Shape Functions - Meshless Methods in Biomechanics - page 99

u ð x I Þ ; ng ¼ p ð x I Þ ; n A ð x I Þ 1 B ð x I Þþ p ð x I Þ T A ð x I Þ 1

; n B ð x I Þþ p ð x I Þ T A ð x I Þ 1 B ð x I Þ ; n

; g

¼ p ð x I Þ ; ng A ð x I Þ 1 B ð x I Þþ p ð x I Þ ; n A ð x I Þ 1

; g B ð x I Þþ p ð x I Þ ; n A ð x I Þ 1 B ð x I Þ ; g

þ p ð x I Þ ; g A ð x I Þ 1

; n B ð x I Þþ p ð x I Þ T A ð x I Þ 1 ; ng B ð x I Þþ p ð x I Þ T A ð x I Þ 1 ; n B ð x I Þ ; g

; g B ð x I Þ ; n þ p ð x I Þ T A ð x I Þ 1 B ð x I Þ ; ng
ð 4 : 41 Þ

þ p ð x I Þ ; g A ð x I Þ 1 B ð x I Þ ; n þ p ð x I Þ T A ð x I Þ 1

Being,

A ð x I Þ 1

; ng ¼ A ð x I Þ 1 A ð x I Þ ; n A ð x I Þ 1

; g

¼ A ð x I Þ 1

; g A ð x I Þ ; n A ð x I Þ 1 A ð x I Þ 1 A ð x I Þ ; nn A ð x I Þ 1 A ð x I Þ 1 A ð x I Þ ; n A ð x I Þ 1 ; g

ð 4 : 42 Þ

and A ð x I Þ ; nn obtained with,

A ð x I Þ ; ng ¼ X

n 2 W ð x i x I Þ

onog

o

p ð x i Þ p ð x i Þ T

ð 4 : 43 Þ

i¼1

The second order partial derivative of matrix B ð x I Þ with respect to n and g is
obtained with,

h i

B ð x I Þ ; ng ¼ o 2 W ð x 1 x I Þ
onog

o 2 W ð x 2 x I Þ
onog

p ð x 2 Þ o 2 W ð x n x I Þ
onog

p ð x 1 Þ p ð x n Þ

ð 4 : 44 Þ

Third, fourth and higher order partial derivatives of the MLS shape functions
can be similarly obtained by repeated differentiations. It is perceptible that the
continuity of the MLS shape function depends on the weight function continuity.

4.3.2 Weight Functions

In the construction of the MLS shape functions the weight function possesses a
crucial importance. The weight function should be positive and bell-shaped, i.e., it
should decrease in magnitude with the increase of the Euclidean norm between the
interest point x I and the ith node x i inside the support-domain,

q

ð x i x I Þ 2 þð y i y I Þ 2 þð z i z I Þ 2

d i ¼ jj x i x I jj ¼ ; 8 x i 2 X I ð 4 : 45 Þ