9 research outputs found
Triangulation of Simple 3D Shapes with Well-Centered Tetrahedra
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 fields. 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.Comment: Accepted at the conference "17th International Meshing Roundtable",
Pittsburgh, Pennsylvania, October 12-15, 2008. Will appear in proceedings of
the conference, published by Springer. For this version, we fixed some typo
Simplices rarely contain their circumcenter in high dimensions
summary:Acute triangles are defined by having all angles less than , and are characterized as the triangles containing their circumcenter in the interior. For simplices of dimension , acuteness is defined by demanding that all dihedral angles between -dimensional faces are smaller than . However, there are, in a practical sense, too few acute simplices in general. This is unfortunate, since the acuteness property provides good qualitative features for finite element methods. The property of acuteness is logically independent of the property of containing the circumcenter when the dimension is greater than two. In this article, we show that the latter property is also quite rare in higher dimensions. In a natural probability measure on the set of -dimensional simplices, we show that the probability that a uniformly random -simplex contains its circumcenter is
A Dihedral Acute Triangulation of the Cube
It is shown that there exists a dihedral acute triangulation of the
three-dimensional cube. The method of constructing the acute triangulation is
described, and symmetries of the triangulation are discussed.Comment: Minor edits for journal version. Added some material to the
introductio
Well-Centered Triangulation
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