结构化曲面四边形网格生成理论及算法

Structured surface quadrilateral mesh generation is a necessary step for the simulation technology, such as iso-geometry and finite element analysis, hence it plays an important role in the development of high-level manufactures. Structured surface quadrilateral mesh also has a close relationship with the development of computer-aided design, mixed-reality, animation and special effects industry, since it can be used for spline construction, modeling, texture mapping and compression. The generation of quadrilateral mesh on surface remains a huge challenge under the requirements of feature preservation, structural tessellation and singularity optimization. The research on the theoretical foundation is not quite sufficient. Based on the observation of the connection between holomorphic quartic differential and quadrilateral mesh with singularities of degree 5, this project first plans to prove that the connect is actually an equivalence relation, and state that all the quad mesh with singularities of degree 5 form a linear space. Second, this project will propose algorithm to compute and represent quad mesh with linear combination of finite meshes, and eventually give an efficient algorithm to compute quad mesh on surfaces with the minimum number of singularities of degree 5. In the end, the project will extend the theory and algorithm to the feature-preserving mesh generation problem. My research experience on Riemann geometry, algorithm design and mesh generation could help this project achieve its goal and the achievements of this project can be applied in high-level manufactures and computer graphics.

结构化曲面四边形网格与等几何分析、有限元等模拟仿真技术密切相关，是发展高端制造业的重要一环；同时可用于样条构造、建模、贴图和压缩等，影响着计算机辅助设计、混合现实、动画及特效制作等产业的发展。曲面四边形网格生成在保特征、结构化、奇异点优化等要求下仍是个巨大的挑战，相关基础理论研究不够充分。观察到曲面全纯四次微分和奇异点度均为五的四边形网格之间的联系，本项目将通过证明两者间的等价关系，给出所有奇异点度均为五的四边形网格构成线性空间的结论，并设计算法生成网格，将任意网格用有限多个网格的线性组合表示，得到无特征约束下奇异点度均为五的结构化曲面四边形网格生成的高效算法；最后将相关成果推广到特征约束情形，为其网格生成和优化提供新的理论依据和算法。申请人在黎曼几何、算法设计和网格生成方面具有较为充分的研究和实践积累，能够保证项目顺利推进；项目成果可应用于高端制造与图形学领域，产生实际应用价值。

本项目面向我国大型工业软件的迫切发展需求，针对其中的关键核心问题—四边形网格生成，基于现代拓扑和几何理论，开展新理论、新算法和软件开发的系统性研究。在该项目的支持下，本项目首次提出并证明曲面结构化网格和亚纯微分的等价性，证明了结构化网格空间的代数结构，开创性地建立了曲面结构化网格的统一理论体系，如奇异点分布的Abel-Jacobi理论，网格的黎曼度量理论等；解决了长期缺乏理论基础导致的自动化程度低，工程难度高的问题；从本质上回答了1973年提出的奇异点分布开放问题；并形成系列突破性算法，包括奇异点分布的合法性判断和优化算法，网格度量和参数化算法，基于最优传输的网格密度控制算法，基于度量空间的网格质量组合优化算法等。相关成果发表论文4篇，A类2篇，申请和授权专利7项，完成具有自主知识产权的四边形网格生成软件一套，培养研究生7名。本项目的实施突破了网格领域的基础理论，形成一系列高质量算法，为CAE大型工业软件的发展提供核心支撑。