Biomedical Engineering Reference
In-Depth Information
2
4
3
5
100000
101001
102004
110100
111111
112124
2
4
3
5
p
1
ð
x
1
Þ
p
2
ð
x
1
Þ
p
m
ð
x
1
Þ
p
1
ð
x
2
Þ
p
2
ð
x
2
Þ
p
m
ð
x
2
Þ
F ¼
¼
ð
4
:
100
Þ
.
.
.
.
.
.
;
p
1
ð
x
n
Þ
p
2
ð
x
n
Þ
p
m
ð
x
n
Þ
It is clear that the obtained moment matrix singular. It is not possible to obtain a
PIM shape function.
Several techniques are available to avoid the singularity of F [
4
,
21
], such as
moving the field nodes, transformation of the coordinate system, algorithms for
matrix triangulation and the inclusion of radial basis functions in the PIM
formulation.
4.4.2 RPI Shape Functions
This section presents the Radial Point Interpolators (RPI). Consider again the
function space T on X
d
. The finite dimensional space T
H
T which dis-
cretizes the domain X is defined by,
R
T
H
:¼ r x
i
x
h
ð
Þ
: i
2
N
^
i
N
i þ
p
k
ð
x
Þ
ð
4
:
101
Þ
d
at least a C
1
-function. The polynomial function p
k
:
d
R
is defined in the space of polynomials of degree less than k. The d-dimensional
spatial domain is discretized in a set of N nodes, whose coordinates are defined as
X ¼ x
1
;
x
2
;
...
;
x
N
Being r :
R
7!
R
R
7!
d
. The density of the nodal distributions is
f
g 2
X
^
x
i
2
R
defined by h, Eq. (
4.4
).
Considering a continuous scalar function u
ð
x
Þ
, with u
2
T , it is possible to
define for an interest point x
I
2
d
, not necessarily coincident with X, the RPIM
R
interpolation function of u
ð
x
I
Þ
as,
u
h
ð
x
I
Þ
¼
X
r
i
ð
x
i
x
I
Þ
a
i
ð
x
I
Þþ
X
n
m
p
j
ð
x
I
Þ
b
j
ð
x
I
Þ
¼r
ð
x
I
Þ
T
a
ð
x
I
Þþ
p
ð
x
I
Þ
T
b
ð
x
I
Þ
i¼1
j¼1
ð
4
:
102
Þ
Being
a
i
ð
x
I
Þ
and
b
j
ð
x
I
Þ
the
non-constant
coefficients
of
r
ð
x
I
Þ
and
p
ð
x
I
Þ
respectively, which can be defined as,
g
T
a
ð
x
I
Þ
¼ a
1
ð
x
I
Þ
f
a
2
ð
x
I
Þ
... a
n
ð
x
I
Þ
ð
4
:
103
Þ
g
T
b
ð
x
I
Þ
¼ b
1
ð
x
I
Þ
f
b
2
ð
x
I
Þ
... b
m
ð
x
I
Þ
ð
4
:
104
Þ
