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

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