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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 jgs 102
10    
11 gross 625
12 jgs 102 \chapter{ The module \finley}
13     \label{CHAPTER ON FINLEY}
14    
15     \begin{figure}
16 gross 599 \centerline{\includegraphics[width=\figwidth]{figures/FinleyMesh.eps}}
17 jgs 102 \caption{Subdivision of an Ellipse into triangles order 1 (\finleyelement{Tri3})}
18     \label{FINLEY FIG 0}
19     \end{figure}
20    
21     \begin{figure}
22 gross 599 \centerline{\includegraphics[width=\figwidth]{figures/FinleyContact.eps}}
23 jgs 102 \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 jgs 107 \index{FEM!mesh} are defined. \fig{FINLEY FIG 0} shows an example of the
40 jgs 102 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 jgs 107 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 jgs 102 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 jgs 107 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 jgs 102 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 jgs 107 modified form is applied. The boundary of the domain is also subdivided into elements. \index{element!face} In \fig{FINLEY FIG 0}
60 jgs 102 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 jgs 107 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 jgs 102 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 jgs 107 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 jgs 102 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 jgs 107 of an interior element or, in case of a rich face element, it must be identical to an interior element.
76 jgs 102 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 jgs 107 The nodes $9$, $12$, $6$, $5$ define contact element $4$, where the coordinates of nodes $12$ and $5$ and
89 jgs 102 nodes $4$ and $6$ are identical with the idea that nodes $12$ and $9$ are located above and
90 jgs 107 nodes $5$ and $6$ below the contact region.
91 jgs 102 Again, the order of the nodes within an element is crucial. There is also the option of using rich elements
92 jgs 107 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 jgs 102 the 'first' face of the element above and the 'first' face of the element below the
95 jgs 107 contact regions line up. The rich version of element
96 jgs 102 $4$ 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 jgs 107 on the face and contacts. \fig{FINLEY.FIG:1}, \fig{FINLEY.FIG:2} and \fig{FINLEY.FIG:4} show the ordering of
101 jgs 102 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 jgs 107 The rich types have to be used if the gradient of function is to be calculated on faces and contacts, resepctively.}
119 jgs 102 \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 jgs 107 In most cases \var{Node_DOF[i]}=\var{Node_ref[i]} however, for periodic boundary conditions,
126 jgs 102 \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 jgs 107 a contact region are marked with $2$ and elements below a contact region are marked with $1$.
132 jgs 102 \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 jgs 107 In the case that rich contact elements are used the contact element section gets
201     the form
202 jgs 102 \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 jgs 107 It allows identification of nodes even if they have different physical locations. For instance, to
210 jgs 102 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 jgs 107 Currently \finley supports the linear solvers \PCG, \GMRES, \PRESTWENTY and \BiCGStab.
238 jgs 102 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 jgs 107 The default solver is \BiCGStab but if the symmetry flag is set \PCG is the default solver.
242 jgs 102 \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 jgs 107 In some installations \finley supports the \Direct solver and the
245 jgs 102 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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