Civil Engineering Reference
In-Depth Information
Table 4.2
CPU time and statistics of example meshes
2 . '
α
' curve
1. 'X' curve
Origin outside
α
Origin inside
α
3. Two peaks
4. Klein bottle
5. Wavy surface
N
4000
3091
3117
7171
23991
29362
M
7865
6106
6159
14185
46624
58117
Max1
2.386
2.533
2.583
2.591
2.660
2.628
Avg1
1.163
1.157
1.151
1.120
1.120
1.121
Max2
4.882
3.237
3.196
3.057
4.003
2.750
Avg2
1.133
1.121
1.121
1.107
1.060
2.750
Max3
6.372
2.985
2.955
2.399
4.189
6.564
Avg3
1.611
1.499
1.485
1.099
1.046
1.135
Max4
100.0
49.91
49.95
10.24
6.653
6.723
Avg4
25.22
8.738
8.453
3.155
4.817
3.193
Iteration
9
7
7
4
4
4
CPU(s)
1.495
1.126
1.137
3.953
13.52
15.98
Note:
N
4
hh
d
4
hh
d
d
hh
L
d
hh
1
L
∑
ij
ij
ij
ij
Max
1
=
max
,
;
Avg
1
=
max
,
N
4
4
kN
=
1
L
ij
ij
ij
ij
k
=
N
L
λ
λ
λ
jj
λ
λ
1
λ
λ
∑
1
j
i
Max
2
=
max,
i
j
;
Avg
2
=
max
i
j
,
N
λ
kN
=
1
L
L
i
k
=
N
L
1
∑
1
j
i
Max
3
=
max,
i
j
;
Avg
3
=
max
i
j
,
N
kN
=
1
L
L
i
k
=
N
λ
1
λ
∑
1
Max
4
=
max
k
k
,
k
k
;
Avg
4
=
max,
k
k
k
λ
N
λ
k
kN
=
1
,
k
=
where
i and j are ellipses connected to edge k.
N = number of nodes, N
L
= Number of edges.
d
ij
= the distance of neighbouring nodes Ci
i
and C
j
.
h
i
, h
j
= Euclidean lengths in the direction of the neighbouring ellipses i and j.
λ
i
,
μ
i
= lengths of principal axes of ellipse i.
Iteration = average number of iterations for ellipse packing.
is given in Table 4.2. A tighter control on the condition of touching between ellipses would
require more iterations; however, a tolerance of 0.9 or above adopted in the examples is usu-
ally sufficient to produce a pack of ellipses virtually touching one another.
Overlapping between the proposed ellipse and the existing ellipses on the front has to be
checked. If all the ellipses on the front are verified, the cost could be very high. However,
because of the convex shape of the generation front, the number of frontal ellipses involved
in the overlapping check could be reduced to a fixed number n, resulting in a linear time
complexity in the checking process. In the examples presented in Section 4.3.3, n has been
set to 10, i.e. checking of overlapping is conducted for 10 ellipses on the left and 10 ellipses
on the right of the insertion site. Indeed, in 90% of the cases, there is no overlapping. The
justification for the use of a small number n = 10 is that the front is more or less convex and
is of the shape of a circular ring with occasional minor concave parts. Hence, the newly
inserted ellipse does not overlap with ellipses far away from the insertion site. The result,
however, is a great reduction of mesh generation time, and a linear relationship between the
