Page 51 - IJB-9-6

International Journal of Bioprinting Sub-regional design of the bionic bone scaffolds

Figure 2. Generation of the graded Voronoi nucleating points where one can see (a) design parameters of the combined probability sphere model,
} is marked green.
} is marked red while {P
(b) combined probability sphere model and Voronoi nucleating points, where {P 1 S N 2 S M
i i 1=
j j 1=
where i = 1, 2, …, n represents the volume mesh
number of the macroscopic model and ρ is a binary design
i
variable representing the cell density of the i-th volume
mesh. A mesh cell is considered deleted when ρ = ρ , and
min
i
it is considered reserved when ρ = 1. The variables U, K,
i
δ, and P represent, respectively, the strain energy, the total
stiffness matrix, the displacement vector, and the external Figure 3. Definition of the scale coefficient where one can see (a) C
face
load vector acting on the structure. V is the volume of the and (b) C .
cell
i
i-th volume mesh cell while V is the initial volume of the
*
macroscopic model. f is the volume fraction coefficient, of a and r are indeterminate, and the specific values will
v
t
t
representing the ratio of the target volume to the initial be discussed in section 3. For random points P , generated
S
1
*
volume. In addition, E and E represent, respectively, the from the probability sphere model with dot pitch of a ,
i
i
1
elastic modulus of the i-th volume mesh cell before and the Boolean operation was used to get a point set {P S N
}
after the topological optimization. Finally, the penalty 1 i i 1=
coefficient p has a fixed value (equal to 3) in this work in sub-region B. In contrast, only the interior and the
[38]
referring to previous studies . surface points in sub-region A were retained as a point set
}
} for P . By combining these two sets of points {P
It is obvious that the 20 × 20 × 20 mm design domain {P 2 S M = S 2 1 S N
3
j j 1
i i 1=
was divided into two sub-regions by the topological model. and {P S M NM+ was defined
} , a set of irregular points {}S
=
After extensive tests, it was found that arranging nucleating 2 j j 1 kk=1
points only in sub-region A leads to extremely poor as the nucleating points of the bionic bone scaffolds. The
geometric continuity. Meanwhile, open meshes were even irregularity ε of this scaffold is defined by the distance from
observed, leading to forming failure. Therefore, this study a random point P to its corresponding probability sphere
S
improved the methodology of generating controllable center P : t
nucleating points that was adopted in previous studies , t S
[12]
(
S
(,
M
proposing a combined probability sphere model where ε= 1 N dist PP ) + ∑ dist PP , 2 j ) (II)
2 j
∑
1i
1i
a and a are the dot pitch of sphere centers (a > a ). It N + M =1 a 1 j =1 a 2
i
1
2
1
2
is worth noting that the following constraints were added:
where N and M represent, respectively, the quantities
(i) r / a = r / a for the purpose of ensuring geometric of P and P after the Boolean operations. The number
S
S
2
1
2
1
2
1
continuity and design controllability, where r and of nucleating points (NNP) is equal to the sum value of
1
r represent the radius of spheres generated from N and M (NNP = N + M). The variables dist(P , P ) and
S
2
regular dot matrices P and P , respectively; S 1i 1i
1i 2j dist(P , P ) denote the distance from the random point
2j
2j
(ii) 0 < r < a / 2 where t ∈ {1,2} is a binary design P to the center of its corresponding probability sphere P t
S
t
t
t
variable, representing two different types of the in 3D Euclidean space. Since the radius of the probability
regular dot matrix. sphere was restricted, ε can be easily deduced (0 < ε < 0.5).
The same r / a value provided an extremely high The point set {}S NM+ was processed using the Voronoi
kk=1
controllability and consistency to the design procedure, 3D Grasshopper™ plugin to obtain the 3D Voronoi cell
effectively alleviating the possible stress concentration and structure. The scale coefficient C was introduced to realize
geometric mutation phenomenon. The Boolean operation the deflation of Voronoi faces and the Voronoi cells. As
process based on the combined probability sphere model is shown in Figure 3, C can be further divided into two
shown in Figure 2. It is worth pointing out that the values variables, C and C , where they represent, respectively,
face
cell
Volume 9 Issue 6 (2023) 43 https://doi.org/10.36922/ijb.0222