中文摘要：

准确求解边界元方法中的近奇异积分是一个非常重要的问题。一般情况下,分析中涉及到的常规积分采用高斯方法即可获得较高的精度。但当源点位于边界附近时,采用高斯积分就会使计算结果精度大大降低,甚至得出错误的结果。对于平面问题,以源点作为原点,以所积分单元的切向和法向为坐标轴建立局部坐标系,对于线性单元可以得到所有积分的解析解。基于除角点外的所有边界点的场变量在边界上连续且有界的特点,所有在边界上引起场变量奇异的项之和必为零,故对于边界上的点可以直接在解析解中删除这些奇异项即可。算例表明,该方法可大大提高边界元的计算精度和效率。

英文摘要：

The exact evaluation of nearly singular integrals is a crucial task in Boundary Element Method（BEM）. Usually, the regular integrals arising from BEM implementation can be evaluated by standard Gaussian quadrature accurately. However, when the source point is close to the boundary, conventional method （Gaussian quadrature） gives a lower numerical precision, even wrong result. For a plane problem, using the source point serving as origin, the tangent and normal direction of integral elements to establish local coordinate system, analytical solutions are obtained for all integrals of linear elements. The field variables of all boundary points except for the corner are continuous and bounded, so the sum of corresponding singular terms are zero and these terms can be ignored directly. The accuracy and efficiency of solutions obtained by this method are verified by several numerical examples.