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

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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.