Shape Functions - Meshless Methods in Biomechanics - page 107

is the ability to reproduce an unknown function that is included as a basis function
in the shape functions construction [ 5 ]. In opposition to the consistency property,
which focuses only on the reproducibility of complete polynomial functions, in
reproducibility the unknown function may and may not be a polynomial function.
The reproducibility property can be proved with the following argument [ 18 ].
Consider a field defined by the following expression,
u ð x Þ ¼ X

k

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

ð 4 : 62 Þ

j¼1

d

where f ð x Þ is defined in a functional space f k : and c j ð x Þ are arbitrary

coefficients of f j ð x Þ . The approximation function is defined as,

R 7! R

u h ð x Þ ¼ X

k

f j ð x Þ a j ð x Þþ X

m

p i ð x Þ b i ð x Þ ¼f ð x Þ T a ð x Þþ p ð x Þ T b ð x Þ ð 4 : 63 Þ

j¼1 i¼1

It is possible to rewrite Eq. ( 4.63 ) as,

¼ g ð x Þ T d ð x Þ

n o a ð x Þ

b ð x Þ

u h ð x Þ ¼ f ð x Þ T ; p ð x Þ T

ð 4 : 64 Þ

Minimizing the quadratic norm presented in Eq. ( 4.11 ) it is possible to obtain a
similar expression to Eq. ( 4.27 ),

u h ð x Þ ¼ X

n

g ð x Þ A ð x Þ 1 W ð x Þ g ð x Þ T u ð x Þ ð 4 : 65 Þ

i¼1

The field functional defined in Eq. ( 4.62 ) can be written as,

() ¼ g ð x Þ T e ð x Þ

c ð x Þ
0
½m 1

u ð x Þ ¼f ð x Þ T c ð x Þ ¼ f f ð x Þ T ; p ð x Þ T g ð 4 : 66 Þ

Substituting the field functional defined in Eq. ( 4.66 ) in the approximation
functions, Eq. ( 4.65 ), it possible to obtain,

u h ð x Þ ¼ X

n

g ð x Þ T A ð x Þ 1 W ð x x i Þ g ð x i Þ g ð x i Þ T e ð x Þ ð 4 : 67 Þ

i¼1

Being the weighted moment matrix A ð x Þ defined as in Eq. ( 4.21 ),

A ð x Þ ¼ X

n

W ð x x i Þ g ð x i Þ g ð x i Þ T ð 4 : 68 Þ

i¼1