Biomedical Engineering Reference
In-Depth Information
A minimum is obtained when b
1
ð
x
Þ
¼a
1
ð
x
Þ
¼c and b
j
ð
x
Þ
¼0
;
j ¼
f
2
;
...
;
m
g
.
Therefore the approximation function can be presented as,
u
h
ð
x
Þ
¼c
ð
4
:
73
Þ
Since the field variable for an interest point x
I
is approximated using the shape
function values obtained at the nodes inside the support-domain of x
I
,
u
h
ð
x
I
Þ
¼
X
u
i
ð
x
I
Þ
u
ð
x
i
Þ
¼
X
u
i
ð
x
I
Þ
c
¼
c
X
n
n
n
u
i
ð
x
I
Þ
ð
4
:
74
Þ
i¼1
i¼1
i¼1
and the approximation function is defined as u
h
ð
x
Þ
¼c, then,
c ¼ c
X
u
i
ð
x
I
Þ,
X
n
n
u
i
ð
x
I
Þ
¼1
ð
4
:
75
Þ
i¼1
i¼1
Proving that if the basis contains a constant term, then the MLS shape function
is of the partition unity.
4.3.3.4 Kronecker Delta
In general, the approximation function obtained with MLS approximants is a
smooth functional unable to pass through the nodal values. Hence, the MLS shape
functions do not possess the Kronecker delta property,
1
;
i ¼ j
u
i
ð
x
j
Þ 6
¼ d
ij
¼
ð
4
:
76
Þ
0
;
i
6
¼ j
This property is demonstrated with the following one-dimensional example.
Consider a one-dimensional domain discretized by a set of 5 nodes defined by
X
¼
x
1
;
x
2
;
x
3
;
x
4
;
x
5
1
. The mesh density parameter of X is iden-
tified by Eq. (
4.4
), being h considered constant in this example. The domain is
represented in Fig.
4.11
. Consider now an interest point x
I
2
X coincident with x
3
possessing an influence-domain containing all the nodes of the domain, X. If the
MLS approximation is able to construct interpolation functions, then the MLS
shape function for the interest point x
I
2
X should resembles the shape function of
node x
3
, u
I
(x)=u
3
(x) presented in Fig.
4.11
,
f
g 2
X
^
x
i
2
R
u
I
(x
Þ
¼u
3
(x
Þ
¼
f
00100
g
T
ð
4
:
77
Þ
The construction of the MLS shape function respect Eq. (
4.31
). To simplify the
present demonstration consider the following spatial coordinates for each node of
the one-dimensional domain,
