Analytical evaluation of nearly singular integrals in the heat conduction boundary integral equations with heat source

Purpose There are domain integrals in the integral equations for analyzing the heat conduction with a heat source. Design/methodology/approach In order to highlight the advantages of the boundary element method, the dual reciprocity method is introduced to transform the domain integral into the boundary integral through the special interpolation function. When the internal points near the boundary are selected as the nodes for constructing the approximating functions in the dual reciprocity method, the nearly singular integrals will occur in calculating the physical quantities of these internal points. The nearly singular integrals cannot be effectively evaluated by the Gaussian numerical integration, which will reduce the calculation accuracy of the temperature and heat flux at the internal points near the boundary. Herein, a set of analytical formulas is proposed to evaluate the nearly singular integrals in the boundary integral equations for the heat conduction problem with a heat source. Findings Thus, the accurate temperature and heat flux at the internal points closer to the boundary can be evaluated by comparing with the conventional boundary element method. The characteristic of the “boundary only” of the boundary element method is maintained for analyzing the temperature field in the heat conduction problem with a heat source. Originality/value The boundary-domain integral equation for the heat conduction problem with a heat source is transformed into the boundary integral equation by the dual reciprocity method. Then, the analytical algorithm is introduced to evaluate the nearly singular integral in the heat conduction boundary integral equations with a heat source. The present method retains the “boundary only” characteristic of the boundary element method and effectively evaluates the nearly singular integrals when the internal point is close to the boundary.