Search

I present an algorithm that can create a mesh that is free of slivers away from the boundary, or that can eliminate such slivers from a pre-existing mesh by refining it. The resulting tetrahedra have dihedral angles between 30 and 135 degrees and radius-edge ratios of at most 1.368, except near the boundary. In comparison, previous bounds on dihedral angles were microscopic. The final mesh can respect specified input vertices and a user-defined sizing function. The algorithm comes with a bound on the sizes of the features it creates, and can provably grade from small to large tetrahedra.

We introduce anisotropic Voronoi diagrams, a generalization of multiplicatively weighted Voronoi diagrams suitable for generating guaranteed-quality meshes of domains in which long, skinny triangles are required, and where the desired anisotropy varies over the domain. We discuss properties of anisotropic Voronoi diagrams of arbitrary dimensionality---most notably circumstances in which a site can see its entire Voronoi cell. In two dimensions, the anisotropic Voronoi diagram dualizes to a triangulation under these same circumstances. We use these properties to develop an algorithm for anisotropic triangular mesh generation in which no triangle has an angle smaller than 20A, as measured from the skewed perspective of any point in the triangle.

The isosurface stuffing algorithm fills an isosurface with a uniformly sized tetrahedral mesh whose dihedral angles are bounded between 10.7° and 164.8°, or (with a change in parameters) between 8.9° and 158.8°. The algorithm is whip fast, numerically robust, and easy to implement because, like Marching Cubes, it generates tetrahedra from a small set of precomputed stencils. A variant of the algorithm creates a mesh with internal grading: on the boundary, where high resolution is generally desired, the elements are fine and uniformly sized, and in the interior they may be coarser and vary in size. This combination of features makes isosurface stuffing a powerful tool for dynamic fluid simulation, large-deformation mechanics, and applications that require interactive remeshing or use objects defined by smooth implicit surfaces. It is the first algorithm that rigorously guarantees the suitability of tetrahedra for finite element methods in domains whose shapes are substantially more challenging than boxes. Our angle bounds are guaranteed by a computer-assisted proof. If the isosurface is a smooth 2-manifold with bounded curvature, and the tetrahedra are sufficiently small, then the boundary of the mesh is guaranteed to be a geometrically and topologically accurate approximation of the isosurface.