Sliver-free three dimensional delaunay mesh generation

A key step in the finite element method is to generate well-shaped meshes in 3D. A mesh is well-shaped if every tetrahedron element has a small aspect ratio. It is an old outstanding problem to generate well-shaped Delaunay meshes in three or more dimensions. Existing algorithms do not completely solve this problem, primarily because they can not eliminate all slivers. A sliver is a tetrahedron whose vertices are almost coplanar and whose circumradius is not much larger than its shortest edge length. We present two new algorithms to generate sliver-free Delaunay meshes. The first algorithm locally moves the vertices of an almost-good mesh, whose tetrahedra have small circumradius to shortest edge length ratio. We show that the Delaunay triangulation of the perturbed mesh vertices is still almost good. Furthermore, most slivers disappear after a mild perturbation of the mesh vertices. The remaining slivers migrate to the boundary where they can be peeled off or can be treated with boundary enforcement heuristics. The second algorithm adds points to generate well-shaped meshes. It is based on the following observations. Any tetrahedron will disappear from the Delaunay triangulation if a point is added inside the circumsphere of the tetrahedron. Among the tetrahedra created by inserting this new point there could be tetrahedra with large radius-edge ratios, or slivers, or both. However, the new point is incident to every new tetrahedron. We first eliminate tetrahedra with large radius-edge ratios. We then select the point that avoids creating any small slivers when inserting point inside the circumsphere of slivers. We show that the algorithm will not introduce short edges to the Delaunay triangulation. A simple volume argument implies that the algorithm terminates and generates a well-shaped Delaunay mesh. The generated mesh has a good grading. The number of mesh elements is within a small constant factor of any almost-good mesh for that given domain. We also describe some variations of this refinement-based algorithm. In particular, we show that inserting points near sinks instead of circumcenters of bad tetrahedra also generates sliver-free Delaunay meshes.