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are to be established. For example, for
m
=
2,
u
=
N
200
v
200
+
N
110
v
110
+···+
N
101
v
101
1 at point
pqr
and equal to
zero elsewhere at other nodes. In the one-dimensional case, this was accomplished using the
product of Eq. (6.71a) with Lagrangian interpolation employed in each direction. Similarly,
for the two-dimensional case,
The shape functions
N
pqr
must be defined such that
N
pqr
=
N
pqr
(
L
1
,L
2
,L
3
)
=
N
p
(
L
1
)
N
q
(
L
2
)
N
r
(
L
3
)
(6.77)
where
N
i
p, q, r
are given by Eq. (6.71).
There is a significant integral property of the triangular coordinates which is useful in
the derivation of stiffness matrices. This is the closed form integral
(
L
j
)
,j
=
1
,
2
,
3 and
i
=
a
!
b
!
c
!
L
1
L
2
L
3
dA
=
!
2
A
(6.78)
(
a
+
b
+
c
+
2
)
A
where
a
is the power to which
L
1
is raised, and so forth. Thus, for example,
A
L
j
dA
=
1
1
3
A
.
These triangular coordinates are quite useful. They can be used to automatically position
interior nodes. Use of these natural coordinates has the advantage that a complete polyno-
mial of the same order as the interpolation is produced. This is quite significant and is one
reason for the widespread use of triangular elements.
3!
2
A
=
Three-Dimensional Case in Natural Coordinates
The natural coordinates for the three-dimensional case are volume ratios or volume
coordinates. Consider the four-node tetrahedron
T
with the volume
V
shown in Fig. 6.33a.
Define the side opposite to node
i
as
S
i
. Let
P
be a point inside the tetrahedron. Each side
S
i
and the point
P
can define an internal tetrahedron
T
i
with the volume
V
i
. Then the natural
coordinate can be defined as the ratio of
V
i
and
V,
i.e.,
V
i
V
L
i
=
(6.79)
Note that
i
=
1
L
i
1. The relationship between the Cartesian coordinates
x, y
and
z
and
L
1
,L
2
,L
3
, and
L
4
is found to be
=
=
1
x
y
z
1111
x
1
L
1
L
2
L
3
L
4
x
2
x
3
x
4
(6.80)
y
1
y
2
y
3
y
4
z
1
z
2
z
3
z
4
N
T
X
=
N
u
where
x
i
,y
i
,z
i
,i
,
4 are the Cartesian coordinates of the nodal points.
From Eq. (6.80), the natural coordinates
L
i
in terms of
x, y, z
are
=
1
,
2
,
...
N
T
N
−
1
u
=
X
(6.81)
or in component form
1
6
V
(α
i
+
β
i
x
L
i
=
+
γ
i
y
+
δ
i
z
)
(6.82)
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