  <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 The paper was withdrawn from submission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.
