On Combining Laplacian and Optimization-based Mesh Smoothing Techniques

Local mesh smoothing algorithms have been shown to be eeective in repairing distorted elements in automatically generated meshes. The simplest such algorithm is Laplacian smoothing , which moves grid points to the geometric center of incident vertices. Unfortunately, this method operates heuristically and can create invalid meshes or elements of worse quality than those contained in the original mesh. In contrast, optimization-based methods are designed to maximize some measure of mesh quality and are very eeective at eliminating extremal angles in the mesh. These improvements come at a higher computational cost, however. In this article we propose four smoothing techniques that combine a smart variant of Laplacian smoothing with an optimization-based approach. Several numerical experiments are performed that compare the mesh quality and computational cost for each of the methods in two and three dimensions. We nd that the combined approaches are very cost eeective and yield high-quality meshes. INTRODUCTION The nite element and nite volume solution methods have proven to be eeective tools in the numerical solution of many sci-entiic and engineering applications. Both techniques require a spatial decomposition of the computational domain into a union of simple geometric elements such as triangles or quadrilaterals in two dimensions and tetrahedra or hexahedra in three dimensions. If the geometry is complex, automatic mesh generation tools are used to facilitate this decomposition (Lo, 1985, Shep-hard and Georges, 1991). A problem with these meshes is that the shape of the elements in the mesh can vary signiicantly. For nite element techniques using simplicial meshes, poorly shaped or distorted elements can result in numerical diiculties during the solution process, particularly as the elements approach the limits of 0