边界元方法中超奇异积分的计算方法 献给林群教授80华诞

Numerical methods to compute hypersingular integral in boundary element methods

超奇异积分的近似计算是边界元方法,特别是自然边界元理论中必须面对的难题之一.经典的数值方法,如Gauss求积公式和Newton-Cotes积分公式等数值方法,都不能直接用于超奇异积分的近似计算.本文将介绍超奇异积分基于不同定义的Gauss积分公式、S型变换公式、Newton-Cotes积分公式和外推法近似计算超奇异积分的思路,重点阐述Newton-Cotes积分公式和基于有限部分积分定义的外推法近似计算超奇异积分的主要结论.

The computation of hypersingular integral is one of the important subjects in boundary element methods especially in natural boundary element methods. Classical numerical methods such as Gauss methods, Newton-Cotes methods cannot be used to approximate the hypersingular integral directly. In this paper, we introduce the numerical methods such as Gauss methods, Newton-Cotes methods, S transformation methods and extrapolation methods which are based on different definitions; then we mainly present the results of Newton-Cotes methods and extrapolation methods which are used to compute the hypersingular integral.