Biomedical Engineering Reference
In-Depth Information
The average nodal spacing d
a
of the n nodes inside the support-domain of x
I
can
be determined using a simple expression [
5
],
ðÞ
1
D
d
a
¼
ð
4
:
2
Þ
ðÞ
1
1
n
d
. The physical size of the
support-domain is represented by D, which for the one-dimensional case defines
the length of the support-domain, for the two-dimensional case represents the
support-domain area and for the three-dimensional case symbolizes the volume of
the support-domain.
The NNRPIM does not require the determination of the support-domain. The
nodal set that constitutes the influence-cell of interest point x
I
is the same nodal set
used to construct the RPI shape functions for the interest point x
I
.
where d is the dimension of the problem domain X
R
4.3 Moving Least Squares
The moving least squares (MLS) approximation was developed by Lancaster and
Salkauskas [
6
] to smoothly approximate scattered data. Due to the MLS simplicity
and low computational effort, several meshless methods use the MLS approxi-
mation to construct the shape functions [
7
-
10
]. Additionally, the use of MLS
shape functions permit to approximate smoothly and continuously the field vari-
ables along the entire discretized domain.
The MLS approximation is constructed using three components: a weight
function with compact support associated to each interest point; a basis, which
usually consists of polynomial functions, and; a set of coefficients dependent on
the interest point position.
4.3.1 MLS Shape Functions
d
. The finite dimensional function space
T
H
, T which discretize the domain X is defined by,
Consider the function space T on X
R
T
H
:¼ p
k
ð
x
Þ
ð
4
:
3
Þ
d
where p
k
:
R
7!
R
is defined in the space of polynomials of degree less than
k.
The
set
of
N
nodes
discretizing
the
space
domain
is
defined
by
d
. The mesh density of X is identified by,
X ¼
f
x
1
;
x
2
;
...
;
x
N
g2
X
^
x
i
2
R
