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This paper describes the design and implementation of a practical parallel algorithm for Delaunay triangulation that works well on general distributions. Although there have been many theoretical parallel algorithms for the problem, and some implementations based on bucketing that work well for uniform distributions, there has been little work on implementations for general distributions. We use the well known reduction of 2D Delaunay triangulation to find the 3D convex hull of points on a paraboloid. Based on this reduction we developed a variant of the Edelsbrunner and Shi 3D convex hull algorithm, specialized for the case when the point set lies on a paraboloid. This simplification reduces the work required by the algorithm (number of operations) from O(n log 2 n) to O(n log n). The depth (parallel time) is O(log 3 n) on a CREW PRAM. The algorithm is simpler than previous O(n log n) work parallel algorithms leading to smaller constants.Initial experiments using a variety of distributions showed that our parallel algorithm was within a factor of 2 in work from the best sequential algorithm. Based on these promising results, the algorithm was implemented using C and an MPI-based toolkit. Compared with previous work, the resulting implementation achieves significantly better speedups over good sequential code, does not assume a uniform distribution of points, and is widely portable due to its use of MPI as a communication mechanism. Results are presented for the IBM SP2, Cray T3D, SGI Power Challenge, and DEC AlphaCluster.

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The decomposition of an arbitrary polyhedral domain into tetrahedra is currently more tractable than its decomposition into hexahedra. However, for some engineering applications, a mesh composed of hexahedra, or even a mixture of hexahedra, pentahedra and tetrahedra, is preferable. One such application is the p-type ÿnite element method, where the total number of elements should be as small as possible. We show in this paper that, given a tetrahedral decomposition, some of the tetrahedra can be e ciently combined into hexahedra and pentahedra. The basis of the method is a classiÿcation, using a generalized graph representation, of all possible tetrahedral decompositions of pentahedra and hexahedra. We then present a tetrahedral merge algorithm that utilizes this result to search for the subgraphs of hexahedra and pentahedra in a tetrahedral mesh. The problem of ÿnding an optimal solution is NP-complete. We present heuristics to increase the number of hexahedra and pentahedra, within a reasonable amount of computation time. The algorithm has been implemented in the PolyFEM mesher, and examples showing the typical merge success of the algorithm are included.

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