Page 92 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

THE FINITE ELEMENT METHOD
84
Substituting Equation 3.177 into Equation 3.224 and integrating, we have
2

2
4
2
E = 4B − 4Bµ 1 − B + µ − 2Bµ 4 + B 2 8 µ 4 (3.225)
3 3 15
The error is minimized by satisfying ∂E/∂B =0,thatis,
∂E 4µ 4 16Bµ 4 2 16Bµ 2
= 8B − + − 4µ + = 0 (3.226)
∂B 3 15 3
which gives
µ 2 µ 2
1 +
2 3
B = 2 (3.227)
1 µ
1 + 2µ 2 +
3 15
Therefore, the solution is given by
µ 2 µ 2
1 +
θ(ζ) 2 2 3
= 1 − (1 − ζ ) (3.228)
2
θ b 2 1 µ
1 + 2µ +
3 15
2
For the particular problem where µ = 3, then
θ(ζ) 15 2
= 1 − (1 − ζ ) (3.229)
θ b 24
Figure 3.27 shows the comparison between all the different weighted residual methods.
As seen, the Galerkin method is the most accurate method.
1
Exact
Collocation
0.8 Sub-domain
Galerkin
Non-dimensional temperature 0.6
Least-squares

0.4

0.2

0
0 0.2 0.4 0.6 0.8 1
Distance from fin tip
Figure 3.27 Comparison between various weighted residual methods and exact solution