Biomedical Engineering Reference
In-Depth Information
Table 4.3
Error obtained in each node using the irregular 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
2.00E-08
0.00E+00
x
3
0.00E+00
0.00E+00
1.00E-08
-5.30E-06
2.69E-04
x
4
0.00E+00
0.00E+00
0.00E+00
1.83E-06
-6.31E-04
x
5
0.00E+00
0.00E+00
0.00E+00
1.39E-05
3.92E-04
x
6
0.00E+00
0.00E+00
-3.75E-08
1.49E-04
1.22E-03
x
7
0.00E+00
0.00E+00
0.00E+00
1.73E-06
-3.88E-04
x
8
0.00E+00
0.00E+00
-7.69E-08
-3.41E-05
1.31E-03
x
9
0.00E+00
0.00E+00
-1.11E-08
-8.44E-07
-5.43E-05
x
10
0.00E+00
0.00E+00
0.00E+00
-2.50E-08
-1.50E-07
x
11
-
-
-
-
-
(a)
(b)
Fig. 4.22 Obtained RPI shape function for node 6 using the a regular nodal distribution and the
b irregular nodal distribution
values for the shape parameter c allow to obtain smooth RPI shape functions
possessing the Kronecker delta property. In Table
4.4
are presented the obtained
values for node 8, of the RPI shape function constructed in node 6. It is visible that
the u
6
(x
8
) values are very close to zero when c \ 0.5. This simple example
demonstrates that the RPI shape function, constructed using the MQ-RBF, require
a low shape parameter c in order to assure the Kronecker delta property.
The use of the correct shape parameter c assumes greater importance in higher
dimensional spaces. Consider a two-dimensional domain X
2
discretized by
the eight nodes represented in Fig.
4.23
a. In order to constructed the RPI shape
functions of the central node, x
1
, using the other seven nodes, it was considered the
MQ-RBF and a linear polynomial basis. Three values for the shape parameter c
were assumed c = {0.01, 0.50, 2.00} and a permanent exponential shape
parameter p = 1.001 was considered. The obtained two-dimensional RPI shape
functions are presented in Fig.
4.23
b, c, d. As it is possible to observe the shape
parameter c affects significantly the RPI shape function silhouette.
R
