Calculation of three-dimensional nearly singular boundary element integrals for steady-state heat conduction

Abstract

In this work, a novel approach is presented for three-dimensional nearly singular boundary element integrals for steady-state heat conduction. Accurate evaluation of the nearly singular integrals is an important issue in the implementation of boundary element method (BEM). In this paper, an exponential transformation is introduced to deal with the nearly singular integrals in three-dimensional BEM. First, a triangle polar coordinate system is introduced. Then, the exponential transformation is performed by five steps. For each step, a new transformation is proposed based on the distance from the source point to surface elements which is expressed as r2=O(Ak2(θ)ρ2+r02), and all steps can finally be unified into a uniform formation. Moreover, to perform integrations on irregular elements, an adaptive integration scheme considering both the element shape and the projection point associated with the proposed transformation is introduced. Numerical examples are presented to verify the proposed method. Results demonstrate the accuracy and efficiency of our method.