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arXiv · 2507.17184

A Scientist Question: Research on the Impact of Super Structured Quadrilateral Meshes on Convergence and Accuracy of Finite Element Analysis

Hui Zhao
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Abstract

In the current practices of both industry and academia, the convergence and accuracy of finite element calculations are closely related to the methods and quality of mesh generation. For years, the research on high-quality mesh generation in the domestic academic field has mainly referred to the local quality of quadrilaterals and hexahedrons approximating that of squares and cubes. The main contribution of this paper is to propose a brand-new research direction and content: it is necessary to explore and study the influence of the overall global arrangement structure and pattern of super structured quadrilateral meshes on the convergence and calculation accuracy of finite element calculations. Through the research in this new field, it can help solve the non-rigorous state of serious reliance on "experience" in the mesh generation stage during simulation in the current industry and academia, and make clear judgments on which global arrangements of mesh generation can ensure the convergence of finite element calculations. In order to generate and design super-structured quadrilateral meshes with controllable overall arrangement structures, a large number of modern two-dimensional and three-dimensional geometric topology theories are required, such as moduli space, Teichmüller space, harmonic foliations, dynamical systems, surface mappings, meromorphic quadratic differentials, surface mappings, etc.

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Hui Zhao. 2025-07-23. A Scientist Question: Research on the Impact of Super Structured Quadrilateral Meshes on Convergence and Accuracy of Finite Element Analysis. https://arxiv.org/abs/2507.17184

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