## 2 Materials and Methods
### Theoretical Background  
Topology Optimization has its roots in the study of Michel trusses and the usage of a ground structure approach, where a dense truss simulates the continuum, and in the course of some structural optimization process regarding the cross-sections of the bars, unloaded elements are eliminated and the optimal topology arises. TO can solve the problem of finding the proper distribution for a limited amount of materials within a specified domain. In current study, the SIMP algorithm is utilized. In total, two TO studies were carried out, and an equal number of structures emerged.  
The history of SIMP takes us back to the 1980s when [PERSON] first conceived the idea of parameterizing the design domain and not proceeding to the solution of an on-off problem, the TO. It was until the 1990s that this method was highlighted. The SIMP method is able to maintain a fixed FE mesh and associate each of the Finite Elements with a density function \(\rho\)(x) whose values range from 0 (void) to 1 (solid). As presented by [PERSON] et al., an artificial material that can also be conceived as a mesostructure material with holes can be created by the intermediate values. The last design solution has as its feature black regions (indicating solid regions), white regions (indicating voids) and grey regions (indicating intermediate densities). In order to subdue the grey areas and to eliminate, because of their impracticality in fabricating with traditional manufacturing methods, the intermediate densities from the last design solution, a lot of techniques emerged [27; 28; 29; 30]. According to the SIMP method, the formula for the stiffness matrix of the optimized structure is:  
\[K_{SIMP\ (\rho)}=\sum_{e=1}^{N}\left[\rho_{min}+(1-\rho_{min})\rho_{e}^{p} \right]K_{e} \tag{1}\]  
where:  
\(K_{e}\) is the elements stiffness matrix;  
\(\rho_{min}\) is the minimum penalty factor;  
\(N\) is the number of total elements in the design domain.  
During the TO study, a main goal is to maximize the stiffness of a structure or minimize its compliance with a given amount of the total mass, which is usually defined as a percentage of the available design area. The total compliance is equal to the cumulative deformation energy of the elements. Minimizing the total compliance, \(C\), is equivalent to maximizing the overall stiffness. The optimization algorithm, through an iterative process,  
Figure 1: Adopted methodology flowchart.  
seeks to solve the element densities that minimize the overall compliance of the structure. The mathematical formula to calculate the \(C\) is:  
\[\text{min}C(\{\rho\})=\sum_{e=1}^{N}(\rho_{e}\ )^{p}\ [\mathcal{U}_{e}\ ]^{T}\ [K_{e} ]\ [\mathcal{U}_{e}] \tag{2}\]  
where:  
(\(U_{e}\)) is the displacement vector of element \(e\);  
(\(K_{e}\)) is the stiffness of element \(e\);  
the vector \(\{\rho\}\) consists of the relative density of the \(\rho_{e}\) elements.  
The behavior of a lattice structure can be described using the generalized Hooke's law: \(\sigma_{ij}=c_{ijkl}\varepsilon_{kl}\). The anisotropic form of [PERSON]'s law can be expressed by the stiffness matric [c]:  
\[\begin{bmatrix}C_{11}&C_{12}&C_{13}&C_{14}&C_{15}&C_{16}\\ C_{12}&C_{22}&C_{23}&C_{24}&C_{25}&C_{26}\\ C_{13}&C_{23}&C_{33}&C_{34}&C_{35}&C_{36}\\ C_{14}&C_{24}&C_{34}&C_{44}&C_{45}&C_{46}\\ C_{15}&C_{25}&C_{35}&C_{45}&C_{55}&C_{56}\\ C_{16}&C_{26}&C_{36}&C_{46}&C_{56}&C_{66}\end{bmatrix}\]  
The Zener ratio is a dimensionless number that is used to quantify the anisotropy for cubic structures [31]. Conceptually, it quantifies how far a material is from being isotropic (where the value of 1 means an isotropic material). The calculation of the Zener ratio is derived from the mathematical formula:  
\[A=\frac{2C_{44}}{C_{11}-C_{12}} \tag{3}\]