Diff of /trunk/doc/user/finley.tex

revision 625 by gross, Thu Mar 23 00:41:25 2006 UTC revision 2417 by gross, Wed May 13 08:18:47 2009 UTC
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5  %               \url{http://www.access.edu.au  % Earth Systems Science Computational Center (ESSCC)
6  %         Primary Business: Queensland, Australia.  % http://www.uq.edu.au/esscc
7  %  %
8    % Primary Business: Queensland, Australia
11    %
12    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
13
14
15    \chapter{ The Module \finley}
\chapter{ The module \finley}
16   \label{CHAPTER ON FINLEY}   \label{CHAPTER ON FINLEY}
17
18  \begin{figure}  \begin{figure}
19  \centerline{\includegraphics[width=\figwidth]{figures/FinleyMesh.eps}}  \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]{figures/FinleyContact.eps}}  \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 34  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    \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
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 64  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 115  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 159  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 230  $7$, $10$, $15$ 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  \include{finleyelements}  \input{finleyelements}
255    \clearpage
256
257  \subsection{Linear Solvers in \LinearPDE}  \subsection{Linear Solvers in \LinearPDE}
258  Currently \finley supports the linear solvers \PCG, \GMRES, \PRESTWENTY and \BiCGStab.  Currently \finley supports the linear solvers \PCG, \GMRES, \PRESTWENTY and \BiCGStab.
259  For \GMRES the options \var{trancation} and \var{restart} of the \method{getSolution} can be  For \GMRES the options \var{truncation} and \var{restart} of the \method{getSolution} can be
260  used to control the trunction and restart during iteration. Default values are  used to control the truncation and restart during iteration. Default values are
261  \var{truncation}=5 and \var{restart}=20.  \var{truncation}=5 and \var{restart}=20.
262  The default solver is \BiCGStab  but if the symmetry flag is set \PCG is the default solver.  The default solver is \BiCGStab  but if the symmetry flag is set \PCG is the default solver.
263  \finley supports the solver options \var{iter_max} which specifies the maximum number of iterations steps,  \finley supports the solver options \var{iter_max} which specifies the maximum number of iterations steps,
# Line 245  In some installations \finley supports t Line 266  In some installations \finley supports t
266  solver options \var{reordering}=\constant{util.NO_REORDERING},  solver options \var{reordering}=\constant{util.NO_REORDERING},
267  \constant{util.MINIMUM_FILL_IN} or \constant{util.NESTED_DISSECTION} (default is \constant{util.NO_REORDERING}),  \constant{util.MINIMUM_FILL_IN} or \constant{util.NESTED_DISSECTION} (default is \constant{util.NO_REORDERING}),
268  \var{drop_tolerance} specifying the threshold for values to be dropped in the  \var{drop_tolerance} specifying the threshold for values to be dropped in the
269  incomplete elimation process (default is 0.01) and \var{drop_storage} specifying the maximum increase  incomplete elimination process (default is 0.01) and \var{drop_storage} specifying the maximum increase
270  in storage allowed in the  in storage allowed in the
271  incomplete elimation process (default is 1.20).  incomplete elimination process (default is 1.20).
272
273  \subsection{Functions}  \subsection{Functions}
275  creates a \Domain object form the FEM mesh defined in  creates a \Domain object form the FEM mesh defined in
276  file \var{fileName}. The file must be given the \finley file format.  file \var{fileName}. The file must be given the \finley file format.
277  If \var{integrationOrder} is positive, a numerical integration scheme  If \var{integrationOrder} is positive, a numerical integration scheme
# Line 259  degree \var{integrationOrder} \index{int Line 280  degree \var{integrationOrder} \index{int
280  an appropriate integration order is chosen independently.  an appropriate integration order is chosen independently.
281  \end{funcdesc}  \end{funcdesc}
282
284    periodic0=\False,useElementsOnFace=\False}  recovers a \Domain object from a dump file created by the \
285  Generates a \Domain object representing a interval $[0,l0]$. The interval is filled with  eateseates a \Domain object form the FEM mesh defined in
286  \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.
287  If \var{integrationOrder} is positive, a numerical integration scheme  If \var{integrationOrder} is positive, a numerical integration scheme
288  chosen which is accurate on each element up to a polynomial of  chosen which is accurate on each element up to a polynomial of
289  degree \var{integrationOrder} \index{integration order}. Otherwise  degree \var{integrationOrder} \index{integration order}. Otherwise
290  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 $x_0=\var{l0}$.
291  \end{funcdesc}  \end{funcdesc}
292
293  \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, \\
294    periodic0=\False,periodic1=\False,useElementsOnFace=\False}    periodic0=\False,periodic1=\False,useElementsOnFace=\False,optimize=\False}
295  Generates a \Domain object representing a two dimensional rectangle between  Generates a \Domain object representing a two dimensional rectangle between
296  $(0,0)$ and $(l0,l1)$ with orthogonal edges. The rectangle is filled with  $(0,0)$ and $(l0,l1)$ with orthogonal edges. The rectangle is filled with
297  \var{n0} elements along the $x_0$-axis and  \var{n0} elements along the $x_0$-axis and
# Line 307  the value on the line $x_0=0$ will be id Line 316  the value on the line $x_0=0$ will be id
316  Correspondingly,  Correspondingly,
317  \var{periodic1}=\False sets periodic boundary conditions  \var{periodic1}=\False sets periodic boundary conditions
318  in $x_1$-direction.  in $x_1$-direction.
319    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.
320  \end{funcdesc}  \end{funcdesc}
321
322  \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, \\
323    periodic0=\False,periodic1=\False,periodic2=\False,useElementsOnFace=\False}    periodic0=\False,periodic1=\False,periodic2=\False,useElementsOnFace=\False,optimize=\False}
324  Generates a \Domain object representing a three dimensional brick between  Generates a \Domain object representing a three dimensional brick between
325  $(0,0,0)$ 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
326  \var{n0} elements along the $x_0$-axis,  \var{n0} elements along the $x_0$-axis,
# Line 335  along the $x_0$-directions are enforced. Line 345  along the $x_0$-directions are enforced.
345  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,
346  \var{periodic1}=\False and \var{periodic2}=\False sets periodic boundary conditions  \var{periodic1}=\False and \var{periodic2}=\False sets periodic boundary conditions
347  in $x_1$-direction and $x_2$-direction, respectively.  in $x_1$-direction and $x_2$-direction, respectively.
348    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.
349  \end{funcdesc}  \end{funcdesc}
350
351  \begin{funcdesc}{GlueFaces}{meshList,safetyFactor=0.2,tolerance=1.e-13}  \begin{funcdesc}{GlueFaces}{meshList,safetyFactor=0.2,tolerance=1.e-13}
352  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.
353  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
354  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.
355
# Line 346  TODO: explain \var{safetyFactor} and sho Line 357  TODO: explain \var{safetyFactor} and sho
357  \end{funcdesc}  \end{funcdesc}
358
359  \begin{funcdesc}{JoinFaces}{meshList,safetyFactor=0.2,tolerance=1.e-13}  \begin{funcdesc}{JoinFaces}{meshList,safetyFactor=0.2,tolerance=1.e-13}
360  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.
361  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
362  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}
363  The corresponding face elements are removed from the mesh.    The corresponding face elements are removed from the mesh.

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