Civil Engineering Reference
In-Depth Information
a general transformation operation involving an auxiliary polyhedron defined by connecting
the centroids of the faces of the 3D solid elements will be introduced.
6.3.3.1 QL smoothing (3D)
Given a 3D mesh of N polyhedral elements,
M
= {E
i
, i = 1,N}, for a given node
x
, the patch
of polyhedral elements surrounding node
x
, as shown in Figure 6.24, is given by
P
(
x
) = {E
k
∈
M
;
x
∈ E
k
},
x
∈ E
k
means that
x
is one of the vertices of element E
k
The
centroid
of polygon
P
is given by
1
mn
∑
∑
k
c
=
x
−
n
x
i
−
E
∈
P
i
k
x
∈
E
k
k
∑
where
m
=
(
numberof nodesinE
k
)
is the sum of the number of nodes in each element
E
k
∈
P
of
P
, and n is the number of elements in
P
. This is an approximation formula for the average
of the nodes on the surface of polyhedron
P
.
However, this
discrepancy may not be a draw-
back, as
c
will automatically lean towards nodes shared by many elements, hence enhancing
the quality of most of the elements in
P
.
By the QL smoothing for polyhedral meshes, the shape qualities of the elements in
P
at
more than one point are evaluated. Points on the line joining
x
and
c
can be conveniently
specified by parameter λ:
x
λ
= (1 − λ)
x
+ λ
c
such that
x
0
=
x
and
x
1
=
c
Let
η
*
be the geometric mean η-quality of the elements in
P
corresponding to a shift of
node
x
to
x
λ
; then by maximising
η
*
, the best location of the shift for
x
can be determined.
To limit the points to be evaluated to within a reasonable number, only λ = 0.9, 1.0 and 1.1
are tested. In the light of more experience and at the expense of more computations, sophis-
ticated schemes can be formulated by evaluating more points around centroid
c
.
6.3.3.2 LO of polyhedral mesh
The geometric mean
η
can be used as the cost function of a polyhedral mesh
M
to be
optimised by shifting of interior nodes subject to the constraint that nodal shifting should
Trial points
x
λ
Patch
P
c
x
E
k
Figure 6.24
A patch of tetrahedral elements around node
x
.
