Abstract
A new approach is presented for the numerical evaluation of arbitrary singular domain integrals. In this method, singular domain integrals are transformed into a boundary integral and a radial integral which contains singularities by using the radial integration method. The analytical elimination of singularities condensed in the radial integral formulas can be accomplished by expressing the nonsingular part of the integration kernels as a series of cubic B-spline basis functions of the distancerand using the intrinsic features of the radial integral. In the proposed method, singularities involved in the domain integrals are explicitly transformed to the boundary integrals, so no singularities exist at internal points. A few numerical examples are provided to verify the correctness and robustness of the presented method.
Highlights
When the boundary element method (BEM) [1, 2] is used to solve complicated engineering problems [3, 4], such as compressible potential flows, heat conduction with heat generation, seepage problems with sources, electromagnetic field problems with electric charge, and elastostatics with body forces, a large number of regular and/or singular domain integrals may appear in the basic integral equations [2]
The most extensively used technique is the dual reciprocity method (DRM) [12], which transforms the domain integrals to the boundary by approximating the given body force effect quantities as a series of prescribed basis functions and employing particular solutions corresponding to these basis functions
As an extension of the idea of DRM, the multiple reciprocity method (MRM) [13, 14] was presented by Nowak and Brebbia [13] and the triple-reciprocity method was developed by Ochiai and Sekiya [15]
Summary
When the boundary element method (BEM) [1, 2] is used to solve complicated engineering problems [3, 4], such as compressible potential flows, heat conduction with heat generation, seepage problems with sources, electromagnetic field problems with electric charge, and elastostatics with body forces, a large number of regular and/or singular domain integrals may appear in the basic integral equations [2]. An efficient approach was presented by Gao and Peng [7] for evaluating arbitrarily high-order singular domain integrals in a unified way based on the radial integration method. The radial integral can be integrated analytically by expressing the nonsingular part of integrand as polynomials of the global distance r This approach can deal with any type of regular, weakly, and high-order singular domain integrals. The nonsingular part of the integrand involved in the domain integral is expressed as a series of third-degree B-spline basis functions [19] of the distance r. A number of singular domain integrals are given to verify the correctness and validity of the presented method
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