Civil Engineering Reference
In-Depth Information
■
Solution
The partial derivatives of
x
and
y
with respect to
r
and
s
per Equations 6.56 and 6.80 are
4
∂
x
∂
∂
N
i
∂
1
4
[
−
(1
−
s
)
x
1
+
(1
−
s
)
x
2
+
(1
+
s
)
x
3
−
(1
+
s
)
x
4
]
r
=
x
i
=
r
i
=
1
4
∂
y
∂
r
=
∂
N
i
∂
r
1
4
[
y
i
=
−
(1
−
s
)
y
1
+
(1
−
s
)
y
2
+
(1
+
s
)
y
3
−
(1
+
s
)
y
4
]
i
=
1
4
∂
x
∂
s
=
∂
N
i
∂
s
1
4
[
−
(1
−
r
)
x
1
−
(1
+
r
)
x
2
+
(1
+
r
)
x
3
+
(1
−
r
)
x
4
]
x
i
=
i
=
1
4
∂
y
∂
s
=
∂
N
i
∂
s
1
4
[
−
(1
−
r
)
y
1
−
(1
+
r
)
y
2
+
(1
+
r
)
y
3
+
(1
−
r
)
y
4
]
y
i
=
i
=
1
The Jacobian matrix is then
(1
−
s
)(
x
2
−
x
1
)
+
(1
+
s
)(
x
3
−
x
4
)(1
−
s
)(
y
2
−
y
1
)
+
(1
+
s
)(
y
3
−
y
4
)
(1
1
4
[
J
]
=
−
r
)(
x
4
−
x
1
)
+
+
r
)(
x
3
−
x
2
)(1
−
r
)(
y
4
−
y
1
)
+
+
r
)(
y
3
−
y
2
)
(1
(1
Note that finding the inverse of this Jacobian matrix in explicit form is not an envi-
able task. The task is impossible except in certain special cases. For this reason, isopara-
metric element formulation is carried out using numerical integration, as discussed in
Section 6.10.
The isoparametric formulation is by no means limited to linear parent ele-
ments. Many higher-order isoparametric elements have been formulated and
used successfully [1]. Figure 6.23 depicts the isoparametric elements corre-
sponding to the six-node triangle and the eight-node rectangle. Owing to the
mapping being described by quadratic functions of the parent elements, the
resulting elements have curved boundaries, which are also described by qua-
dratic functions of the global coordinates. Such elements can be used to closely
approximate irregular boundaries. However, note that curved elements do not,
in general, exactly match a specified boundary curve.
(a)
(b)
Figure 6.23
Isoparametric mapping of quadratic elements into curved elements:
(a) Six-node triangle. (b) Eight-node rectangle.
