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

Title: 科学家的问题：超结构四边形网格对有限元分析收敛性和准确性影响的研究

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, Teichm\"uller space, harmonic foliations, dynamical systems, surface mappings, meromorphic quadratic differentials, surface mappings, etc. Show Chinese abstract

Abstract: 在当前工业界和学术界的实践中，有限元计算的收敛性和准确性与网格生成的方法和质量密切相关。 多年来，国内学术界对高质量网格生成的研究主要参考了四边形和六边形局部质量，使其接近正方形和立方体的特性。 本文的主要贡献是提出一个全新的研究方向和内容：有必要探索和研究超结构四边形网格的整体全局排列结构和模式对有限元计算收敛性和计算精度的影响。 通过在这一新领域的研究，可以有助于解决当前工业界和学术界在仿真过程中网格生成阶段严重依赖“经验”的非严谨状态，并明确判断哪些网格生成的全局安排可以保证有限元计算的收敛性。 为了生成和设计具有可控整体排列结构的超结构四边形网格，需要大量现代二维和三维几何拓扑理论，如模空间、Teichmüller空间、调和叶状结构、动力系统、曲面映射、亚纯二次微分、曲面映射等。