Slow convergence of the BEM with constant elements in solving beam bending problems

Abstract

Constant elements offer many advantages as compared with other higher-order elements in the boundary element method (BEM). With the use of constant elements, integrals in the BEM can be calculated accurately with analytical integrations and no corner problems need to be addressed. These features can make fast solution methods for the BEM (such as the fast multipole, adaptive cross approximation, and pre-corrected fast Fourier transform methods) especially efficient in computation. However, it is well known that the collocation BEM with constant elements is not adequate for solving beam bending problems due to the slow convergence or lack of convergence in the BEM solutions. In this study, we quantify this assertion using simple beam models and applying the fast multipole BEM code so that a large number of elements can be used. It is found that the BEM solutions do converge numerically to analytical solutions. However, the convergence rate is very slow, in the order of h to the power of 0.55–0.63, where h is the element size. Some possible reasons for the slow convergence are discussed in this paper.