Page 63 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

THE FINITE ELEMENT METHOD
method of establishing the shape functions exists, which is based on local coordinates. The
rationale behind this is given by Silvester (Silvester 1969) and can also be used to find the
shape functions for a cubic triangular element. 55
Silvester introduced a triple-index numbering scheme αβγ , which satisfies the following
expression,
α + β + γ = n (3.67)
where n is the order of the interpolation polynomial used. We can write N αβγ to denote
the interpolation function for a node as a function of the area coordinates L i ,L j and L k ,
namely,
N αβγ (L i ,L j ,L k ) = N α (L i )N β (L j )N γ (L k ) (3.68)

where
nL i − i + 1
N α (L i ) = α i=1 if α ≥ 1 (3.69)
i

N α (L i ) = 1 if α = 0 (3.70)
Similarly, we can write relations for N β and N γ in terms of L j and L k respectively.
For a quadratic triangular element, as shown in Figure 3.11, the shape functions are
designated as
Corner nodes: N 1 = N 200 ; N 3 = N 020 ; N 5 = N 002
Mid-side nodes: N 2 = N 110 ; N 4 = N 011 ; N 6 = N 101
Let us calculate typical terms, for example, N 200 and N 110 .

N 200 = N 2 (L i )N 0 (L j )N 0 (L k ) (3.71)
In the above equation, α = 2,β = 0and γ = 0 and therefore, from equation 3.69 we
have

nL i − i + 1 2L i − 1 + 1 2L i − 2 + 1
2
N α = N 2 (L i ) = = = L i (2L i − 1)
i=1
i 1 2
(3.72)
y
5 N 002
a = 0
b = 0
N 101
6 4 N 011

3
2
1
N 200 N 110 N 020
x
g = 0
Figure 3.11 Shape function designations of a quadratic triangular element