Shape Functions - Meshless Methods in Biomechanics - page 96

The non-constant coefficients b ð x I Þ can be obtained with Eq. ( 4.19 ),

b ð x I Þ ¼A ð x I Þ 1 B ð x I Þ u s ð 4 : 25 Þ

By back substitution in Eq. ( 4.6 ) it is possible to write,

u h ð x I Þ ¼p ð x I Þ A ð x I Þ 1 B ð x I Þ u s

ð 4 : 26 Þ

Recovering the summation which originates the B ð x I Þ u s operation, Eq. ( 4.26 )
can be represented as,

u h ð x I Þ ¼p ð x I Þ T A ð x I Þ 1 X

n

W ð x i x I Þ p ð x i Þ T u i ð 4 : 27 Þ

i¼1

Notice that interest point polynomial vector p ð x I Þ and the moment matrix A ð x I Þ
can be moved inside the summation,

u h ð x I Þ ¼ X

n

p ð x I Þ T A ð x I Þ 1 W ð x i x I Þ p ð x i Þ T u i ð 4 : 28 Þ

i ¼ 1

Since the field variable for an interest point x I is approximated using shape
function values obtained at the nodes inside the support-domain of the interest
point x I ,

u h ð x I Þ ¼ X

n

u i ð x I Þ u i ¼ u ð x I Þ T u s ð 4 : 29 Þ

i¼1

It is possible to recognize the MLS shape function u i ð x I Þ ,

u i ð x I Þ ¼p ð x I Þ T A ð x I Þ 1 W ð x i x I Þ p ð x i Þ T ð 4 : 30 Þ

being u i ð x I Þ the shape function value of interest point x I on the ith node. u i ð x I Þ is
obtained considering the nodes inside the support-domain of interest point x I . The
MLS shape function vector for the n nodes inside the support-domain of x I is
defined as,

u ð x I Þ T ¼ f u 1 ð x I Þ ... u n ð x I Þg ¼p ð x I Þ T A ð x I Þ 1 B ð x I Þ u 2 ð x I Þ ð 4 : 31 Þ

Notice that the approximation function u h ð x Þ is defined for a specific interest
point x I possessing a particular support-domain with n nodes. Consider two distinct
interest points

d , being x J 6 ¼ x I . Both interest points shape
functions possess the same support-domain nodal set N I ¼ N J ¼ n 1 ; n 2 ; ... ; n f g
N, being N the complete nodal set discretizing the problem domain. Although

f x I ; x J g 2 X ^ x i 2 R