Shape Functions - Meshless Methods in Biomechanics - page 102

Fig. 4.4 1D weight
functions. W3 cubic spline.
W4 quartic spline

!

2

2 n i n I
d i d I

a n s i

o

n i n I
d i d I

d 2
I

s n 1
i

¼ a n þ n 1 ð Þ ð 4 : 56 Þ

on 2

The second order cross partial derivative with respect to variables n and g of the
same s i dependent polynomial term of Eq. ( 4.55 ) can be determined with,

2 n i n I
d i d I

o a n s i

ð n i n I Þ g i g I ð Þ

s n 1
i

¼ a n n 1 ð Þ ð 4 : 57 Þ

d i d I

onog

Both cubic and quartic spline weight functions are continuous in the entire
support-domain, Fig. 4.4 . However the cubic spline weight function does not
present a smooth second order derivative and it is unable to provide a continuous
third order derivative. The quartic spline weight function is numerically more
efficient, it provides smooth and continuous first and second order derivatives.
These features are presented in Figs. 4.5 and 4.6 .
If a two-dimensional domain is considered the obtained cubic and quartic spline
weight functions assume the bell-shaped surfaces indicated in Fig. 4.7 . The
respective first order derivatives are presented in Fig. 4.8 and the second order
derivatives in Fig. 4.9 . The second order derivative two-dimensional representa-
tion repeats the one-dimensional observation, the cubic spline weight function is
not able to produce a smooth second order derivative surface, Fig. 4.10 .
With the cubic and the quartic spline weight functions, the function itself and
the respective first and the second order partial derivatives are exactly zero on the
support-domain boundary. Therefore, both weight functions are capable to provide
compatibility up to the second order. The quartic spline weight functions allow
compatibility up to the third order.