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1 % $Id$
2 %
3 % Copyright © 2006, 2007 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 \section{Formulation}
38
39 For a single PDE with a solution with a single component the linear PDE is defined in the
40 following form:
41 \begin{equation}\label{FINLEY.SINGLE.1}
42 \begin{array}{cl} &
43 \displaystyle{
44 \int\hackscore{\Omega}
45 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 } \\
46 + & \displaystyle{\int\hackscore{\Gamma} d \cdot vu \; d{\Gamma} }
47 + \displaystyle{\int\hackscore{\Gamma^{contact}} d^{contact} \cdot [v][u] \; d{\Gamma} } \\
48 = & \displaystyle{\int\hackscore{\Omega} X\hackscore{j} \cdot v\hackscore{,j}+ Y \cdot v \; d\Omega }\\
49 + & \displaystyle{\int\hackscore{\Gamma} y \cdot v \; d{\Gamma}} +
50 \displaystyle{\int\hackscore{\Gamma^{contact}} y^{contact}\cdot [v] \; d{\Gamma}} \\
51 \end{array}
52 \end{equation}
53
54 \section{Meshes}
55 To understand the usage of \finley one needs to have an understanding of how the finite element meshes
56 \index{FEM!mesh} are defined. \fig{FINLEY FIG 0} shows an example of the
57 subdivision of an ellipse into so called elements \index{FEM!elements} \index{element}.
58 In this case, triangles have been used but other forms of subdivisions
59 can be constructed, e.g. into quadrilaterals or, in the three dimensional case, into tetrahedrons
60 and hexahedrons. The idea of the finite element method is to approximate the solution by a function
61 which is a polynomial of a certain order and is continuous across it boundary to neighbour elements.
62 In the example of \fig{FINLEY FIG 0} a linear polynomial is used on each triangle. As one can see, the triangulation
63 is quite a poor approximation of the ellipse. It can be improved by introducing a midpoint on each element edge then
64 positioning those nodes located on an edge expected to describe the boundary, onto the boundary.
65 In this case the triangle gets a curved edge which requires a parametrization of the triangle using a
66 quadratic polynomial. For this case, the solution is also approximated by a piecewise quadratic polynomial
67 (which explains the name isoparametrical elements), see \Ref{Zienc,NumHand} for more details.
68
69 The union of all elements defines the domain of the PDE.
70 Each element is defined by the nodes used to describe its shape. In \fig{FINLEY FIG 0} the element,
71 which has type \finleyelement{Tri3},
72 with element reference number $19$ \index{element!reference number} is defined by the nodes
73 with reference numbers $9$, $11$ and $0$ \index{node!reference number}. Notice that the order is counterclockwise.
74 The coefficients of the PDE are evaluated at integration nodes with each individual element.
75 For quadrilateral elements a Gauss quadrature scheme is used. In the case of triangular elements a
76 modified form is applied. The boundary of the domain is also subdivided into elements. \index{element!face} In \fig{FINLEY FIG 0}
77 line elements with two nodes are used. The elements are also defined by their describing nodes, e.g.
78 the face element reference number $20$ which has type \finleyelement{Line2} is defined by the nodes
79 with the reference numbers $11$ and $0$. Again the order is crucial, if moving from the first
80 to second node the domain has to lie on the left hand side (in the case of a two dimension surface element
81 the domain has to lie on the left hand side when moving counterclockwise). If the gradient on the
82 surface of the domain is to be calculated rich face elements face to be used. Rich elements on a face
83 are identical to interior elements but with a modified order of nodes such that the 'first' face of the element aligns
84 with the surface of the domian. In \fig{FINLEY FIG 0}
85 elements of the type \finleyelement{Tri3Face} are used.
86 The face element reference number $20$ as a rich face element is defined by the nodes
87 with reference numbers $11$, $0$ and $9$. Notice that the face element $20$ is identical to the
88 interior element $19$ except that, in this case, the order of the node is different to align the first
89 edge of the triangle (which is the edge starting with the first node) with the boundary of the domain.
90
91 Be aware that face elements and elements in the interior of the domain must match, i.e. a face element must be the face
92 of an interior element or, in case of a rich face element, it must be identical to an interior element.
93 If no face elements are specified
94 \finley implicitly assumes homogeneous natural boundary conditions \index{natural boundary conditions!homogeneous},
95 i.e. \var{d}=$0$ and \var{y}=$0$, on the entire boundary of the domain. For
96 inhomogeneous natural boundary conditions \index{natural boundary conditions!inhomogeneous},
97 the boundary must be described by face elements.
98
99 If discontinuities of the PDE solution are considered contact elements
100 \index{element!contact}\index{contact conditions} are introduced to describe the contact region $\Gamma^{contact}$
101 even if $d^{contact}$ and $y^{contact}$ are zero. \fig{FINLEY FIG 01} shows a simple example of a mesh
102 of rectangular elements around a contact region $\Gamma^{contact}$ \index{element!contact}.
103 The contact region is described by the
104 elements $4$, $3$ and $6$. Their element type is \finleyelement{Line2_Contact}.
105 The nodes $9$, $12$, $6$, $5$ define contact element $4$, where the coordinates of nodes $12$ and $5$ and
106 nodes $4$ and $6$ are identical with the idea that nodes $12$ and $9$ are located above and
107 nodes $5$ and $6$ below the contact region.
108 Again, the order of the nodes within an element is crucial. There is also the option of using rich elements
109 if the gradient is to be calculated on the contact region. Similarly to the rich face elements
110 these are constructed from two interior elements by reordering the nodes such that
111 the 'first' face of the element above and the 'first' face of the element below the
112 contact regions line up. The rich version of element
113 $4$ is of type \finleyelement{Rec4Face_Contact} and is defined by the nodes $9$, $12$, $16$, $18$, $6$, $5$, $0$ and
114 $2$.
115
116 \tab{FINLEY TAB 1} shows the interior element types and the corresponding element types to be used
117 on the face and contacts. \fig{FINLEY.FIG:1}, \fig{FINLEY.FIG:2} and \fig{FINLEY.FIG:4} show the ordering of
118 the nodes within an element.
119
120 \begin{table}
121 \begin{tablev}{l|llll}{textrm}{interior}{face}{rich face}{contact}{rich contact}
122 \linev{\finleyelement{Line2}}{\finleyelement{Point1}}{\finleyelement{Line2Face}}{\finleyelement{Point1_Contact}}{\finleyelement{Line2Face_Contact}}
123 \linev{\finleyelement{Line3}}{\finleyelement{Point1}}{\finleyelement{Line3Face}}{\finleyelement{Point1_Contact}}{\finleyelement{Line3Face_Contact}}
124 \linev{\finleyelement{Tri3}}{\finleyelement{Line2}}{\finleyelement{Tri3Face}}{\finleyelement{Line2_Contact}}{\finleyelement{Tri3Face_Contact}}
125 \linev{\finleyelement{Tri6}}{\finleyelement{Line3}}{\finleyelement{Tri6Face}}{\finleyelement{Line3_Contact}}{\finleyelement{Tri6Face_Contact}}
126 \linev{\finleyelement{Rec4}}{\finleyelement{Line2}}{\finleyelement{Rec4Face}}{\finleyelement{Line2_Contact}}{\finleyelement{Rec4Face_Contact}}
127 \linev{\finleyelement{Rec8}}{\finleyelement{Line3}}{\finleyelement{Rec8Face}}{\finleyelement{Line3_Contact}}{\finleyelement{Rec8Face_Contact}}
128 \linev{\finleyelement{Rec9}}{\finleyelement{Line3}}{\finleyelement{Rec9Face}}{\finleyelement{Line3_Contact}}{\finleyelement{Rec9Face_Contact}}
129 \linev{\finleyelement{Tet4}}{\finleyelement{Tri6}}{\finleyelement{Tet4Face}}{\finleyelement{Tri6_Contact}}{\finleyelement{Tet4Face_Contact}}
130 \linev{\finleyelement{Tet10}}{\finleyelement{Tri9}}{\finleyelement{Tet10Face}}{\finleyelement{Tri9_Contact}}{\finleyelement{Tet10Face_Contact}}
131 \linev{\finleyelement{Hex8}}{\finleyelement{Rec4}}{\finleyelement{Hex8Face}}{\finleyelement{Rec4_Contact}}{\finleyelement{Hex8Face_Contact}}
132 \linev{\finleyelement{Hex20}}{\finleyelement{Rec8}}{\finleyelement{Hex20Face}}{\finleyelement{Rec8_Contact}}{\finleyelement{Hex20Face_Contact}}
133 \end{tablev}
134 \caption{Finley elements and corresponding elements to be used on domain faces and contacts.
135 The rich types have to be used if the gradient of function is to be calculated on faces and contacts, resepctively.}
136 \label{FINLEY TAB 1}
137 \end{table}
138
139 The native \finley file format is defined as follows.
140 Each node \var{i} has \var{dim} spatial coordinates \var{Node[i]}, a reference number
141 \var{Node_ref[i]}, a degree of freedom \var{Node_DOF[i]} and tag \var{Node_tag[i]}.
142 In most cases \var{Node_DOF[i]}=\var{Node_ref[i]} however, for periodic boundary conditions,
143 \var{Node_DOF[i]} is chosen differently, see example below. The tag can be used to mark nodes sharing
144 the same properties. Element \var{i} is defined by the \var{Element_numNodes} nodes \var{Element_Nodes[i]}
145 which is a list of node reference numbers. The order is crucial.
146 It has a reference number \var{Element_ref[i]} and a tag \var{Element_tag[i]}. The tag
147 can be used to mark elements sharing the same properties. For instance elements above
148 a contact region are marked with $2$ and elements below a contact region are marked with $1$.
149 \var{Element_Type} and \var{Element_Num} give the element type and the number of elements in the mesh.
150 Analogue notations are used for face and contact elements. The following Python script
151 prints the mesh definition in the \finley file format:
152 \begin{python}
153 print "%s\n"%mesh_name
154 # node coordinates:
155 print "%dD-nodes %d\n"%(dim,numNodes)
156 for i in range(numNodes):
157 print "%d %d %d"%(Node_ref[i],Node_DOF[i],Node_tag[i])
158 for j in range(dim): print " %e"%Node[i][j]
159 print "\n"
160 # interior elements
161 print "%s %d\n"%(Element_Type,Element_Num)
162 for i in range(Element_Num):
163 print "%d %d"%(Element_ref[i],Element_tag[i])
164 for j in range(Element_numNodes): print " %d"%Element_Nodes[i][j]
165 print "\n"
166 # face elements
167 print "%s %d\n"%(FaceElement_Type,FaceElement_Num)
168 for i in range(FaceElement_Num):
169 print "%d %d"%(FaceElement_ref[i],FaceElement_tag[i])
170 for j in range(FaceElement_numNodes): print " %d"%FaceElement_Nodes[i][j]
171 print "\n"
172 # contact elements
173 print "%s %d\n"%(ContactElement_Type,ContactElement_Num)
174 for i in range(ContactElement_Num):
175 print "%d %d"%(ContactElement_ref[i],ContactElement_tag[i])
176 for j in range(ContactElement_numNodes): print " %d"%ContactElement_Nodes[i][j]
177 print "\n"
178 # point sources (not supported yet)
179 write("Point1 0",face_element_typ,numFaceElements)
180 \end{python}
181
182 The following example of a mesh file defines the mesh shown in \fig{FINLEY FIG 01}:
183 \begin{verbatim}
184 Example 1
185 2D Nodes 16
186 0 0 0 0. 0.
187 2 2 0 0.33 0.
188 3 3 0 0.66 0.
189 7 4 0 1. 0.
190 5 5 0 0. 0.5
191 6 6 0 0.33 0.5
192 8 8 0 0.66 0.5
193 10 10 0 1.0 0.5
194 12 12 0 0. 0.5
195 9 9 0 0.33 0.5
196 13 13 0 0.66 0.5
197 15 15 0 1.0 0.5
198 16 16 0 0. 1.0
199 18 18 0 0.33 1.0
200 19 19 0 0.66 1.0
201 20 20 0 1.0 1.0
202 Rec4 6
203 0 1 0 2 6 5
204 1 1 2 3 8 6
205 2 1 3 7 10 8
206 5 2 12 9 18 16
207 7 2 13 19 18 9
208 10 2 20 19 13 15
209 Line2 0
210 Line2_Contact 3
211 4 0 9 12 6 5
212 3 0 13 9 8 6
213 6 0 15 13 10 8
214 Point1 0
215 \end{verbatim}
216 Notice that the order in which the nodes and elements are given is arbitrary.
217 In the case that rich contact elements are used the contact element section gets
218 the form
219 \begin{verbatim}
220 Rec4Face_Contact 3
221 4 0 9 12 16 18 6 5 0 2
222 3 0 13 9 18 19 8 6 2 3
223 6 0 15 13 19 20 10 8 3 7
224 \end{verbatim}
225 Periodic boundary condition \index{boundary conditions!periodic} can be introduced by altering \var{Node_DOF}.
226 It allows identification of nodes even if they have different physical locations. For instance, to
227 enforce periodic boundary conditions at the face $x_0=0$ and $x_0=1$ one identifies
228 the degrees of freedom for nodes $0$, $5$, $12$ and $16$ with the degrees of freedom for
229 $7$, $10$, $15$ and $20$, respectively. The node section of the \finley mesh gets now the form:
230 \begin{verbatim}
231 2D Nodes 16
232 0 0 0 0. 0.
233 2 2 0 0.33 0.
234 3 3 0 0.66 0.
235 7 0 0 1. 0.
236 5 5 0 0. 0.5
237 6 6 0 0.33 0.5
238 8 8 0 0.66 0.5
239 10 5 0 1.0 0.5
240 12 12 0 0. 0.5
241 9 9 0 0.33 0.5
242 13 13 0 0.66 0.5
243 15 12 0 1.0 0.5
244 16 16 0 0. 1.0
245 18 18 0 0.33 1.0
246 19 19 0 0.66 1.0
247 20 16 0 1.0 1.0
248 \end{verbatim}
249
250
251 \include{finleyelements}
252
253 \subsection{Linear Solvers in \LinearPDE}
254 Currently \finley supports the linear solvers \PCG, \GMRES, \PRESTWENTY and \BiCGStab.
255 For \GMRES the options \var{trancation} and \var{restart} of the \method{getSolution} can be
256 used to control the trunction and restart during iteration. Default values are
257 \var{truncation}=5 and \var{restart}=20.
258 The default solver is \BiCGStab but if the symmetry flag is set \PCG is the default solver.
259 \finley supports the solver options \var{iter_max} which specifies the maximum number of iterations steps,
260 \var{verbose}=\True or \False and \var{preconditioner}=\constant{JACOBI} or \constant {ILU0}.
261 In some installations \finley supports the \Direct solver and the
262 solver options \var{reordering}=\constant{util.NO_REORDERING},
263 \constant{util.MINIMUM_FILL_IN} or \constant{util.NESTED_DISSECTION} (default is \constant{util.NO_REORDERING}),
264 \var{drop_tolerance} specifying the threshold for values to be dropped in the
265 incomplete elimation process (default is 0.01) and \var{drop_storage} specifying the maximum increase
266 in storage allowed in the
267 incomplete elimation process (default is 1.20).
268
269 \subsection{Functions}
270 \begin{funcdesc}{Mesh}{fileName,integrationOrder=-1}
271 creates a \Domain object form the FEM mesh defined in
272 file \var{fileName}. The file must be given the \finley file format.
273 If \var{integrationOrder} is positive, a numerical integration scheme
274 chosen which is accurate on each element up to a polynomial of
275 degree \var{integrationOrder} \index{integration order}. Otherwise
276 an appropriate integration order is chosen independently.
277 \end{funcdesc}
278
279 \begin{funcdesc}{Interval}{n0,order=1,l0=1.,integrationOrder=-1, \\
280 periodic0=\False,useElementsOnFace=\False}
281 Generates a \Domain object representing a interval $[0,l0]$. The interval is filled with
282 \var{n0} elements.
283 For \var{order}=1 and \var{order}=2
284 \finleyelement{Line2} and
285 \finleyelement{Line3} are used, respectively.
286 In the case of \var{useElementsOnFace}=\False,
287 \finleyelement{Point1} are used to describe the boundary points.
288 In the case of \var{useElementsOnFace}=\True (this option should be used if gradients
289 are calculated on domain faces),
290 \finleyelement{Line2} and
291 \finleyelement{Line3} are used on both ends of the interval.
292 If \var{integrationOrder} is positive, a numerical integration scheme
293 chosen which is accurate on each element up to a polynomial of
294 degree \var{integrationOrder} \index{integration order}. Otherwise
295 an appropriate integration order is chosen independently. If
296 \var{periodic0}=\True, periodic boundary conditions \index{periodic boundary conditions}
297 along the $x_0$-directions are enforced. That means when for any solution of a PDE solved by \finley
298 the value at $x_0=0$ will be identical to the values at $x_0=\var{l0}$.
299 \end{funcdesc}
300
301 \begin{funcdesc}{Rectangle}{n0,n1,order=1,l0=1.,l1=1., integrationOrder=-1, \\
302 periodic0=\False,periodic1=\False,useElementsOnFace=\False}
303 Generates a \Domain object representing a two dimensional rectangle between
304 $(0,0)$ and $(l0,l1)$ with orthogonal edges. The rectangle is filled with
305 \var{n0} elements along the $x_0$-axis and
306 \var{n1} elements along the $x_1$-axis.
307 For \var{order}=1 and \var{order}=2
308 \finleyelement{Rec4} and
309 \finleyelement{Rec8} are used, respectively.
310 In the case of \var{useElementsOnFace}=\False,
311 \finleyelement{Line2} and
312 \finleyelement{Line3} are used to subdivide the edges of the rectangle, respectively.
313 In the case of \var{useElementsOnFace}=\True (this option should be used if gradients
314 are calculated on domain faces),
315 \finleyelement{Rec4Face} and
316 \finleyelement{Rec8Face} are used on the edges, respectively.
317 If \var{integrationOrder} is positive, a numerical integration scheme
318 chosen which is accurate on each element up to a polynomial of
319 degree \var{integrationOrder} \index{integration order}. Otherwise
320 an appropriate integration order is chosen independently. If
321 \var{periodic0}=\True, periodic boundary conditions \index{periodic boundary conditions}
322 along the $x_0$-directions are enforced. That means when for any solution of a PDE solved by \finley
323 the value on the line $x_0=0$ will be identical to the values on $x_0=\var{l0}$.
324 Correspondingly,
325 \var{periodic1}=\False sets periodic boundary conditions
326 in $x_1$-direction.
327 \end{funcdesc}
328
329 \begin{funcdesc}{Brick}{n0,n1,n2,order=1,l0=1.,l1=1.,l2=1., integrationOrder=-1, \\
330 periodic0=\False,periodic1=\False,periodic2=\False,useElementsOnFace=\False}
331 Generates a \Domain object representing a three dimensional brick between
332 $(0,0,0)$ and $(l0,l1,l2)$ with orthogonal faces. The brick is filled with
333 \var{n0} elements along the $x_0$-axis,
334 \var{n1} elements along the $x_1$-axis and
335 \var{n2} elements along the $x_2$-axis.
336 For \var{order}=1 and \var{order}=2
337 \finleyelement{Hex8} and
338 \finleyelement{Hex20} are used, respectively.
339 In the case of \var{useElementsOnFace}=\False,
340 \finleyelement{Rec4} and
341 \finleyelement{Rec8} are used to subdivide the faces of the brick, respectively.
342 In the case of \var{useElementsOnFace}=\True (this option should be used if gradients
343 are calculated on domain faces),
344 \finleyelement{Hex8Face} and
345 \finleyelement{Hex20Face} are used on the brick faces, respectively.
346 If \var{integrationOrder} is positive, a numerical integration scheme
347 chosen which is accurate on each element up to a polynomial of
348 degree \var{integrationOrder} \index{integration order}. Otherwise
349 an appropriate integration order is chosen independently. If
350 \var{periodic0}=\True, periodic boundary conditions \index{periodic boundary conditions}
351 along the $x_0$-directions are enforced. That means when for any solution of a PDE solved by \finley
352 the value on the plane $x_0=0$ will be identical to the values on $x_0=\var{l0}$. Correspondingly,
353 \var{periodic1}=\False and \var{periodic2}=\False sets periodic boundary conditions
354 in $x_1$-direction and $x_2$-direction, respectively.
355 \end{funcdesc}
356
357 \begin{funcdesc}{GlueFaces}{meshList,safetyFactor=0.2,tolerance=1.e-13}
358 Generates a new \Domain object from the list \var{mehList} of \finley meshes.
359 Nodes in face elements whose difference of coordinates is less then \var{tolerance} times the
360 diameter of the domain are merged. The corresponding face elements are removed from the mesh.
361
362 TODO: explain \var{safetyFactor} and show an example.
363 \end{funcdesc}
364
365 \begin{funcdesc}{JoinFaces}{meshList,safetyFactor=0.2,tolerance=1.e-13}
366 Generates a new \Domain object from the list \var{mehList} of \finley meshes.
367 Face elements whose nodes coordinates have difference is less then \var{tolerance} times the
368 diameter of the domain are combined to form a contact element \index{element!contact}
369 The corresponding face elements are removed from the mesh.
370
371 TODO: explain \var{safetyFactor} and show an example.
372 \end{funcdesc}

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