Shape Functions - Meshless Methods in Biomechanics - page 144

2
4

3
5

r 23 r 13 r 23 r 12 r 23

1
2r 12 r 13 r 23

M 1
T

¼

r 13 r 23 r 13

ð 4 : 179 Þ

r 12 r 13

r 12 r 13 r 12

r 12 r 23

If an interest point x I is considered coincident with one of the nodes inside the
influence-domain, lets say node 1, x I ¼ x 1 , then with Eq. ( 4.123 ) is possible to
obtain,

u ð x I Þ ¼r ð x I Þ T M 1 g M 1

T

g T ¼ 0 f r 12 r 13 ¼ 100 f ð 4 : 180 Þ

T

showing the Kronecker delta property. This demonstration can be extended to a
support-domain containing n and also for RPI shape functions containing poly-
nomial basis functions. As has been shown in Sect. 4.4.4 , the RPI shape functions
constructed with the MQ-RBF only possess the Kronecker delta property if
cd a % 0.
To determine if the total moment matrix M T is well-conditioned, the condition
number of M T must be determined. The condition number is obtained with
Cond ð M T Þ ¼ jjj M T jjj jjj M 1 jjj , being the matrix norm defined by

P i¼1 jð M T Þ ij j .

T

jjj M T jjj ¼ max

1 j n

The nodal spatial disposition minimizing

Cond ð M T Þ is an equidistant nodal distribution, for which, considering again
Eq. ( 4.178 ), the total moment matrix M T components should be defined with
g 12 = g 13 = g 23 . However in meshless methods, nodes can be arbitrarily distrib-
uted, therefore Cond ð M T Þ is maximized when two nodes are extremely close to
each other when compared with a third node. In this case the total moment matrix
M T components would be defined with g 13 = g 23 g 12 . Therefore,

* +

j g 12 j þ g 13 j j

jjj G jjj ¼ max j g 12 j þ g 23 j j ¼ 2 g 13 j j ð 4 : 181 Þ

j þ g 23 j g 13 j j

and,

þ g 13 g 23

* +

g 23

j j þ g 12 g 23 j j

þ g 12 g 13

1
2g 12 g 13 g 23 max

¼ 1
g 12

1
2g 13

jjj G 1 jjj ¼ j þ g 13

j g 13 g 23 j j þ

j þ g 12 j g 12 g 23 j þ g 12 g 13 j

ð 4 : 182 Þ

the total moment matrix M T condition number is defined with Cond ð M T Þ ¼

1 þ 2g 13 = g 12 .
In this simple example, the lowest condition number is achieved when all nodes
are equidistant, Cond ð M T Þ ¼3, which indicates a well-conditioned matrix.
Notice that Cond ð M T Þ is proportional to the relation between the radial distance
determined for the nodes of the support-domain, Eq. ( 4.108 ). The most important