Deformation Methods for Grid Adaptation

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DMS-9732742Guojun LiaoA key factor to the success of numerical simulation of scientific and engineering problems is that of grid generation of the physical domains. The objective of the project is to develop an adaptive grid generation method that can be used with various numerical schemes such as finite element, finite difference, etc. Our method is based on sophisticated mathematics and its numerical implementation is straightforward. It is the deformation method which redistributes the grid points according to the physical variables that are being simulated. A monitor function is calculated which is small where the error is large. The deformation method then moves each grid point according to the monitor function so that the new grid will be concentrated where the need for better resolution exists to reduce the error. Model problems from fluid dynamics and other important applications will be simulated on such moving grids. We will demonstrate that the method is capable of generating grids that are adapted to the fine features of the physical variables such as shock waves, moving fronts, and boundary layers, etc.This project is aimed to greatly enhance our ability to simulate large scale computational problems that are crucial to the national strategic interests. The advance of computing technology allows us to simulate some real world problem without conducting costly experiments. But the needs for accurate and efficient simulation far exceed the most powerful computers of today. A lot of simplification and upgrading have to be made to stay within the limit of today's computing capacity. Fixed grids are wasteful in computing resource. Our research will lead to software that can automatically put more points to the regions where and when the physical variables change rapidly without adding new points. This would be very useful for high performance computing and the numerical simulation of material science, environment problems, manufacturing process and other fields that require a grid. By working in the project, mathematics and engineering students will gain first hand experience on the application of mathematics to the solution of engineering problem. Advanced degrees will be granted to graduate students who make significant contributions to the project after their completion of course work.