Page 26 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

2

Some Basic Discrete Systems

2.1 Introduction

Many engineering systems may be simplified by subdividing them into components or
elements. These elements can readily be analysed from first principles, and by assembling
these together, the analysis of a full original system can be reconstructed. We refer to
such systems as discrete systems. In a large number of situations, a reasonably adequate
model can be obtained using a finite number of well-defined components. This chapter
discusses the application of such techniques for the formulation of certain heat and fluid
flow problems. The problems presented here provide a valuable basis for the discussion of
the finite element method (Bathe 1982; Huebner and Thornton 1982; Hughes 2000; Reddy
1993; Segerlind 1984; Zienkiewicz and Taylor 2000), which is presented in subsequent
chapters.
In the analysis of a discrete system, the actual system response is described directly
by the solution of a finite number of unknowns. However, a continuous system is one
in which a continuum is described by complex differential equations. In other words, the
system response is described by an infinite number of unknowns. It is often difficult to
obtain an exact solution for a continuum problem and therefore standard numerical methods
are required.
If the characteristics of a problem can be represented by relatively simplified equations,
it can be analysed employing a finite number of components and simple matrices as shown
in the following sections of this chapter. Such procedures reduce the continuous system to
an idealization that can be analysed as a discrete physical system. In reality, an important
preliminary study to be made by the engineer is whether an engineering system can be
treated as discrete or continuous.
If a system is to be analysed using complex governing differential equations, then
one has to make a decision on how these equations can be discretized by an appropriate
numerical method. Such a system is a refined version of discrete systems, and the accuracy
of the solution can be controlled by changing the number of unknowns and elements. The

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)