A low-frequency fast multipole boundary element method based on analytical integration of the hypersingular integral for 3D acoustic problems

A low-frequency fast multipole boundary element method (FMBEM) for 3D acoustic problems is proposed in this paper. The FMBEM adopts the explicit integration of the hypersingular integral in the dual boundary integral equation (BIE) formulation which was developed recently by Matsumoto, Zheng et al. for boundary discretization with constant element. This explicit integration formulation is analytical in nature and cancels out the divergent terms in the limit process. But two types of regular line integrals remain which are usually evaluated numerically using Gaussian quadrature. For these two types of regular line integrals, an accurate and efficient analytical method to evaluate them is developed in the present paper that does not use the Gaussian quadrature. In addition, the numerical instability of the low-frequency FMBEM using the rotation, coaxial translation and rotation back (RCR) decomposing algorithm for higher frequency acoustic problems is reported in this paper. Numerical examples are presented to validate the FMBEM based on the analytical integration of the hypersingular integral. The diagonal form moment which has analytical expression is applied in the upward pass. The improved low-frequency FMBEM delivers an algorithm with efficiency between the low-frequency FMBEM based on the RCR and the diagonal form FMBEM, and can be used for acoustic problems analysis of higher frequency.