# Reliability-Based Optimization of Impeller Structure Using Exponential Function Approximation
## 4 Reliability-Based Optimization of an Impeller Structure
### Reliability-Based Optimization of the Impeller Structure  
As the impeller optimization process is iterative and the impeller reliability analysis is relatively time-consuming, this study used a kriging surrogate model to approximate the reliability analysis model. The training samples for the kriging model were generated via Latin Hypercube Sampling (LHS) to ensure uniform coverage of the input parameter space (Table 1). The initial set of 15 samples was increased by an active learning strategy based on U-functions, which prioritized the addition of samples in regions with high prediction uncertainty near the limit states. Seven random variables were defined as inputs to this surrogate model: rotational speed (rev/min), inlet temperature (\({}^{\circ}\)C), outlet pressure (MPa), inlet pressure (MPa), elastic modulus (GPa), length (mm), and width (mm). These variables were selected based on a sensitivity analysis of the impeller's structural response under typical operating conditions, where inlet temperature and pressure exhibited the highest correlation with stress concentrations (Figure 4). The distribution parameters (mean and standard deviation) were determined from historical operational data and manufacturing tolerances. All variables that followed normal distributions with the parameters are listed in Table 1.  
The limit state equation based on the kriging surrogate model is given by  
\[\hat{g}(x)=5800-\hat{y}(x) \tag{13}\]  
where \(\hat{y}(x)\) is the maximum stress predicted by the surrogate model and the allowable stress threshold is defined as 5800 MPa. This limit state equation was used to evaluate the reliability of the impeller under different operating conditions x. As the stress model is most sensitive to the inlet temperature and inlet pressure, these variables were considered in the reliability-based optimization design using the distribution parameters listed in Table 1. The optimization constraints were defined as the reliability constraints of the impeller, \(\mathrm{P_{F}\leq P_{F}^{tol}}\), where \(\mathrm{P_{F}^{tol}=10^{-4}}\). The objective function was based on cost and normalized to balance the different magnitudes of the design variables \(\theta_{1}\) (inlet temperature) and \(\theta_{2}\) (inlet pressure) as follows:  
\[\mathrm{C(\theta_{1},\theta_{2})=\frac{\theta_{1}}{160.985}-\frac{\theta_{2}} {4.62}} \tag{14}\]  
Thus, the reliability-based optimization design model for the impeller is given by  
\[ \begin{array}{ll}\mathrm{Min}&\mathrm{C(\theta_{1},\theta_{2})=\frac{\theta _{1}}{160.985}-\frac{\theta_{2}}{4.62}}\\ \mathrm{s.t.}&\mathrm{P_{F}<P_{F}^{tol}}\\ &-190.985\leq\theta_{1}\leq-130.985\\ &1.62\leq\theta_{1}\leq 5.62\end{array}  \tag{15}\]  
The effectiveness of the proposed first-order exponential approximation (IS) method was demonstrated by comparing its reliability-based optimization results with those ob \begin{table}
\begin{tabular}{c c c} \hline \hline
**Variable** & **Mean Value** & **Standard Deviation** \\ Rotational speed (rev/min) & 3000 & 200 \\ Inlet temperature (\({}^{\circ}\)C) & \([-190.985,\,-130.985]\) & 10.7317 \\ Outlet pressure (MPa) & 0.18 & 0.01333 \\ Inlet pressure (MPa) & [1.62, 5.62] & 0.3 \\ Elastic modulus (GPa) & 200 & 13.3333 \\ Length (mm) & 112 & 3.6667 \\ Width (mm) & 30 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Distribution information of the variables of the impeller structure. tained using the double-loop importance sampling (IS [31]) and subset simulation (SS [32]) methods, as shown in Table 2. The optimal design and its corresponding objective function value are listed in the table. The objective function values obtained by the three methods were quite similar, indicating consistent optimization results. In terms of the design variables, the optimal inlet temperatures and pressures obtained by the three optimization methods were also quite close, demonstrating that the proposed method can effectively converge to an optimal solution.