Reliability-based Robust Optimization Design of Automobile Components with Non-normal Distribution Parameters

In the reliability designing procedure of the vehicle components, when the distribution styles of the random variables are unknown or non-normal distribution, the result evaluated contains great error or even is wrong if the reliability value R is larger than 1 by using the existent method, in which case the formula is necessary to be revised. This is obviously inconvenient for programming. Combining reliability-based optimization theory, robust designing method and reliability based sensitivity analysis, a new method for reliability robust designing is proposed. Therefore the influence level of the designing parameters’ changing to the reliability of vehicle components can be obtained. The reliability sensitivity with respect to design parameters is viewed as a sub-objective function in the multi-objective optimization problem satisfying reliability constraints. Given the first four moments of basic random variables, a fourth-moment technique and the proposed optimization procedure can obtain reliability-based robust design of automobile components with non-normal distribution parameters accurately and quickly. By using the proposed method, the distribution style of the random parameters is relaxed. Therefore it is much closer to the actual reliability problems. The numerical examples indicate the following: (1) The reliability value obtained by the robust method proposed increases (】0.04%) comparing to the value obtained by the ordinary optimization algorithm; (2) The absolute value of reliability-based sensitivity decreases (】0.01%), and the robustness of the products’ quality is improved accordingly. Utilizing the reliability-based optimization and robust design method in the reliability designing procedure reduces the manufacture cost and provides the theoretical basis for the reliability and robust design of the vehicle components.

Multi-parameter Sensitivity Analysis and Application Research in the Robust Optimization Design for Complex Nonlinear System

The current research of complex nonlinear system robust optimization mainly focuses on the features of design parameters, such as probability density functions, boundary conditions, etc. After parameters study, high-dimensional curve or robust control design is used to find an accurate robust solution. However, there may exist complex interaction between parameters and practical engineering system. With the increase of the number of parameters, it is getting hard to determine high-dimensional curves and robust control methods, thus it＇s difficult to get the robust design solutions. In this paper, a method of global sensitivity analysis based on divided variables in groups is proposed. By making relevant variables in one group and keeping each other independent among sets of variables, global sensitivity analysis is conducted in grouped variables and the importance of parameters is evaluated by calculating the contribution value of each parameter to the total variance of system response. By ranking the importance of input parameters, relatively important parameters are chosen to conduct robust design analysis of the system. By applying this method to the robust optimization design of a real complex nonlinear system-a vehicle occupant restraint system with multi-parameter, good solution is gained and the response variance of the objective function is reduced to 0.01, which indicates that the robustness of the occupant restraint system is improved in a great degree and the method is effective and valuable for the robust design of complex nonlinear system. This research proposes a new method which can be used to obtain solutions for complex nonlinear system robust design.

MULTIDISCIPLINARY ROBUST OPTIMIZATION DESIGN

Because uncertainty factors inevitably exist under multidisciplinary designenvironment, a hierarchical multidisciplinary robust optimization design based on response surfaceis proposed. The method constructs optimization model of subsystem level and system level tocoordinate the coupling among subsystems, and also the response surface based on the artificialneural network is introduced to provide information for system level optimization tool to maintainthe independence of subsystems, i.e. to realize multidisciplinary parallel design. The applicationcase of electrical packaging demonstrates that reasonable robust optimum solution can be yielded andit is a potential and efficient multi-disciplinary robust optimization approach.

Robust Design Optimization Method for Centrifugal Impellers under Surface Roughness Uncertainties Due to Blade Fouling

Blade fouling has been proved to be a great threat to compressor performance in operating stage. The current researches on fouling-induced performance degradations of centrifugal compressors are based mainly on simplified roughness models without taking into account the realistic factors such as spatial non-uniformity and randomness of the fouling-induced surface roughness. Moreover, little attention has been paid to the robust design optimization of centrifugal compressor impellers with considerations of blade fouling. In this paper, a multi-objective robust design optimization method is developed for centrifugal impellers under surface roughness uncertainties due to blade fouling. A three-dimensional surface roughness map is proposed to describe the nonuniformity and randomness of realistic fouling accumulations on blades. To lower computational cost in robust design optimization, the support vector regression（SVR） metamodel is combined with the Monte Carlo simulation（MCS） method to conduct the uncertainty analysis of fouled impeller performance. The analyzed results show that the critical fouled region associated with impeller performance degradations lies at the leading edge of blade tip. The SVR metamodel has been proved to be an efficient and accurate means in the detection of impeller performance variations caused by roughness uncertainties. After design optimization, the robust optimal design is found to be more efficient and less sensitive to fouling uncertainties while maintaining good impeller performance in the clean condition. This research proposes a systematic design optimization method for centrifugal compressors with considerations of blade fouling, providing a practical guidance to the design of advanced centrifugal compressors.

Research on Low Cycle Fatigue Reliability-based Robust Design Optimization of Turbine Blade

针对涡轮叶片低周疲劳可靠性稳健设计优化问题，对叶片材料进行了高温疲劳试验，采用定量方程随机化方法处理试验数据，获得叶片材料的概率-应变-寿命曲线。采用贝塞尔曲线描述叶片型线方程，建立了涡轮叶片结构及流场的参数化模型，采用热-流-固耦合有限元法对涡轮流场和叶片进行了数值分析，得到叶片动能效率和应力应变分布特性。建立了叶片疲劳可靠性稳健设计优化模型，并采用响应面法获得叶片结构性能函数和极限状态函数，将叶片低周疲劳可靠性作为基本约束条件，采用序列二次规划优化法得到设计优化结果。研究结果表明，优化后的叶片低周疲劳可靠性以及稳健性显著提高，模型及方法正确可行，可用于涡轮叶片以及其他复杂结构的低周疲劳可靠性稳健设计优化。

Response surface methodology-based hybrid robust design optimization for complex product under mixed uncertainties

Minimizing the impact of the mixed uncertainties(i.e.,the aleatory uncertainty and the epistemic uncertainty) for a complex product of compliant mechanism(CPCM) quality improvement signifies a fascinating research topic to enhance the robustness.However, most of the existing works in the CPCM robust design optimization neglect the mixed uncertainties, which might result in an unstable design or even an infeasible design. To solve this issue, a response surface methodology-based hybrid robust design optimization(RSM-based HRDO) approach is proposed to improve the robustness of the quality characteristic for the CPCM via considering the mixed uncertainties in the robust design optimization. A bridge-type amplification mechanism is used to manifest the effectiveness of the proposed approach. The comparison results prove that the proposed approach can not only keep its superiority in the robustness, but also provide a robust scheme for optimizing the design parameters.

Robust Design Optimization and Improvement by Metamodel

The robust design optimization(RDO)is an effective method to improve product performance with uncertainty factors.The robust optimal solution should be not only satisfied the probabilistic constraints but also less sensitive to the variation of design variables.There are some important issues in RDO,such as how to judge robustness,deal with multi-objective problem and black-box situation.In this paper,two criteria are proposed to judge the deterministic optimal solution whether satisfies robustness requirment.The robustness measure based on maximum entropy is proposed.Weighted sum method is improved to deal with the objective function,and the basic framework of metamodel assisted robust optimization is also provided for improving the efficiency.Finally,several engineering examples are used to verify the advantages.

Robust Multiobjective and Multidisciplinary Design Optimization of Electrical Drive Systems

Design and optimization of electrical drive systems often involve simultaneous consideration of multiple objectives that usually contradict to each other and multiple disciplines that normally coupled to each other.This paper aims to present efficient system-level multiobjective optimization methods for the multidisciplinary design optimization of electrical drive systems.From the perspective of quality control,deterministic and robust approaches will be investigated for the development of the optimization models for the proposed methods.Meanwhile,two approximation methods,Kriging model and Taylor expansion are employed to decrease the computation/simulation cost.To illustrate the advantages of the proposed methods,a drive system with a permanent magnet synchronous motor driven by a field oriented control system is investigated.Deterministic and robust Pareto optimal solutions are presented and compared in terms of several steady-state and dynamic performances(like average torque and speed overshoot)of the drive system.The robust multiobjective optimization method can produce optimal Pareto solutions with high manufacturing quality for the drive system.

Robust optimization design on impeller of mixed-flow pump

To increase the robustness of the optimization solutions of the mixed-flow pump,the impeller was firstly indirectly parameterized based on the 2D blade design theory.Secondly,the robustness of the optimization solution was mathematically defined,and then calculated by Monte Carlo sampling method.Thirdly,the optimization on the mixed-flow pump′s impeller was decomposed into the optimal and robust sub-optimization problems,to maximize the pump head and efficiency and minimize the fluctuation degree of them under varying working conditions at the same time.Fourthly,using response surface model,a surrogate model was established between the optimization objectives and control variables of the shape of the impeller.Finally,based on a multi-objective genetic optimization algorithm,a two-loop iterative optimization process was designed to find the optimal solution with good robustness.Comparing the original and optimized pump,it is found that the internal flow field of the optimized pump has been improved under various operating conditions,the hydraulic performance has been improved consequently,and the range of high efficient zone has also been widened.Besides,with the changing of working conditions,the change trend of the hydraulic performance of the optimized pump becomes gentler,the flow field distribution is more uniform,and the influence degree of the varia-tion of working conditions decreases,and the operating stability of the pump is improved.It is concluded that the robust optimization method proposed in this paper is a reasonable way to optimize the mixed-flow pump,and provides references for optimization problems of other fluid machinery.

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。

Robust design and optimization for autonomous PV-wind hybrid power systems

This study presents a robust design method for autonomous photovoltaic (PV)-wind hybrid power systems to obtain an optimum system configuration insensitive to design variable variations. This issue has been formulated as a constraint multi-objective optimization problem, which is solved by a multi-objective genetic algorithm, NSGA-II. Monte Carlo Simulation (MCS) method, combined with Latin Hypercube Sampling (LHS), is applied to evaluate the stochastic system performance. The potential of the proposed method has been demonstrated by a conceptual system design. A comparative study between the proposed robust method and the deterministic method presented in literature has been conducted. The results indicate that the proposed method can find a large mount of Pareto optimal system configurations with better compromising performance than the deterministic method. The trade-off information may be derived by a systematical comparison of these configurations. The proposed robust design method should be useful for hybrid power systems that require both optimality and robustness.

Robust Collaborative Optimization Method Based on Dual-response Surface

A novel method for robust collaborative design of complex products based on dual-response surface （DRS-RCO） is proposed to solve multidisciplinary design optimization （MDO） problems under uncertainty. Collaborative optimization （CO） which decomposes the whole system into a double-level nonlinear optimization problem is widely accepted as an efficient method to solve MDO problems. In order to improve the quality of complex product in design process, robust collaborative optimization （RCO） is developed to solve those problems under uncertain conditions. RCO does optimization on the linear sum of mean and standard deviation of objective function and gets an optimal solution with high robustness. Response surfaces method is an important way to do approximation in robust design. DRS-RCO is an improved RCO method in which dual-response surface replaces system uncertainty analysis module of CO. The dual-response surface is the approximate model of mean and standard deviation of objective function respectively. In DRS-RCO, All the information of subsystems is included in dual-response surfaces. As an additional item, the standard deviation of objective function is added to the subsystem optimization. This item guarantee both the mean and standard deviation of this subsystem is reaching the minima at the same time. Finally, a test problem with two coupled subsystems is conducted to verify the feasibility and effectiveness of DRS-RCO.

ROBUST OPTIMUM DESIGN OF LAMINATED COMPOSITE PLATES

A last-ply failure (LPF) analysis method for laminated composite plates is incorpo- rated into the ?nite element code-ANSYS, and a robust optimum design method is presented. The composite structure is analyzed by considering both in-plane and out-of-plane loads. For a lamina, two major failure modes are considered: matrix failure and ?ber breakage that are characterized by the proper strength criteria in the literature. When a lamina has failed, the laminate sti?ness is modi?ed to re?ect the damage, and stresses in the structure are re-analyzed. This procedure is repeatedly performed until the whole structure fails and thus the ultimate strength is determined. A structural optimization problem is solved with the ?ber orientation and the lamina thickness as the design variables and the LPF load as the objective. Finally, the robust optimum design method for laminates is presented and discussed.

Robust Topology Optimization of Periodic Multi-Material Functionally Graded Structures under Loading Uncertainties

This paper presents a robust topology optimization design approach for multi-material functional graded structures under periodic constraint with load uncertainties.To characterize the random-field uncertainties with a reduced set of random variables,the Karhunen-Lo`eve(K-L)expansion is adopted.The sparse grid numerical integration method is employed to transform the robust topology optimization into a weighted summation of series of deterministic topology optimization.Under dividing the design domain,the volume fraction of each preset gradient layer is extracted.Based on the ordered solid isotropic microstructure with penalization(Ordered-SIMP),a functionally graded multi-material interpolation model is formulated by individually optimizing each preset gradient layer.The periodic constraint setting of the gradient layer is achieved by redistributing the average element compliance in sub-regions.Then,the method of moving asymptotes(MMA)is introduced to iteratively update the design variables.Several numerical examples are presented to verify the validity and applicability of the proposed method.The results demonstrate that the periodic functionally graded multi-material topology can be obtained under different numbers of sub-regions,and robust design structures are more stable than that indicated by the deterministic results.

A novel robust design method for improving stability of optimized structures

It is known that structural optimization may lead to designs of structures having low stability and sometimes even kinematically unstable designs. This paper presents a robust design method for improving the stability of opti mized structures. A new approach is proposed, in which cer tain perturbation loads are introduced and the corresponding compliance is added to the objective function as a penaliza tion. The stability of the optimized structures can thus be improved substantially by considering structural responses to the original and the introduced loads. Numerical exam ples show the simplicity and effectiveness of the proposed method.

Non-probabilistic Robust Optimal Design Method

For the purpose of dealing with uncertainty factors in engineering optimization problems, this paper presents a new non-probabilistic robust optimal design method based on maximum variation estimation. The method analyzes the effect of uncertain factors to objective and constraints functions, and then the maximal variations to a solution are calculated. In order to guarantee robust feasibility the maximal variations of constraints are added to original constraints as penalty term; the maximal variation of objective function is taken as a robust index to a solution; linear physical programming is used to adjust the values of quality characteristic and quality variation, and then a bi-level mathematical robust optimal model is constructed. The method does not require presumed probability distribution of uncertain factors or continuous and differentiable of objective and constraints functions. To demonstrate the proposed method, the design of the two-bar structure acted by concentrated load is presented. In the example the robustness of the normal stress, feasibility of the total volume and the buckling stress are studied. The robust optimal design results show that in the condition of maintaining feasibility robustness, the proposed approach can obtain a robust solution which the designer is satisfied with the value of objective function and its variation.

Randomized Algorithms for Probabilistic Optimal Robust Performance Controller Design

Polynomial-time randomized algorithms were constructed to approximately solve optimal robust performance controller design problems in probabilistic sense and the rigorous mathematical justification of the approach was given. The randomized algorithms here were based on a property from statistical learning theory known as (uniform) convergence of empirical means (UCEM). It is argued that in order to assess the performance of a controller as the plant varies over a pre-specified family, it is better to use the average performance of the controller as the objective function to be optimized, rather than its worst-case performance. The approach is illustrated to be efficient through an example.

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.

APPLICATION OF MODERN ROBUST OPTIMAL DESIGN METHOD TO THE SHOCK ABSORBER IN A CAR*

To reduce the variation of velocity characteristic of the shock absorber in acar, a modern robust optimal design method is applied to its structural parameters design. Firstly,the method is used to obtain the robust values which have low sensitivity to velocitycharacteristic and analyze the influences of the parameters on velocity characteristic. Secondly,the method is used to obtain their maximum tolerances under the condition of ensuring productquality. The results obviously improve the velocity characteristic.