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1
2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
3 %
4 % Copyright (c) 2003-2010 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]{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]{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 \label{FINLEY MESHES}
59 To understand the usage of \finley one needs to have an understanding of how the finite element meshes
60 \index{FEM!mesh} are defined. \fig{FINLEY FIG 0} shows an example of the
61 subdivision of an ellipse into so called elements \index{FEM!elements} \index{element}.
62 In this case, triangles have been used but other forms of subdivisions
63 can be constructed, e.g. into quadrilaterals or, in the three dimensional case, into tetrahedrons
64 and hexahedrons. The idea of the finite element method is to approximate the solution by a function
65 which is a polynomial of a certain order and is continuous across it boundary to neighbor elements.
66 In the example of \fig{FINLEY FIG 0} a linear polynomial is used on each triangle. As one can see, the triangulation
67 is quite a poor approximation of the ellipse. It can be improved by introducing a midpoint on each element edge then
68 positioning those nodes located on an edge expected to describe the boundary, onto the boundary.
69 In this case the triangle gets a curved edge which requires a parameterization of the triangle using a
70 quadratic polynomial. For this case, the solution is also approximated by a piecewise quadratic polynomial
71 (which explains the name isoparametrical elements), see \Ref{Zienc,NumHand} for more details.
72 \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.
73
74 The union of all elements defines the domain of the PDE.
75 Each element is defined by the nodes used to describe its shape. In \fig{FINLEY FIG 0} the element,
76 which has type \finleyelement{Tri3},
77 with element reference number $19$ \index{element!reference number} is defined by the nodes
78 with reference numbers $9$, $11$ and $0$ \index{node!reference number}. Notice that the order is counterclockwise.
79 The coefficients of the PDE are evaluated at integration nodes with each individual element.
80 For quadrilateral elements a Gauss quadrature scheme is used. In the case of triangular elements a
81 modified form is applied. The boundary of the domain is also subdivided into elements. \index{element!face} In \fig{FINLEY FIG 0}
82 line elements with two nodes are used. The elements are also defined by their describing nodes, e.g.
83 the face element reference number $20$ which has type \finleyelement{Line2} is defined by the nodes
84 with the reference numbers $11$ and $0$. Again the order is crucial, if moving from the first
85 to second node the domain has to lie on the left hand side (in the case of a two dimension surface element
86 the domain has to lie on the left hand side when moving counterclockwise). If the gradient on the
87 surface of the domain is to be calculated rich face elements face to be used. Rich elements on a face
88 are identical to interior elements but with a modified order of nodes such that the 'first' face of the element aligns
89 with the surface of the domain. In \fig{FINLEY FIG 0}
90 elements of the type \finleyelement{Tri3Face} are used.
91 The face element reference number $20$ as a rich face element is defined by the nodes
92 with reference numbers $11$, $0$ and $9$. Notice that the face element $20$ is identical to the
93 interior element $19$ except that, in this case, the order of the node is different to align the first
94 edge of the triangle (which is the edge starting with the first node) with the boundary of the domain.
95
96 Be aware that face elements and elements in the interior of the domain must match, i.e. a face element must be the face
97 of an interior element or, in case of a rich face element, it must be identical to an interior element.
98 If no face elements are specified
99 \finley implicitly assumes homogeneous natural boundary conditions \index{natural boundary conditions!homogeneous},
100 i.e. \var{d}=$0$ and \var{y}=$0$, on the entire boundary of the domain. For
101 inhomogeneous natural boundary conditions \index{natural boundary conditions!inhomogeneous},
102 the boundary must be described by face elements.
103
104 If discontinuities of the PDE solution are considered contact elements
105 \index{element!contact}\index{contact conditions} are introduced to describe the contact region $\Gamma^{contact}$
106 even if $d^{contact}$ and $y^{contact}$ are zero. \fig{FINLEY FIG 01} shows a simple example of a mesh
107 of rectangular elements around a contact region $\Gamma^{contact}$ \index{element!contact}.
108 The contact region is described by the
109 elements $4$, $3$ and $6$. Their element type is \finleyelement{Line2_Contact}.
110 The nodes $9$, $12$, $6$, $5$ define contact element $4$, where the coordinates of nodes $12$ and $5$ and
111 nodes $4$ and $6$ are identical with the idea that nodes $12$ and $9$ are located above and
112 nodes $5$ and $6$ below the contact region.
113 Again, the order of the nodes within an element is crucial. There is also the option of using rich elements
114 if the gradient is to be calculated on the contact region. Similarly to the rich face elements
115 these are constructed from two interior elements by reordering the nodes such that
116 the 'first' face of the element above and the 'first' face of the element below the
117 contact regions line up. The rich version of element
118 $4$ is of type \finleyelement{Rec4Face_Contact} and is defined by the nodes $9$, $12$, $16$, $18$, $6$, $5$, $0$ and
119 $2$.
120
121
122
123 \tab{FINLEY TAB 1} shows the interior element types and the corresponding element types to be used
124 on the face and contacts. \fig{FINLEY.FIG:1}, \fig{FINLEY.FIG:2} and \fig{FINLEY.FIG:4} show the ordering of
125 the nodes within an element.
126
127 \begin{table}
128 \begin{tablev}{l|llll}{textrm}{interior}{face}{rich face}{contact}{rich contact}
129 \linev{\finleyelement{Line2}}{\finleyelement{Point1}}{\finleyelement{Line2Face}}{\finleyelement{Point1_Contact}}{\finleyelement{Line2Face_Contact}}
130 \linev{\finleyelement{Line3}}{\finleyelement{Point1}}{\finleyelement{Line3Face}}{\finleyelement{Point1_Contact}}{\finleyelement{Line3Face_Contact}}
131 \linev{\finleyelement{Tri3}}{\finleyelement{Line2}}{\finleyelement{Tri3Face}}{\finleyelement{Line2_Contact}}{\finleyelement{Tri3Face_Contact}}
132 \linev{\finleyelement{Tri6}}{\finleyelement{Line3}}{\finleyelement{Tri6Face}}{\finleyelement{Line3_Contact}}{\finleyelement{Tri6Face_Contact}}
133 \linev{\finleyelement{Rec4}}{\finleyelement{Line2}}{\finleyelement{Rec4Face}}{\finleyelement{Line2_Contact}}{\finleyelement{Rec4Face_Contact}}
134 \linev{\finleyelement{Rec8}}{\finleyelement{Line3}}{\finleyelement{Rec8Face}}{\finleyelement{Line3_Contact}}{\finleyelement{Rec8Face_Contact}}
135 \linev{\finleyelement{Rec9}}{\finleyelement{Line3}}{\finleyelement{Rec9Face}}{\finleyelement{Line3_Contact}}{\finleyelement{Rec9Face_Contact}}
136 \linev{\finleyelement{Tet4}}{\finleyelement{Tri6}}{\finleyelement{Tet4Face}}{\finleyelement{Tri6_Contact}}{\finleyelement{Tet4Face_Contact}}
137 \linev{\finleyelement{Tet10}}{\finleyelement{Tri9}}{\finleyelement{Tet10Face}}{\finleyelement{Tri9_Contact}}{\finleyelement{Tet10Face_Contact}}
138 \linev{\finleyelement{Hex8}}{\finleyelement{Rec4}}{\finleyelement{Hex8Face}}{\finleyelement{Rec4_Contact}}{\finleyelement{Hex8Face_Contact}}
139 \linev{\finleyelement{Hex20}}{\finleyelement{Rec8}}{\finleyelement{Hex20Face}}{\finleyelement{Rec8_Contact}}{\finleyelement{Hex20Face_Contact}}
140 \linev{\finleyelement{Hex27}}{\finleyelement{Rec9}}{N\textbackslash A}{N\textbackslash A}{N\textbackslash A}
141 \linev{\finleyelement{Hex27Macro}}{\finleyelement{Rec9Macro}}{N\textbackslash A}{N\textbackslash A}{N\textbackslash A}
142 \linev{\finleyelement{Tet10Macro}}{\finleyelement{Tri6Macro}}{N\textbackslash A}{N\textbackslash A}{N\textbackslash A}
143 \linev{\finleyelement{Rec9Macro}}{\finleyelement{Line3Macro}}{N\textbackslash A}{N\textbackslash A}{N\textbackslash A}
144 \linev{\finleyelement{Tri6Macro}}{\finleyelement{Line3Macro}}{N\textbackslash A}{N\textbackslash A}{N\textbackslash A}
145 \end{tablev}
146 \caption{Finley elements and corresponding elements to be used on domain faces and contacts.
147 The rich types have to be used if the gradient of function is to be calculated on faces and contacts, respectively.}
148 \label{FINLEY TAB 1}
149 \end{table}
150
151 The native \finley file format is defined as follows.
152 Each node \var{i} has \var{dim} spatial coordinates \var{Node[i]}, a reference number
153 \var{Node_ref[i]}, a degree of freedom \var{Node_DOF[i]} and tag \var{Node_tag[i]}.
154 In most cases \var{Node_DOF[i]}=\var{Node_ref[i]} however, for periodic boundary conditions,
155 \var{Node_DOF[i]} is chosen differently, see example below. The tag can be used to mark nodes sharing
156 the same properties. Element \var{i} is defined by the \var{Element_numNodes} nodes \var{Element_Nodes[i]}
157 which is a list of node reference numbers. The order is crucial.
158 It has a reference number \var{Element_ref[i]} and a tag \var{Element_tag[i]}. The tag
159 can be used to mark elements sharing the same properties. For instance elements above
160 a contact region are marked with $2$ and elements below a contact region are marked with $1$.
161 \var{Element_Type} and \var{Element_Num} give the element type and the number of elements in the mesh.
162 Analogue notations are used for face and contact elements. The following Python script
163 prints the mesh definition in the \finley file format:
164 \begin{python}
165 print "%s\n"%mesh_name
166 # node coordinates:
167 print "%dD-nodes %d\n"%(dim,numNodes)
168 for i in range(numNodes):
169 print "%d %d %d"%(Node_ref[i],Node_DOF[i],Node_tag[i])
170 for j in range(dim): print " %e"%Node[i][j]
171 print "\n"
172 # interior elements
173 print "%s %d\n"%(Element_Type,Element_Num)
174 for i in range(Element_Num):
175 print "%d %d"%(Element_ref[i],Element_tag[i])
176 for j in range(Element_numNodes): print " %d"%Element_Nodes[i][j]
177 print "\n"
178 # face elements
179 print "%s %d\n"%(FaceElement_Type,FaceElement_Num)
180 for i in range(FaceElement_Num):
181 print "%d %d"%(FaceElement_ref[i],FaceElement_tag[i])
182 for j in range(FaceElement_numNodes): print " %d"%FaceElement_Nodes[i][j]
183 print "\n"
184 # contact elements
185 print "%s %d\n"%(ContactElement_Type,ContactElement_Num)
186 for i in range(ContactElement_Num):
187 print "%d %d"%(ContactElement_ref[i],ContactElement_tag[i])
188 for j in range(ContactElement_numNodes): print " %d"%ContactElement_Nodes[i][j]
189 print "\n"
190 # point sources (not supported yet)
191 write("Point1 0",face_element_type,numFaceElements)
192 \end{python}
193
194 The following example of a mesh file defines the mesh shown in \fig{FINLEY FIG 01}:
195 \begin{verbatim}
196 Example 1
197 2D Nodes 16
198 0 0 0 0. 0.
199 2 2 0 0.33 0.
200 3 3 0 0.66 0.
201 7 4 0 1. 0.
202 5 5 0 0. 0.5
203 6 6 0 0.33 0.5
204 8 8 0 0.66 0.5
205 10 10 0 1.0 0.5
206 12 12 0 0. 0.5
207 9 9 0 0.33 0.5
208 13 13 0 0.66 0.5
209 15 15 0 1.0 0.5
210 16 16 0 0. 1.0
211 18 18 0 0.33 1.0
212 19 19 0 0.66 1.0
213 20 20 0 1.0 1.0
214 Rec4 6
215 0 1 0 2 6 5
216 1 1 2 3 8 6
217 2 1 3 7 10 8
218 5 2 12 9 18 16
219 7 2 13 19 18 9
220 10 2 20 19 13 15
221 Line2 0
222 Line2_Contact 3
223 4 0 9 12 6 5
224 3 0 13 9 8 6
225 6 0 15 13 10 8
226 Point1 0
227 \end{verbatim}
228 Notice that the order in which the nodes and elements are given is arbitrary.
229 In the case that rich contact elements are used the contact element section gets
230 the form
231 \begin{verbatim}
232 Rec4Face_Contact 3
233 4 0 9 12 16 18 6 5 0 2
234 3 0 13 9 18 19 8 6 2 3
235 6 0 15 13 19 20 10 8 3 7
236 \end{verbatim}
237 Periodic boundary condition \index{boundary conditions!periodic} can be introduced by altering \var{Node_DOF}.
238 It allows identification of nodes even if they have different physical locations. For instance, to
239 enforce periodic boundary conditions at the face $x_0=0$ and $x_0=1$ one identifies
240 the degrees of freedom for nodes $0$, $5$, $12$ and $16$ with the degrees of freedom for
241 $7$, $10$, $15$ and $20$, respectively. The node section of the \finley mesh gets now the form:
242 \begin{verbatim}
243 2D Nodes 16
244 0 0 0 0. 0.
245 2 2 0 0.33 0.
246 3 3 0 0.66 0.
247 7 0 0 1. 0.
248 5 5 0 0. 0.5
249 6 6 0 0.33 0.5
250 8 8 0 0.66 0.5
251 10 5 0 1.0 0.5
252 12 12 0 0. 0.5
253 9 9 0 0.33 0.5
254 13 13 0 0.66 0.5
255 15 12 0 1.0 0.5
256 16 16 0 0. 1.0
257 18 18 0 0.33 1.0
258 19 19 0 0.66 1.0
259 20 16 0 1.0 1.0
260 \end{verbatim}
261
262 \clearpage
263 \input{finleyelements}
264 \clearpage
265
266 \begin{figure}[th]
267 \begin{center}
268 \subfigure[Triangle]{\label{FINLEY MACRO TRI}\includegraphics[scale=0.25]{FinleyMacroTri}}
269 \subfigure[Quadrilateral]{\label{FINLEY MACRO REC}\includegraphics[scale=0.25]{FinleyMacroRec}}
270 \includegraphics[scale=0.2]{FinleyMacroLeg}
271 \end{center}
272 Macro elements in \finley.
273 \end{figure}
274
275 \section{Macro Elements}
276 \label{SEC FINLEY MACRO}
277 \finley supports the usage of macro elements~\index{macro elements} which can be used to
278 achieve LBB compliance when solving incompressible fluid flow problems. LBB compliance is required to
279 get a problem which has a unique solution for pressure and velocity. For macro elements the
280 pressure and velocity are approximated by a polynomial of order 1 but the velocity approximation bases on a refinement of the element. The nodes of a triangle and quadrilateral element is shown in Figures~\ref{FINLEY MACRO TRI} and~\ref{FINLEY MACRO REC}, respectively. In essence, the velocity uses the same nodes like a quadratic polynomial approximation but replaces the quadratic polynomial by piecewise linear polynomials. In fact, this is the
281 way \finley is defining the macro elements. In particular \finley uses the same local ordering of the nodes for the macro element as for the corresponding quadratic element. Another interpretation is that
282 one uses a linear approximation of the velocity together with a linear approximation of the pressure but on elements
283 created by combining elements to macro elements. Notice that the macro elements still use quadratic interpolation to represent the element and domain boundary. However, if elements have linear boundary
284 a macro element approximation for the velocity is equivalent to using a linear approximation on a mesh which is created through a one step, global refinement.
285 Typically macro elements are only required to use when an incompressible fluid flow problem
286 is solved, e.g the Stokes problem in Section \ref{STOKES PROBLEM}. Please see Section~\ref{FINLEY MESHES} for
287 more details on the supported macro elements.
288
289
290
291 \begin{table}
292 {\scriptsize
293 \begin{tabular}{l||c|c|c|c|c|c|c|c}
294 \member{setSolverMethod} & \member{DIRECT}& \member{PCG} & \member{GMRES} & \member{TFQMR} & \member{MINRES} & \member{PRES20} & \member{BICGSTAB} & \member{LUMPING} \\
295 \hline
296 \hline
297 \member{setReordering} & $\checkmark$ & & & & & &\\
298 \hline \member{setRestart} & & & $\checkmark$ & & & $20$ & \\
299 \hline\member{setTruncation} & & & $\checkmark$ & & & $5$ & \\
300 \hline\member{setIterMax} & & $\checkmark$& $\checkmark$ & $\checkmark$& $\checkmark$& $\checkmark$ & $\checkmark$ \\
301 \hline\member{setTolerance} & & $\checkmark$& $\checkmark$ & $\checkmark$& $\checkmark$& $\checkmark$ & $\checkmark$ \\
302 \hline\member{setAbsoluteTolerance} & & $\checkmark$& $\checkmark$ & $\checkmark$& $\checkmark$& $\checkmark$ & $\checkmark$ \\
303 \hline\member{setReordering} & $\checkmark$ & & & & & & & \\
304 \end{tabular}
305 }
306 \caption{Solvers available for
307 \finley
308 and the \PASO package and the relevant options in \class{SolverOptions}.
309 \MKL supports
310 \MINIMUMFILLIN
311 and
312 \NESTEDDESCTION
313 reordering.
314 Currently the \UMFPACK interface does not support any reordering.
315 \label{TAB FINLEY SOLVER OPTIONS 1} }
316 \end{table}
317
318 \begin{table}
319 {\scriptsize
320 \begin{tabular}{l||c|c|c|c|c|c|c|c}
321 \member{setPreconditioner} &
322 \member{NO_PRECONDITIONER} &
323 \member{AMG} &
324 \member{JACOBI} &
325 \member{GAUSS_SEIDEL}&
326 \member{REC_ILU}&
327 \member{RILU} &
328 \member{ILU0} &
329 \member{DIRECT} \\
330 \hline
331 status: &
332 later &
333 later &
334 $\checkmark$ &
335 $\checkmark$&
336 $\checkmark$ &
337 later &
338 $\checkmark$ &
339 later \\
340 \hline
341 \hline
342 \member{setCoarsening}&
343 &
344 $\checkmark$ &
345 &
346 &
347 &
348 &
349 &
350 \\
351
352
353 \hline\member{setLevelMax}&
354 &
355 $\checkmark$ &
356 &
357 &
358 &
359 &
360 &
361 \\
362
363 \hline\member{setCoarseningThreshold}&
364 &
365 $\checkmark$ &
366 &
367 &
368 &
369 &
370 &
371 \\
372
373 \hline\member{setMinCoarseMatrixSize} &
374 &
375 $\checkmark$ &
376 &
377 &
378 &
379 &
380 &
381 \\
382
383 \hline\member{setNumSweeps} &
384 &
385 &
386 $\checkmark$ &
387 $\checkmark$ &
388 &
389 &
390 &
391 \\
392
393 \hline\member{setNumPreSweeps}&
394 &
395 $\checkmark$ &
396 &
397 &
398 &
399 &
400 &
401 \\
402
403 \hline\member{setNumPostSweeps} &
404 &
405 $\checkmark$ &
406 &
407 &
408 &
409 &
410 &
411 \\
412
413 \hline\member{setInnerTolerance}&
414 &
415 &
416 &
417 &
418 &
419 &
420 &
421 \\
422
423 \hline\member{setDropTolerance}&
424 &
425 &
426 &
427 &
428 &
429 &
430 &
431 \\
432
433 \hline\member{setDropStorage}&
434 &
435 &
436 &
437 &
438 &
439 &
440 &
441 \\
442
443 \hline\member{setRelaxationFactor}&
444 &
445 &
446 &
447 &
448 &
449 $\checkmark$ &
450 &
451 \\
452
453 \hline\member{adaptInnerTolerance}&
454 &
455 &
456 &
457 &
458 &
459 &
460 &
461 \\
462
463 \hline\member{setInnerIterMax}&
464 &
465 &
466 &
467 &
468 &
469 &
470 &
471 \\
472 \end{tabular}
473 }
474 \caption{Preconditioners available for \finley and the \PASO package and the relevant options in \class{SolverOptions}. \label{TAB FINLEY SOLVER OPTIONS 2}}
475 \end{table}
476
477 \section{Linear Solvers in \SolverOptions}
478 Table~\ref{TAB FINLEY SOLVER OPTIONS 1} and
479 Table~\ref{TAB FINLEY SOLVER OPTIONS 2} show the solvers and preconditioners supported by
480 \finley through the \PASO library. Currently direct solvers are not supported under MPI.
481 By default, \finley is using the iterative solvers \PCG for symmetric and \BiCGStab for non-symmetric problems.
482 If the direct solver is selected which can be useful when solving very ill-posed equations
483 \finley uses the \MKL \footnote{If the stiffness matrix is non-regular \MKL may return without
484 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
485 a suitable iterative solver from the \PASO is used.
486
487 \section{Functions}
488 \begin{funcdesc}{ReadMesh}{fileName \optional{, \optional{integrationOrder=-1}, optimize=True}}
489 creates a \Domain object form the FEM mesh defined in
490 file \var{fileName}. The file must be given the \finley file format.
491 If \var{integrationOrder} is positive, a numerical integration scheme
492 chosen which is accurate on each element up to a polynomial of
493 degree \var{integrationOrder} \index{integration order}. Otherwise
494 an appropriate integration order is chosen independently.
495 By default the labeling of mesh nodes and element distribution is
496 optimized. Set \var{optimize=False} to switch off relabeling and redistribution.
497 \end{funcdesc}
498
499 \begin{funcdesc}{ReadGmsh}{fileName \optional{, \optional{integrationOrder=-1}, optimize=True\optional{, useMacroElements=False}}}
500 creates a \Domain object form the FEM mesh defined in
501 file \var{fileName}. The file must be given the \gmshextern file format.
502 If \var{integrationOrder} is positive, a numerical integration scheme
503 chosen which is accurate on each element up to a polynomial of
504 degree \var{integrationOrder} \index{integration order}. Otherwise
505 an appropriate integration order is chosen independently.
506 By default the labeling of mesh nodes and element distribution is
507 optimized. Set \var{optimize=False} to switch off relabeling and redistribution.
508 If \var{useMacroElements} is set, second order elements are interpreted as macro elements~\index{macro elements}.
509 Currently \function{ReadGmsh} does not support MPI.
510 \end{funcdesc}
511
512 \begin{funcdesc}{MakeDomain}{design\optional{, integrationOrder=-1\optional{, optimizeLabeling=True\optional{, useMacroElements=False}}}}
513 Creates a Finley \Domain from a \class{Design} object from \pycad using \gmshextern.
514 The \class{Design} \var{design} defines the geometry.
515 If \var{integrationOrder} is positive, a numerical integration scheme
516 chosen which is accurate on each element up to a polynomial of
517 degree \var{integrationOrder} \index{integration order}. Otherwise
518 an appropriate integration order is chosen independently.
519 Set \var{optimizeLabeling=False} to switch off relabeling and redistribution (not recommended).
520 If \var{useMacroElements} is set, macro elements~\index{macro elements} are used.
521 Currently \function{MakeDomain} does not support MPI.
522 \end{funcdesc}
523
524
525 \begin{funcdesc}{load}{fileName}
526 recovers a \Domain object from a dump file created by the \
527 \function{dump} method of a \Domain object defined in
528 file \var{fileName}.
529 \end{funcdesc}
530
531
532 \begin{funcdesc}{Rectangle}{n0,n1,order=1,l0=1.,l1=1., integrationOrder=-1, \\
533 periodic0=\False, periodic1=\False, useElementsOnFace=\False, useMacroElements=\False, optimize=\False}
534 Generates a \Domain object representing a two dimensional rectangle between
535 $(0,0)$ and $(l0,l1)$ with orthogonal edges. The rectangle is filled with
536 \var{n0} elements along the $x_0$-axis and
537 \var{n1} elements along the $x_1$-axis.
538 For \var{order}=1 and \var{order}=2
539 \finleyelement{Rec4} and
540 \finleyelement{Rec8} are used, respectively.
541 In the case of \var{useElementsOnFace}=\False,
542 \finleyelement{Line2} and
543 \finleyelement{Line3} are used to subdivide the edges of the rectangle, respectively.
544 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}.
545 In the case of \var{useElementsOnFace}=\True (this option should be used if gradients
546 are calculated on domain faces),
547 \finleyelement{Rec4Face} and
548 \finleyelement{Rec8Face} are used on the edges, respectively.
549 If \var{integrationOrder} is positive, a numerical integration scheme
550 chosen which is accurate on each element up to a polynomial of
551 degree \var{integrationOrder} \index{integration order}. Otherwise
552 an appropriate integration order is chosen independently. If
553 \var{periodic0}=\True, periodic boundary conditions \index{periodic boundary conditions}
554 along the $x_0$-directions are enforced. That means when for any solution of a PDE solved by \finley
555 the value on the line $x_0=0$ will be identical to the values on $x_0=\var{l0}$.
556 Correspondingly,
557 \var{periodic1}=\False sets periodic boundary conditions
558 in $x_1$-direction.
559 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.
560 \end{funcdesc}
561
562 \begin{funcdesc}{Brick}{n0,n1,n2,order=1,l0=1.,l1=1.,l2=1., integrationOrder=-1, \\
563 periodic0=\False,periodic1=\False,periodic2=\False,useElementsOnFace=\False, useMacroElements=\False, optimize=\False}
564 Generates a \Domain object representing a three dimensional brick between
565 $(0,0,0)$ and $(l0,l1,l2)$ with orthogonal faces. The brick is filled with
566 \var{n0} elements along the $x_0$-axis,
567 \var{n1} elements along the $x_1$-axis and
568 \var{n2} elements along the $x_2$-axis.
569 For \var{order}=1 and \var{order}=2
570 \finleyelement{Hex8} and
571 \finleyelement{Hex20} are used, respectively.
572 In the case of \var{useElementsOnFace}=\False,
573 \finleyelement{Rec4} and
574 \finleyelement{Rec8} are used to subdivide the faces of the brick, respectively.
575 In the case of \var{useElementsOnFace}=\True (this option should be used if gradients
576 are calculated on domain faces),
577 \finleyelement{Hex8Face} and
578 \finleyelement{Hex20Face} are used on the brick faces, respectively.
579 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}.
580 If \var{integrationOrder} is positive, a numerical integration scheme
581 chosen which is accurate on each element up to a polynomial of
582 degree \var{integrationOrder} \index{integration order}. Otherwise
583 an appropriate integration order is chosen independently. If
584 \var{periodic0}=\True, periodic boundary conditions \index{periodic boundary conditions}
585 along the $x_0$-directions are enforced. That means when for any solution of a PDE solved by \finley
586 the value on the plane $x_0=0$ will be identical to the values on $x_0=\var{l0}$. Correspondingly,
587 \var{periodic1}=\False and \var{periodic2}=\False sets periodic boundary conditions
588 in $x_1$-direction and $x_2$-direction, respectively.
589 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.
590 \end{funcdesc}
591
592 \begin{funcdesc}{GlueFaces}{meshList,safetyFactor=0.2,tolerance=1.e-13}
593 Generates a new \Domain object from the list \var{meshList} of \finley meshes.
594 Nodes in face elements whose difference of coordinates is less then \var{tolerance} times the
595 diameter of the domain are merged. The corresponding face elements are removed from the mesh.
596
597 TODO: explain \var{safetyFactor} and show an example.
598 \end{funcdesc}
599
600 \begin{funcdesc}{JoinFaces}{meshList,safetyFactor=0.2,tolerance=1.e-13}
601 Generates a new \Domain object from the list \var{meshList} of \finley meshes.
602 Face elements whose nodes coordinates have difference is less then \var{tolerance} times the
603 diameter of the domain are combined to form a contact element \index{element!contact}
604 The corresponding face elements are removed from the mesh.
605
606 TODO: explain \var{safetyFactor} and show an example.
607 \end{funcdesc}

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