# Reliability-Based Optimization of Impeller Structure Using Exponential Function Approximation
## 3 Proposed Reliability-Based Optimization Method
### Decoupling and Sequential Approximation Framework  
The failure probability function obtained using the first-order exponential function approximation expresses a local failure probability because constructing a global approximate failure probability function over the entire range of design variables can incur high computational costs and is frequently impractical. The global accuracy of the reliability-based optimization results was ensured in this study by employing an iterative reliability analysis and optimization process that continued until the design vector and objective function values converged. This decoupling iterative reliability approximation optimization method is described in this section.  
First, the design values \(\theta^{(k)}\) at the current iteration step  
An appropriate optimization interval must be selected as it affects the efficiency of the proposed iterative method. When the local approximation of the failure probability function is sufficiently accurate, a large local optimization interval can be adopted to facilitate the rapid convergence of the optimization process; otherwise, a smaller local approximation interval should be used to ensure the accuracy of the local approximation. Therefore, the local optimization interval used in this study was selected as follows:  
\[\theta_{i}^{k}=\max\left\{\theta_{i}^{k}(1-R_{k}).\underline{\theta}_{i} \right\},\delta^{k}=\min\left\{\theta_{i}^{k}(1+R_{k}),\bar{\theta}_{i}\right\} \tag{11}\]  
where \(\mathrm{R_{k}}(\mathrm{k}=1,2,\ldots)\) controls the size of the local optimization interval, with a typical value range of [10%, 50%].  
The accuracy of the optimal solution was ensured by reducing the optimization search area during iteration using the following gradient strategy:  
\[\mathrm{R_{k}}=\mathrm{R_{0}}\times\mathrm{r^{k}} \tag{12}\]  
where \(\mathrm{R_{0}}\) is the initial value, typically 20%, and \(\mathrm{r}\) is a reduction factor with a value range of \([0.8,1]\). The choice of values for the initial interval parameter \(\mathrm{R_{0}}\) and the reduction factor \(\mathrm{r}\) based on a larger initial interval, i.e., \(\mathrm{R_{0}}=20\%\), helps to explore the design space quickly in early iterations to avoid falling into a local optimum solution, and a slow reduction (\(\mathrm{r}\geq 0.8\) ) balances the accuracy of the local approximation with the speed of convergence and avoids precocious convergence due to the interval shrinking too fast. This strategy performs robustly in similar reliability optimization problems such as those in the literature [17; 27]. However, future research could further explore adaptive tuning strategies, such as dynamically updating \(\mathrm{R_{k}}\) based on the rate of change of the objective function or the approximation model error, e.g., if the difference in the optimization results of neighboring iteration steps exceeds a predefined threshold, \(\mathrm{R_{k}}\) could be automatically narrowed down to improve the accuracy of the local approximation, and vice versa to expand the interval to accelerate the convergence. Such adaptive mechanisms will enhance the generalizability of the method but require additional computational cost and a trade-off between efficiency and robustness.  
From Equations (11) and (12), it can be concluded that the iterative narrowing of the optimization interval ensures that the first-order assumptions remain valid in the local region, thus correcting the inaccuracies that may be introduced by the initial approximation.  
Figure 1: Schematic diagram of sequential approximate optimization with decoupling.  
A flowchart of the iterative decoupling reliability approximation optimization method is presented in Figure 2. The outline of the step-by-step algorithm for reliability-based decoupling optimization is shown in Figure 3.  
Figure 2: Flowchart of the decoupling optimization method.