A semi-analytical expression for calculating conductivity matrices of finite element in heat conduction problems

Abstract

The heat transfer equation by conduction is not more than a mathematical expression of the energy conservation law for a given solid. Solving the equation which model this problem is generally very dificult or impossible in an analytical way, so it is necessary to make a discrete approximation of the continuous problem. In this paper, we present a methodology applied to the quadrilateral finite elements in problems of heat transfer by conduction, where the components of the thermal conductivity matrix are obtained by a semi-analytical expression and simple algebraic manipulations. This technique has been used successfully in stiffness arrays' integrations of bidimensional and tridimensional finite elements, reporting substantial improvements of CPU times compared with the Gaussian integration.