Civil Engineering Reference
In-Depth Information
(cf. Problem 5.16). There are eight degrees of freedom. The first four are deter-
mined by the values at the vertices. The remaining parameters
e, f, g
and
h
can be
computed directly from the values at the midpoints of the sides. This element is
called the
eight node element
or the
serendipity element
. If we add the term
k(x
2
1
)(y
2
−
−
1
),
we get one more degree of freedom, and can then interpolate a value at the center
of the rectangle. By dropping some degrees of freedom, we can also get useful six
node elements (with
e
=
f
=
0or
g
=
h
=
0, respectively), as shown in Fig. 18.
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Fig. 18.
Rectangular elements with 6, 8, or 9 nodes for a rectangle with edges
on the lines
|
x
|=
1
,
|
y
|=
1.
Affine Families
In the above discussion of special finite element spaces, we have implicitly made
use of the following formal construction; cf. Ciarlet [1978].
5.8 Definition.
A
finite element
is a triple
(T,,)
with the following properties:
(i)
T
is a polyhedron in
d
. (The parts of the surface
∂T
lie on hyperplanes and
R
are called
faces
.)
(ii)
is a subspace of
C(T )
with finite dimension
s
. (Functions in
are called
shape functions
if they form a basis of
.)
(iii)
is a set of
s
linearly independent functionals on
. Every
p
∈
is uniquely
defined by the values of the
s
functionals in
. - Since usually the functionals
involve point evaluation of a function or its derivatives at points in
T
, we call
these
(generalized) interpolation conditions
.
In (ii)
s
is the
number of local degrees of freedom
or
local dimension
.
Although generally
consists of polynomials, it is not enough to look only
at polynomial spaces, since otherwise we would exclude piecewise polynomial
elements such as the Hsieh-Clough-Tocher element. In fact, there are even finite
elements consisting of piecewise rational functions; see Wachspress [1971].
As a first example consider the finite element families
M
0
.Wehave
k
0
k
,
k
),
M
=
(T ,
P
1
,
2
,...,
(k
+
1
)(k
+
2
)
k
:
={
p(z
i
)
;
i
=
}
,
2
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