The Dual Reciprocity Boundary Element Method Applied To A Classical Transient Heat Conduction Problem

Abstract

The Dual Reciprocity Method (DRM) is the most powerful and versatile technique which has appeared to date for taking domain integrals to the boundary in the boundary element analysis. In this study it was applied to the solution of a classical transient,one-dimensional heat conduction problem. The DRM results were validated by comparing with the analytical solutions. Introduction Since 1978 the Boundary Element Method (BEM) has developed considerably and been applied to solve a wide range of engineering problems, particularly those involving linear analysis. Further increase in the number of applications of the method has been hampered by the need to operate with relatively complex fundamental solutions or the difficulties encountered when trying to extend the BEM to non-linear and time dependent problems. The Dual Reciprocity Method (DRM) was proposed by Nardini and Brebbia [1] in 1982 for the solution of elastodynamic problems. It is essentially a generalized way of constructing particular solutions and it can be used to solve non-linear and time dependent problems. DRM is the most powerful and versatile technique which has appeared to date to handle domain integrals without the need to use internal cells. It will be employed to solve a classical one-space-dimension,transient heat conduction problem. A majority of the most important problems in fluid mechanics involve transport of a scalar quantity by convection,as well as diffusion. Convective effects introduce special difficulties in the finite element approximation [5]. The study of species transport crosses many disciplines,e.g. ,heat transfer and combustion processes in fluid dynamics,chemical kinetics,mass transport, pollutants in lakes and oceans,aerosol dispersion in the atmosphere,contaminant transport in groundwater flow,etc. The processes are modeled by the diffusion convection equation which can be written as Transactions on Modelling and Simulation vol 7, © 1994 WIT Press, www.witpress.com, ISSN 1743-355X 102 Boundary Element Method XVI Where u is a temperature, a concentration of a substance etc, D* and Dy are diffusion coefficients, v% and Vy are velocities, k is a decay parameter and 0, T = T,, x L, t > 0, T f(x), 0< x < L, t = 0, the solution for one-dimensional, non-steady state heat conduction in the form of a trigonometrical series is