# Reliability-Based Optimization of Impeller Structure Using Exponential Function Approximation
## 3 Proposed Reliability-Based Optimization Method  
### Failure Probability Function Approximation Using First-Order Exponential Function  
Decoupling methods approximate the failure probability function using a surrogate model \(\hat{P}_{F}(\theta)\) to directly express the relationship between the design variables and failure probability [26]. A first-order exponential function was used in this study to approximate the failure probability function and thereby realize an efficient decoupling optimization. Define \(x=[x_{1},x_{2},\ldots,x_{n}]\) as the \(n\) random variables describing a structural system and \(\theta=[\theta_{1},\theta_{2},\ldots,\theta_{n_{0}}]\) as the \(n_{0}\) design parameters of the system. The first-order exponential function for the approximation of the failure probability function can be expressed as  
\[\hat{P}_{F}(\theta)=e^{C_{0}+\sum_{k=1}^{n_{0}}C_{i}\cdot\theta_{i}} \tag{2}\]  
where \(C_{0},C_{i}(i=1,2,\ldots,n_{0})\) are the undetermined coefficients of the first-order exponential function. As Equation (2) contains \(n_{0}+1\) unknowns, it requires \(n_{0}+1\) equations to solve. Conventional reliability analysis is performed by using different values of \(|\theta|\) to obtain the corresponding failure probability values, and then establishing \(n_{0}+1\) equations to solve the \(n_{0}+1\) undetermined coefficients. In contrast, this study used sensitivity information to establish these equations. Under this method, the \(n_{0}+1\) coefficients for the first-orderexponential function in Equation (2) [27] can be determined based on the known failure probability value at a certain interpolated point and the reliability sensitivity values of each design variable \(\mathrm{n_{0}}\). As the samples used to obtain the reliability sensitivity are the same as those used to obtain the failure probability, only one reliability analysis is required to obtain both the failure probability and \(\mathrm{n_{0}}\) sensitivity values, which can in turn establish the \(\mathrm{n_{0}}+1\) equations [28] required to solve the corresponding undetermined coefficients. Approximation accuracy can be improved by taking the midpoint \(\mathrm{\theta_{0}}=\left(\overline{\mathrm{\theta}}+\underline{\mathrm{ \theta}}\right)/2\) of the design variable range as the initial design point (pre-interpolation point) [29] and determining the failure probability \(\mathrm{\hat{P}_{F}}(\mathrm{\theta_{0}})\) and sensitivity at this point using the following reliability analysis:  
\[\left. \begin{array}{ll}\left.\frac{\partial\mathrm{\hat{P}_{F}}}{\partial \mathrm{\theta_{1}}}\right|_{\mathrm{\theta}=\mathrm{\theta_{0}}}&=\mathrm{ C_{i}}\cdot\mathrm{e}^{\mathrm{C_{0}}+\sum_{i=1}^{\mathrm{n_{0}}}\mathrm{C_{i}} \cdot\mathrm{\theta_{1}}}\right|_{\mathrm{\theta}=\mathrm{\theta_{0}}}\\ &=\mathrm{C_{i}}\cdot\mathrm{\hat{P}_{F}}(\mathrm{\theta_{0}})(\mathrm{i}=1, \ldots,\mathrm{n_{0}})\end{array} \right. \tag{3}\]  
Taking the logarithm of both sides of Equation (2) yields  
\[\mathrm{ln}\big{(}\mathrm{\hat{P}_{F}}(\mathrm{\theta_{0}})\big{)}=\mathrm{C_{ 0}}+\sum_{i=1}^{\mathrm{n_{0}}}\mathrm{C_{i}}\cdot\mathrm{\theta_{1}}\Big{|} _{\mathrm{\theta}=\mathrm{\theta_{0}}} \tag{4}\]  
Simplifying Equations (3) and (4) results in  
\[\mathrm{C_{i}}=\left.\frac{\left.\frac{\partial\mathrm{\hat{P}_{F}}}{\partial \mathrm{\theta_{1}}}\right|_{\mathrm{\theta}=\mathrm{\theta_{0}}}}{\mathrm{ \hat{P}_{F}}(\mathrm{\theta_{0}})}(\mathrm{i}=1,\ldots,\mathrm{n_{0}})\right. \tag{5}\]  
\[\mathrm{C_{0}}=\mathrm{ln}\big{(}\mathrm{\hat{P}_{F}}(\mathrm{\theta_{0}}) \big{)}-\sum_{i=1}^{\mathrm{n}}\mathrm{C_{i}}\cdot\mathrm{\theta_{i}}\Big{|} _{\mathrm{\theta}=\mathrm{\theta_{0}}} \tag{6}\]  
The undetermined coefficients of the first-order exponential function can be obtained using Equations (5) and (6) and then substituted into Equation (2) to derive the expression for the local failure probability function approximation.