12 research outputs found
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
Delaunay Hodge Star
We define signed dual volumes at all dimensions for circumcentric dual
meshes. We show that for pairwise Delaunay triangulations with mild boundary
assumptions these signed dual volumes are positive. This allows the use of such
Delaunay meshes for Discrete Exterior Calculus (DEC) because the discrete Hodge
star operator can now be correctly defined for such meshes. This operator is
crucial for DEC and is a diagonal matrix with the ratio of primal and dual
volumes along the diagonal. A correct definition requires that all entries be
positive. DEC is a framework for numerically solving differential equations on
meshes and for geometry processing tasks and has had considerable impact in
computer graphics and scientific computing. Our result allows the use of DEC
with a much larger class of meshes than was previously considered possible.Comment: Corrected error in Figure 1 (columns 3 and 4) and Figure 6 and a
formula error in Section 2. All mathematical statements (theorems and lemmas)
are unchanged. The previous arXiv version v3 (minus the Appendix) appeared in
the journal Computer-Aided Desig
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 meshing
A well-centered simplex is a simplex whose circumcenter lies in its interior, and a well-centered mesh is a simplicial mesh in which every simplex is well-centered. We examine properties of the well-centered simplex and well-centered meshes, present experimental results from an optimization method designed to make meshes well-centered, and give examples of well-centered tetrahedral meshes of a variety of three-dimensional regions
Well-centered Planar Triangulation -- An Iterative Approach
We present an iterative algorithm to transform a given planar triangle mesh into a well-centered one by moving the interior vertices while keeping the connectivity fixed. A well-centered planar triangulation is one in which all angles are acute. Our approach is based on minimizing a certain energy that we propose. Well-centered meshes have the advantage of having nice orthogonal dual meshes (the dual Voronoi diagram). This may be useful in scientific computing, for example, in discrete exterior calculus, in covolume method, and in space-time meshing. For some connectivities with no well-centered configurations, we present preprocessing steps that increase the possibility of finding a well-centered configuration. We show the results of applying our energy minimization approach to small and large meshes, with and without holes and gradations. Results are generally good, but in certain cases the method might result in inverted elements