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Mesh improvement by minimizing a weighted sum of squared element volumes
Author(s) -
Renka Robert J.
Publication year - 2014
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.4845
Subject(s) - polygon mesh , laplacian smoothing , smoothing , mathematics , triangulation , triangle mesh , vertex (graph theory) , sizing , volume mesh , algorithm , topology (electrical circuits) , planar , mesh generation , finite element method , mathematical optimization , computer science , geometry , combinatorics , graph , engineering , art , statistics , computer graphics (images) , structural engineering , visual arts
Summary We describe a new mesh smoothing method that consists of minimizing the sum of squared element volumes over the free vertex positions. To the extent permitted by the fixed vertices and mesh topology, the resulting mesh elements have uniformly distributed volumes. In the case of a triangulation, uniform volume implies well‐shaped triangles. If a graded mesh is required, the element volumes may be weighted by centroidal values of a sizing function, resulting in a mesh that conforms to the required vertex density. The method has both a local and a global implementation. In addition to smoothing, the method serves as a simple parameter‐free means of untangling a mesh with inverted elements. It applies to all types of meshes, but we present test results here only for planar triangle meshes. Our test results show the new method to be fast, superior in uniformity or conformity to a sizing function, and among the best methods in terms of triangle shape quality. We also present a new angle‐based method that is simpler and more effective than alternatives. This method is directly aimed at producing well‐shaped triangles and is particularly effective when combined with the volume‐based method. It also generalizes to anisotropic mesh smoothing. Copyright © 2014 John Wiley & Sons, Ltd.