Civil Engineering Reference
In-Depth Information
6.3.1 Optimisation of triangular meshes
Three optimisation schemes for triangular meshes by node shifting, namely, QL smooth-
ing, local quality optimisation and GETMeT3, will be presented in this section. Examples
of various characteristics are given, and their performance will be assessed in terms of the
shape quality of the meshes and the CPU time taken in the optimisation process.
6.3.1.1 QL smoothing
Given a triangular mesh of N triangle,
T
= {Δ
i
, i = 1,N}, for a given node
x
, the patch of
triangles surrounding node
x
, as shown in Figure 6.10, is given by
P
(
x
) = {∆
k
∊
T
;
x
∊ ∆
k
},
x
∊ ∆
k
means that
x
is one of the vertices of triangle ∆
k
The
centroid
of polygon
P
is given by
∈
∑
k
(
)
k
k
xxxx
++−
2
3
Δ
k
P
c
=
2
where n is the number of triangles in
P
. Here,
centroid
does not mean the centre of gravity of
polygon
P
, but the average of the co-ordinates of the nodes on the boundary of
P
. By the
classical Laplace smoothing, node
x
is shifted to
c
, and this node-smoothing process can
be applied sequentially to each node in turn until all the nodes in the mesh are treated. A
number of cycles of Laplace smoothing can be applied to a triangular mesh until no further
improvement can be made. However, node shifting to the centroid of the surrounding poly-
gon does not guarantee that all triangles are valid elements, especially for concave polygon
P
; some of the triangles may become inverted by the nodal displacement. In order to prevent
inverted elements from being formed in the Laplace smoothing process, the α-qualities of
the triangles in
P
are computed, which are to be compared to those after shifting node
x
to
centroid
c
. The displacement of
x
to
c
is only executed when there is an improvement in the
overall quality of the triangles in
P
. As a quality measure of the triangles in
P
, the geometric
mean α-quality and the minimum α-value of the triangles can be used.
1/
n
∏
k
α
=
α
;
α
=
min
α
min
k
Δ
∈
P
k
Δ
∈
P
k
Trial points
x
λ
Polygon
P
c
∆
k
x
Figure 6.10
Quality Laplace smoothing.
