## 4 Numerical Examples
### Example 2--MBB Beam  
We performed the RBTO analysis using ESO, SESO, SERA and SIMP evolutionary methods for a MBB-3D beam with a minimum radius of 1.5. The dimensions shown in Figure 8 are length \(L_{x}=120\), height \(L_{y}=20\) and thickness \(L_{z}=20\). The design domain is discretized with 48,000 8-node hexahedral finite elements. Material properties are a [PERSON]'s modulus of 100 E3, and a [PERSON]'s ratio of 0.3. The load applied at the center of the top face is 100 N. It is worth highlighting that symmetry is not used for this problem. Figure 7: Objective function and performance index (PI): (**a**) ESO, (**b**) SERA, (**c**) SESO and (**d**) SIMP. The Figure 9 shows the optimal topology obtained using the DTO based on the evolutionary methods, presenting a side and a front view. The displacements of this analysis are used to calculate the prescribed average displacement applied in the RBTO. The ESO, SERA, and SESO methods have similar optimal configurations, with small differences in their optimal topology, as can be seen in the front view in Figure 9. SIMP proposes a different and more complex topology for manufacturing than other methods. Furthermore, in all case studies, the SIMP model has the highest value for the OF. However, it is worth noting that the SIMP model presents greater stability during the iterative process. During the optimization procedure, it starts from high compliance, and the function is always decreasing without jumps, while in the other methods, compliance presents at least one peak during the optimization process. The answer may lie in the removal heuristics of each method or in the fact that ESO, SERA, and SESO methods are discretized. For the RBTO model, parameters relating to geometry, load, and volume are assembled into the averages vector given by \(m=\{nelx,nely,nelz,V,F\}\). Given that the standard deviations' values are \(\sigma_{i}=0.1\ m_{i},\ \forall\ i=1,2,\ldots,M\), the random vector \(Y\) is normally  
Figure 8: Design domain. Figure 9: Optimal topology (DTO)—side view and front view: (**a**) ESO, (**b**) SERA, (**c**) SESO and (**d**) SIMP. distributed and given by (\(y=m+u\sigma\)). Thus, when the reliability analysis is completed, satisfying the constraints, the optimal values of the normalized vector \(u^{*}\) are obtained, which lead to the input random vector \(Y\). The limit state functions are the same as defined in Equation (20), with displacement equal to \(\delta=0.00695\). Figure 10 illustrates the optimal topologies for the MBB-3D beam, using the proposed RBTO analysis with \(\beta_{t}=3.805\), which corresponds to a failure probability of \(7.2348\times 10^{-5}\). In addition, it presents surface graphics that provide greater clarity for identifying small topological differences. Figure 10: Optimal topology (RBTO), side view and surface: (**a**) ESO, (**b**) SERA, (**c**) SESO and (**d**) SIMP. The ESO, SERA and SESO methods allow the removal of elements with low compliance to improve the structure's performance. Thus, a characteristic performance curve for this continuum structure is shown in Figure 11. The weight of a structure is gradually reduced during the optimization process as the compliance increases. In this way, the characteristic curve of a structure's performance during the optimization process can be expressed by the structure's weight and its strain energy [50]. Therefore, in the RBTO analysis, the optimized structure's performance provides a designer with valuable information on the structural design. It not only indicates whether the optimized design is feasible, but the optimized design's success for stiffness. When analyzing the above curves, Figure 10(b), the designer should note that the optimal structure achieved with a compliance 20% higher than the initial compliance, in the graphic \(\frac{C_{i}}{C_{0}}=1.20\), is obtained with an approximate volume \(\frac{V_{i}}{V_{0}}=0.48\).