Civil Engineering Reference
In-Depth Information
a
macro-element
. Another macro-element is the Powell-Sabin element; see Powell
and Sabin [1977].
It should be noted that the continuity of derivatives along the element bound-
aries is easy to handle in terms of the Bernstein-Bezier representation of polyno-
mials.
Bilinear Elements
The polynomial families
P
t
are not used on rectangular partitions of a domain. We
can see why by looking at the simplest example, the bilinear element. Instead of
using
P
t
as we did for triangles, on rectangular elements we use the polynomial
family which contains
tensor products
:
c
ik
x
i
y
k
Q
t
:
={
u(x, y)
=
}
.
(
5
.
5
)
0
≤
i,k
≤
t
If more general quadrilateral elements are involved, we can use appropriately trans-
formed families.
We consider first a rectangular grid whose grid lines run parallel to the coor-
dinate axes. On each rectangle we use
u(x, y)
=
a
+
bx
+
cy
+
dxy,
(
5
.
6
)
where the four parameters are uniquely determined by the values of
u
at the four
vertices of the rectangle. Although
u
is a polynomial of degree 2, its restriction
to each edge is a linear function. Because of this, we automatically get global
continuity of the elements since neighboring bilinear pieces share the same node
information.
Fig. 17.
A rectangle rotated by 45
◦
, and a parallelogram element
The polynomial form (5.6) is not usable on a grid which has been rotated by
45
◦
as in Fig. 17. Indeed, the term
dxy
in (5.6) vanishes at all of the vertices of the
rotated square.
We can get the correct polynomial form for general parallelograms (and thus
for the rotated elements shown in Fig. 17) by means of a linear transformation.
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