Robust Design Optimization Method Research Articles

Sensitivity Analysis and Robust Optimization to High-Dimensional Uncertainties of Compressors with Active Subspace method

The performance of compressors often deviates significantly from the design value due to the substantial geometric deviations arising from manufacturing errors and in-service degradation. Robust design optimization (RDO) plays a crucial role in ensuring the high performance and reliability of compressors. The study proposes an active subspace-based dimensionality-reduction RDO method for high-dimensional uncertainties of compressors. By employing the active subspace (AS) method, a dimensionality-reduction artificial neural network (ANN) is raised to solve the "dimension disaster" problem for high-dimensional uncertainty quantification (UQ) analysis. Additionally, to decrease the computation cost of the original AS method, a data-driven method is utilized to calculate the gradient of the quantity of interest (QoI) within the AS dimensionality-reduction framework. Furthermore, Shapley method is applied to study the sensitivity of geometric variables and their variation direction on the mean and variance of the performance, and the robustness improvement method by only modifying key geometric variables based on Shapley results is discussed. The RDO of high dimensional geometrical uncertainties of a transonic compressor stage is studied by using this method. The results demonstrate a reduction from the original 24-dimensional uncertainties to a 4-dimensional space, achieved through this method. Consequently, the required sample size is reduced by 75% while maintaining nearly unchanged model accuracy. Furthermore, the findings reveal that after RDO, the mean efficiency increases by 1.04%, with a 42% reduction in variance. Compared to RDO, which requires simultaneous optimization of all 24 geometric variables, the robustness improvement based on Shapley only necessitates modifying 4 key geometric variables, resulting in a substantial increase in both the mean and variance of efficiency.

Robust optimal design of UFLS schemes in island power systems

Since settings of conventional underfrequency load shedding (UFLS) schemes cannot be adjusted in real-time, this paper proposes a novel robust optimal design method of the UFLS scheme for an island power system (IPS) that once fixed, gives an adequate performance in all possible operating conditions in the IPS. The core of the novel robust optimal design method is the implementation of a synthetic data generation algorithm that covers all the feasible generator operating points, which are further classified into a set of large number (but computationally manageable) representative generation dispatches using clustering techniques. A real-world Spanish island will be used to illustrate the method, where the UFLS robust design will be tested and compared with non-robust UFLS designs tuned using generation dispatches of an actual day of operation provided by both standard and frequency-constrained unit commitment algorithms. Results will show that the proposed robust design avoids frequency constraints violations in contingencies and obtains better frequency metrics.

Multidisciplinary Robust Design Optimization Incorporating Extreme Scenario in Sparse Samples

Abstract Robust design optimization (RDO) is a potent methodology that ensures stable performance in designed products during their operational phase. However, there remains a scarcity of robust design optimization methods that account for the intricacies of multidisciplinary coupling. In this article, we propose a multidisciplinary robust design optimization (MRDO) framework for physical systems under sparse samples containing the extreme scenario. The collaboration model is used to select samples that comply with multidisciplinary feasibility, avoiding time-consuming multidisciplinary decoupling analyses. To assess the robustness of sparse samples containing the extreme scenario, linear moment estimation is employed as the evaluation metric. The comparative analysis of MRDO results is conducted across various sample sizes, with and without the presence of the extreme scenario. The effectiveness and reliability of the proposed method are demonstrated through a mathematical case, a conceptual aircraft sizing design, and an energy efficiency optimization of a hobbing machine tool.

Robust design optimization of a multi-body system with aleatory and epistemic uncertainty

There are several issues in robust design optimization of the multi-body system with aleatory and epistemic uncertainty, such as the difficulty in the multi-source uncertainty characterization and low efficiency in dynamic modeling, analysis and the corresponding solving strategy for its robust design optimization. To address these problems, this paper firstly constructs a unified analysis framework for aleatory and epistemic uncertainty for the multi-body system based on probability theory and evidence theory. Taking a multi-body system of a shipborne artillery model as an example, the transfer matrix method is next employed to rapidly calculate the dynamic response of the muzzle. The results are compared with those by the finite element method to verify the accuracy and efficiency of the transfer matrix method. A decision model of the robust design optimization with both aleatory and epistemic uncertainty is proposed. Based on the polynomial chaos expansion-Kriging model, subset simulation optimization method and a weighted single-objective solving strategy, the multi-objective robust design optimization problem involving aleatory and epistemic uncertainty is then transformed into a single-objective optimization one. Finally, results of both the multi-objective and weighted single-objective robust design optimization methods for the shipborne artillery model are compared to demonstrate the performance of the proposed method.

Development of a deviation package method for low-cost robust optimization in compressor blade design

Manufacture variations can greatly increase the performance variability of compressor blades. Current robust design optimization methods have a critical role in reducing the adverse impact of the variations, but can be affected by errors if the assumptions of the deviation models and distribution parameters are inaccurate. A new approach for robust design optimization without the employment of the deviation models is proposed. The deviation package method and the interval estimation method are exploited in this new approach. Simultaneously, a stratified strategy is used to reduce the computational cost and assure the optimization accuracy. The test case employed for this study is a typical transonic compressor blade profile, which resembles most of the manufacture features of modern compressor blades. A set of 96 newly manufactured blades was measured using a coordinate measurement machine to obtain the manufacture variations and produce a deviation package. The optimization results show that the scatter of the aerodynamic performance for the optimal robust design is 20% less than the baseline value. By comparing the optimization results obtained from the deviation package method with those obtained from widely-used methods employing the deviation model, the efficiency and accuracy of the deviation package method are demonstrated. Finally, the physical mechanisms that control the robustness of different designs were further investigated, and some statistical laws of robust design were extracted.

A novel robust design optimization method combining improved flower pollination algorithm and dual Kriging

PurposeThis study aims to propose a more precise method for robust design optimization of mechanical structures with black-box problems, while also considering the efficiency of uncertainty analysis.Design/methodology/approachThe method first introduces a dual adaptive chaotic flower pollination algorithm (DACFPA) to overcome the shortcomings of the original flower pollination algorithm (FPA), such as its susceptibility to poor accuracy and convergence efficiency when dealing with complex optimization problems. Furthermore, a DACFPA-Kriging model is developed by optimizing the relevant parameter of Kriging model via DACFPA. Finally, the dual Kriging model is constructed to improve the efficiency of uncertainty analysis, and a robust design optimization method based on DACFPA-Dual-Kriging is proposed.FindingsThe DACFPA outperforms the FPA, particle swarm optimization and gray wolf optimization algorithms in terms of solution accuracy, convergence speed and capacity to avoid local optimal solutions. Additionally, the DACFPA-Kriging model exhibits superior prediction accuracy and robustness contrasted with the original Kriging and FPA-Kriging. The proposed method for robust design optimization based on DACFPA-Dual-Kriging is applied to the motor hanger of the electric multiple units as an engineering case study, and the results confirm a significant reduction in the fluctuation of the maximum equivalent stress.Originality/valueThis study represents the initial attempt to enhance the prediction accuracy of the Kriging model using the improved FPA and to combine the dual Kriging model for uncertainty analysis, providing an idea for the robust optimization design of mechanical structure with black-box problem.

EVTOL Tilt-Wing Aircraft Design under Uncertainty Using a Multidisciplinary Possibilistic Approach

Recent development in Electric Vertical Take-off and Landing (eVTOL) aircraft makes it a popular design approach for urban air mobility (UAM). When designing these configurations, due to the uncertainty present in semi-empirical estimations, often used for aerodynamic characteristics during the conceptual design phase, results can only be trusted to approximately 80% accuracy. Accordingly, an optimized aircraft using semi-empirical estimations and deterministic multi-disciplinary design optimization (MDO) approaches can be at risk of not being certifiable in the detailed design phase of the life cycle. The focus of this study was to implement a robust and efficient possibility-based design optimization (PBDO) method for the MDO of an eVTOL tilt-wing aircraft in the conceptual design phase, using existing conventional designs as an initial configuration. As implemented, the optimization framework utilizes a deterministic gradient-based optimizer, run sequentially with a possibility assessment algorithm, to select an optimal design. To achieve this, the uncertainties which arise from multi-fidelity calculations, such as semi-empirical methods, are considered and used to modify the final design such that its viability is guaranteed in the detailed design phase. With respect to various requirements, including trim, stability, and control behaviors, the optimized eVTOL tilt-wing aircraft design offers the preferred results which ensure that airworthiness criteria are met whilst complying with predefined constraints. The proposed approach may be used to revise currently available light aircraft and develop eVTOL versions from the original light aircraft. The resulting aircraft is not only an optimized layout but one where the stability of the eVTOL tilt-wing aircraft has been guaranteed.

Adaptive decoupled robust design optimization

Robust design optimization (RDO) is a valuable technique in the design of engineering structures as it can provide an optimum design solution that is relatively insensitive to input uncertainties. However, the nested double-loop estimation process required in RDO often results in significant computational costs. To address this issue, we propose an adaptive decoupled RDO method based on the Kriging surrogate model. This method transforms the nested double-loop estimation process into a traditional deterministic optimization procedure, thus reducing computational costs. Furthermore, a novel estimation expression for the performance standard deviation that can simultaneously reflect the uncertainties in both the prediction and the performance mean is established. The closed-form expressions of the performance mean and performance standard deviation under different design parameters are deduced, which are further implemented to the uncertainty propagation during the design optimization. Moreover, an adaptive framework is introduced to improve the computational accuracy of uncertainty propagation as well as optimization procedure to guarantee the estimation accuracy of RDO problems. Several numerical examples along with engineering cases are introduced to illustrate the effectiveness of the established adaptive decoupled adaptive RDO method, and the results demonstrate that the proposed method can effectively optimize the design of structures while reducing computational costs.

Deterministic-based robust design optimization of composite structures under material uncertainty

We propose a new deterministic robust design optimization method for composite laminate structures under worst-case material uncertainty. The method is based on a simultaneous parametrization of topology and material and combines a design problem and a material uncertainty problem into a single min–max optimization problem which provides an efficient approach to handle variation of material properties in stiffness driven design optimization problems. An analysis is performed using a design problem based on a failure criterion formulation to evaluate the ability of the proposed method to generate robust composite designs. The design problem is solved using various loads, boundary conditions and manufacturing constraints. The designs generated with the proposed method have improved objective responses compared to the worst-case response of designs generated with nominal material properties and are less sensitive to the variation of material properties. The analysis indicates that the proposed method can be efficiently applied in a robust structural optimization framework.

A Reliability-Based Robust Design Optimization Method for Rolling Bearing Fatigue under Cyclic Load Spectrum

Reliability-based robust design methods have been widely used in the field of product design; however, they are difficult to apply to the fatigue reliability design process of rolling bearings due to the problems of determining fatigue accumulated damage caused by the internal cyclic time-varying load distribution of rolling bearings and the computational cost of time-varying reliability. Therefore, a reliability-based robust design method for rolling bearing fatigue failure is proposed, which derives the formula for fatigue accumulated damage of a rolling bearing under cyclic load spectrum and significantly reduces the computational cost of rolling bearing time-varying reliability compared with existing methods. First, the state response of a rolling bearing under random design parameters is obtained by finite element simulation. Then, the adaptive kriging method is used to characterize the correlation between the random parameters and the state response. The Miner fatigue cumulative damage theory is improved and the rolling bearing fatigue time-varying equation of state under cyclic load spectrum is derived. Subsequently, a fatigue time-varying reliability model based on an improved fourth-order moment method is developed, and a reliability robust optimization design method is proposed. Finally, a rolling bearing example is presented to demonstrate that the method achieves time-varying fatigue reliability design under cyclic load spectrum and effectively improves the reliability and robustness of the product design.

A data-driven robust design optimization method and its application in compressor blade

The probability-based robust optimization methods require a large amount of sample data to build probability distribution models of uncertain parameters. However, it is a common situation that only scarce sampled data are available in practice due to expensive tests. This study proposes a data-driven robust optimization framework by embedding a novel uncertainty quantification (UQ) method, which can quantify the uncertainty based on the statistical moments of scarce input data. The computational robustness and accuracy of the developed UQ methods are validated. Then, the data-driven multi-objective optimization framework is applied to improve the mean performance and aerodynamic robustness of a two-dimensional compressor blade with real stagger angle errors. Uncertainty analysis shows that there is a probability of 47.55% to deviate from the nominal total pressure loss coefficient by more than 1% for the actual performance values at high positive incidence i=7° condition. In the optimization process, the total pressure loss coefficient is selected as the objective function, while the static pressure ratio is used as a constraint. The Gaussian process regression model is trained to improve the robust optimization efficiency. The robust optimization is conducted under the most sensitive conditions. Optimized results indicate that compared with the nominal blade, the mean performance of the selected robust blades is increased by 10.9%, 8.56%, and 0.83%; the performance dispersion is decreased by 19.0%, 24.8%, and 35.3%, respectively. The optimized results can provide useful references for the robust design of compressor blades.

System-Level Robust Design Optimization of a Permanent Magnet Motor Under Design Parameter Uncertainties

In the presence of design parameter uncertainties, a system-level robust design optimization method for a permanent magnet motor is proposed to enhance the dynamic and steady performances of a whole motor drive system. To achieve the goal, the influence of individual motor parameters on transient system responses is first investigated with the method of Taguchi experimental planning. The temperature-dependent material property of each motor component is then examined around an operating temperature. A conventionally customized motor drive system is fully simulated based on the finite element method. Finally, it is optimized by the univariate dimension reduction method to ensure the robustness of system performances against the manufacturing tolerance and operating temperature fluctuation.

Robust optimization design method for structural reliability based on active-learning MPA-BP neural network

PurposeMechanical products usually require deterministic finite element analysis in the design phase to determine whether their structures meet the requirements. However, deterministic design ignores the influence of uncertainties in the design and manufacturing process of mechanical products, leading to the problem of a lack of design safety or excessive redundancy in the design. In order to improve the accuracy and rationality of the design results, a robust design method for structural reliability based on an active-learning marine predator algorithm (MPA)–backpropagation (BP) neural network is proposed.Design/methodology/approachThe MPA was used to obtain the optimal weights and thresholds of a BP neural network, and an active-learning function applicable to neural networks was proposed to efficiently improve the prediction performance of the BP neural network. On this basis, a robust optimization design method for mechanical product reliability based on the active-learning MPA-BP model was proposed. Random moving quadrilateral sampling was used to obtain the sample points required for training and testing of the neural network, and the reliability sensitivity corresponding to each sample point was calculated by subset simulated significant sampling (SSIS). The total mass of the mechanical product and the structural reliability sensitivity of the trained active-learning MPA-BP model output were taken as the optimization objectives, and a multi-objective reliability-robust optimization design model was constructed, which was solved by the second-generation non-dominated ranking genetic algorithm (NSGA-II). Then, the dominance function was used in the obtained Pareto solution set to make a dominance-seeking decision to obtain the final reliability-robust optimization design solution. The feasibility of the proposed method was verified by a reliability-robust optimization design example of the bogie frame.FindingsThe prediction error of the active-learning MPA-BP neural network was smaller than those of the particle swarm optimization (PSO)-BP, marine predator algorithm (MPA)-BP and genetic algorithm (GA)-BP neural networks under the same basic parameter settings of the algorithm, which indicated that the improvement strategy proposed in this paper improved the prediction accuracy of the BP neural network. To ensure the reliability of the bogie frame, the reliability sensitivity and total mass of the bogie frame were reduced, which not only realized the lightweight design of the bogie frame, but also improved the reliability and robustness of the bogie.Originality/valueThe MPA algorithm with a higher optimization efficiency was introduced to find the weights and thresholds of the BP neural network. A new active-learning function was proposed to improve the prediction accuracy of the MPA-BP neural network.

Robust Design Optimization of the Cogging Torque for a PMSM Based on Manufacturing Uncertainties Analysis and Approximate Modeling

A permanent magnet synchronous motor (PMSM) is a crucial device for power conversion in an energy system. The cogging torque of the PMSM is a crucial output characteristic, the robustness of which affects the operational reliability of the energy system. Therefore, the robust design optimization (RDO) of cogging torque has aroused widespread concern. There are several challenges in designing a robust cogging torque PMSM. In particular, some design parameters contain repetitive units, and the finite element analysis (FEA) method is time-consuming. State-of-the-art RDO methods usually treat these uncertainties from repetitive units as the same parameter, which neglects the fluctuation of the manufacturing process and cannot obtain a robust solution for the cogging torque of the motor efficiently and accurately. In order to solve this issue, an approximate modeling method based on manufacturing uncertainties analysis for RDO is proposed in this paper. First, the peak-to-peak value of cogging torque (Tcpp) is used to characterize the cogging torque, which is decoupled to an ideal component and fluctuation component produced by the center values and manufacturing tolerances of design parameters. The design of experiments (DoE) and simulation of the two components are carried out. Then, these two components are approximated separately, and the approximate model of Tcpp is obtained by adding the two components. Finally, the proposed approximate model is embedded into the RDO algorithm, and the PMSM design scheme for good Tcpp robustness is obtained. The effectiveness of the proposed method is verified through a case study of the PMSM.

ROBUST OPTIMAL DESIGN BASED ON STORY DUCTILITY RATIO OF ELASTIC-PLASTIC BUILDING STRUCTURES WITH UNCERTAIN OIL DAMPERS SUBJECTED TO CRITICAL DOUBLE IMPULSE

A new robustness evaluation method based on the specified story ductility ratio is proposed for a building model with uncertain oil dampers. Since the influence of input motion uncertainty on the response variation can be simply taken into account, the critical double impulse input is used instead of input ground motions. In the proposed method, the robustness function is derived as a function of the input velocity amplitude for a specified ductility ratio. This procedure is suitable for the performance-based design. A robust optimal design method on damper placement is also presented based on the proposed new robustness function. Numerical examples of robust optimal design of dampers are presented for a 12-story elasto-plastic building model with linear oil dampers.

Robust Optimization of a Rare-Earth-Reduced High-Torque-Density PM Motor for Electric Vehicles Based on Parameter Sensitivity Region

In this paper, a robust optimization design method is proposed and investigated, where a rare-earth-reduced high-torque-density permanent magnet (RH-PM) motor is selected as a research object. The key of this study is to introduce a new concept of parameter sensitivity region (PSR) into the process of optimization, aiming at comprehensively considering the fluctuation of multiple motor parameters into the motor design. To determine the motor parameters related to the sensitivity region, a parameters anti-disturbance analysis is carried out based on analysis of variance. According to the analysis, motor design parameters are classified into two groups, where the PSR-based uncertainty optimization method is purposely utilized in the group with highly robustness-related parameters. Then, a deterministic optimization is employed for the group with weakly robustness-related parameters. After optimization, the motor performances are analyzed in detail, including torque, torque ripple, and so on, as well as their corresponding analysis of performance consistency. Finally, a prototype is manufactured and tested to further verify the proposed method and RH-PM motor.

Robust Design Optimization of a Permanent Magnet Motor from the System-Level Design Perspective

From the system-level design perspective, a robust design optimization method for a permanent magnetic motor is proposed to enhance the dynamic performance of a drive system while maintaining its steady performance. To achieve the goal, an elaborate numerical model for the whole drive system is first constructed by incorporating a control circuit simulator into a finite element analysis tool, and then the influence of motor parameter variations on transient system responses is investigated by means of the method of Taguchi experimental planning. A conventionally customized motor is optimized by the univariate dimension reduction method to ensure the robustness of system performances against manufacturing tolerances. Finally, a comparative performance analysis between two motor drive systems is provided to demonstrate the validity of the proposed method.

Robust Design Optimization of Component Parameters for DMDEB Powertrain System Based on Taguchi Method

Abstract The selection of component parameters plays an important role in the energy optimization of dual-motor drive electric buses (DMDEBs). To design a set of parameters that can be robust to different road slopes, passenger masses, and bus routes, a Taguchi-based robust design optimization method is proposed in this paper for DMDEBs. First, the dynamic programming (DP) is designed as a part of the Taguchi method to explore the optimal solution (electricity consumption) corresponding to each set of component parameters. Then, by taking the optimal electricity consumption obtained by DP as the response, the Taguchi method is utilized to find the robust component parameters that are insensitive to the noise factors. To evaluate the robustness and reliability of the designed component parameters, a six sigma analysis based on Monte Carlo sampling is employed considering three kinds of stochastic disturbances. Finally, simulation results demonstrate that the proposed robust design optimization method is effective and the designed component parameters have better robustness than the initial component parameters.

Bayesian reliability-based robust design optimization of mechanical systems under both aleatory and epistemic uncertainties

Uncertainties can be divided into two general categories: aleatory and epistemic. Conventional reliability-based robust design optimization approaches, which disregard epistemic uncertainties due to lack of knowledge about the physical nature of systems, have previously been developed. To overcome this weakness, unlike previous methods, a Bayesian reliability-based robust design optimization method is proposed in the presence of both aleatory and epistemic uncertainties. The proposed formulation is presented as a multi-objective optimization problem. The univariate dimension reduction method is used to approximate the mean and variance of the design function. The non-dominated sorting genetic algorithm-II is used to solve the multi-objective optimization problem. To find the final optimum design from the Pareto front, Shannon’s entropy-based technique for order of preference by similarity to ideal solution (TOPSIS) algorithm is applied. Finally, to demonstrate the applicability of the proposed method, two case studies are considered and the results are compared and discussed.

An improved reliability-based robust design optimization method using Bayesian seemingly unrelated regression and multivariate loss function

An improved reliability-based robust design optimization method using Bayesian seemingly unrelated regression and multivariate loss function