Element differential method for solving transient heat conduction problems

In this paper, a new numerical method, Element Differential Method (EDM), is developed for solving transient heat conduction problems with variable conductivity. The key point of this method is based on the direct differentiation of shape functions of isoparametric elements used to evaluate the geometry and physical variables. A new collocation method is proposed for establishing the system of equations, in which the governing differential equation is collocated at nodes inside elements, and the flux equilibrium equation is collocated at interface nodes between elements and outer surface nodes of the problem. Attributed to the use of the Lagrange elements that can guarantee the variation of physical variables consistent through all elemental nodes, EDM has higher stability than the traditional collocation method. The other main characteristics of EDM are that no variational principle or a control volume are required to set up the system of equations and no integrals are included to form the coefficients of the system. Based on the implicit backward differentiation scheme, an unconditionally stable and non-oscillatory time marching solution scheme is developed for solving the time-dependent system equations. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.