Civil Engineering Reference
In-Depth Information
Similar to the parallelisation in 2D, the most important requirement in a spatial partition
of points into cells is to ensure that each point belongs to one and only one cell, and the sum
of points in all the cells is equal to the total number of points, i.e.
N
c
∑
1
Nn
=
wheren
=
numberof pointsin celli
i
i
i
=
7.4.2 Grouping cells into zones
Let N
p
= D
x
D
y
D
z
be the number of processors available for parallel insertion, where D
x
, D
y
and D
z
are the zonal divisions along the x-, y- and z-axes, respectively, as shown in Figure 7.17.
Then cells are grouped into zones such that each zone will consist of m cells given by
N
D
N
D
N
D
y
mNINT
=
x
x
NINT
NINT
z
z
y
For best performance of parallel insertion, the number of zones has to be an integral mul-
tiple of the number of processors available; for example, division into 2 × 2 × 3 = 12 zones
for four or six processors is a sound division. The number of cells in some zones near the
boundary may have fewer or more cells if N
x
/D
x
, N
y
/D
y
or N
z
/D
z
is not a whole number.
Such a variation would not cause any problem in the subsequent operations, as a zone is
identified by the bounding cells in x-, y- and z-directions, i.e. a zone I, I = 1 ~ Np, is specified
by Zone I = Zone (N
x1
, N
x2
, N
y1
, N
y2
, N
z1
, N
z2
), where (N
x1
, N
x2
), (N
y1
, N
y2
) and (N
z1
, N
z2
)
are, respectively, the starting and ending cell division lines along the x-, y- and z-directions.
7.4.3 Simultaneous insertion in 3D
Analogous to the simultaneous insertion in 2D, the procedure can be applied in exactly
the same manner in 3D by replacing the rectangle by a rectangular cuboid, the triangles by
tetrahedra, the circle by a sphere, etc. A division into 2 × 2 × 2 = 8 zones for the insertion
of 2000 3D points is shown in Figure 7.18, in which the average number of points in a cell
D
z
= 3
D
y
= 4
D
x
= 2
Figure 7.17
Partitioned into 2 × 4 × 3 = 24 zones.
