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New variable transformations for evaluating nearly singular integrals in 3D boundary element method. (English) Zbl 1287.65127
Summary: This work presents new variable transformations for accurate evaluation of the nearly singular integrals arising in the 3D boundary element method (BEM). The proposed method is an extension of the variable transformation method in [G. Xie et al., Eng. Anal. Bound. Elem. 35, No. 6, 811–817 (2011; Zbl 1259.65185)] for 2D BEM to 3D BEM. In this paper, first a new system denoted as $$(\alpha,\beta)$$ is introduced compared with the polar coordinate system. So the original transformations in [loc. cit.] can be developed to 3D in $$(\alpha,\beta)$$ or the polar coordinate system. Then, the new transformation is performed by four steps in case the source point coincides with the projection point or five steps otherwise. For each step, a new transformation is proposed based on the approximate distance function, so that all steps can finally be unified into a uniform formation. To perform integration on irregular elements, an adaptive integration scheme combined with the transformations is applied. Numerical examples compared with other methods are presented. The results demonstrate that our method is accurate and effective.

##### MSC:
 65N38 Boundary element methods for boundary value problems involving PDEs 65D30 Numerical integration 45E05 Integral equations with kernels of Cauchy type
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##### References:
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