Abstract: By analyzing the geometric feature of 8-noded quadrilateral element in three dimensional boundary element method (3D BEM), the relative distance is first defined as the approach degree from a source point to the high order surface element. And then a local polar coordinate ρθ is built which origin point is the project point of the source point on the element surface. The approximate singular kernel function is constructed corresponding to the nearly singular integral on high order surface elements in 3D potential BEM by a series of deduction, which has the same singularity as the nearly singular kernel function. The leading singular part is separated by subtracting the approximate kernel function from the original kernel function. Thus the nearly singular surface integrals on high order elements are transformed into the sum of both the non-singular integrals and singular integrals. The former can be efficiently computed by the Gaussian quadrature. The integral variables ρ and θ of the later are separated in the local polar coordinate. The singular surface integrals with respect to polar variable ρ are firstly expressed by the analytic formulations. Then the surface integrals are transformed into the line integrals with respect to variable θ, which can be evaluated by the Gaussian quadrature. Consequently, the new semi-analytic algorithm is established to calculate the nearly strongly and hyper-singular surface integrals on high order element in 3D potential BEM. Some numerical examples about the high order BE analysis for 3D heat conduction problems are given to demonstrate the efficiency and accuracy of the present semi-analytic algorithm. In comparison with the published regularization algorithm which is applied to calculating the nearly singular integrals on 3-noded triangular element, the present semi-analytic algorithm with 8-noded quadrilateral element can evaluate the potentials and potential gradients of inner points more close to the boundary. Moreover, the semi-analytic algorithm can be applied to more efficiently analyze thin structures in 3D potentials.

Key words: 3D|potential|BEM|high order element|nearly singular integrals|semi-analytic method