Scaled Boundary Finite Element Method Coupled with Equilibrated Basis Functions for Heat Transfer Problems

Abstract

The Scaled Boundary Finite Element Method (SBFEM) discretizes only the boundary by using a technique for scaling the domain response onto its boundary. In this research, heat transfer problems in two-dimensional space are solved with a new approach based on combining the scaled boundary finite element method and the equilibrated basis functions. The SBFEM develops its relations in radial and circumferential coordinate systems, but only discretizes the boundary of the problem through development of a semi-analytical solution in radial direction. So the challenges of appropriate elemental grid for the solution domain, or the need for fundamental solutions of the equation, as usual in the finite element method or the boundary element method respectively, do not appear. In this research, after scaling the boundary in the scaled boundary finite element method and extracting the related equations, the equilibrated basis functions are used to approximate the semi-analytical solution in radial direction. After estimating the radial solution by the first kind Chebyshev polynomials, the weighted residual form of the governing equation is applied for approximately satisfaction. Finally, the unknown degrees of freedom of the boundary are derived, and there will be no need for the usual eigenvalue solution of the SBFEM. It will be shown that this approach benefits good accuracy and convergence rate.