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Analytic formulations for calculating nearly singular integrals in two-dimensional BEM

机译:二维BEM中几乎奇异积分的解析公式

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There exist the nearly singular integrals in the boundary integral equations when a source point is close to an integration element but not on the element, such as the field problems with thin domains. In this paper, the analytic formulations are achieved to calculate the nearly weakly singular, strongly singular and hyper-singular integrals on the straight elements for the two-dimensional (2D) boundary element methods (BEM). The algorithm is performed after the BIE are discretized by a set of boundary elements. The singular factor, which is expressed by the minimum relative distance from the source point to the closer element, is separated from the nearly singular integrands by the use of integration by parts. Thus, it results in exact integrations of the nearly singular integrals for the straight elements, instead of the numerical integration. The analytic algorithm is also used to calculate nearly singular integrals on the curved element by subdividing it into several linear or sub-parametric elements only when the nearly singular integrals need to be determined. The approach can achieve high accuracy in cases of the curved elements without increasing other computational efforts. As an application, the technique is employed to analyze the 2D elasticity problems, including the thin-walled structures. Some numerical results demonstrate the accuracy and effectiveness of the algorithm.
机译:当源点靠近积分元素但不在积分元素上时,边界积分方程中几乎存在奇异积分,例如,具有薄域的场问题。在本文中,为二维(2D)边界元方法(BEM),获得了解析公式来计算直线元素上的几乎弱奇异,强奇异和超奇异积分。在将BIE通过一组边界元素离散化之后执行该算法。奇异因子由从源点到近距离元素的最小相对距离表示,通过使用部分积分与几乎奇异的被积分数分开。因此,它导致直线元素的近似奇异积分的精确积分,而不是数值积分。解析算法还用于仅在需要确定近似奇异积分时,通过将其细分为几个线性或子参数元素,来计算弯曲元素上的近似奇异积分。该方法在弯曲元件的情况下可以实现高精度,而无需增加其他计算工作。作为一种应用,该技术被用来分析二维弹性问题,包括薄壁结构。一些数值结果证明了该算法的准确性和有效性。

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