Meshless Methods Introduction - Meshless Methods in Biomechanics - page 60

Voronoï cell subdivision in hexahedral sub-cells. Then each partition, Fig. 3.14 c,
is isoparameterized, Fig. 3.14 d, and the Gauss-Legendre quadrature scheme is
applied. Afterwards the obtained integrations points spatial location and respective
integration weights are transformed again to the Cartesian coordinate system,
Fig. 3.14 e.
The 3D analysis is computational very demanding. Regarding the NNRPIM and
the NREM meshless methods, research works available in the literature [ 19 , 43 ]
show that, although it is possible to apply all the integration schemes referred in
Sect. 3.3.2.1 , there is no need to use a Gauss-Legendre quadrature scheme higher
than 1 1 1 per hexahedron. This integration scheme is sufficient for the 3D
NNRPIM and 3D NREM meshless formulations, since higher integration schemes
raise enormously the computational cost. Additionally it was noticed that the 3D
nodal integration without stabilization is not sufficient to integrate the NNRPIM or
the NREM interpolation functions.

3.4 Numerical Implementation

This topic presents two meshless methods based on the Galerkin weak formula-
tion, defined over the global problem domain, using the locally supported meshless
shape functions that will be introduced in Chap. 4 . In this section it is presented
the generic numerical implementation of approximation and interpolation mesh-
less methods based on the Galerkin weak formulation.
Consider the solid domain defined by X

3 and bounded by C, Fig. 3.15 . The
solid domain is made from a generic heterogeneous material, being the density of
such material defined by the functional q ð x Þ2

R

.
Besides the body forces caused by the material density, the solid is submitted to
the external forces t C ð x Þ and t S ð x Þ applied on the boundary C, respectively over the
curve C t ð C Þ 2 C and over the surface, C t ð S Þ 2 C. The solid displacements are
constrained on the curve C u ð C Þ 2 C and on the surface, C u ð S Þ 2 C.
First, the solid domain is discretized by a irregularly nodal distribution
X ¼ x 1 ; x 2 ; ... ; x f g 2 X, being N the total number of nodes discretizing the
problem domain and x 2

R

3 , Fig. 3.16 a. These field nodes permit to approximate

R

u ¼ f u 1 ; u 2 ; ... ; u N g T , the field variable—the displacement field being

u i ¼ f u i ; v i ; w i g T 2

3 ^ i 2f 1 ; 2 ; ... ; N g .
With the nodal distribution it is possible to obtain the background integration
mesh, using for instance one of the integration schemes presented in Sect. 3.3 .In
the

R

end, an integration mesh discretizing the problem domain is obtained,

2 X, being Q the total number of integration points dis-
cretizing the problem domain and q 2

Q ¼ q 1 ; q 2 ; ... ; q Q

3 the spatial coordinates. Notice that
Q X n C, Fig. 3.16 b, i.e., the integration points are defined only inside the solid

R