并行Delaunay加细：算法与分析
Parallel Delaunay Refinement: Algorithms and Analyses
摘要：
本文分析了Delaunay网格剖分方法自然并行化的复杂性。并行化采用一个简单的策略：在每次迭代中，它们选择一组"独立“点插入域中，然后更新德洛内三角剖分。我们表明，这样一组独立的点可以有效地并行构造，所需的迭代次数是$O(\log^2(L/s))$，其中$L$是域的直径，和$s$是在输出网格中的最小边。此外，我们表明，插入的每个独立集的点可以实现顺序由Ruppert的方法在二维和Shewchuk的在三维。因此，我们的并行Delaunay细化方法提供了相同的元素质量和网格大小的保证，在二维和三维的顺序算法。对于准均匀网格，如那些由Chew的方法，我们表明，迭代次数可以减少到$O(\log(L/s))$。据我们所知，这些是第一个可证明polylog $(L/s)$并行时间Delaunay网格算法，生成良好的形状网格的大小最佳的一个常数。
英文摘要：
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 $O(\log^2(L/s))$, 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.
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