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Snow Globe: An Advancing-Front 3D Delaunay Mesh Refinement Algorithm | | Shankar Prasad Sastry
; | | Date: |
1 Aug 2015 | | Abstract: | 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. | | Source: | arXiv, 1508.0060 | | Services: | Forum | Review | PDF | Favorites | |
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