# Reliability-Based Optimization of Impeller Structure Using Exponential Function Approximation
## 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, [PERSON] 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: [PERSON] and [PERSON] [7] proposed the initial single-loop strategy, [PERSON] et al. [8] described the single-vector method, [PERSON] and [PERSON] [9] detailed the semi-single-loop method, [PERSON] et al. [10] introduced the chaotic single-loop method, [PERSON] et al. [11] demonstrated the adaptive hybrid single-loop method, and [PERSON] and [PERSON] [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.