二维Helmholtz边界超奇异积分方程解析研究

Analytical Study of Hypersinglar Integral Equations with Constant Element for 2D Helmholtz Problems

基于常规边界元法及超奇异边界积分方程复线性耦合的Burton-Miller方法应用于无限域声学问题的最大难点在于处理超奇异积分(二维问题).目前,此类超奇异积分主要使用各种弱奇异/正则化方法求解,而这些弱奇异/正则化方法具有时间消耗大等弱点.基于围道积分定理,本文给出一种使用常值单元的二维Helmholtz边界超奇异积分的解析表达式.在有限部分积分意义下,所有的奇异和超奇异积分可以解析表达.数值算例表明该解析表达式是有效的.

Burton-Miller method,a complex linear combination of conventional boundary element method（CBIE） and hypersinglar boundary element method（HBIE）,is widely used to deal with exterior acoustic problems.The difficult in implementing Burton-Miller method is computing strongly singular integrals（2D problems）.Although,many weakly singular/regularization methods have been presented to evaluate these integrals,these methods are still difficult or extremely time consuming.In this paper,analytical integration of strongly singular boundary integral equations discretized with constant element for 2D Helmholtz problems is presented.All singular and strongly singular integrals are analytically evaluated in finite part sense as constant elements are applied to discretize boundary.Contour integral is used for singular and strongly integrals.Validity of formulas is demonstrated with numerical examples.