Page 47 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

THE FINITE ELEMENT METHOD
methods (Ozisik and Czisik 1994), as soon as irregular geometries or an unusual speci-
fication of boundary conditions are encountered, the finite difference technique becomes
difficult to use. 39
The finite volume method is a further refined version of the finite difference method and
has become popular in computational fluid dynamics (Patankar 1980). The vertex-centred
finite volume technique is very similar to the linear finite element method (Malan et al.
2002).
The finite element method (Baker 1985; Bathe 1982; Chandrupatla and Belegundu 1991;
Huebner and Thornton 1982; Hughes 2000; Lewis et al. 1996; Rao 1989; Reddy 1993;
Segerlind 1984; Zienkiewicz and Morgan 1983; Zienkiewicz and Taylor 2000) considers
that the solution region comprises many small, interconnected, sub-regions or elements and
gives a piece-wise approximation to the governing equations, that is, the complex partial
differential equations are reduced to either linear or nonlinear simultaneous equations. Thus,
the finite element discretization (i.e., dividing the region into a number of smaller regions)
procedure reduces the continuum problem, which has an infinite number of unknowns, to
one with a finite number of unknowns at specified points referred to as nodes. Since the
finite element method allows us to form the elements, or sub-regions, in an arbitrary sense,
a close representation of the boundaries of complicated domains is possible.
Most of the finite difference schemes used in fluid dynamics and heat transfer problems
can be viewed as special cases within a weighted residual framework. For weighted residual
procedures, the error in the approximate solution of the conservation equations is not set to
zero, but instead its integral, with respect to selected ‘weights’, is required to vanish. Within
this family, the collocation method reproduces the classical finite difference equations,
whereas the finite volume algorithm is obtained by using constant weights.
For engineers whose expertise lies in fluid dynamics and heat transfer, the finite element
approaches introduced by mathematicians or structural analysts, may be difficult to follow.
Therefore, in this book we intend to present a step-by-step procedure of the finite element
method as applied to heat transfer problems. In doing so, we intend to present the topic
in as simplified a form as possible so that both students and practising engineers can
benefit.
A numerical model for a heat transfer problem starts with the physical model of the
problem, an example of which is shown in Figure 3.1. As can be seen, one part of the model
deals with the discretization of the domain and the other carries out the discrete approxima-
tion of the partial differential equations. Finally, by combining both, the numerical solution
to the problem is achieved.
The solution of a continuum problem by the finite element method is approximated by
1
the following step-by-step process .
1. Discretize the continuum
Divide the solution region into non-overlapping elements or sub-regions. The finite
element discretization allows a variety of element shapes, for example, triangles, quadrilat-
erals. Each element is formed by the connection of a certain number of nodes (Figure 3.2).

1 It should be noted that on first reading, these steps may not be very obvious to beginners. However, these
steps will be clear as we go through the details in the following sections