Finite Element Mesh Optimization in Three Dimensions

We discuss an optimization procedure for improving three-dimensional finite element meshes. Our method can be effectively coupled with any automatic mesh generator in order to obtain a triangulation without badly distorted elements. In this way, we are able to weaken the requirements on the generator, allowing it to produce singular elements, and post-process the result to get a valid mesh. Even though several other methods to improve finite element meshes already exist, the proposed one is the first to incorporate what finite element practitioners have for long known: One bad element renders the mesh useless. We report some very encouraging results with a quite crude algorithm. Our conclusion is that building meshes combining an initial generation step followed by an optimization step makes the meshing of arbitrary domains much more reliable, as compared to the usual approach of precluding distorted elements during the initial generation stage.