Biomedical Engineering Reference
In-Depth Information
(a)
(b)
Fig. 4.21
Obtained
approximation
solutions
for
the
a
regular
nodal
distribution
and
the
b irregular nodal distribution
Table 4.2
Error obtained in each node using the regular nodal distribution
Node
Shape parameter
c = 0.001
c = 0.251
c = 0.501
c = 1.001
c = 1.501
x
1
-
-
-
-
-
x
2
0.00E+00
0.00E+00
0.00E+00
-5.80E-07
-1.11E-05
x
3
0.00E+00
0.00E+00
1.00E-08
3.05E-06
1.22E-04
x
4
0.00E+00
0.00E+00
0.00E+00
1.12E-05
4.07E-04
x
5
0.00E+00
0.00E+00
-1.11E-08
2.29E-05
-6.97E-03
x
6
0.00E+00
0.00E+00
-2.50E-08
-1.67E-06
-2.72E-03
x
7
0.00E+00
0.00E+00
0.00E+00
2.77E-05
4.44E-03
x
8
0.00E+00
0.00E+00
0.00E+00
-1.42E-05
9.47E-03
x
9
0.00E+00
0.00E+00
1.11E-08
-4.22E-07
1.49E-04
x
10
0.00E+00
0.00E+00
0.00E+00
-4.00E-07
-4.43E-05
x
11
-
-
-
-
-
the regular nodal distribution the value of the shape parameter c does not influ-
ences significantly the approximation. It is also perceptible that regardless the
considered shape parameter c value the approximation function interpolates per-
fectly the nodal values. However these observations are not true.
In Tables
4.2
and
4.3
are presented the local errors for each node discretizing
the problem domain. The error is obtained with: error ¼
ð
u
ð
x
i
Þ
u
h
ð
x
i
ÞÞ=ð
u
ð
x
i
ÞÞ
.
For both the regular and irregular nodal distributions, Tables
4.2
and
4.3
show that
in fact the RPI approximation is not capable to interpolate accurately the nodal
values when c [ 0.5. Increasing the shape parameter value leads to RPI shape
functions without the Kronecker delta property.
In Fig.
4.22
a it is presented the RPI shape function of the central node x
6
obtained with the regular nodal distribution. The RPI shape function obtained with
the irregular nodal distribution is presented in Fig.
4.22
b. These figures permit to
visualize directly the effects of the variation of the shape parameter. Using low
