A method of treating the nearly singular integral in calculation of sound radiation with BEM

Abstract

For the nearly singular integral of three-dimensional acoustic boundary element method (BEM), based on the 6-noded triangular isoparametric element, a new semi-analytical algorithm of three-dimensional high order element is proposed in this paper. Using Taylor expansion of trigonometric functions in the three dimensional acoustic fundamental solutions, the singular part of the fundamental solutions is separated. Based on the geometric characteristics of the 6-noded triangular element, an approximate singular kernel function is constructed which has the same singularity as singular integral kernel function. Subtracting the approximate kernel function from the kernel function of the singular integral, the latter is decomposed into a regular kernel function and an approximate singular kernel function. The integral of the regular kernel function can be calculated accurately by using the conventional Gauss numerical quadrature. The integral of the new singular part is calculated by the semi-analytic formula derived in this paper. In the surface of the integral element, the local coordinate system ρθ is established and the approximate singular integral is transformed into the integrals of variables ρ and θ which are already separated in ρθ system. The integral with respect to polar variable ρ is expressed by the analytic formulations first. Then the new singular integral which is a surface integral is transformed into the line integral with respect to variable θ , which can be evaluated by the Gaussian quadrature. Consequently, the new semi-analytic algorithm is established to calculate the nearly singular surface integrals in 3D acoustic BEM. Some examples are given in the last part of this paper to show the accuracy and the effectiveness of the present algorithm. The computed results demonstrate that the semi-analytic algorithm with high order element presented in this paper is more effective than linear regularization BEM to solve nearly singular integrals for 3D acoustic BEM.