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Multidisciplinary design optimization (MDO) is concerned with solving design problems involving coupled numerical models of complex engineering systems. While various MDO software frameworks exist, none of them take full advantage of state-of-the-art algorithms to solve coupled models efficiently. Furthermore, there is a need to facilitate the computation of the derivatives of these coupled models for use with gradient-based optimization algorithms to enable design with respect to large numbers of variables. In this paper, we present the theory and architecture of OpenMDAO, an open-source MDO framework that uses Newton-type algorithms to solve coupled systems and exploits problem structure through new hierarchical strategies to achieve high computational efficiency. OpenMDAO also provides a framework for computing coupled derivatives efficiently and in a way that exploits problem sparsity. We demonstrate the framework’s efficiency by benchmarking scalable test problems. We also summarize a number of OpenMDAO applications previously reported in the literature, which include trajectory optimization, wing design, and structural topology optimization, demonstrating that the framework is effective in both coupling existing models and developing new multidisciplinary models from the ground up. Given the potential of the OpenMDAO framework, we expect the number of users and developers to continue growing, enabling even more diverse applications in engineering analysis and design.

Robust optimization design is significant and urgently required for the fly wings, owing to its unique characteristics. However, there is a lack of efficient tools for performing shape optimization which considers multiple uncertainties. This is in part because implementing robust design in the widely used and very efficient adjoint-based optimization method is challenging. This paper addresses this need by developing an uncertainty-based optimization design framework where the gradient-enhanced polynomial chaos expansion and discrete, adjoint-based optimization framework are coupled to perform shape optimization under multiple uncertainties. The gradient information from adjoint equation is applied to improve the computation efficiency. The objective function is the statistic moment, consisting of mean and standard deviation. The gradients of the statistic moment are computed using the adjoint-based system and reconstructing a regression algorithm. A flying wing configuration with deterministic and two uncertainty-based optimizations is performed. The first uncertainty-based optimization considers flight conditions, Mach and angle of attack, and the second one added the planform uncertainty parameters, i.e., inner and outer wing sweep angle. The uncertainty-based optimizations gain reductions of statistic moments by 8.58% and 5.3%, respectively. Compared with the deterministic optimization, the uncertainty-based optimizations behave much better in robustness but sacrifice a small aerodynamic performance. The successful uncertainty-based optimization enables acceptable risks of fly wing design in the development process and indicates that our established framework can be applied for future aircraft robust optimization design.

Multiobjective reliability-based design optimization (RBDO) is a research area, which has not been investigated in the literatures comparing with single-objective RBDO. This work conducts an exhaustive study of fifteen new and popular metaheuristic multiobjective RBDO algorithms, including non-dominated sorting genetic algorithm II, differential evolution for multiobjective optimization, multiobjective evolutionary algorithm based on decomposition, multiobjective particle swarm optimization, multiobjective flower pollination algorithm, multiobjective bat algorithm, multiobjective gray wolf optimizer, multiobjective multiverse optimization, multiobjective water cycle optimizer, success history-based adaptive multiobjective differential evolution, success history-based adaptive multiobjective differential evolution with whale optimization, multiobjective salp swarm algorithm, real-code population-based incremental learning and differential evolution, unrestricted population size evolutionary multiobjective optimization algorithm, and multiobjective jellyfish search optimizer. In addition, the adaptive chaos control method is employed for the above-mentioned algorithms to estimate the probabilistic constraints effectively. This comparative analysis reveals the critical technologies and enormous challenges in the RBDO field. It also offers new insight into simultaneously dealing with the multiple conflicting design objectives and probabilistic constraints. Also, this study presents the advantage and future development trends or incurs the increased challenge of researchers to put forward an effective multiobjective RBDO algorithm that assists the complex engineering system design.

To teach multidisciplinary design optimization (MDO) to students effectively, it is useful to have accessible software that runs quickly, allowing hands-on exploration of coupled systems and optimization methods. Open-source software exists for low-fidelity aerodynamic or structural analysis, but there is no existing software for fast tightly coupled aerostructural analysis and design optimization. To address this need, we present OpenAeroStruct, an open-source low-fidelity aerostructural analysis and optimization tool developed in NASA’s OpenMDAO framework. It uses the coupled adjoint method to compute the derivatives required for efficient gradient-based optimization. OpenAeroStruct combines a vortex lattice method and 1-D finite-element analysis to model lifting surfaces, such as aircraft wings and tails, and uses the coupled-adjoint method to compute the aerostructural derivatives. We use the Breguet range equation to compute the fuel burn as a function of structural weight and aerodynamic performance. OpenAeroStruct has proved effective both as an educational tool and as a benchmark for researching new MDO methods. There is much more potential to be exploited as the research community continues to develop and use this tool.

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.

With the rapid development of the economy, the disparity between supply and demand of resources is becoming increasingly prominent in engineering design. In this paper, an improved gray wolf optimization algorithm is proposed (IGWO) to optimize engineering design problems. First, a tent map is used to generate the initial location of the gray wolf population, which evenly distributes the gray wolf population and lays the foundation for a diversified global search process. Second, Gaussian mutation perturbation is used to perform various operations on the current optimal solution to avoid the algorithm falling into local optima. Finally, a cosine control factor is introduced to balance the global and local exploration capabilities of the algorithm and to improve the convergence speed. The IGWO algorithm is applied to four engineering optimization problems with different typical complexity, including a pressure vessel design, a tension spring design, a welding beam design and a three-truss design. The experimental results show that the IGWO algorithm is superior to other comparison algorithms in terms of optimal performance, solution stability, applicability and effectiveness; and can better solve the problem of resource waste in engineering design. The IGWO also optimizes 23 different types of function problems and uses Wilcoxon rank-sum test and Friedman test to verify the 23 test problems. The results show that the IGWO algorithm has higher convergence speed, convergence precision and robustness compared with other algorithms.

In the multi-objective optimization design of automotive seats based on Approximation-Based Design Optimization, a single approximation model may not adequately address the requirement of accurately fitting highly nonlinear feature data. For this reason, the Hybrid Approximation Models based on the Multi-Species Approximation Model (HAM-MSAM) is proposed to meet the requirement for high fitting accuracy. Subsequently, this study introduces a HAM-MSAM-based Approximation-Based Global Multi-Objective Optimization Design (ABGMOOD) strategy. This strategy is employed in the multi-objective optimization of the rear seat of a passenger car. HAM-MSAM was constructed from an experimentally validated finite element model and a training set generated through experimental design. The advantages of HAM-MSAM in capturing the highly nonlinear response under seat crash conditions were validated through comparison with hybrid model construction methods reported in existing literature. Finally, the optimization results obtained by the ABGMOOD strategy were compared to those of the classical local multi-objective optimization strategy, demonstrating the substantial advantages of the ABGMOOD optimization scheme in economy and weight reduction. In addition, the safety of the rear seats is slightly lower than that of the local optimization scheme but remains in compliance with regulatory requirements. The final optimized rear seat demonstrates notable improvements in safety, economy, and weight reduction, validating the feasibility of the ABGMOOD strategy and providing valuable insights for similar engineering optimization challenges.

Design optimization of structural and multidisciplinary systems under uncertainty has been an active area of research due to its evident advantages over deterministic design optimization. In deterministic design optimization, the uncertainties of a structural or multidisciplinary system are taken into account by using safety factors specified in the regulations or design codes. This uncertainty treatment is a subjective and indirect way of dealing with uncertainty. On the other hand, design under uncertainty approaches provide an objective and direct way of dealing with uncertainty. This paper provides a review of the uncertainty treatment practices in design optimization of structural and multidisciplinary systems under uncertainties. To this end, the activities in uncertainty modeling are first reviewed, where theories and methods on uncertainty categorization (or classification), uncertainty handling (or management), and uncertainty characterization are discussed. Second, the tools and techniques developed and used for uncertainty modeling and propagation are discussed under the broad two classes of probabilistic and non-probabilistic approaches. Third, various design optimization methods under uncertainty which incorporate all the techniques covered in uncertainty modeling and analysis are reviewed. In addition to these in-depth reviews on uncertainty modeling, uncertainty analysis, and design optimization under uncertainty, some real-life engineering applications and benchmark test examples are provided in this paper so that readers can develop an appreciation on where and how the discussed techniques can be applied and how to compare them. Finally, concluding remarks are provided, and areas for future research are suggested.

This research explores the evaluation methods for waterfront public spaces based on different project scales. It categorises the evaluation methods into three levels: large-scale, medium-scale, and small-scale, and proposes the most effective evaluation strategies for each. This study selected waterfront spaces in 38 Chinese cities and 17 international cities as case studies to analyse spatial evaluation methodologies and high-frequency keywords across varying spatial scales. Based on a systematic review of literature published in the past five years, the research employed term frequency analysis to investigate narrative patterns in academic discourse, supplemented by a data-driven analysis of 120 papers for keyword extraction and thematic categorisation. The results indicate that at the macro scale, design evaluation optimises the placement of planning points; at the medium-scale, feedback can be used to adjust spatial layouts and functions; and at the micro scale, dynamic updates of service facilities are possible. This study provides effective methods for more precise analysis of user needs and design optimization strategies.

The bio-inspired algorithms are novel, modern, and efficient tools for the design of electrical machines. However, from the mathematical point of view, these problems belong to the most general branch of non-linear optimization problems, where these tools cannot guarantee that a global minimum is found. The numerical cost and the accuracy of these algorithms depend on the initialization of their internal parameters, which may themselves be the subject of parameter tuning according to the application. In practice, these optimization problems are even more challenging, because engineers are looking for robust designs, which are not sensitive to the tolerances and the manufacturing uncertainties. These criteria further increase these computationally expensive problems due to the additional evaluations of the goal function. The goal of this paper is to give an overview of the widely used optimization techniques in electrical machinery and to summarize the challenges and open problems in the applications of the robust design optimization and the prospects in the case of the newly emerging technologies.