5.3. Finite Element Methods for Three-Dimensional Problems
129
FIGURE 5.17. Isoparameter element in the
global coordinate system.
z
8
2
O,r-----------------__
x
T ABLE 5.5. Values of coefficient ß in Eq. (5.3.17).
Order of sides
First
Second
Third
1/3
~~i - 2/3
(9/8)~ - 19/24
Values of P
p~
1/3
cei - 2/3
(9/8),.,2 - 19/24
1/3
cei - 2/3
(9/8)(2 - 19/24
y
taken as nodes (nodes 5 and 6 in Figure 5.16). For edges of third order,
points at one-third and two-third ofthe edges are taken as nodes (nodes 9,10,
11, 12 in Figure 5.16). The corresponding nodes in the global coordinate
system are shown in Figure 5.17. The associated basis functions can be
defined as folIows:
(1) For node i located at a corner point, we define
where
{
"': ~(I + «,)(1 + ~,)(I + ce"
Pi - P~ + P" + Pe·
(5.3.16)
(5.3.17)
P~, p", Pe depend on the orders of intersection sides of the node, as given in
Table 5.5. For example, let us consider node 8 in Figure 5.16. Its local
coordinates are (-1, -1,1). The orders ofits three intersecting sides are 2, 3,
and 1, respectively. From Table 5.5 we find
Ps = (-~ -~) + (~'12 _ 19) + ~
3
8
24
3'