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

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

Velocity Level Approximation of Pressure Field Contact Patches

Pressure Field Contact (PFC) was recently introduced as a method for detailed modeling of contact interface regions at rates much faster than elasticity-theory models, while at the same time predicting essential trends and capturing rich contact behavior. The PFC model was designed to work in conjunction with error-controlled integration at the acceleration level. Therefore a vast majority of existent multibody codes using solvers at the velocity level cannot incorporate PFC in its original form. In this work we introduce a discrete in time approximation of PFC making it suitable for use with existent velocity-level time steppers and enabling execution at real-time rates. We evaluate the accuracy and performance gains of our approach and demonstrate its effectiveness in simulating relevant manipulation tasks. The method is available in open source as part of Drake's Hydroelastic Contact model.Comment: 8 pages, 10 figures. Supplementary video can be found at https://youtu.be/AdCnTyqqQP

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

Tetrahedral Mesh Improvement, Algorithms and Experiments

111 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2001.The persistent appearance of slivers in large three-dimensional Delaunay meshes has been reported as early as 1985. They persist even after treatment with the Delaunay refinement algorithm. Cheng et al. proposed to remove slivers by assigning real weights to the points and change the Delaunay to the weighted Delaunay mesh. This is referred to as the sliver exudation algorithm. Their theoretical bound on the achieved minimum mesh quality is a constant that is positive but exceedingly small. We perform computational experiments to testify the practical effectiveness of sliver exudation.U of I OnlyRestricted to the U of I community idenfinitely during batch ingest of legacy ETD

Tetrahedral Mesh Improvement, Algorithms and Experiments

111 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2001.The persistent appearance of slivers in large three-dimensional Delaunay meshes has been reported as early as 1985. They persist even after treatment with the Delaunay refinement algorithm. Cheng et al. proposed to remove slivers by assigning real weights to the points and change the Delaunay to the weighted Delaunay mesh. This is referred to as the sliver exudation algorithm. Their theoretical bound on the achieved minimum mesh quality is a constant that is positive but exceedingly small. We perform computational experiments to testify the practical effectiveness of sliver exudation.U of I OnlyRestricted to the U of I community idenfinitely during batch ingest of legacy ETD