Robust Design Optimization Research Articles

An efficient and non-intrusive approach for robust design optimization with the first-order second-moment method

A modified robust design optimization approach is presented, which uses the first-order second-moment method to compute the mean value and the standard deviation for arbitrary objective functions. Existing approaches compute the gradient of the variance using the adjoint method, direct differentiation or finite differences, respectively. These approaches either access to the FE-code and/or have high computational cost. In this paper, a new approach for the computation of the gradient of the variance is provided. It can be easily implemented as a non-intrusive method, which behaves similar to finite differences with the cost of only one additional objective evaluation, independent of the number of variables. Here, a step-size has to be chosen carefully and therefore, a procedure to determine a problem-independent step-size is provided. As an alternative, the approach can be implemented as an analytic method with the same cost like the adjoint method, but providing wider applicability (e.g. eigenvalue problems). The provided approach is derived, analyzed and applied to several benchmark examples.

Robust aerodynamic optimization and design exploration of a wide-chord transonic fan under geometric and operational uncertainties

Axial compressors are inevitably affected by various uncertain factors in the process of manufacture and operation. These uncertainties obviously lead to reduced efficiency and large performance dispersion. However, researches on uncertainty quantification and robust design of compressors still faces severe difficulties due to the complexity of compressor structure and internal flow. This paper aims to present an automated and effective framework for uncertainty quantification and aerodynamic robustness optimization of axial compressor. The manufacturing error distribution is derived from the measurement data of machined fan blades, and the sparse grid-based polynomial chaos expansion method is used to propagate the uncertain factors and predict the probability density distribution of the fan performance. A novel surrogate model that combines a self-organizing mapping and a back-propagation neural network is constructed to explore and visualize the correlation between uncertainty parameters and performance responses. Robust aerodynamic design optimization is achieved based on the genetic algorithm. The results indicate that the coupled neural network model exhibits good accuracy for uncertain approximate modeling. Compared with the prototype fan, the optimized fan's mean isentropic efficiency and pressure ratio increase by 0.97% and 0.72%, respectively. The standard deviation of isentropic efficiency, pressure ratio, and mass flow rate decrease by 46.3%, 21.4%, and 15.2%, respectively. The present study provides a reference and exploration for uncertainty quantification and robust optimization of advanced refined multi-stage turbomachinery.

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.

Robust optimization design of permanent magnet synchronous motors for a solar airplane based on a lightweight structure

Due to the characteristics of high-altitude solar airplanes, the permanent magnet synchronous motors (PMSMs) require not only high torque density, but also high reliability. The balance of quality, reliability, and performance in a design proposal is particularly important. In this paper, a robust mechanical–electrical integrated optimization method for PMSMs based on 6σ theory is proposed. According to the fatigue characteristics of materials under the propeller working conditions and the uncertainty factors caused by the actual machining dimension tolerance, the electromagnetic dimension, mechanical dimension and reliability of the PMSM are connected. And a robust optimization model for the reliability and performance of the PMSMs is established. The Monte Carlo method and finite element simulation method are used to verify the correctness of the established optimization model. According to the comparison between the deterministic optimization method and the proposed optimization model, this method can significantly improve the output performance and reliability of the PMSMs.

Swarm intelligence hybridized with genetic search in multi-objective design optimization under constrained-Pareto dominance

The inclusion of uncertainty in structural design optimization led to more complex design optimization formulations, where uncertainty quantification has significant influence on solution methods and computing times. Designs maintaining steady levels of performance, under uncertainty, are called robust and the combination of both robustness and performance optimality leads to a methodology called Robust Design Optimization (RDO). In this work a new approach to the RDO of angle-ply composite laminate structures is proposed. The key concept of this methodology is the hybridization between the principles of Particle Swarm Optimization and Genetic Algorithms, through the evolution of multiple populations based on local and global constrain-dominance. The RDO problem is here defined as the bi-objective minimization of the structural weight (optimality) and the determinant of the variance–covariance matrix (robustness) of the system’s response functionals, subject to stress and displacement constraints. A numerical structural design problem, representing a composite laminate engine hood shell, shows the capabilities of the proposed approach. Results display the good convergence properties of the proposed hybridizations both in the search and objective spaces.

Where to build the ideal solar-powered ammonia plant? Design optimization of a Belgian and Moroccan power-to-ammonia plant for covering the Belgian demand under uncertainties

Regions with abundantly available renewable energy are not necessarily the same as those with a high population density and energy consumption. Therefore, renewable energy can be produced in optimal climate conditions with a remote renewable hub and transported to these population-dense regions. To establish this energy transport to these regions, ammonia provides a flexible, easy-to-handle energy carrier. However, current literature rarely considers the impact of techno-economic uncertainty on the feasibility of this transport. Using those uncertainties, we performed a robust design optimization on the levelized cost of ammonia and the power-to-ammonia efficiency to compare the local (Belgium) and remote (Morocco) ammonia production and transport to Belgium. This paper provides the robust designs (i.e. least sensitive to uncertainty) for local and remote renewable ammonia production and the advantages of both approaches on the levelized cost and energy efficiency. The results confirm that ammonia production in regions with high solar irradiance followed by the transport of ammonia is cost-effective and robust (601 euro/tonneNH3 in mean and 98 euro/tonneNH3 in standard deviation) over local production (852 euro/tonneNH3 in mean and 139 euro/tonneNH3 in standard deviation). However, local ammonia production provides for more efficient (54.8% in mean) and less sensitive power-to-ammonia plant designs (0.16% in standard deviation), while the production in Morocco is less efficient (52.2% in mean) and more sensitive to uncertainties (0.39% in standard deviation). The capacity of the photovoltaic arrays and the electrolyzers highly influences both objectives. The sensitivity analysis shows that capital and operational expenses of the photovoltaics and electrolyzer stack dominate the designs with the lowest levelized cost in mean and standard deviation. However, the energy consumption uncertainty of the Haber–Bosch also impacts the cost of the lowest mean levelized cost. This uncertainty also dominates the designs with the highest energy efficiency in mean and lowest standard deviation.

The latent failure probability: A conceptual basis for robust, reliability-based and risk-based design optimization

The safety of structural systems depends on social, political, financial, organizational, behavioral, and other non-structural factors. These factors are related to use of the structure and its environment, malevolent and other unanticipated accidental loading, construction method and experience of construction crews, gross errors, monitoring to prevent abuse, etc. The above can broadly be classified as non-technical factors, involving significant epistemic uncertainty. In this manuscript, we formalize a procedure to handle non-technical factors and/or epistemic uncertainties in Robust, Reliability-based and Risk-based structural Design Optimization (R3DO). We propose that latent failure probabilities, subjective point-estimates reflecting non-technical factors and/or epistemic uncertainties, be added to the nominal member failure probabilities, calculated from objective aleatory uncertainty (technical factors). Latent failure probabilities are subjectively estimated by the analysis team, based on a holistic risk analysis addressing the technical and non-technical factors above. The relevance and impact of latent failure probabilities in R3DO have already been demonstrated elsewhere; these results are briefly recapped herein. We demonstrate that structural systems should be made redundant, not because of objective aleatory uncertainties in loads and material strengths, but to cope with the large impact of subjective epistemic uncertainties related to non-technical factors.

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 of a flying wing using adjoint and uncertainty-based aerodynamic optimization approach

Robust optimization design of a flying wing using adjoint and uncertainty-based aerodynamic optimization approach

Deterministic vs. robust design optimization using DEM-based metamodels

In design optimization of bulk handling equipment (BHE) we generally focus on the mean performance of the equipment. However, granular materials behave stochastic due to irregularities in particle shape and size which leads to stochastic performance of the equipment. To include the stochastic performance we propose robust metamodel-based design optimization (MBDO). The used metamodels are trained with stochastic performance data from randomly repeated discrete element method (DEM) simulations and predict mean and variance of the equipment performance. This method is compared to the conventional deterministic optimization method by means of a case study of a discharging hopper including verification and validation. The robust MBDO shows more distinctive optimal designs compared to the deterministic approach. In addition, the DEM-based metamodel is a relatively accurate method to predict DEM-model simulation results. However, the validation indicates that differences between DEM-model and experimental results highly affect the reliability of the found optima.

A generalized approach for robust topology optimization using the first-order second-moment method for arbitrary response functions

The paper presents a rigorous formulation of adjoint systems to be solved for a robust design optimization using the first-order second-moment method. This formulation allows to apply the method for any objective function, which is demonstrated by considering deformation at certain point and maximum stress as objectives subjected to random material stiffness and geometry, respectively. The presented approach requires the solution of at most three additional adjoint systems per uncertain system response, when compared to the deterministic case. Hence, the number of adjoint systems to be solved is independent of the number of random variables. This comes at the expense of accuracy, since the objective functions are assumed to be linear with respect to random parameters. However, the application to two standard cases and the validation with Monte Carlo simulations show that the approach is still able to find robust designs.

A response band-based method for time-dependent reliability-based robust design optimization

In this paper, a response band-based method for time-dependent reliability-based robust design optimization is proposed. The proposed method provides a novel alternative framework, consist of a two-step transformation stage and a solving stage, to solve the time-dependent reliability-based robust design optimization problem. The original time-dependent reliability-based robust design optimization problem is transformed into an equivalent deterministic robust design optimization problem in the transformation stage, and the equivalent problem is settled in the solving stage. In the transformation stage, the dynamic modal decomposition technique and the kriging technique are combined to overcome the problem that there is no standard for both time division and observation sampling in the commonly used transformation methods. In the solving stage, an approach for constructing the response band of the objective function is presented, which significantly reduces the computational consumption of the variation evaluation of the objective function. Five cases are employed to verify the effectiveness of the proposed method.

Novel Approach to Robust Concept Development for Offshore Floating Systems

This paper describes a new approach to the application of robust design optimisation to the development and selection of floating systems during concept and early-stage engineering. The basis of the approach is to combine a genetic algorithm with an efficient non-intrusive statistical model to allow both uncertainties in modelling (intrinsic uncertainty) and requirements (extrinsic uncertainty) to be accounted for within a Robust Design Framework that directs optimisation toward unique design solutions. A particular interpretation of the Pareto design objective space enables required levels of performance margin and meaningful measures of confidence in design objectives to be achieved. The approach can be used to provide a systematic series of optimal hull geometries (geosims) that meet the most likely variations in design requirements, and so mitigate the impact of change early in the design cycle. This paper follows on from and earlier publication (IJME648) on multi-objective, multi-discipline, optimisation and uses the same case study.

Robust optimization design for sealing performance of globe-cone joint considering manufacturing and assembly uncertainties

Robust optimization design for sealing performance of globe-cone joint considering manufacturing and assembly uncertainties

Robust Design Optimization of Expensive Stochastic Simulators Under Lack-of-Knowledge

Abstract Robust design optimization of stochastic black-box functions is a challenging task in engineering practice. Crashworthiness optimization qualifies as such problem especially with regards to the high computational costs. Moreover, in early design phases, there may be significant uncertainty about the numerical model parameters. Therefore, this paper proposes an adaptive surrogate-based strategy for robust design optimization of noise-contaminated models under lack-of-knowledge uncertainty. This approach is a significant extension to the robustness under lack-of-knowledge method (RULOK) previously introduced by the authors, which was limited to noise-free models. In this work, it is proposed to use a Gaussian Process as a regression model based on a noisy kernel. The learning process is adapted to account for noise variance either imposed and known or empirically learned as part of the learning process. The method is demonstrated on three analytical benchmarks and one engineering crashworthiness optimization problem. In the case studies, multiple ways of determining the noise kernel are investigated: (1) based on a coefficient of variation, (2) calibration in the Gaussian Process model, (3) based on engineering judgment, including a study of the sensitivity of the result with respect to these parameters. The results highlight that the proposed method is able to efficiently identify a robust design point even with extremely limited or biased prior knowledge about the noise.

Adjoint-based robust optimization design of laminar flow wing under flight condition uncertainties

It is an inherent uncertainty problem that the application of laminar flow technology to the wing of large passenger aircraft is affected by flight conditions. In order to seek a more robust natural laminar flow control effect, it is necessary to develop an effective optimization design method. Meanwhile, attention must be given to the impact of crossflow (CF) instability brought on by the sweep angle. This paper constructs a robust optimization design framework based on discrete adjoint methods and non-intrusive polynomial chaos. Transition prediction is implemented by coupled Reynolds-Averaged Navier-Stokes (RANS) and simplified eN method, which can consider both Tollmien-Schlichting (TS) wave and crossflow vortex instability. We have performed gradient enhancement processing on the general Polynomial Chaos Expansion (PCE), which is advantageous to reduce the computational cost of single uncertainty propagation. This processing takes advantage of the gradient information obtained by solving the coupled adjoint equations considering transition. The statistical moment gradient solution used for the robust optimization design also uses the derivatives of coupled adjoint equations. The framework is applied to the robust design of a 25° swept wing with infinite span in transonic flow. The uncertainty quantification and sensitivity analysis on the baseline wing shows that the uncertainty quantification method in this paper has high accuracy, and qualitatively reveals the factors that dominate in different flow field regions. By the robust optimization design, the mean and standard deviation of the drag coefficient can be reduced by 29% and 45%, respectively, and compared with the deterministic optimization design results, there is less possibility of forming shock waves under flight condition uncertainties. Robust optimization results illustrate the trade-off between the transition delay and the wave drag reduction.

Correction: Robust Optimal Design of a Six-Bar Linkage-Based Finger Clamping Unit for High Durability

Correction: Robust Optimal Design of a Six-Bar Linkage-Based Finger Clamping Unit for High Durability

Uncertainty quantification in inerter-based quasiperiodic lattices

Inerter-based periodic structures have attracted significant interest among the research community due to their wide range of applications. However, existing studies on quasiperiodic structures, especially with inerters, are limited. Therefore, this timely work investigates the dynamics of novel one-dimensional inerter-based quasiperiodic lattices with and without local resonators. The quasiperiodicity is introduced through the modulation of both spring and inerter properties resulting in the well-known Hofstadter-like butterfly spectrum having a fractal structure. Moreover, by considering the appropriate boundary conditions and many mass–spring-inerter subsystems, the bulk spectrum of the system is investigated demonstrating the existence of multiple edge states in lower and higher frequency ranges. The deterministic study showed that the effect of inertia amplification is to reduce the frequencies of the Hofstadter-like butterfly and to widen the low frequency band gaps. The second contribution of this work involves investigating the effect of parametric uncertainty for the acoustic-metamaterial-like chain based on Monte Carlo simulations. Moreover, to address the computational aspect of the problem, grey-box modeling using machine learning was performed. A Gaussian process model was trained on a limited dataset and was found to capture the stochastic responses of the lattices adequately. The statistical variation of parameters with different levels of uncertainty demonstrated significant effects on the Hofstadter-like butterfly, band gaps, corresponding edge states and frequency responses. The sensitivity of the dynamic behavior in quasiperiodic lattices to variabilities reveal the need to account for system uncertainties for their targeted performance as future vibration absorbers and energy harvesters. The study also paves the way to utilize the results as useful prior information for robust design optimization of real inerter-based quasiperiodic lattice devices.

RELIABILITY-BASED ROBUST DESIGN OPTIMIZATION FOR BUILDING-CONNECTING INERTIAL-MASS DAMPER

In this paper, we propose an accurate and efficient reliability evaluation method for robust design optimization using a probabilistic approach. First, we verify the validity of this method using two benchmark functions. Then, we apply this method to the robust design optimization problem of building-connecting damper parameters and verify the effect of robust design. We consider the variation of the natural period and structural damping ratio of buildings, and directly evaluate the criteria satisfaction rate (CSR) using the response value calculated by time-history response analysis.

Task-Driven-Based Robust Control Design and Fuzzy Optimization for Coordinated Robotic Arm Systems

Task-Driven-Based Robust Control Design and Fuzzy Optimization for Coordinated Robotic Arm Systems