Fast Boundary Element Method Solutions For Three Dimensional Large Scale Problems

Abstract

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, may not be suitable for these problems due to, for example, the complexity of mesh generation. The Boundary Element Method (BEM), based on boundary integral formulations (BIE), offers one possible solution to this issue by discretizing only the surface of the domain. 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 (BC) include, but are not limited to, the lack of appropriate BIE and the difficulties in the treatment of the volume integrals that result 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 BC.

For nonlinear problems, a major difficulty in applying the BEM is the treatment of the volume integrals in the BIE. An efficient approach, based on the precorrected-FFT technique, is developed to evaluate the volume integrals. In this approach, the 3-D uniform grid constructed initially to accelerate surface integration is used as the baseline mesh to evaluate volume integrals. The cubes enclosing part of the boundary are partitioned using surface panels. No volume discretization of the interior cubes is necessary. This grid is also used to accelerate volume integration. Based on this approach, accelerated BEM solvers for non-homogeneous and nonlinear problems are developed and tested. Good agreement is achieved between simulation results and analytical results. Qualitative comparison is made with current approaches.

Stokes problems with slip BC are of particular importance in micro gas flows such as those encountered in MEMS devices. An efficient approach based on 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.