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1 ksteube 1811
2     %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
3 ksteube 1316 %
4 jfenwick 2881 % Copyright (c) 2003-2010 by University of Queensland
5 ksteube 1811 % Earth Systems Science Computational Center (ESSCC)
6     % http://www.uq.edu.au/esscc
7 gross 625 %
8 ksteube 1811 % 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 gross 625 %
12 ksteube 1811 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
13 jgs 102
14 ksteube 1811
15 caltinay 3293 \chapter{The \finley Module}\label{CHAPTER ON FINLEY}
16 jgs 102
17     \begin{figure}
18 caltinay 3279 \centerline{\includegraphics[width=\figwidth]{FinleyMesh}}
19 jgs 102 \caption{Subdivision of an Ellipse into triangles order 1 (\finleyelement{Tri3})}
20     \label{FINLEY FIG 0}
21     \end{figure}
22    
23     \begin{figure}
24 caltinay 3279 \centerline{\includegraphics[width=\figwidth]{FinleyContact}}
25 jgs 102 \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 caltinay 3293 \index{partial differential equations} (PDEs) or systems of PDEs using isoparametrical finite
34 jgs 102 elements \index{FEM!isoparametrical}.
35     It supports unstructured, 1D, 2D and 3D meshes. The module \finley provides an access to the
36 caltinay 3293 library through the \LinearPDE class of \escript supporting its full functionality. {\it finley}
37     is parallelized using the OpenMP \index{OpenMP} paradigm.
38 jgs 102
39 gross 993 \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 jfenwick 3295 \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 gross 993 \end{array}
54     \end{equation}
55    
56     \section{Meshes}
57 gross 2793 \label{FINLEY MESHES}
58 jgs 102 To understand the usage of \finley one needs to have an understanding of how the finite element meshes
59 jgs 107 \index{FEM!mesh} are defined. \fig{FINLEY FIG 0} shows an example of the
60 caltinay 3293 subdivision of an ellipse into so called elements \index{FEM!elements} \index{element}.
61 jgs 102 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 lgraham 1700 which is a polynomial of a certain order and is continuous across it boundary to neighbor elements.
65 jgs 107 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 caltinay 3293 In this case the triangle gets a curved edge which requires a parameterization of the triangle using a
69 jgs 102 quadratic polynomial. For this case, the solution is also approximated by a piecewise quadratic polynomial
70 caltinay 3293 (which explains the name isoparametrical elements), see \Ref{Zienc,NumHand} for more details.
71 gross 2748 \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 jgs 102
73     The union of all elements defines the domain of the PDE.
74 jgs 107 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 caltinay 3293 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 jgs 107 modified form is applied. The boundary of the domain is also subdivided into elements. \index{element!face} In \fig{FINLEY FIG 0}
81 jgs 102 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 jgs 107 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 ksteube 1316 with the surface of the domain. In \fig{FINLEY FIG 0}
89 caltinay 3293 elements of the type \finleyelement{Tri3Face} are used.
90 jgs 102 The face element reference number $20$ as a rich face element is defined by the nodes
91 jgs 107 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 jgs 102 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 jgs 107 of an interior element or, in case of a rich face element, it must be identical to an interior element.
97 jgs 102 If no face elements are specified
98     \finley implicitly assumes homogeneous natural boundary conditions \index{natural boundary conditions!homogeneous},
99 caltinay 3293 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 jgs 102
103 caltinay 3293 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 jgs 102 even if $d^{contact}$ and $y^{contact}$ are zero. \fig{FINLEY FIG 01} shows a simple example of a mesh
106 caltinay 3293 of rectangular elements around a contact region $\Gamma^{contact}$ \index{element!contact}.
107 jgs 102 The contact region is described by the
108 caltinay 3293 elements $4$, $3$ and $6$. Their element type is \finleyelement{Line2_Contact}.
109 jgs 107 The nodes $9$, $12$, $6$, $5$ define contact element $4$, where the coordinates of nodes $12$ and $5$ and
110 caltinay 3293 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 jgs 102 Again, the order of the nodes within an element is crucial. There is also the option of using rich elements
113 caltinay 3293 if the gradient is to be calculated on the contact region. Similarly to the rich face elements
114 jgs 107 these are constructed from two interior elements by reordering the nodes such that
115 caltinay 3293 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 jgs 102 $2$.
119    
120 gross 2748
121    
122 jgs 102 \tab{FINLEY TAB 1} shows the interior element types and the corresponding element types to be used
123 jgs 107 on the face and contacts. \fig{FINLEY.FIG:1}, \fig{FINLEY.FIG:2} and \fig{FINLEY.FIG:4} show the ordering of
124 jgs 102 the nodes within an element.
125    
126     \begin{table}
127 caltinay 3293 \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 jgs 102 \caption{Finley elements and corresponding elements to be used on domain faces and contacts.
149 ksteube 1316 The rich types have to be used if the gradient of function is to be calculated on faces and contacts, respectively.}
150 jgs 102 \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 jgs 107 In most cases \var{Node_DOF[i]}=\var{Node_ref[i]} however, for periodic boundary conditions,
157 jgs 102 \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 caltinay 3293 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 jgs 102 \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 caltinay 3293 for i in range(numNodes):
171 jgs 102 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 ksteube 1316 write("Point1 0",face_element_type,numFaceElements)
194 jgs 102 \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 jgs 107 In the case that rich contact elements are used the contact element section gets
232     the form
233 jgs 102 \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 jgs 107 It allows identification of nodes even if they have different physical locations. For instance, to
241 jgs 102 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 caltinay 3293 $7$, $10$, $15$ and $20$, respectively. The node section of the \finley mesh gets now the form:
244 jgs 102 \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 jfenwick 1955 \clearpage
265     \input{finleyelements}
266     \clearpage
267 jgs 102
268 gross 2793 \begin{figure}[th]
269     \begin{center}
270 caltinay 3279 \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 gross 2793 \end{center}
274     Macro elements in \finley.
275     \end{figure}
276    
277 gross 2748 \section{Macro Elements}
278     \label{SEC FINLEY MACRO}
279 gross 2793 \finley supports the usage of macro elements~\index{macro elements} which can be used to
280 caltinay 3293 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 gross 2793 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 caltinay 3293 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 gross 2793 one uses a linear approximation of the velocity together with a linear approximation of the pressure but on elements
285 caltinay 3293 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 gross 2793 is solved, e.g the Stokes problem in Section \ref{STOKES PROBLEM}. Please see Section~\ref{FINLEY MESHES} for
289 caltinay 3293 more details on the supported macro elements.
290 jgs 102
291 gross 2748
292    
293 gross 2558 \begin{table}
294 jfenwick 2651 {\scriptsize
295 gross 2558 \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 gross 2573 \hline\member{setReordering} & $\checkmark$ & & & & & & & \\
306 gross 2558 \end{tabular}
307     }
308 gross 2573 \caption{Solvers available for
309 caltinay 3293 \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 gross 2573 \label{TAB FINLEY SOLVER OPTIONS 1} }
318 caltinay 3293 \end{table}
319 gross 2558
320     \begin{table}
321     {\scriptsize
322     \begin{tabular}{l||c|c|c|c|c|c|c|c}
323 caltinay 3293 \member{setPreconditioner} &
324 gross 2558 \member{NO_PRECONDITIONER} &
325 caltinay 3293 \member{AMG} &
326     \member{JACOBI} &
327     \member{GAUSS_SEIDEL}&
328 gross 2558 \member{REC_ILU}&
329     \member{RILU} &
330     \member{ILU0} &
331     \member{DIRECT} \\
332     \hline
333     status: &
334     later &
335 caltinay 3293 later &
336 gross 2558 $\checkmark$ &
337 caltinay 3293 $\checkmark$&
338     $\checkmark$ &
339 gross 2558 later &
340     $\checkmark$ &
341     later \\
342     \hline
343     \hline
344 caltinay 3293 \member{setCoarsening}&
345 gross 2558 &
346 caltinay 3293 $\checkmark$ &
347 gross 2558 &
348 caltinay 3293 &
349     &
350 gross 2558 &
351     &
352     \\
353    
354    
355 caltinay 3293 \hline\member{setLevelMax}&
356 gross 2558 &
357 caltinay 3293 $\checkmark$ &
358     &
359 gross 2558 &
360 caltinay 3293 &
361 gross 2558 &
362     &
363     \\
364    
365     \hline\member{setCoarseningThreshold}&
366     &
367 caltinay 3293 $\checkmark$ &
368     &
369 gross 2558 &
370 caltinay 3293 &
371 gross 2558 &
372     &
373     \\
374    
375     \hline\member{setMinCoarseMatrixSize} &
376     &
377 caltinay 3293 $\checkmark$ &
378     &
379 gross 2558 &
380 caltinay 3293 &
381 gross 2558 &
382     &
383     \\
384    
385     \hline\member{setNumSweeps} &
386     &
387 caltinay 3293 &
388     $\checkmark$ &
389     $\checkmark$ &
390 gross 2558 &
391     &
392     &
393     \\
394    
395     \hline\member{setNumPreSweeps}&
396     &
397 caltinay 3293 $\checkmark$ &
398     &
399 gross 2558 &
400 caltinay 3293 &
401 gross 2558 &
402     &
403     \\
404    
405     \hline\member{setNumPostSweeps} &
406     &
407 caltinay 3293 $\checkmark$ &
408     &
409 gross 2558 &
410 caltinay 3293 &
411 gross 2558 &
412     &
413     \\
414    
415     \hline\member{setInnerTolerance}&
416     &
417 caltinay 3293 &
418     &
419 gross 2558 &
420 caltinay 3293 &
421 gross 2558 &
422     &
423     \\
424    
425     \hline\member{setDropTolerance}&
426     &
427 caltinay 3293 &
428     &
429 gross 2558 &
430 caltinay 3293 &
431 gross 2558 &
432     &
433     \\
434    
435     \hline\member{setDropStorage}&
436     &
437 caltinay 3293 &
438     &
439 gross 2558 &
440 caltinay 3293 &
441 gross 2558 &
442     &
443     \\
444    
445     \hline\member{setRelaxationFactor}&
446     &
447 caltinay 3293 &
448     &
449 gross 2558 &
450 caltinay 3293 &
451 gross 2558 $\checkmark$ &
452     &
453     \\
454    
455     \hline\member{adaptInnerTolerance}&
456     &
457 caltinay 3293 &
458     &
459 gross 2558 &
460 caltinay 3293 &
461 gross 2558 &
462     &
463     \\
464    
465     \hline\member{setInnerIterMax}&
466     &
467 caltinay 3293 &
468     &
469 gross 2558 &
470 caltinay 3293 &
471 gross 2558 &
472     &
473     \\
474     \end{tabular}
475     }
476 caltinay 3293 \caption{Preconditioners available for \finley and the \PASO package and the relevant options in \class{SolverOptions}. \label{TAB FINLEY SOLVER OPTIONS 2}}
477 gross 2558 \end{table}
478    
479 gross 2793 \section{Linear Solvers in \SolverOptions}
480 gross 2558 Table~\ref{TAB FINLEY SOLVER OPTIONS 1} and
481 caltinay 3293 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 gross 2793 If the direct solver is selected which can be useful when solving very ill-posed equations
485 caltinay 3293 \finley uses the \MKL \footnote{If the stiffness matrix is non-regular \MKL may return without
486 jfenwick 3300 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 gross 2558 a suitable iterative solver from the \PASO is used.
488    
489 gross 2793 \section{Functions}
490 gross 2690 \begin{funcdesc}{ReadMesh}{fileName \optional{, \optional{integrationOrder=-1}, optimize=True}}
491 caltinay 3293 creates a \Domain object form the FEM mesh defined in
492 jgs 102 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 caltinay 3293 an appropriate integration order is chosen independently.
497     By default the labeling of mesh nodes and element distribution is
498 gross 2690 optimized. Set \var{optimize=False} to switch off relabeling and redistribution.
499     \end{funcdesc}
500    
501 gross 2748 \begin{funcdesc}{ReadGmsh}{fileName \optional{, \optional{integrationOrder=-1}, optimize=True\optional{, useMacroElements=False}}}
502 caltinay 3293 creates a \Domain object form the FEM mesh defined in
503 gross 2690 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 jgs 102 an appropriate integration order is chosen independently.
508 caltinay 3293 By default the labeling of mesh nodes and element distribution is
509     optimized. Set \var{optimize=False} to switch off relabeling and redistribution.
510 gross 2793 If \var{useMacroElements} is set, second order elements are interpreted as macro elements~\index{macro elements}.
511 caltinay 3293 Currently \function{ReadGmsh} does not support MPI.
512 jgs 102 \end{funcdesc}
513    
514 gross 2748 \begin{funcdesc}{MakeDomain}{design\optional{, integrationOrder=-1\optional{, optimizeLabeling=True\optional{, useMacroElements=False}}}}
515 caltinay 3293 Creates a Finley \Domain from a \class{Design} object from \pycad using \gmshextern.
516 gross 2748 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 caltinay 3293 Currently \function{MakeDomain} does not support MPI.
524 gross 2748 \end{funcdesc}
525 gross 2690
526 gross 2748
527 gross 2417 \begin{funcdesc}{load}{fileName}
528 caltinay 3293 recovers a \Domain object from a dump file created by the \
529     \function{dump} method of a \Domain object defined in
530 gross 2793 file \var{fileName}.
531 gross 2417 \end{funcdesc}
532    
533 gross 2748
534 jgs 102 \begin{funcdesc}{Rectangle}{n0,n1,order=1,l0=1.,l1=1., integrationOrder=-1, \\
535 jfenwick 3301 periodic0=\False, periodic1=\False, useElementsOnFace=\False, useMacroElements=\False,\\ optimize=\False}
536 jgs 102 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 caltinay 3293 \var{n1} elements along the $x_1$-axis.
540 jgs 102 For \var{order}=1 and \var{order}=2
541 caltinay 3293 \finleyelement{Rec4} and
542     \finleyelement{Rec8} are used, respectively.
543 jgs 102 In the case of \var{useElementsOnFace}=\False,
544 caltinay 3293 \finleyelement{Line2} and
545     \finleyelement{Line3} are used to subdivide the edges of the rectangle, respectively.
546 gross 2748 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 jgs 102 In the case of \var{useElementsOnFace}=\True (this option should be used if gradients
548     are calculated on domain faces),
549 caltinay 3293 \finleyelement{Rec4Face} and
550     \finleyelement{Rec8Face} are used on the edges, respectively.
551 jgs 102 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 ksteube 1459 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 jgs 102 \end{funcdesc}
563    
564 jfenwick 3301 \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 jgs 102 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 caltinay 3293 \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 jgs 102 For \var{order}=1 and \var{order}=2
572 caltinay 3293 \finleyelement{Hex8} and
573     \finleyelement{Hex20} are used, respectively.
574 jgs 102 In the case of \var{useElementsOnFace}=\False,
575 caltinay 3293 \finleyelement{Rec4} and
576     \finleyelement{Rec8} are used to subdivide the faces of the brick, respectively.
577 jgs 102 In the case of \var{useElementsOnFace}=\True (this option should be used if gradients
578     are calculated on domain faces),
579 caltinay 3293 \finleyelement{Hex8Face} and
580     \finleyelement{Hex20Face} are used on the brick faces, respectively.
581 gross 2748 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 jgs 102 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 ksteube 1459 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 jgs 102 \end{funcdesc}
593    
594     \begin{funcdesc}{GlueFaces}{meshList,safetyFactor=0.2,tolerance=1.e-13}
595 ksteube 1316 Generates a new \Domain object from the list \var{meshList} of \finley meshes.
596 caltinay 3293 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 jgs 102
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 ksteube 1316 Generates a new \Domain object from the list \var{meshList} of \finley meshes.
604 caltinay 3293 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 jgs 102
608     TODO: explain \var{safetyFactor} and show an example.
609     \end{funcdesc}

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