An n-simplex is said to be n-well-centered if its circumcenter lies in its interior. We introduce several other geometric conditions and an algebraic condition that can be used to determine whether a simplex is n-well-centered. These conditions, toge
ther with some other observations, are used to describe restrictions on the local combinatorial structure of simplicial meshes in which every simplex is well-centered. In particular, it is shown that in a 3-well-centered (2-well-centered) tetrahedral mesh there are at least 7 (9) edges incident to each interior vertex, and these bounds are sharp. Moreover, it is shown that, in stark contrast to the 2-dimensional analog, where there are exactly two vertex links that prevent a well-centered triangle mesh in R^2, there are infinitely many vertex links that prohibit a well-centered tetrahedral mesh in R^3.
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 p
roperties 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.
A completely well-centered tetrahedral mesh is a triangulation of a three dimensional domain in which every tetrahedron and every triangle contains its circumcenter in its interior. Such meshes have applications in scientific computing and other fiel
ds. We show how to triangulate simple domains using completely well-centered tetrahedra. The domains we consider here are space, infinite slab, infinite rectangular prism, cube and regular tetrahedron. We also demonstrate single tetrahedra with various combinations of the properties of dihedral acuteness, 2-well-centeredness and 3-well-centeredness.
