Some investigations on convergence of GMRES in solving BEM equations for slender beam structures

Abstract

The boundary element method (BEM) has remarkable advantages in dimension reduction and higher accuracy, but the classical direct method (like Gauss elimination) is not efficient for large scale problems and not suitable for the implementation of fast algorithms like Fast-Multipole Method (FMM) or methods base on the hierarchical matrices. Alternatively, some iterative methods for nonsymmetrical systems of linear equations, such as the Generalized Minimum Residual Method (GMRES), has much improvement in convergence and computational efficiency and is well compatible with fast algorithms. However, in our recent research, we found that the convergence of conventional GMRES is slow for thin-walled structure. For example, for the slender beam problem, the convergence is obviously decreased with the increase of the slenderness ratio of the beam. In this paper, by taking the slender beam as an example, we investigate the convergence and efficiency of GMRES method dealing with BEM equations. Some influences about slenderness ratio, the behavior of the orthogonalization process and restart strategy, and preconditioning process are under consideration. A new preconditioner is proposed based on the geometry of the slender beam, and numerical results have shown that it can bring much improvement in convergence and computational efficiency.