Shape Functions - Meshless Methods in Biomechanics - page 106

Fig. 4.10 Comparison between the cubic spline (W3) and the quartic spline (W4) weight
functions second order partial derivatives

u ð x Þ ¼ X p i ð x Þ a i ð x Þþ X

k m

p j ð x Þ 0 ð 4 : 59 Þ

i¼1 j¼k þ 1

The non-constants coefficients b i ð x Þ can be obtained with the minimization of
the quadratic norm presented in Eq. ( 4.11 ). Then, substituting in Eq. ( 4.11 ) the
approximation function, u h ð x Þ , from Eq. ( 4.6 ) and the field function, u ð x Þ , defined
by Eq. ( 4.59 ), it is possible to write,

" ! X ! # 2

J ¼ X W ð x Þ X p i ð x Þ b i ð x Þþ X p i ð x Þ a i ð x Þþ X

n k m k m

p j ð x Þ b j ð x Þ p j ð x Þ 0

i¼1 i¼1 j¼k þ 1 i¼1 j¼k þ 1

ð 4 : 60 Þ

It is visible that J = 0 (and consequently qJ/qb = 0) if b i ð x Þ ¼a j ð x Þ; i ¼
f 1 ; 2 ; ... ; k g and b j ð x Þ ¼0 ; j ¼ f k þ 1 ; ... ; m g , leading to,

u h ð x Þ ¼ X

k

p i ð x Þ a i ð x Þ ¼u ð x Þ ð 4 : 61 Þ

i¼1

which proves the MLS approximation is capable to reproduce any set of mono-
mials included in the polynomial basis of the MLS formulation.

4.3.3.2 Reproducibility

Since in meshless methods distinct types of basis functions can be used in the
construction of the shape functions, within meshless methods the reproducibility
property is not included in the consistency property. The reproducibility property