Biomedical Engineering Reference
In-Depth Information
construct interpolation shape functions for meshless methods. The PIM [
21
] was
originally developed using uniquely polynomial basis functions. The PIM for-
mulation permits to easily construct shape functions and allows to obtain solutions
with high accuracy [
21
]. However this polynomial version of the PIM presents a
heavy disadvantage, the construction of the PIM shape function it is not always
possible. In some cases, the perfect alignment of the nodes lead to singular
moment matrices precluding to obtain the shape function.
In order to solve this PIM drawback, radial basis functions were included in the
PIM formulation [
22
,
23
]. This PIM version is entitled Radial Point Interpolation
Method (RPIM) and it permits to obtain always a non-singular moment matrix and
consequently it allows to construct consistently the interpolation shape function.
Comparing with the polynomial PIM formulation, the radial point interpolation
(RPI) formulation is more complex however it permits to obtain numerical solu-
tions with higher accuracy.
4.4.1 PIM Generic Shape Functions
To understand the generic PIM construction, consider a function space T on
X
d
. It is possible to define the finite dimensional function space T
H
T
discretizing the domain X with: T
H
:¼ f
k
ð
x
Þ
, being f
k
:
R
d
defined in the
functional space. It is assumed that the d-dimensional spatial domain is discretized
in N nodes: X ¼ x
1
;
x
2
;
...
;
x
N
R
7!
R
d
.
Considering a continuous scalar function u
ð
x
Þ
, being u
2
T, it is possible to
define for an interest point x
I
2
f
g 2
X
^
x
i
2
R
d
, not necessarily coincident with X, the PIM
R
interpolation function of u
ð
x
I
Þ
as,
u
h
ð
x
I
Þ
¼
X
m
f
i
ð
x
I
Þ
b
i
ð
x
I
Þ
¼f
ð
x
I
Þ
T
b
ð
x
I
Þ
ð
4
:
87
Þ
i¼1
Being b
i
ð
x
I
Þ
the non-constant coefficients of f
i
ð
x
I
Þ
and m the number of func-
tions f
i
ð
x
I
Þ
used as basis. Notice that if the m functions, f
i
ð
x
I
Þ
, of Eq. (
4.87
) are
substituted by m monomial terms p
i
ð
x
I
Þ
of a complete polynomial basis obtained
from the triangle of Pascal, Fig.
4.2
, Eq. (
4.87
) equalizes Eq. (
4.6
), which leads to
the classic polynomial PIM formulation [
21
].
In opposition to the MLS approximation function, the PIM interpolation
function u
h
ð
x
Þ
match the continuous scalar function u
ð
x
Þ
, Fig.
4.19
, therefore
u
h
ð
x
Þ
¼u
ð
x
Þ
. It is possible to obtain u
h
ð
x
Þ
¼u
ð
x
Þ
because within the PIM inter-
polation the number of nodes n on the support-domain of the interest point x
I
is
equal to m, the number of unknowns coefficients of b
ð
x
I
Þ
.
The non-constant coefficients b
ð
x
I
Þ
can be obtained enforcing u
h
ð
x
I
Þ
to pass
through all the n nodal values on the support-domain of x
I
. Thus, Eq. (
4.87
) must
