一种处理弹性动力边界元法中奇异积分的方法

利用边界元法求解瞬态弹性动力学问题时，时域基本解函数的分段连续性和奇异性为该问题的求解带来很大的困难。为了解决时域基本解中的奇异性问题，本文依据柯西主值的定义，对经过时间解析积分之后的时域基本解进行奇异值分解，将其分成奇异和正则积分两部分；其中正则部分可通过采用常规高斯积分方法来计算，而奇异部分具有简单的形式，可以利用解析积分计算。经过上述操作之后，就可以达到直接消除时域基本解中奇异积分的目的。和传统方法相比，本文方法并不依赖静力学基本解来消除奇异性，是一种直接求解方法。最后给定两个数值算例来验证本文提出方法的正确性和可行性，结果表明使用本文算法可以解决弹性动力学边界积分方程中的奇异性问题。

A method for dealing with singular integrals in the elastodynamics boundary element method

When the boundary element method is used to analyze transient elastodynamics problems, the piecewise continuity and the singularity of the time-domain fundamental solution function present an enormous difficulty for their solution.To tackle the singularity of the fundamental solution, the definition of a Cauchy principal value is employed to decompose the time-domain fundamental solution into the singular part and the regular part in this paper.The regular part can be calculated by using the conventional Gaussian integral method, and the singular part which has a simple form can be calculated by analytical integrations.After the above operations, the singular integral in the time-domain fundamental solution can be eliminated directly.Compared with the traditional method, the proposed method is a direct method for solving singular integrals, and does not rely on the static fundamental solution to eliminate the singularity.Finally, several numerical examples are given to verify the accuracy and feasibility of the proposed method.The numerical results show that the proposed method can solve the singularity problem in the boundary integral equation of elastodynamics.