Civil Engineering Reference
In-Depth Information
(a
max
, b
max
, c
max
)
Cell k
I
z2
P
3
r
d
I
z1
I
x2
P
2
P
1
I
y2
I
x1
I
y1
(a
min
, b
min
, c
min
)
Figure 2.45
Cells intersected by P
1
P
2
P
3
.
where
I
x1
= [a
min
/d
x
] + 1, I
x2
= [a
max
/d
x
] + 1,
I
y1
= [b
min
/d
y
] + 1, I
y2
= [b
max
/d
y
] + 1,
I
z1
= [c
min
/d
z
] + 1, I
z2
= [c
max
/d
z
] + 1.
ii. Cells too far away from the triangle are eliminated, i.e. d > r, where d is the distance
from the centre of the cell to triangular facet P
1
P
2
P
3
(Section 2.4.2), and r is the radius
of the enclosing sphere, as shown in Figure 2.45.
2.7.5 Irregular grid
Rectangular grids of a set of N points with irregular (unequal) spacing along the x- and
y-direction can be constructed to cope with a non-uniform distribution of points. In an
irregular grid, points are put to a cell according to its rank along the x- and y-axes rather
than on their x- and y-values. To define irregular grid spacing along the x-direction, the
points have to be sorted (ranked) in the x-direction, and similarly, points are also sorted
along the y-direction to determine the grid spacing in the y-direction.
Let I
i
and J
i
be, respectively, the rank of point i along the x-direction and y-direction, i.e.
I
1
is the point with the smallest x-value, I
N
is the point with the largest x-value, etc. The
following
pseudo-code
will arrange sorted points into an irregular grid of N
c
= N
x
N
y
cells.
Given the average number of points in a cell, n, calculate the number of divisions N
x
and N
y
.
R
x
= x
max
− x
min
, R
y
= y
max
− y
min
,
α = R
x
/R
y
The number of divisions in the x-direction is given by
(
)
N
x
=
INTN n
/(
α
)
The number of divisions in the y-direction is
N
y
= NINT(αN
x
) where NINT(.) = earest integer
The number of points in a strip along the x-axis is N
1
= N/N
x
, and along the y-axis,
N
2
= N/N
y
.
