The Regularized Boundary Element Method For The 3D Problems

The boundary element method (BEM), which has the advantages of high accuracy and reducing the dimension of the solving problem, is an important numerical algorithm applied to various engineering problems. However, unlike the domain-type numerical method, it is required to perform the singular integrals in the BEM, thus the key for the analysis of BEM is the accuracy evaluation of these integrals.The wide practice had been focused on the research of direct regularized BEM, but in this paper, indirect regularized one is studied. Based on the theory of 2D indirect regularized BEM, the new regularized method of 3D problems is proposed. Compared with direct regularized BEM, the proposed method has many advantages:a) the continuity requirements of density functions are reduced, i.e. it is only required the density function belongs to C0,a but not C1,a; b) It has advantage for solving the thin body problems because the solution process doesn’t involve the HFP integrals and nearly HFP integrals, so regularize computation of integrals is more precise and effective; c) the proposed regularized BIEs can be used for the computation of any potential gradients on the boundary, and but not limited to the normal gradients. Moreover, they are independent of the potential BIEs.In the first chapter, the development status of BEM, particularly for the algorithms for dealing with the singular and domain integrals, is summarized.The basic theory and method of regularized BEM for 3D problems is discussed in the second chapter. Two special tangential vectors that are linearly independent and associated with the normal vectors are constructed firstly. Then, characteristics theorems about contour integrations of normal and tangential gradients, related with the fundamental solutions of governing differential equations, are established. Meanwhile, a limit theorem for the transformation from domain integral equations into boundary integral equations (BIEs) is proposed. At last, Continuous and discontinuous quadratic elements, which used generally in the numerical calculation, are introduced in this chapter. Besides, for the special boundary surfaces such as spherical (ellipsoidal) surface, an isogeometric exact element is developed.In the third and fourth chapters, the indirect regularized BIEs of 3D potential and elastic problems are developed respectively. The indirect regularized ones of 2D and 3D thermoelastic problems are presented in the fifth chapter, in which the domain integrals of temperature are transformed into boundary integrals by the radial integration method (RIM).Numerical examples show that the present method is stability and efficiency, and numerical results are in good agreement with the exact solutions.