Research On Computational Methods Of Singular Integrals Over Narrow Strip Elements In Boundary Element Analysis Of Elastic Thin Plate

As modern engineering applications require higher performance of structures, structure lightening has become one of the most necessary indexes. Under this circumstance, the application of thin-walled structure is more and more widespread. Compared with Finite Element Method, Boundary Element Method (BEM) has an advantage in analyzing thin-walled structure problems that only the boundary of the computational domain needs to be discretized. However, due to the special geometric shape of the thin-walled structures, the BEM analysis of such structures is one of the most difficult tasks, among which the nearly singular boundary integral is a key issue. Domestic and international scholars have been focusing on the accurate calculation of nearly-singular integrals for a long time, and a number of computational methods on the treatment of nearly singular integrals have been proposed, such as the virtual boundary element method, the regularization method, and the interval segmentation method. However, these methods still can't get satisfying results for some cases. In fact, apart from the need of treating nearly singular integrals occurred when an element under integration is located on the opposite side of the source point, a special integration scheme is also needed for treating the singular integrals over the flank elements when the source point is located on these elements.In this thesis, 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 firstly 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 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. 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.At the end, 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.