Mesh shape-quality optimization using the inverse mean-ratio metric |
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Authors: | Todd Munson |
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Institution: | (1) Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL 60439, USA |
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Abstract: | Meshes containing elements with bad quality can result in poorly conditioned systems of equations that must be solved when
using a discretization method, such as the finite-element method, for solving a partial differential equation. Moreover, such
meshes can lead to poor accuracy in the approximate solution computed. In this paper, we present a nonlinear fractional program
that relocates the vertex coordinates of a given mesh to optimize the average element shape quality as measured by the inverse
mean-ratio metric. To solve the resulting large-scale optimization problems, we apply an efficient implementation of an inexact
Newton algorithm that uses the conjugate gradient method with a block Jacobi preconditioner to compute the direction. We show
that the block Jacobi preconditioner is positive definite by proving a general theorem concerning the convexity of fractional
functions, applying this result to components of the inverse mean-ratio metric, and showing that each block in the preconditioner
is invertible. Numerical results obtained with this special-purpose code on several test meshes are presented and used to
quantify the impact on solution time and memory requirements of using a modeling language and general-purpose algorithm to
solve these problems. |
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Keywords: | |
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