Shape Functions - Meshless Methods in Biomechanics - page 129

¼ X

n

u i þ X

m

2 u h ð x I Þ
onog

2 u i ð x I Þ
onog

2 w i ð x I Þ
onog z i
|{z}
0

o o o

¼ u ð x I Þ ; ng u s ð 4 : 135 Þ

i¼1 i¼1

Following the same argument used to obtain Eq. ( 4.131 ), the spatial second
order partial derivative of the RPI shape function with respect to n and g, is defined
as,

n o ¼ r ð x I Þ ; ng n o M 1

T

u ð x I Þ ; ng w ð x I Þ ; ng p ð x I Þ ; ng ð 4 : 136 Þ

in which the second order cross partial derivative of the RBF vector is defined as,

T

r ð x I Þ ; ng ¼ r 1 ð x I Þ ; ng

r 2 ð x I Þ ; ng ... r n ð x I Þ ; ng

n o T

¼ o 2 r 1 ð x I Þ
onog

o 2 r 2 ð x I Þ
onog

o 2 r n ð x I Þ
onog

ð 4 : 137 Þ

being,

p 2

o 2 r i ð x I Þ
onog

Þ d iI þ cd ð 2 ¼ 4pp 1 ð Þ n i n I ð Þ g i g I ð ð 4 : 138 Þ

The respective second order cross partial derivative of the polynomial basis
vector is obtained with,

T

p ð x I Þ ; ng ¼ p 1 ð x I Þ ; ng

p 2 ð x I Þ ; ng ... p m ð x I Þ ; ng

n o T

¼ o 2 p 1 ð x I Þ
onog

o 2 p 2 ð x I Þ
onog

o 2 p n ð x I Þ
onog

ð 4 : 139 Þ

The second order partial derivative of the interpolated field function with
respect to the generic variable n can be determined with,

¼ X u i þ X

o 2 u h ð x I Þ
on 2

n

o 2 u i ð x I Þ
on 2

o 2 w i ð x I Þ
on 2 z i
|{z}
0

m

¼ u ð x I Þ ; nn u s ð 4 : 140 Þ

i¼1 i¼1

Following the previous argument, the spatial second order partial derivative of
the PIM shape function with respect to n is obtained.

n o ¼ r ð x I Þ ; nn n o M 1

T

u ð x I Þ ; nn w ð x I Þ ; nn p ð x I Þ ; nn

ð 4 : 141 Þ