Civil Engineering Reference
In-Depth Information
By construction,
N
N
N
∑
∑
c
N
∑
1
n
=
=
m
=
N
k
m
2
1
2
ρ
∑
N
ρ
k
k
k
=
1
k
=
1
k
=
1
2
ρ
k
=
1
k
However, usually it is necessary to allow for a minimum number of line segments, say
n
0
= 10, in a cell, and Equation 3.4 has to be modified as follows:
*
N
c
*
c
=
m
and
n
=
+
n
with
NN
=
−
n
(3.5)
k
0
mm
0
1
2
2
∑
N
ρ
k
ρ
k
=
1
k
where
N
=
N
x
×
N
y
is the number of cells in the background grid.
To complete the grid system with cells of variable size, an auxiliary integer array P of size
N + 1 is also needed as a pointer to mark off the number of line segments in each cell, i.e.
the set of line segments contained in cell k, k = 1 ~ N, is given by
{
L
i
,
i
=
P
k
+ 1,
P
k
+1
} with
P
k
+1
=
P
k
+
n
k
=
∑
and
P
N
+1
=
N
m
, the length of the integer array
L
holding all the boundary segments recorded in the cells, {
L
i
,
i
= 1,
N
m
}. In case ρ is not
known for the first mesh, the line segments could be evenly distributed to all the cells, as a
relatively coarse mesh is used for the first preliminary analysis. Hence, the number of seg-
ments allowed for cell k in the initial mesh
n
Set
P
1
= 0. It is easy to see that
P
n
k
+
1
k
N
N
0
=
.
m
k
463
478
302
83
68
361
302
302
141
785
419
1209
536
1575
522
302
112
1004
24
83
53
68
302
302
Figure 3.47
Rectangular domain partitioned into cells,
N
e
= 10,054.
