Biomedical Engineering Reference
In-Depth Information
where n is the number of field nodes inside the support-domain of interest point x
I
and m is the number of monomials of the complete polynomial basis p
j
ð
x
I
Þ
, which
can be defined by the triangle of Pascal, Fig.
4.2
, presenting the following vector
form,
g
T
p
ð
x
I
Þ
¼ p
1
ð
x
I
Þ
f
p
2
ð
x
I
Þ
... p
m
ð
x
I
Þ
ð
4
:
105
Þ
The Radial Basis Function (RBF) can be defined as,
g
T
r
ð
x
I
Þ
¼ r
1
ð
x
I
Þ
f
r
2
ð
x
I
Þ
r
n
ð
x
I
Þ
ð
4
:
106
Þ
g
T
¼ r
ð
x
1
x
I
Þ
f
r
ð
x
2
x
I
Þ
r
ð
x
n
x
I
Þ
The only variable in the RBF is the Euclidean norm between the field nodes and
the interest point, d
iI
, which can be defined for a three-dimensional space,
x ¼
f
x
;
y
;
z
g
, as,
q
ð
x
i
x
I
Þ
2
þð
y
i
y
I
Þ
2
þð
z
i
z
I
Þ
2
d
iI
¼
ð
4
:
107
Þ
In the literature it is possible to find several appropriate RBF to incorporate the
RPI formulation [
22
-
25
]. As indicated in [
26
], within the meshless RPI methods
the most frequently used globally supported RBFs are the multi-quadrics (MQ)
function,
p
r
i
ð
x
I
Þ
¼ d
iI
þ
cd
ð
2
ð
4
:
108
Þ
the Gaussian function,
d
ðÞ
2
d
iI
r
i
ð
x
I
Þ
¼e
c
ð
4
:
109
Þ
and the thin plate spline function,
r
i
ð
x
I
Þ
¼
d
iI
ð
4
:
110
Þ
being c and p the RBF shape parameters. The coefficient d
a
is a size coefficient,
indicating the influence size of the interest x
I
. For meshless methods using the
classical influence-domain concept, such as the EFGM and the RPIM, d
a
is the
average nodal spacing of the n nodes inside the support-domain of x
I
defined by
Eq. (
4.2
). However, for the NNRPIM, which uses the influence-cell concept, the
coefficient d
a
can be considered as the size of the Voronoï cell of interest point x
I
.
For the MQ-RBF, if x
I
is an integration point, then the coefficient d
a
can be
considered d
a
¼
_
I
, being
_
I
integration weight of x
I
. The main drawback of
MQ-RBF is that the shape parameters c and p need to be determined and optimize
to obtain accurate results.
