A general algorithm for the numerical evaluation of nearly singular integrals on 3D boundary element. (English) Zbl 1219.65031

Summary: A general numerical method is proposed to compute nearly singular integrals arising in the boundary integral equations. The method provides a new implementation of the conventional distance transformation technique to make the result stable and accurate no matter where the projection point is located. The distance functions are redefined in two local coordinate systems. A new system denoted as \((\alpha ,\beta )\) is introduced here firstly. Its implementation is simpler than that of the polar system and it also performs efficiently. Then a new distance transformation is developed to remove or weaken the near singularities. To perform integration on irregular elements, an adaptive integration scheme is applied. Numerical examples are presented for both planar and curved surface elements. The results demonstrate that our method can provide accurate results even when the source point is very close to the integration element, and can keep reasonable accuracy on very irregular elements. Furthermore, the accuracy of our method is much less sensitive to the position of the projection point than the conventional method.


65D32 Numerical quadrature and cubature formulas
41A55 Approximate quadratures
65N38 Boundary element methods for boundary value problems involving PDEs
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