Biomedical Engineering Reference
In-Depth Information
Table 4.1 Spatial location of each node discretizing the problem domain and the respective
nodal value
Regular distribution
x
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
u
ð
x
Þ
0.00
0.05
0.10
0.12
0.09
0.08
0.11
0.13
0.09
0.04
0.00
Irregular distribution
x
0.00
0.12
0.19
0.31
0.43
0.49
0.58
0.71
0.79
0.92
1.00
u
ð
x
Þ
0.00
0.05
0.10
0.12
0.09
0.08
0.11
0.13
0.09
0.04
0.00
location of each node x
i
discretizing the problem domain and the respective nodal
values u
ð
x
i
Þ
are presents in Table
4.1
.
To
u
h
ð
x
Þ
the
obtain
the
approximation
function
following
procedure
is
performed:
1. It is created a background mesh of interest points covering the problem domain,
Q ¼ q
1
;
q
2
;
...
;
q
101
1
. This background mesh is equivalent to
the integration mesh that will be required to numerically integrate the integro-
differential equations governing the studied physical phenomenon.
2. The
f
g 2
X
^
q
i
2
R
size
of
the
support-domain
of
the
shape
functions
is
defined
as:
d
s
= s
h. In this case it is considered h = 1/10.
3. Based on the support-domain it is defined the size of the influence-domain of
each interest point q
I
, d
I
= d
s
. Then, each interest point q
I
searches for the
n field nodes within the radial distance d
I
, establishing the individual influence-
domains of interest points q
I
.
4. The MLS shape functions are constructed for each interest point q
I
following
the procedure indicated in
Sect. 4.3.5
. In this example it is used the cubic spline
weight function and the linear polynomial basis.
5. For each interest point q
I
it is obtained the approximation field value with:
u
h
ð
q
I
Þ
¼
P
i¼1
u
i
ð
q
I
Þ
u
ð
x
i
Þ
The previously described procedure is performed for five distinct support-
domains, considering s
¼
1
:
02
:
03
:
04
:
05
:f g
.
For the regular nodal distribution, the results of each analysis are presented in
Fig.
4.17
a. In Fig.
4.17
b it is possible to observe the effect of s on the weight
function shape. Notice that the use of support-domains with reduce size, s = 1,
permitted to interpolate the data. However this effect it is not universal, it was only
possible because the nodal distribution is uniform and the number of nodes inside
each influence-domain fulfil the minimum to create a non-singular momentum
matrix. The increase of the size of the support-domain leads to a smoother
approximated solution u
h
ð
x
Þ
.
The results regarding the irregular nodal distribution are presented in
Fig.
4.18
a. The effect of the support-domain size on the weight function shape is
presented in Fig.
4.18
b. It is visible that the support-domain with reduce size,
s = 1, does not provide an acceptable approximation solution. In this case the size
of the support-domain is not enough to guarantee the construction of non-singular
