A new multi-level strategy of numerical integration in the fast multipole BEM for analyzing 3D potential problems

A multi-level strategy of numerical integration, incorporating Hammer-type quadrature, adaptive Gauss quadrature, and the improved semi-analytic method, is developed to evaluate the boundary integrals on triangular elements for three dimensional potential problems. The boundary integrals are classified into three categories according to the relative distance between the source point and the integral element. Hammer-type quadrature is used to evaluate the integrals on the element that is far from the source point, while adaptive..point Gauss quadrature is proposed to evaluate the integrals on the element that is closer to the source point, the number of quadrature points varies with the change of the relative distance. The improved semi-analytic method is proposed for the nearly singular integral on the element that is very close to the source point, where the conformal and sigmoidal transformations are introduced to mitigate the adverse effects caused by the distorted shape of the element. The effective ranges of Hammer-type quadrature and adaptive Gauss quadrature are investigated and a reasonable scheme for their activation is provided. The present multi-level strategy exhibits high accuracy, robustness, and efficiency for the integrals with 1/r, 1/r(3), and 1/r(5). It makes the fast multipole boundary element method more competitive and applicable in the analysis of large complex structures.