Civil Engineering Reference
In-Depth Information
Since the one-dimensional Hermite interpolation problem for two points and
cubic polynomials has a unique solution, we can compute
q
. Hence, the values at the
ten nodes shown in Fig. 16 for the Lagrange interpolation are uniquely determined.
Thus we have reduced the interpolation problem for the Hermite triangle to the
usual Lagrange interpolation problem which is known to be solvable.
We emphasize that the derivatives are continuously joined only at the vertices
[but not along the edges]. The cubic Hermite triangle is
not
a
C
1
element. Never-
theless, we will see in Chapter VI that it provides appropriate nonconforming
H
2
elements for the treatment of plates.
•
••
•
•••
••••
Fig. 20.
Interpolation points for two elements with
=
P
3
, i.e. with piecewise
cubic polynomials
5.10 The Bogner-Fox-Schmit rectangle.
On the other hand there is a
C
1
-element
with bicubic functions. It is called the Bogner-Fox-Schmit element and depicted
in Fig. 21.
ref
:
16
,
:
=
p(a
i
), ∂
x
p(a
i
), ∂
y
p(a
i
), ∂
xy
p(a
i
), i
=
1
,
2
,
3
,
4
.
=
Q
3
,
dim
ref
=
(
5
.
10
)
Since the data in (5.10) refer to the tensor products of one-dimensional Hermite
interpolation, the 16 functionals in
are linearly independent on
Q
3
.
To verify
C
1
continuity of the Bogner-Fox-Schmit element, consider the uni-
variate polynomial on a vertical edge of the rectangle. Its restriction to the edge is a
cubic polynomial in
y
which is determined by
p
and
∂
y
p
at the two vertices. Sim-
ilarly the normal derivative
∂
x
p
is also a cubic polynomial and determined by
∂
x
p
and
∂
xy
p
at the vertices. Thus we have continuity of
p
and
∂
x
p
, i.e.
C
1
continuity.
Fig. 21.
C
1
-element of Bogner-Fox-Schmit. The symbol
refers to the mixed
second derivative
∂
xy
p
.
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