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1 ksteube 1811
2 jfenwick 3989 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
3 jfenwick 6651 % Copyright (c) 2003-2018 by The University of Queensland
4 jfenwick 3989 % http://www.uq.edu.au
5 gross 625 %
6 ksteube 1811 % Primary Business: Queensland, Australia
7 jfenwick 6112 % Licensed under the Apache License, version 2.0
8     % http://www.apache.org/licenses/LICENSE-2.0
9 gross 625 %
10 jfenwick 3989 % Development until 2012 by Earth Systems Science Computational Center (ESSCC)
11 jfenwick 4657 % Development 2012-2013 by School of Earth Sciences
12     % Development from 2014 by Centre for Geoscience Computing (GeoComp)
13 jfenwick 3989 %
14     %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
15 jgs 102
16 caltinay 5295 \chapter{The \finley Module}\label{chap:finley}
17 caltinay 3306 %\declaremodule{extension}{finley}
18     %\modulesynopsis{Solving linear, steady partial differential equations using finite elements}
19 jgs 102
20 jfenwick 5658 The \finley library allows the creation of domains for solving
21     linear, steady partial differential
22 caltinay 4891 equations\index{partial differential equations} (PDEs) or systems
23 caltinay 3330 of PDEs using isoparametrical finite elements\index{FEM!isoparametrical}.
24     It supports unstructured 1D, 2D and 3D meshes.
25 jfenwick 5658 The PDEs themselves are represented by the \LinearPDE class
26     of \escript.
27     \finley is parallelized under both \OPENMP and \MPI.
28     A more restricted form of this library ({\it dudley}) is described in
29     Section~\ref{sec:dudley}.
30 jgs 102
31 gross 993 \section{Formulation}
32 caltinay 3330 For a single PDE that has a solution with a single component the linear PDE is
33     defined in the following form:
34 gross 993 \begin{equation}\label{FINLEY.SINGLE.1}
35     \begin{array}{cl} &
36     \displaystyle{
37 jfenwick 3295 \int_{\Omega}
38     A_{jl} \cdot v_{,j}u_{,l}+ B_{j} \cdot v_{,j} u+ C_{l} \cdot v u_{,l}+D \cdot vu \; d\Omega } \\
39     + & \displaystyle{\int_{\Gamma} d \cdot vu \; d{\Gamma} }
40     + \displaystyle{\int_{\Gamma^{contact}} d^{contact} \cdot [v][u] \; d{\Gamma} } \\
41     = & \displaystyle{\int_{\Omega} X_{j} \cdot v_{,j}+ Y \cdot v \; d\Omega }\\
42     + & \displaystyle{\int_{\Gamma} y \cdot v \; d{\Gamma}} +
43     \displaystyle{\int_{\Gamma^{contact}} y^{contact}\cdot [v] \; d{\Gamma}} \\
44 gross 993 \end{array}
45     \end{equation}
46    
47     \section{Meshes}
48 gross 2793 \label{FINLEY MESHES}
49 jgs 102
50 caltinay 3330 \begin{figure}
51     \centerline{\includegraphics{FinleyMesh}}
52     \caption{Subdivision of an Ellipse into triangles order 1 (\finleyelement{Tri3})}
53     \label{FINLEY FIG 0}
54     \end{figure}
55    
56     To understand the usage of \finley one needs to have an understanding of how
57     the finite element meshes\index{FEM!mesh} are defined.
58     \fig{FINLEY FIG 0} shows an example of the subdivision of an ellipse into
59     so-called elements\index{FEM!elements}\index{element}.
60     In this case, triangles have been used but other forms of subdivisions can be
61     constructed, e.g. quadrilaterals or, in the three-dimensional case, into
62     tetrahedra and hexahedra. The idea of the finite element method is to
63     approximate the solution by a function which is a polynomial of a certain order
64     and is continuous across its boundary to neighbour elements.
65     In the example of \fig{FINLEY FIG 0} a linear polynomial is used on each
66     triangle. As one can see, the triangulation is quite a poor approximation of
67     the ellipse. It can be improved by introducing a midpoint on each element edge
68     then positioning those nodes located on an edge expected to describe the
69     boundary, onto the boundary.
70     In this case the triangle gets a curved edge which requires a parameterization
71     of the triangle using a quadratic polynomial.
72     For this case, the solution is also approximated by a piecewise quadratic
73     polynomial (which explains the name isoparametrical elements),
74     see \Ref{Zienc,NumHand} for more details.
75     \finley also supports macro elements\index{macro elements}.
76     For these elements a piecewise linear approximation is used on an element which
77     is further subdivided (in the case of \finley halved).
78     As such, these elements do not provide more than a further mesh refinement but
79     should be used in the case of incompressible flows, see \class{StokesProblemCartesian}.
80     For these problems a linear approximation of the pressure across the element is
81     used (use the \ReducedSolutionFS) while the refined element is used to
82     approximate velocity. So a macro element provides a continuous pressure
83     approximation together with a velocity approximation on a refined mesh.
84     This approach is necessary to make sure that the incompressible flow has a
85     unique solution.
86    
87 jgs 102 The union of all elements defines the domain of the PDE.
88 caltinay 3330 Each element is defined by the nodes used to describe its shape.
89     In \fig{FINLEY FIG 0} the element, which has type \finleyelement{Tri3}, with
90     element reference number $19$\index{element!reference number} is defined by the
91     nodes with reference numbers $9$, $11$ and $0$\index{node!reference number}.
92     Notice that the order is counterclockwise.
93     The coefficients of the PDE are evaluated at integration nodes with each
94     individual element.
95     For quadrilateral elements a Gauss quadrature scheme is used.
96     In the case of triangular elements a modified form is applied.
97     The boundary of the domain is also subdivided into elements\index{element!face}.
98     In \fig{FINLEY FIG 0} line elements with two nodes are used.
99     The elements are also defined by their describing nodes, e.g. the face element
100     with reference number $20$, which has type \finleyelement{Line2}, is defined by
101     the nodes with the reference numbers $11$ and $0$.
102     Again the order is crucial, if moving from the first to second node the domain
103     has to lie on the left hand side (in the case of a two-dimensional surface
104     element the domain has to lie on the left hand side when moving
105     counterclockwise). If the gradient on the surface of the domain is to be
106     calculated rich face elements need to be used. Rich elements on a face are
107     identical to interior elements but with a modified order of nodes such that the
108     'first' face of the element aligns with the surface of the domain.
109     In \fig{FINLEY FIG 0} elements of the type \finleyelement{Tri3Face} are used.
110     The face element reference number $20$ as a rich face element is defined by the
111     nodes with reference numbers $11$, $0$ and $9$.
112     Notice that the face element $20$ is identical to the interior element $19$
113     except that, in this case, the order of the node is different to align the first
114     edge of the triangle (which is the edge starting with the first node) with the
115     boundary of the domain.
116 jgs 102
117 caltinay 3330 Be aware that face elements and elements in the interior of the domain must
118     match, i.e. a face element must be the face of an interior element or, in case
119     of a rich face element, it must be identical to an interior element.
120     If no face elements are specified \finley implicitly assumes homogeneous
121     natural boundary conditions\index{natural boundary conditions!homogeneous},
122     i.e. \var{d}=$0$ and \var{y}=$0$, on the entire boundary of the domain.
123     For inhomogeneous natural boundary conditions\index{natural boundary conditions!inhomogeneous},
124 caltinay 3293 the boundary must be described by face elements.
125 jgs 102
126 caltinay 3330 \begin{figure}
127     \centerline{\includegraphics{FinleyContact}}
128     \caption{Mesh around a contact region (\finleyelement{Rec4})}
129     \label{FINLEY FIG 01}
130     \end{figure}
131 jgs 102
132 caltinay 3330 If discontinuities of the PDE solution are considered, contact
133     elements\index{element!contact}\index{contact conditions} are introduced to
134     describe the contact region $\Gamma^{contact}$ even if $d^{contact}$ and
135     $y^{contact}$ are zero.
136     \fig{FINLEY FIG 01} shows a simple example of a mesh of rectangular elements
137     around a contact region $\Gamma^{contact}$\index{element!contact}.
138     The contact region is described by the elements $4$, $3$ and $6$.
139     Their element type is \finleyelement{Line2_Contact}.
140     The nodes $9$, $12$, $6$ and $5$ define contact element $4$, where the
141     coordinates of nodes $12$ and $5$ and nodes $4$ and $6$ are identical, with the
142     idea that nodes $12$ and $9$ are located above and nodes $5$ and $6$ below the
143     contact region.
144     Again, the order of the nodes within an element is crucial.
145     There is also the option of using rich elements if the gradient is to be
146     calculated on the contact region. Similarly to the rich face elements these
147     are constructed from two interior elements by reordering the nodes such that
148     the 'first' face of the element above and the 'first' face of the element below
149     the contact regions line up. The rich version of element $4$ is of type
150     \finleyelement{Rec4Face_Contact} and is defined by the nodes $9$, $12$, $16$,
151     $18$, $6$, $5$, $0$ and $2$.
152     \tab{FINLEY TAB 1} shows the interior element types and the corresponding
153     element types to be used on the face and contacts.
154     \fig{FINLEY.FIG:1}, \fig{FINLEY.FIG:2} and \fig{FINLEY.FIG:4} show the ordering
155     of the nodes within an element.
156 gross 2748
157 jgs 102 \begin{table}
158 caltinay 3293 \centering
159     \begin{tabular}{l|llll}
160 caltinay 3330 \textbf{interior}&\textbf{face}&\textbf{rich face}&\textbf{contact}&\textbf{rich contact}\\
161 caltinay 3293 \hline
162     \finleyelement{Line2} & \finleyelement{Point1} & \finleyelement{Line2Face} & \finleyelement{Point1_Contact} & \finleyelement{Line2Face_Contact}\\
163     \finleyelement{Line3} & \finleyelement{Point1} & \finleyelement{Line3Face} & \finleyelement{Point1_Contact} & \finleyelement{Line3Face_Contact}\\
164     \finleyelement{Tri3} & \finleyelement{Line2} & \finleyelement{Tri3Face} & \finleyelement{Line2_Contact} & \finleyelement{Tri3Face_Contact}\\
165     \finleyelement{Tri6} & \finleyelement{Line3} & \finleyelement{Tri6Face} & \finleyelement{Line3_Contact} & \finleyelement{Tri6Face_Contact}\\
166     \finleyelement{Rec4} & \finleyelement{Line2} & \finleyelement{Rec4Face} & \finleyelement{Line2_Contact} & \finleyelement{Rec4Face_Contact}\\
167     \finleyelement{Rec8} & \finleyelement{Line3} & \finleyelement{Rec8Face} & \finleyelement{Line3_Contact} & \finleyelement{Rec8Face_Contact}\\
168     \finleyelement{Rec9} & \finleyelement{Line3} & \finleyelement{Rec9Face} & \finleyelement{Line3_Contact} & \finleyelement{Rec9Face_Contact}\\
169     \finleyelement{Tet4} & \finleyelement{Tri6} & \finleyelement{Tet4Face} & \finleyelement{Tri6_Contact} & \finleyelement{Tet4Face_Contact}\\
170     \finleyelement{Tet10} & \finleyelement{Tri9} & \finleyelement{Tet10Face} & \finleyelement{Tri9_Contact} & \finleyelement{Tet10Face_Contact}\\
171     \finleyelement{Hex8} & \finleyelement{Rec4} & \finleyelement{Hex8Face} & \finleyelement{Rec4_Contact} & \finleyelement{Hex8Face_Contact}\\
172     \finleyelement{Hex20} & \finleyelement{Rec8} & \finleyelement{Hex20Face} & \finleyelement{Rec8_Contact} & \finleyelement{Hex20Face_Contact}\\
173 caltinay 3330 \finleyelement{Hex27} & \finleyelement{Rec9} & N/A & N/A & N/A\\
174     \finleyelement{Hex27Macro} & \finleyelement{Rec9Macro} & N/A & N/A & N/A\\
175     \finleyelement{Tet10Macro} & \finleyelement{Tri6Macro} & N/A & N/A & N/A\\
176     \finleyelement{Rec9Macro} & \finleyelement{Line3Macro} & N/A & N/A & N/A\\
177     \finleyelement{Tri6Macro} & \finleyelement{Line3Macro} & N/A & N/A & N/A\\
178 caltinay 3293 \end{tabular}
179 caltinay 3330 \caption{Finley elements and corresponding elements to be used on domain faces
180     and contacts.
181     The rich types have to be used if the gradient of the function is to be
182     calculated on faces and contacts, respectively.}
183 jgs 102 \label{FINLEY TAB 1}
184     \end{table}
185    
186     The native \finley file format is defined as follows.
187 caltinay 3330 Each node \var{i} has \var{dim} spatial coordinates \var{Node[i]}, a reference
188     number \var{Node_ref[i]}, a degree of freedom \var{Node_DOF[i]} and a tag
189     \var{Node_tag[i]}.
190     In most cases \var{Node_DOF[i]}=\var{Node_ref[i]} however, for periodic
191     boundary conditions, \var{Node_DOF[i]} is chosen differently, see example below.
192     The tag can be used to mark nodes sharing the same properties.
193     Element \var{i} is defined by the \var{Element_numNodes} nodes
194     \var{Element_Nodes[i]} which is a list of node reference numbers.
195     The order of these is crucial. Each element has a reference number
196     \var{Element_ref[i]} and a tag \var{Element_tag[i]}.
197     The tag can be used to mark elements sharing the same properties.
198     For instance elements above a contact region are marked with tag $2$ and
199     elements below a contact region are marked with tag $1$.
200     \var{Element_Type} and \var{Element_Num} give the element type and the number
201     of elements in the mesh.
202     Analogue notations are used for face and contact elements.
203 acodd 6932 %The following \PYTHON script prints the mesh definition in the \finley file
204     %format:
205     %\begin{python}
206     % print("%s\n"%mesh_name)
207     % # node coordinates:
208     % print("%dD-nodes %d\n"%(dim, numNodes))
209     % for i in range(numNodes):
210     % print("%d %d %d"%(Node_ref[i], Node_DOF[i], Node_tag[i]))
211     % for j in range(dim): print(" %e"%Node[i][j])
212     % print("\n")
213     % # interior elements
214     % print("%s %d\n"%(Element_Type, Element_Num))
215     % for i in range(Element_Num):
216     % print("%d %d"%(Element_ref[i], Element_tag[i]))
217     % for j in range(Element_numNodes): print(" %d"%Element_Nodes[i][j])
218     % print("\n")
219     % # face elements
220     % print("%s %d\n"%(FaceElement_Type, FaceElement_Num))
221     % for i in range(FaceElement_Num):
222     % print("%d %d"%(FaceElement_ref[i], FaceElement_tag[i]))
223     % for j in range(FaceElement_numNodes): print(" %d"%FaceElement_Nodes[i][j])
224     % print("\n")
225     % # contact elements
226     % print("%s %d\n"%(ContactElement_Type, ContactElement_Num))
227     % for i in range(ContactElement_Num):
228     % print("%d %d"%(ContactElement_ref[i], ContactElement_tag[i]))
229     % for j in range(ContactElement_numNodes):
230     % print(" %d"%ContactElement_Nodes[i][j])
231     % print("\n")
232     % # point sources (not supported yet)
233     % print("Point1 0")
234     %\end{python}
235 jgs 102
236 acodd 6932 %The following example of a mesh file defines the mesh shown in \fig{FINLEY FIG 01}:
237     %\begin{verbatim}
238     %Example 1
239     %2D Nodes 16
240     %0 0 0 0. 0.
241     %2 2 0 0.33 0.
242     %3 3 0 0.66 0.
243     %7 4 0 1. 0.
244     %5 5 0 0. 0.5
245     %6 6 0 0.33 0.5
246     %8 8 0 0.66 0.5
247     %10 10 0 1.0 0.5
248     %12 12 0 0. 0.5
249     %9 9 0 0.33 0.5
250     %13 13 0 0.66 0.5
251     %15 15 0 1.0 0.5
252     %16 16 0 0. 1.0
253     %18 18 0 0.33 1.0
254     %19 19 0 0.66 1.0
255     %20 20 0 1.0 1.0
256     %Rec4 6
257     % 0 1 0 2 6 5
258     % 1 1 2 3 8 6
259     % 2 1 3 7 10 8
260     % 5 2 12 9 18 16
261     % 7 2 13 19 18 9
262     %10 2 20 19 13 15
263     %Line2 0
264     %Line2_Contact 3
265     % 4 0 9 12 6 5
266     % 3 0 13 9 8 6
267     % 6 0 15 13 10 8
268     %Point1 0
269     %\end{verbatim}
270     %Notice that the order in which the nodes and elements are given is arbitrary.
271     %In the case that rich contact elements are used the contact element section
272     %gets the form
273     %\begin{verbatim}
274     %Rec4Face_Contact 3
275     % 4 0 9 12 16 18 6 5 0 2
276     % 3 0 13 9 18 19 8 6 2 3
277     % 6 0 15 13 19 20 10 8 3 7
278     %\end{verbatim}
279     %Periodic boundary conditions\index{boundary conditions!periodic} can be
280     %introduced by altering \var{Node_DOF}.
281     %It allows identification of nodes even if they have different physical locations.
282     %For instance, to enforce periodic boundary conditions at the face $x_0=0$ and
283     %$x_0=1$ one identifies the degrees of freedom for nodes $0$, $5$, $12$ and $16$
284     %with the degrees of freedom for $7$, $10$, $15$ and $20$, respectively.
285     %The node section of the \finley mesh now reads:
286     %\begin{verbatim}
287     %2D Nodes 16
288     %0 0 0 0. 0.
289     %2 2 0 0.33 0.
290     %3 3 0 0.66 0.
291     %7 0 0 1. 0.
292     %5 5 0 0. 0.5
293     %6 6 0 0.33 0.5
294     %8 8 0 0.66 0.5
295     %10 5 0 1.0 0.5
296     %12 12 0 0. 0.5
297     %9 9 0 0.33 0.5
298     %13 13 0 0.66 0.5
299     %15 12 0 1.0 0.5
300     %16 16 0 0. 1.0
301     %18 18 0 0.33 1.0
302     %19 19 0 0.66 1.0
303     %20 16 0 1.0 1.0
304     %\end{verbatim}
305 jgs 102
306 jfenwick 1955 \clearpage
307     \input{finleyelements}
308     \clearpage
309 jgs 102
310 caltinay 3330 \section{Macro Elements}
311     \label{SEC FINLEY MACRO}
312    
313 gross 2793 \begin{figure}[th]
314     \begin{center}
315 caltinay 3330 \includegraphics{FinleyMacroLeg}\\
316     \subfigure[Triangle]{\label{FINLEY MACRO TRI}\includegraphics{FinleyMacroTri}}\quad
317     \subfigure[Quadrilateral]{\label{FINLEY MACRO REC}\includegraphics{FinleyMacroRec}}
318 gross 2793 \end{center}
319 caltinay 3330 \caption{Macro elements in \finley}
320 gross 2793 \end{figure}
321    
322 caltinay 3330 \finley supports the usage of macro elements\index{macro elements} which can be
323     used to achieve LBB compliance when solving incompressible fluid flow problems.
324     LBB compliance is required to get a problem which has a unique solution for
325     pressure and velocity. For macro elements the pressure and velocity are
326     approximated by a polynomial of order 1 but the velocity approximation bases on
327     a refinement of the elements. The nodes of a triangle and quadrilateral element
328     are shown in Figures~\ref{FINLEY MACRO TRI} and~\ref{FINLEY MACRO REC},
329     respectively. In essence, the velocity uses the same nodes like a quadratic
330     polynomial approximation but replaces the quadratic polynomial by piecewise
331     linear polynomials. In fact, this is the way \finley defines the macro elements.
332     In particular \finley uses the same local ordering of the nodes for the macro
333     element as for the corresponding quadratic element. Another interpretation is
334     that one uses a linear approximation of the velocity together with a linear
335     approximation of the pressure but on elements created by combining elements to
336     macro elements. Notice that the macro elements still use quadratic
337     interpolation to represent the element and domain boundary.
338     However, if elements have linear boundaries a macro element approximation for
339     the velocity is equivalent to using a linear approximation on a mesh which is
340     created through a one-step global refinement.
341     Typically macro elements are only required to use when an incompressible fluid
342     flow problem is solved, e.g. the Stokes problem in \Sec{STOKES PROBLEM}.
343     Please see \Sec{FINLEY MESHES} for more details on the supported macro elements.
344 jgs 102
345 caltinay 3330 \section{Linear Solvers in \SolverOptions}
346 acodd 6928 If available, Trilinos sovers are used by default, see Chapter \ref{TRILINOS}.
347 caltinay 3330 Table~\ref{TAB FINLEY SOLVER OPTIONS 1} and
348 acodd 6928 %Table~\ref{TAB FINLEY SOLVER OPTIONS 2}
349     show the solvers %and preconditioners
350 caltinay 3330 supported by \finley through the \PASO library.
351 acodd 6928 %Currently direct solvers are not supported under \MPI.
352     \finley uses the iterative solvers \PCG for symmetric and \BiCGStab
353 caltinay 3330 for non-symmetric problems.
354 acodd 6928 %If the direct solver is selected, which can be useful when solving very
355     %ill-posed equations, \finley uses the \MKL\footnote{If the stiffness matrix is
356     %non-regular \MKL may return without a proper error code. If you observe
357     %suspicious solutions when using \MKL, this may be caused by a non-invertible
358     %operator.} solver package. If \MKL is not available \UMFPACK is used.
359     %If \UMFPACK is not available a suitable iterative solver from \PASO is used.
360 gross 2748
361 gross 2558 \begin{table}
362 caltinay 3330 \centering
363 jfenwick 2651 {\scriptsize
364 gross 2558 \begin{tabular}{l||c|c|c|c|c|c|c|c}
365 gross 3379 \member{setSolverMethod} & \member{DIRECT}& \member{PCG} & \member{GMRES} & \member{TFQMR} & \member{MINRES} & \member{PRES20} & \member{BICGSTAB} & lumping \\
366 gross 2558 \hline
367     \hline
368     \member{setReordering} & $\checkmark$ & & & & & &\\
369     \hline \member{setRestart} & & & $\checkmark$ & & & $20$ & \\
370     \hline\member{setTruncation} & & & $\checkmark$ & & & $5$ & \\
371     \hline\member{setIterMax} & & $\checkmark$& $\checkmark$ & $\checkmark$& $\checkmark$& $\checkmark$ & $\checkmark$ \\
372     \hline\member{setTolerance} & & $\checkmark$& $\checkmark$ & $\checkmark$& $\checkmark$& $\checkmark$ & $\checkmark$ \\
373     \hline\member{setAbsoluteTolerance} & & $\checkmark$& $\checkmark$ & $\checkmark$& $\checkmark$& $\checkmark$ & $\checkmark$ \\
374 gross 2573 \hline\member{setReordering} & $\checkmark$ & & & & & & & \\
375 gross 2558 \end{tabular}
376     }
377 caltinay 3330 \caption{Solvers available for \finley and the \PASO package and the relevant
378     options in \class{SolverOptions}.
379     \MKL supports \member{MINIMUM_FILL_IN}\index{linear solver!minimum fill-in ordering}\index{minimum fill-in ordering}
380     and \member{NESTED_DISSECTION}\index{linear solver!nested dissection ordering}\index{nested dissection}
381 caltinay 3293 reordering.
382     Currently the \UMFPACK interface does not support any reordering.
383 caltinay 3330 \label{TAB FINLEY SOLVER OPTIONS 1}}
384 caltinay 3293 \end{table}
385 gross 2558
386 acodd 6928 %\begin{table}
387     %\begin{center}
388     %{\scriptsize
389     %\begin{tabular}{l||c|c|c|c|c|c|c}
390     %\member{NO_PRECONDITIONER}&
391     %\member{AMG}&
392     %\member{JACOBI}&
393     %\member{GAUSS_SEIDEL}&
394     %\member{REC_ILU}&
395     %\member{RILU}&
396     %\member{ILU0}&
397     %\member{DIRECT}\\
398     %\hline
399     %status:& $\checkmark$ &$\checkmark$&$\checkmark$&$\checkmark$&later&$\checkmark$&later\\
400     %\hline
401     %\hline
402     %\member{setLevelMax}&$\checkmark$& & & & & &\\
403     %\hline
404     %\member{setCoarseningThreshold}&$\checkmark$& & & & & &\\
405     %\hline
406     %\member{setMinCoarseMatrixSize}&$\checkmark$& & & & & &\\
407     %\hline
408     %\member{setMinCoarseMatrixSparsity}&$\checkmark$& & & & & &\\
409     %\hline
410     %\member{setNumSweeps}& &$\checkmark$&$\checkmark$& & & &\\
411     %\hline
412     %\member{setNumPreSweeps}&$\checkmark$& & & & & &\\
413     %\hline
414     %\member{setNumPostSweeps}&$\checkmark$& & & & & &\\
415     %\hline
416     %\member{setDiagonalDominanceThreshold}&$\checkmark$& & & & & &\\
417     %\hline
418     %\member{setAMGInterpolation}&$\checkmark$& & & & & &\\
419     %\hline
420     %\member{setRelaxationFactor}& & & & &$\checkmark$& &\\
421     %\end{tabular}
422     %}
423     %\caption{Preconditioners available for \finley and the \PASO package and the
424     %relevant options in \class{SolverOptions}.
425     %\label{TAB FINLEY SOLVER OPTIONS 2}}
426     %\end{center}
427     %\end{table}
428 gross 2558
429 gross 2793 \section{Functions}
430 gross 2690 \begin{funcdesc}{ReadMesh}{fileName \optional{, \optional{integrationOrder=-1}, optimize=True}}
431 caltinay 3330 creates a \Domain object from the FEM mesh defined in file \var{fileName}.
432     The file must be in the \finley file format.
433     If \var{integrationOrder} is positive, a numerical integration scheme is chosen
434     which is accurate on each element up to a polynomial of degree
435     \var{integrationOrder}\index{integration order}.
436     Otherwise an appropriate integration order is chosen independently.
437     By default the labeling of mesh nodes and element distribution is optimized.
438     Set \var{optimize=False} to switch off relabeling and redistribution.
439 gross 2690 \end{funcdesc}
440    
441 gross 3625 \begin{funcdesc}{ReadGmsh}{fileName, numDim, \optional{, \optional{integrationOrder=-1}, optimize=True\optional{, useMacroElements=False}}}
442     creates a \Domain object from the FEM mesh defined in file \var{fileName} for
443     a domain of dimension \var{numDim}.
444 caltinay 3330 The file must be in the \gmshextern file format.
445     If \var{integrationOrder} is positive, a numerical integration scheme is chosen
446     which is accurate on each element up to a polynomial of degree
447     \var{integrationOrder}\index{integration order}.
448     Otherwise an appropriate integration order is chosen independently.
449     By default the labeling of mesh nodes and element distribution is optimized.
450     Set \var{optimize=False} to switch off relabeling and redistribution.
451     If \var{useMacroElements} is set, second order elements are interpreted as
452     macro elements\index{macro elements}.
453 jgs 102 \end{funcdesc}
454    
455 gross 2748 \begin{funcdesc}{MakeDomain}{design\optional{, integrationOrder=-1\optional{, optimizeLabeling=True\optional{, useMacroElements=False}}}}
456 caltinay 3330 creates a \finley \Domain from a \pycad \class{Design} object using \gmshextern.
457 gross 2748 The \class{Design} \var{design} defines the geometry.
458 caltinay 3330 If \var{integrationOrder} is positive, a numerical integration scheme is chosen
459     which is accurate on each element up to a polynomial of degree
460     \var{integrationOrder}\index{integration order}.
461     Otherwise an appropriate integration order is chosen independently.
462     Set \var{optimizeLabeling=False} to switch off relabeling and redistribution
463     (not recommended).
464     If \var{useMacroElements} is set, macro elements\index{macro elements} are used.
465     Currently \function{MakeDomain} does not support \MPI.
466 gross 2748 \end{funcdesc}
467 gross 2690
468 gross 2417 \begin{funcdesc}{load}{fileName}
469 caltinay 3330 recovers a \Domain object from a dump file \var{fileName} created by the
470     \function{dump} method of a \Domain object.
471 gross 2417 \end{funcdesc}
472    
473 jgs 102 \begin{funcdesc}{Rectangle}{n0,n1,order=1,l0=1.,l1=1., integrationOrder=-1, \\
474 gross 3904 periodic0=\False, periodic1=\False, useElementsOnFace=\False, optimize=\False}
475 caltinay 3330 generates a \Domain object representing a two-dimensional rectangle between
476     $(0,0)$ and $(l0,l1)$ with orthogonal edges.
477     The rectangle is filled with \var{n0} elements along the $x_0$-axis and
478 caltinay 3293 \var{n1} elements along the $x_1$-axis.
479 caltinay 3330 For \var{order}=1 and \var{order}=2, elements of type \finleyelement{Rec4} and
480 caltinay 3293 \finleyelement{Rec8} are used, respectively.
481 caltinay 3330 In the case of \var{useElementsOnFace}=\False, \finleyelement{Line2} and
482 caltinay 3293 \finleyelement{Line3} are used to subdivide the edges of the rectangle, respectively.
483 caltinay 3330 If \var{order}=-1, \finleyelement{Rec8Macro} and \finleyelement{Line3Macro}\index{macro elements}
484     are used. This option should be used when solving incompressible fluid flow
485     problems, e.g. \class{StokesProblemCartesian}.
486     In the case of \var{useElementsOnFace}=\True (this option should be used if
487     gradients are calculated on domain faces), \finleyelement{Rec4Face} and
488 caltinay 3293 \finleyelement{Rec8Face} are used on the edges, respectively.
489 caltinay 3330 If \var{integrationOrder} is positive, a numerical integration scheme is chosen
490     which is accurate on each element up to a polynomial of degree
491     \var{integrationOrder}\index{integration order}.
492     Otherwise an appropriate integration order is chosen independently.
493     If \var{periodic0}=\True, periodic boundary conditions\index{periodic boundary conditions}
494     along the $x_0$-direction are enforced.
495     That means for any solution of a PDE solved by \finley the values on the line
496     $x_0=0$ will be identical to the values on $x_0=\var{l0}$.
497     Correspondingly, \var{periodic1}=\True sets periodic boundary conditions in the
498     $x_1$-direction.
499     If \var{optimize}=\True mesh node relabeling will be attempted to reduce the
500     computation and also ParMETIS will be used to improve the mesh partition if
501     running on multiple CPUs with \MPI.
502 jgs 102 \end{funcdesc}
503    
504 jfenwick 3301 \begin{funcdesc}{Brick}{n0,n1,n2,order=1,l0=1.,l1=1.,l2=1., integrationOrder=-1,
505 jduplessis 5677 periodic0=\False, periodic1=\False, \\ periodic2=\False, useElementsOnFace=\False,useFullElementOrder=\False, optimize=\False}
506 caltinay 3330 generates a \Domain object representing a three-dimensional brick between
507 jgs 102 $(0,0,0)$ and $(l0,l1,l2)$ with orthogonal faces. The brick is filled with
508 caltinay 3293 \var{n0} elements along the $x_0$-axis,
509     \var{n1} elements along the $x_1$-axis and
510     \var{n2} elements along the $x_2$-axis.
511 caltinay 3330 For \var{order}=1 and \var{order}=2, elements of type \finleyelement{Hex8} and
512 caltinay 3293 \finleyelement{Hex20} are used, respectively.
513 caltinay 3330 In the case of \var{useElementsOnFace}=\False, \finleyelement{Rec4} and
514 caltinay 3293 \finleyelement{Rec8} are used to subdivide the faces of the brick, respectively.
515 caltinay 3330 In the case of \var{useElementsOnFace}=\True (this option should be used if
516     gradients are calculated on domain faces), \finleyelement{Hex8Face} and
517 caltinay 3293 \finleyelement{Hex20Face} are used on the brick faces, respectively.
518 caltinay 3330 If \var{order}=-1, \finleyelement{Hex20Macro} and \finleyelement{Rec8Macro}\index{macro elements}
519     are used. This option should be used when solving incompressible fluid flow
520     problems, e.g. \class{StokesProblemCartesian}.
521     If \var{integrationOrder} is positive, a numerical integration scheme is chosen
522     which is accurate on each element up to a polynomial of degree
523     \var{integrationOrder}\index{integration order}.
524     Otherwise an appropriate integration order is chosen independently.
525     If \var{periodic0}=\True, periodic boundary conditions\index{periodic boundary conditions}
526     along the $x_0$-direction are enforced.
527     That means for any solution of a PDE solved by \finley the values on the plane
528     $x_0=0$ will be identical to the values on $x_0=\var{l0}$.
529     Correspondingly, \var{periodic1}=\True and \var{periodic2}=\True sets periodic
530     boundary conditions in the $x_1$-direction and $x_2$-direction, respectively.
531     If \var{optimize}=\True mesh node relabeling will be attempted to reduce the
532     computation and also ParMETIS will be used to improve the mesh partition if
533     running on multiple CPUs with \MPI.
534 jgs 102 \end{funcdesc}
535    
536 gross 3567 \begin{funcdesc}{GlueFaces}{meshList, tolerance=1.e-13}
537 caltinay 3330 generates a new \Domain object from the list \var{meshList} of \finley meshes.
538     Nodes in face elements whose difference of coordinates is less than
539     \var{tolerance} times the diameter of the domain are merged.
540     The corresponding face elements are removed from the mesh.
541 gross 3567 \function{GlueFaces} is not supported under \MPI with more than one rank.
542 jgs 102 \end{funcdesc}
543    
544 gross 3567 \begin{funcdesc}{JoinFaces}{meshList, tolerance=1.e-13}
545 caltinay 3330 generates a new \Domain object from the list \var{meshList} of \finley meshes.
546     Face elements whose node coordinates differ by less than \var{tolerance} times
547     the diameter of the domain are combined to form a contact element\index{element!contact}.
548 caltinay 3293 The corresponding face elements are removed from the mesh.
549 gross 3567 \function{JoinFaces} is not supported under \MPI with more than one rank.
550 caltinay 3330 \end{funcdesc}
551 jgs 102
552 jfenwick 5658 \section{\dudley}
553     \label{sec:dudley}
554     The {\it dudley} library is a restricted version of {\it finley}.
555     So in many ways it can be used as a ``drop-in'' replacement.
556     Dudley domains are simpler in that only triangular (2D), tetrahedral (3D) and line elements are supported.
557     Note, this also means that dudley does not support:
558     \begin{itemize}
559     \item dirac delta functions
560     \item contact elements
561     \item macro elements
562     \end{itemize}
563    

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