## 3 Design of Topologically Optimized Lattice Structures  
The design process began from a cube with dimensions 15 \(\times\) 15 \(\times\) 15 mm; the topology optimization block of nTopology was used to carry out the TO studies. The boundary conditions involve a constraint restraint applied to the bottom face of the cube, with an exclusion of all degrees of freedom. The initial cube has been assigned with an isotropic elastic material (Table 1). During the meshing process, a solid mesh model consisting of 25,781 tetrahedral elements with an average edge length of 1 mm is created. The satisfaction of the objective function is achieved with the minimization of the material within the specified design domain. The fulfillment of the volume fraction restraint plays a decisive role in the configuration of the optimized model. The value, which is required for the volume fraction restraint, may range from 0 to 1, while the optimized volume will have a volume fraction equal to or less than the required value. The creation of different structures results from the various ways of imposing the loads, which, in both studies, are 300 N. In the first study, the load case is applied in plane to all model vertices. On the contrary, in the second study, the loads are applied in plane but only on the vertices of the top cube surface. Two topologically optimized structures have been created,'structure #1' and'structure #2'. The first structure is performed after twenty-three iterations; the final and intermediate shapes of TO study appear in Figure 3. Respectively, Figure 4 shows the progressive shape of the second structure in different iterations, the algorithm terminated after thirty-three iterations. In both studies, the volume fraction decreased sharply in the first 10 steps until asymptotically approaching the minimum (Figures 5a and 6a). After the evaluation, the target value of the volume fraction constraint for'structure #1' is 0.4 and, for'structure #2', is 0.1 (Figures 5b and 6b). Topology Optimization is correlated with the material distribution in a specific domain in order to design constraints to be satisfied. In essence, a more rigid and stiff structure combined with material minimization is sought. Therefore, the problem goal is to minimize its compliance in order to increase the stiffness of the optimal structures. The optimized structures are refined and reconstructed in a specific design volume. After this process, two new custom unit cell lattices are created (Figure 7a). From these lattice cells, new lattice structures are designed (Figure 7b). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Young’s Modulus** & **Density** & **Poisson’s Ratio** \\
**(MPa)** & **(kg/mm\({}^{3}\))** & **Poisson’s Ratio** \\ \hline
187,000 & 0.0079 & 0.33 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Linear isotropic material properties. Figure 3: Intermediate shapes. Different iteration stages (**b–e**) from the initial design area (**a**) of Topology Optimized ‘structure #1’: (**a**) design area, (**b**) step 5, (**c**) step 10, (**d**) step 18 and (**e**) step 23. After evaluating the Topology Optimization results, the stage of creating the cell by utilizing the optimized structures took place. The homogenized cells were created in a design volume with dimensions 15 \(\times\) 15 \(\times\) 15 mm. In order to execute the homogenization study, two solid mesh models were created, with an edge length of 0.7 mm and 75,489, as well as 62,595 tetrahedral elements, respectively. As a result of using the homogenization unit cell block of nTopology software, new homogenized structures are emerging and can now be utilized as materials that will be evaluated by performing a structural static analysis. Figure 4: Iterations steps (**a**–**e**) of the Topology Optimization study for ‘structure 2’: (**a**) design area, (**b**) step 10, (**c**) step 20, (**d**) step 25 and (**e**) step 33. Figure 5: (**a**) Volume fraction per each step (**b**), and the value of the volume fraction constraints on each iteration step of ‘structure #1’.