Biomedical Engineering Reference
In-Depth Information
Therefore, if the MLS approximation is able to construct interpolation func-
tions, then Eq. (
4.30
) must be verified,
8
<
9
=
0
0
1
0
0
w
1
a
11
a
12
w
2
1
w
2
w
1
¼
f
1
x
I
g
ð
4
:
84
Þ
a
12
a
22
2w
1
w
2
0
w
2
2w
1
:
;
Which can be developed to obtain five equations,
8
<
9
=
8
<
9
=
0
0
1
0
0
w
1
ð
a
11
þ
a
12
x
I
Þ
2w
1
ð
a
12
þ
a
22
x
I
Þ
w
2
ð
a
11
þ
a
12
x
I
Þ
w
2
ð
a
12
þ
a
22
x
I
Þ
a
11
þ
a
12
x
I
w
2
ð
a
11
þ
a
12
x
I
Þþ
w
2
ð
a
12
þ
a
22
x
I
Þ
w
1
ð
a
11
þ
a
12
x
I
Þþ
2w
1
ð
a
12
þ
a
22
x
I
Þ
¼
ð
4
:
85
Þ
:
;
:
;
Adding the first equation to the fifth equation,
a
11
¼
a
12
x
I
ð
4
:
86
Þ
Substituting in third equation gives 1 = 0, indicating that equation system
presented in Eq. (
4.85
) is not true. It is not possible to define a A
ð
x
Þ
1
matrix
capable to produce the interpolation function presented in Eq. (
4.77
). Therefore,
the MLS shape functions do not possess the Kronecker delta property.
4.3.3.5 Compact Support
The MLS shape functions is obtained considering only the nodes inside an initially
defined compact support-domain. Since the value of u x
ðÞ
outside the support-
domain is zero, the MLS shape functions possess compact support. This property
permits to create sparse and banded discretized systems of equations.
4.3.3.6 Compatibility
The MLS shape functions are compatible in a local support-domain because the
support-domain movement does not perturb the continuity of the approximated
field function. The bell-shaped weight function, used to construct the MSL shape
function, permit to smoothly update the nodes entering and leaving the support-
domain.
Consider
the
one-dimensional
domain
described
in
Fig.
4.12
,
being
X ¼
1
the set of nodes discretizing the problem domain.
In this example the average nodal spacing is considered equal to the mesh density
f
x
1
;
x
2
;
...
;
x
12
g 2
X
^
x
i
2
R
