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A novel semi-analytical algorithm of nearly singular integrals on higher order elements in two dimensional BEM

机译：二维BEM中高阶元素近奇异积分的一种新型半解析算法

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摘要

In this paper, a novel semi-analytical algorithm is developed to evaluate the nearly strong and hyper-singular integrals on higher order elements in two dimensional (2-D) BEM. By analyzing the geometrical feature of higher order elements in the intrinsic coordinates, the relative distance from a source point to the element of integration is defined to describe the character of the nearly singular integrals. By a series of deduction, the leading singular part of the integral kernel functions on the higher order elements is separated from each of the nearly singular integrals. Then the nearly singular integrals on the higher order elements close to the source point are transformed to the sum of both the non-singular parts and nearly singular parts by the subtraction, in which the former are calculated by the conventional numerical quadratures and the latter are evaluated by the resulting analytical formulations. Furthermore, the BEM with the quadratic elements was used to analyze the displacements and stresses near the boundary as well as thin-walled structures in 2-D elasticity. The numerical results from three examples demonstrate that the quadratic BE analysis with the semi-analytical algorithm is more accurate and efficient than the Linear BE analysis with the analytical algorithm for the nearly singular integrals. In fact, the Linear BE analysis has been greatly more advantageous compared with the finite element analysis for the thin-walled structures.
机译：在本文中，开发了一种新颖的半解析算法来评估二维（2-D）BEM中高阶元素上的近强和超奇异积分。通过分析本征坐标中高阶元素的几何特征，定义了从源点到积分元素的相对距离，以描述近似奇异积分的特征。通过一系列的推论，将高阶元素上积分核函数的前导奇异部分与每个几乎奇异的积分分开。然后，通过减法将接近源点的高阶元素上的近似奇异积分转换为非奇异部分和近似奇异部分的总和，其中前者由常规数值正交计算而后者为通过所得的分析配方进行评估。此外，具有二次元的边界元法被用于分析边界附近的位移和应力以及二维弹性中的薄壁结构。三个例子的数值结果表明，用半解析算法进行二次BE分析比使用近似算法的线性BE分析对奇异积分更为准确和高效。实际上，与薄壁结构的有限元分析相比，线性BE分析具有更大的优势。

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