Mathematical Sciences: Deformation of Domains, Jacobian Determinant and Grid Generation

数学科学：域变形、雅可比行列式和网格生成

• 批准号: 9106942
• 负责人: Dale Anderson
• 金额: $ 2.88万
• 依托单位: University of Texas at Arlington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1991
• 资助国家: 美国
• 起止时间: 1991-08-01 至 1993-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9106942&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Deformation; Domains; Jacobian

The objective of this research is to provide a new technique, the deformation method, of generating grids in solving physical and engineering problems. The technique is based on solid mathematical foundation. Each grid point in an existing grid is to be moved according to a decoupled system of n ordinary differential equations (n is the number of space dimensions). The technique will provide a one-to-one correspondence of the old and new grid points and thus folding of the grid will definitely be eliminated. Moreover, it will provide direct and precise control of the cell size. Thus, it will be adaptive. Finally, the deformation method will be computationally efficient. In particular it will be suitable for parallel computing. The research project is aimed to greatly enhance our ability to solve large scale scientific computational problems that are crucial to our economy, environment, and security. For example, to model the air flow about an airplane, millions of grid points must be chosen outside the airplane. In biology, the order of components in the human nervous system is a trillion. In physics, millions of particles must be analyzed in a supercollider. Even simple models of these problems typically require months of manpower to generate grids before computers can perform the computation. The research project will result in an improved, powerful algorithm for automatic grid generation by computer.

本研究的目的是提供一种生成网格的新技术——变形方法来解决物理和工程问题。 该技术基于坚实的数学基础。 现有网格中的每个网格点将根据 n 个常微分方程的解耦系统（n 是空间维数）进行移动。 该技术将提供新旧网格点的一一对应，因此网格的折叠肯定会被消除。 此外，它将提供对单元尺寸的直接和精确的控制。 因此，它将具有适应性。 最后，变形方法将具有计算效率。 特别是它将适用于并行计算。 该研究项目旨在大大提高我们解决对我们的经济、环境和安全至关重要的大规模科学计算问题的能力。 例如，要对飞机周围的气流进行建模，必须在飞机外部选择数百万个网格点。 在生物学中，人类神经系统的组成部分有万亿级。 在物理学中，必须在超级对撞机中分析数百万个粒子。 即使是这些问题的简单模型通常也需要数月的人力来生成网格，然后计算机才能执行计算。 该研究项目将产生一种改进的、强大的计算机自动网格生成算法。