Meshless Methods Introduction - Meshless Methods in Biomechanics - page 67

to be determined on the surface domain C t ð S Þ using only the spatial information of
the field nodes belonging to C t ð S Þ . Thus, an additional set of integration points
Q C ¼ f q 1 ; q 2 ; ... ; q QC g2 C t ð S Þ is obtained just to integrate numerically the
external forces functional t S ð x Þ on the boundary surface C t ð S Þ . Notice that now the
integration points on set Q C represent infinitesimal surfaces, which means that the
integration weight _ C

I of each integration point q I 2 Q C represent a surface area. A
schematic example of the field nodes and integration points along the boundary
surface C t ð S Þ is presented in Fig. 3.18 .
Once again, using only the field nodes X C on the boundary surface C t ð S Þ , new
influence-domains have to be determined for each q I 2 Q C . Then, it is possible to
construct the meshless shape functions of each integration point q I and to obtain

the global force vector on the boundary surface C t ð S Þ ,

¼ Z H T t S ð x Þ dC t ð S Þ ¼ X

Q

_ C
I

H I

t S ð q I Þ
|{z}
½3 1

f S
|{z}
½3N 1

ð 3 : 26 Þ

|{z}
½3N 3

I¼1

C t ð S Þ

being the generic vector t S ð q I Þ defined by t S ð q I Þ ¼ f t S ð q I Þ x ; t S ð q I Þ y ; t S ð q I Þ z g and
the global diagonal shape function matrix H I defined as in Sect. 3.4.2 .
After the definition of all force vectors it is possible to combine the three load
vectors in a global force vector,

f
|{z}
½3N 1

¼ f b þ f C þ f S ð 3 : 27 Þ

|{z}
½3N 1

|{z}
½3N 1

|{z}
½3N 1

3.4.5 Essential Boundary Conditions

In meshless methods based on the Galerkin weak formulation, the body forces and
the external forces are naturally formulated into the discretized system of equa-
tions. Thus, the body forces and the external forces are generally named as natural
boundary conditions. On the other hand, the boundary conditions regarding dis-
placement constrains are not included in the weak form. Thus, it is essential to
impose explicitly the displacement constrains on the discretized system of equa-
tions. This is the reason why the displacement constrains are called essential
boundary conditions.
In the case of meshless methods approximations, the essential conditions can be
exactly satisfied but the natural conditions can only be satisfied up to the order of
the shape function of the meshless method.