Element differential method for solving general heat conduction problems

In this paper, a new numerical method, Element Differential Method (EDM), is proposed for solving general heat conduction problems with variable conductivity and heat source subjected to various boundary conditions. The key aspect of this method is based on the direct differentiation of shape functions of isoparametric elements used to characterize the geometry and physical variables. A set of analytical expressions for computing the first and second order partial derivatives of the shape functions with respect to global coordinates are derived, which can be directly applied to governing differential equations and boundary conditions. A new collocation method is proposed to form 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. EDM is a strong-form numerical method. It doesn't require a variational principle or a control volume to set up the computational scheme, and no integration is involved. A number of numerical examples of two- and three-dimensional problems are given to demonstrate the correctness and efficiency of the proposed method.