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Embarrassingly parallel mesh refinement by edge subdivision

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Abstract

We have previously proposed a new technique for the communication-free adaptive refinement of tetrahedral meshes that works for all configurations. Implementations of the scheme must deal with all possible geometric configurations, which results in a large number of cases that in turn result in practical programming issues. In this article, we address this issue with a Python script that generates C++ code using the symmetric group \({\mathfrak{S}}_{4}\) acting over canonical topological and geometric configurations. We then analyze the performance of the technique by characterizing (a) mesh quality, (b) execution time and parallel speedup, and (c) traits of the algorithm that could affect quality or execution time differently for different meshes and different mesh refinement strategies. This article also details the method used to debug the many subdivision templates that the algorithm relies upon. Mesh quality is on par with other similar refinement schemes, and we suggest a more elaborate technique that may substantially improve mesh quality. We show that throughput on modern hardware can exceed 600,000 output tetrahedra per second per processor, and that the method is embarrassingly parallel—assuming the application has partitioned the input properly.

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Notes

  1. In the online (color) version, all figures will display edges interior to a displayed face in red, edges interior to a tetrahedron in green, and others in black; black and red respectively turn into grey and pink when the edge is not in the foreground. In the print version, tetrahedra will be illustrated with unobscured bounding edges in bold lines, obscured bounding edges in narrow lines, and interior edges in dashed lines.

  2. The average in our implementation is 5.6 tetrahedra per configuration.

  3. Cases 0, 1, 2b, 3b and 6 are exempt from the longest-edge criterion because they have no ambiguous faces.

  4. our slightly different implementation, as will be explained further, actually regroups the tetrahedra in 50 sets.

  5. and, more generally, about groups acting over sets.

  6. http://www.vtk.org/doc/nightly/html/classvtkMeshQuality.html

  7. millions of tetrahedra

  8. http://www.vtk.org/doc/nightly/html/classvtkStreamingTessellator.html

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Acknowledgements

David C. Thompson and Philippe P. Pébay were supported by the United States Department of Energy, Office of Defense Programs. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed-Martin Company, for the United States Department of Energy under contract DE-AC04-94-AL85000.

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  1. Sandia National Laboratories, MS 9152, P.O. Box 969, Livermore, CA, 94550, USA

    David C. Thompson & Philippe P. Pébay

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  1. David C. Thompson
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  2. Philippe P. Pébay
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Correspondence to David C. Thompson.

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Thompson, D.C., Pébay, P.P. Embarrassingly parallel mesh refinement by edge subdivision. Engineering with Computers 22, 75–93 (2006). https://doi.org/10.1007/s00366-006-0020-3

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