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The user guide now builds under pdflatex (if you have converted the figures). Unfortunately, it has a really ugly title.
1
2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
3 %
4 % Copyright (c) 2003-2008 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
15 \chapter{ The Module \finley}
16 \label{CHAPTER ON FINLEY}
17
18 \begin{figure}
19 \centerline{\includegraphics[width=\figwidth]{figures/FinleyMesh}}
20 \caption{Subdivision of an Ellipse into triangles order 1 (\finleyelement{Tri3})}
21 \label{FINLEY FIG 0}
22 \end{figure}
23
24 \begin{figure}
25 \centerline{\includegraphics[width=\figwidth]{figures/FinleyContact}}
26 \caption{Mesh around a contact region (\finleyelement{Rec4})}
27 \label{FINLEY FIG 01}
28 \end{figure}
29
30 \declaremodule{extension}{finley} \modulesynopsis{Solving linear, steady partial differential equations using
31 finite elements}
32
33 {\it finley} is a library of C functions solving linear, steady partial differential equations
34 \index{partial differential equations} (PDEs) or systems of PDEs using isoparametrical finite
35 elements \index{FEM!isoparametrical}.
36 It supports unstructured, 1D, 2D and 3D meshes. The module \finley provides an access to the
37 library through the \LinearPDE class of \escript supporting its full functionality. {\it finley}
38 is parallelized using the OpenMP \index{OpenMP} paradigm.
39
40 \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
59 \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}.
61 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 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
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 In this case the triangle gets a curved edge which requires a parametrization of the triangle using a
69 quadratic polynomial. For this case, the solution is also approximated by a piecewise quadratic polynomial
70 (which explains the name isoparametrical elements), see \Ref{Zienc,NumHand} for more details.
71
72 The union of all elements defines the domain of the PDE.
73 Each element is defined by the nodes used to describe its shape. In \fig{FINLEY FIG 0} the element,
74 which has type \finleyelement{Tri3},
75 with element reference number $19$ \index{element!reference number} is defined by the nodes
76 with reference numbers $9$, $11$ and $0$ \index{node!reference number}. Notice that the order is counterclockwise.
77 The coefficients of the PDE are evaluated at integration nodes with each individual element.
78 For quadrilateral elements a Gauss quadrature scheme is used. In the case of triangular elements a
79 modified form is applied. The boundary of the domain is also subdivided into elements. \index{element!face} In \fig{FINLEY FIG 0}
80 line elements with two nodes are used. The elements are also defined by their describing nodes, e.g.
81 the face element reference number $20$ which has type \finleyelement{Line2} is defined by the nodes
82 with the reference numbers $11$ and $0$. Again the order is crucial, if moving from the first
83 to second node the domain has to lie on the left hand side (in the case of a two dimension surface element
84 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
86 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 domain. In \fig{FINLEY FIG 0}
88 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
90 with reference numbers $11$, $0$ and $9$. Notice that the face element $20$ is identical to the
91 interior element $19$ except that, in this case, the order of the node is different to align the first
92 edge of the triangle (which is the edge starting with the first node) with the boundary of the domain.
93
94 Be aware that face elements and elements in the interior of the domain must match, i.e. a face element must be the face
95 of an interior element or, in case of a rich face element, it must be identical to an interior element.
96 If no face elements are specified
97 \finley implicitly assumes homogeneous natural boundary conditions \index{natural boundary conditions!homogeneous},
98 i.e. \var{d}=$0$ and \var{y}=$0$, on the entire boundary of the domain. For
99 inhomogeneous natural boundary conditions \index{natural boundary conditions!inhomogeneous},
100 the boundary must be described by face elements.
101
102 If discontinuities of the PDE solution are considered contact elements
103 \index{element!contact}\index{contact conditions} are introduced to describe the contact region $\Gamma^{contact}$
104 even if $d^{contact}$ and $y^{contact}$ are zero. \fig{FINLEY FIG 01} shows a simple example of a mesh
105 of rectangular elements around a contact region $\Gamma^{contact}$ \index{element!contact}.
106 The contact region is described by the
107 elements $4$, $3$ and $6$. Their element type is \finleyelement{Line2_Contact}.
108 The nodes $9$, $12$, $6$, $5$ define contact element $4$, where the coordinates of nodes $12$ and $5$ and
109 nodes $4$ and $6$ are identical with the idea that nodes $12$ and $9$ are located above and
110 nodes $5$ and $6$ below the contact region.
111 Again, the order of the nodes within an element is crucial. There is also the option of using rich elements
112 if the gradient is to be calculated on the contact region. Similarly to the rich face elements
113 these are constructed from two interior elements by reordering the nodes such that
114 the 'first' face of the element above and the 'first' face of the element below the
115 contact regions line up. The rich version of element
116 $4$ is of type \finleyelement{Rec4Face_Contact} and is defined by the nodes $9$, $12$, $16$, $18$, $6$, $5$, $0$ and
117 $2$.
118
119 \tab{FINLEY TAB 1} shows the interior element types and the corresponding element types to be used
120 on the face and contacts. \fig{FINLEY.FIG:1}, \fig{FINLEY.FIG:2} and \fig{FINLEY.FIG:4} show the ordering of
121 the nodes within an element.
122
123 \begin{table}
124 \begin{tablev}{l|llll}{textrm}{interior}{face}{rich face}{contact}{rich contact}
125 \linev{\finleyelement{Line2}}{\finleyelement{Point1}}{\finleyelement{Line2Face}}{\finleyelement{Point1_Contact}}{\finleyelement{Line2Face_Contact}}
126 \linev{\finleyelement{Line3}}{\finleyelement{Point1}}{\finleyelement{Line3Face}}{\finleyelement{Point1_Contact}}{\finleyelement{Line3Face_Contact}}
127 \linev{\finleyelement{Tri3}}{\finleyelement{Line2}}{\finleyelement{Tri3Face}}{\finleyelement{Line2_Contact}}{\finleyelement{Tri3Face_Contact}}
128 \linev{\finleyelement{Tri6}}{\finleyelement{Line3}}{\finleyelement{Tri6Face}}{\finleyelement{Line3_Contact}}{\finleyelement{Tri6Face_Contact}}
129 \linev{\finleyelement{Rec4}}{\finleyelement{Line2}}{\finleyelement{Rec4Face}}{\finleyelement{Line2_Contact}}{\finleyelement{Rec4Face_Contact}}
130 \linev{\finleyelement{Rec8}}{\finleyelement{Line3}}{\finleyelement{Rec8Face}}{\finleyelement{Line3_Contact}}{\finleyelement{Rec8Face_Contact}}
131 \linev{\finleyelement{Rec9}}{\finleyelement{Line3}}{\finleyelement{Rec9Face}}{\finleyelement{Line3_Contact}}{\finleyelement{Rec9Face_Contact}}
132 \linev{\finleyelement{Tet4}}{\finleyelement{Tri6}}{\finleyelement{Tet4Face}}{\finleyelement{Tri6_Contact}}{\finleyelement{Tet4Face_Contact}}
133 \linev{\finleyelement{Tet10}}{\finleyelement{Tri9}}{\finleyelement{Tet10Face}}{\finleyelement{Tri9_Contact}}{\finleyelement{Tet10Face_Contact}}
134 \linev{\finleyelement{Hex8}}{\finleyelement{Rec4}}{\finleyelement{Hex8Face}}{\finleyelement{Rec4_Contact}}{\finleyelement{Hex8Face_Contact}}
135 \linev{\finleyelement{Hex20}}{\finleyelement{Rec8}}{\finleyelement{Hex20Face}}{\finleyelement{Rec8_Contact}}{\finleyelement{Hex20Face_Contact}}
136 \end{tablev}
137 \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, respectively.}
139 \label{FINLEY TAB 1}
140 \end{table}
141
142 The native \finley file format is defined as follows.
143 Each node \var{i} has \var{dim} spatial coordinates \var{Node[i]}, a reference number
144 \var{Node_ref[i]}, a degree of freedom \var{Node_DOF[i]} and tag \var{Node_tag[i]}.
145 In most cases \var{Node_DOF[i]}=\var{Node_ref[i]} however, for periodic boundary conditions,
146 \var{Node_DOF[i]} is chosen differently, see example below. The tag can be used to mark nodes sharing
147 the same properties. Element \var{i} is defined by the \var{Element_numNodes} nodes \var{Element_Nodes[i]}
148 which is a list of node reference numbers. The order is crucial.
149 It has a reference number \var{Element_ref[i]} and a tag \var{Element_tag[i]}. The tag
150 can be used to mark elements sharing the same properties. For instance elements above
151 a contact region are marked with $2$ and elements below a contact region are marked with $1$.
152 \var{Element_Type} and \var{Element_Num} give the element type and the number of elements in the mesh.
153 Analogue notations are used for face and contact elements. The following Python script
154 prints the mesh definition in the \finley file format:
155 \begin{python}
156 print "%s\n"%mesh_name
157 # node coordinates:
158 print "%dD-nodes %d\n"%(dim,numNodes)
159 for i in range(numNodes):
160 print "%d %d %d"%(Node_ref[i],Node_DOF[i],Node_tag[i])
161 for j in range(dim): print " %e"%Node[i][j]
162 print "\n"
163 # interior elements
164 print "%s %d\n"%(Element_Type,Element_Num)
165 for i in range(Element_Num):
166 print "%d %d"%(Element_ref[i],Element_tag[i])
167 for j in range(Element_numNodes): print " %d"%Element_Nodes[i][j]
168 print "\n"
169 # face elements
170 print "%s %d\n"%(FaceElement_Type,FaceElement_Num)
171 for i in range(FaceElement_Num):
172 print "%d %d"%(FaceElement_ref[i],FaceElement_tag[i])
173 for j in range(FaceElement_numNodes): print " %d"%FaceElement_Nodes[i][j]
174 print "\n"
175 # contact elements
176 print "%s %d\n"%(ContactElement_Type,ContactElement_Num)
177 for i in range(ContactElement_Num):
178 print "%d %d"%(ContactElement_ref[i],ContactElement_tag[i])
179 for j in range(ContactElement_numNodes): print " %d"%ContactElement_Nodes[i][j]
180 print "\n"
181 # point sources (not supported yet)
182 write("Point1 0",face_element_type,numFaceElements)
183 \end{python}
184
185 The following example of a mesh file defines the mesh shown in \fig{FINLEY FIG 01}:
186 \begin{verbatim}
187 Example 1
188 2D Nodes 16
189 0 0 0 0. 0.
190 2 2 0 0.33 0.
191 3 3 0 0.66 0.
192 7 4 0 1. 0.
193 5 5 0 0. 0.5
194 6 6 0 0.33 0.5
195 8 8 0 0.66 0.5
196 10 10 0 1.0 0.5
197 12 12 0 0. 0.5
198 9 9 0 0.33 0.5
199 13 13 0 0.66 0.5
200 15 15 0 1.0 0.5
201 16 16 0 0. 1.0
202 18 18 0 0.33 1.0
203 19 19 0 0.66 1.0
204 20 20 0 1.0 1.0
205 Rec4 6
206 0 1 0 2 6 5
207 1 1 2 3 8 6
208 2 1 3 7 10 8
209 5 2 12 9 18 16
210 7 2 13 19 18 9
211 10 2 20 19 13 15
212 Line2 0
213 Line2_Contact 3
214 4 0 9 12 6 5
215 3 0 13 9 8 6
216 6 0 15 13 10 8
217 Point1 0
218 \end{verbatim}
219 Notice that the order in which the nodes and elements are given is arbitrary.
220 In the case that rich contact elements are used the contact element section gets
221 the form
222 \begin{verbatim}
223 Rec4Face_Contact 3
224 4 0 9 12 16 18 6 5 0 2
225 3 0 13 9 18 19 8 6 2 3
226 6 0 15 13 19 20 10 8 3 7
227 \end{verbatim}
228 Periodic boundary condition \index{boundary conditions!periodic} can be introduced by altering \var{Node_DOF}.
229 It allows identification of nodes even if they have different physical locations. For instance, to
230 enforce periodic boundary conditions at the face $x_0=0$ and $x_0=1$ one identifies
231 the degrees of freedom for nodes $0$, $5$, $12$ and $16$ with the degrees of freedom for
232 $7$, $10$, $15$ and $20$, respectively. The node section of the \finley mesh gets now the form:
233 \begin{verbatim}
234 2D Nodes 16
235 0 0 0 0. 0.
236 2 2 0 0.33 0.
237 3 3 0 0.66 0.
238 7 0 0 1. 0.
239 5 5 0 0. 0.5
240 6 6 0 0.33 0.5
241 8 8 0 0.66 0.5
242 10 5 0 1.0 0.5
243 12 12 0 0. 0.5
244 9 9 0 0.33 0.5
245 13 13 0 0.66 0.5
246 15 12 0 1.0 0.5
247 16 16 0 0. 1.0
248 18 18 0 0.33 1.0
249 19 19 0 0.66 1.0
250 20 16 0 1.0 1.0
251 \end{verbatim}
252
253 \clearpage
254 \input{finleyelements}
255 \clearpage
256
257 \subsection{Linear Solvers in \LinearPDE}
258 Currently \finley supports the linear solvers \PCG, \GMRES, \PRESTWENTY and \BiCGStab.
259 For \GMRES the options \var{truncation} and \var{restart} of the \method{getSolution} can be
260 used to control the truncation and restart during iteration. Default values are
261 \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.
263 \finley supports the solver options \var{iter_max} which specifies the maximum number of iterations steps,
264 \var{verbose}=\True or \False and \var{preconditioner}=\constant{JACOBI} or \constant {ILU0}.
265 In some installations \finley supports the \Direct solver and the
266 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}),
268 \var{drop_tolerance} specifying the threshold for values to be dropped in the
269 incomplete elimination process (default is 0.01) and \var{drop_storage} specifying the maximum increase
270 in storage allowed in the
271 incomplete elimination process (default is 1.20).
272
273 \subsection{Functions}
274 \begin{funcdesc}{Mesh}{fileName,integrationOrder=-1}
275 creates a \Domain object form the FEM mesh defined in
276 file \var{fileName}. The file must be given the \finley file format.
277 If \var{integrationOrder} is positive, a numerical integration scheme
278 chosen which is accurate on each element up to a polynomial of
279 degree \var{integrationOrder} \index{integration order}. Otherwise
280 an appropriate integration order is chosen independently.
281 \end{funcdesc}
282
283 \begin{funcdesc}{Rectangle}{n0,n1,order=1,l0=1.,l1=1., integrationOrder=-1, \\
284 periodic0=\False,periodic1=\False,useElementsOnFace=\False,optimize=\False}
285 Generates a \Domain object representing a two dimensional rectangle between
286 $(0,0)$ and $(l0,l1)$ with orthogonal edges. The rectangle is filled with
287 \var{n0} elements along the $x_0$-axis and
288 \var{n1} elements along the $x_1$-axis.
289 For \var{order}=1 and \var{order}=2
290 \finleyelement{Rec4} and
291 \finleyelement{Rec8} are used, respectively.
292 In the case of \var{useElementsOnFace}=\False,
293 \finleyelement{Line2} and
294 \finleyelement{Line3} are used to subdivide the edges of the rectangle, respectively.
295 In the case of \var{useElementsOnFace}=\True (this option should be used if gradients
296 are calculated on domain faces),
297 \finleyelement{Rec4Face} and
298 \finleyelement{Rec8Face} are used on the edges, respectively.
299 If \var{integrationOrder} is positive, a numerical integration scheme
300 chosen which is accurate on each element up to a polynomial of
301 degree \var{integrationOrder} \index{integration order}. Otherwise
302 an appropriate integration order is chosen independently. If
303 \var{periodic0}=\True, periodic boundary conditions \index{periodic boundary conditions}
304 along the $x_0$-directions are enforced. That means when for any solution of a PDE solved by \finley
305 the value on the line $x_0=0$ will be identical to the values on $x_0=\var{l0}$.
306 Correspondingly,
307 \var{periodic1}=\False sets periodic boundary conditions
308 in $x_1$-direction.
309 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.
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,optimize=\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 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.
339 \end{funcdesc}
340
341 \begin{funcdesc}{GlueFaces}{meshList,safetyFactor=0.2,tolerance=1.e-13}
342 Generates a new \Domain object from the list \var{meshList} of \finley meshes.
343 Nodes in face elements whose difference of coordinates is less then \var{tolerance} times the
344 diameter of the domain are merged. The corresponding face elements are removed from the mesh.
345
346 TODO: explain \var{safetyFactor} and show an example.
347 \end{funcdesc}
348
349 \begin{funcdesc}{JoinFaces}{meshList,safetyFactor=0.2,tolerance=1.e-13}
350 Generates a new \Domain object from the list \var{meshList} of \finley meshes.
351 Face elements whose nodes coordinates have difference is less then \var{tolerance} times the
352 diameter of the domain are combined to form a contact element \index{element!contact}
353 The corresponding face elements are removed from the mesh.
354
355 TODO: explain \var{safetyFactor} and show an example.
356 \end{funcdesc}

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