Reliability-Based Optimization of Impeller Structure Using Exponential Function Approximation

Abstract

This study proposes a decoupling approach based on first-order exponential function approximation for reliability-based optimization. It establishes an explicit relationship between the failure probability and design parameters within only one implementation of reliability analysis, without the repeated reliability analysis involved in traditional approaches. Then, the obtained explicit exponential function approximation is used to decouple the reliability-based optimization problem and transform it into an equivalent deterministic optimization problem, where a general optimization algorithm can be adopted to solve steadily with a neglectable computational cost. The proposed approach is applied to conduct the reliability-based optimization of a centrifugal compressor impeller design. The results indicate that the proposed method can provide significantly improved computational efficiency while maintaining optimization accuracy comparable to that obtained by traditional double-loop optimization methods. Meanwhile, sensitivity analysis reveals the relative importance of design parameters to structural reliability.

1. Introduction

The increasing demand for high-performance equipment in aerospace, energy, and industrial machinery has highlighted reliability as a core issue to be addressed during the engineering design process. In aerospace applications, turbine engines and hypersonic vehicle components require extreme reliability under high thermal and mechanical loads. Similarly, energy systems such as wind turbines and nuclear reactors demand robust designs to prevent catastrophic failures under cyclic stress. To ensure operational safety, modern industries increasingly rely on condition monitoring systems (CMS) to track real-time performance degradation. For example, Bono et al. [1] developed neural network-based soft sensors for detecting anomalies in rotating machinery, demonstrating the critical role of CMS in predictive maintenance. Structural components in such applications must operate under harsh conditions, including high temperatures, pressures, and speeds as well as complex loads, and their failures can lead to catastrophic consequences. Therefore, efficient reliability-based optimization methods are required to improve component designs. Current approaches for reliability-based optimization can be categorized according to their solution strategies as double-loop, single-loop, or decoupled methods [2].

Double-loop methods [3,4,5,6] represent direct approaches for solving reliability-based optimization problems. They use an outer loop to optimize the design variables and an inner loop to solve the reliability problem, which includes both design and random variables. These methods typically incur high computational costs and rely upon the selection of design point solutions when combined with approximate analytical methods [6].

Single-loop methods [7,8,9,10,11,12,13] transform the reliability-based optimization process into a deterministic problem that can be solved using standard optimization algorithms. This approach simplifies the solution process from a double loop to a single loop by approximating deterministic constraints to replace the original reliability constraints. Various single-loop conversion strategies have been developed: Madsen and Hansen [7] proposed the initial single-loop strategy, Chen et al. [8] described the single-vector method, Lim and Lee [9] detailed the semi-single-loop method, Zeng et al. [10] introduced the chaotic single-loop method, Jiang et al. [11] demonstrated the adaptive hybrid single-loop method, and Keshtegar and Hao [14] proposed the enhanced single-loop method. The core function of all single-loop methods is to convert probabilistic constraints—using, for example, the Karush–Kuhn–Tucker optimization conditions [12] or limit state sensitivity [13]—into a form that can be handled by standard optimization algorithms. However, such methods require probability conversion and derivative solving, which are often impractical in engineering applications.

Decoupled methods [15,16,17,18,19,20,21] use explicit expressions for the design variables to approximate the probabilities considered in reliability-based optimization problems. This approach transforms the reliability-based optimization problem into a deterministic optimization problem that can be solved using conventional optimization algorithms. The successful application of a decoupled method relies upon the selection of an appropriate approximation of the failure probability function, i.e., the failure probability solution. Failure probability functions can be approximated using the exponential function approximation method or the extended reliability method [19,20], or by combining design variable sensitivity information.

Given the high computational costs and limited application of the existing double-loop methods and single-loop methods for reliability-based optimization in practical engineering, this paper adopts a decoupling approach based on first-order exponential function approximation to solve the RBDO problem of a rotor turbine structure.

Unlike adaptive surrogate methods that require iterative model updates or hybrid single-loop methods dependent on derivative calculations, the proposed first-order exponential approximation decouples reliability analysis and optimization by constructing an explicit failure probability function through a single reliability evaluation. This eliminates repeated reliability analyses and sensitivity iterations, making it computationally efficient and practical for complex engineering systems. It first expresses the relationship between the relevant design variables and failure probability using a first-order exponential function approximation, and then applies this approximated function in an iterative decoupling process in optimization to refine the design solution. The proposed optimization method was applied to the design of a centrifugal compressor impeller; the results indicate significant improvements in computational efficiency using the adopted method when compared to traditional double-loop methods.

The remainder of this paper is organized as follows: Section 2 details the proposed reliability-based optimization methodology, Section 3 applies the proposed optimization method to design an impeller, and Section 4 concludes the study.

2. Reliability-Based Optimization

Reliability-based optimization design methods attempt to minimize structural costs by optimizing design variables while satisfying the given reliability and other deterministic constraints [2]. This is accomplished using the following mathematical expressions:

$$\begin{array}{lll}Min& C(\theta ,d)& \\ \mathit{s}.\mathit{t}. & P_{Fl}\left(\theta \right)\le P_{Fl}^{tol}& \left(l=\mathrm{1,2},\dots ,n_P\right)\\ & L_j(\theta ,d)\le 0& \left(j=\mathrm{1,2},\dots ,n_D\right)\\ & {\underset{\_}{\theta }}_j\le {\theta }_i\le {\bar{\theta }}_i& \left(i=\mathrm{1,2},\dots ,n_{\theta }\right)\\ & {\underset{\_}{d}}_k\le d_k\le {\bar{d}}_k& \left(k=\mathrm{1,2},\dots ,n_d\right)\end{array}$$

(1)

where $C(\theta ,d)$ is the objective function, in which $\theta =\left[{\theta }_1,\dots ,{\theta }_{n_{\theta }}\right]$ are the design parameters related to the distribution of the system’s basic random variables (e.g., mean values) and $d=\left[d_1,\dots ,d_n\right]$ are the other deterministic design variables; $L_j(\theta ,d)$ represents the $j$-th deterministic constraint; $P_{Fl}(x,\theta )$ is the $l$-th failure probability constraint (i.e., reliability constraint), which is a function of $\theta$ and $P_{Fl}\left(\theta \right)$, and $P_{Fl}^{\mathrm{tol} }$ is the corresponding failure probability constraint threshold; and $n_P,n_D,n_g$, and $n_d$ are the number of probabilistic constraints, number of deterministic constraints, number of design parameters, and number of deterministic parameters..

The objective function $\mathrm{C}(\mathsf{\theta },\mathrm{d})$ typically includes the material, manufacturing, and maintenance costs [22]. In general equipment systems, material costs $\mathsf{\theta }$ and $\mathrm{d}$ are directly related to design parameters and can be calculated using the volume and density of the material [23]. Manufacturing costs are related to the processing technology employed and complexity of the intended design; they can be estimated using the process parameters. Maintenance costs are related to structural reliability and expected life and can be predicted using a reliability analysis [24].

Critically, the deterministic constraints ${\mathrm{L}}_{\mathrm{j}}(\mathsf{\theta },\mathrm{d})$ include geometric and mechanical constraints that must be satisfied during the design process. These constraints can be mathematically expressed in forms such as ${\mathrm{L}}_{\mathrm{j}}(\mathsf{\theta },\mathrm{d})={\mathrm{g}}_{\mathrm{j}}(\mathsf{\theta },\mathrm{d})-{\mathrm{g}}_{\mathrm{j}}^{\mathrm{l}\mathrm{i}\mathrm{m}}\le 0$, where ${\mathrm{g}}_{\mathrm{j}}(\mathsf{\theta },\mathrm{d})$ is a function of the design variables and ${\mathrm{g}}_{\mathrm{j}}^{\mathrm{l}\mathrm{i}\mathrm{m}}$ is the corresponding upper limit constraint [25].

3. Proposed Reliability-Based Optimization Method

3.1. Failure Probability Function Approximation Using First-Order Exponential Function

Decoupling methods approximate the failure probability function using a surrogate model ${\stackrel{\mathrm{ˆ}}{\mathrm{P}}}_{\mathrm{F}}(\mathsf{\theta })$ 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 $\mathrm{x}=\left[{\mathrm{x}}_1,{\mathrm{x}}_2,\dots ,{\mathrm{x}}_{\mathrm{n}}\right]$ as the $\mathrm{n}$ random variables describing a structural system and $\mathsf{\theta }=\left[{\mathsf{\theta }}_1,{\mathsf{\theta }}_2,\dots ,{\mathsf{\theta }}_{{\mathrm{n}}_{\mathsf{\theta }}}\right]$ as the ${\mathrm{n}}_{\mathsf{\theta }}$ design parameters of the system. The first-order exponential function for the approximation of the failure probability function can be expressed as

$${\stackrel{\mathrm{ˆ}}{\mathrm{P}}}_{\mathrm{F}}(\mathsf{\theta })={\mathrm{e}}^{{\mathrm{C}}_0+{\sum }_{\mathrm{i}=1}^{{\mathrm{n}}_{\mathsf{\theta }}} {\mathrm{C}}_{\mathrm{i}}\cdot {\mathsf{\theta }}_{\mathrm{i}}}$$

(2)

where ${\mathrm{C}}_0,{\mathrm{C}}_{\mathrm{i}}\left(\mathrm{i}=\mathrm{1,2},\dots ,{\mathrm{n}}_{\mathsf{\theta }}\right)$ are the undetermined coefficients of the first-order exponential function.

As Equation (2) contains ${\mathrm{n}}_{\mathsf{\theta }}+1$ unknowns, it requires ${\mathrm{n}}_{\mathsf{\theta }}+1$ equations to solve. Conventional reliability analysis is performed by using different values of $\left|\mathsf{\theta }\right|$ to obtain the corresponding failure probability values, and then establishing ${\mathrm{n}}_{\mathsf{\theta }}+1$ equations to solve the ${\mathrm{n}}_{\mathsf{\theta }}+1$ undetermined coefficients. In contrast, this study used sensitivity information to establish these equations. Under this method, the ${\mathrm{n}}_{\mathsf{\theta }}+1$ coefficients for the first-order exponential 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 ${\mathrm{n}}_{\mathsf{\theta }}$. 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 ${\mathrm{n}}_{\mathsf{\theta }}$ sensitivity values, which can in turn establish the ${\mathrm{n}}_{\mathsf{\theta }}+1$ equations [28] required to solve the corresponding undetermined coefficients. Approximation accuracy can be improved by taking the midpoint ${\mathsf{\theta }}_0=(\overline{\mathsf{\theta }}+\underset{\_}{\mathsf{\theta }})/2$ of the design variable range as the initial design point (pre-interpolation point) [29] and determining the failure probability ${\stackrel{\mathrm{ˆ}}{\mathrm{P}}}_{\mathrm{F}}\left({\mathsf{\theta }}_0\right)$ and sensitivity at this point using the following reliability analysis:

$$\begin{array}{ll}{\left.{\displaystyle \frac{\partial {\stackrel{\mathrm{ˆ}}{\mathrm{P}}}_{\mathrm{F}}}{\partial {\mathsf{\theta }}_{\mathrm{i}}}}\right|}_{\mathsf{\theta }={\mathsf{\theta }}_0}& ={\left.{\mathrm{C}}_{\mathrm{i}}\cdot {\mathrm{e}}^{{\mathrm{C}}_0+\sum _{\mathrm{i}=1}^{{\mathrm{n}}_{\mathsf{\theta }}} {\mathrm{C}}_{\mathrm{i}}\cdot {\mathsf{\theta }}_{\mathrm{i}}}\right|}_{\mathsf{\theta }={\mathsf{\theta }}_0}\\ & ={\mathrm{C}}_{\mathrm{i}}\cdot {\stackrel{\mathrm{ˆ}}{\mathrm{P}}}_{\mathrm{F}}\left({\mathsf{\theta }}_0\right)\left(\mathrm{i}=1,\dots ,{\mathrm{n}}_{\mathsf{\theta }}\right)\end{array}$$

(3)

Taking the logarithm of both sides of Equation (2) yields

$$\mathrm{l}\mathrm{n}\left({\stackrel{\mathrm{ˆ}}{\mathrm{P}}}_{\mathrm{F}}\left({\mathsf{\theta }}_0\right)\right)={\mathrm{C}}_0+{\left.{\sum }_{\mathrm{i}=1}^{{\mathrm{n}}_{\mathsf{\theta }}} {\mathrm{C}}_{\mathrm{i}}\cdot {\mathsf{\theta }}_{\mathrm{i}}\right|}_{\mathsf{\theta }={\mathsf{\theta }}_0}$$

(4)

Simplifying Equations (3) and (4) results in

$${\mathrm{C}}_{\mathrm{i}}={\displaystyle \frac{{\left.\frac{\partial {\stackrel{\mathrm{ˆ}}{\mathrm{P}}}_{\mathrm{F}}}{\partial {\mathsf{\theta }}_{\mathrm{i}}}\right|}_{\mathsf{\theta }={\mathsf{\theta }}_0}}{{\stackrel{\mathrm{ˆ}}{\mathrm{P}}}_{\mathrm{F}}\left({\mathsf{\theta }}_0\right)}}\left(\mathrm{i}=1,\dots ,{\mathrm{n}}_{\mathsf{\theta }}\right)$$

(5)

$${\mathrm{C}}_0=\ln \left({\stackrel{\mathrm{ˆ}}{\mathrm{P}}}_{\mathrm{F}}\left({\mathsf{\theta }}_0\right)\right)-{\left.{\sum }_{\mathrm{i}=1}^{\mathrm{n}} {\mathrm{C}}_{\mathrm{i}}\cdot {\mathsf{\theta }}_{\mathrm{i}}\right|}_{\mathsf{\theta }={\mathsf{\theta }}_0}$$

(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.

The first-order exponential function is chosen for its computational efficiency in decoupling reliability analysis from optimization. This form requires only one reliability analysis to approximate the relationship between design parameters and failure probability, making it particularly suitable for problems with low-to-moderate nonlinearity. However, for systems with highly nonlinear limit states (e.g., threshold effects) or strong interactions between variables, higher-order terms or adaptive approximation functions may be necessary. The sequential approximation framework in the next section mitigates these limitations by iteratively refining the local region of validity.

In reliability-based optimization, it is crucial to estimate both the failure probability and the sensitivity of failure probability with respect to various design parameters. These quantities are essential for solving the undetermined coefficients of a polynomial response surface model, which is used to approximate the system’s behavior. In the approach discussed here, the failure probability at an interpolation point and the reliability sensitivity values for each design parameter are computed in order to solve for the coefficients $C_0,C_i\left(i=\mathrm{1,2},\dots ,n_{\theta }\right)$ [30]. The calculation process is facilitated using importance sampling (IS) [31], a well-established method that significantly improves computational efficiency, especially for rare-event probability estimations like failure probability.

Importance sampling (IS) is a statistical technique used to estimate the failure probability by generating samples from a different, auxiliary distribution $q\left(x\right)$, which is chosen to better capture the tail behavior of the distribution, where failures are more likely to occur. The method is particularly useful when failure events are rare, which is typically the case in reliability analysis. Instead of sampling directly from the original distribution $f_X\left(x\right)$, which would require a prohibitively large number of samples to observe failures, IS samples more frequently from regions of the design space where failures are expected to occur, thus reducing the variance in the estimated failure probability.

Let $\mathrm{N}$ be the total number of samples drawn from $q\left(x\right)$, and let $x^{\left(j\right)}$ represent the $j$-th sample. The failure probability ${\hat{P}}_f$ is then estimated using the following equation:

$${\hat{P}}_f={\displaystyle \frac{1}{N}} {\sum }_{j=1}^N {\displaystyle \frac{f_X\left(x^{\left(j\right)}\right)}{q\left(x^{\left(j\right)}\right)}}\cdot I\left(g\left(x^{\left(j\right)}\right)\le 0\right)$$

(7)

Here, $f_X\left(x\right)$ is the original probability density function of the input variables, $q\left(x\right)$ is the proposal sampling distribution, and $I(\cdot )$ is the indicator function, which is 1 if the failure criterion $g\left(x\right)\le 0$ is met and 0 otherwise. This formulation essentially weights the failure events according to the likelihood of the sample under the auxiliary distribution, correcting for the over-sampling in the failure region.

Once the failure probability is computed, the next step is to evaluate how sensitive the failure probability is with respect to each design parameter ${\theta }_i$. This is essential in identifying which parameters most influence the reliability of the system, which can inform design improvements or modifications. The sensitivity of the failure probability is computed using the same IS samples, which ensures computational efficiency by avoiding the need for a separate sampling procedure.

The sensitivity of ${\hat{P}}_f$ with respect to the $i$-th design parameter ${\theta }_i$ is estimated using the score function method. This method involves taking the derivative of the log of the probability density function with respect to ${\theta }_i$. The sensitivity is given by

$${\displaystyle \frac{\partial {\hat{P}}_f}{\partial {\theta }_i}}={\displaystyle \frac{1}{N}}{\sum }_{j=1}^N {\displaystyle \frac{f_X\left(x^{\left(j\right)}\right)}{q\left(x^{\left(j\right)}\right)}}\cdot I\left(g\left(x^{\left(j\right)}\right)\le 0\right)\cdot {\displaystyle \frac{\partial ln{f}_X\left(x^{\left(j\right)};\theta \right)}{\partial {\theta }_i}}$$

(8)

In this equation, the derivative term ${\displaystyle \frac{\partial ln{f}_X\left(x^{\left(j\right)};\theta \right)}{\partial {\theta }_i}}$ is the score function for the $j$-th sample. This term captures how the probability density changes with respect to the design parameter ${\theta }_i$, and it is used to adjust the sensitivity estimation accordingly. The sensitivity calculation can be performed without the need to differentiate the limit state function $g\left(x\right)$, which simplifies the process considerably and reduces the computational cost.

One of the key advantages of using importance sampling (IS) in combination with the score function method is the significant reduction in computational effort. By reusing the same set of IS samples for both the failure probability and its sensitivity calculations, the method avoids the need for multiple sampling processes, which can be computationally expensive. This is particularly beneficial when the failure probability is low and requires a large number of samples to estimate accurately.

Moreover, the sensitivity formulation based on the score function method does not require the explicit calculation of gradients or higher-order derivatives of the limit state function g(x). Instead, it relies on the score function, which is computationally simpler and more efficient. This approach makes it possible to compute both the failure probability and its sensitivities with a minimal increase in computational cost, providing a significant advantage in high-dimensional reliability problems.

In summary, this method offers an efficient way to calculate the failure probability and the sensitivity of the failure probability with respect to design parameters. By leveraging the power of importance sampling and the score function method, it provides a reliable and computationally efficient approach for reliability analysis in optimization problems. This approach is particularly valuable in situations where the failure event is rare and traditional Monte Carlo methods would require an impractically large number of samples. Additionally, the ability to compute sensitivities without the need for additional sampling or differentiation of the limit state function further enhances its efficiency.

3.2. 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 ${\mathsf{\theta }}^{\left(\mathrm{k}\right)}$ at the current iteration step are used to obtain the corresponding local approximate failure probability ${\stackrel{\mathrm{ˆ}}{\mathrm{P}}}_{\mathrm{F}}^{\mathrm{k}}\left({\mathsf{\theta }}^{\left(\mathrm{k}\right)}\right)$ as

$${\mathrm{P}}_{\mathrm{F}}(\mathsf{\theta })\approx {\stackrel{\mathrm{ˆ}}{\mathrm{P}}}_{\mathrm{F}}^{\mathrm{k}}(\mathsf{\theta })$$

(9)

The local failure probability function constructed based on Equation (9) is applied to transform the reliability-based optimization problem for the current iteration step into the following deterministic optimization problem:

$$\begin{array}{lll} \mathrm{Min} & \mathrm{C}\left({\mathsf{\theta }}^{(\mathrm{k}+1)},{\mathrm{d}}^{(\mathrm{k}+1)}\right)& \\ \mathrm{s}.\mathrm{t}. & {\stackrel{\mathrm{ˆ}}{\mathrm{P}}}_{\mathrm{F}}^{\mathrm{k}}\left({\mathsf{\theta }}^{(\mathrm{k}+1)}\right)\le {\mathrm{P}}_{\mathrm{F}}^{\mathrm{t}\mathrm{o}\mathrm{l}}& \\ & {\mathrm{L}}_{\mathrm{j}}\left({\mathsf{\theta }}^{(\mathrm{k}+1)},{\mathrm{d}}^{(\mathrm{k}+1)}\right)\le 0& \left(\mathrm{j}=\mathrm{1,2},\dots ,{\mathrm{n}}_{\mathrm{D}}\right)\\ & {\underset{\_}{\mathrm{d}}}_{\mathrm{i}}\le {\mathrm{d}}_{\mathrm{i}}^{(\mathrm{k}+1)}\le {\stackrel{\mathrm{‾}}{\mathrm{d}}}_{\mathrm{i}}& \left(\mathrm{i}=\mathrm{1,2},\dots ,{\mathrm{n}}_{\mathrm{d}}\right)\\ & {\underset{\_}{\mathsf{\theta }}}^{(\mathrm{k}+1)}\le {\mathsf{\theta }}_{\mathrm{i}}^{(\mathrm{k}+1)}\le {\stackrel{\mathrm{‾}}{\mathsf{\theta }}}_{\mathrm{i}}^{(\mathrm{k}+1)}& \left(\mathrm{i}=\mathrm{1,2},\dots ,{\mathrm{n}}_{\mathsf{\theta }}\right)\end{array}$$

(10)

where ${\underset{\_}{\mathsf{\theta }}}_{\mathrm{i}}^{\mathrm{k}+1}$ and ${\stackrel{\mathrm{‾}}{\mathsf{\theta }}}_{\mathrm{i}}^{\mathrm{k}+1}$ represent the bounds of the local optimization region at the $\mathrm{k}+1$ iteration.

Next, the deterministic optimization problem is solved to obtain the optimal design value solution ${\mathsf{\theta }}^{(\mathrm{k}+1)}$ at the current iteration step. This optimal solution is subsequently applied to establish an approximation of the failure probability function using Equation (2), leading to the next iteration step.

Each iteration of the approximate optimization method requires solving the deterministic optimization problem, the calculation cost of which is quite low, permitting the use of most available optimization algorithms. This process is repeated to produce the optimal solution sequence ${\mathsf{\theta }}^{\left(\mathrm{k}\right)}(\mathrm{k}=\mathrm{1,2},\dots )$, which gradually approaches the optimal solution of the original reliability-based optimization problem. The general solution sequence of this optimization process is illustrated in Figure 1.

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:

$$\underset{\_}{{\theta }_i^k}=\mathrm{m}\mathrm{a}\mathrm{x}\left\{{\theta }_i^k\left(1-R_k\right),{\underset{\_}{\theta }}_i\right\},{\bar{\theta }}^k=\mathrm{m}\mathrm{i}\mathrm{n}\left\{{\theta }_i^k\left(1+R_k\right),{\bar{\theta }}_i\right\}$$

(11)

where ${\mathrm{R}}_{\mathrm{k}}(\mathrm{k}=\mathrm{1,2},\dots )$ 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:

$${\mathrm{R}}_{\mathrm{k}}={\mathrm{R}}_0\times {\mathrm{r}}^{\mathrm{k}}$$

(12)

where ${\mathrm{R}}_0$ is the initial value, typically $20\mathrm{\%}$, and $\mathrm{r}$ is a reduction factor with a value range of $\left[\mathrm{0.8,1}\right]$. The choice of values for the initial interval parameter ${\mathrm{R}}_0$ and the reduction factor r based on a larger initial interval, i.e., ${\mathrm{R}}_0=20\mathrm{\%}$, helps to explore the design space quickly in early iterations to avoid falling into a local optimum solution, and a slow reduction ($\mathrm{r}\ge 0.8$ ) 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 ${\mathrm{R}}_{\mathrm{k}}$ 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, ${\mathrm{R}}_{\mathrm{k}}$ 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.

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.

4. Reliability-Based Optimization of an Impeller Structure

4.1. Centrifugal Compressor Impeller Structure

4.1.1. Modeling Assumptions and Limitations

In this study, the fluid–structure decoupled analysis method is used, i.e., the pressure distribution is first obtained from the CFD analysis, and then it is used as a boundary condition for the structural analysis. Although this method greatly reduces the computational cost, it also brings certain limitations. The accurate transfer of hydrodynamic loads depends on fluid–structure coupling analysis. Neglecting FSI may lead to underestimation or overestimation of local stress concentrations, especially at the leading and trailing edges of the blades, failure to capture the nonlinear coupling effects between fluid flow and structural deformation, and inaccurate estimation of the probability of failure. If the FSI effect causes the actual stresses to be lower than those predicted by the decoupling analysis, then ignoring the FSI will result in an overestimation of the probability of failure and an overly conservative design. On the contrary, if the FSI effect enhances the stresses, ignoring the FSI may underestimate the probability of failure, resulting in a potential safety issue. Future research should incorporate FSI coupling to improve the accuracy and reliability of the optimization framework for high-performance applications where fluid–structure coupling effects are significant.

4.1.2. Structural Analysis Methodology

The impeller is a critical component that directly affects the efficiency and stability of rotating machinery such as fans and pumps. Therefore, this study used the proposed iterative decoupling reliability-based optimization method based on first-order exponential approximation to design a centrifugal compressor impeller blade.

First, a 3D model of the considered centrifugal compressor impeller was completed using SOLIDWORKS, as shown in Figure 3. The process for analyzing the rotor blade structure without considering the coupling of the fluid–structure interaction is simple. First, the pressure distribution on the blade surface was solved within the fluid domain using computational fluid dynamics. Next, the 3D model was imported into ANSYS 2024 R1 for meshing using a hexahedral-dominated hybrid meshing scheme with local refinements on the blade surfaces and near stress concentrations. A comparative analysis of incrementally denser meshes indicated that the relative change in the maximum stress was less than 1% once the number of mesh elements exceeded 150,000, satisfying the mesh independence requirement. In the structural analysis of the impeller, the boundary conditions are set as follows: A fixed support constraint is applied to the bottom end face of the impeller to simulate its connection to the shaft; a rotational speed of −3000 r/min is applied to the entire structural domain, and the influence of the Earth’s gravitational field (9.81 m/s²) on the structure is considered. Meanwhile, the pressure loads obtained from the CFD analysis of the fluid domain are mapped onto the blade surfaces to simulate the fluid–structure interaction effects. The combined action of these boundary conditions enables the structural analysis to capture the comprehensive effects of rotational forces, gravitational loads, and fluid pressure on the impeller blades, ultimately determining the actual stress and strain conditions of the blade structure. Schematic diagrams of the physics, 3D model, and boundary conditions are shown in Figure 4, and the analytical stress results are presented in Figure 5.

4.2. 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. Distribution information of the variables of the impeller structure.

Variable	Mean Value	Standard Deviation
Rotational speed (rev/min)	3000	200
Inlet temperature (°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

). 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 (°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

$$\stackrel{\mathrm{ˆ}}{\mathrm{g}}\left(\mathrm{x}\right)=5800-\stackrel{\mathrm{ˆ}}{\mathrm{y}}\left(\mathrm{x}\right)$$

(13)

where $\stackrel{\mathrm{ˆ}}{\mathrm{y}}\left(\mathrm{x}\right)$ 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 $\mathrm{x}$.

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, ${\mathrm{P}}_{\mathrm{F}}\le {\mathrm{P}}_{\mathrm{F}}^{\mathrm{tol} }$, where ${\mathrm{P}}_{\mathrm{F}}^{\mathrm{tol} }={10}^{-4}$. The objective function was based on cost and normalized to balance the different magnitudes of the design variables ${\mathsf{\theta }}_1$ (inlet temperature) and ${\mathsf{\theta }}_2$ (inlet pressure) as follows:

$$\mathrm{C}\left({\mathsf{\theta }}_1,{\mathsf{\theta }}_2\right)={\displaystyle \frac{{\mathsf{\theta }}_1}{160.985}}-{\displaystyle \frac{{\mathsf{\theta }}_2}{4.62}}$$

(14)

Thus, the reliability-based optimization design model for the impeller is given by

$$\begin{array}{ll} \mathrm{Min} & \mathrm{C}\left({\mathsf{\theta }}_1,{\mathsf{\theta }}_2\right)={\displaystyle \frac{{\mathsf{\theta }}_1}{160.985}}-{\displaystyle \frac{{\mathsf{\theta }}_2}{4.62}}\\ \mathrm{s}.\mathrm{t}. & {\mathrm{P}}_{\mathrm{F}}<{\mathrm{P}}_{\mathrm{F}}^{\mathrm{tol} }\\ & -190.985\le {\mathsf{\theta }}_1\le -130.985\\ & 1.62\le {\mathsf{\theta }}_1\le 5.62\end{array}$$

(15)

The effectiveness of the proposed first-order exponential approximation (IS) method was demonstrated by comparing its reliability-based optimization results with those obtained using the double-loop importance sampling (IS [31]) and subset simulation (SS [32]) methods, as shown in

Table 2. Reliability-based optimization results of the proposed decoupling first-order approximation method and double-loop methods.

Method	Inlet Temperature (°C)	Inlet Pressure (MPa)	Objective Function	Failure Probability	Coefficient of Variation	Number of Function Evaluations
Double-loop IS	−158.68	5.6200	−2.2291	1.00 × 10−4	0.0292	4.7 × 105
Double-loop SS	−161.16	5.4153	−2.1991	9.44 × 10−5	0.0965	2.0 × 106
First-order exponential approximation (IS)	−153.65	5.6200	−2.1978	3.20 × 10−5	0.0709	2.4 × 104

. 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. Notably, the failure probabilities obtained by all three optimization methods were less than 10⁻⁴, satisfying the reliability constraint conditions; the LS method exhibited the smallest failure probability, indicating its superior ability to ensure structural reliability. The coefficients of variation for the IS, SS, and LS methods were 0.0292, 0.0965, and 0.0709, respectively, indicating that the computational accuracy of each method was within the acceptable range.

In terms of computational efficiency, the IS, SS, and LS methods, respectively, required 4.7 × 10⁵, 2.0 × 10⁶, and 2.4 × 10⁴ iterations. Clearly, the proposed method exhibited significantly improved computational efficiency, requiring an order of magnitude fewer iterations than the conventional double-loop methods.

The proposed method yields a failure probability lower than the target, which may suggest conservatism in the final design. This occurs because the first-order exponential approximation ensures strict satisfaction of the reliability constraint within each local optimization interval during the sequential optimization process. However, the objective function value of the proposed method, −2.1978, is only 1.4% higher, i.e., less optimal than the double-loop IS result, −2.2291, while achieving a 95% reduction in computational cost. Such a minor sacrifice in optimality is acceptable in engineering practice, given the drastic efficiency improvement.

4.3. Sensitivity Analysis of Design Parameters

The sensitivity coefficients were computed for the key design parameters of the impeller structure.

Table 3. Sensitivity analysis results for impeller design parameters.

Design Parameter	Current Value	Sensitivity Coefficient	Normalized Sensitivity	Ranking
Inlet Temperature Mean (°C)	−160.985	−2.34 × 10−6	0.452	1
Inlet Pressure Mean (MPa)	3.62	1.87 × 10−5	0.318	2
Temperature Std. Deviation (°C)	10.73	−8.92 × 10−7	0.165	3
Pressure Std. Deviation (MPa)	0.30	3.45 × 10−6	0.065	4

Note: Normalized sensitivities are calculated as the ratio of absolute sensitivity to the sum of all absolute sensitivities.

presents the sensitivity analysis results, including both absolute sensitivity coefficients and normalized values for comparison.

The results indicate that the inlet temperature mean has the highest sensitivity coefficient (0.452), followed by the inlet pressure mean (0.318). The standard deviations of both parameters show relatively lower sensitivities, suggesting that the mean values of the operating conditions are more critical for structural reliability than their uncertainties.

Sensitivity analysis provides valuable insights into the actual design and operation of impellers. The high sensitivity to the mean inlet temperature indicates that precise temperature control is essential to maintain structural reliability. Lower inlet temperatures typically result in reduced thermal stresses, which improves reliability. Design engineers should prioritize robust temperature control systems and insulation. Second, the high sensitivity to mean inlet pressure emphasizes the importance of pressure regulation systems. Higher inlet pressures increase the mechanical load on the impeller structure, which can lead to a higher probability of failure. Proper pressure control valves and monitoring systems are critical. Further, the relatively low sensitivity to standard deviation suggests that while controlling average operating conditions is critical, reducing the impact of operating uncertainty on reliability is secondary, but still relevant. This finding supports prioritizing investment in control systems over uncertainty reduction measures.

In summary, the first-order exponential approximation method not only achieved optimization results comparable to those obtained by traditional methods, but it also offered significant advantages in terms of computational efficiency. Indeed, the application of the proposed iterative decoupling reliability approximation optimization method provided an impeller design capable of exhibiting suitable performance and safety in the presence of various uncertainties.

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.

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.