Evaluation Of High Order Singular Boundary Integrals In BEM And Its Application In Multi-medium Mechanical Problems

Boundary element method (BEM) based on boundary integral equation (BIE) is developed as a comparatively accurate and efficient numerical method. It has many advantages such as easily simulating complex boundary shape, high computational accuracy and dimension reduction. In addition, it has particular advantages in solving infinite field, stress concentration, and crack problems. In BEM, the evaluation of singular integrations is a challenge task, which influence the accuracy of the results and the efficiency of the analysis procedure. Some existing methods are effective in solving weak and strong singular integrals, but unstable in solving hyper or super line/surface singular integrals.In this thesis, beginning with the definition of Cauchy Principal Value (CPV) integral and Hadamard finite-part (HFP) integral, the limiting process of singular boundary integral equations and the theory of eliminating of infinite divergent items are described, and then it will be shown that the integral form of the free items can regulate the singular integral equations.A novel direct method for numerical evaluation of all kinds of singular curved boundary integrals in2D/3D BEM analysis is proposed based on a global size projection method. Firstly, geometry variables defined on a curved line/surface element are expressed in terms of parameters on the projection line/plane. Then, for2D problems, after the nonsingular part of the integrand is expanded as power series in local distance in projection line, various singularities are removed analytically on the projection line; for3D problems, the Radical Integral Method (RIM)[1]is employed to transform the integral over the projection plane to a closed line integral over the counter of the projection plane, and then singularities are removed through power series expansion and analytical manipulation in a similar process to the2D problems. Compared to existing methods for removing singularities on the parameter plane of intrinsic coordinates, the new proposed method performs the manipulation utterly in the real spatial scale, and therefore the operation is straightforward and convenient, and the developed method can be applied to treat arbitrary high order of singular integrals with different integrands.Classic approach for solving multi-medium mechanical problems is MDBEM[2,3],which may result in large scale systems of equations, and low computational efficiency. The interface integral BEM (IIBEM)[4] derived from the boundary-domain integral equations for single medium elasticity problems is a method to use a single equation to solve multi-medium mechanics problems. IIBEM has the advantage of easy coding, but the existence of hyper-singular integrals in the stress interface integral equations, make it very difficult to evaluate the stresses for points located on interfaces. In the last chapter of the thesis, a method will be presented for computing interfacial stresses through the combination of the newly proposed singular integral method and interface integral equation method.