Shape Functions - Meshless Methods in Biomechanics - page 92

Fig. 4.2 Triangle of Pascal
for polynomial monomials

; 8 i ; fg2

h ¼ min x j x i N : i ; fg N ^ i 6 ¼ j ð 4 : 4 Þ

where || || is the L 2 -norm, i.e., the Euclidean norm,

! 1 = 2

jj x jj ¼ X

d

x j 2 ð 4 : 5 Þ

i¼1

being d the space dimension. Consider now a continuous scalar function u ð x Þ ,
being u 2 T. It is possible to define for an interest point x I 2

d , not necessarily

R

coincident with X, the MLS approximation of u ð x I Þ as,

u h ð x I Þ ¼ X

m

p i ð x I Þ b i ð x I Þ ¼p ð x I Þ T b ð x I Þ ð 4 : 6 Þ

i¼1

being b i ð x I Þ the non-constant coefficients of p i ð x I Þ ,

b ð x I Þ T ¼ b 1 ð x I Þ

f b 2 ð x I Þ ... b m ð x I Þ g ð 4 : 7 Þ

The monomials of the polynomial basis are defined by p i ð x I Þ and m is the basis
monomial number. The polynomial basis p ð x I Þ can be constructed using mono-
mials from the triangle of Pascal, Fig. 4.2 .
Therefore, for a one-dimensional space, considering an interest point x I ¼ f x I g ,
a quadratic polynomial basis is defined as,