Efficient and Accurate Unstructured Mesh Generation

The goal of this project is to create a quality unstructured mesh generation tool. To this end, we have developed a C++ code that combines unconstrained Delaunay triangulation with spring dynamics in order to achieve a high-quality mesh. Our code will be made publicly available as part of the RedPack software package. As a test of this code, we consider in this paper the problem of generating a mesh on the surface of a sphere. We worked on three different methods to generate the mesh. We have implemented a C++ version of Persson’s distmesh program [7], which we have used in two different ways: First, we created a full three-dimensional mesh of the unit sphere, then selected only the points and elements that lie on the sphere surface. Second, we used the two-dimensional version of the code to generate a square patch to use in the cubed sphere method of Ronchi, et al. [8]. We also implemented the icosahedral mesh generation method of Heikes and Randall [2], [3]. We examine the quality and efficiency in generation of these meshes, as well as examining ways to improve their quality by doing post-processing work on the meshes. Finally, as a test, we attempt to solve the advection equation upon our meshes, following the technique of Hubbard and Roe [5], and describe the results. 1. Structured and Unstructured Meshes In order to numerically solve partial differential equations on a surface, it is necessary to first have a mesh of the surface. This mesh specifies the points where the equation will be solved as well as the connections between these points. One category of meshes are the so-called structured meshes. The Cartesian grid is the simplest such example (Figure 1a). The solution points are located at the intersection of the grid lines, while the elements are the rectangles determined by the intersecting lines. This type of mesh is extremely simple and quick to generate. However, if the problem to be solved has curved internal and/or external boundaries, solving on a structured Cartesian grid requires one to modify the numerical scheme near these boundaries. This is usually quite difficult. See, for example, Helzel, et al., [4]. Another type of structured mesh is the body-fitted structured mesh. Here, the grid is still essentially Cartesian, but the quadrilateral cells are shaped to fit the boundary (Figure 1b). Numerical methods that were originally developed for Cartesian grids can be modified to body-fitted grids through grid mappings that do not change the essence of the numerical scheme (see LeVeque, Chapter 23 [6]). The difficulty in implementing this method lies in generating the mesh, particularly if there are multiple objects in the interior.