10.1.1.65.8226

# 10.1.1.65.8226 - Locally Optimal Unstructured Finite...

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Locally Optimal Unstructured Finite Element Meshes in 3 Dimensions Rashid Mahmood and Peter K. Jimack School of Computing, University of Leeds, Leeds LS2 9JT, UK. Abstract This paper investigates the adaptive finite element solution of a general class of variational problems in three dimensions using a combination of node movement, edge swapping, face swapping and node insertion. The adaptive strategy proposed is a generalization of previous work in two dimensions and is based upon the construction of a hierarchy of locally optimal meshes. Results presented, both for a single equation and a system of coupled equations, sug- gest that this approach is able to produce better meshes of tetrahedra than those obtained by more conventional adaptive strategies and in a relatively efficient manner. Keywords: finite elements, variational problems, mesh optimization, tetrahedral elements, node movement, edge swapping, node insertion. 1 Introduction In this paper we present an extension of our previous work on mesh optimization, presented in [7, 8], 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 1

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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. 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): (1) for some energy density function . Here is the dimension of the problem and is the dimension of the dependent variable . Physically this variational form may be used to model problems in linear and nonlinear elasticity, heat and electrical conduction, motion by mean curvature and many more. Throughout this paper we restrict our attention to the three-dimensional case where . For variational problems of the form (1), the fact that the exact solution minimizes the en- ergy functional provides a natural optimality criterion for the design of computational grids using -refinement (defined here to include both node relocation and mesh reconnection). In- deed, the idea of locally minimising the energy with respect to the location of the vertices of a mesh of fixed topology has been considered by a number of authors (e.g. [2],[16]), as has the approach of locally minimising the energy with respect to the connectivity of a mesh with fixed vertices (e.g. [14]). All of this work has been undertaken in only two space dimensions however and, to our knowledge, this is the first work in which mesh optimization with respect to the solution energy has been attempted for unstructured tetrahedral meshes in three space dimensions. The algorithm that we use consists of a number of sweeps through each of the nodes in
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## This note was uploaded on 05/12/2011 for the course INDUSTRIAL 321 taught by Professor Memet during the Spring '11 term at Mitchell Technical Institute.

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10.1.1.65.8226 - Locally Optimal Unstructured Finite...

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