## 4 Numerical Examples  
### Example 1--Cantilever Beam  
The ESO, SERA, SESO and SIMP methods are used to perform a RBTO analysis in a 3D cantilever beam. This structure's dimensions, Figure 4, are length \(L_{x}=80\), height \(L_{y}=50\) and thickness \(L_{z}=50\). The material properties are [PERSON]'s modulus \(E=1E5\), and [PERSON]'s ratio of \(\
u=0.30\). The load of \(F=100N\) is applied to the center of the free end. The domain is discretized with 200,000 eight-node hexahedral finite elements. \[ \begin{split}&\text{\emph{Minimize}}\ C=U^{T}KU\\ &\text{Subject to}\ g=1-PI\leq 0\\ &\ h=\delta-\delta\leq 0\\ &\beta(u)=\beta_{t}\\ & KU=F\\ & V=\sum_{i=1}^{N}\frac{x_{i}V_{i}(x_{i})}{V_{0}(x_{i})}\leq V^{*} \\ & x_{i}=1\text{\emph{or}}\ x_{i}=10^{-9}\forall\ i=1,2,\ldots,N \end{split}  \tag{20}\]  
All random variables are considered with normal distribution and standard deviation of 10% of the average value. The limit state functions are defined as the PI for the structure and the difference between the allowable displacements and the prescribed displacement, imposed as constraints in the topological analysis. The prescribed displacement is obtained from the DTO analysis by calculating the average displacement (\(\overline{\delta}\)) and standard deviation (\(\overline{\sigma}\)) of the maximum allowed displacements using the Kolmogorov-Smirnov test. This test's statistics quantifies a distance between the empirical distribution function of the sample (allowed displacements) and the cumulative distribution function of the reference distribution. In this article, the cumulative standard normal distribution function is used. Thus, an empirical sample is used, \(n=50\), of the maximum displacements from the DTO analysis to define the displacements' limit values. Thus, in the RBTO analysis, the prescribed values for the average displacement as the maximum allowable displacement in the TO procedure is used. It is noteworthy that a stopping criterion is used that compares the difference between the allowed and prescribed displacements with the standard deviation obtained from the Kolmogorov-Smirnov test in the RBTO procedure. Figure 5 shows the contour graphics for a plane section, the optimal topology for the DTO, and the surface graphics to illustrate the main differences between the presented models. The average displacement \(\overline{\delta}=0.002957\) is considered in this case. Additionally, Figure 5 shows the contour graphics, optimal topologies, and surface graphics for the 3D cantilever  
Figure 4: Design domain. beam, using a target reliability index, \(\beta_{t}=3.0\), which corresponds to a failure probability of 0.135% for the RBTO analysis of the presented evolutionary procedures. Figure 6a-c shows the optimal topologies presented, respectively, by ESO, SERA and SESO. These configurations are similar, although the design variable in SERA is the element density, while, in ESO and SESO, the variables are the extended fixed domain finite elements. These results can be compared with those presented by [24], using the standard response surface method (SRSM). The optimal topology achieved by SIMP, Figure 6d, is different from the other formulations because convergence occurs at intermediate densities in a continuous optimization process unlike the other formulations that have a discrete procedure. Figure 5: DTO: contour, topology and surface: (**a**) ESO, (**b**) SERA, (**c**) SESO and (**d**) SIMP. Figure 6: _Cont_. For the 3D results' comparison, the volumes, displacements, OF, and computation time due to each method, for the cantilever, are listed in Table 1. From the comparison of the RBTO with the DTO of the 3D cantilever beam, note that the optimal RBTO volume is about 15.38% smaller than the SRSM method proposed by [24]. It is worth highlighting that ESO, SERA and SESO converge to the same value of the OF in the DTO and RBTO analyses and are computationally more efficient than SIMP, respectively, 16.3% and 11.4% on average.