Shape Functions - Meshless Methods in Biomechanics - page 138

Then, substituting the equations on Eq. ( 4.152 )on( 4.153 ) it is possible to
obtain the coefficients a ð x I Þ and b ð x I Þ . Equation ( 4.153 ) is true only if a i ð x I Þ ¼0,
with i = {1, 2, …, n}, and b j ð x I Þ ¼c j ð x I Þ , with j = {1, 2, …, t}, and b k ð x I Þ ¼0,
with k = {t + 1, t + 2, …, m}. Thus,

u h ð x Þ ¼ X

t

p j ð x Þ c j ð x Þ ¼p ð x Þ T c ð x Þ ¼u ð x Þ ð 4 : 154 Þ

j¼1

Proving that the RPI approximation is capable to reproduce any set of mono-
mials included in the polynomial basis of the RPI formulation.

4.4.5.2 Reproducibility

As mentioned before, in meshless methods the reproducibility property is not
included in the consistency property because in meshless methods it is possible to
include in the shape function construction procedure distinct types of basis
functions. If the meshless shape function is capable to reproduce an unknown
function which is included as a basis function in the meshless shape functions
formulation, then the meshless shape function possess the reproducibility property.
While the consistency property focuses only on the reproducibility of complete
polynomial functions, the reproducibility property is concern with any kind of
functional besides polynomial functions.
The argument used to prove the consistency can be used to demonstrate the
reproducibility properties of the RPI shape function. Consider a field defined by
the following function,

u ð x Þ ¼ X

k

f j ð x Þ c j ð x Þ ¼f ð x Þ T c ð x Þ

ð 4 : 155 Þ

j¼1

Being c j ð x Þ arbitrary coefficients of f j ð x Þ , which is a function defined in a
functional space f k :

d

R 7! R . The RPI approximation function can be defined as,

u h ð x I Þ ¼ X r i ð x i x I Þ a i ð x I Þþ X p j ð x I Þ b j ð x I Þþ X

n m t

f k ð x I Þ d k ð x I Þ 4 : 156 Þ

i¼1 j¼1 k¼1

and in the matrix form as,

8
<
:

9
=
;

a ð x Þ
b ð x Þ
d ð x Þ

u h ð x Þ ¼r ð x Þ T a ð x Þþ p ð x Þ T b ð x Þþ f ð x Þ T d ð x Þ ¼ r ð x Þ T

p ð x Þ T f ð x Þ T

ð 4 : 157 Þ