## 4 Conclusions  
Topology optimization focuses on the percentage of maintained mass to ensure compliance with a set of objectives and restrictions due to the mechanical stress to which a piece is subjected. If restrictions are met, the optimum lightweight solution maintains the lowest percentage of mass, which is related to a reduction of the material cost. In these cases, the material is efficiently distributed throughout the design space in order to sustain and resist mechanical stresses. However, the parameter percentage of maintained mass does not provide information on the efficiency of the material in distributing itself through the geometric space. Furthermore, it is not possible to discern which optimized piece best distributes its mass when several solutions with similar maintained mass are obtained. In this work, fractal dimension was first tested as a parameter in a topology optimization framework. It was evaluated for both characterizing complex geometries in structural pieces and providing complementary information to the traditional maintained mass. A cantilever beam with a single force applied at the end was employed as an example in this work. Initially, a lightening based on a Menger sponge was carried out on the beam to test the method for estimating the fractal dimension, as well as the procedure for calculating the mechanical parameters of the study (von Mises stress, factor of safety, and maximum displacement). Regarding topology optimization, some findings were remarkable. When the objective was to maximize the stiffness (OT1), a relationship among the geometry, fractal dimension and maximum displacement was observed. The fractal dimension increased with the percentage of maintained mass until around 70%, at which point the fractal dimension began to decrease. This is because fractal dimension not only depends on the amount of material but on how that material is distributed in space. Therefore, the percentage of maintained mass for which the fractal dimension reached a maximum value was found. This maximum was associated with an asymptotic behavior in the maximum displacement. Thus, the results suggest that when the objective of the optimization is to maximize stiffness, fractal dimension is a useful index for determining the optimum percentage of maintained mass, which is also related to stable maximum displacement. However, when the objective is to minimize the mass (OT3), the relationship between fractal dimension and maximum displacement was a decreasing curve. Variations in the thickness of the topology optimization solutions influenced neither the fractal dimension nor the maximum displacement (OT2 and OT4). In summary, our results indicate that when the objective of the topology optimization is to maximize the stiffness, fractal dimension is related to the amount of maintained mass as well as the geometric distribution of the material. Therefore, we suggest that fractal dimension is a useful parameter for describing the geometric efficiency of the lightened beam. By contrast, when minimizing mass is used as a design objective, our results indicate  
Figure 14: Fractal dimension vs. maximum displacement for experiment OT3. that fractal dimension is only a parameter linked to the amount of mass held in the piece. Note that the main limitations of this study are conditioned by the typology of beam chosen (a cantilever beam). In future work, it might be interesting to develop similar studies on other types of beams (double-embedded, bi-supported, etc. ), with different loads (point, uniformly distributed, etc. ), as well as different sections and geometries. Overall, the results of this study suggest that it could be promising to add the fractal dimension as an additional parameter for making decisions in topology optimization software.