Page 88 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

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Non-dimensional temperature 0.8 1 Variational method THE FINITE ELEMENT METHOD
Exact
0.6
0.4

0.2
0
0 0.2 0.4 0.6 0.8 1
Distance from fin tip
Figure 3.26 Comparison between variational method and exact solution

or
µ 2
2
B = (3.197)
2 2
1 + µ
5
Substituting into Equation 3.177 gives the solution as
µ 2
θ(ζ) 2 2
= 1 − (1 − ζ ) (3.198)
θ 2 2
1 + µ
5
2
2
For the fin problem of the previous subsection with µ = 3and m = 300, the com-
parison between the variational method and the exact solution is shown in Figure 3.26. As
seen, the agreement between the solutions is better than the agreement between the exact
and Ritz solutions.
It can be observed from the variational Integral Equation 3.192 that it contains only a
first-order derivative even though the original differential Equation 3.185 contains a second-
order derivative.
If a body has two materials, the second derivative of the temperature, required by the
original differential equation at the point where the two materials meet, may not exist.
In this case, the variational formulation of the problem would readily yield an accurate
solution, since the second derivative in this example is not needed in the formulation. For
this reason, the variational formulation of a physical problem is often referred to as the
weak formulation.

3.3.3 The method of weighted residuals

For those differential equations for which we cannot write a variational formulation, there
is a need to find an alternative method of formulation. The method of weighted residuals
provides a very powerful approximate solution procedure that is applicable to a wide variety
of problems and thus makes it unnecessary to search for variational formulations in order
to apply the finite element method for these problems.