Treatment Of Singularities In Boundary Integral Equation And Its Application In Fracture Mechanics

CAE analysis technology plays an important role in the machinery industry. The mainstream numerical method of CAE analysis technology is finite element method (FEM). However, there are some inherent flaws with FEM and these flaws can be avoided by the boundary integral equation method (BIEM). In the numerical implementation of the BIEM, nearly singular integrals and singular integrals are important factors which affect the accuracy of BIEM. Thus this work focuses on the nearly singular integrals and singular integrals which arise in the BIEM and their applications in thin structures and fracture mechanics. Moreover, in order to broaden the application of BIEM, this work also includes a series of treatment for nearly singular and singular volume integrals. Treating these core contents, the work of this paper is as follows:(1) This paper presents new transformations for nearly singular integrals in two and three dimensional BIEM. Compared with the traditional methods for nearly singular integrals, the proposed transformations focus on the core issues of nearly singular integrals in nature, namely, the location of the projection point and distance function. In this paper, employing the Taylor expansion, the distance function is obtained. Considering the location of the projection point and the distance function, the nearly singular integrals are divided into three categories. Using the distance function and the idea to lower the gradient of integrands, three different transformations for nearly singular integrals are constructed. And different from that in two dimensional BIEM, a new coordinate system is introduced and new transformation is simpler in the new system. Moreover, due to the uncertainty of the location of the projection point, another nearest point is introduced. An element subdivision technique based on the projection point and the nearest point is developed. With the help of element subdivision technique, subelements of fine shape are obtained, which can improve the integration accuracy. Numerical examples demonstrate that the proposed method can be successfully applied for problems on the thin structures.(2) A comprehensive and systematic analysis of singular integrals in BIEM. Directly from the definition of the Cauchy principal value (C.P.V) and Hadamard finite part integrals (H.F.P), the local coordinate approximate expansion is employed to analyze the nature of weakly singular integrals, strong singular integrals, and hypersingular integrals. And the corresponding treatment is obtained. To meet the symmetry requirement of integration interval for C.P.V and H.F.P, an adaptive subdivision technique is developed to deal with singular integrals in three dimensional BIEM, which can efficiently improve the integration accuracy. These methods are successfully applied for two and three dimensional fracture mechanics.(3) BIEM for two and three dimensional fracture mechanics. For two dimensional fracture mechanics, crack opening displacements (CODs) are introduced. Using the nature of the fundamental solutions and the traction equilibrium boundary conditions of crack faces, the improved dual boundary integral equations can be obtained. The dual boundary integral equations are collocated only on one of the crack faces and the external boundary, thus reducing the matrix size and more efficient. Furthermore, for three dimensional fracture mechanics, only the traction boundary integral equation is needed to collocate on one of the crack faces and the external boundary. Thus the advantages for two dimensional fracture mechanics are retained. Moreover, it is easy for code programming to deal with singular integrals. To capture the nature of the displacement of the crack tips, special crack-front elements are proposed. Combined with the asymptotic properties of the crack tip displacement, crack tip stress intensity factors are obtained by the linear interpolation formula of the crack opening displacements. The proposed method can be successfully applied for two and three dimensional fracture mechanics. Numerical results are in good agreement with existing analytical solutions or numerical results.(4) Reasonable schemes for nearly singular and singular domain integrals. To ensure the universality of BIEM, nearly singular and singular domain integrals techniques are developed for the three common domain cells. An adaptive cell subdivision which is based on the proportion between the distance from the source point to the domain cell and the size of domain cell for nearly singular domain integrals is proposed. For singular integrals, the domain cell is divided into several cone and pyramid subcells. Singular integrals on these subcells are regularized by transformations. The proposed method is successfully applied for the domain integrals. Numerical examples demonstrate the effectiveness of our method.