薄体位势问题边界元法中的解析积分算法

An Analytical Integral Algorithm in BEM for Potential Problems with Thin Bodies

薄体结构的数值分析是边界元法的难点问题之一。该文导出了一种完全解析积分算法,用这种算法计算了薄体平面位势问题边界元法中出现的几乎弱奇异、强奇异和超奇异积分。当边界离散为一系列线性单元,边界积分方程离散计算的积分可归纳为三种形式。对薄体问题,源点与积分单元距离通常相距很近,这些积分产生显著几乎奇异性,直接采用常规高斯积分不能有效计算。为此该文导出了这些几乎奇异积分的全解析计算公式。按源点与单元的距离是否为零,公式分两种情况。新算法采用全解析积分公式处理几乎奇异积分,首先精确计算出薄体问题边界未知位势和法向位势梯度,然后再进一步计算了域内点的物理参量。算例表明该文算法可处理狭长比为1.E-08的薄体问题,显示了边界元法分析薄体问题具有独特的优势。

The numerical analysis of structures with thin bodies is one of the difficult problems in the boundary element method. A new completely analytical integral algorithm was proposed and applied to the evaluation of the nearly weakly-singular, nearly strongly-singular and nearly hypersingular integrals in the boundary element method of planar potential problems with thin bodies. When the boundary was dis-cretized by a set of linear elements, the integrals of the discretized boundary integral equations could be classified into three forms. For problems with thin bodies, a source point is very close to some integral element commonly. Due to the nearly singularity, these integrals are not effectively and directly computed by conventional Gaussian integral. In order to accurately calculate these integrals, the completely analytical integral formulas were presented, which include two cases according that the distance between the source point and the integral element is equal to zero or not. The present algorithm was applied these analytical formulas to treat nearly singular integrals. Firstly, the unknown potentials and potentials' gradients with respect to outward normal to surface were calculated accurately. Then the physical quantities of the internal points were computed also. Numerical examples demonstrate the present algorithm can handle the thin structures with thickness-to-length ratios down to 1. E-08, which illustrates the boundary element method is especially accurate and efficient in the analysis of problems with thin bodies.