Civil Engineering Reference
In-Depth Information
Use the element as the parent element in the isoparametric mapping
x
=
N
1
(
r
)
x
1
+
N
2
(
r
)
x
2
+
N
3
(
r
)
x
3
with
x
1
<
x
2
<
x
3
but otherwise arbitrary nodal coordinates.
a.
How does the
x
coordinate vary between nodes of the isoparametric element?
b.
Has the basic element geometry changed from that of the parent element?
c.
Determine the Jacobian matrix for the transformation.
d.
Find the inverse of the Jacobian matrix.
r
1
2
3
r
1
0
r
2
0.5
r
3
1
Figure P6.31
Consider again the three-node line element of Figure P6.31 as the parent element
for the two-dimensional mapping defined by
6.32
x
=
N
1
(
r
)
x
1
+
N
2
(
r
)
x
2
+
N
2
(
r
)
x
3
y
=
N
1
(
r
)
y
1
+
N
2
(
r
)
y
2
+
N
2
(
r
)
y
3
where
(
x
i
,
y
i
)
are the coordinates of node
i
.
a.
Let
(
x
1
,
y
1
)
=
(1, 1), (
x
2
,
y
2
)
=
(2, 3), (
x
3
,
y
3
)
=
(4, 2)
and plot the
geometry of the isoparametric element to scale.
b.
Could the resulting isoparametric element be used in a finite element analysis
of heat conduction (refer to Chapter 5) through a curved solid with ideally
insulated surfaces? Explain your answer.
Show by analytical integration that the result given in Example 6.9 is exact.
6.33
Use Gaussian quadrature to obtain exact values for the following integrals.
Verify exactness by analytical integration.
6.34
3
(
x
2
a.
−
1) d
x
0
6
(
y
3
b.
+
2
y
)d
y
1
1
(4
r
3
c.
+
r
)d
r
−
1
1
(
r
4
3
r
2
)d
r
d.
+
−
1
1
(
r
4
+
r
3
+
r
2
e.
+
r
+
1) d
r
−
1
