Civil Engineering Reference
In-Depth Information
* >
and
α
mi
*
is greater than α
min
, where α* is
the α-quality after node shifting. Element inversion can be prevented as some triangles will
then have negative α* values after node shifting. This Laplace smoothing scheme, which
involves the checking of the shape quality of the elements before and after node shifting,
is known as
smart Laplace smoothing
. This idea can be further refined to check the shape
quality of the triangles in
P
at more than one point. Points on the line joining
x
and
c
can be
conveniently specified by parameter λ.
Node shifting will be carried out only if
αα
x
λ
= (1 − λ)
x
+ λ
c
such that
x
0
=
x
and
x
1
=
c
Let
α
*
be the geometric mean α-quality of the triangles in
P
corresponding to a shift of
node
x
to
x
λ
; then by maximising
α
*
, 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. The enhanced Laplace smoothing scheme in which the node shift is based on the
quality assessment at some strategic points is known as
QL smoothing
.
6.3.1.2 LO of triangular mesh
Based on shape measures such as mean ratio η or α-quality coefficient on triangular ele-
ments, the quality of a triangular mesh
T
= {Δ
i
, i = 1,N} can be evaluated. Let α
i
be the
α-quality of triangle Δ
i
; then a measure of the quality of triangular mesh
T
is given by
∏
1/
N
α
=
α
;
α
=
min
α
i
min
i
Δ
∈
T
i
Δ
∈
T
i
The geometric mean
α
can be used as the cost function of the triangular mesh to be
optimised by shifting of interior nodes subject to the constraint that nodal shifting should
not reduce the α-quality of the triangles in
T
lower than α
min
. The condition that
T
being
optimal with respect to α-quality measure is that for each point
x
surrounded by polygon
P
,
∂
∂
α
∂
∂
α
1
/
n
∏
P
P
=
0
and
=
0
,
where
α
=
α
n
= numberoftriangles in
P
P
k
x
y
Δ
∈
P
k
As a global optimisation on α is very costly, this local condition for being optimal in α
measure can be used to improve the quality of a triangular mesh. Analytical expressions for
the derivatives of α
P
in closed form are not available and, in general, rather tedious. Hence,
numerical differentiation is applied to obtain approximate solutions for
∂
∂
α
∂
∂
α
P
=
0
and
P
=
0
.
x
y
Let
x
* =
x
+ (Δx, Δy) be the new position of node
x
; then the α
P
value at
x
* is given by
+
∂
∂
α
+
∂
∂
α
x
*
P
P
α
()
≈
α
()
x
Δ
x
Δ
y
P
P
x
y
λα λ
αα
P
∂
∂
∂
∂
α
P
can always be improved by setting
(
P
P
for some positive λ:
ΔΔ
x,
y
)
=
=
,
xy
2
2
∂
∂
α
∂
∂
α
α
()
x
*
=
α
()
x
+
λ
P
+
P
P
P
x
y
