Page 74 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

THE FINITE ELEMENT METHOD
66
1
N 3 =
4
1 (1 + ζ)(1 − η)(ζ − η − 1)
2
N 4 = (1 + ζ)(1 − η )
2
1
N 5 = (1 + ζ)(1 + η)(ζ + η − 1)
4
1 2
N 6 = (1 − ζ )(1 + η)
2
1
N 7 = (1 − ζ)(1 + η)(−ζ + η − 1)
2
1 2
N 8 = (1 − ζ)(1 − η ) (3.119)
2
The ζ and η variables are curvilinear coordinates and as such their direction will vary
with position. The nodes of the element are input in an anticlockwise sequence starting from
any corner node. The directions of ζ and η are indicated on Figure 3.18, that is, positive
ζ in the direction from nodes 1 to 3 and positive η in the direction from nodes 3 to 5.
Example 3.2.5 Evaluate the partial derivatives of the shape functions at ζ = 1/2, η = 1/2
of a quadrilateral element, assuming that the temperature is approximated by (a) bilinear
and (b) quadratic interpolating polynomials.
(a) Bilinear
The shape function derivatives in local coordinates are
∂N 1 1 − η ∂N 1 1 − ζ
=− ; =−
∂ζ 4 ∂η 4
1 − η 1 + ζ
∂N 2 ∂N 2
= ; =−
∂ζ 4 ∂η 4
∂N 3 1 + η ∂N 3 1 + ζ
= ; =
∂ζ 4 ∂η 4
∂N 4 1 + η ∂N 4 1 − ζ
=− ; = (3.120)
∂ζ 4 ∂η 4
The Jacobian matrix and its inverse are calculated from Equations 3.114 and 3.116,
that is,
 
4 4
∂N i ∂N i
x i y i
 
∂ζ ∂ζ
  1 25 4
 i=1 i=1  (3.121)
4 4  =
[J] = 
∂N i ∂N i
  8 514
 x i y i 
∂η ∂η
i=1 i=1
The determinant of the Jacobian matrix is
(25)(14) (5)(4) 330
det [J] = − = (3.122)
(8)(8) (8)(8) 64