三维位势问题新的规则化边界元法

Abstract

This presentation is mainly devoted to the research on the regularization of indirect boundary integral equations (IBIEs) for three-dimensional problems and establishes the new theory and method of the regularized BEM. The two special tangential vectors, which are linearly independent and associated with the normal vectors, are constructed, and then a characteristics theorem for the contour integrations of the normal and tangential gradients of some quantities, related with the fundamental solutions for 3D potential problems, is presented. A limit theorem for the transformation from domain integral equations into boundary integral equations (BIEs) is also proposed. Based on this, together with a novel decomposition technique to the fundamental solution, the regularized BIEs with indirect unknowns, which don’t involve the direct calculation of CPV and HFP integrals, are derived for 3D potential problems. Compared with the widely practiced direct regularized BEMs, the presented method has many advantages. First, the continuity requirement for density function in the direct formulation can be reduced here. Second, it is more suitable for solving the structures of thin bodies, considering the solution process for boundary or field quantities doesn’t involve the HFP integrals and nearly HFP integrals so the regularization algorithm to the considered singular or nearly singular integrals is more effective. Third, the proposed regularized BIEs can calculate the any potential gradients on the boundary, but not limited to the normal fluxes, and also independent of the potential BIEs. A systematic approach for implementing numerical solutions is proposed by adopting the ℃ continuous elements to depict the boundary surface and the discontinuous interpolation to approximate the boundary quantities. Especially, for the boundary value problems with elliptic surfaces or piecewise plane surfaces boundary, the exact elements are developed to model their boundaries with almost no error. The validity of the proposed scheme is demonstrated by several benchmark examples. Excellent agreement between the numerical results and exact solutions is obtained even with using small amounts of element.