Page 158 - IJB-7-3

Topology Optimized Locking Compression Plates to Minimize Stress Shielding
modifications (e.g. lattice structure, internal hollow, or porous min Z ρ f ⋅ u
T
( ) =
structures) [10,11] . However, the fabrication of functionally e (2.3)
graded structures is complex and expensive, while design  N
v ≤
modifications usually require laborious and costly iterative  V = ∑ ρ ee V ,
steps performed by expert technicians. Recently, we st . .  e= 1 (2.3a)

proposed a novel design method to minimize stress shielding  ρ = 0 1 , or e = 1,… , ,N
e
using topology optimization (TO) . Results-focused on where V is the initial volume and v is volume of
[12]
two-dimensional (2D) topologies, considering tensile loads each element. e
on the plates, loads on the holes simulating the screw and The Solid Isotropic Microstructure with Penalization
combination of both, and a maximum volume reduction of
75%. The approach allowed to obtain designs of lightweight (SIMP), a gradient-based TO approach, was considered to
. SIMP forces the design variable,
solve the TO problem
[13,14]
plates with reduced equivalent stiffness compared to to have a penalized continuous convergence as follows:
the proposed original design domains. In this paper, we 1
extended the study to three-dimensional (3D) topologies, 0 < ρ ≤ρ ≤ (2.4)
e
0
investigating the use of 3D TO to redesign novel LCPs to E (x,y,z ρ (x,y,z p i
minimize the stress shielding phenomenon. Three initial ) [= )] E⋅ (2.5)
plate designs with different screw hole numbers (four-, six-, N p
and eight-hole plates) were redesigned considering different K ( ) ρ= ∑ ρ ⋅ K e (2.6)
e
loading conditions (compression, bending, torsion, and e 1 =
a combination of these loads) imposing different volume where ρ is the non-zero minimum density, p is the
0
reductions to the initial designs (25, 45, and 75%). The penalization factor (recommended to have a value of
i
effect of mesh density on the TO is also investigated and three for a better convergence), and E is the initial Elastic
discussed. Topology-optimized plate design considering modulus at ρ=1 of the material.
bending load presented bending elastic modulus equivalent Therefore, SIMP solves equation (2.3) as follows:
to native bone. min T
( ) =
2. TO ρ e Z ρ e f ⋅ u (2.7)
TO is an automatic iterative design technique to obtain  N
optimal structural topologies based on the user-defined  ∑ (ρ e )v ≤ V ,
e

loading and boundary conditions. The goal is to achieve  e= 1
N

a design with an optimal material distribution (Ω*) . st ∑ (ρ p )K u = , f (2.7a) (2.7b) (2.7c)
from an initially given design domain (Ω), which is  e e
discretized into number of finite elements (N). The aim  e= 1
was to redesign the domain by minimizing compliance  0 ρ < 0 ρ ≤ e ≤ 1,
(maximizing stiffness): 

T
min f  u (2.1) The iterative steps to achieve optimality through TO
e ρ ,u are shown in Figure 1. The solution starts with the given
where u is the displacement vectors and f is load design domain, Ω, comprising the user-defined material
vectors governed by: being discretized into a set of finite elements, with each
N element consisting of a density value contributing to
f = Kk e ∑  K e ()k e (2.2) the overall design density. Then, sensitivity breakdown
( ) u =
e= 1 is defined across the displacement field according to
where K is the global stiffness matrix, K is the the loading and boundary conditions, which means that
e
matrices of the element stiffness, and k is the stiffness of the density is updated toward optimality, satisfying the
e
each element. objective function and constraints in each iteration.
The density (ρ) is considered as a function to modify The first step in each computational iteration
the design’s stiffness matrix and treated as a design corresponds to the calculation of element sensitivity
variable. Therefore, the density imposed on each element based on deriving Eq. 2.7 in respect to the density:
(ρ ) of the design domain follows a binary variation ∂ Z p T
k
e
resulting in either ρ =1 (i.e., keeping the element) or ρ =0 ∂ ρ = − ρ p [ ] u [ ][ ] u (2.8)
e
e
e
(i.e., removing the element) values. Solving this problem e e
through mathematical programming algorithms require Sensitivity filtering determines the element
the rewriting of optimization problem (2.1) as follows: distribution passing through filtering formulas (e.g., mesh-
154 International Journal of Bioprinting (2021)–Volume 7, Issue 3