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6.6.3
Numerical Integration Over Triangular and Tetrahedral Regions
It is convenient to employ the natural coordinates defined by Eq. (6.75) and (6.82) in perform-
ing integration over triangular and tetrahedral domains. Particular closed form integrals
are given by Eqs. (6.78) and (6.85). Some integrals, based on Gaussian quadrature, that are
useful in finite element analyses are listed in this section. For the triangle,
1
(
1
−
L
1
)
n
W
(
n
)
i
I
=
F
(
L
1
,L
2
,L
3
)
dL
1
dL
2
=
F
(
L
1
i
,L
2
i
,L
3
i
)
(6.127)
0
0
=
i
1
Note that the limits of integration involve the coordinate variable. Some triangular coordi-
nates and weighting coefficients are given in Table 6.8.
TABLE 6.8
Numerical Integration Formulas for Triangles
Triangular
Location
Coordinates
Weights
Errors
)
∗
a
1
/
3
,
1
/
3
,
1
/
3
1
R
n
=
O
(
h
2
a
1
/
2
,
1
/
2
,
0
1
/
3
b
0
,
1
/
2
,
1
/
2
1
/
3
R
n
=
O
(
h
3
)
c
1
/
2
,
0
,
1
/
2
1
/
3
a
1
/
3
,
1
/
3
,
1
/
3
27
/
60
b
1
/
2
,
1
/
2
,
0
"
c
0
,
1
/
2
,
1
/
2
8
/
60
R
n
=
O
(
h
4
)
#
d
1
/
2
,
0
,
1
/
2
e
1
,
0
,
0
"
f
0
,
1
,
0
3
/
60
#
g
0
,
0
,
1
a
1
/
3
,
1
/
3
,
1
/
3
,
0.225
b
a
1
,a
2
,a
2
c
a
2
,a
1
,a
2
0.13239415
d
a
2
,a
2
,a
1
R
n
=
O
(
h
6
)
e
a
3
,a
4
,a
4
f
a
4
,a
3
,a
4
0
.
12593918
g
a
4
,a
4
,a
3
with
a
1
=
.
0
05971587
a
2
=
.
0
47014206
a
3
=
0
.
79742699
a
4
=
0
.
10128651
h
n
means that when
h
n
*
O
(
)
→
0
,R
n
→
0
.
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