Performance of BiCG-like iterative solvers with fast multipole boundary element method and improvement of convergence

摘要

Iterative solvers are widely used for solving large systems of linear equations, and these are also applied to the systems obtained with the acoustical BEM. Since the convergence rate of the solvers directly affects the computational time, it is desirable to know the convergence behavior. In this paper, we investigate the convergence behavior of BiCG-like iterative solvers towards the systems with the 3-D acoustical BEM. Instead of the BEM, we use the fast multipole BEM (FMBEM), based on the fast multipole algorithm, which enables us to reduce large amount of computational time and memory requirements for large-scale problems such as concert halls and long noise barriers. Objects with simple shapes and with complex shapes are used for analyses. We consider as factors affecting the convergence, (1) boundary conditions and shapes, (2) kinds of formulations such as regular, hypersingular, and Burton-Miller formulation for avoiding the fictitious eigenfrequency difficulties, and (3) kinds of iterative solvers such as CGS, BiCGStab, BiCGStab2, and GPBiCG. We also investigate the effects of preconditioning and the initial shadow residual used in BiCG-like solvers. Recommended solvers and settings for improvement of the convergence through numerical results are briefly summarized as follows: (1) for internal problems, regular formulation with GPBiCG or BiCGStab2, and (2) for external problems, Burton-Miller formulation with CGS with pseudorandom numbers as the initial shadow residual.