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

TL;DR

This paper introduces a new 3D Delaunay mesh refinement algorithm using 'snow globes' constraints to improve mesh quality and size, inspired by advancing-front methods, but the paper was withdrawn due to incorrect proofs.

Contribution

It develops a deterministic optimization routine for Steiner vertex placement within 'snow globes' constraints, enhancing mesh quality and size optimality in 3D Delaunay refinement.

Findings

Smaller high-quality meshes achieved with new constraints.

Improved size optimality constant for generated meshes.

Algorithm behaves like an advancing-front method.

Abstract

<incorrect proofs; does not consider an important case because of which the proofs are wrong. The paper was withdrawn from submission> 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.