Todd Phillips

Sparse Voronoi Refinement

We present a new algorithm, Sparse Voronoi Refinement, that produces a conformal Delaunay mesh in arbitrary dimension with guaranteed mesh size and quality. Our algorithm runs in output-sensitive time O(n log(L/s)+m), with constants depending only on dimension and on prescribed element shape quality bounds. For a large class of inputs, including integer coordinates, this matches the optimal time bound of Θ(n log n + m). Our new technique uses interleaving: we maintain a sparse mesh as we mix the recovery of input features with the addition of Steiner vertices for quality improvement. This technical report is the long version of an article [HMP06] presented at IMR 2006, and contains full proofs.

Sparse parallel Delaunay mesh refinement

The authors recently introduced the technique of sparse mesh refinement to produce the first near-optimal sequential time bounds of O(n lg L/s+m) for inputs in any fixed dimension with piecewiselinear constraining (PLC) features. This paper extends that work to the parallel case, refining the same inputs in time O(lg(L/s) lgm) on an EREW PRAM while maintaining the work bound; in practice, this means we expect linear speedup for any practical number of processors. This is faster than the best previously known parallel Delaunay mesh refinement algorithms in two dimensions. It is the first technique with work bounds equal to the sequential case. In higher dimension, it is the first provably fast parallel technique for any kind of quality mesh refinement with PLC inputs. Furthermore, the algorithms implementation is straightforward enough that it is likely to be extremely fast in practice.

SVR: Practical Engineering of a Fast 3D Meshing Algorithm*

The recent Sparse Voronoi Refinement (SVR) Algorithm for mesh generation has the fastest theoretical bounds for runtime and memory usage. We present a robust practical software implementation of the SVR for meshing a piecewise linear complex in 3 dimensions. Our software is competitive in runtime with state of the art freely available packages on generic inputs, and on pathological worse cases inputs, we show SVR indeed leverages its theoretical guarantees to produce vastly superior runtime and memory usage. The theoretical algorithm description of SVR leaves open several data structure design options, especially with regard to point location strategies. We show that proper strategic choices can greatly effect constant factors involved in runtime.

A bézier-based approach to unstructured moving meshes

We present a new framework for maintaining the quality of two dimensional triangular moving meshes. The use of curved elements is the key idea that allows us to avoid excessive refinement and still obtain good quality meshes consisting of a low number of well shaped elements. We use B-splines curves to model object boundaries, and objects are meshed with second order Bézier triangles. As the mesh evolves according to a non-uniform flow velocity field, we keep track of object boundaries and, if needed, carefully modify the mesh to keep it well shaped by applying a combination of vertex insertion and deletion, edge flipping, and edge smoothing operations at each time step. Our algorithms for these tasks are extensions of known algorithms for meshes built of straight--sided elements and are designed for any fixed-order Bézier elements and B-splines. Although in this work we have concentrated on quadratic elements, most of the operations are valid for elements of any order and they generalize well to higher dimensions. We present results of our scheme for a set of objects mimicking red blood cells subject to a precomputed flow velocity field.

Representing topological structures using cell-chains

A new topological representation of surfaces in higher dimensions, “cell-chains” is developed. The representation is a generalization of Brissons cell-tuple data structure. Cell-chains are identical to cell-tuples when there are no degeneracies: cells or simplices with identified vertices. The proof of correctness is based on axioms true for maps, such as those in Brissons cell-tuple representation. A critical new condition (axiom) is added to those of Lienhardts n-G-maps to give “cell-maps”. We show that cell-maps and cell-chains characterize the same topological representations.

Size competitive meshing without large angles

We present a new meshing algorithm for the plane, Overlay Stitch Meshing (OSM), accepting as input an arbitrary Planar Straight Line Graph and producing a triangulation with all angles smaller than 170°. The output triangulation has competitive size with any optimal size mesh having equally bounded largest angle. The competitive ratio is O(log(L/s)) where L and s are respectively the largest and smallest features in the input. OSM runs in O(n log(L/s) +m) time/work where n is the input size and m is the output size. The algorithm first uses Sparse Voronoi Refinement to compute a quality overlay mesh of the input points alone. This triangulation is then combined with the input edges to give the final mesh.