The Thermal-Mass-Redistributed Finite Element Method Combined with the Houbolt Method for 2D Transient Heat Conduction Problems

Abstract

The paper proposes a finite element method with high precision and convergence for solving 2D transient heat conduction problems. The temperature gradient with respect to time is approximated using the Houbolt method, a third-order difference scheme, which improves the accuracy and convergence rate and permits the use of large time steps for long-term prediction. The proposed finite element method redistributes the thermal mass matrix to balance it with the thermal stiffness matrix, further enhancing the precision of solutions. Besides, the effective condition number is introduced to determine the position of optimal flexible integration points, ensuring that the error of the solutions is minimal. Finally, intensive transient heat conduction problems are analyzed using the present method to demonstrate good performance in precision and convergence. Compared to FEM using the backward finite difference method, the proposed method achieves more accurate and super convergent temperature solutions. In comparison to FEM with the forward finite difference method, it can use larger time steps while maintaining the same level of accuracy.