Uncertainty‐based robust aerodynamic optimization of rotor blades

The aerodynamic performance of a compressor is highly sensitive to uncertain working conditions. This paper presents an efficient robust aerodynamic optimization method on the basis of nondeterministic computational fluid dynamic (CFD) simulation and multi‐objective genetic algorithm (MOGA). A nonintrusive polynomial chaos method is used in conjunction with an existing well‐verified CFD module to quantify the uncertainty propagation in the flow field. This method is validated by comparing with a Monte Carlo method through full 3D CFD simulations on an axial compressor (National Aeronautics and Space Administration rotor 37). On the basis of the validation, the nondeterministic CFD is coupled with a surrogate‐based MOGA to search for the Pareto front. A practical engineering application is implemented to the robust aerodynamic optimization of rotor 37 under random outlet static pressure. Two curve angles and two sweep angles at tip and hub are used as design variables. Convergence analysis shows that the surrogate‐based MOGA can obtain the Pareto front properly. Significant improvements of both mean and variance of the efficiency are achieved by the robust optimization. The comparison of the robust optimization results with that of the initial design, and a deterministic optimization demonstrate that the proposed method can be applied to turbomachinery successfully. Copyright © 2012 John Wiley & Sons, Ltd.     

基于转静干涉效应的压气机叶片气动载荷分析

转静叶排的相互作用会使压气机内部流场存在复杂的非定常性。为深入研究压气机叶片的气动载荷特性,以某型航空发动机压气机为研究对象,考虑叶排间的转静干涉效应,利用滑移网格技术对整个叶盘的三维流场展开模拟,求解干涉周期T_b内压气机转子内部的流动规律。同时对叶片气动载荷的非定常特性进行进一步分析,讨论了不同压比、转速对压气机叶片气动载荷的影响。结果表明叶片压力面和吸力面气动载荷波动峰值的主导频率皆为转静干涉频率f₀的倍频,其中一倍频（1×f₀）分量占主导地位。在干涉周期T_b内,叶片表面压力涡发生周期性的迁移与耗散,压力面和吸力面气动载荷的变化呈相反趋势。随着压比的增加,压气机叶片气动载荷逐渐增大,但其脉动幅值和频谱峰值基本不变。转速的升高使得转静干涉的频率增大,增强了压气机叶片气动载荷的非定常特性。研究结果能够应用于叶盘结构的气动优化设计,可为高性能航空发动机压气机的研制提供支持和参考。     

一种基于响应面的翼型鲁棒优化设计方法

针对不确定因素引起的翼型气动性能波动现象,探讨了翼型几何形状随机不确定波动对翼型气动特性的影响,并进行了减小此影响的鲁棒优化设计。引入类别形状函数变换(CST)方法可以大大降低设计变量自由度数目,以NACA0012翼型为例,进行了考虑翼型几何形状随机扰动的气动不确定性分析,发现几何形状的微小波动对升力特性影响很小。发展了一种基于响应面和遗传算法的鲁棒优化设计方法,能够高效的减小阻力及其波动的影响。计算结果显示尽管相对于确定性优化结果鲁棒优化阻力略有增加,但波动更小,气动性能具有更好的鲁 棒性。     

Achieving robust design through statistical effect screening

This work proposes a method for statistical effect screening to identify design parameters of a numerical simulation that are influential to performance while simultaneously being robust to epistemic uncertainty introduced by calibration variables. Design parameters are controlled by the analyst, but the optimal design is often uncertain, while calibration variables are introduced by modeling choices. We argue that uncertainty introduced by design parameters and calibration variables should be treated differently, despite potential interactions between the two sets. Herein, a robustness criterion is embedded in our effect screening to guarantee the influence of design parameters, irrespective of values used for calibration variables. The Morris screening method is utilized to explore the design space, while robustness to uncertainty is quantified in the context of info‐gap decision theory. The proposed method is applied to the National Aeronautics and Space Administration Multidisciplinary Uncertainty Quantification Challenge Problem, which is a black‐box code for aeronautic flight guidance that requires 35 input parameters. The application demonstrates that a large number of variables can be handled without formulating simplifying assumptions about the potential coupling between calibration variables and design parameters. Because of the computational efficiency of the Morris screening method, we conclude that the analysis can be applied to even larger‐dimensional problems. (Approved for unlimited, public release on October 9, 2013, LA‐UR‐13‐27839, Unclassified.) Copyright © 2015 John Wiley & Sons, Ltd.     

Aerodynamic optimization of turbomachinery blades using evolutionary methods and ANN-based surrogate models

A fast, flexible, and robust simulation-based optimization scheme using an ANN-surrogate model was developed, implemented, and validated. The optimization method uses Genetic Algorithm (GA), which is coupled with an Artificial Neural Network (ANN) that uses a back propagation algorithm. The developed optimization scheme was successfully applied to single-point aerodynamic optimization of a transonic turbine stator and multi-point optimization of a NACA65 subsonic compressor rotor in two-dimensional flow, both were represented by 2D linear cascades. High fidelity CFD flow simulations, which solve the Reynolds-Averaged Navier-Stokes equations, were used in generating the data base used in building the ANN low fidelity model. The optimization objective is a weighted sum of the performance objectives and is penalized with the constraints; it was constructed so as to achieve a better aerodynamic performance at the design point or over the full operating range by reshaping the blade profile. The latter is represented using NURBS functions, whose coefficients are used as the design variables. Parallelizing the CFD flow simulations reduced the turn-around computation time at close to 100% efficiency. The ANN model was able to approximate the objective function rather accurately and to reduce the optimization computing time by ten folds. The chosen objective function and optimization methodology result in a significant and consistent improvement in blade performance.     

Robust optimal-parameter design approach for tolerance design problems

A robust optimal-parameter design method, henceforth called the TGAOA, to solve tolerance design problems has been investigated. Tolerance affects system performance and leads to violation of design constraints. The TGAOA approach conducts global exploration by using a genetic algorithm and exploits optimal offspring via the Taguchi method. It is able to effectively reduce the impact of parameter variations in reaching robust optimal-solutions as allowed by the tolerance. Two design examples are employed to evaluate the performance of the new method. The first is for a mixed H₂/H_∞ optimal PID controller under varying PID component specifications, plant uncertainty, and other external unknown disturbance. The second involves a 13-variable test function, which includes quadratic, linear, and polynomial forms to illustrate the general robustness and computational efficiency, for which comparisons are also made with its predecessors of genetic algorithm and hybrid Taguchi-genetic algorithm.     

Response surface approach to aerodynamic optimization design of helicopter rotor blade

This paper describes a hovering rotor blade design through the suitable combination of flow analysis and optimization technique. It includes a parametric study concerned with the influence of design variables and different design conditions such as objective functions and constraints on the rotor performance. Navier–Stokes analysis is employed to compute the hovering rotor performance in subsonic and transonic operating conditions. Response surface method based on D‐optimal 3‐level factorial design and genetic algorithm are applied to obtain the optimum solution of a defined objective function including the penalty terms of constraints. The designs of the rotor airfoil geometry and the rotor tip shape are performed in subsonic and transonic conditions, and it is observed that the new rotor blades optimized by various objective functions and constraints have better aerodynamic characteristics than the baseline rotor blade. The influence of design variables and their mutual interactions on the rotor performance is also examined through the optimization process. Copyright © 2005 John Wiley & Sons, Ltd.     

Structural reliability analysis and uncertainties‐based collaborative design and optimization of turbine blades using surrogate model

In power and energy systems, both the aerodynamic performance and the structure reliability of turbine equipment are affected by utilized blades. In general, the design process of blade is high dimensional and nonlinear. Different coupled disciplines are also involved during this process. Moreover, unavoidable uncertainties are transported and accumulated between these coupled disciplines, which may cause turbine equipment to be unsafe. In this study, a saddlepoint approximation reliability analysis method is introduced and combined with collaborative optimization method to address the above challenge. During the above reliability analysis and design optimization process, surrogate models are utilized to alleviate the computational burden for uncertainties‐based multidisciplinary design and optimization problems. Smooth response surfaces of the performance of turbine blades are constructed instead of expensively time‐consuming simulations. A turbine blade design problem is solved here to validate the effectiveness and show the utilization of the given approach.     

Development of a simple and efficient method for robust optimization

Robust optimization problems are newly formulated and an efficient computational scheme is proposed. Both design variables and design parameters are considered as random variables about their nominal values. To ensure the robustness of objective performance, we introduce a new performance index bounding the performance together with a constraint limiting the performance variation. The constraint variations are regulated by considering the probability of feasibility. Each probability constraint is transformed into a sub‐optimization problem by the advanced first‐order second moment (AFOSM) method for computational efficiency. The proposed robust optimization method has the advantages that the mean value and the variation of the performance function are controlled simultaneously and rationally and the second‐order sensitivity information is not required even in case of gradient‐based optimization process. The suggested method is examined by solving three examples and the results are compared with those for the deterministic case and those available in the literature. Copyright © 2002 John Wiley & Sons, Ltd.     

Bad-scenario-set robust scheduling for a job shop to hedge against processing time uncertainty

This paper proposed two robust scheduling formulations in real manufacturing systems based on the concept of bad scenario set to hedge against processing time uncertainty, which is described by discrete scenarios. Two proposed robust scheduling formulations are applied to an uncertain job-shop scheduling problem with the makespan as the performance criterion. The united-scenario neighbourhood (UN) structure is constructed based on bad scenario set for the scenario job-shop scheduling problem. A tabu search (TS) algorithm with the UN structure is developed to solve the proposed robust scheduling problem. An extensive experiment was conducted. The computational results show that the first robust scheduling formulation could be preferred to the second one for the discussed problem. It is also verified that the obtained robust solutions could hedge against the processing time uncertainty through decreasing the number of bad scenarios and the degree of performance degradation on bad scenarios. Moreover, the computational results demonstrate that the developed TS algorithm is competitive for the proposed robust scheduling formulations.     

A computational approach for robust nondestructive test design maximizing characterization capabilities for solids and structures subject to uncertainty

A robust approach to nondestructive test (NDT) design for material characterization and damage identification in solids and structures is presented and numerically evaluated. The generally applicable approach combines maximization of test sensitivity with minimization of test information redundancy, while simultaneously minimizing the effects of uncertain system parameters to determine optimal NDT parameters for robust nondestructive evaluation. In addition, to maintain reasonable computational expense while also allowing for general applicability, a stochastic collocation technique is presented for the quantification of uncertainty in the robust design metrics. The robust NDT design approach was tested through simulated case studies based on the characterization of localized variations in Young's modulus distributions in aluminum structural components utilizing steady‐state dynamic surface excitation and localized measurements of displacement and compared with a purely deterministic NDT design approach. The robust design approach is shown to produce substantially different NDT designs than the analogous deterministic strategy. More importantly, the robust NDT designs are shown to provide significant improvements in the ability to accurately nondestructively evaluate structural properties for the cases considered, when there is significant uncertainty in system parameters and/or aspects of the NDT implementation. Copyright © 2015 John Wiley & Sons, Ltd.     

A regional genetic algorithm for the discrete optimal design of truss structures

A regional genetic algorithm (R‐GA) is used for the discrete optimal design of truss structures. The chromosomes are selected from a sub‐region centred on the continuous optimum. This approach replaces genetic rebirth as previously proposed by other authors, thereby significantly reducing computational costs. As a pure discrete method, the R‐GA method does not require heuristic arguments or approximations. This makes the algorithm highly effective when buckling and slenderness constraints with scatter in the data are introduced. A large set of numerical test examples is used to illustrate the capabilities of the method. The algorithm is shown to be effective and robust, making it suitable for the optimal design of very large truss structures. Copyright © 1999 John Wiley & Sons, Ltd.     

A robust design method using variable transformation and Gauss–Hermite integration

Robust design seeks an optimal solution where the design objective is insensitive to the variations of input variables while the design feasibility under the variations is maintained. Accurate robustness assessment for both design objective and feasibility usually requires an intensive computational effort. In this paper, an accurate robustness assessment method with a moderate computational effort is proposed. The numerical Gauss–Hermite integration technique is employed to calculate the mean and standard deviation of the objective and constraint functions. To effectively use the Gauss–Hermite integration technique, a transformation from a general random variable into a normal variable is performed. The Gauss–Hermite integration and the transformation result in concise formulas and produce an accurate approximation to the mean and standard deviation. This approach is then incorporated into the framework of robust design optimization. The design of a two‐bar truss and an automobile torque arm is used to demonstrate the effectiveness of the proposed method. The results are compared with the commonly used Taylor expansion method and Monte Carlo simulation in terms of accuracy and efficiency. Copyright © 2005 John Wiley & Sons, Ltd.     

Hybrid surrogate-model-based multi-fidelity efficient global optimization applied to helicopter blade design

A multi-fidelity optimization technique by an efficient global optimization process using a hybrid surrogate model is investigated for solving real-world design problems. The model constructs the local deviation using the kriging method and the global model using a radial basis function. The expected improvement is computed to decide additional samples that can improve the model. The approach was first investigated by solving mathematical test problems. The results were compared with optimization results from an ordinary kriging method and a co-kriging method, and the proposed method produced the best solution. The proposed method was also applied to aerodynamic design optimization of helicopter blades to obtain the maximum blade efficiency. The optimal shape obtained by the proposed method achieved performance almost equivalent to that obtained using the high-fidelity, evaluation-based single-fidelity optimization. Comparing all three methods, the proposed method required the lowest total number of high-fidelity evaluation runs to obtain a converged solution.     

Wind turbine design through evolutionary algorithms based on surrogate CFD methods

An automatic design environment using evolutionary techniques has been developed for the design of Horizontal Axis Wind Turbine (HAWT) blades taking into account both the aerodynamics and the structural constraints of the blades. This design environment is based on a simple, fast, and robust aerodynamic simulator intended for the prediction of the performance of any turbine blade produced as an intermediate individual of the evolutionary search process. In order to reduce the computational costs, the results obtained by this simple simulator have been corrected through the application of Artificial Neural Network (ANN) based approximations. An additional stress analysis stage has been introduced in order to take into account the structural resistance of the blades as a constraint throughout the optimization process. Results of the simulations obtained using this technique, of the application of the automatic design procedure and of the performance of the wind turbines thus produced are presented. In a first test case the rotor designed through the proposed method is compared to the one obtained analytically for a given air speed. This comparison to an analytical optimum provides a good index in order to evaluate the performance of the method. In a second case the rotor is optimized for a typical wind speed distribution, which is a more realistic scenario where the cost performance of the resulting turbines needs to be evaluated.     

Mathematical modelling of a robust inspection process plan: Taguchi and Monte Carlo methods

This study develops a new optimisation framework for process inspection planning of a manufacturing system with multiple quality characteristics, in which the proposed framework is based on a mixed-integer mathematical programming (MILP) model. Due to the stochastic nature of production processes and since their production processes are sensitive to manufacturing variations; a proportion of products do not conform the design specifications. A common source of these variations is maladjustment of each operation that leads to a higher number of scraps. Therefore, uncertainty in maladjustment is taken into account in this study. A twofold decision is made on the subject that which quality characteristic needs what kind of inspection, and the time this inspection should be performed. To cope with the introduced uncertainty, two robust optimisation methods are developed based on Taguchi and Monte Carlo methods. Furthermore, a genetic algorithm is applied to the problem to obtain near-optimal solutions. To validate the proposed model and solution approach, several numerical experiments are done on a real industrial case. Finally, the conclusion is provided.     

Dynamic and fatigue compressor blade characteristics during fluid-structure interaction: Part I—Blade modelling and vibration analysis

Fatigue failure in blades resulting from alternating loads is a primary cause of aircraft engine failure. In this study, the dynamic and fatigue performance of a compressor blade, which is prone to fatigue failure in practical use, is verified through numerical analysis and tests to identify specific failure causes and ultimately prevent blade failure. Two companion papers describe this work. Part I—blade modelling and vibration analysis establishes a compressor blade model based on reverse engineering and investigates the blade's dynamic characteristics during aerodynamic and structural behaviour interactions under various flight conditions. To accurately describe the dynamic performance of a real blade, three blade models with different fitting precisions are constructed to support dynamic response analysis. In addition, the flow field of each model is constructed to simulate the aerodynamic loads that cause blade vibration. A numerically based analysis of the fluid-structure interaction is subsequently performed. A reasonable calculation model is selected by comparing the dynamic characteristics among these models. The Campbell diagram is analysed to determine resonance probability, and blade failure cause is ultimately verified based on the vibration analysis. The results show that different accuracies in the blade calculation model produce varying degrees of error; the calculation model's ability to reflect the dynamic and fatigue behaviours of a real blade is recommended to be used as a critical evaluation index in the numerical study. The first-order harmonic resonance occurring at critical speeds may contribute to blade failure.     

Robust concurrent topology optimization of multiscale structure under single or multiple uncertain load cases

Concurrent topology optimization of macrostructure and material microstructure has attracted significant interest in recent years. However, most of the existing works assumed deterministic load conditions, thus the obtained design might have poor performance in practice when uncertainties exist. Therefore, it is necessary to take uncertainty into account in structural design. This article proposes an efficient method for robust concurrent topology optimization of multiscale structure under single or multiple load cases. The weighted sum of the mean and standard deviation of the structural compliance is minimized and constraints are imposed to both the volume fractions of macrostructure and microstructure. The effective properties of the microstructure are calculated via the homogenization method. An efficient sensitivity analysis method is proposed based on the superposition principle and orthogonal similarity transformation of real symmetric matrices. To further reduce the computational cost, an efficient decoupled sensitivity analysis method for microscale design variables is proposed. The bidirectional evolutionary structural optimization method is employed to obtain black and white designs for both macrostructure and microstructure. Several two-dimensional and three-dimensional numerical examples are presented to demonstrate the effectiveness of the proposed approach and the effects of load uncertainty on the optimal design of both macrostructure and microstructure.     

ROBUST DESIGN FOR IMPROVED VEHICLE HANDLING UNDER A RANGE OF MANEUVER CONDITIONS

In this article, a robust design procedure is applied to achieve improved vehicle handling performance as an integral part of simulation-based vehicle design. Recent developments in the field of robust design optimization and the techniques for creating global approximations of design behaviors are applied to improve the computational efficiency of robust vehicle design built upon sophisticated vehicle dynamic simulations. The approach is applied to the design of a M916A1 6-wheel tractor/M870A2 3-axle semi-trailer. The results illustrate that the proposed procedure is effective for preventing the rollover of ground vehicles as well as for identifying a design that is not only optimal against the worst maneuver condition but is also robust with respect to a range of maneuver inputs. Furthermore, a comparison is made between a statistical approach and a bi-level optimization approach in terms of their effectiveness in solving robust design problems     

基于MLS-SVM的结构整体可靠度与全局灵敏度分析

全局灵敏度分析,旨在考量结构系统中各输入随机变量对输出响应不确定性及风险水平影响的重要度。它能为后续的可靠度评估、故障诊断、系统设计、预测及优化等提供重要参考。尽管各类全局灵敏度分析方法不断涌现,但高维复杂结构(如风机叶片结构)灵敏度分析仍是目前的难题。该文针对空间分割全局灵敏度分析方法的三种可能计算形式及最优分割方案展开研究,通过标准算例分析和误差理论推导,提出能充分利用样本信息、有效减轻计算负担的求解形式及分割方案,并将其应用于风机叶片极限载荷工况下的全局灵敏度分析中,同时为未来设计更为高效、经济和可靠的风机结构提供参考。