Biomedical Engineering Reference
In-Depth Information
Fig. 4.3 The discrete nodal
parameters u
ð
x
i
Þ
and the
MLS approximation function
u
h
ð
x
Þ
p
ð
x
I
Þ
T
¼
f
1
x
I
g ;
x
I
m ¼ 3
ð
4
:
8
Þ
For a two-dimensional space, x
I
¼
f
x
i
;
y
I
g
, the quadratic polynomial basis is
obtained,
p
ð
x
I
Þ
T
¼
f
1
x
I
y
I
g ;
x
I
y
I
x
I
y
I
m ¼ 6
ð
4
:
9
Þ
and for a three-dimensional space, x
I
¼
f
x
I
;
y
I
;
z
I
g
,
p
ð
x
I
Þ
T
¼
f
1
x
I
y
I
z
I
x
I
y
I
z
I
x
I
y
I
y
I
z
I
z
I
x
I
g ;
m ¼ 10
ð
4
:
10
Þ
In order to improve the MLS approximation function performance, Eq. (
4.6
)
can be enriched with additional functions [
11
]. This enrichment technique permit
to capture with a higher accuracy the stress fields in the vicinity of crack tips and in
the interface of distinct materials. In this topic only pure polynomial basis are
considered.
Ideally the MLS approximation function u
h
ð
x
i
Þ
should match the continuous
scalar function u
ð
x
Þ
, however generally u
h
ð
x
Þ 6
¼ u
ð
x
Þ
. This feature is represented
in Fig.
4.3
. Usually the MLS approximation function it is not capable to achieve
the discrete values of u
ð
x
Þ
, since the number of nodes n used in the MLS
approximation is generally much larger than m, the number of unknowns coeffi-
cients of b
ð
x
I
Þ
.
In order to adjust the approximation function u
h
ð
x
i
Þ
to the n discrete nodal
values u
ð
x
i
Þ
¼u
i
inside the influence-domain of interest point x
I
, the following
weighted residual functional is established,
J ¼
X
n
2
W
ð
x
i
x
I
Þ
u
h
ð
x
i
Þ
u
ð
x
i
Þ
ð
4
:
11
Þ
i¼1
being W
ð
x
i
x
I
Þ
the weight function, which is presented in detail in
Sect. 4.3.2
.
The inclusion of the weight function permits to attribute distinct weights to the
