Anisotropic Delaunay Mesh Generation

Abstract

Anisotropic meshes are triangulations of a given domain in the plane or in higher dimensions, with elements elongated along prescribed directions. Anisotropic triangulations are known to be well suited for interpolation of functions or solving PDEs. Assuming that the anisotropic shape requirements for mesh elements are given through a metric field varying over the domain, we propose a new approach to anisotropic mesh generation, relying on the notion of anisotropic Delaunay meshes. An anisotropic Delaunay mesh is defined as a mesh in which the star of each vertex v consists of simplices that are Delaunay for the metric associated to vertex v. This definition works in any dimension and allows to define a simple refinement algorithm. The algorithm takes as input a domain and a metric field and provides, after completion, an anisotropic mesh whose elements are shaped according to the metric field.

Geometric and Topological Inference

“…It is also possible to remove flakes and inconsistencies by perturbing the position of the points. This kind of perturbation may be prefered to the weighting mechanism in the context of mesh generation [99,24,74].…”

“…The approach followed in this chapter that defines local triangulations and remove inconsistencies among the local triangulations has been pioneered by Shewchuk to maintain triangulations of moving points [122] and by Boissonnat, Wormser and Yvinec to generate anisotropic meshes [24]. The central question behind this approach is the stability of Delaunay triangulations and the existence and construction of Delaunay triangulations on manifolds [14,13,27].…”

Anisotropic Delaunay Meshes of Surfaces

“…In Boissonnat et al [2008Boissonnat et al [ , 2011, we showed it is possible to refine the set V of sites until there are no more inconsistencies among the set T (V ) of 3D stars. This leads to a locally uniform anisotropic 3D mesh in which the star of each vertex is Delaunay with respect to the metric of this vertex.…”

“…It can be shown that the distortion radius satisfies the following continuity property [Boissonnat et al 2011] .…”

“…The proof, only sketched here, is given in Boissonnat et al [2011]. It uses a volume argument based on the two following facts: (1) the number of hitting sets of a picking region is bounded; (2) For given values of parameters α, β, δ, and ρ 0 , the volume of the forbidden region associated to a hitting set tends to zero when γ 0 tends to 1.…”