Shape Functions - Meshless Methods in Biomechanics - page 120

2
4

3
5

f 1 ð x 1 Þ f 2 ð x 1 Þ f m ð x 1 Þ

f 1 ð x 2 Þ f 2 ð x 2 Þ f m ð x 2 Þ

F ¼ ð 4 : 90 Þ

. .

.

.

. .

f 1 ð x n Þ f 2 ð x n Þ f m ð x n Þ

Solving Eq. ( 4.89 ) it is possible to obtain the coefficients b ð x I Þ ,

b ð x I Þ ¼F 1 u s

ð 4 : 91 Þ

The obtained coefficients b ð x I Þ are in fact constant as long as the same n nodes
inside the support-domain of the interest point x I are maintained. The PIM shape
functions can be obtained substituting Eq. ( 4.91 )in( 4.87 ),

u h ð x I Þ ¼f ð x I Þ T F 1 u s ¼ X

n

u i ð x I Þ u ð x i Þ ¼u ð x I Þ T u s ¼ u ð x I Þ ð 4 : 92 Þ

i¼1

being u i ð x I Þ the shape function value of interest point x I on the ith node, obtained
considering the nodes inside the support-domain of interest point x I . The PIM
shape function vector for the n nodes inside the support-domain of x I is defined as,

u ð x I Þ T ¼ f u 1 ð x I Þ ... u n ð x I Þg ¼ f ð x I Þ T F 1 u 2 ð x I Þ ð 4 : 93 Þ

Notice that in opposition to the MLS shape functions, in which the moment
matrix A(x I Þ and the weighted polynomial matrix B(x I Þ , used to construct the
approximation function u h ð x Þ in Eq. ( 4.26 ), have to be defined for a specific
interest point x I possessing a particular support-domain with n nodes, in the PIM
shape functions construction the moment matrix F does not depend on the interest
point x I spatial position, therefore F is valid for other interest points possessing the
same support-domain.
It is necessary to determine the PIM shape functions partial derivatives in order
to obtain the partial derivatives of the interpolated field function, Eq. ( 4.92 ).
Compared with the MLS shape functions, the PIM shape functions partial deriv-
atives are much more simple to obtain. The first order partial derivatives of
interpolated field function, with respect to a generic variable n, can be obtained
with,

¼ X

n

o u h ð x I Þ
on

o u i ð x I Þ
on

u i ¼ u ð x I Þ ; n u s ð 4 : 94 Þ

i¼1

being the first order partial derivative with respect to n of the PIM shape function
defined as,