Boundary element method (BEM) is a numerical technique based on integral equations. BEM has found to be very efficient for boundary value problems where the domains are infinite or singularities are involved. As a result, the use of BEM has become very popular in areas like acoustics and fracture mechanics. The standard formulation for BEM involves solution to a linear system of equations Ax=b where A is a dense and nonsymmetrical matrix. The computational time is seen to be order O(N²) with iterative solvers. The Fast multipole method (FMM) and Adaptive cross approximation (ACA) methods are two methods that have been developed in recent years to reduce the order of computation to O(N). In this research we investigate the ACA and FMM, and conduct a comparison study between the two.
This research discusses the FMM in details. The FMM is a tree based algorithm to do fast matrix vector multiplication. Over the years, a variety of techniques have been developed to make FMM more efficient. Some of these techniques are studied and implemented. The research considers three different types of problems, viz. potential, acoustic and elastostatic problems for the purpose of comparison. A new L2 modification has been introduced in this research. An efficient usage of adaptive tree is also presented. Finally, a composite technique encompassing the existing strategies with the new proposed modifications has been presented as the new adaptive algorithm. An implementation of this technique, coded in FORTRAN, has yielded the capacity of solving very large engineering problems (up to 1 million DOFs) on a desktop PC in small amount of time. Numerical results have been presented which emphasize the efficiency of this new implementation.
The second part of this research is to study ACA and compare with FMM. A study of ACA has been done and a modified implementation of the ACA has been proposed. This implementation has enabled one to solve a few large problems (more than a half million DOFs) on a desktop PC.
Finally a comparison study has been conducted between the FMM and ACA for simple acoustic problems. It was found that the ACA is much more efficient for solving small to medium sized problems (up to 200,000 DOFs) while the FMM was found to be more efficient for large sized problems (more than 200,000 DOFs). This thesis research has significantly improved the computational efficiencies of both the FMBEM and ACA BEM towards applications of the BEM in solving real engineering problems on desktop PCs.