Civil Engineering Reference
In-Depth Information
3.6.3.1 Construction of the background grid
In 2D, a background grid of
N
=
N
x
×
N
y
cells (boxes, zones, bins) subdivides regularly the
domain of interest into a number of regions, such that for any given point, the cell that con-
tains it can be determined rapidly without much calculation. As shown in Figure 3.46, the
dimensions of a typical cell are given by
D
N
D
N
y
x
d
=
,
d
=
x
y
x
y
where
D
x
and
D
y
are the lengths of the domain along the x- and y-directions, respectively.
N
x
and
N
y
are the numbers of division along the x- and y-directions, respectively.
N
=
N
x
×
N
y
is the number of cells in the background grid.
However, in the actual applications, in particular, for adaptive meshing, elements are not
evenly distributed, which means that an even distribution of line segments in the cells is not
an efficient scheme. A variable allocation scheme of line segments into cells is preferred in
which the number of line segments contained in each cell varies from cell to cell dependent
on the node spacing function ρ over the given domain.
3.6.3.2 Setting the size of each cell in the grid
Let
L
be an integer array of size
N
m
assigned for recording line segments in the cells, and ρ
be the node spacing function over domain Ω to be meshed such that
ρ: Ω →
R
+
The problem is how to determine the size (number of segments to be stored) in each cell k,
k = 1 ~ N, according to the given node spacing function ρ in the most optimal manner. The
size of cell k,
n
k
, can be estimated based on a weight inversely proportional to the square of
the spacing function, i.e.
1
2
c
n
k
∝=
ρρ
forsomeconstant
c
2
ρ
2
has to be taken simply because ρ only prescribes the linear distance between nodes, whereas
the number of line segments in a cell should be related to the area of the cell. Having adopted
the weighting relationship, the following can be written:
N
N
1
N
∑∑
∑
nN c
=
=
N
or
c
=
m
(3.4)
k
m
m
2
1
ρ
N
k
k
=
1
k
=
1
2
ρ
k
=
1
k
where ρ
k
is the node spacing function computed at the centre of cell k, and a more accurate
estimation based on more Gaussian points should be employed if the cells of a large area
are used. Hence,
c
n
=
ρ
2
k
k
