Shape Functions - Meshless Methods in Biomechanics - page 95

since b ð x I Þ does not depend on x i ,
X

!

n

b ð x I Þ ¼ X

n

W ð x i x I Þ p ð x i Þ p ð x i Þ T W ð x i x I Þ p ð x i Þ T u i ð 4 : 18 Þ

i¼1 i¼1

Leading to the following linear system of equations,
A ð x I Þ b ð x I Þ ¼B ð x I Þ u s

ð 4 : 19 Þ

being u s the vector with the field function nodal parameters for each node inside
the support-domain of the MLS shape function,

u s ¼ f u 1 u 2 ... u n g ð 4 : 20 Þ

The weighted moment matrix A ð x I Þ can be defined as,
A ð x I Þ ¼ X

n

W ð x i x I Þ p ð x i Þ p ð x i Þ T ð 4 : 21 Þ

i¼1

Considering the three-dimensional space, x ¼ f x ; y ; z g , and using a linear
polynomial basis, p ð x Þ T ¼ f 1 xyz g with m ¼ 4, the moment matrix A ð x I Þ

can be explicitly defined as,

2
4

3
5

1
x i
y i
z i

A ð x I Þ ¼ X

n

W ð x i x I Þ 1 x i y i z i

½

i¼1

2
4

3
5

2
4

3
5

1 x 1 y 1 z 1 1 x n y n z n

x 1 x n

x 1 x 1 y 1 x 1 z 1 x n x n y n x n z n

¼ W ð x 1 x I Þ þþ W ð x n x 1 Þ

y 1 y n y 1 y 1 x 1 y 1 z 1 y n y n x n y n z n

z 1 z 1 x 1 z 1 y 1 z 1 z n z n x n z n y n z n

ð 4 : 22 Þ

The weighted polynomial matrix, B ð x I Þ , can be defined as,

B ð x I Þ ¼½W ð x 1 x I Þ p ð x 1 Þ W ð x 2 x I Þ p ð x 2 Þ W ð x n x I Þ p ð x n Þ ð 4 : 23 Þ

Therefore, considering the same previous spatial conditions and polynomial
basis, it is possible to defined B ð x I Þ explicitly as,

2
4

2
4

3
5

2
4

3
5

2
4

3
5

3
5

1
x 1
y 1
z 1

1
x 2
y 2
z 2

1
x n
y n
z n

B ð x I Þ ¼ W ð x 1 x I Þ W ð x 2 x I Þ ... W ð x n x I Þ

ð 4 : 24 Þ