Page 89 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

81
THE FINITE ELEMENT METHOD
Let the governing equations be represented by
L(T ) = 0in
(3.199)
Let
n

T ≈ T = a i N i (x) (3.200)
i=1
Substitution of the above equation into Equation 3.199 results in
L(T) = 0
= R(residual) (3.201)
The method of weighted residual requires that the parameters a 1 ,a 2 ,... ,a n be deter-
mined by satisfying

w i (x)R dx = 0 with i = 1, 2,... ,n (3.202)

where the functions w i (x) are the n arbitrary weighting functions. There are an infinite
number of choices for w i (x) but four particular functions are most often used. Depending
on the choice of the weighting functions, different names are given

Collocation: w i = δ(x − x i )

= 0 (3.203)
Rδ(x − x i )dx = R x=x i

Sub-domain: w i = 1 (Note the sub-domain
i in the integration)

R dx = 0 with i = 1, 2,... ,n (3.204)

i
Galerkin: w i (x) = N i (x), that is, the same trial functions as used in T(x)

N i (x)R dx = 0 with i = 1, 2,... ,n (3.205)

Least Squares: w i = ∂R/∂a i

∂R
dx = 0 with i = 1, 2,... ,n (3.206)

∂a i
For illustration purposes the fin problem is re-solved with each of the above methods.

Collocation method
The weight is w i = δ(x − x i )
Let ζ i = 1/2 as there is only one unknown in the fin problem. Rewriting the equation
in collocation form in the non-dimensional coordinates gives the following:

2
1 d θ
2
− µ θ δ(ζ − ζ i )dζ = 0 (3.207)
0 dζ 2