CAREER: Computational Geometry, Mesh Generation, Geometric Modeling

职业：计算几何、网格生成、几何建模

• 批准号: 0846872
• 负责人: Alper Ungor
• 金额: $ 40.06万
• 依托单位: University of Florida
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-05-01 至 2014-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0846872&HistoricalAwards=false
• 关键词: CAREER; Computational; Geometry; Mesh; Generation

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.

复杂物理现象模拟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。 我们研究空间域的几何离散化问题的解决方案（在传统的科学计算中遇到的）和时空域（由下一代数值方法直接在时空连续体中求解偏微分方程的动机）。 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）和实际的算法（易于实现，在实践中的性能，可扩展性）的优势。 研究了两大类问题：（i）我们开发了快速，顺序和并行算法和软件来计算优质，尺寸最优，简单和立方体网格的空间域，可以演变为各向同性和各向异性问题的模拟进展。 (ii)我们开发了可扩展的，证明良好的网格算法和软件来计算时空三角剖分，使我们能够直接在时空域进行模拟。 该项目开发的算法和软件工具正在与应用程序集成，并有助于工程，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。因此，该项目在许多科学，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，重点是将图论，几何和拓扑学等理论领域的基本概念与生物学和工程学中的应用问题联系起来，从而产生学术影响。