Helmholtz边界积分方程中奇异积分间接求解方法

提出了间接求解传统Helmholtz边界积分方程CBIE的强奇异积分和自由项系数,以及Burton-Miller边界积分方程BMBIE中的超强奇异积分的特解法。对于声场的内域问题,给出了满足Helmholtz控制方程的特解,间接求出了CBIE中的强奇异积分和自由项系数。对于声场外域对应的BMBIE中的超强奇异积分,按Guiggiani方法计算其柯西主值积分需要进行泰勒级数展开的高阶近似,公式繁复,实施困难。本文给出了满足Helmholtz控制方程和Sommerfeld散射条件的特解,提出了间接求出超强奇异积分的方法。推导了轴对称结构外场问题的强奇异积分中的柯西主值积分表达式,并通过轴对称问题算例证明了本文方法的高效性。数值结果表明,对于内域问题,采用本文特解法的计算结果优于直接求解强奇异积分和自由项系数的结果,且本文的特解法可避免针对具体几何信息计算自由项系数,因而具有更好的适用性。对于外域问题,两者精度相当,但本文的特解法可避免对核函数进行高阶泰勒级数展开,更易于数值实施。

Indirect computation of singular integrals in Helmholtz boundary integral equation

A new particular solution method is proposed to indirectly calculate the strong singular integrals and free terms in conventional Helmholtz boundary integral equation (CBIE) and hyper-strong singular integrals in Burton-Miller boundary integral equation (BMBIE).For the acoustic problem of interior field,the particular solution satisfying Helmholtz governing equation is given,and the strong singular integral and free terms in CBIE are obtained indirectly.For an exterior field problem,however,calculation of its Cauchy principal value (CPV) for hyper-strong singular integral needs higher-order approximation of the kernel function through Taylor series expansion,which makes numerical implementation quite complex.In this paper,the particular solution satisfying Helmholtz governing equation and Sommerfeld radiation condition is given,and the hyper-strong singular integrals are obtained using a proposed new particular solution method.Also,the CPV of the strongly singular integral for an axisymmetric structure is derived.The high efficiency of the method is demonstrated with axisymmetric examples.The numerical results show that for the interior domain problem,the accuracy obtained by the proposed particular solution method is superior to that of directly calculating the strongly singular integral and the free term coefficient.Furthermore,the particular solution method can avoid calculating the free term with consideration of specific geometric information,and thus is of more general applicability.For an exterior domain problem,both methods provide almost the same accuracy,however,the proposed particular method can avoid expanding the kernel function to higher order and is easier to implement numerically.