This research targets investigation of the internal elastic field near the boundaries of 3D anisotropic bodies by the boundary element method (BEM). This analysis appears to be most important, especially for the interest in the internal solutions near corners or crack tips, where structural failure is most likely to occur. Although the BEM is very efficient for analyzing the problem, nearly singular integration will arise when the internal point of interest stays near the boundary. The present work is to show how the self-regularized boundary integral equation (BIE) can be applied to the interior analysis for 3D generally anisotropic bodies. So far, to the authors' best knowledge, no implementation work has been reported in the literature for successfully treating the near boundary solutions in 3D generally anisotropic solids. In the end, a few benchmark examples are presented to demonstrate the veracity of the present approach for the interior BEM analysis of 3D anisotropic elasticity.

We present a simple, accurate method for computing singular or nearly singular integrals on a smooth, closed surface, such as layer potentials for harmonic functions evaluated at points on or near the surface. The integral is computed with a regularized kernel and corrections are added for regularization and discretization, which are found from analysis near the singular point. The surface integrals are computed from a new quadrature rule using surface points which project onto grid points in coordinate planes. The method does not require coordinate charts on the surface or special treatment of the singularity other than the corrections. The accuracy is about O(h3), where h is the spacing in the background grid, uniformly with respect to the point of evaluation, on or near the surface. Improved accuracy is obtained for points on the surface. The treecode of Duan and Krasny for Ewald summation is used to perform sums. Numerical examples are presented with a variety of surfaces.

The natural boundary integral equation (NBIE) is developed to calculate potential derivatives for potential problems with anisotropic media. Firstly, the governing equation of the two-dimensional anisotropic potential problem is transformed into standard Laplace equation by a coordinate transformation method. Then a potential derivative boundary integral equation named as NBIE is extended to solve the anisotropic potential problem. The most important virtue of the NBIE is that the singularity of the integral kernel function is reduced by one order in comparison with the conventional potential derivative boundary integral equation(CDBIE). Therefore the new potential derivative boundary integral equation only contains strongly singular integrals rather than hyper-singular integrals. Thus the NBIE can calculate more accurate potential derivative results for both boundary nodes and interior points. Moreover, in combination with the analytical integral regularization algorithm of nearly singular integrals, the NBIE can obtain more accurate potential derivatives of interior points very close to the boundary than the CDBIE. Numerical examples on heat conduction in anisotropic media demonstrate the accuracy and efficiency of the NBIE.

A straightforward method is presented for computing three-dimensional Stokes flow, due to forces on a surface, with high accuracy at points near the surface. The flow quantities are written as boundary integrals using the free-space Green’s function. To evaluate the integrals near the boundary, the singular kernels are regularized and a simple quadrature is applied in coordinate charts. High order accuracy is obtained by adding special corrections for the regularization and discretization errors, derived here using local asymptotic analysis. Numerical tests demonstrate the uniform convergence rates of the method.