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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% Copyright 2003-2007 by ACceSS MNRF |
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% Copyright 2007 by University of Queensland |
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% |
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% http://esscc.uq.edu.au |
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% Primary Business: Queensland, Australia |
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% Licensed under the Open Software License version 3.0 |
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% http://www.opensource.org/licenses/osl-3.0.php |
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% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% |
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\chapter{ The Module \finley} |
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\label{CHAPTER ON FINLEY} |
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\begin{figure} |
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\centerline{\includegraphics[width=\figwidth]{figures/FinleyMesh.eps}} |
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\caption{Subdivision of an Ellipse into triangles order 1 (\finleyelement{Tri3})} |
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\label{FINLEY FIG 0} |
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\end{figure} |
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\begin{figure} |
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\centerline{\includegraphics[width=\figwidth]{figures/FinleyContact.eps}} |
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\caption{Mesh around a contact region (\finleyelement{Rec4})} |
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\label{FINLEY FIG 01} |
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\end{figure} |
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\declaremodule{extension}{finley} \modulesynopsis{Solving linear, steady partial differential equations using |
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finite elements} |
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{\it finley} is a library of C functions solving linear, steady partial differential equations |
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\index{partial differential equations} (PDEs) or systems of PDEs using isoparametrical finite |
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elements \index{FEM!isoparametrical}. |
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It supports unstructured, 1D, 2D and 3D meshes. The module \finley provides an access to the |
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library through the \LinearPDE class of \escript supporting its full functionality. {\it finley} |
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is parallelized using the OpenMP \index{OpenMP} paradigm. |
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\section{Formulation} |
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For a single PDE with a solution with a single component the linear PDE is defined in the |
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following form: |
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\begin{equation}\label{FINLEY.SINGLE.1} |
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\begin{array}{cl} & |
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\displaystyle{ |
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\int\hackscore{\Omega} |
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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 } \\ |
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+ & \displaystyle{\int\hackscore{\Gamma} d \cdot vu \; d{\Gamma} } |
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+ \displaystyle{\int\hackscore{\Gamma^{contact}} d^{contact} \cdot [v][u] \; d{\Gamma} } \\ |
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= & \displaystyle{\int\hackscore{\Omega} X\hackscore{j} \cdot v\hackscore{,j}+ Y \cdot v \; d\Omega }\\ |
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+ & \displaystyle{\int\hackscore{\Gamma} y \cdot v \; d{\Gamma}} + |
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\displaystyle{\int\hackscore{\Gamma^{contact}} y^{contact}\cdot [v] \; d{\Gamma}} \\ |
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\end{array} |
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\end{equation} |
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\section{Meshes} |
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To understand the usage of \finley one needs to have an understanding of how the finite element meshes |
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\index{FEM!mesh} are defined. \fig{FINLEY FIG 0} shows an example of the |
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subdivision of an ellipse into so called elements \index{FEM!elements} \index{element}. |
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In this case, triangles have been used but other forms of subdivisions |
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can be constructed, e.g. into quadrilaterals or, in the three dimensional case, into tetrahedrons |
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and hexahedrons. The idea of the finite element method is to approximate the solution by a function |
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which is a polynomial of a certain order and is continuous across it boundary to neighbour elements. |
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In the example of \fig{FINLEY FIG 0} a linear polynomial is used on each triangle. As one can see, the triangulation |
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is quite a poor approximation of the ellipse. It can be improved by introducing a midpoint on each element edge then |
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positioning those nodes located on an edge expected to describe the boundary, onto the boundary. |
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In this case the triangle gets a curved edge which requires a parametrization of the triangle using a |
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quadratic polynomial. For this case, the solution is also approximated by a piecewise quadratic polynomial |
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(which explains the name isoparametrical elements), see \Ref{Zienc,NumHand} for more details. |
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The union of all elements defines the domain of the PDE. |
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Each element is defined by the nodes used to describe its shape. In \fig{FINLEY FIG 0} the element, |
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which has type \finleyelement{Tri3}, |
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with element reference number $19$ \index{element!reference number} is defined by the nodes |
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with reference numbers $9$, $11$ and $0$ \index{node!reference number}. Notice that the order is counterclockwise. |
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The coefficients of the PDE are evaluated at integration nodes with each individual element. |
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For quadrilateral elements a Gauss quadrature scheme is used. In the case of triangular elements a |
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modified form is applied. The boundary of the domain is also subdivided into elements. \index{element!face} In \fig{FINLEY FIG 0} |
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line elements with two nodes are used. The elements are also defined by their describing nodes, e.g. |
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the face element reference number $20$ which has type \finleyelement{Line2} is defined by the nodes |
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with the reference numbers $11$ and $0$. Again the order is crucial, if moving from the first |
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to second node the domain has to lie on the left hand side (in the case of a two dimension surface element |
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the domain has to lie on the left hand side when moving counterclockwise). If the gradient on the |
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surface of the domain is to be calculated rich face elements face to be used. Rich elements on a face |
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are identical to interior elements but with a modified order of nodes such that the 'first' face of the element aligns |
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with the surface of the domain. In \fig{FINLEY FIG 0} |
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elements of the type \finleyelement{Tri3Face} are used. |
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The face element reference number $20$ as a rich face element is defined by the nodes |
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with reference numbers $11$, $0$ and $9$. Notice that the face element $20$ is identical to the |
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interior element $19$ except that, in this case, the order of the node is different to align the first |
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edge of the triangle (which is the edge starting with the first node) with the boundary of the domain. |
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Be aware that face elements and elements in the interior of the domain must match, i.e. a face element must be the face |
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of an interior element or, in case of a rich face element, it must be identical to an interior element. |
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If no face elements are specified |
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\finley implicitly assumes homogeneous natural boundary conditions \index{natural boundary conditions!homogeneous}, |
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i.e. \var{d}=$0$ and \var{y}=$0$, on the entire boundary of the domain. For |
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inhomogeneous natural boundary conditions \index{natural boundary conditions!inhomogeneous}, |
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the boundary must be described by face elements. |
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If discontinuities of the PDE solution are considered contact elements |
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\index{element!contact}\index{contact conditions} are introduced to describe the contact region $\Gamma^{contact}$ |
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even if $d^{contact}$ and $y^{contact}$ are zero. \fig{FINLEY FIG 01} shows a simple example of a mesh |
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of rectangular elements around a contact region $\Gamma^{contact}$ \index{element!contact}. |
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The contact region is described by the |
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elements $4$, $3$ and $6$. Their element type is \finleyelement{Line2_Contact}. |
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The nodes $9$, $12$, $6$, $5$ define contact element $4$, where the coordinates of nodes $12$ and $5$ and |
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nodes $4$ and $6$ are identical with the idea that nodes $12$ and $9$ are located above and |
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nodes $5$ and $6$ below the contact region. |
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Again, the order of the nodes within an element is crucial. There is also the option of using rich elements |
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if the gradient is to be calculated on the contact region. Similarly to the rich face elements |
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these are constructed from two interior elements by reordering the nodes such that |
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the 'first' face of the element above and the 'first' face of the element below the |
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contact regions line up. The rich version of element |
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$4$ is of type \finleyelement{Rec4Face_Contact} and is defined by the nodes $9$, $12$, $16$, $18$, $6$, $5$, $0$ and |
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$2$. |
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\tab{FINLEY TAB 1} shows the interior element types and the corresponding element types to be used |
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on the face and contacts. \fig{FINLEY.FIG:1}, \fig{FINLEY.FIG:2} and \fig{FINLEY.FIG:4} show the ordering of |
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the nodes within an element. |
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\begin{table} |
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\begin{tablev}{l|llll}{textrm}{interior}{face}{rich face}{contact}{rich contact} |
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\linev{\finleyelement{Line2}}{\finleyelement{Point1}}{\finleyelement{Line2Face}}{\finleyelement{Point1_Contact}}{\finleyelement{Line2Face_Contact}} |
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\linev{\finleyelement{Line3}}{\finleyelement{Point1}}{\finleyelement{Line3Face}}{\finleyelement{Point1_Contact}}{\finleyelement{Line3Face_Contact}} |
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\linev{\finleyelement{Tri3}}{\finleyelement{Line2}}{\finleyelement{Tri3Face}}{\finleyelement{Line2_Contact}}{\finleyelement{Tri3Face_Contact}} |
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\linev{\finleyelement{Tri6}}{\finleyelement{Line3}}{\finleyelement{Tri6Face}}{\finleyelement{Line3_Contact}}{\finleyelement{Tri6Face_Contact}} |
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\linev{\finleyelement{Rec4}}{\finleyelement{Line2}}{\finleyelement{Rec4Face}}{\finleyelement{Line2_Contact}}{\finleyelement{Rec4Face_Contact}} |
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\linev{\finleyelement{Rec8}}{\finleyelement{Line3}}{\finleyelement{Rec8Face}}{\finleyelement{Line3_Contact}}{\finleyelement{Rec8Face_Contact}} |
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\linev{\finleyelement{Rec9}}{\finleyelement{Line3}}{\finleyelement{Rec9Face}}{\finleyelement{Line3_Contact}}{\finleyelement{Rec9Face_Contact}} |
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\linev{\finleyelement{Tet4}}{\finleyelement{Tri6}}{\finleyelement{Tet4Face}}{\finleyelement{Tri6_Contact}}{\finleyelement{Tet4Face_Contact}} |
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\linev{\finleyelement{Tet10}}{\finleyelement{Tri9}}{\finleyelement{Tet10Face}}{\finleyelement{Tri9_Contact}}{\finleyelement{Tet10Face_Contact}} |
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\linev{\finleyelement{Hex8}}{\finleyelement{Rec4}}{\finleyelement{Hex8Face}}{\finleyelement{Rec4_Contact}}{\finleyelement{Hex8Face_Contact}} |
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\linev{\finleyelement{Hex20}}{\finleyelement{Rec8}}{\finleyelement{Hex20Face}}{\finleyelement{Rec8_Contact}}{\finleyelement{Hex20Face_Contact}} |
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\end{tablev} |
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\caption{Finley elements and corresponding elements to be used on domain faces and contacts. |
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The rich types have to be used if the gradient of function is to be calculated on faces and contacts, respectively.} |
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\label{FINLEY TAB 1} |
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\end{table} |
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The native \finley file format is defined as follows. |
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Each node \var{i} has \var{dim} spatial coordinates \var{Node[i]}, a reference number |
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\var{Node_ref[i]}, a degree of freedom \var{Node_DOF[i]} and tag \var{Node_tag[i]}. |
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In most cases \var{Node_DOF[i]}=\var{Node_ref[i]} however, for periodic boundary conditions, |
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\var{Node_DOF[i]} is chosen differently, see example below. The tag can be used to mark nodes sharing |
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the same properties. Element \var{i} is defined by the \var{Element_numNodes} nodes \var{Element_Nodes[i]} |
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which is a list of node reference numbers. The order is crucial. |
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It has a reference number \var{Element_ref[i]} and a tag \var{Element_tag[i]}. The tag |
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can be used to mark elements sharing the same properties. For instance elements above |
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a contact region are marked with $2$ and elements below a contact region are marked with $1$. |
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\var{Element_Type} and \var{Element_Num} give the element type and the number of elements in the mesh. |
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Analogue notations are used for face and contact elements. The following Python script |
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prints the mesh definition in the \finley file format: |
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\begin{python} |
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print "%s\n"%mesh_name |
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# node coordinates: |
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print "%dD-nodes %d\n"%(dim,numNodes) |
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for i in range(numNodes): |
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print "%d %d %d"%(Node_ref[i],Node_DOF[i],Node_tag[i]) |
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for j in range(dim): print " %e"%Node[i][j] |
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print "\n" |
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# interior elements |
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print "%s %d\n"%(Element_Type,Element_Num) |
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for i in range(Element_Num): |
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print "%d %d"%(Element_ref[i],Element_tag[i]) |
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for j in range(Element_numNodes): print " %d"%Element_Nodes[i][j] |
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print "\n" |
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# face elements |
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print "%s %d\n"%(FaceElement_Type,FaceElement_Num) |
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for i in range(FaceElement_Num): |
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print "%d %d"%(FaceElement_ref[i],FaceElement_tag[i]) |
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for j in range(FaceElement_numNodes): print " %d"%FaceElement_Nodes[i][j] |
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print "\n" |
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# contact elements |
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print "%s %d\n"%(ContactElement_Type,ContactElement_Num) |
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for i in range(ContactElement_Num): |
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print "%d %d"%(ContactElement_ref[i],ContactElement_tag[i]) |
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for j in range(ContactElement_numNodes): print " %d"%ContactElement_Nodes[i][j] |
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print "\n" |
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# point sources (not supported yet) |
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write("Point1 0",face_element_type,numFaceElements) |
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\end{python} |
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The following example of a mesh file defines the mesh shown in \fig{FINLEY FIG 01}: |
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\begin{verbatim} |
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Example 1 |
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2D Nodes 16 |
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0 0 0 0. 0. |
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2 2 0 0.33 0. |
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3 3 0 0.66 0. |
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7 4 0 1. 0. |
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5 5 0 0. 0.5 |
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6 6 0 0.33 0.5 |
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8 8 0 0.66 0.5 |
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10 10 0 1.0 0.5 |
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12 12 0 0. 0.5 |
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9 9 0 0.33 0.5 |
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13 13 0 0.66 0.5 |
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15 15 0 1.0 0.5 |
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16 16 0 0. 1.0 |
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18 18 0 0.33 1.0 |
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19 19 0 0.66 1.0 |
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20 20 0 1.0 1.0 |
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Rec4 6 |
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0 1 0 2 6 5 |
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1 1 2 3 8 6 |
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2 1 3 7 10 8 |
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5 2 12 9 18 16 |
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7 2 13 19 18 9 |
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10 2 20 19 13 15 |
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Line2 0 |
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Line2_Contact 3 |
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4 0 9 12 6 5 |
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3 0 13 9 8 6 |
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6 0 15 13 10 8 |
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Point1 0 |
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\end{verbatim} |
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Notice that the order in which the nodes and elements are given is arbitrary. |
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In the case that rich contact elements are used the contact element section gets |
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the form |
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\begin{verbatim} |
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Rec4Face_Contact 3 |
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4 0 9 12 16 18 6 5 0 2 |
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3 0 13 9 18 19 8 6 2 3 |
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6 0 15 13 19 20 10 8 3 7 |
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\end{verbatim} |
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Periodic boundary condition \index{boundary conditions!periodic} can be introduced by altering \var{Node_DOF}. |
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It allows identification of nodes even if they have different physical locations. For instance, to |
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enforce periodic boundary conditions at the face $x_0=0$ and $x_0=1$ one identifies |
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the degrees of freedom for nodes $0$, $5$, $12$ and $16$ with the degrees of freedom for |
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$7$, $10$, $15$ and $20$, respectively. The node section of the \finley mesh gets now the form: |
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\begin{verbatim} |
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2D Nodes 16 |
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0 0 0 0. 0. |
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2 2 0 0.33 0. |
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3 3 0 0.66 0. |
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7 0 0 1. 0. |
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5 5 0 0. 0.5 |
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6 6 0 0.33 0.5 |
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8 8 0 0.66 0.5 |
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10 5 0 1.0 0.5 |
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12 12 0 0. 0.5 |
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9 9 0 0.33 0.5 |
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13 13 0 0.66 0.5 |
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15 12 0 1.0 0.5 |
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16 16 0 0. 1.0 |
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18 18 0 0.33 1.0 |
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19 19 0 0.66 1.0 |
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20 16 0 1.0 1.0 |
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\end{verbatim} |
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\include{finleyelements} |
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\subsection{Linear Solvers in \LinearPDE} |
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Currently \finley supports the linear solvers \PCG, \GMRES, \PRESTWENTY and \BiCGStab. |
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For \GMRES the options \var{truncation} and \var{restart} of the \method{getSolution} can be |
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used to control the truncation and restart during iteration. Default values are |
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\var{truncation}=5 and \var{restart}=20. |
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The default solver is \BiCGStab but if the symmetry flag is set \PCG is the default solver. |
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\finley supports the solver options \var{iter_max} which specifies the maximum number of iterations steps, |
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\var{verbose}=\True or \False and \var{preconditioner}=\constant{JACOBI} or \constant {ILU0}. |
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In some installations \finley supports the \Direct solver and the |
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solver options \var{reordering}=\constant{util.NO_REORDERING}, |
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\constant{util.MINIMUM_FILL_IN} or \constant{util.NESTED_DISSECTION} (default is \constant{util.NO_REORDERING}), |
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\var{drop_tolerance} specifying the threshold for values to be dropped in the |
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incomplete elimination process (default is 0.01) and \var{drop_storage} specifying the maximum increase |
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in storage allowed in the |
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incomplete elimination process (default is 1.20). |
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\subsection{Functions} |
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\begin{funcdesc}{Mesh}{fileName,integrationOrder=-1} |
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creates a \Domain object form the FEM mesh defined in |
| 277 |
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file \var{fileName}. The file must be given the \finley file format. |
| 278 |
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If \var{integrationOrder} is positive, a numerical integration scheme |
| 279 |
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chosen which is accurate on each element up to a polynomial of |
| 280 |
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degree \var{integrationOrder} \index{integration order}. Otherwise |
| 281 |
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an appropriate integration order is chosen independently. |
| 282 |
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\end{funcdesc} |
| 283 |
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|
| 284 |
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\begin{funcdesc}{Rectangle}{n0,n1,order=1,l0=1.,l1=1., integrationOrder=-1, \\ |
| 285 |
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periodic0=\False,periodic1=\False,useElementsOnFace=\False} |
| 286 |
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Generates a \Domain object representing a two dimensional rectangle between |
| 287 |
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$(0,0)$ and $(l0,l1)$ with orthogonal edges. The rectangle is filled with |
| 288 |
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\var{n0} elements along the $x_0$-axis and |
| 289 |
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\var{n1} elements along the $x_1$-axis. |
| 290 |
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For \var{order}=1 and \var{order}=2 |
| 291 |
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\finleyelement{Rec4} and |
| 292 |
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\finleyelement{Rec8} are used, respectively. |
| 293 |
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In the case of \var{useElementsOnFace}=\False, |
| 294 |
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\finleyelement{Line2} and |
| 295 |
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\finleyelement{Line3} are used to subdivide the edges of the rectangle, respectively. |
| 296 |
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In the case of \var{useElementsOnFace}=\True (this option should be used if gradients |
| 297 |
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are calculated on domain faces), |
| 298 |
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\finleyelement{Rec4Face} and |
| 299 |
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\finleyelement{Rec8Face} are used on the edges, respectively. |
| 300 |
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If \var{integrationOrder} is positive, a numerical integration scheme |
| 301 |
|
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chosen which is accurate on each element up to a polynomial of |
| 302 |
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degree \var{integrationOrder} \index{integration order}. Otherwise |
| 303 |
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an appropriate integration order is chosen independently. If |
| 304 |
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\var{periodic0}=\True, periodic boundary conditions \index{periodic boundary conditions} |
| 305 |
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along the $x_0$-directions are enforced. That means when for any solution of a PDE solved by \finley |
| 306 |
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the value on the line $x_0=0$ will be identical to the values on $x_0=\var{l0}$. |
| 307 |
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Correspondingly, |
| 308 |
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\var{periodic1}=\False sets periodic boundary conditions |
| 309 |
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in $x_1$-direction. |
| 310 |
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\end{funcdesc} |
| 311 |
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|
| 312 |
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\begin{funcdesc}{Brick}{n0,n1,n2,order=1,l0=1.,l1=1.,l2=1., integrationOrder=-1, \\ |
| 313 |
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periodic0=\False,periodic1=\False,periodic2=\False,useElementsOnFace=\False} |
| 314 |
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Generates a \Domain object representing a three dimensional brick between |
| 315 |
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$(0,0,0)$ and $(l0,l1,l2)$ with orthogonal faces. The brick is filled with |
| 316 |
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\var{n0} elements along the $x_0$-axis, |
| 317 |
|
|
\var{n1} elements along the $x_1$-axis and |
| 318 |
|
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\var{n2} elements along the $x_2$-axis. |
| 319 |
|
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For \var{order}=1 and \var{order}=2 |
| 320 |
|
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\finleyelement{Hex8} and |
| 321 |
|
|
\finleyelement{Hex20} are used, respectively. |
| 322 |
|
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In the case of \var{useElementsOnFace}=\False, |
| 323 |
|
|
\finleyelement{Rec4} and |
| 324 |
|
|
\finleyelement{Rec8} are used to subdivide the faces of the brick, respectively. |
| 325 |
|
|
In the case of \var{useElementsOnFace}=\True (this option should be used if gradients |
| 326 |
|
|
are calculated on domain faces), |
| 327 |
|
|
\finleyelement{Hex8Face} and |
| 328 |
|
|
\finleyelement{Hex20Face} are used on the brick faces, respectively. |
| 329 |
|
|
If \var{integrationOrder} is positive, a numerical integration scheme |
| 330 |
|
|
chosen which is accurate on each element up to a polynomial of |
| 331 |
|
|
degree \var{integrationOrder} \index{integration order}. Otherwise |
| 332 |
|
|
an appropriate integration order is chosen independently. If |
| 333 |
|
|
\var{periodic0}=\True, periodic boundary conditions \index{periodic boundary conditions} |
| 334 |
|
|
along the $x_0$-directions are enforced. That means when for any solution of a PDE solved by \finley |
| 335 |
|
|
the value on the plane $x_0=0$ will be identical to the values on $x_0=\var{l0}$. Correspondingly, |
| 336 |
|
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\var{periodic1}=\False and \var{periodic2}=\False sets periodic boundary conditions |
| 337 |
|
|
in $x_1$-direction and $x_2$-direction, respectively. |
| 338 |
|
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\end{funcdesc} |
| 339 |
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|
| 340 |
|
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\begin{funcdesc}{GlueFaces}{meshList,safetyFactor=0.2,tolerance=1.e-13} |
| 341 |
ksteube |
1316 |
Generates a new \Domain object from the list \var{meshList} of \finley meshes. |
| 342 |
jgs |
102 |
Nodes in face elements whose difference of coordinates is less then \var{tolerance} times the |
| 343 |
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diameter of the domain are merged. The corresponding face elements are removed from the mesh. |
| 344 |
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|
| 345 |
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TODO: explain \var{safetyFactor} and show an example. |
| 346 |
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\end{funcdesc} |
| 347 |
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|
| 348 |
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\begin{funcdesc}{JoinFaces}{meshList,safetyFactor=0.2,tolerance=1.e-13} |
| 349 |
ksteube |
1316 |
Generates a new \Domain object from the list \var{meshList} of \finley meshes. |
| 350 |
jgs |
102 |
Face elements whose nodes coordinates have difference is less then \var{tolerance} times the |
| 351 |
|
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diameter of the domain are combined to form a contact element \index{element!contact} |
| 352 |
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The corresponding face elements are removed from the mesh. |
| 353 |
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|
| 354 |
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TODO: explain \var{safetyFactor} and show an example. |
| 355 |
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\end{funcdesc} |
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