Analyze the singularity of the heat flux with a singular boundary element

Abstract In the steady state heat conduction problem, the heat flux could be infinite at the re-entrant corner, at the point where the boundary condition is discontinuous, or at the place where the material properties are changing abruptly. The conventional numerical methods, such as the finite element method (FEM) and the boundary element method (BEM), have difficulties in analyzing the singular heat flux field. Herein, a new singular element employed in the boundary integral equation is developed to interpolate the heat flux field near the singular point. The shape function of the singular element is set as an asymptotic expansion model with respect to the distance from the singular point. The singular order in the asymptotic expansion can be determined by solving the singularity eigen equation on the basis of the interpolating matrix method. The self-adaptive co-ordinate transformation method is introduced to deal with the weakly singular integral in the proposed method. Benefited from the singular element, more accurate temperature and heat flux results near the singular point are obtained with fewer elements. In addition, the proposed method is easy to be coupled with the conventional BEM without too much modifications.