A New Formulation of the Scaled Boundary Finite Element Method for Heterogeneous Media: Application to Heat Transfer Problems

The solution to heat transfer problems in two-dimensional heterogeneous media is attended based on the scaled boundary finite element method (SBFEM) coupled with equilibrated basis functions (EqBFs). The SBFEM reduces the model order by scaling the boundary solution onto the inner element. To this end, tri-lateral elements are emanated from a scaling center, followed by the development of a semi-analytical solution along the radial direction and a finite element solution along the circumferential/boundary direction. The discretization is thus limited to the boundaries of the model, and the semi-analytical radial solution is found through the solution of an eigenvalue problem, which restricts the methods’ applicability to heterogeneous media. In this research, we first extracted the SBFEM formulation considering the heterogeneity of the media. Then, we replaced the semi-analytical radial solution with the EqBFs and removed the eigenvalue solution step from the SBFEM. The varying coefficients of the partial differential equation (PDE) resulting from the heterogeneity of the media are replaced by a finite series in the radial and circumferential directions of the element. A weighted residual approach is applied to the radial equation. The equilibrated radial solution series is used in the new formulation of the SBFEM.