Well-Centered Triangulation

Abstract

Meshes composed of well-centered simplices have nice orthogonal dual meshes (the dual Voronoi diagram). This is useful for certain numerical algorithms that prefer such primal-dual mesh pairs. We prove that well-centered meshes also have optimality properties and relationships to Delaunay and minmax angle triangulations. We present an iterative algorithm that seeks to transform a given triangulation in two or three dimensions into a well-centered one by minimizing a cost function and moving the interior vertices while keeping the mesh connectivity and boundary vertices fixed. The cost function is a direct result of a new characterization of well-centeredness in arbitrary dimensions that we present. Ours is the first optimization-based heuristic for well-centeredness, and the first one that applies in both two and three dimensions. We show the results of applying our algorithm to small and large two-dimensional meshes, some with a complex boundary, and obtain a well-centered tetrahedralization of the cube. We also show numerical evidence that our algorithm preserves gradation and that it improves the maximum and minimum angles of acute triangulations created by the best known previous method.Comment: Content has been added to experimental results section. Significant edits in introduction and in summary of current and previous results. Minor edits elsewher

Optimizing Voronoi Diagrams for Polygonal Finite Element Computations

“…Although motivated by a different downstream application, we realize that our energy is closely related to the concept of well-centered triangulations [41]. A triangulation is called well-centered if each of its triangles contains its circumcenter, a property important for simulations based on discrete exterior calculus [16].…”

“…VanderZee and colleagues compute well-centered triangulations by iteratively maximizing the (signed) distance of mesh vertices to the opposite edges of their incident triangles, normalized by circumradii [41]. This per-vertex measure is minimized with respect to an L p -norm, with p typically being 4, 6, 8, or 10, or a combination thereof.…”

“…For triangular and tetrahedral meshes quality criteria for element shapes are well understood [30], and successful mesh generation and optimization techniques based on Delaunay refinement [28,29,7] or variational optimization [38,39,41] have been proposed.…”

“…To this end, a number of very successful mesh improvement approaches have been proposed, being based on either Delaunay refinement [28,29,9] or some variational optimization [1,38,22,39,10,41]. So-called pliant methods, which combine local topological changes (e.g., Delaunay refinement) and vertex relocation (e.g., Laplacian smoothing or Lloyd relaxation) have been found to be superior over methods involving one of the techniques only [5,22,38].…”

Triangulation of Simple 3D Shapes with Well-Centered Tetrahedra

“…This definition is further refined to that of a k-well-centered simplex which is one whose k-dimensional faces have the well-centeredness property. An n-dimensional simplex which is k-well-centered for all 1 ≤ k ≤ n is called completely well-centered [15]. These properties extend to simplicial complexes, i.e.…”