## 2 Materials and Method
### Reliability Evaluation  
Within the scope of topology optimization designs, the reliability criteria incorporate the uncertainties associated with random variables. This methodology is embedded within the optimization process to identify a dependable and ideal configuration for a structure in a specified area. In conducting a reliability analysis, a normalized vector of the random variable (\(u\)) and a reliability index (\(\beta\)) are introduced [41]. Although the design variable \(\rho\) does not appear explicitly in Equation (7), it influences the structural response and performance functions used in the reliability analysis, thereby indirectly affecting the value of \(u\) through the limit-state function. When dealing with a normal distribution, the expressions for \(u\) and \(\beta\) are defined as follows:  
\[ \begin{split} u_{j}=\frac{y_{j}-m_{y_{j}}}{\sigma_{y_{j}}},\\ \beta=\min\Bigl{(}\sqrt{\mathrm{u}^{T}\mathrm{u}}\Bigr{)},\end{split}  \tag{7}\]  
where \(y_{j}\) is the \(j\)'th random variable with a mean value of \(m_{y_{j}}\) and a standard deviation \(\sigma_{y_{j}}\). The vector (\(u\)) represents the standard normal transformation of random variables, which are implicitly dependent on design variables through the reliability analysis process. Therefore, an RBTO (Reliability-Based Topology Optimization) problem, formulated by integrating the reliability analysis into the topology optimization problem, can be described as follows:  
\[ \begin{split}\min& C(\rho)=\mathbf{d}^{T}\mathbf{Kd} =\sum_{\mathrm{e}=1}^{N}{(\rho_{e})^{p}d_{e}^{T}k_{0}d_{e}}\\ \text{subject to}&\beta(\mathbf{u})\geq\beta_{t}\\ &\frac{V(\rho,\mathbf{y},\mathbf{u})}{V_{0}}\leq f\\ \mathbf{K}(\rho,\mathbf{y},\mathbf{u})\cdot\mathbf{d}(\rho, \mathbf{y},\mathbf{u})=\mathbf{f}(\mathbf{y},\mathbf{u})\\ 0<\rho_{min}\leq\rho_{e}\leq 1.\end{split}  \tag{8}\]  
Here, \(\beta\) and \(\beta_{t}\) represent the system's reliability index and target reliability index, respectively, and \(\mathbf{u}\) represents random variables described in the standard normal space. In the expression found in Equation (8), there are two nested optimization loops. For every value that the design variables assume in the solution phase of the outer loop, the value of the reliability index described in the inner loop is computed. In the RBTO problem, where the outer loop is directed towards a minimum compliance goal and constrained by reliability, the inner loop is characterized as a reliability problem. Within the inner loop, the failure region is denoted as \(G(\mathbf{u})\leq 0\), while the limit state function is expressed as \(G(\mathbf{u})=0\). The probability of failure, as defined in this context, is given as follows:  
\[P_{f_{i}}=Pr[G_{i}(\mathbf{u})<0]=\int\ldots\int_{G_{i}(\mathbf{u})<0}f_{ \mathbf{u}}(u)du. \tag{9}\]  
Here, \(f_{\mathbf{u}}(u)\) shows the combined probability density function of random variables. Therefore, to solve the integral given in Equation (9), analytical methods such as the First- and Second-Order Reliability Method (FORM and SORM) shown in Figure 2[42] or simulation-based methods such as Monte Carlo can be considered [43]. The Reliability Index Approach (RIA) and the Performance Measurement Approach (PMA) are the two prevalent methods for evaluating reliability using FORM. In RIA, the calculation of the reliability index involves solving a minimization problem in the variable space \(\mathbf{u}\), being the vector of standard normal variables. Conversely, PMA aims to identify a probabilistic measure of performance by approaching the problem as an inverse reliability problem. The PMA method has found widespread application in reliability assessment [44; 45]. The equation employed in the inner loop for the TO solution, making use of PMA, is articulated in Equation (10) for scenarios where displacement is taken into account:  
\[ \begin{split}\min& G(\mathbf{u})=\delta_{max}- \delta_{allow}\\ \text{subject to}&\beta=\|\mathbf{u}\|=\sqrt{u_{1}^{2 }+\dots+u_{j}^{2}+\dots+u_{j}^{2}}\\ &\delta_{max}=|\mathbf{u}_{max}|.\end{split}