Parallel Delaunay Refinement: Algorithms and Analyses

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Parallel Delaunay Refinement: Algorithms and Analyses (2002)

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by Daniel A. Spielman , Shang-Hua Teng , Alper Üngör , I Shang-hua , Teng Alper

Venue:	In Proceedings, 11th International Meshing Roundtable
Citations:	25 - 4 self

BibTeX

@INPROCEEDINGS{Spielman02paralleldelaunay,
    author = {Daniel A. Spielman and Shang-Hua Teng and Alper Üngör and I Shang-hua and Teng Alper},
    title = {Parallel Delaunay Refinement: Algorithms and Analyses},
    booktitle = {In Proceedings, 11th International Meshing Roundtable},
    year = {2002},
    pages = {205--217}
}

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Abstract

In this paper, we analyze the complexity of natural parallelizations of Delaunay refinement methods for mesh generation. The parallelizations employ a simple strategy: at each iteration, they choose a set of "independent" points to insert into the domain, and then update the Delaunay triangulation. We show that such a set of independent points can be constructed efficiently in parallel and that the number of iterations needed is O(log²(L/s)), where L is the diameter of the domain, and s is the smallest edge in the output mesh. In addition, we show that the insertion of each independent set of points can be realized sequentially by Ruppert's method in two dimensions and Shewchuk's in three dimensions. Therefore, our parallel Delaunay refinement methods provide the same element quality and mesh size guarantees as the sequential algorithms in both two and three dimensions. For quasi-uniform meshes, such as those produced by Chew's method, we show that the number of iterations can be reduced to O(log(L/s)). To the best of our knowledge, these are the first provably polylog(L/s) parallel time Delaunay meshing algorithms that generate well-shaped meshes of size optimal to within a constant.

Keyphrases

parallel delaunay refinement independent point log sup2 independent set size optimal well-shaped mesh quasi-uniform mesh natural parallelizations mesh generation delaunay refinement method parallel time delaunay simple strategy delaunay triangulation output mesh sequential algorithm element quality mesh size guarantee parallel delaunay refinement method

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