A new semi-analytic algorithm of nearly singular integrals on higher order element in 3D potential BEM☆

Abstract By analyzing the geometric characteristics of 8-noded quadrilateral surface elements in three dimensional boundary element method (3D BEM), the relative distance from a source point to the integral element is defined. For the nearly singular integrals on higher order elements in 3D potential BEM, the equivalent integral kernels are constructed by the geometric analysis between the source point and the element in ρ θ system. Subtracting the equivalent kernels from and adding them back to the nearly singular kernels, the nearly singular surface integrals are transformed into the sum of both the non-singular integrals and the singular integrals. So the leading singular parts are separated. The former are computed efficiently by the Gaussian quadrature and the latter are performed with respect to the integral variables ρ and θ , respectively, in which the integrations with respect to ρ are expressed by analytical formulations. Consequently, a new semi-analytic algorithm is established to calculate the nearly strongly singular and hyper-singular surface integrals on higher order element in 3D BEM. Several examples about 3D heat conduction are given to demonstrate the efficiency and accuracy of the present semi-analytic algorithm in BE analysis. Moreover, the present algorithm is used to analyze very thin structures in 3D BEM.