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Finished user's guide chapters up to Appendix.

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 \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} & \member{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:&later&later&$\checkmark$&$\checkmark$&$\checkmark$&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{setNumSweeps}& & &$\checkmark$&$\checkmark$& & & &\\
404 \hline
405 \member{setNumPreSweeps}& &$\checkmark$& & & & & &\\
406 \hline
407 \member{setNumPostSweeps}& &$\checkmark$& & & & & &\\
408 \hline
409 \member{setInnerTolerance}& & & & & & & &\\
410 \hline
411 \member{setDropTolerance}& & & & & & & &\\
412 \hline
413 \member{setDropStorage}& & & & & & & &\\
414 \hline
415 \member{setRelaxationFactor}& & & & & &$\checkmark$& &\\
416 \hline
417 \member{adaptInnerTolerance}& & & & & & & &\\
418 \hline
419 \member{setInnerIterMax}& & & & & & & &\\
420 \end{tabular}
421 }
422 \caption{Preconditioners available for \finley and the \PASO package and the
423 relevant options in \class{SolverOptions}.
424 \label{TAB FINLEY SOLVER OPTIONS 2}}
425 \end{table}
426
427 \section{Functions}
428 \begin{funcdesc}{ReadMesh}{fileName \optional{, \optional{integrationOrder=-1}, optimize=True}}
429 creates a \Domain object from the FEM mesh defined in file \var{fileName}.
430 The file must be in the \finley file format.
431 If \var{integrationOrder} is positive, a numerical integration scheme is chosen
432 which is accurate on each element up to a polynomial of degree
433 \var{integrationOrder}\index{integration order}.
434 Otherwise an appropriate integration order is chosen independently.
435 By default the labeling of mesh nodes and element distribution is optimized.
436 Set \var{optimize=False} to switch off relabeling and redistribution.
437 \end{funcdesc}
438
439 \begin{funcdesc}{ReadGmsh}{fileName \optional{, \optional{integrationOrder=-1}, optimize=True\optional{, useMacroElements=False}}}
440 creates a \Domain object from the FEM mesh defined in file \var{fileName}.
441 The file must be in the \gmshextern file format.
442 If \var{integrationOrder} is positive, a numerical integration scheme is chosen
443 which is accurate on each element up to a polynomial of degree
444 \var{integrationOrder}\index{integration order}.
445 Otherwise an appropriate integration order is chosen independently.
446 By default the labeling of mesh nodes and element distribution is optimized.
447 Set \var{optimize=False} to switch off relabeling and redistribution.
448 If \var{useMacroElements} is set, second order elements are interpreted as
449 macro elements\index{macro elements}.
450 Currently \function{ReadGmsh} does not support \MPI.
451 \end{funcdesc}
452
453 \begin{funcdesc}{MakeDomain}{design\optional{, integrationOrder=-1\optional{, optimizeLabeling=True\optional{, useMacroElements=False}}}}
454 creates a \finley \Domain from a \pycad \class{Design} object using \gmshextern.
455 The \class{Design} \var{design} defines the geometry.
456 If \var{integrationOrder} is positive, a numerical integration scheme is chosen
457 which is accurate on each element up to a polynomial of degree
458 \var{integrationOrder}\index{integration order}.
459 Otherwise an appropriate integration order is chosen independently.
460 Set \var{optimizeLabeling=False} to switch off relabeling and redistribution
461 (not recommended).
462 If \var{useMacroElements} is set, macro elements\index{macro elements} are used.
463 Currently \function{MakeDomain} does not support \MPI.
464 \end{funcdesc}
465
466 \begin{funcdesc}{load}{fileName}
467 recovers a \Domain object from a dump file \var{fileName} created by the
468 \function{dump} method of a \Domain object.
469 \end{funcdesc}
470
471 \begin{funcdesc}{Rectangle}{n0,n1,order=1,l0=1.,l1=1., integrationOrder=-1, \\
472 periodic0=\False, periodic1=\False, useElementsOnFace=\False, useMacroElements=\False,\\ optimize=\False}
473 generates a \Domain object representing a two-dimensional rectangle between
474 $(0,0)$ and $(l0,l1)$ with orthogonal edges.
475 The rectangle is filled with \var{n0} elements along the $x_0$-axis and
476 \var{n1} elements along the $x_1$-axis.
477 For \var{order}=1 and \var{order}=2, elements of type \finleyelement{Rec4} and
478 \finleyelement{Rec8} are used, respectively.
479 In the case of \var{useElementsOnFace}=\False, \finleyelement{Line2} and
480 \finleyelement{Line3} are used to subdivide the edges of the rectangle, respectively.
481 If \var{order}=-1, \finleyelement{Rec8Macro} and \finleyelement{Line3Macro}\index{macro elements}
482 are used. This option should be used when solving incompressible fluid flow
483 problems, e.g. \class{StokesProblemCartesian}.
484 In the case of \var{useElementsOnFace}=\True (this option should be used if
485 gradients are calculated on domain faces), \finleyelement{Rec4Face} and
486 \finleyelement{Rec8Face} are used on the edges, respectively.
487 If \var{integrationOrder} is positive, a numerical integration scheme is chosen
488 which is accurate on each element up to a polynomial of degree
489 \var{integrationOrder}\index{integration order}.
490 Otherwise an appropriate integration order is chosen independently.
491 If \var{periodic0}=\True, periodic boundary conditions\index{periodic boundary conditions}
492 along the $x_0$-direction are enforced.
493 That means for any solution of a PDE solved by \finley the values on the line
494 $x_0=0$ will be identical to the values on $x_0=\var{l0}$.
495 Correspondingly, \var{periodic1}=\True sets periodic boundary conditions in the
496 $x_1$-direction.
497 If \var{optimize}=\True mesh node relabeling will be attempted to reduce the
498 computation and also ParMETIS will be used to improve the mesh partition if
499 running on multiple CPUs with \MPI.
500 \end{funcdesc}
501
502 \begin{funcdesc}{Brick}{n0,n1,n2,order=1,l0=1.,l1=1.,l2=1., integrationOrder=-1,
503 periodic0=\False, periodic1=\False, \\ periodic2=\False, useElementsOnFace=\False, useMacroElements=\False, optimize=\False}
504 generates a \Domain object representing a three-dimensional brick between
505 $(0,0,0)$ and $(l0,l1,l2)$ with orthogonal faces. The brick is filled with
506 \var{n0} elements along the $x_0$-axis,
507 \var{n1} elements along the $x_1$-axis and
508 \var{n2} elements along the $x_2$-axis.
509 For \var{order}=1 and \var{order}=2, elements of type \finleyelement{Hex8} and
510 \finleyelement{Hex20} are used, respectively.
511 In the case of \var{useElementsOnFace}=\False, \finleyelement{Rec4} and
512 \finleyelement{Rec8} are used to subdivide the faces of the brick, respectively.
513 In the case of \var{useElementsOnFace}=\True (this option should be used if
514 gradients are calculated on domain faces), \finleyelement{Hex8Face} and
515 \finleyelement{Hex20Face} are used on the brick faces, respectively.
516 If \var{order}=-1, \finleyelement{Hex20Macro} and \finleyelement{Rec8Macro}\index{macro elements}
517 are used. This option should be used when solving incompressible fluid flow
518 problems, e.g. \class{StokesProblemCartesian}.
519 If \var{integrationOrder} is positive, a numerical integration scheme is chosen
520 which is accurate on each element up to a polynomial of degree
521 \var{integrationOrder}\index{integration order}.
522 Otherwise an appropriate integration order is chosen independently.
523 If \var{periodic0}=\True, periodic boundary conditions\index{periodic boundary conditions}
524 along the $x_0$-direction are enforced.
525 That means for any solution of a PDE solved by \finley the values on the plane
526 $x_0=0$ will be identical to the values on $x_0=\var{l0}$.
527 Correspondingly, \var{periodic1}=\True and \var{periodic2}=\True sets periodic
528 boundary conditions in the $x_1$-direction and $x_2$-direction, respectively.
529 If \var{optimize}=\True mesh node relabeling will be attempted to reduce the
530 computation and also ParMETIS will be used to improve the mesh partition if
531 running on multiple CPUs with \MPI.
532 \end{funcdesc}
533
534 \begin{funcdesc}{GlueFaces}{meshList,safetyFactor=0.2,tolerance=1.e-13}
535 generates a new \Domain object from the list \var{meshList} of \finley meshes.
536 Nodes in face elements whose difference of coordinates is less than
537 \var{tolerance} times the diameter of the domain are merged.
538 The corresponding face elements are removed from the mesh.
539 %TODO: explain \var{safetyFactor} and show an example.
540 \end{funcdesc}
541
542 \begin{funcdesc}{JoinFaces}{meshList,safetyFactor=0.2,tolerance=1.e-13}
543 generates a new \Domain object from the list \var{meshList} of \finley meshes.
544 Face elements whose node coordinates differ by less than \var{tolerance} times
545 the diameter of the domain are combined to form a contact element\index{element!contact}.
546 The corresponding face elements are removed from the mesh.
547 %TODO: explain \var{safetyFactor} and show an example.
548 \end{funcdesc}
549

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