Civil Engineering Reference
In-Depth Information
5.11 Standard Elements.
Strictly speaking the diagrams show the sets
T
and
from the triple
(T,,)
. Nevertheless, in many cases the associated family
is
considered clear and is sometimes just mentioned without a detailed specification.
For example, diagrams as in Fig. 16 refer to
0
. If a point evaluation at the
=
M
1
2
center of a triangle is added to
0
, then the space is augmented by a bubble
function as for the MINI element in Ch. III, §7 or the plate elements in Ch. VI, §6.
Figures 17 and 18 show further standard elements with
=
P
1
and
=
P
3
.
The functionals which are encountered with elements for scalar equations, are
found in Table 2 and Fig. 21. The specification of vector valued elements often
contains normal components or tangential components on the edges. Motivation is
given in Problem 5.13, but applications are only contained in Chapter III and VI.
Definition 5.8 refers to a single element. The analysis of the finite element
spaces can be obtained from results for a reference element, if all elements are
constructed by affine transformations.
M
0
or
M
d
5.12 Definition.
A family of finite element spaces
S
h
for partitions
T
h
of
⊂ R
is called an
affine family
provided there exists a finite element
(T
ref
,
ref
,)
called
the
reference element
with the following properties:
(iv) For every
T
j
∈
T
h
, there exists an affine mapping
F
j
:
T
ref
−→
T
j
such that
for every
v
∈
S
h
, its restriction to
T
j
has the form
v(x)
=
p(F
−
1
x)
with
p
∈
ref
.
j
We have already encountered several examples of affine families. The families
0
and the rectangular elements considered so far are affine families. For example,
M
is defined by the triple
( T,
P
k
,
k
)
, where
k
0
M
T
:
2
={
(ξ, η)
∈ R
;
ξ
≥
0
,η
≥
0
,
1
−
ξ
−
η
≥
0
}
(
5
.
11
)
is the unit triangle and
k
:
={
p(z
i
)
;
i
=
1
,
2
,...,s
:
=
k(k
+
1
)/
2
}
is the set of
nodal basis points
z
i
in Remark 5.6.
In our above discussion of the bilinear rectangular elements and the analogous
biquadratic ones, it is clear how the transformation (iv) works (cf. (5.8)), but this is
not the case for the complete polynomials on triangles. For rectangular elements,
the unit square [
1]
2
is the natural reference quadrilateral.
On the other hand, whenever conditions on the normal derivatives enter into
the definition (e.g., in the definition of the Argyris triangle), then we do not have an
affine family; see Fig. 22. This can be remedied by combining the normal derivatives
with the tangential ones in the analysis. This has led to the theory of
almost-affine
families
; see Ciarlet [1978].
−
1
,
+
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