Civil Engineering Reference
In-Depth Information
P
3
I´
P
n
I
2
n
1
P
2
P
1
Figure 4.41
Shifting of interior node I to I
′
.
1
/
n
ψψψψ ψ
=
(
)
and
=
min(
ψ ψψ
,
,
)
12
n
min
12
n
1.
Laplacian smoothing.
The node is shifted to the centre of the polygon if there is
improvement both for
ψ
and ψ
min
; otherwise, the shifting is ignored for this interior
node.
2.
Gradient-guided shifting.
Compute
ψ
ψψ
xy
=
∂
∂
∂
∂
by numerical differentiation; the
,
node is shifted along
ψ
a small
d
istance (about 5% of the size of the polygon) if
there is an improvement both for
ψ
and ψ
min
; otherwise, the movement of the node
is reduced. This node-shifting process can be repeated a number of times until an
improvement can no longer be made for this node.
Either by Laplacian smoothing or gradient-guided shifting, each interior node is pro-
cessed in turn until all the interior nodes are treated. Usually, two cycles of smoothing can
be carried out before moving on to the next phase of diagonal swapping.
4.2.13.2 Diagonal swapping
Diagonal swapping is a topological operation in which the diagonal of a quadrilateral formed
by two adjacent triangles is swapped to the other position to improve the overall quality of
the two triangles. The unique lines or edges of the mesh are extracted, as described in
Section 2.5.4, and each interior edge shared by two triangles is examined in turn. For each
edge, compute the overall quality of the triangles sharing it to see if any improvement can
be made by a diagonal swap. Diagonal BD is swapped to AC if ψ(ABD)ψ(BCD) < ψ(ACD)
ψ(ABC), and the minimum ψ-value is not compromised in the swapping process, as shown
in Figure 4.42. Although the best results are expected by working with edges in the sequence
C
B
D
A
Figure 4.42
Diagonal swap.
