Civil Engineering Reference
In-Depth Information
c
8
c
7
c
5
c
6
c
4
c
3
c
2
c
1
Figure 6.28
Transformation of H by the auxiliary octahedron.
{
}
As shown in Figure 6.28, hexahedron H is transformed to
18
, whose vertices
Hi
=
x
,
=
are given by
1
2
H
=
x
=
c
+
λ
nn
/
i
=
18
,;
λ
∈
,
1
i
i
i
i
The properties of the GETMe transformation, i.e. invariant with respect to translation,
rotation and scaling, and preserves the centroid of the initial hexahedron, are discussed in
Vartziotis and Wipper (2011).
6.3.3.3.3 Pentahedral (wedge) element
Let {
x
i
, i = 1,6} be the vertices of a pentahedral element P labelled in a usual manner, as
shown in Figure 6.29a. An auxiliary dual polyhedron {
y
i
, i = 1,5} can be defined by con-
necting the centroids of the five faces of pentahedron P, as shown in Figure 6.29b, such that
y
y
y
y
y
xxx
xxxx
xxxx
x
(
++
+++
+++
)
/
3
1
1
2
3
(
)
/
/
4
4
2
1
2
5
4
auxiliary polyhedron
=
(
)
3
2
3
6
5
(
xxx
xxx
++++
++
)
/
4
4
3
1
4
6
(
)
/
3
5
4
5
6
(a)
(b)
(c)
x
6
x
5
6
5
y
5
5
n
4
y
5
n
6
n
5
4
x
4
y
3
y
3
3
y
4
y
4
y
2
2
1-τ
τ
y
2
4
n
3
n
2
x
3
x
2
3
2
y
1
y
1
1
n
1
Pentahedron P
Pentahedron P´
x
1
1
Figure 6.29
GETMe transformation of a pentahedral element: (a) centroids of faces; (b) dual polyhedron;
(c) special base points.
