Page 64 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

THE FINITE ELEMENT METHOD
56
and similarly
Hence, N β = N 0 (L j ) = 1and N γ = N 0 (L j ) = 1 (3.73)
N 200 = N 2 (L i )N 0 (L j )N 0 (L k ) = L i (2L i − 1) = N 1 (3.74)
is the shape function for node 1. Similarly,
N 3 = N 020 = L j (2L j − 1) and
N 5 = N 002 = L k (2L k − 1) (3.75)

For a middle node, with shape function N 110 ,wehave
N 110 = N 1 (L i )N 1 (L j )N 0 (L k )
2L i − i + 1 1 2L j − i + 1

1
=
i=1 i=1
i i

2L i − 1 + 1 2L j − 1 + 1
= (3.76)
1 1
Thus,
N 2 = N 110 = 4L i L j (3.77)
Similarly,
N 4 = N 011 = 4L j L k
N 6 = N 101 = 4L k L i (3.78)
We can summarize the nodal shape functions for a quadratic triangle as follows:
For corner nodes,
N m = L n (2L n − 1) with m = 1, 3, 5and n = i, j, k (3.79)

and for nodes at centres,
N 2 = 4L i L j
N 4 = 4L j L k
N 6 = 4L k L i (3.80)

In a similar way, we can show that the interpolation functions for a 10-node cubic
triangle are (see Figure 3.12) as follows:
For corner nodes,
1
N m = L n (3L n − 1)(3L n − 2) with m = 1, 4, 7and n = i, j, k (3.81)
2
Side ij
9
N 2 = L i L j (3L i − 1)
2
9
N 3 = L i L j (3L j − 1) (3.82)
2