Shape Functions - Meshless Methods in Biomechanics - page 139

It is possible to construct a system of n equations with n + m + t unknowns
satisfying Eq. ( 4.156 ) for all n nodes on the support-domain of the interest point
x I ,

<
:

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 m t

f k ð x 1 Þ d k ð x I Þ

i¼1 j¼1 k¼1

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 m t

f k ð x 2 Þ d k ð x I Þ

ð 4 : 158 Þ

i¼1 j¼1 k¼1

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

n m t

f k ð x n Þ d k ð x I Þ

i¼1 j¼1 k¼1

Since Eq. ( 4.158 ) has n + m + t unknowns it is require to consider extra sets of
equations in order to obtain an unique solution [ 36 ]. Therefore, the following
supplementary equations can be can be added to the initial equation system,

8
<

X

n

p 1 ð x i Þ a i ð x I Þ ¼0

i¼1

X

n

p 2 ð x i Þ a i ð x I Þ ¼0

ð 4 : 159 Þ

i¼1

:

.
X

n

p m ð x i Þ a i ð x I Þ ¼0

i¼1

and,

8
<

X

n

f 1 ð x i Þ a i ð x I Þ ¼0

i¼1

X

n

f 2 ð x i Þ a i ð x I Þ ¼0

ð 4 : 160 Þ

i¼1

:

.
X

n

f t ð x i Þ a i ð x I Þ ¼0

i¼1

The extra system of equations can be presented in the following matrix form,