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1 % $Id$
2 %
3 % Copyright © 2006 by ACcESS MNRF
4 % \url{http://www.access.edu.au
5 % Primary Business: Queensland, Australia.
6 % Licensed under the Open Software License version 3.0
7 % http://www.opensource.org/licenses/osl-3.0.php
8 %
9
10
11
12 \chapter{ The module \finley}
13 \label{CHAPTER ON FINLEY}
14
15 \begin{figure}
16 \centerline{\includegraphics[width=\figwidth]{figures/FinleyMesh.eps}}
17 \caption{Subdivision of an Ellipse into triangles order 1 (\finleyelement{Tri3})}
18 \label{FINLEY FIG 0}
19 \end{figure}
20
21 \begin{figure}
22 \centerline{\includegraphics[width=\figwidth]{figures/FinleyContact.eps}}
23 \caption{Mesh around a contact region (\finleyelement{Rec4})}
24 \label{FINLEY FIG 01}
25 \end{figure}
26
27 \declaremodule{extension}{finley} \modulesynopsis{Solving linear, steady partial differential equations using
28 finite elements}
29
30 {\it finley} is a library of C functions solving linear, steady partial differential equations
31 \index{partial differential equations} (PDEs) or systems of PDEs using isoparametrical finite
32 elements \index{FEM!isoparametrical}.
33 It supports unstructured, 1D, 2D and 3D meshes. The module \finley provides an access to the
34 library through the \LinearPDE class of \escript supporting its full functionality. {\it finley}
35 is parallelized using the OpenMP \index{OpenMP} paradigm.
36
37 \subsection{Meshes}
38 To understand the usage of \finley one needs to have an understanding of how the finite element meshes
39 \index{FEM!mesh} are defined. \fig{FINLEY FIG 0} shows an example of the
40 subdivision of an ellipse into so called elements \index{FEM!elements} \index{element}.
41 In this case, triangles have been used but other forms of subdivisions
42 can be constructed, e.g. into quadrilaterals or, in the three dimensional case, into tetrahedrons
43 and hexahedrons. The idea of the finite element method is to approximate the solution by a function
44 which is a polynomial of a certain order and is continuous across it boundary to neighbour elements.
45 In the example of \fig{FINLEY FIG 0} a linear polynomial is used on each triangle. As one can see, the triangulation
46 is quite a poor approximation of the ellipse. It can be improved by introducing a midpoint on each element edge then
47 positioning those nodes located on an edge expected to describe the boundary, onto the boundary.
48 In this case the triangle gets a curved edge which requires a parametrization of the triangle using a
49 quadratic polynomial. For this case, the solution is also approximated by a piecewise quadratic polynomial
50 (which explains the name isoparametrical elements), see \Ref{Zienc,NumHand} for more details.
51
52 The union of all elements defines the domain of the PDE.
53 Each element is defined by the nodes used to describe its shape. In \fig{FINLEY FIG 0} the element,
54 which has type \finleyelement{Tri3},
55 with element reference number $19$ \index{element!reference number} is defined by the nodes
56 with reference numbers $9$, $11$ and $0$ \index{node!reference number}. Notice that the order is counterclockwise.
57 The coefficients of the PDE are evaluated at integration nodes with each individual element.
58 For quadrilateral elements a Gauss quadrature scheme is used. In the case of triangular elements a
59 modified form is applied. The boundary of the domain is also subdivided into elements. \index{element!face} In \fig{FINLEY FIG 0}
60 line elements with two nodes are used. The elements are also defined by their describing nodes, e.g.
61 the face element reference number $20$ which has type \finleyelement{Line2} is defined by the nodes
62 with the reference numbers $11$ and $0$. Again the order is crucial, if moving from the first
63 to second node the domain has to lie on the left hand side (in the case of a two dimension surface element
64 the domain has to lie on the left hand side when moving counterclockwise). If the gradient on the
65 surface of the domain is to be calculated rich face elements face to be used. Rich elements on a face
66 are identical to interior elements but with a modified order of nodes such that the 'first' face of the element aligns
67 with the surface of the domian. In \fig{FINLEY FIG 0}
68 elements of the type \finleyelement{Tri3Face} are used.
69 The face element reference number $20$ as a rich face element is defined by the nodes
70 with reference numbers $11$, $0$ and $9$. Notice that the face element $20$ is identical to the
71 interior element $19$ except that, in this case, the order of the node is different to align the first
72 edge of the triangle (which is the edge starting with the first node) with the boundary of the domain.
73
74 Be aware that face elements and elements in the interior of the domain must match, i.e. a face element must be the face
75 of an interior element or, in case of a rich face element, it must be identical to an interior element.
76 If no face elements are specified
77 \finley implicitly assumes homogeneous natural boundary conditions \index{natural boundary conditions!homogeneous},
78 i.e. \var{d}=$0$ and \var{y}=$0$, on the entire boundary of the domain. For
79 inhomogeneous natural boundary conditions \index{natural boundary conditions!inhomogeneous},
80 the boundary must be described by face elements.
81
82 If discontinuities of the PDE solution are considered contact elements
83 \index{element!contact}\index{contact conditions} are introduced to describe the contact region $\Gamma^{contact}$
84 even if $d^{contact}$ and $y^{contact}$ are zero. \fig{FINLEY FIG 01} shows a simple example of a mesh
85 of rectangular elements around a contact region $\Gamma^{contact}$ \index{element!contact}.
86 The contact region is described by the
87 elements $4$, $3$ and $6$. Their element type is \finleyelement{Line2_Contact}.
88 The nodes $9$, $12$, $6$, $5$ define contact element $4$, where the coordinates of nodes $12$ and $5$ and
89 nodes $4$ and $6$ are identical with the idea that nodes $12$ and $9$ are located above and
90 nodes $5$ and $6$ below the contact region.
91 Again, the order of the nodes within an element is crucial. There is also the option of using rich elements
92 if the gradient is to be calculated on the contact region. Similarly to the rich face elements
93 these are constructed from two interior elements by reordering the nodes such that
94 the 'first' face of the element above and the 'first' face of the element below the
95 contact regions line up. The rich version of element
96 $4$ is of type \finleyelement{Rec4Face_Contact} and is defined by the nodes $9$, $12$, $16$, $18$, $6$, $5$, $0$ and
97 $2$.
98
99 \tab{FINLEY TAB 1} shows the interior element types and the corresponding element types to be used
100 on the face and contacts. \fig{FINLEY.FIG:1}, \fig{FINLEY.FIG:2} and \fig{FINLEY.FIG:4} show the ordering of
101 the nodes within an element.
102
103 \begin{table}
104 \begin{tablev}{l|llll}{textrm}{interior}{face}{rich face}{contact}{rich contact}
105 \linev{\finleyelement{Line2}}{\finleyelement{Point1}}{\finleyelement{Line2Face}}{\finleyelement{Point1_Contact}}{\finleyelement{Line2Face_Contact}}
106 \linev{\finleyelement{Line3}}{\finleyelement{Point1}}{\finleyelement{Line3Face}}{\finleyelement{Point1_Contact}}{\finleyelement{Line3Face_Contact}}
107 \linev{\finleyelement{Tri3}}{\finleyelement{Line2}}{\finleyelement{Tri3Face}}{\finleyelement{Line2_Contact}}{\finleyelement{Tri3Face_Contact}}
108 \linev{\finleyelement{Tri6}}{\finleyelement{Line3}}{\finleyelement{Tri6Face}}{\finleyelement{Line3_Contact}}{\finleyelement{Tri6Face_Contact}}
109 \linev{\finleyelement{Rec4}}{\finleyelement{Line2}}{\finleyelement{Rec4Face}}{\finleyelement{Line2_Contact}}{\finleyelement{Rec4Face_Contact}}
110 \linev{\finleyelement{Rec8}}{\finleyelement{Line3}}{\finleyelement{Rec8Face}}{\finleyelement{Line3_Contact}}{\finleyelement{Rec8Face_Contact}}
111 \linev{\finleyelement{Rec9}}{\finleyelement{Line3}}{\finleyelement{Rec9Face}}{\finleyelement{Line3_Contact}}{\finleyelement{Rec9Face_Contact}}
112 \linev{\finleyelement{Tet4}}{\finleyelement{Tri6}}{\finleyelement{Tet4Face}}{\finleyelement{Tri6_Contact}}{\finleyelement{Tet4Face_Contact}}
113 \linev{\finleyelement{Tet10}}{\finleyelement{Tri9}}{\finleyelement{Tet10Face}}{\finleyelement{Tri9_Contact}}{\finleyelement{Tet10Face_Contact}}
114 \linev{\finleyelement{Hex8}}{\finleyelement{Rec4}}{\finleyelement{Hex8Face}}{\finleyelement{Rec4_Contact}}{\finleyelement{Hex8Face_Contact}}
115 \linev{\finleyelement{Hex20}}{\finleyelement{Rec8}}{\finleyelement{Hex20Face}}{\finleyelement{Rec8_Contact}}{\finleyelement{Hex20Face_Contact}}
116 \end{tablev}
117 \caption{Finley elements and corresponding elements to be used on domain faces and contacts.
118 The rich types have to be used if the gradient of function is to be calculated on faces and contacts, resepctively.}
119 \label{FINLEY TAB 1}
120 \end{table}
121
122 The native \finley file format is defined as follows.
123 Each node \var{i} has \var{dim} spatial coordinates \var{Node[i]}, a reference number
124 \var{Node_ref[i]}, a degree of freedom \var{Node_DOF[i]} and tag \var{Node_tag[i]}.
125 In most cases \var{Node_DOF[i]}=\var{Node_ref[i]} however, for periodic boundary conditions,
126 \var{Node_DOF[i]} is chosen differently, see example below. The tag can be used to mark nodes sharing
127 the same properties. Element \var{i} is defined by the \var{Element_numNodes} nodes \var{Element_Nodes[i]}
128 which is a list of node reference numbers. The order is crucial.
129 It has a reference number \var{Element_ref[i]} and a tag \var{Element_tag[i]}. The tag
130 can be used to mark elements sharing the same properties. For instance elements above
131 a contact region are marked with $2$ and elements below a contact region are marked with $1$.
132 \var{Element_Type} and \var{Element_Num} give the element type and the number of elements in the mesh.
133 Analogue notations are used for face and contact elements. The following Python script
134 prints the mesh definition in the \finley file format:
135 \begin{python}
136 print "%s\n"%mesh_name
137 # node coordinates:
138 print "%dD-nodes %d\n"%(dim,numNodes)
139 for i in range(numNodes):
140 print "%d %d %d"%(Node_ref[i],Node_DOF[i],Node_tag[i])
141 for j in range(dim): print " %e"%Node[i][j]
142 print "\n"
143 # interior elements
144 print "%s %d\n"%(Element_Type,Element_Num)
145 for i in range(Element_Num):
146 print "%d %d"%(Element_ref[i],Element_tag[i])
147 for j in range(Element_numNodes): print " %d"%Element_Nodes[i][j]
148 print "\n"
149 # face elements
150 print "%s %d\n"%(FaceElement_Type,FaceElement_Num)
151 for i in range(FaceElement_Num):
152 print "%d %d"%(FaceElement_ref[i],FaceElement_tag[i])
153 for j in range(FaceElement_numNodes): print " %d"%FaceElement_Nodes[i][j]
154 print "\n"
155 # contact elements
156 print "%s %d\n"%(ContactElement_Type,ContactElement_Num)
157 for i in range(ContactElement_Num):
158 print "%d %d"%(ContactElement_ref[i],ContactElement_tag[i])
159 for j in range(ContactElement_numNodes): print " %d"%ContactElement_Nodes[i][j]
160 print "\n"
161 # point sources (not supported yet)
162 write("Point1 0",face_element_typ,numFaceElements)
163 \end{python}
164
165 The following example of a mesh file defines the mesh shown in \fig{FINLEY FIG 01}:
166 \begin{verbatim}
167 Example 1
168 2D Nodes 16
169 0 0 0 0. 0.
170 2 2 0 0.33 0.
171 3 3 0 0.66 0.
172 7 4 0 1. 0.
173 5 5 0 0. 0.5
174 6 6 0 0.33 0.5
175 8 8 0 0.66 0.5
176 10 10 0 1.0 0.5
177 12 12 0 0. 0.5
178 9 9 0 0.33 0.5
179 13 13 0 0.66 0.5
180 15 15 0 1.0 0.5
181 16 16 0 0. 1.0
182 18 18 0 0.33 1.0
183 19 19 0 0.66 1.0
184 20 20 0 1.0 1.0
185 Rec4 6
186 0 1 0 2 6 5
187 1 1 2 3 8 6
188 2 1 3 7 10 8
189 5 2 12 9 18 16
190 7 2 13 19 18 9
191 10 2 20 19 13 15
192 Line2 0
193 Line2_Contact 3
194 4 0 9 12 6 5
195 3 0 13 9 8 6
196 6 0 15 13 10 8
197 Point1 0
198 \end{verbatim}
199 Notice that the order in which the nodes and elements are given is arbitrary.
200 In the case that rich contact elements are used the contact element section gets
201 the form
202 \begin{verbatim}
203 Rec4Face_Contact 3
204 4 0 9 12 16 18 6 5 0 2
205 3 0 13 9 18 19 8 6 2 3
206 6 0 15 13 19 20 10 8 3 7
207 \end{verbatim}
208 Periodic boundary condition \index{boundary conditions!periodic} can be introduced by altering \var{Node_DOF}.
209 It allows identification of nodes even if they have different physical locations. For instance, to
210 enforce periodic boundary conditions at the face $x_0=0$ and $x_0=1$ one identifies
211 the degrees of freedom for nodes $0$, $5$, $12$ and $16$ with the degrees of freedom for
212 $7$, $10$, $15$ and $20$, respectively. The node section of the \finley mesh gets now the form:
213 \begin{verbatim}
214 2D Nodes 16
215 0 0 0 0. 0.
216 2 2 0 0.33 0.
217 3 3 0 0.66 0.
218 7 0 0 1. 0.
219 5 5 0 0. 0.5
220 6 6 0 0.33 0.5
221 8 8 0 0.66 0.5
222 10 5 0 1.0 0.5
223 12 12 0 0. 0.5
224 9 9 0 0.33 0.5
225 13 13 0 0.66 0.5
226 15 12 0 1.0 0.5
227 16 16 0 0. 1.0
228 18 18 0 0.33 1.0
229 19 19 0 0.66 1.0
230 20 16 0 1.0 1.0
231 \end{verbatim}
232
233
234 \include{finleyelements}
235
236 \subsection{Linear Solvers in \LinearPDE}
237 Currently \finley supports the linear solvers \PCG, \GMRES, \PRESTWENTY and \BiCGStab.
238 For \GMRES the options \var{trancation} and \var{restart} of the \method{getSolution} can be
239 used to control the trunction and restart during iteration. Default values are
240 \var{truncation}=5 and \var{restart}=20.
241 The default solver is \BiCGStab but if the symmetry flag is set \PCG is the default solver.
242 \finley supports the solver options \var{iter_max} which specifies the maximum number of iterations steps,
243 \var{verbose}=\True or \False and \var{preconditioner}=\constant{JACOBI} or \constant {ILU0}.
244 In some installations \finley supports the \Direct solver and the
245 solver options \var{reordering}=\constant{util.NO_REORDERING},
246 \constant{util.MINIMUM_FILL_IN} or \constant{util.NESTED_DISSECTION} (default is \constant{util.NO_REORDERING}),
247 \var{drop_tolerance} specifying the threshold for values to be dropped in the
248 incomplete elimation process (default is 0.01) and \var{drop_storage} specifying the maximum increase
249 in storage allowed in the
250 incomplete elimation process (default is 1.20).
251
252 \subsection{Functions}
253 \begin{funcdesc}{Mesh}{fileName,integrationOrder=-1}
254 creates a \Domain object form the FEM mesh defined in
255 file \var{fileName}. The file must be given the \finley file format.
256 If \var{integrationOrder} is positive, a numerical integration scheme
257 chosen which is accurate on each element up to a polynomial of
258 degree \var{integrationOrder} \index{integration order}. Otherwise
259 an appropriate integration order is chosen independently.
260 \end{funcdesc}
261
262 \begin{funcdesc}{Interval}{n0,order=1,l0=1.,integrationOrder=-1, \\
263 periodic0=\False,useElementsOnFace=\False}
264 Generates a \Domain object representing a interval $[0,l0]$. The interval is filled with
265 \var{n0} elements.
266 For \var{order}=1 and \var{order}=2
267 \finleyelement{Line2} and
268 \finleyelement{Line3} are used, respectively.
269 In the case of \var{useElementsOnFace}=\False,
270 \finleyelement{Point1} are used to describe the boundary points.
271 In the case of \var{useElementsOnFace}=\True (this option should be used if gradients
272 are calculated on domain faces),
273 \finleyelement{Line2} and
274 \finleyelement{Line3} are used on both ends of the interval.
275 If \var{integrationOrder} is positive, a numerical integration scheme
276 chosen which is accurate on each element up to a polynomial of
277 degree \var{integrationOrder} \index{integration order}. Otherwise
278 an appropriate integration order is chosen independently. If
279 \var{periodic0}=\True, periodic boundary conditions \index{periodic boundary conditions}
280 along the $x_0$-directions are enforced. That means when for any solution of a PDE solved by \finley
281 the value at $x_0=0$ will be identical to the values at $x_0=\var{l0}$.
282 \end{funcdesc}
283
284 \begin{funcdesc}{Rectangle}{n0,n1,order=1,l0=1.,l1=1., integrationOrder=-1, \\
285 periodic0=\False,periodic1=\False,useElementsOnFace=\False}
286 Generates a \Domain object representing a two dimensional rectangle between
287 $(0,0)$ and $(l0,l1)$ with orthogonal edges. The rectangle is filled with
288 \var{n0} elements along the $x_0$-axis and
289 \var{n1} elements along the $x_1$-axis.
290 For \var{order}=1 and \var{order}=2
291 \finleyelement{Rec4} and
292 \finleyelement{Rec8} are used, respectively.
293 In the case of \var{useElementsOnFace}=\False,
294 \finleyelement{Line2} and
295 \finleyelement{Line3} are used to subdivide the edges of the rectangle, respectively.
296 In the case of \var{useElementsOnFace}=\True (this option should be used if gradients
297 are calculated on domain faces),
298 \finleyelement{Rec4Face} and
299 \finleyelement{Rec8Face} are used on the edges, respectively.
300 If \var{integrationOrder} is positive, a numerical integration scheme
301 chosen which is accurate on each element up to a polynomial of
302 degree \var{integrationOrder} \index{integration order}. Otherwise
303 an appropriate integration order is chosen independently. If
304 \var{periodic0}=\True, periodic boundary conditions \index{periodic boundary conditions}
305 along the $x_0$-directions are enforced. That means when for any solution of a PDE solved by \finley
306 the value on the line $x_0=0$ will be identical to the values on $x_0=\var{l0}$.
307 Correspondingly,
308 \var{periodic1}=\False sets periodic boundary conditions
309 in $x_1$-direction.
310 \end{funcdesc}
311
312 \begin{funcdesc}{Brick}{n0,n1,n2,order=1,l0=1.,l1=1.,l2=1., integrationOrder=-1, \\
313 periodic0=\False,periodic1=\False,periodic2=\False,useElementsOnFace=\False}
314 Generates a \Domain object representing a three dimensional brick between
315 $(0,0,0)$ and $(l0,l1,l2)$ with orthogonal faces. The brick is filled with
316 \var{n0} elements along the $x_0$-axis,
317 \var{n1} elements along the $x_1$-axis and
318 \var{n2} elements along the $x_2$-axis.
319 For \var{order}=1 and \var{order}=2
320 \finleyelement{Hex8} and
321 \finleyelement{Hex20} are used, respectively.
322 In the case of \var{useElementsOnFace}=\False,
323 \finleyelement{Rec4} and
324 \finleyelement{Rec8} are used to subdivide the faces of the brick, respectively.
325 In the case of \var{useElementsOnFace}=\True (this option should be used if gradients
326 are calculated on domain faces),
327 \finleyelement{Hex8Face} and
328 \finleyelement{Hex20Face} are used on the brick faces, respectively.
329 If \var{integrationOrder} is positive, a numerical integration scheme
330 chosen which is accurate on each element up to a polynomial of
331 degree \var{integrationOrder} \index{integration order}. Otherwise
332 an appropriate integration order is chosen independently. If
333 \var{periodic0}=\True, periodic boundary conditions \index{periodic boundary conditions}
334 along the $x_0$-directions are enforced. That means when for any solution of a PDE solved by \finley
335 the value on the plane $x_0=0$ will be identical to the values on $x_0=\var{l0}$. Correspondingly,
336 \var{periodic1}=\False and \var{periodic2}=\False sets periodic boundary conditions
337 in $x_1$-direction and $x_2$-direction, respectively.
338 \end{funcdesc}
339
340 \begin{funcdesc}{GlueFaces}{meshList,safetyFactor=0.2,tolerance=1.e-13}
341 Generates a new \Domain object from the list \var{mehList} of \finley meshes.
342 Nodes in face elements whose difference of coordinates is less then \var{tolerance} times the
343 diameter of the domain are merged. The corresponding face elements are removed from the mesh.
344
345 TODO: explain \var{safetyFactor} and show an example.
346 \end{funcdesc}
347
348 \begin{funcdesc}{JoinFaces}{meshList,safetyFactor=0.2,tolerance=1.e-13}
349 Generates a new \Domain object from the list \var{mehList} of \finley meshes.
350 Face elements whose nodes coordinates have difference is less then \var{tolerance} times the
351 diameter of the domain are combined to form a contact element \index{element!contact}
352 The corresponding face elements are removed from the mesh.
353
354 TODO: explain \var{safetyFactor} and show an example.
355 \end{funcdesc}

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