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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 % http://www.uq.edu.au/esscc
7 %
8 % Primary Business: Queensland, Australia
9 % Licensed under the Open Software License version 3.0
10 % http://www.opensource.org/licenses/osl-3.0.php
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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