Civil Engineering Reference
In-Depth Information
The normal and the base point on each face of the dual polyhedron are given by
n
n
n
n
n
n
yy
×
114
y
y
b
b
b
b
b
b
(
1
−++
−
τ
)
y
τ
((
y
y
)
/
2
1
12
1
1
4
2
yy
×
×
×
×
×
yy
(
1
++
−++
−++
τ
)
y
τ
(
yy
)
/
/
/
/
/
2
2
13
12
2
1
2
3
yy
yy
(
1
τ
)
y
τ
(
y
y
)
2
3
14
13
3
1
3
4
normals:
=
,
ase points:
=
yy
yy
(
1
τ
)
y
τ
(
y
y
)
2
4
54
52
4
5
4
2
yy
yy
(
1
−++
−++
τ
)
y
τ
(
y
y
)
2
5
52
53
5
5
2
3
yy
y
y
(
1
τ
)
y
τ
(
y
y
)
2
6
53
5
4
6
5
3
4
The centroids on the faces of the dual polyhedron cannot be used as base points for the
face normals in the transformation as this will not satisfy the basic requirement that regular
elements should be reproduced in the transformation. Special base points located between
the triangular face apex and the mid-point of the opposite edge could be used for which the
parameter τ is given by Vartziotis and Wipper (2012)
4
5
4
39
τ
=
1
−
λ
4
{
}
As shown in Figure 6.29c, pentahedron P is transformed to
=
P
x
,
i
=
16
such that
1
2
P
=
x
=
b
+
λ
nn
/
,
i
=
16
, ;
λ
∈
,
1
i
i
i
i
6.3.3.3.4 Pyramid element
Let {
x
i
, i = 1,5} be the vertices of a pyramid element Y with nodes properly labelled, as shown in
Figure 6.30a. An auxiliary dual pyramid {
y
i
, i = 1,5} can be defined by connecting the centroids
of the five faces of pyramid Y (Vartziotis and Wipper 2012), as shown in Figure 6.30b, such that
y
y
y
y
y
(
xxxx
xxx
xxx
xx
+++
++
++
+
)
/
4
1
1
2
3
4
(
)
/
/
3
3
2
1
2
5
auxiliary polyhedron
=
(
)
3
2
3
5
(
x
xxx
+
++
)
/
/
3
3
4
3
4
5
(
)
5
4
1
5
(a)
(b)
(c)
x
5
5
Pyramid Y
n
5
y
4
y
5
y
5
y
4
4
x
4
y
2
1
y
2
y
3
3
y
3
x
1
n
3
y
1
n
1
y
1
x
3
Pyramid Y´
n
2
2
x
2
Figure 6.30
GETMe transformation of a pyramid element: (a) centroids of faces; (b) dual polyhedron;
(c) base points.
