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 1 ksteube 1811 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3 ksteube 1316 % 4 jfenwick 2881 % Copyright (c) 2003-2010 by University of Queensland 5 ksteube 1811 % Earth Systems Science Computational Center (ESSCC) 6 7 gross 625 % 8 ksteube 1811 % Primary Business: Queensland, Australia 9 % Licensed under the Open Software License version 3.0 10 11 gross 625 % 12 ksteube 1811 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 13 jgs 102 14 ksteube 1811 15 caltinay 3293 \chapter{The \finley Module}\label{CHAPTER ON FINLEY} 16 jgs 102 17 \begin{figure} 18 caltinay 3279 \centerline{\includegraphics[width=\figwidth]{FinleyMesh}} 19 jgs 102 \caption{Subdivision of an Ellipse into triangles order 1 (\finleyelement{Tri3})} 20 \label{FINLEY FIG 0} 21 \end{figure} 22 23 \begin{figure} 24 caltinay 3279 \centerline{\includegraphics[width=\figwidth]{FinleyContact}} 25 jgs 102 \caption{Mesh around a contact region (\finleyelement{Rec4})} 26 \label{FINLEY FIG 01} 27 \end{figure} 28 29 \declaremodule{extension}{finley} \modulesynopsis{Solving linear, steady partial differential equations using 30 finite elements} 31 32 {\it finley} is a library of C functions solving linear, steady partial differential equations 33 caltinay 3293 \index{partial differential equations} (PDEs) or systems of PDEs using isoparametrical finite 34 jgs 102 elements \index{FEM!isoparametrical}. 35 It supports unstructured, 1D, 2D and 3D meshes. The module \finley provides an access to the 36 caltinay 3293 library through the \LinearPDE class of \escript supporting its full functionality. {\it finley} 37 is parallelized using the OpenMP \index{OpenMP} paradigm. 38 jgs 102 39 gross 993 \section{Formulation} 40 41 For a single PDE with a solution with a single component the linear PDE is defined in the 42 following form: 43 \begin{equation}\label{FINLEY.SINGLE.1} 44 \begin{array}{cl} & 45 \displaystyle{ 46 jfenwick 3295 \int_{\Omega} 47 A_{jl} \cdot v_{,j}u_{,l}+ B_{j} \cdot v_{,j} u+ C_{l} \cdot v u_{,l}+D \cdot vu \; d\Omega } \\ 48 + & \displaystyle{\int_{\Gamma} d \cdot vu \; d{\Gamma} } 49 + \displaystyle{\int_{\Gamma^{contact}} d^{contact} \cdot [v][u] \; d{\Gamma} } \\ 50 = & \displaystyle{\int_{\Omega} X_{j} \cdot v_{,j}+ Y \cdot v \; d\Omega }\\ 51 + & \displaystyle{\int_{\Gamma} y \cdot v \; d{\Gamma}} + 52 \displaystyle{\int_{\Gamma^{contact}} y^{contact}\cdot [v] \; d{\Gamma}} \\ 53 gross 993 \end{array} 54 \end{equation} 55 56 \section{Meshes} 57 gross 2793 \label{FINLEY 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 caltinay 3293 subdivision of an ellipse into so called elements \index{FEM!elements} \index{element}. 61 jgs 102 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 caltinay 3293 In this case the triangle gets a curved edge which requires a parameterization of the triangle using a 69 jgs 102 quadratic polynomial. For this case, the solution is also approximated by a piecewise quadratic polynomial 70 caltinay 3293 (which explains the name isoparametrical elements), see \Ref{Zienc,NumHand} for more details. 71 gross 2748 \finley supports macro elements\index{macro elements}. For these elements a piecewise linear approximation is used on an element which is further subdivided (in the case \finley halved). As such these elements do not provide more than a further mesh refinement but should be used in the case of incompressible flows, see \class{StokesProblemCartesian}. For these problems a linear approximation of the pressure across the element is used (use the \ReducedSolutionFS \FunctionSpace) while the refined element is used to approximate velocity. So a macro element provides a continuous pressure approximation together with a velocity approximation on a refined mesh. This approach is necessary to make sure that the incompressible flow has a unique solution. 72 jgs 102 73 The union of all elements defines the domain of the PDE. 74 jgs 107 Each element is defined by the nodes used to describe its shape. In \fig{FINLEY FIG 0} the element, 75 which has type \finleyelement{Tri3}, 76 with element reference number $19$ \index{element!reference number} is defined by the nodes 77 caltinay 3293 with reference numbers $9$, $11$ and $0$ \index{node!reference number}. Notice that the order is counterclockwise. 78 The coefficients of the PDE are evaluated at integration nodes with each individual element. 79 For quadrilateral elements a Gauss quadrature scheme is used. In the case of triangular elements a 80 jgs 107 modified form is applied. The boundary of the domain is also subdivided into elements. \index{element!face} In \fig{FINLEY FIG 0} 81 jgs 102 line elements with two nodes are used. The elements are also defined by their describing nodes, e.g. 82 the face element reference number $20$ which has type \finleyelement{Line2} is defined by the nodes 83 with the reference numbers $11$ and $0$. Again the order is crucial, if moving from the first 84 jgs 107 to second node the domain has to lie on the left hand side (in the case of a two dimension surface element 85 the domain has to lie on the left hand side when moving counterclockwise). If the gradient on the 86 surface of the domain is to be calculated rich face elements face to be used. Rich elements on a face 87 are identical to interior elements but with a modified order of nodes such that the 'first' face of the element aligns 88 ksteube 1316 with the surface of the domain. In \fig{FINLEY FIG 0} 89 caltinay 3293 elements of the type \finleyelement{Tri3Face} are used. 90 jgs 102 The face element reference number $20$ as a rich face element is defined by the nodes 91 jgs 107 with reference numbers $11$, $0$ and $9$. Notice that the face element $20$ is identical to the 92 interior element $19$ except that, in this case, the order of the node is different to align the first 93 jgs 102 edge of the triangle (which is the edge starting with the first node) with the boundary of the domain. 94 95 Be aware that face elements and elements in the interior of the domain must match, i.e. a face element must be the face 96 jgs 107 of an interior element or, in case of a rich face element, it must be identical to an interior element. 97 jgs 102 If no face elements are specified 98 \finley implicitly assumes homogeneous natural boundary conditions \index{natural boundary conditions!homogeneous}, 99 caltinay 3293 i.e. \var{d}=$0$ and \var{y}=$0$, on the entire boundary of the domain. For 100 inhomogeneous natural boundary conditions \index{natural boundary conditions!inhomogeneous}, 101 the boundary must be described by face elements. 102 jgs 102 103 caltinay 3293 If discontinuities of the PDE solution are considered contact elements 104 \index{element!contact}\index{contact conditions} are introduced to describe the contact region $\Gamma^{contact}$ 105 jgs 102 even if $d^{contact}$ and $y^{contact}$ are zero. \fig{FINLEY FIG 01} shows a simple example of a mesh 106 caltinay 3293 of rectangular elements around a contact region $\Gamma^{contact}$ \index{element!contact}. 107 jgs 102 The contact region is described by the 108 caltinay 3293 elements $4$, $3$ and $6$. Their element type is \finleyelement{Line2_Contact}. 109 jgs 107 The nodes $9$, $12$, $6$, $5$ define contact element $4$, where the coordinates of nodes $12$ and $5$ and 110 caltinay 3293 nodes $4$ and $6$ are identical with the idea that nodes $12$ and $9$ are located above and 111 nodes $5$ and $6$ below the contact region. 112 jgs 102 Again, the order of the nodes within an element is crucial. There is also the option of using rich elements 113 caltinay 3293 if the gradient is to be calculated on the contact region. Similarly to the rich face elements 114 jgs 107 these are constructed from two interior elements by reordering the nodes such that 115 caltinay 3293 the 'first' face of the element above and the 'first' face of the element below the 116 contact regions line up. The rich version of element 117 $4$ is of type \finleyelement{Rec4Face_Contact} and is defined by the nodes $9$, $12$, $16$, $18$, $6$, $5$, $0$ and 118 jgs 102 $2$. 119 120 gross 2748 121 122 jgs 102 \tab{FINLEY TAB 1} shows the interior element types and the corresponding element types to be used 123 jgs 107 on the face and contacts. \fig{FINLEY.FIG:1}, \fig{FINLEY.FIG:2} and \fig{FINLEY.FIG:4} show the ordering of 124 jgs 102 the nodes within an element. 125 126 \begin{table} 127 caltinay 3293 \centering 128 \begin{tabular}{l|llll} 129 \bfseries interior & face & rich face & contact & rich contact\\ 130 \hline 131 \finleyelement{Line2} & \finleyelement{Point1} & \finleyelement{Line2Face} & \finleyelement{Point1_Contact} & \finleyelement{Line2Face_Contact}\\ 132 \finleyelement{Line3} & \finleyelement{Point1} & \finleyelement{Line3Face} & \finleyelement{Point1_Contact} & \finleyelement{Line3Face_Contact}\\ 133 \finleyelement{Tri3} & \finleyelement{Line2} & \finleyelement{Tri3Face} & \finleyelement{Line2_Contact} & \finleyelement{Tri3Face_Contact}\\ 134 \finleyelement{Tri6} & \finleyelement{Line3} & \finleyelement{Tri6Face} & \finleyelement{Line3_Contact} & \finleyelement{Tri6Face_Contact}\\ 135 \finleyelement{Rec4} & \finleyelement{Line2} & \finleyelement{Rec4Face} & \finleyelement{Line2_Contact} & \finleyelement{Rec4Face_Contact}\\ 136 \finleyelement{Rec8} & \finleyelement{Line3} & \finleyelement{Rec8Face} & \finleyelement{Line3_Contact} & \finleyelement{Rec8Face_Contact}\\ 137 \finleyelement{Rec9} & \finleyelement{Line3} & \finleyelement{Rec9Face} & \finleyelement{Line3_Contact} & \finleyelement{Rec9Face_Contact}\\ 138 \finleyelement{Tet4} & \finleyelement{Tri6} & \finleyelement{Tet4Face} & \finleyelement{Tri6_Contact} & \finleyelement{Tet4Face_Contact}\\ 139 \finleyelement{Tet10} & \finleyelement{Tri9} & \finleyelement{Tet10Face} & \finleyelement{Tri9_Contact} & \finleyelement{Tet10Face_Contact}\\ 140 \finleyelement{Hex8} & \finleyelement{Rec4} & \finleyelement{Hex8Face} & \finleyelement{Rec4_Contact} & \finleyelement{Hex8Face_Contact}\\ 141 \finleyelement{Hex20} & \finleyelement{Rec8} & \finleyelement{Hex20Face} & \finleyelement{Rec8_Contact} & \finleyelement{Hex20Face_Contact}\\ 142 \finleyelement{Hex27} & \finleyelement{Rec9} & N\textbackslash A & N\textbackslash A & N\textbackslash A\\ 143 \finleyelement{Hex27Macro} & \finleyelement{Rec9Macro} & N\textbackslash A & N\textbackslash A & N\textbackslash A\\ 144 \finleyelement{Tet10Macro} & \finleyelement{Tri6Macro} & N\textbackslash A & N\textbackslash A & N\textbackslash A\\ 145 \finleyelement{Rec9Macro} & \finleyelement{Line3Macro} & N\textbackslash A & N\textbackslash A & N\textbackslash A\\ 146 \finleyelement{Tri6Macro} & \finleyelement{Line3Macro} & N\textbackslash A & N\textbackslash A & N\textbackslash A\\ 147 \end{tabular} 148 jgs 102 \caption{Finley elements and corresponding elements to be used on domain faces and contacts. 149 ksteube 1316 The rich types have to be used if the gradient of function is to be calculated on faces and contacts, respectively.} 150 jgs 102 \label{FINLEY TAB 1} 151 \end{table} 152 153 The native \finley file format is defined as follows. 154 Each node \var{i} has \var{dim} spatial coordinates \var{Node[i]}, a reference number 155 \var{Node_ref[i]}, a degree of freedom \var{Node_DOF[i]} and tag \var{Node_tag[i]}. 156 jgs 107 In most cases \var{Node_DOF[i]}=\var{Node_ref[i]} however, for periodic boundary conditions, 157 jgs 102 \var{Node_DOF[i]} is chosen differently, see example below. The tag can be used to mark nodes sharing 158 the same properties. Element \var{i} is defined by the \var{Element_numNodes} nodes \var{Element_Nodes[i]} 159 which is a list of node reference numbers. The order is crucial. 160 caltinay 3293 It has a reference number \var{Element_ref[i]} and a tag \var{Element_tag[i]}. The tag 161 can be used to mark elements sharing the same properties. For instance elements above 162 a contact region are marked with $2$ and elements below a contact region are marked with $1$. 163 jgs 102 \var{Element_Type} and \var{Element_Num} give the element type and the number of elements in the mesh. 164 Analogue notations are used for face and contact elements. The following Python script 165 prints the mesh definition in the \finley file format: 166 \begin{python} 167 print "%s\n"%mesh_name 168 # node coordinates: 169 print "%dD-nodes %d\n"%(dim,numNodes) 170 caltinay 3293 for i in range(numNodes): 171 jgs 102 print "%d %d %d"%(Node_ref[i],Node_DOF[i],Node_tag[i]) 172 for j in range(dim): print " %e"%Node[i][j] 173 print "\n" 174 # interior elements 175 print "%s %d\n"%(Element_Type,Element_Num) 176 for i in range(Element_Num): 177 print "%d %d"%(Element_ref[i],Element_tag[i]) 178 for j in range(Element_numNodes): print " %d"%Element_Nodes[i][j] 179 print "\n" 180 # face elements 181 print "%s %d\n"%(FaceElement_Type,FaceElement_Num) 182 for i in range(FaceElement_Num): 183 print "%d %d"%(FaceElement_ref[i],FaceElement_tag[i]) 184 for j in range(FaceElement_numNodes): print " %d"%FaceElement_Nodes[i][j] 185 print "\n" 186 # contact elements 187 print "%s %d\n"%(ContactElement_Type,ContactElement_Num) 188 for i in range(ContactElement_Num): 189 print "%d %d"%(ContactElement_ref[i],ContactElement_tag[i]) 190 for j in range(ContactElement_numNodes): print " %d"%ContactElement_Nodes[i][j] 191 print "\n" 192 # point sources (not supported yet) 193 ksteube 1316 write("Point1 0",face_element_type,numFaceElements) 194 jgs 102 \end{python} 195 196 The following example of a mesh file defines the mesh shown in \fig{FINLEY FIG 01}: 197 \begin{verbatim} 198 Example 1 199 2D Nodes 16 200 0 0 0 0. 0. 201 2 2 0 0.33 0. 202 3 3 0 0.66 0. 203 7 4 0 1. 0. 204 5 5 0 0. 0.5 205 6 6 0 0.33 0.5 206 8 8 0 0.66 0.5 207 10 10 0 1.0 0.5 208 12 12 0 0. 0.5 209 9 9 0 0.33 0.5 210 13 13 0 0.66 0.5 211 15 15 0 1.0 0.5 212 16 16 0 0. 1.0 213 18 18 0 0.33 1.0 214 19 19 0 0.66 1.0 215 20 20 0 1.0 1.0 216 Rec4 6 217 0 1 0 2 6 5 218 1 1 2 3 8 6 219 2 1 3 7 10 8 220 5 2 12 9 18 16 221 7 2 13 19 18 9 222 10 2 20 19 13 15 223 Line2 0 224 Line2_Contact 3 225 4 0 9 12 6 5 226 3 0 13 9 8 6 227 6 0 15 13 10 8 228 Point1 0 229 \end{verbatim} 230 Notice that the order in which the nodes and elements are given is arbitrary. 231 jgs 107 In the case that rich contact elements are used the contact element section gets 232 the form 233 jgs 102 \begin{verbatim} 234 Rec4Face_Contact 3 235 4 0 9 12 16 18 6 5 0 2 236 3 0 13 9 18 19 8 6 2 3 237 6 0 15 13 19 20 10 8 3 7 238 \end{verbatim} 239 Periodic boundary condition \index{boundary conditions!periodic} can be introduced by altering \var{Node_DOF}. 240 jgs 107 It allows identification of nodes even if they have different physical locations. For instance, to 241 jgs 102 enforce periodic boundary conditions at the face $x_0=0$ and $x_0=1$ one identifies 242 the degrees of freedom for nodes $0$, $5$, $12$ and $16$ with the degrees of freedom for 243 caltinay 3293 $7$, $10$, $15$ and $20$, respectively. The node section of the \finley mesh gets now the form: 244 jgs 102 \begin{verbatim} 245 2D Nodes 16 246 0 0 0 0. 0. 247 2 2 0 0.33 0. 248 3 3 0 0.66 0. 249 7 0 0 1. 0. 250 5 5 0 0. 0.5 251 6 6 0 0.33 0.5 252 8 8 0 0.66 0.5 253 10 5 0 1.0 0.5 254 12 12 0 0. 0.5 255 9 9 0 0.33 0.5 256 13 13 0 0.66 0.5 257 15 12 0 1.0 0.5 258 16 16 0 0. 1.0 259 18 18 0 0.33 1.0 260 19 19 0 0.66 1.0 261 20 16 0 1.0 1.0 262 \end{verbatim} 263 264 jfenwick 1955 \clearpage 265 \input{finleyelements} 266 \clearpage 267 jgs 102 268 gross 2793 \begin{figure}[th] 269 \begin{center} 270 caltinay 3279 \subfigure[Triangle]{\label{FINLEY MACRO TRI}\includegraphics[scale=0.25]{FinleyMacroTri}} 271 \subfigure[Quadrilateral]{\label{FINLEY MACRO REC}\includegraphics[scale=0.25]{FinleyMacroRec}} 272 \includegraphics[scale=0.2]{FinleyMacroLeg} 273 gross 2793 \end{center} 274 Macro elements in \finley. 275 \end{figure} 276 277 gross 2748 \section{Macro Elements} 278 \label{SEC FINLEY MACRO} 279 gross 2793 \finley supports the usage of macro elements~\index{macro elements} which can be used to 280 caltinay 3293 achieve LBB compliance when solving incompressible fluid flow problems. LBB compliance is required to 281 get a problem which has a unique solution for pressure and velocity. For macro elements the 282 gross 2793 pressure and velocity are approximated by a polynomial of order 1 but the velocity approximation bases on a refinement of the element. The nodes of a triangle and quadrilateral element is shown in Figures~\ref{FINLEY MACRO TRI} and~\ref{FINLEY MACRO REC}, respectively. In essence, the velocity uses the same nodes like a quadratic polynomial approximation but replaces the quadratic polynomial by piecewise linear polynomials. In fact, this is the 283 caltinay 3293 way \finley is defining the macro elements. In particular \finley uses the same local ordering of the nodes for the macro element as for the corresponding quadratic element. Another interpretation is that 284 gross 2793 one uses a linear approximation of the velocity together with a linear approximation of the pressure but on elements 285 caltinay 3293 created by combining elements to macro elements. Notice that the macro elements still use quadratic interpolation to represent the element and domain boundary. However, if elements have linear boundary 286 a macro element approximation for the velocity is equivalent to using a linear approximation on a mesh which is created through a one step, global refinement. 287 Typically macro elements are only required to use when an incompressible fluid flow problem 288 gross 2793 is solved, e.g the Stokes problem in Section \ref{STOKES PROBLEM}. Please see Section~\ref{FINLEY MESHES} for 289 caltinay 3293 more details on the supported macro elements. 290 jgs 102 291 gross 2748 292 293 gross 2558 \begin{table} 294 jfenwick 2651 {\scriptsize 295 gross 2558 \begin{tabular}{l||c|c|c|c|c|c|c|c} 296 \member{setSolverMethod} & \member{DIRECT}& \member{PCG} & \member{GMRES} & \member{TFQMR} & \member{MINRES} & \member{PRES20} & \member{BICGSTAB} & \member{LUMPING} \\ 297 \hline 298 \hline 299 \member{setReordering} & $\checkmark$ & & & & & &\\ 300 \hline \member{setRestart} & & & $\checkmark$ & & & $20$ & \\ 301 \hline\member{setTruncation} & & & $\checkmark$ & & & $5$ & \\ 302 \hline\member{setIterMax} & & $\checkmark$& $\checkmark$ & $\checkmark$& $\checkmark$& $\checkmark$ & $\checkmark$ \\ 303 \hline\member{setTolerance} & & $\checkmark$& $\checkmark$ & $\checkmark$& $\checkmark$& $\checkmark$ & $\checkmark$ \\ 304 \hline\member{setAbsoluteTolerance} & & $\checkmark$& $\checkmark$ & $\checkmark$& $\checkmark$& $\checkmark$ & $\checkmark$ \\ 305 gross 2573 \hline\member{setReordering} & $\checkmark$ & & & & & & & \\ 306 gross 2558 \end{tabular} 307 } 308 gross 2573 \caption{Solvers available for 309 caltinay 3293 \finley 310 and the \PASO package and the relevant options in \class{SolverOptions}. 311 \MKL supports 312 \MINIMUMFILLIN 313 and 314 \NESTEDDESCTION 315 reordering. 316 Currently the \UMFPACK interface does not support any reordering. 317 gross 2573 \label{TAB FINLEY SOLVER OPTIONS 1} } 318 caltinay 3293 \end{table} 319 gross 2558 320 \begin{table} 321 {\scriptsize 322 \begin{tabular}{l||c|c|c|c|c|c|c|c} 323 caltinay 3293 \member{setPreconditioner} & 324 gross 2558 \member{NO_PRECONDITIONER} & 325 caltinay 3293 \member{AMG} & 326 \member{JACOBI} & 327 \member{GAUSS_SEIDEL}& 328 gross 2558 \member{REC_ILU}& 329 \member{RILU} & 330 \member{ILU0} & 331 \member{DIRECT} \\ 332 \hline 333 status: & 334 later & 335 caltinay 3293 later & 336 gross 2558 $\checkmark$ & 337 caltinay 3293 $\checkmark$& 338 $\checkmark$ & 339 gross 2558 later & 340 $\checkmark$ & 341 later \\ 342 \hline 343 \hline 344 caltinay 3293 \member{setCoarsening}& 345 gross 2558 & 346 caltinay 3293 $\checkmark$ & 347 gross 2558 & 348 caltinay 3293 & 349 & 350 gross 2558 & 351 & 352 \\ 353 354 355 caltinay 3293 \hline\member{setLevelMax}& 356 gross 2558 & 357 caltinay 3293 $\checkmark$ & 358 & 359 gross 2558 & 360 caltinay 3293 & 361 gross 2558 & 362 & 363 \\ 364 365 \hline\member{setCoarseningThreshold}& 366 & 367 caltinay 3293 $\checkmark$ & 368 & 369 gross 2558 & 370 caltinay 3293 & 371 gross 2558 & 372 & 373 \\ 374 375 \hline\member{setMinCoarseMatrixSize} & 376 & 377 caltinay 3293 $\checkmark$ & 378 & 379 gross 2558 & 380 caltinay 3293 & 381 gross 2558 & 382 & 383 \\ 384 385 \hline\member{setNumSweeps} & 386 & 387 caltinay 3293 & 388 $\checkmark$ & 389 $\checkmark$ & 390 gross 2558 & 391 & 392 & 393 \\ 394 395 \hline\member{setNumPreSweeps}& 396 & 397 caltinay 3293 $\checkmark$ & 398 & 399 gross 2558 & 400 caltinay 3293 & 401 gross 2558 & 402 & 403 \\ 404 405 \hline\member{setNumPostSweeps} & 406 & 407 caltinay 3293 $\checkmark$ & 408 & 409 gross 2558 & 410 caltinay 3293 & 411 gross 2558 & 412 & 413 \\ 414 415 \hline\member{setInnerTolerance}& 416 & 417 caltinay 3293 & 418 & 419 gross 2558 & 420 caltinay 3293 & 421 gross 2558 & 422 & 423 \\ 424 425 \hline\member{setDropTolerance}& 426 & 427 caltinay 3293 & 428 & 429 gross 2558 & 430 caltinay 3293 & 431 gross 2558 & 432 & 433 \\ 434 435 \hline\member{setDropStorage}& 436 & 437 caltinay 3293 & 438 & 439 gross 2558 & 440 caltinay 3293 & 441 gross 2558 & 442 & 443 \\ 444 445 \hline\member{setRelaxationFactor}& 446 & 447 caltinay 3293 & 448 & 449 gross 2558 & 450 caltinay 3293 & 451 gross 2558 $\checkmark$ & 452 & 453 \\ 454 455 \hline\member{adaptInnerTolerance}& 456 & 457 caltinay 3293 & 458 & 459 gross 2558 & 460 caltinay 3293 & 461 gross 2558 & 462 & 463 \\ 464 465 \hline\member{setInnerIterMax}& 466 & 467 caltinay 3293 & 468 & 469 gross 2558 & 470 caltinay 3293 & 471 gross 2558 & 472 & 473 \\ 474 \end{tabular} 475 } 476 caltinay 3293 \caption{Preconditioners available for \finley and the \PASO package and the relevant options in \class{SolverOptions}. \label{TAB FINLEY SOLVER OPTIONS 2}} 477 gross 2558 \end{table} 478 479 gross 2793 \section{Linear Solvers in \SolverOptions} 480 gross 2558 Table~\ref{TAB FINLEY SOLVER OPTIONS 1} and 481 caltinay 3293 Table~\ref{TAB FINLEY SOLVER OPTIONS 2} show the solvers and preconditioners supported by 482 \finley through the \PASO library. Currently direct solvers are not supported under MPI. 483 By default, \finley is using the iterative solvers \PCG for symmetric and \BiCGStab for non-symmetric problems. 484 gross 2793 If the direct solver is selected which can be useful when solving very ill-posed equations 485 caltinay 3293 \finley uses the \MKL \footnote{If the stiffness matrix is non-regular \MKL may return without 486 returning a proper error code. If you observe suspicious solutions when using MKL, this may cause by a non-invertible operator. } solver package. If \MKL is not available \UMFPACK is used. If \UMFPACK is not available 487 gross 2558 a suitable iterative solver from the \PASO is used. 488 489 gross 2793 \section{Functions} 490 gross 2690 \begin{funcdesc}{ReadMesh}{fileName \optional{, \optional{integrationOrder=-1}, optimize=True}} 491 caltinay 3293 creates a \Domain object form the FEM mesh defined in 492 jgs 102 file \var{fileName}. The file must be given the \finley file format. 493 If \var{integrationOrder} is positive, a numerical integration scheme 494 chosen which is accurate on each element up to a polynomial of 495 degree \var{integrationOrder} \index{integration order}. Otherwise 496 caltinay 3293 an appropriate integration order is chosen independently. 497 By default the labeling of mesh nodes and element distribution is 498 gross 2690 optimized. Set \var{optimize=False} to switch off relabeling and redistribution. 499 \end{funcdesc} 500 501 gross 2748 \begin{funcdesc}{ReadGmsh}{fileName \optional{, \optional{integrationOrder=-1}, optimize=True\optional{, useMacroElements=False}}} 502 caltinay 3293 creates a \Domain object form the FEM mesh defined in 503 gross 2690 file \var{fileName}. The file must be given the \gmshextern file format. 504 If \var{integrationOrder} is positive, a numerical integration scheme 505 chosen which is accurate on each element up to a polynomial of 506 degree \var{integrationOrder} \index{integration order}. Otherwise 507 jgs 102 an appropriate integration order is chosen independently. 508 caltinay 3293 By default the labeling of mesh nodes and element distribution is 509 optimized. Set \var{optimize=False} to switch off relabeling and redistribution. 510 gross 2793 If \var{useMacroElements} is set, second order elements are interpreted as macro elements~\index{macro elements}. 511 caltinay 3293 Currently \function{ReadGmsh} does not support MPI. 512 jgs 102 \end{funcdesc} 513 514 gross 2748 \begin{funcdesc}{MakeDomain}{design\optional{, integrationOrder=-1\optional{, optimizeLabeling=True\optional{, useMacroElements=False}}}} 515 caltinay 3293 Creates a Finley \Domain from a \class{Design} object from \pycad using \gmshextern. 516 gross 2748 The \class{Design} \var{design} defines the geometry. 517 If \var{integrationOrder} is positive, a numerical integration scheme 518 chosen which is accurate on each element up to a polynomial of 519 degree \var{integrationOrder} \index{integration order}. Otherwise 520 an appropriate integration order is chosen independently. 521 Set \var{optimizeLabeling=False} to switch off relabeling and redistribution (not recommended). 522 If \var{useMacroElements} is set, macro elements~\index{macro elements} are used. 523 caltinay 3293 Currently \function{MakeDomain} does not support MPI. 524 gross 2748 \end{funcdesc} 525 gross 2690 526 gross 2748 527 gross 2417 \begin{funcdesc}{load}{fileName} 528 caltinay 3293 recovers a \Domain object from a dump file created by the \ 529 \function{dump} method of a \Domain object defined in 530 gross 2793 file \var{fileName}. 531 gross 2417 \end{funcdesc} 532 533 gross 2748 534 jgs 102 \begin{funcdesc}{Rectangle}{n0,n1,order=1,l0=1.,l1=1., integrationOrder=-1, \\ 535 gross 2748 periodic0=\False, periodic1=\False, useElementsOnFace=\False, useMacroElements=\False, optimize=\False} 536 jgs 102 Generates a \Domain object representing a two dimensional rectangle between 537 $(0,0)$ and $(l0,l1)$ with orthogonal edges. The rectangle is filled with 538 \var{n0} elements along the $x_0$-axis and 539 caltinay 3293 \var{n1} elements along the $x_1$-axis. 540 jgs 102 For \var{order}=1 and \var{order}=2 541 caltinay 3293 \finleyelement{Rec4} and 542 \finleyelement{Rec8} are used, respectively. 543 jgs 102 In the case of \var{useElementsOnFace}=\False, 544 caltinay 3293 \finleyelement{Line2} and 545 \finleyelement{Line3} are used to subdivide the edges of the rectangle, respectively. 546 gross 2748 If \var{order}=-1, \finleyelement{Rec8Macro} and \finleyelement{Line3Macro}~\index{macro elements}. This option should be used when solving incompressible fluid flow problem, e.g. \class{StokesProblemCartesian}. 547 jgs 102 In the case of \var{useElementsOnFace}=\True (this option should be used if gradients 548 are calculated on domain faces), 549 caltinay 3293 \finleyelement{Rec4Face} and 550 \finleyelement{Rec8Face} are used on the edges, respectively. 551 jgs 102 If \var{integrationOrder} is positive, a numerical integration scheme 552 chosen which is accurate on each element up to a polynomial of 553 degree \var{integrationOrder} \index{integration order}. Otherwise 554 an appropriate integration order is chosen independently. If 555 \var{periodic0}=\True, periodic boundary conditions \index{periodic boundary conditions} 556 along the $x_0$-directions are enforced. That means when for any solution of a PDE solved by \finley 557 the value on the line $x_0=0$ will be identical to the values on $x_0=\var{l0}$. 558 Correspondingly, 559 \var{periodic1}=\False sets periodic boundary conditions 560 in $x_1$-direction. 561 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. 562 jgs 102 \end{funcdesc} 563 564 \begin{funcdesc}{Brick}{n0,n1,n2,order=1,l0=1.,l1=1.,l2=1., integrationOrder=-1, \\ 565 gross 2748 periodic0=\False,periodic1=\False,periodic2=\False,useElementsOnFace=\False, useMacroElements=\False, optimize=\False} 566 jgs 102 Generates a \Domain object representing a three dimensional brick between 567 $(0,0,0)$ and $(l0,l1,l2)$ with orthogonal faces. The brick is filled with 568 caltinay 3293 \var{n0} elements along the $x_0$-axis, 569 \var{n1} elements along the $x_1$-axis and 570 \var{n2} elements along the $x_2$-axis. 571 jgs 102 For \var{order}=1 and \var{order}=2 572 caltinay 3293 \finleyelement{Hex8} and 573 \finleyelement{Hex20} are used, respectively. 574 jgs 102 In the case of \var{useElementsOnFace}=\False, 575 caltinay 3293 \finleyelement{Rec4} and 576 \finleyelement{Rec8} are used to subdivide the faces of the brick, respectively. 577 jgs 102 In the case of \var{useElementsOnFace}=\True (this option should be used if gradients 578 are calculated on domain faces), 579 caltinay 3293 \finleyelement{Hex8Face} and 580 \finleyelement{Hex20Face} are used on the brick faces, respectively. 581 gross 2748 If \var{order}=-1, \finleyelement{Hex20Macro} and \finleyelement{Rec8Macro}~\index{macro elements}. This option should be used when solving incompressible fluid flow problem, e.g. \class{StokesProblemCartesian}. 582 jgs 102 If \var{integrationOrder} is positive, a numerical integration scheme 583 chosen which is accurate on each element up to a polynomial of 584 degree \var{integrationOrder} \index{integration order}. Otherwise 585 an appropriate integration order is chosen independently. If 586 \var{periodic0}=\True, periodic boundary conditions \index{periodic boundary conditions} 587 along the $x_0$-directions are enforced. That means when for any solution of a PDE solved by \finley 588 the value on the plane $x_0=0$ will be identical to the values on $x_0=\var{l0}$. Correspondingly, 589 \var{periodic1}=\False and \var{periodic2}=\False sets periodic boundary conditions 590 in $x_1$-direction and $x_2$-direction, respectively. 591 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. 592 jgs 102 \end{funcdesc} 593 594 \begin{funcdesc}{GlueFaces}{meshList,safetyFactor=0.2,tolerance=1.e-13} 595 ksteube 1316 Generates a new \Domain object from the list \var{meshList} of \finley meshes. 596 caltinay 3293 Nodes in face elements whose difference of coordinates is less then \var{tolerance} times the 597 diameter of the domain are merged. The corresponding face elements are removed from the mesh. 598 jgs 102 599 TODO: explain \var{safetyFactor} and show an example. 600 \end{funcdesc} 601 602 \begin{funcdesc}{JoinFaces}{meshList,safetyFactor=0.2,tolerance=1.e-13} 603 ksteube 1316 Generates a new \Domain object from the list \var{meshList} of \finley meshes. 604 caltinay 3293 Face elements whose nodes coordinates have difference is less then \var{tolerance} times the 605 diameter of the domain are combined to form a contact element \index{element!contact} 606 The corresponding face elements are removed from the mesh. 607 jgs 102 608 TODO: explain \var{safetyFactor} and show an example. 609 \end{funcdesc}

## Properties

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