Solving Steady-State Heat Conduction Problems by Means of Numerical Methods

Abstract

This chapter is devoted to numerical methods, which are used to determine steady-state temperature fields. It contains detailed description of the following numerical methods: finite-difference method, finite-volume method (control volume), finite element method (FEM) and pseudo-transient method for solving stationary problems, based on the method of lines. Linear and non-linear problems, both simple and inverse, are solved here. Specific computational programs are developed for determining steady-state temperature fields, while Gauss elimination method, Gauss-Seidel iterative method or over-relaxation method are applied to integrate an algebraic equation system. Ordinary differential equation system in the pseudo-transient method is solved using Rung-Kutta method of 4th order. Finite element method, based on Galerkin method, is discussed in great detail, as well as the two methods for creating global equation system in FEM. Basic matrixes and vectors, which occur in FEM for one-dimensional and two-dimensional triangular and rectangular elements, are also developed. Furthermore, authors present their own solutions to FEM problems. The obtained results are compared with analytical solutions or the solutions acquired by means of finite volume method. The application of the ANSYS program is presented in Ex. 11.20, 11.21 and 11.22.