## 3 Results and Discussion
### Beam Based on the Topology Optimization  
According to the methodology described in Section 2.4, we evaluated the potential relationship between the complex geometries obtained from topology optimization and the factors derived from static simulation. Note that identical boundary conditions and design parameters were applied in order to obtain comparable results among optimized beams. A representation of some optimized beams of the OT1 experiment is depicted in Figure 8. Notice that the objective in OT1 was to maximize the stiffness by varying the percentage of maintained mass. Lightened solutions with a percentage of maintained mass lower than 20% were rejected because the beams exhibited discontinuities and they would not be a valid solution for realistic mechanical studies. Figure 9 shows that the relationship between the maintained mass and the fractal dimension was a convex parabola, which fit to a third-degree polynomial with a goodness of fit (\(R^{2}\)) greater than 0.96. The fractal dimension reached a maximum value when the percentage of maintained mass was around 70%. Thus, percentages of maintained mass  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Iteration** & \(F_{D}\) & _Nodes_ & _Mass (\%)_ & \(\sigma_{von\_Mises}\) (MPa) & _FOS_ & \(Disp_{max}\) (mm) \\ \hline
0 & 3.000 & 70,785 & 100\% & 308 & 0.65 & 0.96 \\
1 & 2.918 & 52,272 & 74\% & 286 & 0.70 & 1.76 \\
2 & 2.874 & 37,728 & 53\% & 298 & 0.67 & 2.80 \\
3 & 2.657 & 24,317 & 30\% & 409 & 0.49 & 4.66 \\ \hline \hlinegreater than 70% resulted in lower fractal dimensions. Thus, the relationship between the maintained mass and the fractal dimension was not always direct. Lower fractal dimensions were obtained for higher percentages of maintained mass--that is, inefficient geometric configurations for filling the space were obtained when the percentage of maintained mass was greater than 70%. In this case, fractal dimension might serve as an indicator of how the geometry of the beam is distributed in space. This finding should be analyzed along with Figure 10. In Figure 10, an inversely proportional relationship between the maximum displacement and the percentage of  
Figure 8: Representation of beams for experiment OT1 maintaining a mass of (**a**) 30%, (**b**) 40%, (**c**) 50% and (**d**) 70%. Figure 7: Fractal dimension vs. maximum displacement for different iterations of the Menger sponge. Figure 9: Maintained mass vs. fractal dimension for experiment OT1. maintained mass can be observed. Thus, maintaining a low percentage of mass caused very high displacements in the beam. This decreasing relationship tended towards an asymptotic behavior above a percentage of maintained mass around 70%. Therefore, maintaining masses greater than 70% led to constant maximum displacements (close to 1 mm). This relationship could be fit to a polynomial of degree three with a goodness of \(R^{2}>0.99\). Therefore, to ensure reasonable displacements, percentages of maintained mass in the range 80-90% are geometrically more inefficient than a maintained mass of 70%. Thus, to ensure acceptable displacements, the fractal dimension can be used as an index of the geometric efficiency of the distribution of a certain mass in space. Von Mises effort and _FOS_ remained practically independent from fractal dimension. In experiment OT2 the objective was to maximize the stiffness using the thickness as a variable. The maintained mass was fixed at 50% and the thickness was evaluated from 200 to 500 mm, with increments of 100 mm. The solution for a thickness of 200 mm was discarded due to discontinuities in the beam. Figure 11 displays the dependence of the fractal dimension on the thickness. For a fixed maintained mass, the influence of thickness on the fractal dimension was negligible because the slope was close to zero in the relationship between both variables. Therefore, fractal dimension cannot be related to mechanical resistance parameters when the thickness of the optimized beams changes. Figure 11: Thickness vs. fractal dimension for experiment OT2.