Civil Engineering Reference
In-Depth Information
Optimal point
Polygon
P
h
u
x
Contours of α
P
Trial points
Figure 6.11
Local optimisation scheme.
α
P
always increases unless the derivatives both vanish. By numerical differentiation, we have
∂
∂
αα α
(
xh
+−
)
()
x
∂
∂
α
α
(
yh
+−
)
α
()
y
P
P
P
P
P
P
≈
and
≈
x
h
y
h
where h can be set equal to the nodal shift, which is approximately 10% of the local element size.
2
2
λ
αα
λ
∂
∂
∂
∂
∂
∂
α
∂
∂
α
P
P
2
P
P
2
2
2
(,
ΔΔ
xy=
)
,
+
=
() ()
ΔΔ
x
+
y
=
h
xy
x
y
h
α
α
λ
=
;
(,
ΔΔ
xy
)
=
λα
=
h
u
h
ere unitvector
u
=
P
P
2
2
∂
∂
α
∂
∂
α
P
P
P
+
x
y
As numerical differentiation has been used, λ has to be varied in order to optimise α
P
(x*).
Typically, we can adjust λ in 0.1λ intervals between [a, b], with a = 0.8λ and b = 1.2λ, as shown in
Figure 6.11. Unless Δα
P
= (α
P
(x + h) − α
P
(x))
2
+ (α
P
(y + h) − α
P
(x))
2
is greater than some threshold
ε = 10
-6
, node shifting will not take place to enhance numerical stability; and this node is deemed
to be optimised since α
P
can hardly be increased any more. For small ∆α
P
< 10
-4
the shift h
u
can
also be reduced from 10% to say 1%. Obviously, the interval [a, b] and the increment of 0.1λ
are rather arbitrary without any theoretical basis, and readers can formulate their own schemes
in the light of more experience. Like the QL smoothing, each interior node in the mesh will be
processed in turn following the natural order of the nodes until all the nodes are processed. A
number of iteration cycles can be performed until no more improvement could be made.
6.3.1.3 GETMe (2D)
The GETMe is a geometry-based smoothing method in which nodes are shifted entirely
based on the geometry of the polygon (polyhedron) in such a way that in each cycle of
node shifting, the polygon (polyhedron) will become more and more regular. A generic
scheme has been proposed for node shifting of a general polygon (polyhedron) in which
the directions for node shifting are based on an auxiliary polygon (polyhedron) formed by
