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.     

Confidence structural robust optimization by non‐linear semidefinite programming‐based single‐level formulation

Structural robust optimization problems are often solved via the so‐called Bi‐level approach. This solution procedure often involves large computational efforts and sometimes its convergence properties are not so good because of the non‐smooth nature of the Bi‐level formulation. Another problem associated with the traditional Bi‐level approach is that the confidence of the robustness of the obtained solutions cannot be fully assured at least theoretically. In the present paper, confidence single‐level non‐linear semidefinite programming (NLSDP) formulations for structural robust optimization problems under stiffness uncertainties are proposed. This is achieved by using some tools such as S‐procedure and quadratic embedding for convex analysis. The resulted NLSDP problems are solved using the modified augmented Lagrange multiplier method which has sound mathematical properties. Numerical examples show that confidence robust optimal solutions can be obtained with the proposed approach effectively. Copyright © 2010 John Wiley & Sons, Ltd.     

Worst‐case robust design optimization under distributional assumptions

Presented in this paper is a novel robust design optimization (RDO) methodology. The problem is reformulated in order to relax, when required, the assumption of normality of objectives and constraints, which often underlies RDO. In the second place, taking into account engineering considerations concerning the risk associated with constraint violation, suitable estimates of tail conditional expectations are introduced in the set of robustness metrics. A computationally affordable yet accurate implementation of the proposed formulation is guaranteed by the adoption of a reduced quadrature technique to perform the uncertainty propagation. The methodology is successfully demonstrated with the aid of an industrial test case performing the sizing of a mid‐range passenger aircraft. Copyright © 2011 John Wiley & Sons, Ltd.     

Dynamic reliability-based robust design optimization with time-variant probabilistic constraints

With the increasing complexity of engineering systems, ensuring high system reliability and system performance robustness throughout a product life cycle is of vital importance in practical engineering design. Dynamic reliability analysis, which is generally encountered due to time-variant system random inputs, becomes a primary challenge in reliability-based robust design optimization (RBRDO). This article presents a new approach to efficiently carry out dynamic reliability analysis for RBRDO. The key idea of the proposed approach is to convert time-variant probabilistic constraints to time-invariant ones by efficiently constructing a nested extreme response surface (NERS) and then carry out dynamic reliability analysis using NERS in an iterative RBRDO process. The NERS employs an efficient global optimization technique to identify the extreme time responses that correspond to the worst case scenario of system time-variant limit state functions. With these extreme time samples, a kriging-based time prediction model is built and used to estimate extreme responses for any given arbitrary design in the design space. An adaptive response prediction and model maturation mechanism is developed to guarantee the accuracy and efficiency of the proposed NERS approach. The NERS is integrated with RBRDO with time-variant probabilistic constraints to achieve optimum designs of engineered systems with desired reliability and performance robustness. Two case studies are used to demonstrate the efficacy of the proposed approach.     

Integrated multi-objective robust optimization and sensitivity analysis with irreducible and reducible interval uncertainty

Uncertainty considered in robust optimization is usually treated as irreducible since it is not reduced during an optimization procedure. In contrast, uncertainty considered in sensitivity analysis is treated as partially or fully reducible in order to quantify the effect of input uncertainty on the outputs of the system. Considering this, and the usual existence of both reducible and irreducible uncertainty, an approach that can perform robust optimization and sensitivity analysis simultaneously is of much interest. This article presents such an integrated optimization model that can be used for both robust optimization and sensitivity analysis for problems that have irreducible and reducible interval uncertainty, multiple objective functions and mixed continuous-discrete design variables. The proposed model is demonstrated by two engineering examples with differing complexity to demonstrate its applicability.     

Robust optimization of structures subjected to stochastic earthquake with limited information on system parameter uncertainty

The optimization of structures subjected to stochastic earthquake and characterized by uncertain parameters is usually posed in the form of non-linear programming with stochastic performance measures where the uncertain parameters are modelled as random variables. Such an approach, however, cannot be adopted in many real life situations when the limited information about uncertainty can be only modelled as of the uncertain but bounded (UBB) type. A robust optimization strategy for stochastic dynamic systems characterized by UBB parameters is proposed in the present study in the framework of the response surface method (RSM). In evaluating the stochastic constraints, repeated computations of the dynamic responses are avoided by applying an adaptive RSM based on the moving least squares method. Numerical results are presented to highlight the effectiveness of the proposed procedure. The effect of parameter uncertainty is also studied by comparing the results obtained from the proposed optimization approach with the conventional stochastic optimization results.     

一种考虑概率分布的鲁棒优化模型

文章以随机规划中的机会约束思想为指导,根据随机参数的概率分布情况,提出了两种鲁棒性条件约束,并在此基础上建立了一种新的鲁棒优化模型,使模型的可行解控制在一定的鲁棒性指标的范围内。该模型不但可处理约束两端同时含有随机参数的情况,还可以方便地推广到非线性模型中。仿真实例说明了模型的有效性。     

Adaptive gradient-assisted robust design optimization under interval uncertainty

Robust optimization techniques attempt to find a solution that is both optimum and relatively insensitive to input uncertainty. In general, these techniques are computationally more expensive than their deterministic counterparts. In this article two new robust optimization methods are presented. The first method is called gradient-assisted robust optimization (GARO). In GARO, a robust optimization problem is first converted to a deterministic one by using a gradient-based approximation technique. After solving this deterministic problem, the solution robustness and the accuracy of the approximation are checked. If the accuracy meets a threshold, a robust optimum solution is found; otherwise, the approximation is adaptively modified until the threshold is met and a solution, if it exists, is obtained. The second method is a faster version of GARO called quasi-concave gradient-assisted robust optimization (QC-GARO). QC-GARO is for problems with quasi-concave objective and constraint functions. The difference between GARO and QC-GARO is in the way that they check the approximation accuracy. Both GARO and QC-GARO methods are applied to a set of six engineering design test problems and the results are compared with a few related previous methods. It was found that, compared to the methods considered, GARO could solve all test problems but with a higher computational effort compared to QC-GARO. However, QC-GARO was computationally much faster when it was able to solve the problems.     

Distributionally robust optimization for engineering design under uncertainty

This paper addresses the challenge of design optimization under uncertainty when the designer only has limited data to characterize uncertain variables. We demonstrate that the error incurred when estimating a probability distribution from limited data affects the out-of-sample performance (ie, performance under the true distribution) of optimized designs. We demonstrate how this can be mitigated by reformulating the engineering design problem as a distributionally robust optimization (DRO) problem. We present computationally efficient algorithms for solving the resulting DRO problem. The performance of the DRO approach is explored in a practical setting by applying it to an acoustic horn design problem. The DRO approach is compared against traditional approaches to optimization under uncertainty, namely, sample-average approximation and multiobjective optimization incorporating a risk reduction objective. In contrast with the multiobjective approach, the proposed DRO approach does not use an explicit risk reduction objective but rather specifies a so-called ambiguity set of possible distributions and optimizes against the worst-case distribution in this set. Our results show that the DRO designs, in some cases, significantly outperform those designs found using the sample-average or the multiobjective approach.     

A robust optimization methodology for preliminary aircraft design

This article focuses on a robust optimization of an aircraft preliminary design under operational constraints. According to engineers' know-how, the aircraft preliminary design problem can be modelled as an uncertain optimization problem whose objective (the cost or the fuel consumption) is almost affine, and whose constraints are convex. It is shown that this uncertain optimization problem can be approximated in a conservative manner by an uncertain linear optimization program, which enables the use of the techniques of robust linear programming of Ben-Tal, El Ghaoui, and Nemirovski [Robust Optimization, Princeton University Press, 2009]. This methodology is then applied to two real cases of aircraft design and numerical results are presented.     

Efficient infill sampling for unconstrained robust optimization problems

A novel infill sampling criterion is proposed for efficient estimation of the global robust optimum of expensive computer simulation based problems. The algorithm is especially geared towards addressing problems that are affected by uncertainties in design variables and problem parameters. The method is based on constructing metamodels using Kriging and adaptively sampling the response surface via a principle of expected improvement adapted for robust optimization. Several numerical examples and an engineering case study are used to demonstrate the ability of the algorithm to estimate the global robust optimum using a limited number of expensive function evaluations.     

Reliability‐based robust design optimization: A multi‐objective framework using hybrid quality loss function

In this globally competitive business environment, design engineers are constantly striving to establish new and effective tools and techniques to ensure a robust and reliable product design. Robust design (RD) and reliability‐based design approaches have shown the potential to deal with variability in the life cycle of a product. This paper explores the possibilities of combining both approaches into a single model and proposes a hybrid quality loss function‐based multi‐objective optimization model. The model is unique because it uses a hybrid form of quality loss‐based objective function that is defined in terms of desirable as well as undesirable deviations to obtain efficient design points with minimum quality loss. The proposed approach attempts to optimize the product design by addressing quality loss, variability, and life‐cycle issues simultaneously by combining both reliability‐based and RD approaches into a single model with various customer aspirations. The application of the approach is demonstrated using a leaf spring design example. Copyright © 2009 John Wiley & Sons, Ltd.     

Generalized information reuse for optimization under uncertainty with non‐sample average estimators

In optimization under uncertainty for engineering design, the behavior of the system outputs due to uncertain inputs needs to be quantified at each optimization iteration, but this can be computationally expensive. Multifidelity techniques can significantly reduce the computational cost of Monte Carlo sampling methods for quantifying the effect of uncertain inputs, but existing multifidelity techniques in this context apply only to Monte Carlo estimators that can be expressed as a sample average, such as estimators of statistical moments. Information reuse is a particular multifidelity method that treats previous optimization iterations as lower fidelity models. This work generalizes information reuse to be applicable to quantities whose estimators are not sample averages. The extension makes use of bootstrapping to estimate the error of estimators and the covariance between estimators at different fidelities. Specifically, the horsetail matching metric and quantile function are considered as quantities whose estimators are not sample averages. In an optimization under uncertainty for an acoustic horn design problem, generalized information reuse demonstrated computational savings of over 60% compared with regular Monte Carlo sampling.     

A unified stochastic framework for robust topology optimization of continuum and truss-like structures

In this article, a unified framework is introduced for robust structural topology optimization for 2D and 3D continuum and truss problems. The uncertain material parameters are modelled using a spatially correlated random field which is discretized using the Karhunen–Loève expansion. The spectral stochastic finite element method is used, with a polynomial chaos expansion to propagate uncertainties in the material characteristics to the response quantities. In continuum structures, either 2D or 3D random fields are modelled across the structural domain, while representation of the material uncertainties in linear truss elements is achieved by expanding 1D random fields along the length of the elements. Several examples demonstrate the method on both 2D and 3D continuum and truss structures, showing that this common framework provides an interesting insight into robustness versus optimality for the test problems considered.     

A kriging metamodel-assisted robust optimization method based on a reverse model

The goal of robust optimization methods is to obtain a solution that is both optimum and relatively insensitive to uncertainty factors. Most existing robust optimization approaches use outer–inner nested optimization structures where a large amount of computational effort is required because the robustness of each candidate solution delivered from the outer level should be evaluated in the inner level. In this article, a kriging metamodel-assisted robust optimization method based on a reverse model (K-RMRO) is first proposed, in which the nested optimization structure is reduced into a single-loop optimization structure to ease the computational burden. Ignoring the interpolation uncertainties from kriging, K-RMRO may yield non-robust optima. Hence, an improved kriging-assisted robust optimization method based on a reverse model (IK-RMRO) is presented to take the interpolation uncertainty of kriging metamodel into consideration. In IK-RMRO, an objective switching criterion is introduced to determine whether the inner level robust optimization or the kriging metamodel replacement should be used to evaluate the robustness of design alternatives. The proposed criterion is developed according to whether or not the robust status of the individual can be changed because of the interpolation uncertainties from the kriging metamodel. Numerical and engineering cases are used to demonstrate the applicability and efficiency of the proposed approach.     

An enhanced fuzzy robust optimization model for regional solid waste management under uncertainty

In this study, an enhanced fuzzy robust optimization (EFRO) model is proposed for supporting regional solid waste management under uncertainty. This model is an extended version of robust optimization from a stochastic to a fuzzy environment, and novel in the following two aspects: (1) it uses multiple algorithms to tackle fuzzy constraints according to their characteristics; and (2) it incorporates fuzzy violation variables into the model, which could effectively reflect the trade-off between system economy and reliability. The regional waste management of the City of Dalian, China, was used as a case study for demonstration. A variety of solutions was obtained under various weight coefficients and confidence levels. From the case study, it was found that EFRO could help decision makers to design desired waste management alternatives under complex uncertainties. The successful application of EFRO in the studied real case is expected to be a good example for solid waste management in many other cities.     

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.     

Multi-objective robust optimization design of a front-end underframe structure for a high-speed train

A multi-objective robust design optimization of a front-end underframe structure for application in high-speed trains is proposed and the structural parameter uncertainty is considered. A finite element model of the structure is developed and verified by dynamic impact experiments. The sensitivity analysis demonstrates that the thicknesses of the centre sill have significant influences on structural crushing behaviours. The specific energy absorption and the initial peak crushing force (F_p) are taken as optimization objectives. Compared with the baseline structure, the 6-sigma robust design shows that the F_p and the structural mass are reduced by 54.86% and 13.06%, respectively, and the robust optimum is more reliable. The 6-sigma robust optimal solution has an efficient energy-absorbing capacity while satisfying the design constraint. Thus, 6-sigma robust optimization can be applied for high-speed trains.     

Level-set topology optimization for robust design of structures under hybrid uncertainties

This paper will develop a new robust topology optimization (RTO) method based on level sets for structures subject to hybrid uncertainties, with a more efficient Karhunen-Loève hyperbolic Polynomial Chaos–Chebyshev Interval method to conduct the hybrid uncertain analysis. The loadings and material properties are considered hybrid uncertainties in structures. The parameters with sufficient information are regarded as random fields, while the parameters without sufficient information are treated as intervals. The Karhunen-Loève expansion is applied to discretize random fields into a finite number of random variables, and then, the original hybrid uncertainty analysis is transformed into a new process with random and interval parameters, to which the hyperbolic Polynomial Chaos–Chebyshev Interval is employed for the uncertainty analysis. RTO is formulated to minimize a weighted sum of the mean and standard variance of the structural objective function under the worst-case scenario. Several numerical examples are employed to demonstrate the effectiveness of the proposed RTO, and Monte Carlo simulation is used to validate the numerical accuracy of our proposed method.