Page 75 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

67
THE FINITE ELEMENT METHOD
Employing Equation 3.116
[J] −1 = 8 14 −4 (3.123)
330 −525
Substituting ζ = 1/2 and η = 1/2 into Equation 3.120
1 1
∂N 1 ∂N 1
=− and =− (3.124)
∂ζ 8 ∂η 8
Substituting into Equation 3.115
 
 ∂N 1 
 
∂x −10
  1
= (3.125)
330 −20
 ∂N 1 
 
 
∂y
In a similar fashion, all other nodal derivatives can be calculated.
(b) Quadratic variation
The shape function at node 1 is
1
N 1 =− (1 − ζ)(1 − η)(ζ + η + 1) (3.126)
4
The derivatives with respect to the transformed coordinates are
∂N 1 1 ∂N 1 3
= and = (3.127)
∂ζ 16 ∂η 16
The derivatives with respect to the global coordinates are
 
 ∂N 1 
 
∂x 30
  1
= (3.128)
660 60
 ∂N 1 
 
∂y
 
Other derivatives can be established in a similar manner.
It is a simple matter to transform the area coordinate system for triangular elements
(L i ,i = 1, 2, 3) to the ζ − η coordinates.
The shape functions for the three-node linear triangle can be expressed in the ζ and η
coordinate system as shown in Figure 3.19, that is,
N 1 = L 1 = 1 − ζ − η

N 2 = L 2 = ζ; 0 ≤ ζ ≤ 1
N 3 = L 3 = η; 0 ≤ η ≤ 1 (3.129)
For a quadratic triangle with six nodes, the shape functions at the corner codes are
N 1 = L 1 (2L 1 − 1) = [2(1 − ζ − η) − 1](1 − ζ − η)

N 3 = L 2 (2L 2 − 1) = ζ(2ζ − 1)
N 5 = L 3 (2L 3 − 1) = η(2η − 1) (3.130)