Revisiting Optimal Delaunay Triangulation for 3D Graded Mesh Generation

Abstract

Abstract. This paper proposes a new algorithm to generate a graded three-dimensional tetrahedral mesh. It revisits the class of methods based on optimal Delaunay triangulation (ODT) and proposes a proper way of injecting a background density function into the objective function minimized by ODT. This continuous/analytic point of view leads to an objective function that is continuous and Delaunay consistent, in contrast with the discrete/geometrical point of view developed in previous work. To optimize the objective function, this paper proposes a hybrid algorithm that combines a local search (quasi-Newton) with a global optimization (simulated annealing). The benefits of the method are both improved performances and an improved quality of the result in terms of dihedral angles. This results from the combination of two effects. First, the local search has a faster speed of convergence than previous work due to the better behavior of the objective function, and second, the algorithm avoids getting stuck in a poor local minimum. Experimental results are evaluated and compared using standard metrics.

Curved optimal delaunay triangulation

“…Its implementation in 3D along with details on the sizing field computations and boundary handling was studied by Alliez et al [2005], and a hybrid approach mixing Delaunay refinement and ODT optimization was later introduced [Tournois et al 2009] to accelerate convergence and improve results. Recent improvements include an alternative boundary treatment [Gao et al 2012] and various numerical accelerations [Chen and Holst 2011;Chen et al 2014]. Today, the isotropic meshes obtained via ODT optimization consistently outperform other approaches (based on advancing front, bubble packing, etc) in terms of element quality (dihedral angles, aspect ratio, etc) for a given vertex budget.…”

“…The effect on the mesh is very visible: all elements quickly tend to become equally sized and equally shaped, as predicted by approximation theory. Faster convergence can also be obtained using variants of Newton's method, where the energy Hessian is approximated through either previous values of the gradient [Chen et al 2014] or a graph-Laplacian [Chen and Holst 2011].…”

“…Another property of ODT making it particularly appropriate for isotropic meshing is that its formulation can be trivially altered to handle arbitrary mesh density: given a positive scalar field ρ : Ω → R + , we can modulate the L 1 norm in the definition of E ODT in Eq. (1) by substituting a modulated local volume form ρ(x) dx for the original dx as introduced in [Chen and Xu 2004] and exploited in [Chen et al 2014]. Modifying the volume form does not affect the isotropic properties of formulation, but the resulting minimizer of the ODT energy now has elements with edges proportional to ρ −1/(d +2) .…”

“…Depending on the application for which a mesh is constructed, vertices may have to be precisely on the boundary, or the mesh boundary just needs to approximate ∂Ω. While the former can be achieved using methods as in [Alliez et al 2005;Chen et al 2014;Gao et al 2012], we derive a simple approach to the latter by exploiting the stability of our non-shrinking minimization.…”

Anisotropic and feature sensitive triangular remeshing using normal lifting

“…Chen et al [16] use the concept of optimal Delaunay triangulation to generate volume meshes, which is somehow dual to our approach. An objective function is defined over the restricted Delaunay triangulation of a set of samples, measuring the quality of the output elements.…”