Robust Design Optimization Method Research Articles

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

Robust optimization of rolling parameters of coarse aggregates based on improved response surface method using satisfaction function method based on entropy and adaptive chaotic gray wolf optimization

Determining the rolling parameter control standard for coarse aggregates by robust optimization is of great significance in reducing the sensitivity to noise in data on site and improving the measurement accuracy and stability of the compaction quality. The existing rolling parameter control standard is mainly determined by field rolling tests, which results in draining human labor and financial resources and makes it difficult to ensure the global optimum. Existing studies on the multiobjective optimization of rolling parameters usually ignore robustness, which in detail is represented by neglecting the overlap width of the rolling strip in the decision variables and rolling efficiency in optimization objectives. To overcome this limitation, a robust optimization design method for the rolling parameters based on the improved response surface method (RSM) using the satisfaction function method based on entropy (SFME) and adaptive chaotic gray wolf optimization (ACGWO) is proposed. First, the mean and standard deviation (SD) of the rolling quality and efficiency are selected as the response objectives, with the rolling thickness, rolling passes, rolling speed, and overlap width as the influencing factors. Then, RSM is used to perform a robust optimization design on the rolling parameters, in which the SFME is used to determine the weight of each response. Finally, an improved ACGWO algorithm with less calculation time, faster convergence speed, and higher accuracy is proposed to establish a robust optimization model of the rolling parameters. The engineering application results indicate that the most robust rolling parameters obtained by the proposed method are applicable and accurate. The method can improve the efficiency of the compaction while maintaining high compaction quality.

Multi-objective robust optimization of a solar power tower plant under uncertainty

The optimal design of a molten salt solar power tower (SPT) plant is sensitive to the variations of uncertainties, such as solar radiation, which result in dispersion of the model output. To mitigate the impacts of uncertainties on the thermo-economic performance of SPT plant, this study develops an uncertainty-based multi-objective robust optimization design method for the case of a SPT plant in Sevilla with the expected value (i.e. the average energy cost) and the standard deviation (i.e. the dispersion of the model output) of the levelized cost of energy (LCOE) as the objectives. The Monte Carlo (MC) simulation and simulated annealing (SA) algorithm are combined to solve the robust optimization problem. The results of Pareto frontier indicate that a trade-off is needed through decision-making. The final optimal solution is determined with expectation of LCOE of 23.09 c/kWhe and standard deviation of LCOE of 1.25 c/kWhe. Compared with the deterministic optimal design, the standard deviation of LCOE of the multi-objective robust optimum is reduced by 17.22 %, which turns out to be less sensitive to the uncertainties. Moreover, the Sobol’ global sensitivity analysis results show that the direct solar radiation, heliostat field cost and receiver cost are the most sensitive to LCOE.

Lightweight robust optimization design of mechanical claws of intelligent sanitation vehicle

As the dustbin grabbing device of the intelligent sanitation vehicle, the dimension errors caused by the manufacturing process should be considered in the lightweight design process of the mechanical claws. In this study, we used the mechanical claws as the carrier. The maximum deformation and the maximum equivalent stress of them were taken as the design responses and the plate thicknesses of different parts were taken as the design variables. A light-weight robust optimization design method for the mechanical claws was proposed by combining the optimal latin hypercube experimental design with high-precision Kriging approximate model and using Particle Swarm optimization (PSO) as the optimization solver. The optimization results show that this method has higher robustness than traditional deterministic methods although it has lower lightweight requirements. The final structural design of this study has higher advantages in preventing the fluctuation of uncertain parameters.

A robust metamodel-based optimization design method for improving pedestrian wind comfort in an infill development project

Constructing a new building inevitably modifies the microclimate in its vicinity. We propose a metamodel-based optimization method to devise near optimal designs that minimize building’s adverse impacts on nearby pedestrians’ wind comfort in hot and cold seasons. Four design parameters, i.e., building width, building depth, building height, and building orientation, are considered for a greatly simplified building design task. Specifically, computational fluid dynamics (CFD) analyses are performed to calculate the mean wind velocity, while all CFD experiment samples are determined through the Box-Behnken design of experiment method. Based on 54 CFD experiments, the relationships between building design variables and the summer and winter mean wind velocities within a specified assessment area are learned using the response surface methodology. Finally, a genetic algorithm with the desirability function as the objective function is applied to identify near optimal design options under the robustness control (i.e., family error rate λ) for response surface models. The framework is applied to an infill development project to highlight its suitability in a real application. The experiment results show that as λ gradually decreases from 1 to 0.05, the overall desirability index dwindles from 0.33 to 0.22, consequently generating a more conservative optimal design decision.

Robust Optimization Design for Permanent Magnet Machine Considering Magnet Material Uncertainties

In this article, a robust optimization design method considering magnet material uncertainties is proposed for permanent magnet (PM) machines. The key of this method is to improve the machine design robustness to resist the uncertainties influence from PM material, including material diversity, manufacture error, and assembly accuracy. To verify the proposed optimization method, a double PM machine is selected as an example. During the implementation of optimization design, first, the robust optimization model is constructed based on the analysis of PM uncertainties. Then, the Kriging surrogate model and non-dominated sorting genetic algorithm II (NSGA-II) are adopted to solve the optimization model. Finally, to verify the effectiveness of the method, the machine performance and corresponding reliability are evaluated, and a prototype is manufactured and tested. Both simulation and experimental results indicate that the proposed robust optimization method can effectively reduce the influence of magnet material uncertainties on PM machine performance.

Surrogate Model–Based Robust Multidisciplinary Design Optimization of an Unmanned Aerial Vehicle

Despite the many advantages of the design optimization technique, this method is costly for real engineering problems. This cost will increase sharply for issues with a multidisciplinary and uncertain nature and more than one objective function. In this paper, the metamodel concept has been used to overcome this problem. Because of the ability of neural networks to approximate the behavior of complex engineering systems, this tool has been used to create a surrogate model. Because multidisciplinary design optimization and robust design optimization methods have been used in this study and according to the high cost of the multidisciplinary analysis module, a surrogate model of this module has been made to reduce the imposed costs. To show the capability of the considered approach, robust multidisciplinary design optimization of an unmanned aerial vehicle (UAV) has been done. Take-off weight and cruise drag are the considered objective functions in this study, and the nondominated sorting genetic algorithm (NSGA-I) has been used for minimization of them. The optimization results show that the use of the metamodeling concept has reduced computational costs by 94.1%.

A hierarchical optimization approach to robust design of energy supply systems based on a mixed-integer linear model

In designing energy supply systems, designers should heighten the robustness in performance criteria against the uncertainty in energy demands. In this paper, a robust optimal design method using a hierarchical mixed-integer linear programming (MILP) method is proposed to maximize the robustness of energy supply systems under uncertain energy demands based on a mixed-integer linear model. A robust optimal design problem is formulated as a three-level min-max-min MILP one by expressing uncertain energy demands by intervals, evaluating the robustness in a performance criterion based on the minimax regret criterion, and considering relationships among integer design variables, uncertain energy demands, and integer and continuous operation variables. This problem is solved by evaluating upper and lower bounds for the minimum of the maximum regret of the performance criterion repeatedly outside, and evaluating lower and upper bounds for the maximum regret repeatedly inside. Different types of optimization problems are solved by applying a hierarchical MILP method developed for ordinary optimal design problems without and with its modifications. In a case study, the proposed approach is applied to the robust optimal design of a cogeneration system. Through the study, its validity and effectiveness are ascertained, and some features of the obtained robust designs are clarified.

Enhanced Robust Design Optimization in Seat Belt Anchorage Strength for Front Crash Safety of Multi-Purpose Vehicle

This paper deals with an enhanced robust design optimization (RDO) method and its application to the strength design problem of seat belt anchorage, related to the front crash safety of multi-purpose vehicles. In order to determine the rational design safety of the newly developed automotive part, such as the seat, in which the reliability of the evaluation data is not sufficient at the design stage, it is necessary to implement a probabilistic design considering uncertainties. Thickness size variables of the seat frame structure’s members were considered random design variables, including uncertainties such as manufacturing tolerance, which are an inevitable hazard in the design of automotive parts. Probabilistic constraints were selected from the strength performances of the seat belt anchorage test, which are regulated in Economic Commission for Europe (ECE) and Federal Motor Vehicle Safety Standard (FMVSS), and the strength performances were evaluated by finite element analyses. The RDO problem was formulated such that the random design variables were determined by minimizing the seat frame weight subject to the probabilistic strength performance constraints evaluated from the reliability analyses. Three sigma level quality was considered for robustness in side constraints. The mean value reliability method (MVRM) and adaptive importance sampling method (AISM) were used for the reliability analyses in the RDO, and reliability probabilities from the MVRM and the AISM on the probabilistic optimum design were assessed by Monte Carlo simulation (MCS). The RDO results according to the reliability analysis methods were compared to determine the optimum design results. In the case of the RDO with the AISM, the structure reliability was fully satisfied for all the constraint functions, so the most reliable structural safety was guaranteed for the seat frame design.

Influence of the Hook Position on the Vertical Vibrations of an Automobile Exhaust System: Application of the Robust Optimization Design

A robust optimization design method is proposed to investigate the influence of the hook position on the vertical vibration (bending) of an automobile exhaust system. A block diagram for the robustness analysis of the exhaust system is initially constructed from the major affecting factors. Secondly, the second-order inertia force is set as the vibration excitation source of the exhaust system and the displacement of four hooks of the exhaust system is selected as the variable factor. Then tests are carried out to investigate the resulting vertical bending considering four influencing factors and three levels of analysis. Finally, a variance analysis of the vertical bending is performed. The present study provides a set of guidelines to control the key factors affecting the vibration of vehicle exhaust systems while proposing an effective method to reduce vehicle vibration and improve noise analysis.

An interval robust design optimization method and its application in heat transfer problems

ABSTRACT An interval robust optimization method is suggested to improve the robustness and performance of the process or product in design. A general uncertain robust optimization problem is considered in which the objective function and constraints are both nonlinear and uncertain, and the uncertainties of design variables and uncertain parameters are all included and depicted by the interval model. A nonlinear interval number programming method is introduced to solve the robust optimization model, in which the uncertain objective function is described by a pair of deterministic functions, and the uncertain constraints are converted to deterministic constraints on the basis of the reliability-based possibility degree of interval. The method is then applied to several heat transfer problems for performance optimization of the heat transfer processes while guaranteeing the robustness of the optimal design.

System-Level Robust Design Optimization of a Switched Reluctance Motor Drive System Considering Multiple Driving Cycles

In this article, a novel system-level robust design optimization method is presented to improve the performance of switched reluctance motor (SRM) drive systems under multiple operating conditions. Based on typical driving cycles of electric vehicles (EVs), five typical driving modes of the SRM are determined. The optimization objectives in each driving mode are established. The significant parameters of the motor and controller of each driving mode are selected as the optimization variables by using the sensitivity analysis. In order to simplify the optimization process, correlation analysis is performed to determine the coherence of the objective functions of all driving modes. Then, a sequential Taguchi method is applied to find an optimal design which is less sensitive to the noise factors. To verify the effectiveness of the proposed method, an SRM drive system applied in EVs with a 12/10 SRM and angle position control method is investigated. It is found that the proposed method can significantly reduce the torque ripple and improve the comprehensive performance. Finally, a 12/10 SRM is prototyped and tested to validate the simulation results.

Optimal Distribution of Renewable Energy Systems Considering Aging and Long-Term Weather Effect in Net-Zero Energy Building Design

Generation system interruptions in net-zero energy buildings (NZEBs) may result in missing the net-zero targets by a great margin. Consequently, it is significant to incorporate a realistic reliability model for renewable energy systems (RESs) that considers aging and long-term weather conditions. This study proposed a robust design optimization method that deals with the selection of RES to achieve NZEB. Different case studies were evaluated: 1. Deterministic approach; 2. Markov chain-based reliability without the aging effect; 3. Markov chain-based reliability with the aging effect. The results showed that the optimal sizes of RES, considering the aging effect, were much larger than the other two cases based on the annual energy balance. Moreover, the consideration of the aging effect on the reliability assessment of the generation system for NZEB opens a pathway for a more robust and economic design of RES.

Multi-levels Kriging surrogate model-based robust aerodynamics optimization design method

Robust design optimization has a great potential application in many engineering fields. In the conventional robust aerodynamics design optimization method, the main difficulty is expensive computational cost related to a large number of function evaluations for uncertainty quantification (UQ). To alleviate the expensive burden for UQ, two levels Kriging surrogate model was introduced. The first level is for the mean value and the second level is for the variances. Through the second level Kriging surrogate models, the method of Monte Carlo Simulation (MCS), which requires a huge number of function evaluations, can be effectively applied to the analysis of variance. Efficient Global Optimization algorithm (EGO) was employed to achieve the global optimized results. To validate the performance of the design method, both one-dimensional function and two-dimensional function were applied. Finally, robust aerodynamics design optimization was applied for a low-drag airfoil. The results show that the optimal solutions obtained from the uncertainty-based optimization formulation are less sensitive to uncertainties to small manufacturing errors.

Robust Design Optimization of Electrical Machines: A Comparative Study and Space Reduction Strategy

This article presents a comparative study on different types of robust design optimization methods for electrical machines. Three robust design approaches, Taguchi parameter design, worst-case design and design for six-sigma, are compared for low-dimensional and high-dimensional design optimization scenarios, respectively. For the high-dimensional scenario, the computational burden is normally massive due to the robustness evaluation of a huge number of design candidates. To attempt this challenge, as the second aim of this paper, a space reduction optimization (SRO) strategy is proposed for these robust design approaches, yielding three new robust optimization methods. To illustrate and compare the performance of different robust design optimization methods, a permanent magnet motor with soft magnetic composite cores is investigated with the consideration of material diversities and manufacturing tolerances. 3-D finite element model and thermal network model are employed in the optimization process and the accuracy of both models has been verified by experimental results. Based on the theoretical analysis and optimization results, a detailed comparison is provided for all investigated and proposed robust design optimization methods in terms of different aspects. It shows that the proposed SRO strategy can greatly improve the design optimization effectiveness and efficiency of those three conventional robust design methods.

Design for Interface Stiffness of Mechanical Products Using Integrated Simulation and Optimization Under Uncertainty

Abstract Interface stiffness is an important factor influencing the performance of mechanical products. Uncertain factors affect the interface stiffness and stability in the process of product design, manufacture, and operation. How to reduce the impact of uncertain factors on the interface stiffness is a vital problem in interface design. In this paper, a robust optimal design method is proposed for mechanical interfaces considering uncertain factors, which combines the finite element simulation, experiment, and optimization to reduce the sensitivity of interface stiffness to uncertain factors. The proposed interface design method provides an effective way to improve the interface stiffness under uncertain conditions. In order to validate the proposed method, the bolted connection structure of a flange is applied as an example. The interface stiffness of the flange is selected as an optimization target, and the Gaussian process regression is used to construct a two-layer optimal model of the objective function for the design and uncertain parameters. When experimental and optimization results differ significantly, the Kalman filter is used to provide the feedback for the optimization results until the results meet requirements. The final results show that the optimized mechanical interface stiffness is increased by 15.5%, and the error between the optimized prediction and experimental results is within 1% after three times experimental validation and feedback adjustment.

Quality and robustness optimization design method for electromagnetic devices consider manufacturing uncertainties and working point migration of permanent magnets

Electromagnetic devices (EMDs) containing permanent magnets are common devices showing low power con-sumption, small size, and high sensitivity. In batch production of EMDs there is a robustness problem linked to the migration of the working point of permanent magnets and to their manufacturing uncertainties. In turn, the design optimization for EMDs has attracted large attention. Robust design optimization (RDO) is one of the most effective methods to boost quality and robustness of EMDs. However, state-of-art RDO methods ignore the various uncertainties from manufacture processing, production environment, and working point migration of permanent magnets. To address those problems, this study presents an efficient and universal method based on the working point migration and various uncertainties of output characteristics for electromagnetic devices. At first, we introduce a two-dimensional factor design scheme considering the working point migration and manufacturing conditions. Then, a robustness assessment measure suitable for different optimization types is proposed. On these bases, the optimization model is established, considering the permanent working point migration for magnets, manufacturing tolerances and robustness constraints, and a particle swarm algorithm is used to obtain the solution. Finally, the effectiveness of the method is verified by a case study involving a specific type of electromagnetic device with permanent magnets.

Coordinated robust optimal design of building envelope and energy systems for zero/low energy buildings considering uncertainties

Uncertainties exist throughout the life cycle of zero/low energy buildings, which may lead to low energy efficiency and even failure of achieving zero/low energy goal in operation. However, current design practice of the entire zero/low energy buildings, including building envelope and energy systems, seldom considers the uncertainties or roughly considers the uncertainties using safety factors in system sizing. Actually, the computing cost of optimal design of the entire zero/low energy buildings is high as numerous design options and parameters are involved. The consideration of uncertainties in the design would further increase the computing cost significantly, since each design option needs to be evaluated under a large number of uncertain scenarios. Thus, an efficient method is needed. In this study, a coordinated robust optimal design method is proposed to efficiently identify the global optimal design solutions for the entire zero/low energy buildings under uncertainties. The design process is divided into two stages, including robust design optimizations of building envelope and energy systems considering uncertainties. These two stages are coordinated to assure that the optimal design solution obtained is global optimal. Point estimate method is adopted for uncertainty quantification. A case study is performed to test and validate the proposed method using the zero carbon building in Hong Kong as the reference building. Results show that the proposed method is robust and efficient to identify the global optimal design solutions for the entire building under uncertainties. It can provide designs of better performance with reduced cost compared with current design methods.

Uncertainty-based robust optimal design of cleanroom air-conditioning systems considering life-cycle performance

Strict and simultaneous space temperature and humidity controls are often required in many applications, such as hospitals, laboratories, cleanrooms for pharmaceutical and semiconductor manufacturing. The energy intensity in such applications can be up to 100 times than typical office buildings, mainly due to the improper system design and control. Although some uncertainty-based design methods have been developed for air-conditioning systems, most of the existing systems are designed based on a certain ventilation mode while neglecting the life-cycle performance of the components. This study, therefore, proposes a robust optimal design method for cleanroom air-conditioning systems, considering the uncertainties in design parameters for inputs and operation strategies as well as the life-cycle performance of components. An adaptive full-range decoupled ventilation strategy, which incorporated five operation modes, was adopted in the design optimization. Two maintenance modes were adopted and compared to consider the flexibility of maintenance. The proposed design method has been implemented and validated in the design optimization of an existing air-conditioning system. The results showed that, compared with the conventional design, up to 54% reduction of life-cycle costs and superior satisfaction of services could be achieved by using the proposed method.