Shape Functions - Meshless Methods in Biomechanics - page 140

P T a ð x I Þ ¼0 ð 4 : 161 Þ

and

F T a ð x I Þ ¼0

ð 4 : 162 Þ

Combining Eq. ( 4.158 ) with ( 4.159 ) and ( 4.160 ) it is obtained the following
system of equations,

8
<

9
=

8
<

9
=

8
<

9
=

2
4

3
5

u h ð x Þ
z P
z F

RPF
P T

a ð x I Þ
b ð x I Þ
d ð x I Þ

a ð x I Þ
b ð x I Þ
d ð x I Þ

¼ Z 1 Z 2 ¼ M T ð 4 : 163 Þ

: ; : ; : ;

F T Z 2

Z 3

Being (Z 1 ) ij = 0 for {i, j} = 1, 2, …, m, Z 2 ) ij = 0 for i = 1, 2, …,
m ^ j = 1, 2, …, t and (Z 3 ) ij = 0 for {i, j} = 1, 2, …, t. The null vectors are
defined as (z F ) i = 0 with i = {1, 2, …, t} and (z P ) i = 0 with i = {1, 2, …, m}.
As in previous subsection, the field functional u ð x Þ from Eq. ( 4.155 ) can also be
presented as,

u ð x Þ ¼ X r i ð x i x I Þ 0 þ X p j ð x Þ 0 þ X

n m t

f k ð x Þ c k ð x Þ ð 4 : 164 Þ

i¼1 j¼1 k¼1

and in the matrix form as,

8
<

9
= z R

z P
c ð x Þ

u ð x Þ ¼ r ð x Þ T z R þ p ð x Þ T z P þ f ð x Þ T c ð x Þ ¼ r ð x Þ T

p ð x Þ T f ð x Þ T

: ;

ð 4 : 165 Þ

being (z R ) i = 0 with i = {1, 2, …, n} and (z P ) i = 0 with i = {1, 2, …, m}. Thus,
it is possible to obtain the n nodal values on the support-domain of x I with the
following system of equations,

8
<

u ð x 1 Þ ¼ X

n

r i ð x i x 1 Þ 0 þ X

m

p j ð x 1 Þ 0 þ X

t

f k ð x 1 Þ c k ð x I Þ

i¼1 j¼1 k¼1

u ð x 2 Þ ¼ X r i ð x i x 2 Þ 0 þ X p j ð x 2 Þ 0 þ X

n m t

f k ð x 2 Þ c k ð x I Þ

ð 4 : 166 Þ

i¼1 j¼1 k¼1

:

.
u ð x n Þ ¼ X r i ð x i x n Þ 0 þ X p j ð x n Þ 0 þ X

n m t

f k ð x n Þ c k ð x I Þ

i¼1 j¼1 k¼1