中文摘要：

基于间接规则化边界积分方程,有效估计奇异边界积分,准确求得边界量,为场变量的计算奠定了基础。在计算场变量时,针对二维弹性力学边界元法中出现的几乎奇异积分,本文采用一类非线性变量替换法,有效地改善了被积函数的震荡特性,从而消除了核积分的几乎奇异性;在不增加计算量的情况下,极大地改进了几乎奇异积分计算的精度,成功地求解了弹性体近边界点上的力学参量,避免了边界层效应。此外,本文引入一种精确几何单元逼近,对于圆弧边界,这样的插值逼近几乎是精确的,提高了计算精度。数值算例表明,本文算法稳定,效率高,并可达到很高的计算精度,即使场点非常靠近边界,如场点到积分单元的距离小到纳米级,仍可避免边界层效应现象。

英文摘要：

To begin with,a regularized boundary integral equation with indirect formulation is adopted to deal with the singular integrals and the boundary unknown quantities can be calculated accurately.When it comes to the physical quantities at the interior points,an efficient non-linear transformation is utilized to evaluate the nearly singular integrals occurring in two-dimensional（2D）elastic problems.The proposed transformation can remove or damp out the near singularity efficiently and can improve the accuracy of numerical results of nearly singular integrals greatly without increasing other computational efforts.Moreover,an exact geometrical representation,named＂arc element＂,was introduced to remove the errors caused by representing arc geometries using polynomial shape functions,and therefore the computational accuracy can be improved efficiently.Numerical examples show the high efficiency and stability of the present approach,and the＂boundary layer effect＂can be avoided efficiently even when the internal point is very close to the boundary,i.e.,when the distance of the computed point to the boundary as small as 10-9.