边界积分方程中近奇异积分计算的一种变量替换法

THE EVALUATION OF NEARLY SINGULAR INTEGRALS IN THE BOUNDARY INTEGRAL EQUATIONS WITH VARIABLE TRANSFORMATION

准确估计近奇异边界积分是边界元分析中一项很重要的课题,其重要性仅次于对奇异积分的处理.近年来已发展了许多方法,都取得了一定程度的成功,但这个问题至今仍未得到彻底的解决.基于一种新的变量变换的思想和观点,提交了一种通用的积分变换法,它非常有效地改善了被积函数的震荡特性,从而消除了积分的近奇异性,在不增加计算量的情况下,极大地改进了近奇异积分计算的精度.数值算例表明,其算法稳定,效率高,并可达到很高的计算精度,即使区域内点非常地靠近边界,仍可取得很理想的结果.

The numerical solution of boundary value problems using boundary integral equations demands the accurate computation of the integral of the kernels, which occur as the nearly singular integrals when the collocation point is close to the element of integration but not on the element in boundary element method （BEM）. Such integrals are difficult to compute by standard quadrature procedures, since the integrand varies very rapidly within the integration interval, more rapidly the closer the collocation point is to the integration element. Practice shows that we can even obtain the results of superconvergence for the computed point far enough from the boundary; however, using standard quadrature procedures, which neglect the pathological behavior of the integrand as the computed point approaches the integration element, will lead to a degeneracy of accuracy of the solution, even no accuracy, which is the so-called ＂boundary layer effect＂. To avoid the ＂boundary layer effect＂, the accurate computation of the nearly singular boundary integrals would be more crucial to some of the engineering problems, such as the crack-like and thin or shell-like structure problems. , The importance of the accurate evaluation of nearly singular integrals is considered to be next to the singular boundary integrals in BEM, and great attentions have been attracted and many numerical techniques have been proposed for it in recent years. These developed methods can be divided on the whole into two categories： ＂indirct algorithms＂ and ＂direct algorithms＂, which have obtained varying degree of success, but the problem of the nearly singular integrals has not been completely resolved so far. In this paper, a new efficient transformation is proposed based on a new idea of transformation with variables. The proposed transformation can remove the nearly singularity efficiently by smoothing out the rapid variations of the integrand of nearly singular integrals, and improve the accuracy of numerical results of nearly singular integrals greatly without inCreasing the computational effort. Numerical examples of potential problem with their satisfactory results in both curved and straight elements are presented, showing encouragingly the high efficiency and stability of the suggested approach, even when the internal point is very close to the boundary. The suggested algorithm is general and canbe applied to other problems in BEM.