Generation of Tetrahedral Finite Element Meshes: Variational Delaunay Approach

The goal is to generate tetrahedral decomposition of a general solid body, whose surface is given as a collection of triangular facets. The principle idea is that a vertex set in general positions guarantees existence of a unique triangulation which satisses the Delaunay empty-sphere property. (Algorithms for robust, parallel construction of such triangulations are available.) However, all of the input surface facets do not necessarily appear in such a triangulation. In order to represent the boundary of the solid, we iterate two operations, edge ip and edge split with the insertion of additional vertex, until all of the boundary facets are present in the tetrahedral mesh. The outcome of the vertex insertion is another triangulation of the input surfaces, but one which is represented as a subset of the tetrahedral faces.