四边形常数单元离散下的声学非奇异BIE

Nonsingular Acoustic BIE based on Quadrilateral Constant Element Discretization

奇异积分是基于Burton-Miller方程的声学边界元法实现过程的难点之一。关于三角形单元离散的积分单元的已经比较成熟,研究四边形常数单元离散下的声学边界积分方程(BIE),通过构造围绕配点的极小半球面进行积分,求得积分中的发散项,推导四边形常数单元离散下边界积分方程及其法向求导的非奇异表达式,从而得到非奇异Burton-Miller方程。运用Gauss Legendre积分公式计算BIE的S(x)的数值解,对比解析解的计算结果,得出了数值解、解析解以及二者的绝对误差、相对误差随ka的变化规律。实际应用时,当给定精度和ka的值后,可以通过改变所需要的截断项数,使得误差满足给定的精度要求。

Singular integral is one of the difficulties in the realization of acoustic boundary element method based on Burton-Miller formulation. Boundary integral based on triangular elements discretization has been well studied. This article mainly focuses on the calculation of boundary integral equation and singular integral when constant quadrilateral elements are used to discretize the boundary. Firstly, the divergent term in the integral is obtained by constructing the minimal semispherical surface around the collocation point, the boundary integral equation and hypersingular boundary integral equation are written in a form of non-singular expression and the singular integral is calculated in the case of quadrilateral constant element discretization. Then nonsingular expressions of Burton-Miller formulations is derived and the numerical evaluation of S□（x） is calculated using Gaussian quadrature. Comparing the analytic solutions, the change regulation of their absolute errors and relative errors with ka are obtained. In practical application, the error can meet the requirement of given precision by changing the number of truncated terms required upon the given tolerance and ka.