Shape Functions - Meshless Methods in Biomechanics - page 101

d I ¼ max jj x i x I jj; 8 x i 2 X I ð 4 : 50 Þ

However, since the exponential weight functions is computational demanding
and the conical weight function presents a low continuity, other weight functions
were proposed in several EFGM research works [ 13 - 16 ]. Presently the most
popular weight functions used in the construction of the MLS shape function are
the cubic spline weight function,

8
<

2

3 4s i þ 4s i ;

s i 0 : 5

4

3 4s i þ 4s i 3 s i ;

W ð d i Þ ¼W ð s i Þ ¼ 0 : 5\s i 1 ð 4 : 51 Þ

:

0 ; s i [ 1

with a second order continuity, and the quartic spline weight function,

W ð d i Þ ¼ W ð s i Þ ¼ 1 6s i þ 8s i 3s i ;

s i 1

ð 4 : 52 Þ

0 ; s i [ 1

with a third order continuity. Notice that both weight functions are polynomials
only dependent on the s i radial variable. In the three-dimensional space s i can be
explicitly expressed as,

2

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

s i ¼ d i
d I

¼ ð 4 : 53 Þ

d I

The first order partial derivate with respect to a generic variable n, which can be
x, y or z,ofas i dependent polynomial term can be defined as,

na n s n 1

o ð a n s i Þ
on

¼ n i n I
d i d I

ð 4 : 54 Þ

i

a constant. Therefore, it is possible to present the first order
partial derivate, with respect to n, of the cubic spline weight function,

Being a n 2 R

8
<

8s i þ 12s i ;

s i 0 : 5

o W ð d i Þ
on

¼ o W ð s i Þ
on

¼ n i n I
d i d I

4 þ 8s i 4s i ;

ð 4 : 55 Þ

0 : 5\s i 1

:

0 ; s i [ 1

In order to obtain the MLS shape function second order partial derivatives,
Eq. ( 4.41 ), it is required to compute the weight function second order partial
derivatives. Therefore, the second order partial derivative with respect to a stan-
dard variable n of a generic s i dependent polynomial term of Eq. ( 4.55 ) can be
obtained with,