Shape Functions - Meshless Methods in Biomechanics - page 119

Fig. 4.19 The discrete nodal
parameters u ð x i Þ and the
PIM interpolation function
u h ð x Þ

be satisfied to all n nodes on the support-domain of the interest point x I , yielding
n equations with m unknowns,

8
<

X

m

f i ð x 1 Þ b i ð x I Þ ¼u ð x 1 Þ ¼f 1 ð x 1 Þ b 1 ð x I Þþ f 2 ð x 1 Þ b 2 ð x I Þþþ f m ð x 1 Þ b m ð x I Þ

i¼1

X

m

f i ð x 2 Þ b i ð x 2 Þ ¼u ð x 2 Þ ¼f 1 ð x 2 Þ b 1 ð x 2 Þþ f 2 ð x 2 Þ b 2 ð x 2 Þþþ f m ð x 2 Þ b m ð x 2 Þ

i¼1

:

.
X

m

f i ð x n Þ b i ð x n Þ ¼u ð x n Þ ¼f 1 ð x n Þ b 1 ð x n Þþ f 2 ð x n Þ b 2 ð x n Þþþ f m ð x n Þ b m ð x n Þ

i¼1

ð 4 : 88 Þ

Being by imposition m = n, if all the equations are independent, it is possible to
determine uniquely all the b i ð x I Þ unknowns. The equation system in Eq. ( 4.88 ) can
be presented in the following matrix form,

F b ð x I Þ ¼u s ð 4 : 89 Þ

where b ð x I Þ and u s are defined respectively by Eqs. ( 4.7 ) and ( 4.20 ). The moment
matrix F with dimensions n 9 m is in fact square, because m = n. Additionally,
F does not depend on the interest point spatial position x I , it only depends on the
spatial position of the n nodes inside the support-domain of the interest point x I .
Therefore, the components of moment matrix F are constants. The moment matrix
F is defined as,