![]() |
Computational & Technology Resources
an online resource for computational,
engineering & technology publications |
Civil-Comp Proceedings
ISSN 1759-3433 CCP: 76
PROCEEDINGS OF THE THIRD INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY Edited by: B.H.V. Topping and Z. Bittnar
Paper 11
Locally Optimal Unstructured Finite Element Meshes in Three Dimensions R. Mahmood and P.K. Jimack
School of Computing, University of Leeds, United Kingdom Full Bibliographic Reference for this paper
R. Mahmood, P.K. Jimack, "Locally Optimal Unstructured Finite Element Meshes in Three Dimensions", in B.H.V. Topping, Z. Bittnar, (Editors), "Proceedings of the Third International Conference on Engineering Computational Technology", Civil-Comp Press, Stirlingshire, UK, Paper 11, 2002. doi:10.4203/ccp.76.11
Keywords: finite elements, variational problems, mesh optimization, tetrahedral elements, node movement, edge swapping, node insertion.
Summary
In this paper we present an extension of our work on mesh optimization
presented at the last Engineering Computational Technology conference in
September 2000 from two space dimensions to three. The
approach that we follow is to consider the adaptive finite element
solution of a general class of variational problems using a combination
of node movement, edge swapping, face swapping and node insertion. The
particular adaptive scheme that is used is based upon the construction of
a hierarchy of locally optimal tetrahedral meshes starting with a coarse
grid for which the location and connectivity of the nodes is optimized.
This grid is then locally refined and the new mesh is optimized in the
same manner. Results presented indicate that, as with triangles in two
dimensions, this approach is able to produce better meshes of tetrahedra
than those obtained by more conventional adaptive strategies and in a
relatively efficient manner.
The class of problem that we consider in this work may be posed in the following form (or similar, according to the precise nature of the boundary conditions): for some energy density function ![]() ![]() ![]() ![]() ![]()
For variational problems of the form (11.1), the fact that the exact
solution minimises the energy functional provides a natural optimality
criterion for the design of computational grids using The algorithm that we use consists of a number of sweeps through each of the nodes in turn until convergence is achieved. At the beginning of each sweep the gradient, with respect to the position of each node, of the energy functional is found (where ![]() ![]() ![]() is used in order to determine along which line the node should be moved. The distance that the node is moved along this line is computed using a one-dimensional minimization of the energy subject to the constraint that the node should not move more than a proportion ![]() ![]() ![]() Here ![]() ![]() ![]() ![]() Once convergence with respect to the position of each node has been achieved a further reduction in the energy of the solution is sought by the use of edge and face swapping. In three dimensions there are a large number of different ways in which the local connectivity of the nodes may be altered. In this work we use the same edge and face swapping stencils as the public domain software package GRUMMP, whose implementation is restricted to improving the geometric quality of the mesh rather than minimizing energy as we do here. Of course the positions of the nodes are likely to be no longer locally optimal at this point due to the edge/face swapping. Hence it is necessary to alternate between the node movement and the swapping algorithms until the whole process has converged (at least approximately). At this stage we allow the application of local mesh refinement to obtain a new mesh at the next level which must itself now be optimized. The process is complete when either a desired accuracy has been obtained or a maximum number of nodes or elements has been reached. purchase the full-text of this paper (price £20)
go to the previous paper |
|