Civil Engineering Reference
In-Depth Information
shape factor. Consider triangle ABC as shown in Figure 3.45. The geometrical shape
factor α of triangle ABC (Lo 1985) is defined as
signed area
sumofedges squared
AB
×
AC
α(
ABC
)
=
=
23
(3.2)
2
2
2
AB
+
B
CC
+
A
in which
23
is a normalising factor so that an equilateral triangle will have a maxi-
mum α value equal to 1. Node spacing conformity δ of a line segment is measured by
the deviation of the edge relative to the prescribed length as imposed by the node spac-
ing function, i.e.
= min
ρ
δ
,
(3.3)
ρ
where ℓ is the length of the edge, and ρ is the node spacing function evaluated at the
mid-point of the line segment. From Equation 3.3, it is seen that 0 < δ ≤ 1. As for tri-
angle ABC created on the base segment AB, there are two newly created edges, namely,
AC and CB. The coefficients of node spacing conformity of edges AC and CB are,
respectively, given by
AC
ρ
CB
ρ
δ
=
min
,
1
and
δ
=
min
,
2
1
2
ρ
ρ
AC
CB
1
2
in which ρ
1
and ρ
2
are values of node spacing function evaluated at the mid-points of
AC and CB, respectively, as shown in Figure 3.45. Hence, the node spacing conformity
of triangle ABC can now be defined as
δ(ABC) = δ
1
δ
2
An overall optimality coefficient λ combining the effects of shape factor α and the
node spacing conformity δ can be written as
λ = αδ = αδ
1
δ
2
λ will be used for judging the overall quality of a triangle in the adaptive meshing
process.
Step 6
:
Create an interior node at strategic location
Create an interior node I, which has the largest λ value without cutting into the genera-
tion front. In general, the position of node I has to be determined by iteration starting
from some ideal point. As shown in Figure 3.45, locate I on the normal bisector of
segment AB such that λ is maximised. Verify if node I intersects with the generation
front; I is accepted if there is no intersection; otherwise, adjust I downwards towards
line segment AB until there is no more intersection. To ensure good-quality elements to
be generated in the subsequent MG process, the new node I created should not be too
close to the generation front, and this could be easily verified in the checking process
for intersections. As to what threshold should be adopted, a simple rule is that the
potential element likely to be formed with the nearest edge should not be inferior to
the element proposed.
