Shape Functions - Meshless Methods in Biomechanics - page 110

Fig. 4.11 Representation of
the one-dimensional domain,
the interest point weight
function and potential shape
function

1

X ¼ x 1 f x 2 x 3 x 4 x 5 g ¼ 2 1012 f g 2 R ð 4 : 78 Þ

A generic bell-shaped weight function W I (x Þ is considered, Fig. 4.11 . Being
x I x 3 , the weight function is centred in the problem domain. Therefore,

8
<
:

W I ð x 1 Þ ¼W I ð x 5 Þ ¼w 1
W I ð x 2 Þ ¼W I ð x 4 Þ ¼w 2
W I ð x 3 Þ ¼w 3 ¼ 1

ð 4 : 79 Þ

It is assumed a linear polynomial basis, p ð x Þ ¼ f 1 x g , therefore for the

interest point x I it is obtained,

p ð x I Þ ¼p ð x 3 Þ ¼ f 1 x I g ¼ f 10 g ð 4 : 80 Þ

The weighted polynomial matrix, B ð x Þ , defined in Eq. ( 4.23 ), can be obtained
with,

w 2 w 3 w 2 w 1

1
x 1

1
x 2

1
x 3

1
x 4

1
x 5

B ð x Þ ¼ w 1 ð 4 : 81 Þ

and after the substitution with the nodal values,

w 1 w 2 1 w 2 w 1

B ð x Þ ¼ ð 4 : 82 Þ

2w 1 w 2 0 w 2 2w 1

In this demonstration the coefficients of A ð x Þ 1 , which can be obtained
inverting the weighted moment matrix A ð x Þ defined in Eq. ( 4.21 ), are considered
as unknowns,

a 11 a 12

A ð x Þ 1 ¼

ð 4 : 83 Þ

a 12 a 22