Snow Globe: An Advancing-Front 3D Delaunay Mesh Refinement Algorithm

One of the objectives of a Delaunay mesh refinement algorithm is to produce meshes with tetrahedral elements having a bounded aspect ratio, which is the ratio between the radius of the circumscribing and inscribing spheres. The refinement is carried out by inserting additional Steiner vertices inside the circumsphere of a poor-quality tetrahedron (to remove the tetrahedron) at a sufficient distance from existing vertices to guarantee the termination and size optimality of the algorithm. This technique eliminates tetrahedra whose ratio of the radius of the circumscribing sphere and the shortest side, the radius-edge ratio, is large. Slivers, almost-planar tetrahedra, which have a small radius-edge ratio, but a large aspect ratio, are avoided by small random perturbations of the Steiner vertices to improve the aspect ratio. Additionally, geometric constraints, called “petals”, have been shown to produce smaller high-quality meshes in 2D Delaunay refinement algorithms. In this paper, we develop a deterministic nondifferentiable optimization routine to place the Steiner vertex inside geometrical constraints that we call “snow globes” for 3D Delaunay refinement. We explore why the geometrical constraints and an ordering on processing of poor-quality tetrahedra result in smaller meshes. The stricter analysis provides an improved constant associated with the size optimality of the generated meshes. Aided by the analysis, we present a modified algorithm to handle boundary encroachment. The final algorithm behaves like an advancing-front algorithms that are commonly used for quadrilateral and hexahedral meshing. ∗The work of the author was supported in part by the NIH/NIGMS Center for Integrative Biomedical Computing grant 2P41 RR0112553-12 and a grant from ExxonMobil. The author would also like to thank Christine Pickett, an editor at the University of Utah, for proofreading and finding numerous typos in the paper.