CAREER: Dynamics, Domain Conformity, and Anisotropy in the Theory and Implementation of Unstructured Mesh Generation

基于偏微分方程组的复杂科学模拟的解使用离散网格来表示物理(连续)域。同样，复杂场景的计算机渲染使用这些网格进行光能传递计算。现有的基于启发式的方法不能产生对包括移动或复杂曲面在内的三维现实世界问题可靠的网格，也不能产生表现出各向异性(方向相关)行为的网格。这个项目将为这些领域设计、分析和实施网格生成算法，这些算法是实用的，并且在可能的情况下，被证明是好的。这项任务的技术方法是使用基于优化的平滑和拓扑变换。对于各向异性问题，该项目还将为将Delaunay三角剖分扩展到各向异性情况奠定坚实的理论基础。最后，在三维方面，该项目将寻求对约束Delaunay四面体及其构造算法的坚定理解。该项目还将积极将研究与教育相结合。它将有助于课程开发和课程设计，以支持加州大学伯克利分校工程学院正在启动的一个新的计算工程科学计划。这项工作将包括网格生成和共轭梯度类解算器的课程和文本。

Solutions to complex scientific simulations based on partial differential equations use discrete meshes to represent the physical (continuous) domain. Similarly, computer rendering of complex scenes uses these meshes for radiosity calculations. Existing heuristic-based methods do not produce meshes that dependable for 3-dimensional real-world problems that include moving or complex surfaces, or that exhibit anisotropic (direction-dependent) behavior. This project will design, analyze, and implement mesh generation algorithms for these domains that are practical and, where possible, provably good. The technical approach to this task is to use optimization-based smoothing and topological transformations. For anisotropic problems, the project will also develop a firm theoretical basis footing for extending Delaunay triangulations to the anisotropic case. Finally, in three dimensions the project will pursue a firm understanding of constrained Delaunay tetrahedralizations and algorithms for their construction.This project will also be active in integrating research with education. It will contribute to curriculum development and course design in support of a new Computational Engineering Science program being initiated within the College of Engineering at UC Berkeley. This effort will include courses and texts in mesh generation and conjugate gradient-like solvers.