# Contents of /trunk/esys2/doc/user/finley.tex

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 1 % $Id$ 2 3 4 \chapter{ The module \finley} 5 \label{CHAPTER ON FINLEY} 6 7 \begin{figure} 8 \centerline{\includegraphics[width=\figwidth]{FinleyMesh}} 9 \caption{Subdivision of an Ellipse into triangles order 1 (\finleyelement{Tri3})} 10 \label{FINLEY FIG 0} 11 \end{figure} 12 13 \begin{figure} 14 \centerline{\includegraphics[width=\figwidth]{FinleyContact}} 15 \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 \index{FEM!mesh} are defined. \fig{FINLEY FIG 0} shows an example of the 32 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 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 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 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 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 modified form is applied. The boundary of the domain is also subdivided into elements. \index{element!face} In \fig{FINLEY FIG 0} 52 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 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 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 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 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 of an interior element or, in case of a rich face element, it must be identical to an interior element. 68 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 The nodes $9$, $12$, $6$, $5$ define contact element $4$, where the coordinates of nodes $12$ and $5$ and 81 nodes $4$ and $6$ are identical with the idea that nodes $12$ and $9$ are located above and 82 nodes $5$ and $6$ below the contact region. 83 Again, the order of the nodes within an element is crucial. There is also the option of using rich elements 84 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 the 'first' face of the element above and the 'first' face of the element below the 87 contact regions line up. The rich version of element 88 $4$ 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 on the face and contacts. \fig{FINLEY.FIG:1}, \fig{FINLEY.FIG:2} and \fig{FINLEY.FIG:4} show the ordering of 93 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 The rich types have to be used if the gradient of function is to be calculated on faces and contacts, resepctively.} 111 \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 In most cases \var{Node_DOF[i]}=\var{Node_ref[i]} however, for periodic boundary conditions, 118 \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 a contact region are marked with $2$ and elements below a contact region are marked with $1$. 124 \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 In the case that rich contact elements are used the contact element section gets 193 the form 194 \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 It allows identification of nodes even if they have different physical locations. For instance, to 202 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 Currently \finley supports the linear solvers \PCG, \GMRES, \PRESTWENTY and \BiCGStab. 230 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 The default solver is \BiCGStab but if the symmetry flag is set \PCG is the default solver. 234 \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 In some installations \finley supports the \Direct solver and the 237 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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