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Animal Shapes, Modal Analysis, and Visualization of Motion (IV): Geometric Constructions and Implementation

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Abstract

We address computer implementation and technology issues in geometric constructions for mathematical and computational modeling purposes, especially regarding 3D finite-element mesh generation for large scale scientific computation of partial differential equations in applications. This mesh generation process for irregular shapes is time-consuming and cannot be done by hand, and has long been recognized as a major bottle-neck in large-scale scientific computing. Significant computer skills and proficiency are required for the applied mathematicians and numerical analysts engaged in the process of 3D-geometry mesh generations. But these days specialty software packages are available that can expedite geometric constructions. Their applications are now the canonical way to carry out the needed numerics of grids. In fact, new technologies such as laser scan are now available which generate a point cloud that, through further processing to remedy some common types of defects, can further expedite the generation of mesh of high quality for complex geometries. In this paper, we use Geomagic Design X as the basic platform to construct CAD containing the underlying geometry from a point cloud. From this CAD, we can then generate finite element mesh by HyperMesh for solving partial differential equations. This paper is mostly survey and tutorial in nature in the hope of providing a somewhat self-contained account for the researcher working on shape analysis and geometric designs. Step-by-step procedures of computer work are explained, with four instances of concrete examples. Trouble-shooting for common mesh generation problems, whether from a point cloud or CAD such as the repair of overlapping regions, mutually penetrating finite elements, gaps and holes, are also indicated.

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Acknowledgements

Chunqiu Wei is supported by the Research Ability Improvement Program for Young Teachers of BUCEA (Grant No. X21031). Tiexin Guo is supported by the National Natural Science Foundation of China (No. 11671278 and No. 11971483) and the Science and Technology Research Project of Chongqing Municipal Education Commission (KJ1706154). Pengfei Yao is supported in part by the National Science Foundation of China, Grant No. 12071463. Junmin Wang is supported in part by the National Natural Science Foundation of China No. 62073037 and 12131008. Zhichun Yang is supported in part by the National Natural Science Foundation of China Grant No. 11971081, and the Science and Technology Research Project of Chongqing Municipal Education Commission under Grant No. KJZD-M202000502.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, China

    Jing Yang & Junmin Wang

  2. Department of Mathematics, Texas A &M University, College Station, TX, 77843, USA

    Goong Chen, Alexey Sergeev, Jingtong Kaya Huang & Matthew M. Scully

  3. Institute of Quantum Science and Engineering, Texas A &M University, College Station, TX, 77843, USA

    Goong Chen

  4. National Center for Theoretical Sciences, National Tsing Hua University, Hsinchu Branch, Hsinchu, Taiwan, ROC

    Goong Chen

  5. Department of Mathematics and Data Science, Beijing University of Civil Engineering and Architecture, Beijing, 100044, China

    Chunqiu Wei

  6. Department of Mathematics, Washington University at St. Louis, St. Louis, MO, 63130, USA

    Steven G. Krantz

  7. School of Mathematics, Key Laboratory of Complex Systems and Data Science of Ministry of Education, Shanxi University, Taiyuan, 030006, Shanxi, China

    Pengfei Yao

  8. School of Mathematics and Statistice, Central South University, Changsha, 410075, Hunan, China

    Tiexin Guo

  9. School of Mathematics and Statistice, Chongqing Normal University, Chongqing, 401331, China

    Zhichun Yang

  10. Department of Mechanical Engineering, Cheng Shiu University, Kaohsiung, Taiwan, ROC

    Ming-Chieh Chen

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  1. Jing Yang
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  2. Goong Chen
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  3. Chunqiu Wei
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  9. Tiexin Guo
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  10. Junmin Wang
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  11. Zhichun Yang
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  12. Ming-Chieh Chen
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Yang, J., Chen, G., Wei, C. et al. Animal Shapes, Modal Analysis, and Visualization of Motion (IV): Geometric Constructions and Implementation. J Geom Anal 33, 330 (2023). https://doi.org/10.1007/s12220-023-01343-5

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