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1 ksteube 1811
2     %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
3 ksteube 1316 %
4 ksteube 1811 % Copyright (c) 2003-2008 by University of Queensland
5     % Earth Systems Science Computational Center (ESSCC)
6     % http://www.uq.edu.au/esscc
7 gross 625 %
8 ksteube 1811 % 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 gross 625 %
12 ksteube 1811 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
13 jgs 102
14 ksteube 1811
15 ksteube 1318 \chapter{ The Module \finley}
16 jgs 102 \label{CHAPTER ON FINLEY}
17    
18     \begin{figure}
19 gross 599 \centerline{\includegraphics[width=\figwidth]{figures/FinleyMesh.eps}}
20 jgs 102 \caption{Subdivision of an Ellipse into triangles order 1 (\finleyelement{Tri3})}
21     \label{FINLEY FIG 0}
22     \end{figure}
23    
24     \begin{figure}
25 gross 599 \centerline{\includegraphics[width=\figwidth]{figures/FinleyContact.eps}}
26 jgs 102 \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 gross 993 \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 jgs 102 To understand the usage of \finley one needs to have an understanding of how the finite element meshes
59 jgs 107 \index{FEM!mesh} are defined. \fig{FINLEY FIG 0} shows an example of the
60 jgs 102 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 lgraham 1700 which is a polynomial of a certain order and is continuous across it boundary to neighbor elements.
65 jgs 107 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 jgs 102 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 jgs 107 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 jgs 102 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 jgs 107 modified form is applied. The boundary of the domain is also subdivided into elements. \index{element!face} In \fig{FINLEY FIG 0}
80 jgs 102 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 jgs 107 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 ksteube 1316 with the surface of the domain. In \fig{FINLEY FIG 0}
88 jgs 102 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 jgs 107 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 jgs 102 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 jgs 107 of an interior element or, in case of a rich face element, it must be identical to an interior element.
96 jgs 102 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 jgs 107 The nodes $9$, $12$, $6$, $5$ define contact element $4$, where the coordinates of nodes $12$ and $5$ and
109 jgs 102 nodes $4$ and $6$ are identical with the idea that nodes $12$ and $9$ are located above and
110 jgs 107 nodes $5$ and $6$ below the contact region.
111 jgs 102 Again, the order of the nodes within an element is crucial. There is also the option of using rich elements
112 jgs 107 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 jgs 102 the 'first' face of the element above and the 'first' face of the element below the
115 jgs 107 contact regions line up. The rich version of element
116 jgs 102 $4$ 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 jgs 107 on the face and contacts. \fig{FINLEY.FIG:1}, \fig{FINLEY.FIG:2} and \fig{FINLEY.FIG:4} show the ordering of
121 jgs 102 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 ksteube 1316 The rich types have to be used if the gradient of function is to be calculated on faces and contacts, respectively.}
139 jgs 102 \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 jgs 107 In most cases \var{Node_DOF[i]}=\var{Node_ref[i]} however, for periodic boundary conditions,
146 jgs 102 \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 jgs 107 a contact region are marked with $2$ and elements below a contact region are marked with $1$.
152 jgs 102 \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 ksteube 1316 write("Point1 0",face_element_type,numFaceElements)
183 jgs 102 \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 jgs 107 In the case that rich contact elements are used the contact element section gets
221     the form
222 jgs 102 \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 jgs 107 It allows identification of nodes even if they have different physical locations. For instance, to
230 jgs 102 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    
254     \include{finleyelements}
255    
256     \subsection{Linear Solvers in \LinearPDE}
257 jgs 107 Currently \finley supports the linear solvers \PCG, \GMRES, \PRESTWENTY and \BiCGStab.
258 ksteube 1316 For \GMRES the options \var{truncation} and \var{restart} of the \method{getSolution} can be
259     used to control the truncation and restart during iteration. Default values are
260 jgs 102 \var{truncation}=5 and \var{restart}=20.
261 jgs 107 The default solver is \BiCGStab but if the symmetry flag is set \PCG is the default solver.
262 jgs 102 \finley supports the solver options \var{iter_max} which specifies the maximum number of iterations steps,
263     \var{verbose}=\True or \False and \var{preconditioner}=\constant{JACOBI} or \constant {ILU0}.
264 jgs 107 In some installations \finley supports the \Direct solver and the
265 jgs 102 solver options \var{reordering}=\constant{util.NO_REORDERING},
266     \constant{util.MINIMUM_FILL_IN} or \constant{util.NESTED_DISSECTION} (default is \constant{util.NO_REORDERING}),
267     \var{drop_tolerance} specifying the threshold for values to be dropped in the
268 ksteube 1316 incomplete elimination process (default is 0.01) and \var{drop_storage} specifying the maximum increase
269 jgs 102 in storage allowed in the
270 ksteube 1316 incomplete elimination process (default is 1.20).
271 jgs 102
272     \subsection{Functions}
273     \begin{funcdesc}{Mesh}{fileName,integrationOrder=-1}
274     creates a \Domain object form the FEM mesh defined in
275     file \var{fileName}. The file must be given the \finley file format.
276     If \var{integrationOrder} is positive, a numerical integration scheme
277     chosen which is accurate on each element up to a polynomial of
278     degree \var{integrationOrder} \index{integration order}. Otherwise
279     an appropriate integration order is chosen independently.
280     \end{funcdesc}
281    
282     \begin{funcdesc}{Rectangle}{n0,n1,order=1,l0=1.,l1=1., integrationOrder=-1, \\
283 ksteube 1459 periodic0=\False,periodic1=\False,useElementsOnFace=\False,optimize=\False}
284 jgs 102 Generates a \Domain object representing a two dimensional rectangle between
285     $(0,0)$ and $(l0,l1)$ with orthogonal edges. The rectangle is filled with
286     \var{n0} elements along the $x_0$-axis and
287     \var{n1} elements along the $x_1$-axis.
288     For \var{order}=1 and \var{order}=2
289     \finleyelement{Rec4} and
290     \finleyelement{Rec8} are used, respectively.
291     In the case of \var{useElementsOnFace}=\False,
292     \finleyelement{Line2} and
293     \finleyelement{Line3} are used to subdivide the edges of the rectangle, respectively.
294     In the case of \var{useElementsOnFace}=\True (this option should be used if gradients
295     are calculated on domain faces),
296     \finleyelement{Rec4Face} and
297     \finleyelement{Rec8Face} are used on the edges, respectively.
298     If \var{integrationOrder} is positive, a numerical integration scheme
299     chosen which is accurate on each element up to a polynomial of
300     degree \var{integrationOrder} \index{integration order}. Otherwise
301     an appropriate integration order is chosen independently. If
302     \var{periodic0}=\True, periodic boundary conditions \index{periodic boundary conditions}
303     along the $x_0$-directions are enforced. That means when for any solution of a PDE solved by \finley
304     the value on the line $x_0=0$ will be identical to the values on $x_0=\var{l0}$.
305     Correspondingly,
306     \var{periodic1}=\False sets periodic boundary conditions
307     in $x_1$-direction.
308 ksteube 1459 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.
309 jgs 102 \end{funcdesc}
310    
311     \begin{funcdesc}{Brick}{n0,n1,n2,order=1,l0=1.,l1=1.,l2=1., integrationOrder=-1, \\
312 ksteube 1459 periodic0=\False,periodic1=\False,periodic2=\False,useElementsOnFace=\False,optimize=\False}
313 jgs 102 Generates a \Domain object representing a three dimensional brick between
314     $(0,0,0)$ and $(l0,l1,l2)$ with orthogonal faces. The brick is filled with
315     \var{n0} elements along the $x_0$-axis,
316     \var{n1} elements along the $x_1$-axis and
317     \var{n2} elements along the $x_2$-axis.
318     For \var{order}=1 and \var{order}=2
319     \finleyelement{Hex8} and
320     \finleyelement{Hex20} are used, respectively.
321     In the case of \var{useElementsOnFace}=\False,
322     \finleyelement{Rec4} and
323     \finleyelement{Rec8} are used to subdivide the faces of the brick, respectively.
324     In the case of \var{useElementsOnFace}=\True (this option should be used if gradients
325     are calculated on domain faces),
326     \finleyelement{Hex8Face} and
327     \finleyelement{Hex20Face} are used on the brick faces, respectively.
328     If \var{integrationOrder} is positive, a numerical integration scheme
329     chosen which is accurate on each element up to a polynomial of
330     degree \var{integrationOrder} \index{integration order}. Otherwise
331     an appropriate integration order is chosen independently. If
332     \var{periodic0}=\True, periodic boundary conditions \index{periodic boundary conditions}
333     along the $x_0$-directions are enforced. That means when for any solution of a PDE solved by \finley
334     the value on the plane $x_0=0$ will be identical to the values on $x_0=\var{l0}$. Correspondingly,
335     \var{periodic1}=\False and \var{periodic2}=\False sets periodic boundary conditions
336     in $x_1$-direction and $x_2$-direction, respectively.
337 ksteube 1459 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.
338 jgs 102 \end{funcdesc}
339    
340     \begin{funcdesc}{GlueFaces}{meshList,safetyFactor=0.2,tolerance=1.e-13}
341 ksteube 1316 Generates a new \Domain object from the list \var{meshList} of \finley meshes.
342 jgs 102 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 ksteube 1316 Generates a new \Domain object from the list \var{meshList} of \finley meshes.
350 jgs 102 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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