Information Technology Reference
In-Depth Information
TABLE 6.9
Numerical Integration Formulas for Tetrahedrons
Tetrahedral
Location
Coordinates
Weights
Errors
h
2
a
1
/
4
,
1
/
4
,
1
/
4
,
1
/
4
1
R
n
=
O
(
)
a
a
1
,a
2
,a
2
,a
2
1
/
4
b
a
2
,a
1
,a
2
,a
2
1
/
4
h
3
c
a
2
,a
2
,a
1
,a
2
1
/
4
R
n
=
O
(
)
d
a
2
,a
2
,a
2
,a
1
1
/
4
a
1
=
0
.
58541020
a
2
=
0
.
13819660
a
1
/
4
,
1
/
4
,
1
/
4
,
1
/
4
−
4
/
5
b
1
/
3
,
1
/
6
,
1
/
6
,
1
/
69
/
20
c
1
/
6
,
1
/
3
,
1
/
6
,
1
/
69
/
20
R
n
=
O
(
h
4
)
d
1
/
6
,
1
/
6
,
1
/
3
,
1
/
69
/
20
e
1
/
6
,
1
/
6
,
1
/
6
,
1
/
39
/
20
For a tetrahedronal region, the integral appears as
1
1
(
1
−
L
1
−
L
2
)
I
=
F
(
L
1
,L
2
,L
3
,L
4
)
dL
1
dL
2
dL
3
(6.128)
0
0
0
n
W
(
n
)
i
=
F
(
L
1
i
,L
2
i
,L
3
i
,L
4
i
)
i
=
1
with the values of concern given in Table 6.9.
6.7
Isoparametric Elements
The use of standard straight-sided elements to model complicated structures, especially
those with curved boundaries, can be both difficult and inefficient. Many such elements
with the concomitant large number of nodal displacements may be required. An appreciable
reduction in the number of elements can be achieved if irregular shaped elements, such as
irregular triangles, quadrilaterals, or even curved boundary elements, are used.
Although several methods for creating these kinds of elements are available, the most
common approach is to establish them such that they are “parametrically” equivalent to rec-
tilinear counterparts. That is, the irregular shaped elements are generated with a mapping
of regular shaped elements.
Search WWH ::
Custom Search