Meshless Methods Introduction - Meshless Methods in Biomechanics - page 66

Fig. 3.18 Representation of
the natural boundaries

X C ¼ x 1 ; x 2 ; ... ; x N f g 2 C t ð C Þ , being X C X and NC the number of nodes along
the boundary curve C t ð C Þ . Then new integration points have to be determined on
the boundary curve C t ð C Þ , since the integration points discretizing the entire solid
domain, Q, are not valid to numerically integrate a functional along the boundary
curve C t ð C Þ . From Sect. 3.3 it is possible to understand that Q X n C, as illustrated
in Fig. 3.16 b. Notice that the set Q discretizes a volume and now a curve dis-
cretization is required.
Therefore, based on the nodal discretization X C , a new set of integration points
Q C ¼ f q 1 ; q 2 ; ... ; q QC g2 C t ð C Þ is defined, in which QC represents the number of
integration points along the boundary curve C t ð C Þ . In opposition to the set Q,in
which the integration points represent infinitesimal volumes, the integration points
on set Q C represent infinitesimal curves. Thus, the integration weight _ C

I of each
integration point q I represent a length. In Fig. 3.18 are represented the field nodes
and the integration points along the boundary curve C t ð C Þ .
Since new integration points were determined, Q C , new influence-domains have
to be established for each one q I 2 Q C . However the nodes of the new influence-
domains have to belong to the field nodes on the boundary curve C t ð C Þ . It is not
permitted the inclusion of any other field node. Hence, the meshless shape func-
tions determined for each one integration point q I are constructed using only the

nodes on the boundary curve C t ð C Þ .
The global force vector for the boundary curve C t ð C Þ is obtained with the
following expression,

¼ Z

H T t C ð x Þ dC t ð C Þ ¼ X

Q

_ C
I

H I

t C ð q I Þ
|{z}
½3 1

f C ð 3 : 25 Þ

|{z}
½3N 1

|{z}
½3N 3

I¼1

C t ð C Þ

The generic vector t C ð q I Þ depends on the integration point spatial position and it
is defined by t C ð q I Þ ¼ f t C ð q I Þ x ; t C ð q I Þ y ; t C ð q I Þ z g . The global diagonal shape
function matrix H I is obtained following the process described in Sect. 3.4.2 .
If the external forces t S ð x Þ are applied along a boundary surface C t ð S Þ 2 C,
Fig. 3.15 , first the field nodes belonging to C t ð S Þ have to be identified:

X C ¼ x 1 ; x 2 ; ... ; x NC f g 2 C t ð S Þ , being X C X. Next, new integration points have