Biomedical Engineering Reference
In-Depth Information
(a)
(b)
(c)
(d)
Fig. 4.20 a MQ-RBF obtained considering p = 1.001 and c = {0.001, 1.001, 2.001}. b Gauss-
ian-RBF obtained considering c = {0.001, 1.001, 10.001}. c Thin-plate-RBF obtained consid-
ering p = {0.001, 1.001, 2.001}. d Compactly supported RBFs
1. A background mesh of interest points, covering the problem domain, is con-
structed, Q ¼ q
1
;
q
2
;
...
;
q
101
1
.
f
g 2
X
^
q
i
2
R
2. The
size
of
the
support-domain
of
the
shape
functions
is
defined
as:
d
s
= 3.0001
h. In this case it is considered h = 1/10.
3. To simplify the analysis, the size of the influence-domain of each interest point
q
I
is defined as d
I
= d
s
. Next, 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 RPI shape functions are constructed for each interest point q
I
following the
procedure indicated in
Sect. 4.4.3
. In this example it is used the MQ-RBF 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 considering a permanent
shape parameter p = 1.001 and five distinct values for the shape parameter c,
being c = cd
s
, c ¼ 0
:
001 0
:
251 0
:
501 1
:
001 1
:
50
f g
.
The results of each analysis are presented in Fig.
4.21
a, b, respectively for the
regular nodal distribution and for the irregular nodal distribution. It seems that for
