CAREER: Computational Geometry, Mesh Generation, Geometric Modeling

用于复杂物理现象仿真的几何建模提出了许多挑战，包括算法效率、实用性、可扩展性、鲁棒性、理论保证以及与新兴数值方法的兼容性。我们研究了空间域（在传统的科学计算中遇到）和时空域（受下一代数值方法的启发，用于直接在时空连续体中求解偏微分方程）的几何离散化问题的解决方案。我们的方法结合了理论算法（时间复杂度、输出大小最优性和质量保证）和实践启发式（易于实现、实践中的性能、可扩展性）的优势。研究了两大类问题：(i)我们开发了快速，顺序和并行的算法和软件来计算空间域的优质，尺寸最优，简单和立方体网格，这些网格可以随着各向同性和各向异性问题的模拟进展而发展。（ii）我们开发可扩展的，可证明良好的网格算法和软件来计算时空三角，使我们能够直接在时空域进行模拟。在这个项目中开发的算法和软件工具正在与应用程序集成，并为工程、科学计算、实体建模、计算机辅助设计、图形学、地理信息系统、计算生物学、可视化和分子建模的基础研究做出贡献。因此，该项目对许多科学、医学和工业领域产生了更广泛的影响。此外，该项目通过纳入代表性不足的群体，开发跨学科课程，将图论，几何和拓扑等理论领域的基本概念与生物学和工程中的应用问题联系起来，从而产生学术影响。

ABSTRACT Geometric modeling for simulation of complex physical phenomena raises many challenges including algorithmic efficiency, practicality, scalability, robustness, theoretical guarantees, and compatibility with the emerging numerical methods. We study solutions for geometric discretization problems for spatial domains (encountered in conventional scientific computing) and for space-time domains (motivated by the next-generation numerical methods being developed for solving PDEs directly in the space-time continuum). Our approach combines the strengths of theoretical algorithms (time complexity, output size optimality, and quality guarantees) and practical heuristics (ease of implementation, performance in practice, scalability). Two broad classes of problems are studied: (i) We develop fast, sequential and parallel algorithms and software to compute premium-quality, size-optimal, simplicial and cubical meshes of spatial domains which can evolve as the simulation progress for isotropic and anisotropic problems. (ii) We develop scalable, provably-good meshing algorithms and software to compute space-time triangulations which enables us to perform simulations directly in the space-time domain. The algorithms and the software tools developed within this project are being integrated with applications and contributing to the fundamental research in engineering, scientific computing, solid modeling, computer-aided design, graphics, geographic information systems, computational biology, visualization and molecular modeling. As a result, this project has broader impact across a number of scientific, medical and industrial fields. Moreover, the project has academic impact through the inclusion of underrepresented groups, the development of interdisciplinary courses which focus on linking fundamental concepts in theoretical areas such as graph theory, geometry and topology to application problems in biology and engineering.