Shape Functions - Meshless Methods in Biomechanics - page 137

construct the approximation, Eq. ( 4.102 ). The RPI shape functions possess the
consistency property if the polynomial basis is included as basis. Thus, being m the
complete monomial order of the polynomial basis function, then the RPI shape
function possess C m consistency. A field functional defined by a complete poly-
nomial functional, in which t B m, can be expressed by,

u ð x Þ ¼ X

t

p j ð x Þ c j ð x Þ ¼p ð x Þ T c ð x Þ

ð 4 : 150 Þ

j¼1

Consider the approximation function u h ð x Þ defined with Eq. ( 4.102 ). Thus, the
complete polynomial field functional can be conveniently presented as,

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

n t m

p k ð x Þ 0 ð 4 : 151 Þ

i¼1 j¼1 k¼t þ 1

The n nodal values on the support-domain of x I are obtained with,

8
<

u ð x 1 Þ ¼ X

n

r i ð x i x 1 Þ 0 þ X

t

p j ð x 1 Þ c j ð x I Þþ X

m

p k ð x 1 Þ 0

i¼1 j¼1 k¼t þ 1

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

n t m

p k ð x 2 Þ 0

ð 4 : 152 Þ

i¼1 j¼1 k¼t þ 1

:

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

n t m

p k ð x n Þ 0

i¼1 j¼1 k¼t þ 1

Using Eq. ( 4.102 ) it is possible to obtain the coefficients a ð x I Þ and b ð x I Þ by
imposing u h ð x i Þ ¼u ð x i Þ . Therefore, satisfying Eq. ( 4.102 ) for all n nodes on the
support-domain of the interest point x I , a system of n equations with

n + m unknowns is obtained,

<
:

u h ð x 1 Þ ¼ X r i ð x i x 1 Þ a i ð x I Þþ X p j ð x 1 Þ b j ð x I Þþ X

n t m

p k ð x 1 Þ b k ð x I Þ ¼u ð x 1 Þ

i¼1 j¼1 k¼1 þ t

u h ð x 2 Þ ¼ X r i ð x i x 2 Þ a i ð x I Þþ X p j ð x 2 Þ b j ð x I Þþ X

n t m

p k ð x 2 Þ b k ð x I Þ ¼u ð x 2 Þ

i¼1 j¼1 k¼1 þ t

.
u h ð x n Þ ¼ X

n

r i ð x i x n Þ a i ð x I Þþ X

t

p j ð x n Þ b j ð x I Þþ X

m

p k ð x n Þ b k ð x I Þ ¼ u ð x n Þ

i ¼ 1 j ¼ 1 k¼1 þ t

ð 4 : 153 Þ