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Assignment
Finite Element Methods
Problems 4.20 and 4.
Problem 4.
The Cartesian coordinates of the nodes of a quadratic quadrilateral isoparametric element are shown in
Figure 4.26. Determine the coordinate transformation relation between the local and global coordinates.
Using this relation, find the global coordinates corresponding to the point (r. s) = (0.0).
Solution:
As we know
1 2 8 1 2 8
[ ]
x
x
x x
N
y y
y
y
Where
[ ]
N N N
N
N N N
According to page 106
i i i i
Ni rr ss rr ss for i= 1,2,
And
2
i
Ni r ss for i = 5,
2
i
Ni rr s for i=6,
Now the given node points for the elements are
1 1 2 2 3 3 4 4
5 5 6 6 7 7 8 8
r s r s r s r s
r s r s r s r s
Therefore
1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8
x N x n x N x N x N x N x N x N x
And
1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8
y N y n y N y N y N y N y N y N y
Now using the values of xi and yi
1 2 3 4 5 6 7 8
x 2.2 N 3.3 N 3.8 N 1.2 N 0.8 N 3.9 N 1.4 N 0.2 N
1 2 3 4 5 6 7 8
y 1.7 N 1.9 N 2.2 N 3.2 N 2.4 N 0.2 N 2.1 N N
By substituting the values of Ns, we obtain an expression for the relation between the local and global
coordinates for the given quadrilateral element
Now for( , ) r s (0,0)
1 2 3 4
5 6 7 8
N N N N
N N N N
So we get,
x
y
