Bone Tissue - Meshless Methods in Biomechanics - page 242

Fig. 6.10 a Voronoï cell with the quadrature points. b Theoretical trabecular architecture of the

sub-cell and homogenized apparent density. c Voronoï cell with the integration points

homogenized apparent densities

process, all the others maintain the previous density. With this approach the
material properties orientation is continuously optimized and only a small fraction
of bone material have its density actualized each time.
The presented iterative process follows the forward Euler scheme with some
particular adaptations to suit the bone internal remodelling analysis. The inclusion
of the NNRPIM meshless method in the remodelling analysis is an asset and not
just another way to obtain the stress and the strain field, since the accuracy of the
remodelling algorithm depends on the accuracy of the used numerical method.

6.3.5.3 Interpretation of the NNRPIM Results

Since all the bone tissue analyses presented in this topic are performed using only
the NNRPIM, first the analysed problem domain, X

d , is discretized by a nodal

R

d and then the Voronoï diagram,
V ¼ V 1 ; V 2 ; ... ; V f g , is obtained, being X ¼ S i¼1 V i . Using the Voronoï diagram
the integration points are determined, Q ¼ q 1 ; q 2 ; ... ; q Q

distribution, X ¼ x 1 ; x 2 ; ... ; x N f g 2 X with x i 2 R

ffi 2 X with q j 2

d .
Recall that the integrations points are sequentially obtained for each Voronoï cell
V i ,

R

therefore each Voronoï cell V i produces k integration points, being:

f g 2 Q and q 1 ; q 2 ; ... ; q f g V i , Fig. 6.10 a.
In the end of each iteration step the local apparent density of each integration
point q j is obtained, q ð q j Þ app . The infinitesimal subdivisions of each Voronoï cell
(the sub-cells) are the smallest dimensional partition of the domain. Each Voronoï
cell infinitesimal subdivision, S V i , is numerically represented by the respective
integration point. Therefore, it is not possible to obtain the detailed microscale
trabecular arrangement represented in Fig. 6.10 b, in which the bone volume, S b ,
and the void volume, S V , are clearly defined. For the infinitesimal subdivision area
represented by the integration point it is only possible to obtain the volume
porosity, p ð q j Þ , and then the local apparent density q ð q j Þ app ¼ q 0 ð 1 p ð q j ÞÞ , being
q 0 ¼ 2 : 1 g/cm 3

q 1 ; q 2 ; ... ; q k

the compact bone density, Fig. 6.10 b.