Page 87 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

THE FINITE ELEMENT METHOD
Using the relations
d
dζ (δθ) = δ dθ 79
dζ
dθ dθ 1 dθ 2
δ = δ
dζ dζ 2 dζ
and
1 2
θδθ = δθ (3.189)
2
Then, Equation 3.188 is simplified to the following:
1 2
1
dθ 1 dθ 2 2
δθ − δ + µ θ dζ = 0 (3.190)
dζ 0 2 0 dζ
When we apply the boundary conditions (Equation 3.186), the first term of the above
equation becomes zero. Thus, the variational formulation for the given problem is

1 1 dθ 2
2 2
δ + µ θ dζ = 0 (3.191)
0 2 dζ
and the corresponding variational integral is given by

1 dθ 2 2
1 2
I = + µ θ dζ (3.192)
0 2 dζ
Now, the profile that minimizes the integral Equation 3.192 is the solution to the dif-
ferential Equation 3.185 with its boundary conditions given by Equation 3.186.
Let us assume the same profile as before (Equation 3.177) and substitute into
Equation 3.192, that is,

1 1
2
2
2
2
2
I = θ {(2Bζ) + µ [1 − (1 − ζ )B] }dζ (3.193)
b
0 2
After integration and substitution of limits, we have
1 2 4 2 2 2 1 2 2 2 2 2
I = θ b B + µ − µ + µ + µ + B −2µ + µ (3.194)
2 3 3 5 3
For I to be minimum, ∂I = 0, that is,
∂B
∂I 1 4 2 2 2 1 2 2 2 2 2
= θ b 2B + µ − µ + µ + µ + −2µ + µ = 0 (3.195)
∂B 2 3 3 5 3
which gives

4 8 2 4 2
2B + µ = µ (3.196)
3 15 3