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A Comparison of Gradient- and Hessian-BasedOptimization Methods for Tetrahedral MeshQuality Improvement

Sastry, Shankar Prasad and Suzanne M. Shontz

Proceedings, 18th International Meshing Roundtable, Springer-Verlag, pp.631-648, October 25-28 2009

IMR
PROCEEDINGS

18th International Meshing Roundtable
Salt Lake City, UT, USA.
October 25-28, 2009

Department of Computer Science and Engineering,
The Pennsylvania State University
University Park, PA 16802
sps210,shontz@cse.psu.edu

First Paragraph
Discretization methods, such as the finite element method, are commonly used in the solution of partial differential equations (PDEs). The accuracy of the computed solution to the PDE depends on the degree of the approximation scheme, the number of elements in the mesh [1], and the quality of the mesh [2, 3]. More specifically, it is known that as the element dihedral angles become too large, the discretization error in the finite element solution increases [4]. In addition, the stability and convergence of the finite element method is affected by poor quality elements. It is known that as the angles become too small, the condition number of the element matrix increases [5]. Recent research has shown the importance of performing mesh quality improvement before solving PDEs in order to: (1) improve the condition number of the linear systems being solved [6], (2) reduce the time to solution [7], and (3) increase the solution accuracy. Therefore, mesh quality improvement methods are often used as a post-processing step in automatic mesh generation. In this paper, we focus on mesh smoothing methods which relocate mesh vertices, while preserving mesh topology, in order to improve mesh quality.

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