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FIGURE 6.27
Rectangle for two-dimensional Lagrangian interpolation.
the
directions will suffice for such cases. Consider the two-dimensional element
shown in Fig. 6.27a, in which there are
m
equally spaced nodes in each row and
n
equally
spaced nodes in each column. The shape function
u
ξ
and
η
(ξ
,
η)
can be written as
N
ij
n
m
u
(ξ
,
η)
=
v
=
N
j
(ξ )
N
i
(η)v
(6.65a)
k
k
i
=
1
j
=
1
where
k
=
(
i
−
1
)
m
+
j
, and
N
j
(ξ )
and
N
i
(η)
are defined in Eq. (6.64). In matrix form,
v
v
v
(
n
−
1
)
m
+
1
N
1
(η)
N
2
(η)
.
1
m
+
1
v
2
v
m
+
2
v
(
n
−
1
)
m
+
2
u
(ξ
,
η)
=
[
N
1
(ξ )
N
2
(ξ )
···
N
m
(ξ )
]
.
.
v
v
···
v
N
n
(η)
m
2
m
nm
N
T
ξ
=
RN
(6.65b)
η
Take the element in Fig. 6.27b as an example.
u
(ξ
,
η)
=
N
1
ξ
N
1
η
v
1
+
N
2
ξ
N
1
η
v
2
+
N
3
ξ
N
1
η
v
3
+
N
1
ξ
N
2
η
v
4
+
N
2
ξ
N
2
η
v
5
+
N
3
ξ
N
2
η
v
6
+
N
1
ξ
N
3
η
v
7
+
N
2
ξ
N
3
η
v
8
+
N
3
ξ
N
3
η
v
9
=
Nv
(6.65c)
where
N
i
ξ
=
,
9are
nodal displacements. Frequently, the finite element method uses a node numbering scheme
N
i
(ξ )
and
N
i
η
=
N
i
(η)
of Eq. (6.64) with
m
=
3. Also,
v
k
,k
=
1
,
2
,
...
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