Shape Functions - Meshless Methods in Biomechanics - page 132

; [RBF Wu1]

r i ð x I Þ ¼ 1 d iI Þ 5 8 þ 40d iI þ 48d iI þ 25d iI þ 5d iI ð ð 4 : 145 Þ

; [RBF Wu2]
ð 4 : 146 Þ

r i ð x I Þ ¼ 1 d iI Þ 6 6 þ 36d iI þ 82d iI þ 72d iI þ 30d iI þ 5d iI ð

r i ð x I Þ ¼ 1 d iI Þ 4 3 þ 4d iI

ð ð Þ; [RBF We1] ð 4 : 147 Þ

; [RBF We2]

r i ð x I Þ ¼ 1 d iI Þ 6 3 þ 18d iI þ 35d iI

ð ð 4 : 148 Þ

; [RBF We3]

Þ 8

r i ð x I Þ ¼ 1 d iI 1 þ 8d iI þ 25d iI þ 32d iI ð ð 4 : 149 Þ

Being d iI ¼ d iI = d s . The compactly supported RBF proposed by Wu and
Wendland are strictly positive definite for all d iI B d s , being d s the shape function
support-domain. When d iI [ 1 the RBF assume a null value. In the literature [ 5 ]it
is possible to find works assuring that, in comparison with non-compactly sup-
ported RBF, there is no clear advantage of using compactly supported RBF to
solve solid mechanical problems or surface fitting.
The RBFs referred in this text are presented in Fig. 4.20 . In all examples the
RBF application domain is x ¼ 0 : 5 ; 0 : 5 and the centre of the RBF is

x = 0 with a support-domain d s = 0.5. The MQ-RBFs presented in Fig. 4.20 a
were obtained considering a fixed value for the shape parameter p = 1.001 and
varying the shape parameter c, c = {0.001, 1.001, 2.001}. In Fig. 4.20 b are pre-
sented Gaussian-RBFs curves obtained considering c = {0.001, 1.001, 10.001}
and

½ ^ x 2 R

in Fig. 4.20 c are presented thin plate spline RBFs constructed using

p = {0.001, 1.001, 2.001}.
It is possible to observe in Fig. 4.20 a, b, c that the constructed RBFs are not
confined to the support-domain and that the variation of the shape parameters
permit to create distinct curves within the same RBF.
In Fig. 4.20 d are presented the compactly supported RBFs from Eq. ( 4.145 )to
( 4.149 ). Notice that these RBFs curves are bell shaped and assume a null value on
the support-domain limit. Since it is the first term of Eq. ( 4.145 )to( 4.149 ) that
imposes the RBF continuity, these compactly supported RBFs can be constructed
with the desire amount of smoothness.
In this work only the MQ-RBF is used. With the purpose of showing the
importance of the RBF shape parameter value on the constructed RPI shape
function, consider a one-dimensional domain X R discretized by the nodal set

X ¼ x 1 ; x 2 ; ... ; x 11 R : x 2 ½0 ; 1 . Two distinct nodal distribu-

tions are considered in this example: a uniform nodal distribution and an irregular
nodal distribution. As in Sect. 4.3.6 , an approximation function u h ð x Þ will be
adjust to the N discrete nodal values u ð x i Þ . In Table 4.1 are presented the spatial
location of each node x i discretizing the problem domain and the respective nodal
values u ð x i Þ . The described procedure is followed to obtain the approximation
function u h ð x Þ :

f g 2 X, being x 2