calculated on domain faces), 530 \finleyelement{Rec4Face} and 531 \finleyelement{Rec8Face} are used on the edges, respectively. 532 If \var{integrationOrder} is positive, a numerical integration scheme 533 chosen which is accurate on each element up to a polynomial of 534 degree \var{integrationOrder} \index{integration order}. Otherwise 535 an appropriate integration order is chosen independently. If 536 \var{periodic0}=\True, periodic boundary conditions \index{periodic boundary conditions} 537 along the $x_0$-directions are enforced. That means when for any solution of a PDE solved by \finley 538 the value on the line $x_0=0$ will be identical to the values on $x_0=\var{l0}$. 539 Correspondingly, 540 \var{periodic1}=\False sets periodic boundary conditions 541 in $x_1$-direction. 542 If \var{optimize}=\True mesh node relabeling will be attempted to reduce the computation and also ParMETIS will be used to improve the mesh partition if running on multiple CPUs with MPI. 543 \end{funcdesc} 544 545 \begin{funcdesc}{Brick}{n0,n1,n2,order=1,l0=1.,l1=1.,l2=1., integrationOrder=-1, \\ 546 periodic0=\False,periodic1=\False,periodic2=\False,useElementsOnFace=\False, useMacroElements=\False, optimize=\False} 547 Generates a \Domain object representing a three dimensional brick between 548 $(0,0,0)$ and $(l0,l1,l2)$ with orthogonal faces. The brick is filled with 549 \var{n0} elements along the $x_0$-axis, 550 \var{n1} elements along the $x_1$-axis and 551 \var{n2} elements along the $x_2$-axis. 552 For \var{order}=1 and \var{order}=2 553 \finleyelement{Hex8} and 554 \finleyelement{Hex20} are used, respectively. 555 In the case of \var{useElementsOnFace}=\False, 556 \finleyelement{Rec4} and 557 \finleyelement{Rec8} are used to subdivide the faces of the brick, respectively. 558 In the case of \var{useElementsOnFace}=\True (this option should be used if gradients 559 are calculated on domain faces), 560 \finleyelement{Hex8Face} and 561 \finleyelement{Hex20Face} are used on the brick faces, respectively. 562 If \var{order}=-1, \finleyelement{Hex20Macro} and \finleyelement{Rec8Macro}~\index{macro elements}. This option should be used when solving incompressible fluid flow problem, e.g. \class{StokesProblemCartesian}. 563 If \var{integrationOrder} is positive, a numerical integration scheme 564 chosen which is accurate on each element up to a polynomial of 565 degree \var{integrationOrder} \index{integration order}. Otherwise 566 an appropriate integration order is chosen independently. If 567 \var{periodic0}=\True, periodic boundary conditions \index{periodic boundary conditions} 568 along the $x_0$-directions are enforced. That means when for any solution of a PDE solved by \finley 569 the value on the plane $x_0=0$ will be identical to the values on $x_0=\var{l0}$. Correspondingly, 570 \var{periodic1}=\False and \var{periodic2}=\False sets periodic boundary conditions 571 in $x_1$-direction and $x_2$-direction, respectively. 572 If \var{optimize}=\True mesh node relabeling will be attempted to reduce the computation and also ParMETIS will be used to improve the mesh partition if running on multiple CPUs with MPI. 573 \end{funcdesc} 574 575 \begin{funcdesc}{GlueFaces}{meshList,safetyFactor=0.2,tolerance=1.e-13} 576 Generates a new \Domain object from the list \var{meshList} of \finley meshes. 577 Nodes in face elements whose difference of coordinates is less then \var{tolerance} times the 578 diameter of the domain are merged. The corresponding face elements are removed from the mesh. 579 580 TODO: explain \var{safetyFactor} and show an example. 581 \end{funcdesc} 582 583 \begin{funcdesc}{JoinFaces}{meshList,safetyFactor=0.2,tolerance=1.e-13} 584 Generates a new \Domain object from the list \var{meshList} of \finley meshes. 585 Face elements whose nodes coordinates have difference is less then \var{tolerance} times the 586 diameter of the domain are combined to form a contact element \index{element!contact} 587 The corresponding face elements are removed from the mesh. 588 589 TODO: explain \var{safetyFactor} and show an example. 590 \end{funcdesc}

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