Evaluation of nearly singular integrals in transient heat conduction boundary element method

Purpose The nearly singular integrals will occur when the interior points are closing to the boundary in the boundary element method. The conventional Gauss integral cannot effectively deal with the nearly singular integrals, which will lead to inaccurate numerical results for near-boundary interior points. Due to the determination of boundary unknown needing the information of interior points in the transient problem, the accurate numerical results for interior points are vital for the transient heat conduction boundary element method. Design/methodology/approach In this paper, the dual reciprocity method is applied to transform the time-dependent domain integrals in transient heat conduction problems into pure boundary integrals by introducing the radial basis functions. The time-dependent terms are expressed in finite difference form using a set of coefficients that correspond with the radial basis functions. The nearly singular integrals are normalized on the local coordinate system. The analytical integral formulas for nearly singular integrals in boundary integral equations are derived through the Gaussian divergence theorem and integration by parts. Findings By introducing different coefficients that match the radial basis functions, the proposed method can be applied to analyze the transient heat conduction problems with heat sources or with variable heat conductivity coefficient. Three numerical examples show that the proposed method can evaluate the transient temperature field of interior points much closer to the boundary. The proposed method eliminates the need for domain discretization, which will reduce both the difficulty of element discretization and the computational cost. Originality/value In this paper, the dual reciprocity method is applied to transform the time-dependent domain integrals in transient heat conduction problems into pure boundary integrals by introducing the radial basis functions. The time-dependent terms are expressed in finite difference form using a set of coefficients that correspond with the radial basis functions. The nearly singular integrals are normalized on the local coordinate system. The analytical integral formulas for nearly singular integrals in boundary integral equations are derived through the Gaussian divergence theorem and integration by parts.