Shape Functions - Meshless Methods in Biomechanics - page 128

¼ X u i þ X

n m

o u h ð x I Þ
on

o u i ð x I Þ
on

ow i ð x I Þ
on z i
|{z}
0

¼ u ð x I Þ ; n u s ð 4 : 128 Þ

i¼1 i¼1

Thus, with respect to a generic variable n, the first order partial derivatives of
interpolated field function can be determined by,

; n ¼ r ð x I Þ T M 1

T

; n

u ð x I Þ T w ð x I Þ T p ð x I Þ T

ð 4 : 129 Þ

and considering Eq. ( 4.127 ),

M 1
T

; n ¼ r ð x I Þ T ; n M 1 M 1

T ; n
|{z}
0
ð 4 : 130 Þ

r ð x I Þ T p ð x I Þ T p ð x I Þ T

þ r ð x I Þ T

p ð x I Þ T

T

Equation ( 4.129 ) can be presented as,

n o ¼ r ð x I Þ ; n n o M 1

T

u ð x I Þ ; n w ð x I Þ ; n p ð x I Þ ; n ð 4 : 131 Þ

The first order partial derivative of the RBF vector with respect to the same
generic variable n is defined as,

n o T

T ¼ o r 1 ð x I Þ
on

o r 2 ð x I Þ
on

o r n ð x I Þ
on

r ð x I Þ ; n ¼ r 1 ð x I Þ ; n

r 2 ð x I Þ ; n ... r n ð x I Þ ; n

ð 4 : 132 Þ

Being for the MQ-RBF,

p 1

o r i ð x I Þ
on

Þ d iI þ cd ð 2 ¼ 2p n i n I ð ð 4 : 133 Þ

For the same generic variable n, the first order partial derivative of the poly-
nomial basis vector is obtained with,

n o T

T ¼ o p 1 ð x I Þ
on

o p 2 ð x I Þ
on

o p n ð x I Þ
on

p ð x I Þ ; n ¼ p 1 ð x I Þ ; n

p 2 ð x I Þ ; n ... p m ð x I Þ ; n

ð 4 : 134 Þ

It is possible to obtain the second order partial derivative of the interpolated
field function, with respect to the generic variables n and g, with the following
expression,