Discrete-analytical solution of the unsteady-state heat conduction transfer problem based on the finite element method

Abstract

The paper presents a theoretical view of the discrete-analytical method to solving a heat conduction transfer problem. For variation formulation of the process is a functional on the scalar function of temperature u under given boundary conditions was used. Discrete-analytical method, allows us to turn out the mathematical formulation of the initial problem to be Cauchy problem for a linear normal system of differential equations. The main idea of this method is to combine discrete and analytical techniques. Initial problem is dividing to 2 stages: in 1st stage will be applying a discrete technique along ones directions; in 2nd stage will be applying analytic method along other directions. The result will be a discrete set of analytic functions. For “discrete stage” will be used finite element method (FEM), and for “analytical stage” is applying the theory of matrix functions, particularly the properties of matrix exponential. This approach allows us to solve the heat conduction problem with unstationary boundary conditions of different types, defined as time-dependent functions. Such modeling describes the real physical processes in structural materials more accurately.