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PurposeAs an advanced calculation methodology, reliability-based multidisciplinary design optimization (RBMDO) has been widely acknowledged for the design problems of modern complex engineering systems, not only because of the accurate evaluation of the impact of uncertain factors but also the relatively good balance between economy and safety of performance. However, with the increasing complexity of engineering technology, the proposed RBMDO method gradually cannot effectively solve the higher nonlinear coupled multidisciplinary uncertainty design optimization problems, which limits the engineering application of RBMDO. Many valuable works have been done in the RBMDO field in recent decades to tackle the above challenges. This study is to review these studies systematically, highlight the research opportunities and challenges, and attempt to guide future research efforts.Design/methodology/approachThis study presents a comprehensive review of the RBMDO theory, mainly including the reliability analysis methods of different uncertainties and the decoupling strategies of RBMDO.FindingsFirst, the multidisciplinary design optimization (MDO) preliminaries are given. The basic MDO concepts and the corresponding mathematical formulas are illustrated. Then, the procedures of three RBMDO methods with different reliability analysis strategies are introduced in detail. These RBMDO methods were proposed for the design optimization problems under different uncertainty types. Furtherly, an optimization problem for a certain operating condition of a turbine runner blade is introduced to illustrate the engineering application of the above method. Finally, three aspects of future challenges for RBMDO, namely, time-varying uncertainty analysis; high-precision surrogate models, and verification, validation and accreditation (VVA) for the model, are discussed followed by the conclusion.Originality/valueThe scope of this study is to introduce the RBMDO theory systematically. Three commonly used RBMDO-SORA methods are reviewed comprehensively, including the methods' general procedures and mathematical models.

Considering the coupling among aerodynamic, heat transfer and strength, a reliability based multidisciplinary design optimization method for cooling turbine blade is introduced. Multidisciplinary analysis of cooling turbine blade is carried out by sequential conjugated heat transfer analysis and strength analysis with temperature and pressure interpolation. Uncertainty data including the blade wall, rib thickness, elasticity Modulus and rotation speed is collected. Data statistics display the probability models of uncertainty data follow three-parameter Weibull distribution. The thickness of blade wall, thickness and height of ribs are chosen as design variables. Kriging surrogate model is introduced to reduce time-consuming multidisciplinary reliability analysis in RBMDO loop. The reliability based multidisciplinary design optimization of a cooling turbine blade is carried out. Optimization results shows that the RBMDO method proposed in this work improves the performance of cooling turbine blade availably.

To meet the rising demand for high reliability in complex multidisciplinary engineering systems, more attention has been paid to reliability-based multidisciplinary design optimization (RBMDO). In this paper, a sequential optimization and fuzzy reliability analysis (SOFRA) method for multidisciplinary systems is developed to decouple the fuzzy reliability analysis from the optimization. In SOFRA, the multidisciplinary design optimization (MDO) and fuzzy reliability analysis are conducted in a sequential manner. Furthermore, a novel adaptive collocation method (ACM) is proposed to conduct the fuzzy reliability analysis for multidisciplinary systems. The ACM arranges points adaptively at the axis of the membership to obtain more accurate results. The shifting distance of the constraint is calculated by the bi-section method. Both numerical and engineering examples are used to demonstrate the validity of the proposed method.

In uncertainty analysis and reliability-based multidisciplinary design and optimization (RBMDO) of engineering structures, the saddlepoint approximation (SA) method can be utilized to enhance the accuracy and efficiency of reliability evaluation. However, the random variables involved in SA should be easy to handle. Additionally, the corresponding saddlepoint equation should not be complicated. Both of them limit the application of SA for engineering problems. The moment method can construct an approximate cumulative distribution function of the performance function based on the first few statistical moments. However, the traditional moment matching method is not very accurate generally. In order to take advantage of the SA method and the moment matching method to enhance the efficiency of design and optimization, a fourth-moment saddlepoint approximation (FMSA) method is introduced into RBMDO. In FMSA, the approximate cumulative generating functions are constructed based on the first four moments of the limit state function. The probability density function and cumulative distribution function are estimated based on this approximate cumulative generating function. Furthermore, the FMSA method is introduced and combined into RBMDO within the framework of sequence optimization and reliability assessment, which is based on the performance measure approach strategy. Two engineering examples are introduced to verify the effectiveness of proposed method.

The design of aerospace systems is recognized as a complex interdisciplinary process. Many studies have shown that the exchange of information among multiple disciplines often results in strong coupling and nonlinearity characteristics in system optimization. Meanwhile, inevitable multi-source uncertainty factors continuously accumulate during the optimization process, greatly compromising the system’s robustness and reliability. In this context, uncertainty-based multidisciplinary design optimization (UMDO) has emerged and has been preliminarily applied in aerospace practices. However, it still encounters major challenges, including the complexity of multidisciplinary analysis modeling, and organizational and computational complexities of uncertainty analysis and optimization. Extensive research has been conducted recently to address these issues, particularly uncertainty analysis and artificial intelligence strategies. The former further enriches the UMDO technique, while the latter makes outstanding contributions to addressing the computational complexity of UMDO. With the aim of providing an overview of currently available methods, this paper summarizes existing state-of-the art UMDO technologies, with a special focus on relevant intelligent optimization strategies.

To address the reliability-based multidisciplinary design optimization (RBMDO) problem under mixed aleatory and epistemic uncertainties, an RBMDO procedure is proposed in this paper based on combined probability and evidence theory. The existing deterministic multistage-multilevel multidisciplinary design optimization (MDO) procedure MDF-CSSO, which combines the multiple discipline feasible (MDF) procedure and the concurrent subspace optimization (CSSO) procedure to mimic the general conceptual design process, is used as the basic framework. In the first stage, the surrogate based MDF is used to quickly identify the promising reliable regions. In the second stage, the surrogate based CSSO is used to organize the disciplinary optimization and system coordination, which allows the disciplinary specialists to investigate and optimize the design with the corresponding high-fidelity models independently and concurrently. In these two stages, the reliability-based optimization both in the system level and the disciplinary level are computationally expensive as it entails nested optimization and uncertainty analysis. To alleviate the computational burden, the sequential optimization and mixed uncertainty analysis (SOMUA) method is used to decompose the traditional double-level reliability-based optimization problem into separate deterministic optimization and mixed uncertainty analysis sub-problems, which are solved sequentially and iteratively until convergence is achieved. By integrating SOMUA into MDF-CSSO, the Mixed Uncertainty based RBMDO procedure MUMDF-CSSO is developed. The effectiveness of the proposed procedure is testified with one simple numerical example and one MDO benchmark test problem, followed by some conclusion remarks.

Multidisciplinary robust design optimization (MRDO) is a useful tool to improve the stability of the performance of complex engineering systems involving uncertainty. However, the majority of existing MRDO studies only consider the parameter uncertainty. Metamodeling uncertainty, defined as the discrepancy between the computer model and metamodel at un-sampled locations, is often overlooked in MRDO. To solve the multidisciplinary problems under parameter and metamodeling uncertainties, this paper proposes a new framework called MRDO under parameter and metamodeling uncertainties (MRDO-UPM). The collaboration model is used to select the samples which satisfy coupled state equations. The selected samples are employed to construct the Gaussian process metamodels of the objective, constraint, and multidisciplinary coupled functions. Monte Carlo simulation is adopted to quantify the compound impact of parameter and metamodeling uncertainties. The MRDO-UPM framework is employed to explore the optimum. The proposed framework is verified through a numerical example, and the design of a speed reducer and a liquid cooling battery thermal management system.

Probabilistic analytical target cascading (PATC) has been developed to incorporate uncertainty of random variables in a hierarchical multilevel system using the framework of ATC. In the decomposed ATC structure, consistency between linked subsystems has to be guaranteed through individual subsystem optimizations employing special coordination strategies such as augmented Lagrangian coordination (ALC). However, the consistency in PATC has to be attained exploiting uncertainty quantification and propagation of interrelated linking variables that are the major concern of PATC and uncertainty-based multidisciplinary design optimization (UMDO). In previous studies, the consistency of linking variables is assured by matching statistical moments under the normality assumption. However, it can induce significant error when the linking variable to be quantified is highly nonlinear and non-normal. In addition, reliability estimated from statistical moments may be inaccurate in each optimization of the subsystem. To tackle the challenges, we propose the sampling-based PATC using multivariate kernel density estimation (KDE). The framework of reliability-based design optimization (RBDO) using sampling methods is adopted in individual optimizations of subsystems in the presence of uncertainty. The uncertainty quantification of linking variables equivalent to intermediate random responses can be achieved by multivariate KDE to account for correlation between linking variables. The constructed KDE based on finite samples of the linking variables can provide accurate statistical representations to linked subsystems and thus be utilized as probability density function (PDF) of linking variables in individual sampling-based RBDOs. Stochastic sensitivity analysis with respect to multivariate KDE is further developed to provide an accurate sensitivity of reliability during the RBDO. The proposed sampling-based PATC using KDE facilitates efficient and accurate procedures to obtain a system optimum in PATC, and the mathematical examples and roof assembly optimization using finite element analysis (FEA) are used to demonstrate the effectiveness of the proposed approach.

Today, due to the complexity of systems and interactions among their subsystems, the design optimization of a system is highly difficult and costly. Multidisciplinary optimization is an approach, in which interactions among different disciplines are taken into account, and it attempts to optimize all disciplines, simultaneously. In the design process of a system, there is usually some uncertainty in parameters. This uncertainty creates some challenges in the design process and affects the systems performance. To cope with the uncertainty, robust design and reliability-based design approaches are developed. In this paper, a reliability-based multidisciplinary design optimization is presented, in which some of the problem parameters are uncertain. In this regard, it is assumed that some of the problem parameters are in the form of fuzzy numbers. Moreover, in this problem cost is considered as one of the design disciplines, due to its importance in engineering problems. To solve the proposed model, a solution method named the sequential optimization and reliability assessment is presented in which Genetic Algorithm and Particle Swarm Optimization are used to solve the deterministic problem in each iteration. Finally, the design of an autonomous underwater vehicle including cost analysis is investigated and two solution methods are applied. The obtained results from two methods are compared and some conclusion are made. The results show that improving the reliability between 0.5 and 0.85 is more cost-effective. However, some other factors besides the cost play a role in choosing the reliability level that must be considered.