Meshless Methods Introduction - Meshless Methods in Biomechanics - page 76

Being q I an interest point discretizing the natural boundary curve, Q Ct ð C Þ , or the
natural boundary surface, Q Ct ð S Þ , as represented in Fig. 3.18 .
A closer look on Eq. ( 3.40 ) permit find some resemblances between the force
integrals and the Lagrange multipliers integrals. In fact the Lagrange multipliers
can be seen as ''smart forces'' forcing u u ¼ 0. In order to obtain the discretized
formulation of the last two terms in Eq. ( 3.40 ), the Lagrange multipliers have to
be approximated using the respective nodal values and the shape functions for the
field nodes along the essential boundaries, sets X Cu ð C Þ and X Cu ð S Þ . Thus, since
the Lagrange multipliers are unknown functions of the coordinates variables,
the Lagrange multipliers approximation on an interest point q I , belonging to the
essential boundary (q I 2 Q Cu ð C Þ or q I 2 Q Cu ð S Þ ), can be determined with,

dk ð q I Þ ¼ X

n k

U i ð q I Þ dk ð x i Þ ¼ N local

dk ð x Þ
|{z }
½3n k 1

;

x 2 C u

ð 3 : 47 Þ

I

|{z}
½3 3n k

i¼1

being n k the number of field nodes used for this interpolation. The vector of the
virtual Lagrange multipliers nodal values, considering only the nodes on the
essential boundary C u , can be represented for the classical three-dimensional
deformation theory as,

dk ð x Þ ¼ dk ð x 1 Þ x

dk ð x 1 Þ y dk ð x 1 Þ z dk ð x n k Þ x dk ð x n k Þ y dk ð x n k Þ z

ð 3 : 48 Þ

The local diagonal interpolation matrix N local
I

is defined by,

ð 3 : 49 Þ

The interpolation function value U i ð q I Þ is not obtained using the MLS
approximation, alternatively U i ð q I Þ it is obtained using, for example, the Lagrange
interpolants.
This is the point where the Lagrange multipliers methodology becomes more
delicate. Each integration point q I 2 Q Cu possesses two influence-domains: one
with n field nodes belonging to C u used to construct the MLS approximation shape
functions u ð q I Þ , Eq. ( 3.18 ), and another with n k field nodes, also belonging to C u ,
used to construct the interpolation functionU ð q I Þ ,