(2) The coordinates of the 9 nodes of a quadrilateral element in the xy-plane are given by




−1 
0















0
2
















1
0















0
 −1 



e
e
−1/2
1
{x } =
, {y } =
√






3/2 
7/4 
















−1/2
1/2








√
√








2/2
−




2/2
−








0
0
(3)
1.5
1.5
3
1
7
1
3
6
7
0.5
4
0.5
9
2
5
-0.5
η
y
4
0
0
9
8
6
-0.5
8
-1
-1
1
-1.5
-1
-0.5
1
0
0.5
1
1.5
2
-1.5
-1.5
-1
5
-0.5
0
2
0.5
1
ξ
x
(a)
(b)
Figure 6: Mapping of a quadrilateral element. (a) A biquadratic element in the physical domain, (x, y)
coordinates. (b) The corresponding canonical biquadratic square element, (ξ, η) coordinates.
The element is mapped onto the canonical biquadratic square element in the ξη-plane. At the point
(ξ, η) = (0.6, 0.4) compute:
(a) (x, y) coordinate of the corresponding point in the xy-plane; the point is marked by the triangle
symbol.
(b) The Jacobian matrix [J], its inverse [J]−1 , and its determinant ||J||.
(c) The gradient matrix [B].
(d) Repeat parts (a), (b), and (c) for the point (ξ, η) = (0, 0).
8
1.5