Element-subdivision method for evaluation of singular integrals over narrow strip boundary elements of super thin and slender structures

In this paper, based on the numerical investigation of singular integrals over narrow strip boundary elements stemming from BEM analysis of thin and slender structures with different numbers of Gauss points, an efficient method is proposed for evaluating the narrow strip singular boundary integrals using an adaptive unequal interval element-subdivision method in the intrinsic parameter plane. In this method, the size of the sub-element closest to the singular point is determined first in terms of the orders of the shape functions along two intrinsic coordinate directions. Then, the sizes of other sub-elements are computed by employing a criterion proposed by Gao and Davies [1] and [2] for evaluating nearly singular integrals in terms of an allowed number of Gauss points and the distance from the source point to the sub-element. The features of the proposed method are that the computational accuracy of various orders of singular integrals is controlled by the upper bound of the error of Gauss quadrature, rather than through artificially giving the size of the sub-elements and number of Gauss points, and because of using the unequal interval element-subdivision method, the number of required sub-elements is not large even for an element with high aspect ratio, usually less than 10 for a plate with aspect ratio of 100:1. A number of numerical examples for plates and shells with different aspect ratios are analyzed for various orders of integrals to demonstrate the efficiency of the proposed method.