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Simple and effective variational optimization of surface and volume triangulations

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Abstract

Optimizing surface and volume triangulations is critical for many advanced numerical simulation applications. We present a variational approach for smoothing triangulated surface and volume meshes to improve their overall mesh qualities. Our method seeks to reduce the discrepancies between the actual elements and ideal reference elements by minimizing two energy functions based on conformal and isometric mappings. We derive simple, closed-form formulas for the values, gradients, and Hessians of these energy functions, which reveal important connections of our method with some well-known concepts and methods in mesh generation and surface parameterization. We then introduce a simple and efficient iterative algorithm for minimizing the energy functions, including a novel asynchronous step-size control scheme. We demonstrate the effectiveness of our method experimentally and compare it against Laplacian smoothing and some other mesh smoothing techniques.

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Notes

  1. We use \(\vert{\user2{A}}\vert\) to denote the determinant of a matrix \({\user2{A}}\) for brevity.

  2. We label vertices of a tetrahedron by 0, 1, 2, and 3 so that the opposite face of vertex 0 has vertices 1, 2, and 3, leading to formulas that are more concise and more consistent with those for triangles.

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Acknowledgments

This work was supported by National Science Foundation under award number DMS-0809285 and also by a subcontract from the Center for Simulation of Advanced Rockets of the University of Illinois at Urbana-Champaign funded by the U.S. Department of Energy through the University of California under subcontract B523819. We thank Dr. Eric Shaffer of UIUC for pointers to some test meshes. We thank Dr. Patrick Knupp for helpful discussions. We thank the anonymous reviewers for their helpful comments.

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Authors and Affiliations

  1. Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, NY, 11794, USA

    Xiangmin Jiao & Duo Wang

  2. College of Computing, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Hongyuan Zha

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  1. Xiangmin Jiao
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  2. Duo Wang
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  3. Hongyuan Zha
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Correspondence to Xiangmin Jiao.

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Jiao, X., Wang, D. & Zha, H. Simple and effective variational optimization of surface and volume triangulations. Engineering with Computers 27, 81–94 (2011). https://doi.org/10.1007/s00366-010-0180-z

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