# Topology Optimization of an Aerospace Bracket: Numerical and Experimental Investigation
## 3 Results and Discussion
### Optimization Results  
The generated mesh with 10-node tetrahedral elements, shown below in Figure 4a, depicts the geometry discretization used for the topology optimization of the bracket. In the same figure, we include the results of the conducted mesh convergence study; see Figure 4b.  
From the table in Figure 4b, we observe that the initial setting with an element size (maximum edge length) of 2 mm produced a mesh with 20,089 nodes and 11,068 elements, resulting in 204.4 MPa of equivalent stress. The successive refinement process leads to a converged value of slightly above 250 MPa, while the last refinement in the table has a mesh size of 0.5 mm, with 1,758,145 nodes, 1,246,520 elements, and a value of 254.16 MPa for the equivalent stress. Furthermore, it can be noted from the table that the change in the equivalent stress value between the fourth and fifth meshes is approximately 0.63%, which drops down to 0.2% for the last pair of meshes, indicating mesh convergence. We considered that the fourth level of refinement included in the same table achieves a good balance between computational accuracy and cost, and therefore it was selected for the topology optimization process.  
The total deformation and equivalent stress distributions are depicted in Figure 5a and Figure 5b, respectively. The red-colored regions indicate high values, while blue-colored regions correspond to low values.  
The topology optimization settings and parameters within the ANSYS package are included in Figure 6, below. A series of material reduction percentages have been considered to evaluate the extent of material reduction, which would maintain similar levels of compliance with the initial design. Specifically, material reductions of 10, 20, 30, and 40% have been attempted, with only the first two percentages yielding compliance levels equal to the original design. Therefore, a material reduction of 20% has been identified as a feasible reduction which would lead to a corresponding weight reduction, from the 0.608 kg of the original design to 0.486 kg for the optimized one (if stainless steel were to be considered in both cases). For the PLA-printed specimens and a 50% infill density, we achieved an actual weight reduction of approximately 19% (from 0.059 kg to 0.048 kg), since wall thickness printing parameters do not allow for a fully homogeneous material distribution. Note that for both level-set and density-based methods, as well as the four cases of optimization domain, the optimization parameters and conditions were the same. A smaller number  
Figure 4: Resulting mesh and mesh convergence study.  
of specimens (two for each case) were printed and tested with a 100% infill density, but since no significant qualitative difference was observed in the experimental results between the 100% and 50% infill densities, the latter was used. This provides a good basis for the comparison of both methods. The optimized results for both methods and all considered optimization regions are depicted in Figures 7-10. From Figures 7-10, it is evident that when different regions are considered under the same optimization conditions, different results are obtained through both optimization methods. The exhibited differences between the presented cases is mainly due to the different constrained regions, i.e., red regions where no topology optimization is performed. When the same case is considered, the exhibited differences in the resulting optimized design are mainly due to the employed method; e.g., the level-set method permits a smooth description of the boundary which allows relatively large holes with smooth boundaries to be generated, whereas the SIMP density-based method, with the ANSYS filtering algorithm enabled, generates a large number of holes or deformations on the original design with rough, poorly-defined boundaries.  
Figure 8: Optimization region of Case 2.  
Figure 7: Optimization region of Case 1.  
Figure 10: Optimization region of Case 4.  
Figure 9: Optimization region of Case 3.