Civil Engineering Reference
In-Depth Information
The normal and the base point on each face of the dual polyhedron are given by
yy
×
yy
n
n
n
n
n
12
15
b
b
b
b
b
(
1
−++
−++
−
τ
)
y
τ
(
y
y
)
/
/
2
1
1
1
5
2
y
yyy
yy
×
×
×
113
12
(
1
τ
)
y
τ
(
y
y
)
2
2
2
1
2
3
yy
normals:
=
,
ase points:
=
14
13
(
1
++
−++
+++
τ
)
y
τ
(
y
y
)
/
/
2
3
3
1
3
4
yy
yy
15
14
(
1
τ
)
y
τ
(
y
y
)
2
4
4
1
4
5
1
2
(y
yyy
)
/
4
yy
×
yy
5
5
2
3
4
5
24
35
Like the case of a pentahedron, a pyramid is not completely symmetric, and the base
points to support the face normal in the transformation located between the triangular
face apex and the mid-point of the opposite edge have to be used, for which the parameter
τ=+
1
2
.
{
}
As shown in Figure 6.30c, pyramid Y is transformed to
15
such that
Y
=
x
,
i
=
1
2
Y
=
x
=
b
+
λ
nn
/
,
i
=
15
, ;
λ
∈
,
1
i
i
i
i
As polyhedral element E will expand in the GETMe transformation, the element E′ after
transformation has to be scaled down so that its volume remains unchanged.
1
∑
(
)
x
*
=+ −
c
ρ
xc
,
i
=
1
, ( ;
nE
centroid
c
=
x
i
i
i
nE
()
x
∈
E
i
where scaling factor
ρ= volume E/volume E
3
()
()
and n(E) = number of vertices in E.
6.3.3.3.5 GETMe transformations on polyhedra
Poorly shaped tetrahedral, hexahedral, pentahedral and pyramid elements are subjected
to a number of GETMe transformations, as shown in Figure 6.31. The co-ordinates of the
polyhedral elements before transformations are given in Table 6.6. It is seen that the shape
quality of the elements can be rapidly improved by even one single GETMe transformation,
and among all the elements tested, the rate of convergence of the pyramid element is rela-
tively slow in approaching the regular form.
6.3.3.4 Examples: Node smoothing for tetrahedral meshes
Delaunay triangulations of randomly generated points are taken as test meshes for the three
node-smoothing schemes. Meshes are obtained by extracting the
convex hull
of DTs, and
these meshes are easily reproduced to assess the performance of any node-smoothing scheme
that is newly developed. A relatively coarse triangulation of 1000 random points before and
after optimisation is depicted in Figure 6.32. Ten more meshes of 100,000 points are gener-
ated, and the node-smoothing schemes, namely, QL, LO and GETMeT4 parallel smooth-
ing, are applied to these meshes. For each optimisation scheme, ten cycles of smoothing are
