A. Introduction

(a) Preamble

The development of finite-element methods (FEMs) for the solution of conduction heat transfer problems is relatively recent in comparison with their finite-difference method counterparts. The FEM development has followed a philosophically different approach from that followed for finite-difference methods, although both are based on a volumetric discretization of the solution domain. One of the prime motivating factors leading to the development of finite-element methods has been the desire to enhance the compatibility of thermal models for use in thermal stress analysis. In stress analysis, there exists a variational extremum principle such that the minimization of the strain energy within a continuum domain leads naturally to the formulation of a discrete model to determine an approximate displacement field from which the approximate stress distribution can be calculated. An additional motivation for the development of finite-element methods for conduction heat transfer has been the desire to achieve geometric independence of the resulting procedure. Thus, whereas a finite-difference code can be constructed rapidly for the analysis of a Cartesian domain of rectangular configuration, the extension of such a code to arbitrary configurations is painful and complex. In contrast, the construction of a finite-element code entails separate consideration of many different aspects of the discrete problem formulation. Once constructed, however, extension of the code to consider geometries of relatively arbitrary configuration is straightforward.

In conduction heat transfer, as opposed to stress analysis, a natural integral formulation having a clear physical interpretation docs not exist. The search for a variational formulation of the heat conduction problem, apparently in the interest of preserving the underlying principles leading to the development of the method, has, however, led to several variational formulations of the problem. The approach adopted by Visser (1965), Zienkiewicz and Parekh (1970), and Zienkiewicz (1971) considers a particular instant in time such that time derivatives of temperature and of physical parameters can be treated as prescribed functions of the spatial coordinate. In this way, a quasi-variational statement can be formulated and converted to a restricted variational statement as provided by Finlayson and Scriven (1967). Biot (1955 and 1956) has also provided quasi-variational statements for the heat conduction problem. These statements for the conduction problem are in contrast to the true variational statement given by Gurtin (1964) through the use of convolution integrals in time. The true variational approach has been used by Wilson and Nickell (1966) for analysis in a Cartesian coordinate frame.

In contrast to the variational approach, the method of weighted residuals (MWR) Finlayson (1972) is being used with increasing frequency to formulate finite-element models for conduction heat transfer problems (Finlayson and Scriven, 1967; Warzec, 1970; Yalamanchili and Chu, 1973). In particular, the Galerkin method of weighted residuals has received the greatest amount of attention. Finlayson and Scriven (1967) state that while both the quasi-variational and restricted variational approaches retain some variational formalism, and they refer to these methods as ad hoc variational principles, the variational method of approximation to which they lead is indistinguishable from the Galerkin method applied to the original equations.