## 3 Reliability-Based Topology Optimization
### Performance Index and RBTO Formulation for Evolutionary Methods  
The defined performance function is the structure performance index (\(PI\)) proposed by [49], which has the ability to monitor the optimization process and is given by  
\[PI=\frac{C_{0}V_{0}}{C_{k}V_{k}} \tag{14}\]  
Figure 3: Transformation T and FORM, adapted from [48].  
where \(C_{0}\) is the initial compliance, \(V_{0}\) is the structure's initial volume, and \(C_{k}\) and \(V_{k}\) are, respectively, the compliance and the volume at the k-th iteration. Function \(G\) can be written as  
\[G\big{(}x_{i},Y_{j}\big{)}=1-\frac{C_{0}\big{(}x_{i},Y_{j},u\big{)}V_{0}\big{(} x_{i},Y_{j},u\big{)}}{C_{k}\big{(}x_{i},Y_{j},u\big{)}V_{k}\big{(}x_{i},Y_{j},u \big{)}} \tag{15}\]  
The design variables \(x_{i}\) are the finite element dimensions of the discretization of the structure's domain, but the random variables \(Y_{j}\) are the external loads and the geometric dimensions of that structure. To control the topologies obtained by the RBTO model, the reliability index \(\beta(u)\) is introduced, see [46], with a normalized vector \(u\), which, in the case of a normal distribution, is calculated as follows:  
\[u_{j}=\frac{Y_{j}-m_{Y_{j}}}{\sigma_{Y_{j}}} \tag{16}\]  
In general, considering the normal distribution law, the standard deviation \(\sigma_{Y_{j}}\) can be related to the mean \(m_{Y_{j}}\) using \(\sigma_{Y_{j}}=\gamma_{Y_{j}}m_{Y_{j}}\), where \(\gamma_{Y_{j}}\) is a given coefficient. Furthermore, let us define the deterministic design variables \(x_{i}\) as the mean value \(m_{Y_{j}}\) of the random variables \(Y_{j}\). The minimization of the reliability problem can be expressed by  
\[\begin{split} Minimize&\beta(u)=\sqrt{\sum\limits_{j=1}^{n}u_{j}^{2}} \\ \text{subject to }& H(x_{i},u)\leq 0\\ \beta(u)>\beta_{t}\end{split} \tag{17}\]  
Equation (17) applied to the proposed RBTO model for ESO and SESO can be expressed as  
\[\begin{split} Minimize& C(x_{i})=U^{T}KU\\ \text{Subject to}& G\big{(}x_{i},Y_{j},u\big{)}=1-PI \big{(}x_{i},Y_{j},u\big{)}\geq 0\\ \beta(u)=\beta_{t}\\ \text{K}\big{(}x_{i},Y_{j},u\big{)}& U\big{(}x_{i},Y _{j},u\big{)}=F\big{(}Y_{j},u\big{)}\\ V\big{(}x_{i},Y_{j},u\big{)}=\sum\limits_{i=1}^{N}x_{i}V_{i} \big{(}x_{i},Y_{j},u\big{)}-V^{*}\leq 0\\ x_{i}=1\text{ or }x_{i}=10^{-9}i=1,2,\ldots,N\text{ and }j=1,2, \ldots,M\end{split} \tag{18}\]  
The volume \(V\big{(}x_{i},Y_{j},u\big{)}\) is consider a random variable because it depends directly on the structure's geometric dimensions. \(V^{*}\) is the prescribed volume. The RBTO model used for SIMP is the same as the one presented by [16]. We highlight only that the design variables in this case are the finite element densities of the domain, and the procedure is not discrete as in the other models presented in this study. The RBTO for the SERA that also has the element density as design variables can be expressed by  
\[\begin{split} Minimize& C(x_{i})=U^{T}KU\\ \text{Subject to}& G\big{(}x_{i},Y_{j},u\big{)}=1-PI \big{(}x_{i},Y_{j},u\big{)}\geq 0\\ \beta(u)=\beta_{t}\\ \text{K}\big{(}x_{i},Y_{j},u\big{)}& U\big{(}x_{i}, Y_{j},u\big{)}=F\big{(}Y_{j},u\big{)}\\ V\big{(}x_{i},Y_{j},u\big{)}=\sum\limits_{i=1}^{N}\frac{x_{i}V_{i} \big{(}x_{i},Y_{j},u\big{)}}{V_{0}\big{(}x_{i},Y_{j},u\big{)}}\leq V^{*}\\ x_{i}=1\text{ or }x_{i}=10^{-9}i=1,2,\ldots,N\text{ and }j=1,2, \ldots,\text{M}\end{split} \tag{19}\]  
where \(\rho_{\varepsilon}\) represents the density of each finite element.