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revision 155 by jgs, Wed Nov 9 02:02:19 2005 UTC revision 2558 by gross, Mon Jul 27 05:03:32 2009 UTC
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 % $Id$  
1    
2    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
3    %
4    % Copyright (c) 2003-2009 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 module \finley}  
15    \chapter{ The Module \finley}
16   \label{CHAPTER ON FINLEY}   \label{CHAPTER ON FINLEY}
17    
18  \begin{figure}  \begin{figure}
19  \centerline{\includegraphics[width=\figwidth]{FinleyMesh}}  \centerline{\includegraphics[width=\figwidth]{figures/FinleyMesh}}
20  \caption{Subdivision of an Ellipse into triangles order 1 (\finleyelement{Tri3})}  \caption{Subdivision of an Ellipse into triangles order 1 (\finleyelement{Tri3})}
21  \label{FINLEY FIG 0}  \label{FINLEY FIG 0}
22  \end{figure}  \end{figure}
23    
24  \begin{figure}  \begin{figure}
25  \centerline{\includegraphics[width=\figwidth]{FinleyContact}}  \centerline{\includegraphics[width=\figwidth]{figures/FinleyContact}}
26  \caption{Mesh around a contact region (\finleyelement{Rec4})}  \caption{Mesh around a contact region (\finleyelement{Rec4})}
27  \label{FINLEY FIG 01}  \label{FINLEY FIG 01}
28  \end{figure}  \end{figure}
# Line 26  It supports unstructured, 1D, 2D and 3D Line 37  It supports unstructured, 1D, 2D and 3D
37  library through the \LinearPDE class of \escript supporting its full functionality. {\it finley}  library through the \LinearPDE class of \escript supporting its full functionality. {\it finley}
38  is parallelized using the OpenMP \index{OpenMP} paradigm.  is parallelized using the OpenMP \index{OpenMP} paradigm.
39    
40  \subsection{Meshes}  \section{Formulation}
41    
42    For a single PDE with a solution with a single component the linear PDE is defined in the
43    following form:
44    \begin{equation}\label{FINLEY.SINGLE.1}
45    \begin{array}{cl} &
46    \displaystyle{
47    \int\hackscore{\Omega}
48    A\hackscore{jl} \cdot v\hackscore{,j}u\hackscore{,l}+ B\hackscore{j} \cdot v\hackscore{,j} u+ C\hackscore{l} \cdot v u\hackscore{,l}+D \cdot vu \; d\Omega }  \\
49    + & \displaystyle{\int\hackscore{\Gamma} d \cdot vu \; d{\Gamma} }
50    +  \displaystyle{\int\hackscore{\Gamma^{contact}} d^{contact} \cdot [v][u] \; d{\Gamma} } \\
51    = & \displaystyle{\int\hackscore{\Omega}  X\hackscore{j} \cdot v\hackscore{,j}+ Y \cdot v \; d\Omega }\\
52    + & \displaystyle{\int\hackscore{\Gamma} y \cdot v \; d{\Gamma}}  +
53    \displaystyle{\int\hackscore{\Gamma^{contact}} y^{contact}\cdot [v] \; d{\Gamma}} \\
54    \end{array}
55    \end{equation}
56    
57    \section{Meshes}
58  To understand the usage of \finley one needs to have an understanding of how the finite element meshes  To understand the usage of \finley one needs to have an understanding of how the finite element meshes
59  \index{FEM!mesh} are defined. \fig{FINLEY FIG 0} shows an example of the  \index{FEM!mesh} are defined. \fig{FINLEY FIG 0} shows an example of the
60  subdivision of an ellipse into so called elements \index{FEM!elements} \index{element}.  subdivision of an ellipse into so called elements \index{FEM!elements} \index{element}.
61  In this case, triangles have been used but other forms of subdivisions  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  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  and hexahedrons. The idea of the finite element method is to approximate the solution by a function
64  which is a polynomial of a certain order and is continuous across it boundary to neighbour elements.  which is a polynomial of a certain order and is continuous across it boundary to neighbor elements.
65  In the example of \fig{FINLEY FIG 0} a linear polynomial is used on each triangle. As one can see, the triangulation  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  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.  positioning those nodes located on an edge expected to describe the boundary, onto the boundary.
# Line 56  to second node the domain has to lie on Line 84  to second node the domain has to lie on
84  the domain has to lie on the left hand side when moving counterclockwise). If the gradient on the  the domain has to lie on the left hand side when moving counterclockwise). If the gradient on the
85  surface of the domain is to be calculated rich face elements face to be used. Rich elements on a face  surface of the domain is to be calculated rich face elements face to be used. Rich elements on a face
86  are identical to interior elements but with a modified order of nodes such that the 'first' face of the element aligns  are identical to interior elements but with a modified order of nodes such that the 'first' face of the element aligns
87  with the surface of the domian. In \fig{FINLEY FIG 0}  with the surface of the domain. In \fig{FINLEY FIG 0}
88  elements of the type \finleyelement{Tri3Face} are used.  elements of the type \finleyelement{Tri3Face} are used.
89  The face element reference number $20$ as a rich face element is defined by the nodes  The face element reference number $20$ as a rich face element is defined by the nodes
90  with reference numbers $11$, $0$ and $9$. Notice that the face element $20$ is identical to the  with reference numbers $11$, $0$ and $9$. Notice that the face element $20$ is identical to the
# Line 107  the nodes within an element. Line 135  the nodes within an element.
135  \linev{\finleyelement{Hex20}}{\finleyelement{Rec8}}{\finleyelement{Hex20Face}}{\finleyelement{Rec8_Contact}}{\finleyelement{Hex20Face_Contact}}  \linev{\finleyelement{Hex20}}{\finleyelement{Rec8}}{\finleyelement{Hex20Face}}{\finleyelement{Rec8_Contact}}{\finleyelement{Hex20Face_Contact}}
136  \end{tablev}  \end{tablev}
137  \caption{Finley elements and corresponding elements to be used on domain faces and contacts.  \caption{Finley elements and corresponding elements to be used on domain faces and contacts.
138  The rich types have to be used if the gradient of function is to be calculated on faces and contacts, resepctively.}  The rich types have to be used if the gradient of function is to be calculated on faces and contacts, respectively.}
139  \label{FINLEY TAB 1}  \label{FINLEY TAB 1}
140  \end{table}  \end{table}
141    
# Line 151  for i in range(ContactElement_Num): Line 179  for i in range(ContactElement_Num):
179     for j in range(ContactElement_numNodes): print " %d"%ContactElement_Nodes[i][j]     for j in range(ContactElement_numNodes): print " %d"%ContactElement_Nodes[i][j]
180     print "\n"     print "\n"
181  # point sources (not supported yet)  # point sources (not supported yet)
182  write("Point1 0",face_element_typ,numFaceElements)  write("Point1 0",face_element_type,numFaceElements)
183  \end{python}  \end{python}
184    
185  The following example of a mesh file defines the mesh shown in \fig{FINLEY FIG 01}:  The following example of a mesh file defines the mesh shown in \fig{FINLEY FIG 01}:
# Line 222  $7$, $10$, $15$ and $20$, respectively. Line 250  $7$, $10$, $15$ and $20$, respectively.
250  20 16 0 1.0  1.0  20 16 0 1.0  1.0
251  \end{verbatim}  \end{verbatim}
252    
253    \clearpage
254    \input{finleyelements}
255    \clearpage
256    
257    
258    \begin{table}
259    {\small
260    \begin{tabular}{l||c|c|c|c|c|c|c|c}
261    \member{setSolverMethod} & \member{DIRECT}& \member{PCG} & \member{GMRES} & \member{TFQMR} & \member{MINRES} & \member{PRES20} & \member{BICGSTAB} & \member{LUMPING} \\
262    \hline
263     \hline
264     \member{setReordering} & $\checkmark$ & & & & & &\\
265     \hline  \member{setRestart} &  & & $\checkmark$ & & & $20$ & \\
266     \hline\member{setTruncation} &  & & $\checkmark$ & & & $5$ & \\
267       \hline\member{setIterMax} &  & $\checkmark$& $\checkmark$ & $\checkmark$& $\checkmark$& $\checkmark$ & $\checkmark$ \\
268     \hline\member{setTolerance} &  & $\checkmark$& $\checkmark$ & $\checkmark$& $\checkmark$& $\checkmark$ & $\checkmark$ \\
269     \hline\member{setAbsoluteTolerance} &  & $\checkmark$& $\checkmark$ & $\checkmark$& $\checkmark$& $\checkmark$ & $\checkmark$ \\
270    \end{tabular}
271    }
272    \caption{Solvers available for \finley and the \PASO package and the relevant options in \class{SolverOptions}
273     \label{TAB FINLEY SOLVER OPTIONS 1}}
274    \end{table}
275    
276  \include{finleyelements}  \begin{table}
277    {\scriptsize
278    \begin{tabular}{l||c|c|c|c|c|c|c|c}
279    \member{setPreconditioner} &
280    \member{NO_PRECONDITIONER} &
281    \member{AMG} &
282    \member{JACOBI} &
283    \member{GAUSS_SEIDEL}&
284    \member{REC_ILU}&
285    \member{RILU} &
286    \member{ILU0} &
287    \member{DIRECT} \\
288     \hline
289     status: &
290    later &
291    later &
292    $\checkmark$ &
293    $\checkmark$&
294    $\checkmark$ &
295    later &
296    $\checkmark$ &
297    later \\
298    \hline
299    \hline
300    \member{setCoarsening}&
301     &
302    $\checkmark$ &
303    &
304    &
305    &
306     &
307     &
308     \\
309    
310    
311    \hline\member{setLevelMax}&
312     &
313    $\checkmark$ &
314     &
315    &
316    &
317     &
318     &
319     \\
320    
321    \hline\member{setCoarseningThreshold}&
322    &
323    $\checkmark$ &
324     &
325    &
326    &
327     &
328     &
329     \\
330    
331    \hline\member{setMinCoarseMatrixSize} &
332     &
333    $\checkmark$ &
334     &
335    &
336    &
337     &
338     &
339     \\
340    
341    \hline\member{setNumSweeps} &
342     &
343     &
344    $\checkmark$ &
345    $\checkmark$ &
346    &
347     &
348     &
349     \\
350    
351    \hline\member{setNumPreSweeps}&
352     &
353    $\checkmark$ &
354      &
355     &
356     &
357      &
358      &
359      \\
360    
361    \hline\member{setNumPostSweeps} &
362     &
363    $\checkmark$ &
364     &
365    &
366    &
367     &
368    &
369     \\
370    
371    \hline\member{setInnerTolerance}&
372     &
373     &
374     &
375    &
376    &
377     &
378    &
379     \\
380    
381    \hline\member{setDropTolerance}&
382     &
383     &
384     &
385    &
386    &
387     &
388    &
389     \\
390    
391    \hline\member{setDropStorage}&
392     &
393     &
394     &
395    &
396    &
397     &
398    &
399     \\
400    
401    \hline\member{setRelaxationFactor}&
402     &
403     &
404     &
405    &
406    &
407    $\checkmark$  &
408     &
409     \\
410    
411    \hline\member{adaptInnerTolerance}&
412     &
413     &
414     &
415    &
416    &
417     &
418    &
419     \\
420    
421    \hline\member{setInnerIterMax}&
422     &
423     &
424     &
425    &
426    &
427     &
428    &
429     \\
430    \end{tabular}
431    }
432    \caption{Preconditioners available for \finley and the \PASO package and the relevant options in \class{SolverOptions}. \label{TAB FINLEY SOLVER OPTIONS 2}}
433    \end{table}
434    
435  \subsection{Linear Solvers in \LinearPDE}  \subsection{Linear Solvers in \SolverOptions}
436  Currently \finley supports the linear solvers \PCG, \GMRES, \PRESTWENTY and \BiCGStab.  Table~\ref{TAB FINLEY SOLVER OPTIONS 1} and
437  For \GMRES the options \var{trancation} and \var{restart} of the \method{getSolution} can be  Table~\ref{TAB FINLEY SOLVER OPTIONS 2} show the solvers and preconditioners supported by
438  used to control the trunction and restart during iteration. Default values are  \finley through the \PASO library. Currently direct solvers are not supported under MPI.
439  \var{truncation}=5 and \var{restart}=20.  By default, \finley is using the iterative solvers \PCG for symmetric and \BiCGStab for non-symmetric problems.
440  The default solver is \BiCGStab  but if the symmetry flag is set \PCG is the default solver.  If the direct solver is selected which can be useful when solving very ill-posedequations
441  \finley supports the solver options \var{iter_max} which specifies the maximum number of iterations steps,  \finley uses the \MKL solver package. If \MKL is not available \UMFPACK is used. If \UMFPACK is not available
442  \var{verbose}=\True or \False and \var{preconditioner}=\constant{JACOBI} or \constant {ILU0}.  a suitable iterative solver from the \PASO is used.
 In some installations \finley supports the \Direct solver and the  
 solver options \var{reordering}=\constant{util.NO_REORDERING},  
 \constant{util.MINIMUM_FILL_IN} or \constant{util.NESTED_DISSECTION} (default is \constant{util.NO_REORDERING}),  
 \var{drop_tolerance} specifying the threshold for values to be dropped in the  
 incomplete elimation process (default is 0.01) and \var{drop_storage} specifying the maximum increase  
 in storage allowed in the  
 incomplete elimation process (default is 1.20).  
443    
444  \subsection{Functions}  \subsection{Functions}
445  \begin{funcdesc}{Mesh}{fileName,integrationOrder=-1}  \begin{funcdesc}{ReadMesh}{fileName,integrationOrder=-1}
446  creates a \Domain object form the FEM mesh defined in  creates a \Domain object form the FEM mesh defined in
447  file \var{fileName}. The file must be given the \finley file format.  file \var{fileName}. The file must be given the \finley file format.
448  If \var{integrationOrder} is positive, a numerical integration scheme  If \var{integrationOrder} is positive, a numerical integration scheme
# Line 251  degree \var{integrationOrder} \index{int Line 451  degree \var{integrationOrder} \index{int
451  an appropriate integration order is chosen independently.  an appropriate integration order is chosen independently.
452  \end{funcdesc}  \end{funcdesc}
453    
454  \begin{funcdesc}{Interval}{n0,order=1,l0=1.,integrationOrder=-1, \\  \begin{funcdesc}{load}{fileName}
455    periodic0=\False,useElementsOnFace=\False}  recovers a \Domain object from a dump file created by the \
456  Generates a \Domain object representing a interval $[0,l0]$. The interval is filled with  eateseates a \Domain object form the FEM mesh defined in
457  \var{n0} elements.  file \var{fileName}. The file must be given the \finley file format.
 For \var{order}=1 and \var{order}=2  
 \finleyelement{Line2} and    
 \finleyelement{Line3} are used, respectively.  
 In the case of \var{useElementsOnFace}=\False,  
 \finleyelement{Point1} are used to describe the boundary points.  
 In the case of \var{useElementsOnFace}=\True (this option should be used if gradients  
 are calculated on domain faces),  
 \finleyelement{Line2} and    
 \finleyelement{Line3} are used on both ends of the interval.    
458  If \var{integrationOrder} is positive, a numerical integration scheme  If \var{integrationOrder} is positive, a numerical integration scheme
459  chosen which is accurate on each element up to a polynomial of  chosen which is accurate on each element up to a polynomial of
460  degree \var{integrationOrder} \index{integration order}. Otherwise  degree \var{integrationOrder} \index{integration order}. Otherwise
461  an appropriate integration order is chosen independently. If  an appropriate integration order is chosen independently.
 \var{periodic0}=\True, periodic boundary conditions \index{periodic boundary conditions}  
 along the $x_0$-directions are enforced. That means when for any solution of a PDE solved by \finley  
 the value at $x_0=0$ will be identical to the values at $x_0=\var{l0}$.  
462  \end{funcdesc}  \end{funcdesc}
463    
464  \begin{funcdesc}{Rectangle}{n0,n1,order=1,l0=1.,l1=1., integrationOrder=-1, \\  \begin{funcdesc}{Rectangle}{n0,n1,order=1,l0=1.,l1=1., integrationOrder=-1, \\
465    periodic0=\False,periodic1=\False,useElementsOnFace=\False}    periodic0=\False,periodic1=\False,useElementsOnFace=\False,optimize=\False}
466  Generates a \Domain object representing a two dimensional rectangle between  Generates a \Domain object representing a two dimensional rectangle between
467  $(0,0)$ and $(l0,l1)$ with orthogonal edges. The rectangle is filled with  $(0,0)$ and $(l0,l1)$ with orthogonal edges. The rectangle is filled with
468  \var{n0} elements along the $x_0$-axis and  \var{n0} elements along the $x_0$-axis and
# Line 299  the value on the line $x_0=0$ will be id Line 487  the value on the line $x_0=0$ will be id
487  Correspondingly,  Correspondingly,
488  \var{periodic1}=\False sets periodic boundary conditions  \var{periodic1}=\False sets periodic boundary conditions
489  in $x_1$-direction.  in $x_1$-direction.
490    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.
491  \end{funcdesc}  \end{funcdesc}
492    
493  \begin{funcdesc}{Brick}{n0,n1,n2,order=1,l0=1.,l1=1.,l2=1., integrationOrder=-1, \\  \begin{funcdesc}{Brick}{n0,n1,n2,order=1,l0=1.,l1=1.,l2=1., integrationOrder=-1, \\
494    periodic0=\False,periodic1=\False,periodic2=\False,useElementsOnFace=\False}    periodic0=\False,periodic1=\False,periodic2=\False,useElementsOnFace=\False,optimize=\False}
495  Generates a \Domain object representing a three dimensional brick between  Generates a \Domain object representing a three dimensional brick between
496  $(0,0,0)$ and $(l0,l1,l2)$ with orthogonal faces. The brick is filled with  $(0,0,0)$ and $(l0,l1,l2)$ with orthogonal faces. The brick is filled with
497  \var{n0} elements along the $x_0$-axis,  \var{n0} elements along the $x_0$-axis,
# Line 327  along the $x_0$-directions are enforced. Line 516  along the $x_0$-directions are enforced.
516  the value on the plane $x_0=0$ will be identical to the values on $x_0=\var{l0}$. Correspondingly,  the value on the plane $x_0=0$ will be identical to the values on $x_0=\var{l0}$. Correspondingly,
517  \var{periodic1}=\False and \var{periodic2}=\False sets periodic boundary conditions  \var{periodic1}=\False and \var{periodic2}=\False sets periodic boundary conditions
518  in $x_1$-direction and $x_2$-direction, respectively.  in $x_1$-direction and $x_2$-direction, respectively.
519    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.
520  \end{funcdesc}  \end{funcdesc}
521    
522  \begin{funcdesc}{GlueFaces}{meshList,safetyFactor=0.2,tolerance=1.e-13}  \begin{funcdesc}{GlueFaces}{meshList,safetyFactor=0.2,tolerance=1.e-13}
523  Generates a new \Domain object from the list \var{mehList} of \finley meshes.  Generates a new \Domain object from the list \var{meshList} of \finley meshes.
524  Nodes in face elements whose difference of coordinates is less then \var{tolerance} times the  Nodes in face elements whose difference of coordinates is less then \var{tolerance} times the
525  diameter of the domain are merged. The corresponding face elements are removed from the mesh.    diameter of the domain are merged. The corresponding face elements are removed from the mesh.  
526    
# Line 338  TODO: explain \var{safetyFactor} and sho Line 528  TODO: explain \var{safetyFactor} and sho
528  \end{funcdesc}  \end{funcdesc}
529    
530  \begin{funcdesc}{JoinFaces}{meshList,safetyFactor=0.2,tolerance=1.e-13}  \begin{funcdesc}{JoinFaces}{meshList,safetyFactor=0.2,tolerance=1.e-13}
531  Generates a new \Domain object from the list \var{mehList} of \finley meshes.  Generates a new \Domain object from the list \var{meshList} of \finley meshes.
532  Face elements whose nodes coordinates have difference is less then \var{tolerance} times the  Face elements whose nodes coordinates have difference is less then \var{tolerance} times the
533  diameter of the domain are combined to form a contact element \index{element!contact}  diameter of the domain are combined to form a contact element \index{element!contact}
534  The corresponding face elements are removed from the mesh.    The corresponding face elements are removed from the mesh.  

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