# Reliability-Based Optimization of Impeller Structure Using Exponential Function Approximation
## 5 Conclusions  
This study proposes an efficient decoupling reliability-based optimization method based on a first-order exponential function approximation. By explicitly approximating the relationship between design parameters and failure probability through a single reliability analysis, the method eliminates the need for repeated reliability evaluations inherent in traditional double-loop approaches. The integration of this approximation within a sequential optimization framework ensures convergence while maintaining computational efficiency, as demonstrated through its application to a centrifugal compressor impeller design. The results show that the optimized impeller achieved a failure probability below the target threshold with computational costs reduced compared to double-loop methods, validating both the effectiveness and practicality of the proposed approach. Despite these advantages, two limitations should be noted. First, the accuracy of the first-order exponential approximation relies on the local linearity of the failure probability function. In systems exhibiting strong nonlinearities or multimodal failure regions (e.g., discontinuous limit states), higher-order approximations or adaptive surrogate models may be necessary. Second, the impeller case study simplified the analysis by decoupling fluid and structural domains. In practical scenarios, fluid-structure interactions (FSIs) could significantly alter stress distributions and reliability outcomes, particularly under transient operating conditions. \begin{table}
\begin{tabular}{c c c c c} \hline
**Design Parameter** & **Current Value** & \begin{tabular}{c} **Sensitivity** \\ **Coefficient** \\ \end{tabular} & \begin{tabular}{c} **Normalized** \\ **Sensitivity** \\ \end{tabular} & **Ranking** \\ \hline \begin{tabular}{c} Inlet Temperature \\ Mean (\({}^{\circ}\)C) \\ \end{tabular} & \(-160.985\) & \(-2.34\times 10^{-6}\) & \(0.452\) & \(1\) \\ \begin{tabular}{c} Inlet Pressure \\ Mean (MPa) \\ \end{tabular} & \(3.62\) & \(1.87\times 10^{-5}\) & \(0.318\) & \(2\) \\ \
\begin{tabular}{c} Temperature Std. \\ Deviation (\({}^{\circ}\)C) \\ \end{tabular} & \(10.73\) & \(-8.92\times 10^{-7}\) & \(0.165\) & \(3\) \\
\begin{tabular}{c} Pressure Std. \\ Deviation (MPa) \\ \end{tabular} & \(0.30\) & \(3.45\times 10^{-6}\) & \(0.065\) & \(4\) \\ \hline \end{tabular} Note: Normalized sensitivities are calculated as the ratio of absolute sensitivity to the sum of all absolute sensitivities.  
\end{table} 
Table 3: Sensitivity analysis results for impeller design parameters. Future work will focus on enhancing the method's generality and physical fidelity. Hybrid strategies combining local exponential approximations with adaptive surrogate models (e.g., kriging or neural networks) will be explored to address nonlinear dependencies in failure probability functions. The use of surrogate models in future work could include the introduction of more alternative models at the reliability analysis level, such as directly approximating the probability of failure function throughout the design space, bypassing the need for iterative reliability analyses (e.g., IS or SS) and eliminating the computational cost of the derivative calculations in Equation (5). These surrogate parameters will complement the current exponential function approximation approach by implementing a hierarchical modeling framework that approximates the stress response, failure probability, and sensitivity at different stages to maximize efficiency. Additionally, integrating multiphysics coupling mechanisms, such as FSI, into the optimization framework will be prioritized to better capture the dynamic behavior of turbomachinery components. These extensions aim to preserve the method's computational efficiency while expanding its applicability to complex engineering systems. Conceptualization, H.L., F.Z. ([PERSON]) and F.Z. ([PERSON]); methodology, H.L., Y.R., H.B., F.Z. ([PERSON]), F.Z. ([PERSON]) and X.B. ; writing--review and editing, H.L., Y.R., H.B., F.Z. ([PERSON]), F.Z. ([PERSON]) and X.B.