Biomedical Engineering Reference
In-Depth Information
η
3
(-1, 1)
(1, 1)
4
4
3
ξ
e
y
1
2
2
1
e
x
(a) Quadrilateral element with
respect to a
global
coordinate system
(-1, -1)
(1, -1)
(b) Quadrilateral element with
respect to a
local
coordinate system
Figure 16.4
Quadrilateral element with respect to global and local coordinate systems.
1
4
(1
N
1
=
−
ξ
)(1
−
η
)
1
4
(1
N
2
=
+
ξ
)(1
−
η
)
(16.42)
1
4
(1
+
ξ
)(1
+
η
)
N
3
=
1
4
(1
−
ξ
)(1
+
η
).
N
4
=
An element having these shape functions is called a
bi-linear
element. Along the
edges of the element the shape functions are linear with respect to either
ξ
or
η
.
Within the element, however, the shape functions are bi-linear with respect to
ξ
and
η
. For instance:
1
4
(1
−
ξ
−
η
+
ξη
) .
N
1
=
(16.43)
Fig.
16.5
shows
N
1
visualized as a contour plot.
The shape function derivatives with respect to the local coordinates
ξ
and
η
are easily computed. However, the shape function derivatives with respect
to the global coordinates
x
and
y
are needed. For this purpose the concept of
isoparametric elements
is used.
For isoparametric elements the global coordinates within an element are
interpolated based on the nodal coordinates using the shape functions
T
(
T
(
x
|
e
=
∼
ξ
,
η
)
∼
e
,
y
|
e
=
∼
ξ
,
η
)
y
∼
e
,
(16.44)
where
∼
e
and
y
∼
e
contain the nodal
x
- and
y
-coordinates, respectively. These equa-
tions reflect the transformation from the local coordinates (
ξ
η
) to the global
coordinates (
x
,
y
). The derivatives of
N
i
with respect to the Cartesian coordinates
x
and
y
can be evaluated with the aid of the chain rule:
,
