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

本研究的目的是为解决物理和工程问题提供一种新的网格生成技术--变形法。这项技术有坚实的数学基础。现有网格中的每个网格点将根据解耦的n个常微分方程组(n是空间维度的数目)来移动。这项技术将提供新旧网格点的一一对应，从而肯定会消除网格的折叠。此外，它还将提供对细胞大小的直接和精确控制。因此，它将是适应性的。最后，形变方法在计算上是有效的。特别是，它将适用于并行计算。该研究项目旨在极大地提高我们解决大规模科学计算问题的能力，这些问题对我们的经济、环境和安全至关重要。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。在生物学上，人类神经系统中的组成成分的顺序是一万亿。在物理学中，数百万个粒子必须在超级对撞机中进行分析。即使是这些问题的简单模型，在计算机执行计算之前，通常也需要数月的人力来生成网格。该研究项目将产生一种改进的、功能强大的计算机网格自动生成算法。

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.