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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 \finley Module}\label{CHAPTER ON FINLEY}
16
17 \begin{figure}
18 \centerline{\includegraphics[width=\figwidth]{FinleyMesh}}
19 \caption{Subdivision of an Ellipse into triangles order 1 (\finleyelement{Tri3})}
20 \label{FINLEY FIG 0}
21 \end{figure}
22
23 \begin{figure}
24 \centerline{\includegraphics[width=\figwidth]{FinleyContact}}
25 \caption{Mesh around a contact region (\finleyelement{Rec4})}
26 \label{FINLEY FIG 01}
27 \end{figure}
28
29 \declaremodule{extension}{finley} \modulesynopsis{Solving linear, steady partial differential equations using
30 finite elements}
31
32 {\it finley} is a library of C functions solving linear, steady partial differential equations
33 \index{partial differential equations} (PDEs) or systems of PDEs using isoparametrical finite
34 elements \index{FEM!isoparametrical}.
35 It supports unstructured, 1D, 2D and 3D meshes. The module \finley provides an access to the
36 library through the \LinearPDE class of \escript supporting its full functionality. {\it finley}
37 is parallelized using the OpenMP \index{OpenMP} paradigm.
38
39 \section{Formulation}
40
41 For a single PDE with a solution with a single component the linear PDE is defined in the
42 following form:
43 \begin{equation}\label{FINLEY.SINGLE.1}
44 \begin{array}{cl} &
45 \displaystyle{
46 \int_{\Omega}
47 A_{jl} \cdot v_{,j}u_{,l}+ B_{j} \cdot v_{,j} u+ C_{l} \cdot v u_{,l}+D \cdot vu \; d\Omega } \\
48 + & \displaystyle{\int_{\Gamma} d \cdot vu \; d{\Gamma} }
49 + \displaystyle{\int_{\Gamma^{contact}} d^{contact} \cdot [v][u] \; d{\Gamma} } \\
50 = & \displaystyle{\int_{\Omega} X_{j} \cdot v_{,j}+ Y \cdot v \; d\Omega }\\
51 + & \displaystyle{\int_{\Gamma} y \cdot v \; d{\Gamma}} +
52 \displaystyle{\int_{\Gamma^{contact}} y^{contact}\cdot [v] \; d{\Gamma}} \\
53 \end{array}
54 \end{equation}
55
56 \section{Meshes}
57 \label{FINLEY 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 parameterization 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 \centering
128 \begin{tabular}{l|llll}
129 \bfseries interior & face & rich face & contact & rich contact\\
130 \hline
131 \finleyelement{Line2} & \finleyelement{Point1} & \finleyelement{Line2Face} & \finleyelement{Point1_Contact} & \finleyelement{Line2Face_Contact}\\
132 \finleyelement{Line3} & \finleyelement{Point1} & \finleyelement{Line3Face} & \finleyelement{Point1_Contact} & \finleyelement{Line3Face_Contact}\\
133 \finleyelement{Tri3} & \finleyelement{Line2} & \finleyelement{Tri3Face} & \finleyelement{Line2_Contact} & \finleyelement{Tri3Face_Contact}\\
134 \finleyelement{Tri6} & \finleyelement{Line3} & \finleyelement{Tri6Face} & \finleyelement{Line3_Contact} & \finleyelement{Tri6Face_Contact}\\
135 \finleyelement{Rec4} & \finleyelement{Line2} & \finleyelement{Rec4Face} & \finleyelement{Line2_Contact} & \finleyelement{Rec4Face_Contact}\\
136 \finleyelement{Rec8} & \finleyelement{Line3} & \finleyelement{Rec8Face} & \finleyelement{Line3_Contact} & \finleyelement{Rec8Face_Contact}\\
137 \finleyelement{Rec9} & \finleyelement{Line3} & \finleyelement{Rec9Face} & \finleyelement{Line3_Contact} & \finleyelement{Rec9Face_Contact}\\
138 \finleyelement{Tet4} & \finleyelement{Tri6} & \finleyelement{Tet4Face} & \finleyelement{Tri6_Contact} & \finleyelement{Tet4Face_Contact}\\
139 \finleyelement{Tet10} & \finleyelement{Tri9} & \finleyelement{Tet10Face} & \finleyelement{Tri9_Contact} & \finleyelement{Tet10Face_Contact}\\
140 \finleyelement{Hex8} & \finleyelement{Rec4} & \finleyelement{Hex8Face} & \finleyelement{Rec4_Contact} & \finleyelement{Hex8Face_Contact}\\
141 \finleyelement{Hex20} & \finleyelement{Rec8} & \finleyelement{Hex20Face} & \finleyelement{Rec8_Contact} & \finleyelement{Hex20Face_Contact}\\
142 \finleyelement{Hex27} & \finleyelement{Rec9} & N\textbackslash A & N\textbackslash A & N\textbackslash A\\
143 \finleyelement{Hex27Macro} & \finleyelement{Rec9Macro} & N\textbackslash A & N\textbackslash A & N\textbackslash A\\
144 \finleyelement{Tet10Macro} & \finleyelement{Tri6Macro} & N\textbackslash A & N\textbackslash A & N\textbackslash A\\
145 \finleyelement{Rec9Macro} & \finleyelement{Line3Macro} & N\textbackslash A & N\textbackslash A & N\textbackslash A\\
146 \finleyelement{Tri6Macro} & \finleyelement{Line3Macro} & N\textbackslash A & N\textbackslash A & N\textbackslash A\\
147 \end{tabular}
148 \caption{Finley elements and corresponding elements to be used on domain faces and contacts.
149 The rich types have to be used if the gradient of function is to be calculated on faces and contacts, respectively.}
150 \label{FINLEY TAB 1}
151 \end{table}
152
153 The native \finley file format is defined as follows.
154 Each node \var{i} has \var{dim} spatial coordinates \var{Node[i]}, a reference number
155 \var{Node_ref[i]}, a degree of freedom \var{Node_DOF[i]} and tag \var{Node_tag[i]}.
156 In most cases \var{Node_DOF[i]}=\var{Node_ref[i]} however, for periodic boundary conditions,
157 \var{Node_DOF[i]} is chosen differently, see example below. The tag can be used to mark nodes sharing
158 the same properties. Element \var{i} is defined by the \var{Element_numNodes} nodes \var{Element_Nodes[i]}
159 which is a list of node reference numbers. The order is crucial.
160 It has a reference number \var{Element_ref[i]} and a tag \var{Element_tag[i]}. The tag
161 can be used to mark elements sharing the same properties. For instance elements above
162 a contact region are marked with $2$ and elements below a contact region are marked with $1$.
163 \var{Element_Type} and \var{Element_Num} give the element type and the number of elements in the mesh.
164 Analogue notations are used for face and contact elements. The following Python script
165 prints the mesh definition in the \finley file format:
166 \begin{python}
167 print "%s\n"%mesh_name
168 # node coordinates:
169 print "%dD-nodes %d\n"%(dim,numNodes)
170 for i in range(numNodes):
171 print "%d %d %d"%(Node_ref[i],Node_DOF[i],Node_tag[i])
172 for j in range(dim): print " %e"%Node[i][j]
173 print "\n"
174 # interior elements
175 print "%s %d\n"%(Element_Type,Element_Num)
176 for i in range(Element_Num):
177 print "%d %d"%(Element_ref[i],Element_tag[i])
178 for j in range(Element_numNodes): print " %d"%Element_Nodes[i][j]
179 print "\n"
180 # face elements
181 print "%s %d\n"%(FaceElement_Type,FaceElement_Num)
182 for i in range(FaceElement_Num):
183 print "%d %d"%(FaceElement_ref[i],FaceElement_tag[i])
184 for j in range(FaceElement_numNodes): print " %d"%FaceElement_Nodes[i][j]
185 print "\n"
186 # contact elements
187 print "%s %d\n"%(ContactElement_Type,ContactElement_Num)
188 for i in range(ContactElement_Num):
189 print "%d %d"%(ContactElement_ref[i],ContactElement_tag[i])
190 for j in range(ContactElement_numNodes): print " %d"%ContactElement_Nodes[i][j]
191 print "\n"
192 # point sources (not supported yet)
193 write("Point1 0",face_element_type,numFaceElements)
194 \end{python}
195
196 The following example of a mesh file defines the mesh shown in \fig{FINLEY FIG 01}:
197 \begin{verbatim}
198 Example 1
199 2D Nodes 16
200 0 0 0 0. 0.
201 2 2 0 0.33 0.
202 3 3 0 0.66 0.
203 7 4 0 1. 0.
204 5 5 0 0. 0.5
205 6 6 0 0.33 0.5
206 8 8 0 0.66 0.5
207 10 10 0 1.0 0.5
208 12 12 0 0. 0.5
209 9 9 0 0.33 0.5
210 13 13 0 0.66 0.5
211 15 15 0 1.0 0.5
212 16 16 0 0. 1.0
213 18 18 0 0.33 1.0
214 19 19 0 0.66 1.0
215 20 20 0 1.0 1.0
216 Rec4 6
217 0 1 0 2 6 5
218 1 1 2 3 8 6
219 2 1 3 7 10 8
220 5 2 12 9 18 16
221 7 2 13 19 18 9
222 10 2 20 19 13 15
223 Line2 0
224 Line2_Contact 3
225 4 0 9 12 6 5
226 3 0 13 9 8 6
227 6 0 15 13 10 8
228 Point1 0
229 \end{verbatim}
230 Notice that the order in which the nodes and elements are given is arbitrary.
231 In the case that rich contact elements are used the contact element section gets
232 the form
233 \begin{verbatim}
234 Rec4Face_Contact 3
235 4 0 9 12 16 18 6 5 0 2
236 3 0 13 9 18 19 8 6 2 3
237 6 0 15 13 19 20 10 8 3 7
238 \end{verbatim}
239 Periodic boundary condition \index{boundary conditions!periodic} can be introduced by altering \var{Node_DOF}.
240 It allows identification of nodes even if they have different physical locations. For instance, to
241 enforce periodic boundary conditions at the face $x_0=0$ and $x_0=1$ one identifies
242 the degrees of freedom for nodes $0$, $5$, $12$ and $16$ with the degrees of freedom for
243 $7$, $10$, $15$ and $20$, respectively. The node section of the \finley mesh gets now the form:
244 \begin{verbatim}
245 2D Nodes 16
246 0 0 0 0. 0.
247 2 2 0 0.33 0.
248 3 3 0 0.66 0.
249 7 0 0 1. 0.
250 5 5 0 0. 0.5
251 6 6 0 0.33 0.5
252 8 8 0 0.66 0.5
253 10 5 0 1.0 0.5
254 12 12 0 0. 0.5
255 9 9 0 0.33 0.5
256 13 13 0 0.66 0.5
257 15 12 0 1.0 0.5
258 16 16 0 0. 1.0
259 18 18 0 0.33 1.0
260 19 19 0 0.66 1.0
261 20 16 0 1.0 1.0
262 \end{verbatim}
263
264 \clearpage
265 \input{finleyelements}
266 \clearpage
267
268 \begin{figure}[th]
269 \begin{center}
270 \subfigure[Triangle]{\label{FINLEY MACRO TRI}\includegraphics[scale=0.25]{FinleyMacroTri}}
271 \subfigure[Quadrilateral]{\label{FINLEY MACRO REC}\includegraphics[scale=0.25]{FinleyMacroRec}}
272 \includegraphics[scale=0.2]{FinleyMacroLeg}
273 \end{center}
274 Macro elements in \finley.
275 \end{figure}
276
277 \section{Macro Elements}
278 \label{SEC FINLEY MACRO}
279 \finley supports the usage of macro elements~\index{macro elements} which can be used to
280 achieve LBB compliance when solving incompressible fluid flow problems. LBB compliance is required to
281 get a problem which has a unique solution for pressure and velocity. For macro elements the
282 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
283 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
284 one uses a linear approximation of the velocity together with a linear approximation of the pressure but on elements
285 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
286 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.
287 Typically macro elements are only required to use when an incompressible fluid flow problem
288 is solved, e.g the Stokes problem in Section \ref{STOKES PROBLEM}. Please see Section~\ref{FINLEY MESHES} for
289 more details on the supported macro elements.
290
291
292
293 \begin{table}
294 {\scriptsize
295 \begin{tabular}{l||c|c|c|c|c|c|c|c}
296 \member{setSolverMethod} & \member{DIRECT}& \member{PCG} & \member{GMRES} & \member{TFQMR} & \member{MINRES} & \member{PRES20} & \member{BICGSTAB} & \member{LUMPING} \\
297 \hline
298 \hline
299 \member{setReordering} & $\checkmark$ & & & & & &\\
300 \hline \member{setRestart} & & & $\checkmark$ & & & $20$ & \\
301 \hline\member{setTruncation} & & & $\checkmark$ & & & $5$ & \\
302 \hline\member{setIterMax} & & $\checkmark$& $\checkmark$ & $\checkmark$& $\checkmark$& $\checkmark$ & $\checkmark$ \\
303 \hline\member{setTolerance} & & $\checkmark$& $\checkmark$ & $\checkmark$& $\checkmark$& $\checkmark$ & $\checkmark$ \\
304 \hline\member{setAbsoluteTolerance} & & $\checkmark$& $\checkmark$ & $\checkmark$& $\checkmark$& $\checkmark$ & $\checkmark$ \\
305 \hline\member{setReordering} & $\checkmark$ & & & & & & & \\
306 \end{tabular}
307 }
308 \caption{Solvers available for
309 \finley
310 and the \PASO package and the relevant options in \class{SolverOptions}.
311 \MKL supports
312 \MINIMUMFILLIN
313 and
314 \NESTEDDESCTION
315 reordering.
316 Currently the \UMFPACK interface does not support any reordering.
317 \label{TAB FINLEY SOLVER OPTIONS 1} }
318 \end{table}
319
320 \begin{table}
321 {\scriptsize
322 \begin{tabular}{l||c|c|c|c|c|c|c|c}
323 \member{setPreconditioner} &
324 \member{NO_PRECONDITIONER} &
325 \member{AMG} &
326 \member{JACOBI} &
327 \member{GAUSS_SEIDEL}&
328 \member{REC_ILU}&
329 \member{RILU} &
330 \member{ILU0} &
331 \member{DIRECT} \\
332 \hline
333 status: &
334 later &
335 later &
336 $\checkmark$ &
337 $\checkmark$&
338 $\checkmark$ &
339 later &
340 $\checkmark$ &
341 later \\
342 \hline
343 \hline
344 \member{setCoarsening}&
345 &
346 $\checkmark$ &
347 &
348 &
349 &
350 &
351 &
352 \\
353
354
355 \hline\member{setLevelMax}&
356 &
357 $\checkmark$ &
358 &
359 &
360 &
361 &
362 &
363 \\
364
365 \hline\member{setCoarseningThreshold}&
366 &
367 $\checkmark$ &
368 &
369 &
370 &
371 &
372 &
373 \\
374
375 \hline\member{setMinCoarseMatrixSize} &
376 &
377 $\checkmark$ &
378 &
379 &
380 &
381 &
382 &
383 \\
384
385 \hline\member{setNumSweeps} &
386 &
387 &
388 $\checkmark$ &
389 $\checkmark$ &
390 &
391 &
392 &
393 \\
394
395 \hline\member{setNumPreSweeps}&
396 &
397 $\checkmark$ &
398 &
399 &
400 &
401 &
402 &
403 \\
404
405 \hline\member{setNumPostSweeps} &
406 &
407 $\checkmark$ &
408 &
409 &
410 &
411 &
412 &
413 \\
414
415 \hline\member{setInnerTolerance}&
416 &
417 &
418 &
419 &
420 &
421 &
422 &
423 \\
424
425 \hline\member{setDropTolerance}&
426 &
427 &
428 &
429 &
430 &
431 &
432 &
433 \\
434
435 \hline\member{setDropStorage}&
436 &
437 &
438 &
439 &
440 &
441 &
442 &
443 \\
444
445 \hline\member{setRelaxationFactor}&
446 &
447 &
448 &
449 &
450 &
451 $\checkmark$ &
452 &
453 \\
454
455 \hline\member{adaptInnerTolerance}&
456 &
457 &
458 &
459 &
460 &
461 &
462 &
463 \\
464
465 \hline\member{setInnerIterMax}&
466 &
467 &
468 &
469 &
470 &
471 &
472 &
473 \\
474 \end{tabular}
475 }
476 \caption{Preconditioners available for \finley and the \PASO package and the relevant options in \class{SolverOptions}. \label{TAB FINLEY SOLVER OPTIONS 2}}
477 \end{table}
478
479 \section{Linear Solvers in \SolverOptions}
480 Table~\ref{TAB FINLEY SOLVER OPTIONS 1} and
481 Table~\ref{TAB FINLEY SOLVER OPTIONS 2} show the solvers and preconditioners supported by
482 \finley through the \PASO library. Currently direct solvers are not supported under MPI.
483 By default, \finley is using the iterative solvers \PCG for symmetric and \BiCGStab for non-symmetric problems.
484 If the direct solver is selected which can be useful when solving very ill-posed equations
485 \finley uses the \MKL \footnote{If the stiffness matrix is non-regular \MKL may return without
486 returning a proper error code. If you observe suspicious solutions when using MKL, this may be caused by a non-invertible operator. } solver package. If \MKL is not available \UMFPACK is used. If \UMFPACK is not available
487 a suitable iterative solver from the \PASO is used.
488
489 \section{Functions}
490 \begin{funcdesc}{ReadMesh}{fileName \optional{, \optional{integrationOrder=-1}, optimize=True}}
491 creates a \Domain object form the FEM mesh defined in
492 file \var{fileName}. The file must be given the \finley file format.
493 If \var{integrationOrder} is positive, a numerical integration scheme
494 chosen which is accurate on each element up to a polynomial of
495 degree \var{integrationOrder} \index{integration order}. Otherwise
496 an appropriate integration order is chosen independently.
497 By default the labeling of mesh nodes and element distribution is
498 optimized. Set \var{optimize=False} to switch off relabeling and redistribution.
499 \end{funcdesc}
500
501 \begin{funcdesc}{ReadGmsh}{fileName \optional{, \optional{integrationOrder=-1}, optimize=True\optional{, useMacroElements=False}}}
502 creates a \Domain object form the FEM mesh defined in
503 file \var{fileName}. The file must be given the \gmshextern file format.
504 If \var{integrationOrder} is positive, a numerical integration scheme
505 chosen which is accurate on each element up to a polynomial of
506 degree \var{integrationOrder} \index{integration order}. Otherwise
507 an appropriate integration order is chosen independently.
508 By default the labeling of mesh nodes and element distribution is
509 optimized. Set \var{optimize=False} to switch off relabeling and redistribution.
510 If \var{useMacroElements} is set, second order elements are interpreted as macro elements~\index{macro elements}.
511 Currently \function{ReadGmsh} does not support MPI.
512 \end{funcdesc}
513
514 \begin{funcdesc}{MakeDomain}{design\optional{, integrationOrder=-1\optional{, optimizeLabeling=True\optional{, useMacroElements=False}}}}
515 Creates a Finley \Domain from a \class{Design} object from \pycad using \gmshextern.
516 The \class{Design} \var{design} defines the geometry.
517 If \var{integrationOrder} is positive, a numerical integration scheme
518 chosen which is accurate on each element up to a polynomial of
519 degree \var{integrationOrder} \index{integration order}. Otherwise
520 an appropriate integration order is chosen independently.
521 Set \var{optimizeLabeling=False} to switch off relabeling and redistribution (not recommended).
522 If \var{useMacroElements} is set, macro elements~\index{macro elements} are used.
523 Currently \function{MakeDomain} does not support MPI.
524 \end{funcdesc}
525
526
527 \begin{funcdesc}{load}{fileName}
528 recovers a \Domain object from a dump file created by the \
529 \function{dump} method of a \Domain object defined in
530 file \var{fileName}.
531 \end{funcdesc}
532
533
534 \begin{funcdesc}{Rectangle}{n0,n1,order=1,l0=1.,l1=1., integrationOrder=-1, \\
535 periodic0=\False, periodic1=\False, useElementsOnFace=\False, useMacroElements=\False,\\ optimize=\False}
536 Generates a \Domain object representing a two dimensional rectangle between
537 $(0,0)$ and $(l0,l1)$ with orthogonal edges. The rectangle is filled with
538 \var{n0} elements along the $x_0$-axis and
539 \var{n1} elements along the $x_1$-axis.
540 For \var{order}=1 and \var{order}=2
541 \finleyelement{Rec4} and
542 \finleyelement{Rec8} are used, respectively.
543 In the case of \var{useElementsOnFace}=\False,
544 \finleyelement{Line2} and
545 \finleyelement{Line3} are used to subdivide the edges of the rectangle, respectively.
546 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}.
547 In the case of \var{useElementsOnFace}=\True (this option should be used if gradients
548 are calculated on domain faces),
549 \finleyelement{Rec4Face} and
550 \finleyelement{Rec8Face} are used on the edges, respectively.
551 If \var{integrationOrder} is positive, a numerical integration scheme
552 chosen which is accurate on each element up to a polynomial of
553 degree \var{integrationOrder} \index{integration order}. Otherwise
554 an appropriate integration order is chosen independently. If
555 \var{periodic0}=\True, periodic boundary conditions \index{periodic boundary conditions}
556 along the $x_0$-directions are enforced. That means when for any solution of a PDE solved by \finley
557 the value on the line $x_0=0$ will be identical to the values on $x_0=\var{l0}$.
558 Correspondingly,
559 \var{periodic1}=\False sets periodic boundary conditions
560 in $x_1$-direction.
561 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.
562 \end{funcdesc}
563
564 \begin{funcdesc}{Brick}{n0,n1,n2,order=1,l0=1.,l1=1.,l2=1., integrationOrder=-1,
565 periodic0=\False, periodic1=\False, \\ periodic2=\False, useElementsOnFace=\False, useMacroElements=\False, optimize=\False}
566 Generates a \Domain object representing a three dimensional brick between
567 $(0,0,0)$ and $(l0,l1,l2)$ with orthogonal faces. The brick is filled with
568 \var{n0} elements along the $x_0$-axis,
569 \var{n1} elements along the $x_1$-axis and
570 \var{n2} elements along the $x_2$-axis.
571 For \var{order}=1 and \var{order}=2
572 \finleyelement{Hex8} and
573 \finleyelement{Hex20} are used, respectively.
574 In the case of \var{useElementsOnFace}=\False,
575 \finleyelement{Rec4} and
576 \finleyelement{Rec8} are used to subdivide the faces of the brick, respectively.
577 In the case of \var{useElementsOnFace}=\True (this option should be used if gradients
578 are calculated on domain faces),
579 \finleyelement{Hex8Face} and
580 \finleyelement{Hex20Face} are used on the brick faces, respectively.
581 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}.
582 If \var{integrationOrder} is positive, a numerical integration scheme
583 chosen which is accurate on each element up to a polynomial of
584 degree \var{integrationOrder} \index{integration order}. Otherwise
585 an appropriate integration order is chosen independently. If
586 \var{periodic0}=\True, periodic boundary conditions \index{periodic boundary conditions}
587 along the $x_0$-directions are enforced. That means when for any solution of a PDE solved by \finley
588 the value on the plane $x_0=0$ will be identical to the values on $x_0=\var{l0}$. Correspondingly,
589 \var{periodic1}=\False and \var{periodic2}=\False sets periodic boundary conditions
590 in $x_1$-direction and $x_2$-direction, respectively.
591 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.
592 \end{funcdesc}
593
594 \begin{funcdesc}{GlueFaces}{meshList,safetyFactor=0.2,tolerance=1.e-13}
595 Generates a new \Domain object from the list \var{meshList} of \finley meshes.
596 Nodes in face elements whose difference of coordinates is less then \var{tolerance} times the
597 diameter of the domain are merged. The corresponding face elements are removed from the mesh.
598
599 TODO: explain \var{safetyFactor} and show an example.
600 \end{funcdesc}
601
602 \begin{funcdesc}{JoinFaces}{meshList,safetyFactor=0.2,tolerance=1.e-13}
603 Generates a new \Domain object from the list \var{meshList} of \finley meshes.
604 Face elements whose nodes coordinates have difference is less then \var{tolerance} times the
605 diameter of the domain are combined to form a contact element \index{element!contact}
606 The corresponding face elements are removed from the mesh.
607
608 TODO: explain \var{safetyFactor} and show an example.
609 \end{funcdesc}

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