Regularization Algorithm for Three Dimensional Helmholtz Integral Equation Based on Taylor Series Expansion

摘要

Boundary element method (BEM) was used to solve the Helmholtz equation of the acoustic fields since Chen's and Chertock's study [1][2]. Although the BEM was effective and advantageous in dealing with some acoustic field problems, there wear some difficult matters [3][4]. One of them was to compute the singular and nearly singular integrals in boundary element analysis of the acoustic fields. The other was the non-uniqueness of the BE solution of the exterior problem under some wave numbers which corresponded with the interior problem eigenvalue wave numbers.In this paper, the trigonometric function of the fundamental solutions of Helmholtz equation was expanded into Taylor series. Then the nearly singular integral parts and non-singular integral parts were separated from the boundary integrals. The semi-analytical regularization algorithm [5] was used to evaluate the nearly singular integrals. Therefore, the difficulty about the calculation of the nearly singular integral in acoustic BEM was overcome. The non-uniqueness of the BE solution for the exterior problem was due to the incomplete equivalence between the BIE of the interior problem of simple closed surface and the one of the exterior problem. The semi-analytical algorithm was employed to deal with it combined with CHIEF (Combined Helmholtz Integral Equation Formulation) point method. The acoustic pressure of both the inner points of far field and the ones of near field was computed in a wide wave number range. The numerical examples show that the present BEM can not only assure the uniqueness of solution of the Helmholtz equation but also remarkably improve the precision of the result.