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This paper presents a new formulation of the second-order fourth-moment method (sometimes referred to as second-order perturbation method or second-order method of moments). The method allows to efficiently predict the stochastic moments of a response function and is therefore often used within robust design optimization. The new approach allows a non-intrusive implementation at the same cost as existing, highly intrusive formulations. Therefore, the new approach can be applied to any objective function without significant implementation effort. It is based on a few finite difference steps into special directions and hence is dependent on the corresponding step sizes. An automatic step size procedure is supplied beside a detailed convergence analysis. The advantages of the new formulation are demonstrated by robust design optimizations of a 2D and a 3D example using the geometrically nonlinear finite element method.

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.

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.

This paper provides a clear perspective on existing several different approaches to robust design optimization of structures. We primarily consider three approaches: the worst-case optimization, the discrepancy (i.e., the maximum gap between the objective values in a nominal case and a possibly occurring case) minimization, and the variance minimization. Some other formulations can also be linked with one of these three approaches. To investigate how the solutions derived by these three approaches differ from each other, we present two numerical examples. This direct comparison clarifies different features of these approaches.

This paper addresses the dependency of design parameters and random variables within robust design optimization. If the stochastic distributions of random input variables are design-dependent, then this dependency must be included in the gradient, when using gradient-based optimization methods. The paper provides the basic theoretical principles and two approaches for incorporating design-dependent distributions of random variables in robust design optimization: one approach based on Monte Carlo sampling and another based on Taylor series expansions. Both these approaches do not require additional structural analyses (e.g., finite element simulations). Describing the design dependency of input distributions can, however, be a challenging task. Numerical applications to different academic examples are presented, demonstrating the potential of the proposed approaches and several implications that may emerge in the process.

Surrogate-based aerodynamic robust design optimization uses a surrogate model to calculate the robustness indices, which strongly relies on the overall accuracy of the model, as well as the efficient exploration of the design space. For most surrogate modeling approaches, significant inaccuracies are often observed at the outlier region of the design space, where very few samples are spotted. A novel method using Chebyshev transformation is applied to re-allocate the orthogonal Latin hypercube sample set to alleviate the corner errors and eventually improve the overall accuracy. An inner Kriging model is developed using the sampling method, and robustness indices are calculated based on the subspaces adjacent to the sampling points. Subsequently, an outer robust model is constructed with the robustness indices as the target. Ultimately, a combination of the inner and outer models is utilized with the genetic algorithm to accomplish multi-objective robust optimization. Theoretical tests are undertaken for classic test functions, showing the advantage of the proposed approach. Based on this method, aerodynamic robust design optimizations are carried out on the RAE 2822 airfoil, for which the lift coefficient and drag coefficient are optimized for a given range of geometrical parameters. An increase of 1.94% lift coefficient and a reduction of 2.53% drag coefficient are achieved compared to the baseline design without sacrificing the robust performances.

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.

PurposeThe primary aim of this article is to ascertain the modalities of leveraging Lean Six Sigma (LSS) for Industry 4.0 (I4.0) with special reference to the process industries. Moreover, it intends to determine the applicability of simulation-based LSS in the automation of the mineral water industry, with special emphasis on the robust design of the control system to improve productivity and performance.Design/methodology/approachThis study adopts the action research methodology, which is exploratory in nature along with the DMAIC (define-measure-analyze-improve-control) approach to systematically unearth the root causes and to develop robust solutions. The MATLAB simulation software and Minitab statistical software are effectively utilized to draw the inferences.FindingsThe root causes of critical to quality characteristic (CTQ) and variation in purity level of water are addressed through the simulation-based LSS approach. All the process parameters and noise parameters of the reverse osmosis (RO) process are optimized to reduce the errors and to improve the purity of the water. The project shows substantial improvement in the sigma rating from 1.14 to 3.88 due to data-based analysis and actions in the process. Eventually, this assists the management to realize an annual saving of 20% of its production and overhead costs. This study indicates that LSS can be applicable even in the advent of I4.0 by reinforcing the existing approach and embracing data analysis through simulation.Research limitations/implicationsThe limitation of this research is that the inference is drawn based on a single case study confined to process industry automation. Having said that, the methodology deployed, scientific information related to optimization, and technical base established can be generalized.Originality/valueThis article is the first of its kind in establishing the integration of simulation, LSS, and I4.0 with special reference to automation in the process industry. It also delineates the case study in a phase-wise manner to explore the applicability and relevance of LSS with I4.0. The study is archetype in enabling LSS to a new era, and can act as a benchmark document for academicians, researchers, and practitioners for further research and development.

Reliability-based robust design optimization (RBRDO) is a powerful tool for achieving optimal, robust, and reliable products. However, most existing approaches focus only on design specification limits at the design stage and are not applicable to products whose performance degrades over time. In addition, the high computational cost of nested double-loop optimization, especially when finite-element (FE) or computer-aided design models are required to evaluate nonlinear performance functions, limits their use in realistic engineering applications. To overcome these challenges, this paper proposes a machine learning-driven time-dependent RBRDO framework. The design problem is reformulated as a time-independent optimization with two objectives and a probabilistic constraint to account for optimality, reliability, and robustness simultaneously. A multilayer perceptron surrogate model is trained on FE data to approximate performance degradation, significantly reducing the computational burden of repeated reliability evaluations. The surrogate is integrated with Monte Carlo Simulation for reliability assessment, and an evolutionary algorithm is employed for optimization. The framework is demonstrated on two widely used nickel-based superalloy springs, Nimonic 90 and alloy X-750, showing its ability to maintain reliability under stress relaxation while drastically reducing computational cost. The results highlight the potential of the proposed method as a general and efficient tool for designing reliable, degradation-resistant components in high-temperature and safety–critical applications.

This work presents a robust design optimization approach for a char combustion process in a limited-data setting, where simulations of the fluid-solid coupled system are computationally expensive. We integrate a polynomial dimensional decomposition (PDD) surrogate model into the design optimization and induce computational efficiency in three key areas. First, we transform the input random variables to have fixed probability measures, which eliminates the need to recalculate the PDD’s basis functions associated with these probability quantities. Second, using the limited data available from a physics-based high-fidelity solver, we estimate the PDD coefficients via sparsity-promoting diffeomorphic modulation under observable response-preserving homotopy regression. Third, we propose a single-pass surrogate model training that avoids the need to generate new training data and update the PDD coefficients during the derivative-free optimization. The results provide insights for optimizing process parameters to ensure consistently high energy production from char combustion.