Smoothed finite element method with exact solutions in heat transfer problems

Abstract

Node-based smoothed finite element method (NS-FEM) with the triangular elements in 2D and tetrahedral elements in 3D has been found capable to produce upper bound solutions in terms of equivalent energy for heat transfer problems attributable to its monotonic ‘softened’ behavior. In this paper, a hybrid smoothed finite element method (HS-FEM) which combines the temperature gradient of NS-FEM and FEM is further extended to solve heat transfer problems. A parameter α that controls the weight of NS-FEM and FEM is equipped into HS-FEM to ensure the stability and accuracy. The theoretical analysis has proved that the exact equivalent energy in heat transfer problem obtained from HS-FEM lies in between those from the compatible FEM solution and the NS-FEM solution. The numerical results for 2D with triangular elements and 3D with tetrahedral elements confirm that the present method provides the exact solution in terms of equivalent energy using very coarse mesh. In addition, the accuracy of temperature distribution in HS-FEM model is much more accurate compared with the standard FEM with the same number of degrees of freedom.