Ph.D. Dissertation Defense by Jian Ding
Friday, December 17, 2004

(Dr. Wenjing Ye, Chair)

"Fast Boundary Element Method Solutions for Three Dimensional Large Scale Problems "


Efficiency is one of the key issues in numerical simulation of large-scale problems with complex 3-D geometry. Traditional domain based methods, such as finite element methods and finite difference methods may not be suitable for these problems due to, for example, the complexity of mesh generation. The Boundary Element Method (BEM), based on integral formulations, offers one possible solution to address this issue. By discretizing only the surface of the domain, it greatly reduces the complexity of mesh generation as well as the size of the problem. However, to date, successful applications of the BEM are mostly limited to linear and continuum problems. The challenges in the extension of the BEM to nonlinear problems or problems with non-continuum boundary conditions include, but are not limited to, the lack of appropriate boundary integral formulations and the difficulties in the treatment of the volume integrals resulted from the nonlinear terms. In this thesis work, new approaches and techniques based on the BEM have been developed for 3-D nonlinear problems and Stokes problems with slip boundary conditions.

For nonlinear problems, a major difficulty in applying the BEM is the treatment of the volume integrals in the integral formulation. An efficient approach, based on the precorrected-FFT technique, is developed to evaluate the volume integrals. In this approach, the 3-D uniform grid is used as the baseline mesh for the evaluation of volume integrals. The boundary cubes are further discretized by projection of surface panels onto the boundary of corresponding cubes. No additional volume discretization of the interior cubes is necessary. Therefore, complicated volume discretization for interior problem domain is avoided. This grid is also used to accelerate volume integration to further reduce the computational cost. Based on this approach, accelerated BEM solvers for non-homogeneous problems and nonlinear problems are developed and tested on several problems. Results are compared with analytical solutions and good agreement has been achieved.

Stokes problems with slip boundary conditions are of particular importance in micro gas flows such as those encountered in MEMS devices. An efficient approach based the BEM combined with the precorrected-FFT technique has been proposed and various techniques have been developed to solve these problems. As the applications of the developed method, drag forces on oscillating objects immersed in an unbounded slip flow are calculated and validated with either analytic solutions or experimental results.