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

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some clarification on lumping

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

## Properties

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