Biomedical Engineering Reference
In-Depth Information
Fig. 3.11 a Voronoï cell and the respecti
ve P
Ii
inte
rsection points. b Middle points M
Ii
and the
respective generated triangles. c Triangle n
I
P
I8
M
I1
As has been shown, it is always possible to divide the Voronoï cellV
I
in n sub-
cells S
Ii
, being n the total number of natural neighbours of n
I
and V
I
¼
[
i¼1
S
Ii
:
Therefore, the size of the Voronoï cell V
I
can be determined using the size of the
n sub-cells S
Ii
,
A
V
I
¼
X
n
A
S
Ii
; 8
A
S
Ii
0
ð
3
:
10
Þ
i¼1
Being A
V
I
the size of the Voronoï cell V
I
and A
S
Ii
the size of sub-cell S
Ii
. For the
one-dimensional domain A represents lengths, for the two-dimensional domain
A stands for areas and for the three-dimensional domain A is a volume. Notice that,
if the set of Voronoï cells are a partition, without gaps, of the global domain then,
the set of sub-cells are also a partition, without gaps, of the global domain.
It is clear now, with Figs.
3.10
and
3.11
, how the construction of the sub-cells
generates two types of basic shapes - triangles or quadrilaterals. Starting with these
two shapes, numerous integrations schemes can be constructed. In this topic it is
shown an ordered scheme, based on the Gauss-Legendre numerical integration.
Basic Integration Scheme
The simplest integration scheme that can be established, using the sub-cells tri-
angular and quadrilateral shapes, is obtained inserting a single integration point in
the barycentre of the sub-cells. Therefore, spatial location of each integration point
is determined on each sub-cell, as indicated in Fig.
3.12
, being the weight of each
integration point the area of the respective sub-cell.
Considering Fig.
3.12
, the area of the triangle shape sub-cell is defined by,
A
I
¼
1
x
2
x
1
y
2
y
1
det
ð
3
:
11
Þ
x
3
x
1
y
3
y
1
2
