Regularization Of Nearly Singular Integrals In The BEM And Its Application To Thin-body Problems

The application range of boundary element method (BEM) is reduced in engineering for a long time due to the difficulty of evaluation of the nearly singular integrals. For most of the existing numerical methods, the geometry of the boundary element is often depicted by using linear shape functions when nearly singular integrals need to be calculated. Nevertheless, most engineering processes occur mostly in complex geometrical domains, and obviously, higher order geometry elements are expected to be more accurate. However, the forms of the integrands under high-order geometry elements are very complex, and for a long time, it was considered a difficult problem to be solved in the BEM. For the study of this problem, there are only a small amount of works can be found in BEM literatures, and therefore this problem need to be further investigated.The objective of the present work is to perform a numerical analysis of the nearly singular integrals. The essence of the nearly singular integrals occurring in the BEM is analyzed. Two regularized algorithms, named the nonlinear transformation and the exact integration method, are introduced in this paper. Numerous numerical examples demonstrate that the proposed regularized algorithms will be more general and accurate than the previous methods. More importantly, the methods proposed in this study are suitable for not only the linear elements, but also the high-order elements. Owing to the employment of the high-order elements, high accuracy can be achieved without increasing more computational efforts, which make it possible to deal with ultra-thin structures very efficiently. In addition, the difficult problem for calculating the nearly singular integrals under high-order geometry elements using analytical method are solved in this study, and to the author抯best knowledge, no similar work has yet been found in BEM literatures.In this work, the regularized algorithms are applied to deal with thin-body problems occurring in 2D potential and elastic problems. Generally speaking, the thickness of the thin-body structures is in the orders of 1E-06 to 1E-09. Thus, numerical analysis of the behavior of these structures represents a great challenge to researchers in engineering applications due to the small size of its thickness. In this work, the unknown physical quantities on boundary nodes are accurately computed at first, and then the physical quantities at interior points are well studied. The new algorithms can solve problems whose thickness-to-length ratios are smaller compared with the conventional method. The numerical examples demonstrate the high efficiency and stability of the suggested approaches, even when the thickness-to-length is as small as 1E-09.The proposed regularized algorithms are further extended to treat the (multi-)coating systems. The boundary element analysis of the temperature and stress field in coating structures is well studied. The coating structure is divided into the substrate domain and the coating one by using a multi-domain boundary element approach, and then the regularized algorithms have been introduced to remove or damp out the near singularities of the kernel integrals in the coating domain.In summary, the regularized algorithms of the treatment of the nearly singular integrals are systematically studied in this dissertation. The difficult task of calculating the nearly singular integrals in thin body problems can be overcame successfully. Thereafter, it makes that the application of the boundary element method in engineering is widened.