Page 46 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

3

The Finite Element Method

3.1 Introduction

The finite element method is a numerical tool for determining approximate solutions to
a large class of engineering problems. The method was originally developed to study the
stresses in complex air-frame structures (Clough 1960) and was later extended to the gen-
eral field of continuum mechanics (Zienkiewicz and Cheung 1965). There have been many
articles on the history of finite elements written by numerous authors with conflicting opin-
ions on the origins of the technique (Gupta and Meek 1996; Oden 1996; Zienkiewicz 1996).
The finite element method is receiving considerable attention in engineering education and
in industry because of its diversity and flexibility as an analysis tool. It is often necessary
to obtain approximate numerical solutions for complex industrial problems, in which exact
closed-form solutions are difficult to obtain. An example of such a complex situation can be
found in the cooling of electronic equipment (or chips). Also, the dispersion of pollutants
during non-uniform atmospheric conditions, metal wall temperatures in the case of gas
turbine blades in which the inlet gas temperatures exceed the melting point of the material
of the blade, cooling problems in electrical motors, various phase-change problems, and so
on, are a few examples of such complex problems. Although it is possible to derive the
governing equations and boundary conditions from first principles, it is difficult to obtain
any form of analytical solution to such problems. The complexity is due to the fact that
either the geometry, or some other feature of the problem, is irregular or arbitrary. Analyt-
ical solutions rarely exist; yet these are the kinds of problems that engineers and scientists
solve on a day-to-day basis.
Among the various numerical methods that have evolved over the years, the most com-
monly used techniques are the finite difference, finite volume and finite element methods.
The finite difference is a well-established and conceptually simple method that requires a
point-wise approximation to the governing equations. The model, formed by writing the
difference equations for an array of grid points, can be improved by increasing the number
of points. Although many heat transfer problems may be solved using the finite difference

Fundamentals of the Finite Element Method for Heat and Fluid Flow R. W. Lewis, P. Nithiarasu and K. N. Seetharamu
 2004 John Wiley & Sons, Ltd ISBNs: 0-470-84788-3 (HB); 0-470-84789-1 (PB)