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

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 1 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3 % 4 % Copyright (c) 2003-2008 by University of Queensland 5 % Earth Systems Science Computational Center (ESSCC) 6 7 % 8 % Primary Business: Queensland, Australia 9 % Licensed under the Open Software License version 3.0 10 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}{ReadMesh}{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}{load}{fileName} 284 recovers a \Domain object from a dump file created by the \ 285 eateseates a \Domain object form the FEM mesh defined in 286 file \var{fileName}. The file must be given the \finley file format. 287 If \var{integrationOrder} is positive, a numerical integration scheme 288 chosen which is accurate on each element up to a polynomial of 289 degree \var{integrationOrder} \index{integration order}. Otherwise 290 an appropriate integration order is chosen independently. 291 \end{funcdesc} 292 293 \begin{funcdesc}{Rectangle}{n0,n1,order=1,l0=1.,l1=1., integrationOrder=-1, \\ 294 periodic0=\False,periodic1=\False,useElementsOnFace=\False,optimize=\False} 295 Generates a \Domain object representing a two dimensional rectangle between 296 $(0,0)$ and $(l0,l1)$ with orthogonal edges. The rectangle is filled with 297 \var{n0} elements along the $x_0$-axis and 298 \var{n1} elements along the $x_1$-axis. 299 For \var{order}=1 and \var{order}=2 300 \finleyelement{Rec4} and 301 \finleyelement{Rec8} are used, respectively. 302 In the case of \var{useElementsOnFace}=\False, 303 \finleyelement{Line2} and 304 \finleyelement{Line3} are used to subdivide the edges of the rectangle, respectively. 305 In the case of \var{useElementsOnFace}=\True (this option should be used if gradients 306 are calculated on domain faces), 307 \finleyelement{Rec4Face} and 308 \finleyelement{Rec8Face} are used on the edges, respectively. 309 If \var{integrationOrder} is positive, a numerical integration scheme 310 chosen which is accurate on each element up to a polynomial of 311 degree \var{integrationOrder} \index{integration order}. Otherwise 312 an appropriate integration order is chosen independently. If 313 \var{periodic0}=\True, periodic boundary conditions \index{periodic boundary conditions} 314 along the $x_0$-directions are enforced. That means when for any solution of a PDE solved by \finley 315 the value on the line $x_0=0$ will be identical to the values on $x_0=\var{l0}$. 316 Correspondingly, 317 \var{periodic1}=\False sets periodic boundary conditions 318 in $x_1$-direction. 319 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. 320 \end{funcdesc} 321 322 \begin{funcdesc}{Brick}{n0,n1,n2,order=1,l0=1.,l1=1.,l2=1., integrationOrder=-1, \\ 323 periodic0=\False,periodic1=\False,periodic2=\False,useElementsOnFace=\False,optimize=\False} 324 Generates a \Domain object representing a three dimensional brick between 325 $(0,0,0)$ and $(l0,l1,l2)$ with orthogonal faces. The brick is filled with 326 \var{n0} elements along the $x_0$-axis, 327 \var{n1} elements along the $x_1$-axis and 328 \var{n2} elements along the $x_2$-axis. 329 For \var{order}=1 and \var{order}=2 330 \finleyelement{Hex8} and 331 \finleyelement{Hex20} are used, respectively. 332 In the case of \var{useElementsOnFace}=\False, 333 \finleyelement{Rec4} and 334 \finleyelement{Rec8} are used to subdivide the faces of the brick, respectively. 335 In the case of \var{useElementsOnFace}=\True (this option should be used if gradients 336 are calculated on domain faces), 337 \finleyelement{Hex8Face} and 338 \finleyelement{Hex20Face} are used on the brick faces, respectively. 339 If \var{integrationOrder} is positive, a numerical integration scheme 340 chosen which is accurate on each element up to a polynomial of 341 degree \var{integrationOrder} \index{integration order}. Otherwise 342 an appropriate integration order is chosen independently. If 343 \var{periodic0}=\True, periodic boundary conditions \index{periodic boundary conditions} 344 along the $x_0$-directions are enforced. That means when for any solution of a PDE solved by \finley 345 the value on the plane $x_0=0$ will be identical to the values on $x_0=\var{l0}$. Correspondingly, 346 \var{periodic1}=\False and \var{periodic2}=\False sets periodic boundary conditions 347 in $x_1$-direction and $x_2$-direction, respectively. 348 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. 349 \end{funcdesc} 350 351 \begin{funcdesc}{GlueFaces}{meshList,safetyFactor=0.2,tolerance=1.e-13} 352 Generates a new \Domain object from the list \var{meshList} of \finley meshes. 353 Nodes in face elements whose difference of coordinates is less then \var{tolerance} times the 354 diameter of the domain are merged. The corresponding face elements are removed from the mesh. 355 356 TODO: explain \var{safetyFactor} and show an example. 357 \end{funcdesc} 358 359 \begin{funcdesc}{JoinFaces}{meshList,safetyFactor=0.2,tolerance=1.e-13} 360 Generates a new \Domain object from the list \var{meshList} of \finley meshes. 361 Face elements whose nodes coordinates have difference is less then \var{tolerance} times the 362 diameter of the domain are combined to form a contact element \index{element!contact} 363 The corresponding face elements are removed from the mesh. 364 365 TODO: explain \var{safetyFactor} and show an example. 366 \end{funcdesc}

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