| 1 |
% |
| 2 |
% $Id$ |
| 3 |
% |
| 4 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 5 |
% |
| 6 |
% Copyright 2003-2007 by ACceSS MNRF |
| 7 |
% Copyright 2007 by University of Queensland |
| 8 |
% |
| 9 |
% http://esscc.uq.edu.au |
| 10 |
% Primary Business: Queensland, Australia |
| 11 |
% Licensed under the Open Software License version 3.0 |
| 12 |
% http://www.opensource.org/licenses/osl-3.0.php |
| 13 |
% |
| 14 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 15 |
% |
| 16 |
|
| 17 |
\chapter{ The Module \finley} |
| 18 |
\label{CHAPTER ON FINLEY} |
| 19 |
|
| 20 |
\begin{figure} |
| 21 |
\centerline{\includegraphics[width=\figwidth]{figures/FinleyMesh.eps}} |
| 22 |
\caption{Subdivision of an Ellipse into triangles order 1 (\finleyelement{Tri3})} |
| 23 |
\label{FINLEY FIG 0} |
| 24 |
\end{figure} |
| 25 |
|
| 26 |
\begin{figure} |
| 27 |
\centerline{\includegraphics[width=\figwidth]{figures/FinleyContact.eps}} |
| 28 |
\caption{Mesh around a contact region (\finleyelement{Rec4})} |
| 29 |
\label{FINLEY FIG 01} |
| 30 |
\end{figure} |
| 31 |
|
| 32 |
\declaremodule{extension}{finley} \modulesynopsis{Solving linear, steady partial differential equations using |
| 33 |
finite elements} |
| 34 |
|
| 35 |
{\it finley} is a library of C functions solving linear, steady partial differential equations |
| 36 |
\index{partial differential equations} (PDEs) or systems of PDEs using isoparametrical finite |
| 37 |
elements \index{FEM!isoparametrical}. |
| 38 |
It supports unstructured, 1D, 2D and 3D meshes. The module \finley provides an access to the |
| 39 |
library through the \LinearPDE class of \escript supporting its full functionality. {\it finley} |
| 40 |
is parallelized using the OpenMP \index{OpenMP} paradigm. |
| 41 |
|
| 42 |
\section{Formulation} |
| 43 |
|
| 44 |
For a single PDE with a solution with a single component the linear PDE is defined in the |
| 45 |
following form: |
| 46 |
\begin{equation}\label{FINLEY.SINGLE.1} |
| 47 |
\begin{array}{cl} & |
| 48 |
\displaystyle{ |
| 49 |
\int\hackscore{\Omega} |
| 50 |
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 } \\ |
| 51 |
+ & \displaystyle{\int\hackscore{\Gamma} d \cdot vu \; d{\Gamma} } |
| 52 |
+ \displaystyle{\int\hackscore{\Gamma^{contact}} d^{contact} \cdot [v][u] \; d{\Gamma} } \\ |
| 53 |
= & \displaystyle{\int\hackscore{\Omega} X\hackscore{j} \cdot v\hackscore{,j}+ Y \cdot v \; d\Omega }\\ |
| 54 |
+ & \displaystyle{\int\hackscore{\Gamma} y \cdot v \; d{\Gamma}} + |
| 55 |
\displaystyle{\int\hackscore{\Gamma^{contact}} y^{contact}\cdot [v] \; d{\Gamma}} \\ |
| 56 |
\end{array} |
| 57 |
\end{equation} |
| 58 |
|
| 59 |
\section{Meshes} |
| 60 |
To understand the usage of \finley one needs to have an understanding of how the finite element meshes |
| 61 |
\index{FEM!mesh} are defined. \fig{FINLEY FIG 0} shows an example of the |
| 62 |
subdivision of an ellipse into so called elements \index{FEM!elements} \index{element}. |
| 63 |
In this case, triangles have been used but other forms of subdivisions |
| 64 |
can be constructed, e.g. into quadrilaterals or, in the three dimensional case, into tetrahedrons |
| 65 |
and hexahedrons. The idea of the finite element method is to approximate the solution by a function |
| 66 |
which is a polynomial of a certain order and is continuous across it boundary to neighbour elements. |
| 67 |
In the example of \fig{FINLEY FIG 0} a linear polynomial is used on each triangle. As one can see, the triangulation |
| 68 |
is quite a poor approximation of the ellipse. It can be improved by introducing a midpoint on each element edge then |
| 69 |
positioning those nodes located on an edge expected to describe the boundary, onto the boundary. |
| 70 |
In this case the triangle gets a curved edge which requires a parametrization of the triangle using a |
| 71 |
quadratic polynomial. For this case, the solution is also approximated by a piecewise quadratic polynomial |
| 72 |
(which explains the name isoparametrical elements), see \Ref{Zienc,NumHand} for more details. |
| 73 |
|
| 74 |
The union of all elements defines the domain of the PDE. |
| 75 |
Each element is defined by the nodes used to describe its shape. In \fig{FINLEY FIG 0} the element, |
| 76 |
which has type \finleyelement{Tri3}, |
| 77 |
with element reference number $19$ \index{element!reference number} is defined by the nodes |
| 78 |
with reference numbers $9$, $11$ and $0$ \index{node!reference number}. Notice that the order is counterclockwise. |
| 79 |
The coefficients of the PDE are evaluated at integration nodes with each individual element. |
| 80 |
For quadrilateral elements a Gauss quadrature scheme is used. In the case of triangular elements a |
| 81 |
modified form is applied. The boundary of the domain is also subdivided into elements. \index{element!face} In \fig{FINLEY FIG 0} |
| 82 |
line elements with two nodes are used. The elements are also defined by their describing nodes, e.g. |
| 83 |
the face element reference number $20$ which has type \finleyelement{Line2} is defined by the nodes |
| 84 |
with the reference numbers $11$ and $0$. Again the order is crucial, if moving from the first |
| 85 |
to second node the domain has to lie on the left hand side (in the case of a two dimension surface element |
| 86 |
the domain has to lie on the left hand side when moving counterclockwise). If the gradient on the |
| 87 |
surface of the domain is to be calculated rich face elements face to be used. Rich elements on a face |
| 88 |
are identical to interior elements but with a modified order of nodes such that the 'first' face of the element aligns |
| 89 |
with the surface of the domain. In \fig{FINLEY FIG 0} |
| 90 |
elements of the type \finleyelement{Tri3Face} are used. |
| 91 |
The face element reference number $20$ as a rich face element is defined by the nodes |
| 92 |
with reference numbers $11$, $0$ and $9$. Notice that the face element $20$ is identical to the |
| 93 |
interior element $19$ except that, in this case, the order of the node is different to align the first |
| 94 |
edge of the triangle (which is the edge starting with the first node) with the boundary of the domain. |
| 95 |
|
| 96 |
Be aware that face elements and elements in the interior of the domain must match, i.e. a face element must be the face |
| 97 |
of an interior element or, in case of a rich face element, it must be identical to an interior element. |
| 98 |
If no face elements are specified |
| 99 |
\finley implicitly assumes homogeneous natural boundary conditions \index{natural boundary conditions!homogeneous}, |
| 100 |
i.e. \var{d}=$0$ and \var{y}=$0$, on the entire boundary of the domain. For |
| 101 |
inhomogeneous natural boundary conditions \index{natural boundary conditions!inhomogeneous}, |
| 102 |
the boundary must be described by face elements. |
| 103 |
|
| 104 |
If discontinuities of the PDE solution are considered contact elements |
| 105 |
\index{element!contact}\index{contact conditions} are introduced to describe the contact region $\Gamma^{contact}$ |
| 106 |
even if $d^{contact}$ and $y^{contact}$ are zero. \fig{FINLEY FIG 01} shows a simple example of a mesh |
| 107 |
of rectangular elements around a contact region $\Gamma^{contact}$ \index{element!contact}. |
| 108 |
The contact region is described by the |
| 109 |
elements $4$, $3$ and $6$. Their element type is \finleyelement{Line2_Contact}. |
| 110 |
The nodes $9$, $12$, $6$, $5$ define contact element $4$, where the coordinates of nodes $12$ and $5$ and |
| 111 |
nodes $4$ and $6$ are identical with the idea that nodes $12$ and $9$ are located above and |
| 112 |
nodes $5$ and $6$ below the contact region. |
| 113 |
Again, the order of the nodes within an element is crucial. There is also the option of using rich elements |
| 114 |
if the gradient is to be calculated on the contact region. Similarly to the rich face elements |
| 115 |
these are constructed from two interior elements by reordering the nodes such that |
| 116 |
the 'first' face of the element above and the 'first' face of the element below the |
| 117 |
contact regions line up. The rich version of element |
| 118 |
$4$ is of type \finleyelement{Rec4Face_Contact} and is defined by the nodes $9$, $12$, $16$, $18$, $6$, $5$, $0$ and |
| 119 |
$2$. |
| 120 |
|
| 121 |
\tab{FINLEY TAB 1} shows the interior element types and the corresponding element types to be used |
| 122 |
on the face and contacts. \fig{FINLEY.FIG:1}, \fig{FINLEY.FIG:2} and \fig{FINLEY.FIG:4} show the ordering of |
| 123 |
the nodes within an element. |
| 124 |
|
| 125 |
\begin{table} |
| 126 |
\begin{tablev}{l|llll}{textrm}{interior}{face}{rich face}{contact}{rich contact} |
| 127 |
\linev{\finleyelement{Line2}}{\finleyelement{Point1}}{\finleyelement{Line2Face}}{\finleyelement{Point1_Contact}}{\finleyelement{Line2Face_Contact}} |
| 128 |
\linev{\finleyelement{Line3}}{\finleyelement{Point1}}{\finleyelement{Line3Face}}{\finleyelement{Point1_Contact}}{\finleyelement{Line3Face_Contact}} |
| 129 |
\linev{\finleyelement{Tri3}}{\finleyelement{Line2}}{\finleyelement{Tri3Face}}{\finleyelement{Line2_Contact}}{\finleyelement{Tri3Face_Contact}} |
| 130 |
\linev{\finleyelement{Tri6}}{\finleyelement{Line3}}{\finleyelement{Tri6Face}}{\finleyelement{Line3_Contact}}{\finleyelement{Tri6Face_Contact}} |
| 131 |
\linev{\finleyelement{Rec4}}{\finleyelement{Line2}}{\finleyelement{Rec4Face}}{\finleyelement{Line2_Contact}}{\finleyelement{Rec4Face_Contact}} |
| 132 |
\linev{\finleyelement{Rec8}}{\finleyelement{Line3}}{\finleyelement{Rec8Face}}{\finleyelement{Line3_Contact}}{\finleyelement{Rec8Face_Contact}} |
| 133 |
\linev{\finleyelement{Rec9}}{\finleyelement{Line3}}{\finleyelement{Rec9Face}}{\finleyelement{Line3_Contact}}{\finleyelement{Rec9Face_Contact}} |
| 134 |
\linev{\finleyelement{Tet4}}{\finleyelement{Tri6}}{\finleyelement{Tet4Face}}{\finleyelement{Tri6_Contact}}{\finleyelement{Tet4Face_Contact}} |
| 135 |
\linev{\finleyelement{Tet10}}{\finleyelement{Tri9}}{\finleyelement{Tet10Face}}{\finleyelement{Tri9_Contact}}{\finleyelement{Tet10Face_Contact}} |
| 136 |
\linev{\finleyelement{Hex8}}{\finleyelement{Rec4}}{\finleyelement{Hex8Face}}{\finleyelement{Rec4_Contact}}{\finleyelement{Hex8Face_Contact}} |
| 137 |
\linev{\finleyelement{Hex20}}{\finleyelement{Rec8}}{\finleyelement{Hex20Face}}{\finleyelement{Rec8_Contact}}{\finleyelement{Hex20Face_Contact}} |
| 138 |
\end{tablev} |
| 139 |
\caption{Finley elements and corresponding elements to be used on domain faces and contacts. |
| 140 |
The rich types have to be used if the gradient of function is to be calculated on faces and contacts, respectively.} |
| 141 |
\label{FINLEY TAB 1} |
| 142 |
\end{table} |
| 143 |
|
| 144 |
The native \finley file format is defined as follows. |
| 145 |
Each node \var{i} has \var{dim} spatial coordinates \var{Node[i]}, a reference number |
| 146 |
\var{Node_ref[i]}, a degree of freedom \var{Node_DOF[i]} and tag \var{Node_tag[i]}. |
| 147 |
In most cases \var{Node_DOF[i]}=\var{Node_ref[i]} however, for periodic boundary conditions, |
| 148 |
\var{Node_DOF[i]} is chosen differently, see example below. The tag can be used to mark nodes sharing |
| 149 |
the same properties. Element \var{i} is defined by the \var{Element_numNodes} nodes \var{Element_Nodes[i]} |
| 150 |
which is a list of node reference numbers. The order is crucial. |
| 151 |
It has a reference number \var{Element_ref[i]} and a tag \var{Element_tag[i]}. The tag |
| 152 |
can be used to mark elements sharing the same properties. For instance elements above |
| 153 |
a contact region are marked with $2$ and elements below a contact region are marked with $1$. |
| 154 |
\var{Element_Type} and \var{Element_Num} give the element type and the number of elements in the mesh. |
| 155 |
Analogue notations are used for face and contact elements. The following Python script |
| 156 |
prints the mesh definition in the \finley file format: |
| 157 |
\begin{python} |
| 158 |
print "%s\n"%mesh_name |
| 159 |
# node coordinates: |
| 160 |
print "%dD-nodes %d\n"%(dim,numNodes) |
| 161 |
for i in range(numNodes): |
| 162 |
print "%d %d %d"%(Node_ref[i],Node_DOF[i],Node_tag[i]) |
| 163 |
for j in range(dim): print " %e"%Node[i][j] |
| 164 |
print "\n" |
| 165 |
# interior elements |
| 166 |
print "%s %d\n"%(Element_Type,Element_Num) |
| 167 |
for i in range(Element_Num): |
| 168 |
print "%d %d"%(Element_ref[i],Element_tag[i]) |
| 169 |
for j in range(Element_numNodes): print " %d"%Element_Nodes[i][j] |
| 170 |
print "\n" |
| 171 |
# face elements |
| 172 |
print "%s %d\n"%(FaceElement_Type,FaceElement_Num) |
| 173 |
for i in range(FaceElement_Num): |
| 174 |
print "%d %d"%(FaceElement_ref[i],FaceElement_tag[i]) |
| 175 |
for j in range(FaceElement_numNodes): print " %d"%FaceElement_Nodes[i][j] |
| 176 |
print "\n" |
| 177 |
# contact elements |
| 178 |
print "%s %d\n"%(ContactElement_Type,ContactElement_Num) |
| 179 |
for i in range(ContactElement_Num): |
| 180 |
print "%d %d"%(ContactElement_ref[i],ContactElement_tag[i]) |
| 181 |
for j in range(ContactElement_numNodes): print " %d"%ContactElement_Nodes[i][j] |
| 182 |
print "\n" |
| 183 |
# point sources (not supported yet) |
| 184 |
write("Point1 0",face_element_type,numFaceElements) |
| 185 |
\end{python} |
| 186 |
|
| 187 |
The following example of a mesh file defines the mesh shown in \fig{FINLEY FIG 01}: |
| 188 |
\begin{verbatim} |
| 189 |
Example 1 |
| 190 |
2D Nodes 16 |
| 191 |
0 0 0 0. 0. |
| 192 |
2 2 0 0.33 0. |
| 193 |
3 3 0 0.66 0. |
| 194 |
7 4 0 1. 0. |
| 195 |
5 5 0 0. 0.5 |
| 196 |
6 6 0 0.33 0.5 |
| 197 |
8 8 0 0.66 0.5 |
| 198 |
10 10 0 1.0 0.5 |
| 199 |
12 12 0 0. 0.5 |
| 200 |
9 9 0 0.33 0.5 |
| 201 |
13 13 0 0.66 0.5 |
| 202 |
15 15 0 1.0 0.5 |
| 203 |
16 16 0 0. 1.0 |
| 204 |
18 18 0 0.33 1.0 |
| 205 |
19 19 0 0.66 1.0 |
| 206 |
20 20 0 1.0 1.0 |
| 207 |
Rec4 6 |
| 208 |
0 1 0 2 6 5 |
| 209 |
1 1 2 3 8 6 |
| 210 |
2 1 3 7 10 8 |
| 211 |
5 2 12 9 18 16 |
| 212 |
7 2 13 19 18 9 |
| 213 |
10 2 20 19 13 15 |
| 214 |
Line2 0 |
| 215 |
Line2_Contact 3 |
| 216 |
4 0 9 12 6 5 |
| 217 |
3 0 13 9 8 6 |
| 218 |
6 0 15 13 10 8 |
| 219 |
Point1 0 |
| 220 |
\end{verbatim} |
| 221 |
Notice that the order in which the nodes and elements are given is arbitrary. |
| 222 |
In the case that rich contact elements are used the contact element section gets |
| 223 |
the form |
| 224 |
\begin{verbatim} |
| 225 |
Rec4Face_Contact 3 |
| 226 |
4 0 9 12 16 18 6 5 0 2 |
| 227 |
3 0 13 9 18 19 8 6 2 3 |
| 228 |
6 0 15 13 19 20 10 8 3 7 |
| 229 |
\end{verbatim} |
| 230 |
Periodic boundary condition \index{boundary conditions!periodic} can be introduced by altering \var{Node_DOF}. |
| 231 |
It allows identification of nodes even if they have different physical locations. For instance, to |
| 232 |
enforce periodic boundary conditions at the face $x_0=0$ and $x_0=1$ one identifies |
| 233 |
the degrees of freedom for nodes $0$, $5$, $12$ and $16$ with the degrees of freedom for |
| 234 |
$7$, $10$, $15$ and $20$, respectively. The node section of the \finley mesh gets now the form: |
| 235 |
\begin{verbatim} |
| 236 |
2D Nodes 16 |
| 237 |
0 0 0 0. 0. |
| 238 |
2 2 0 0.33 0. |
| 239 |
3 3 0 0.66 0. |
| 240 |
7 0 0 1. 0. |
| 241 |
5 5 0 0. 0.5 |
| 242 |
6 6 0 0.33 0.5 |
| 243 |
8 8 0 0.66 0.5 |
| 244 |
10 5 0 1.0 0.5 |
| 245 |
12 12 0 0. 0.5 |
| 246 |
9 9 0 0.33 0.5 |
| 247 |
13 13 0 0.66 0.5 |
| 248 |
15 12 0 1.0 0.5 |
| 249 |
16 16 0 0. 1.0 |
| 250 |
18 18 0 0.33 1.0 |
| 251 |
19 19 0 0.66 1.0 |
| 252 |
20 16 0 1.0 1.0 |
| 253 |
\end{verbatim} |
| 254 |
|
| 255 |
|
| 256 |
\include{finleyelements} |
| 257 |
|
| 258 |
\subsection{Linear Solvers in \LinearPDE} |
| 259 |
Currently \finley supports the linear solvers \PCG, \GMRES, \PRESTWENTY and \BiCGStab. |
| 260 |
For \GMRES the options \var{truncation} and \var{restart} of the \method{getSolution} can be |
| 261 |
used to control the truncation and restart during iteration. Default values are |
| 262 |
\var{truncation}=5 and \var{restart}=20. |
| 263 |
The default solver is \BiCGStab but if the symmetry flag is set \PCG is the default solver. |
| 264 |
\finley supports the solver options \var{iter_max} which specifies the maximum number of iterations steps, |
| 265 |
\var{verbose}=\True or \False and \var{preconditioner}=\constant{JACOBI} or \constant {ILU0}. |
| 266 |
In some installations \finley supports the \Direct solver and the |
| 267 |
solver options \var{reordering}=\constant{util.NO_REORDERING}, |
| 268 |
\constant{util.MINIMUM_FILL_IN} or \constant{util.NESTED_DISSECTION} (default is \constant{util.NO_REORDERING}), |
| 269 |
\var{drop_tolerance} specifying the threshold for values to be dropped in the |
| 270 |
incomplete elimination process (default is 0.01) and \var{drop_storage} specifying the maximum increase |
| 271 |
in storage allowed in the |
| 272 |
incomplete elimination process (default is 1.20). |
| 273 |
|
| 274 |
\subsection{Functions} |
| 275 |
\begin{funcdesc}{Mesh}{fileName,integrationOrder=-1} |
| 276 |
creates a \Domain object form the FEM mesh defined in |
| 277 |
file \var{fileName}. The file must be given the \finley file format. |
| 278 |
If \var{integrationOrder} is positive, a numerical integration scheme |
| 279 |
chosen which is accurate on each element up to a polynomial of |
| 280 |
degree \var{integrationOrder} \index{integration order}. Otherwise |
| 281 |
an appropriate integration order is chosen independently. |
| 282 |
\end{funcdesc} |
| 283 |
|
| 284 |
\begin{funcdesc}{Rectangle}{n0,n1,order=1,l0=1.,l1=1., integrationOrder=-1, \\ |
| 285 |
periodic0=\False,periodic1=\False,useElementsOnFace=\False,optimize=\False} |
| 286 |
Generates a \Domain object representing a two dimensional rectangle between |
| 287 |
$(0,0)$ and $(l0,l1)$ with orthogonal edges. The rectangle is filled with |
| 288 |
\var{n0} elements along the $x_0$-axis and |
| 289 |
\var{n1} elements along the $x_1$-axis. |
| 290 |
For \var{order}=1 and \var{order}=2 |
| 291 |
\finleyelement{Rec4} and |
| 292 |
\finleyelement{Rec8} are used, respectively. |
| 293 |
In the case of \var{useElementsOnFace}=\False, |
| 294 |
\finleyelement{Line2} and |
| 295 |
\finleyelement{Line3} are used to subdivide the edges of the rectangle, respectively. |
| 296 |
In the case of \var{useElementsOnFace}=\True (this option should be used if gradients |
| 297 |
are calculated on domain faces), |
| 298 |
\finleyelement{Rec4Face} and |
| 299 |
\finleyelement{Rec8Face} are used on the edges, respectively. |
| 300 |
If \var{integrationOrder} is positive, a numerical integration scheme |
| 301 |
chosen which is accurate on each element up to a polynomial of |
| 302 |
degree \var{integrationOrder} \index{integration order}. Otherwise |
| 303 |
an appropriate integration order is chosen independently. If |
| 304 |
\var{periodic0}=\True, periodic boundary conditions \index{periodic boundary conditions} |
| 305 |
along the $x_0$-directions are enforced. That means when for any solution of a PDE solved by \finley |
| 306 |
the value on the line $x_0=0$ will be identical to the values on $x_0=\var{l0}$. |
| 307 |
Correspondingly, |
| 308 |
\var{periodic1}=\False sets periodic boundary conditions |
| 309 |
in $x_1$-direction. |
| 310 |
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. |
| 311 |
\end{funcdesc} |
| 312 |
|
| 313 |
\begin{funcdesc}{Brick}{n0,n1,n2,order=1,l0=1.,l1=1.,l2=1., integrationOrder=-1, \\ |
| 314 |
periodic0=\False,periodic1=\False,periodic2=\False,useElementsOnFace=\False,optimize=\False} |
| 315 |
Generates a \Domain object representing a three dimensional brick between |
| 316 |
$(0,0,0)$ and $(l0,l1,l2)$ with orthogonal faces. The brick is filled with |
| 317 |
\var{n0} elements along the $x_0$-axis, |
| 318 |
\var{n1} elements along the $x_1$-axis and |
| 319 |
\var{n2} elements along the $x_2$-axis. |
| 320 |
For \var{order}=1 and \var{order}=2 |
| 321 |
\finleyelement{Hex8} and |
| 322 |
\finleyelement{Hex20} are used, respectively. |
| 323 |
In the case of \var{useElementsOnFace}=\False, |
| 324 |
\finleyelement{Rec4} and |
| 325 |
\finleyelement{Rec8} are used to subdivide the faces of the brick, respectively. |
| 326 |
In the case of \var{useElementsOnFace}=\True (this option should be used if gradients |
| 327 |
are calculated on domain faces), |
| 328 |
\finleyelement{Hex8Face} and |
| 329 |
\finleyelement{Hex20Face} are used on the brick faces, respectively. |
| 330 |
If \var{integrationOrder} is positive, a numerical integration scheme |
| 331 |
chosen which is accurate on each element up to a polynomial of |
| 332 |
degree \var{integrationOrder} \index{integration order}. Otherwise |
| 333 |
an appropriate integration order is chosen independently. If |
| 334 |
\var{periodic0}=\True, periodic boundary conditions \index{periodic boundary conditions} |
| 335 |
along the $x_0$-directions are enforced. That means when for any solution of a PDE solved by \finley |
| 336 |
the value on the plane $x_0=0$ will be identical to the values on $x_0=\var{l0}$. Correspondingly, |
| 337 |
\var{periodic1}=\False and \var{periodic2}=\False sets periodic boundary conditions |
| 338 |
in $x_1$-direction and $x_2$-direction, respectively. |
| 339 |
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. |
| 340 |
\end{funcdesc} |
| 341 |
|
| 342 |
\begin{funcdesc}{GlueFaces}{meshList,safetyFactor=0.2,tolerance=1.e-13} |
| 343 |
Generates a new \Domain object from the list \var{meshList} of \finley meshes. |
| 344 |
Nodes in face elements whose difference of coordinates is less then \var{tolerance} times the |
| 345 |
diameter of the domain are merged. The corresponding face elements are removed from the mesh. |
| 346 |
|
| 347 |
TODO: explain \var{safetyFactor} and show an example. |
| 348 |
\end{funcdesc} |
| 349 |
|
| 350 |
\begin{funcdesc}{JoinFaces}{meshList,safetyFactor=0.2,tolerance=1.e-13} |
| 351 |
Generates a new \Domain object from the list \var{meshList} of \finley meshes. |
| 352 |
Face elements whose nodes coordinates have difference is less then \var{tolerance} times the |
| 353 |
diameter of the domain are combined to form a contact element \index{element!contact} |
| 354 |
The corresponding face elements are removed from the mesh. |
| 355 |
|
| 356 |
TODO: explain \var{safetyFactor} and show an example. |
| 357 |
\end{funcdesc} |
Parent Directory
| 
Revision Log