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.     

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.     

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.     

A multiple-design-point approach for reliability-based design optimization

The application of design-point-based reliability-based design optimization (RBDO) methods is hindered by the challenge of multiple-design-point problems. In this article, to improve the commonality of design-point-based RBDO methods, a novel multiple-design-point (MDP) approach is developed. The MDP approach uses the trace of the design points from consequent reliability analysis iterations to identify whether there are multiple design points, then all of the design points are used to calculate shifting vectors for the sequential optimization and reliability assessment method, and the corresponding probabilistic constraints are moved to the feasible region along these multiple shifting vectors at the same time. With multiple shifted probabilistic constraints, the design feasibility associated with this probabilistic constraint will be satisfied. Two mathematical examples, a speed reducer design and a honeycomb crashworthiness design, are presented to validate the effectiveness of the MDP method. The results show that the MDP approach is effective for handling multiple-design-point problems.     

An efficient metamodeling approach for uncertainty quantification of complex systems with arbitrary parameter probability distributions

This paper proposes an efficient metamodeling approach for uncertainty quantification of complex system based on Gaussian process model (GPM). The proposed GPM‐based method is able to efficiently and accurately calculate the mean and variance of model outputs with uncertain parameters specified by arbitrary probability distributions. Because of the use of GPM, the closed form expressions of mean and variance can be derived by decomposing high‐dimensional integrals into one‐dimensional integrals. This paper details on how to efficiently compute the one‐dimensional integrals. When the parameters are either uniformly or normally distributed, the one‐dimensional integrals can be analytically evaluated, while when parameters do not follow normal or uniform distributions, this paper adopts the effective Gaussian quadrature technique for the fast computation of the one‐dimensional integrals. As a result, the developed GPM method is able to calculate mean and variance of model outputs in an efficient manner independent of parameter distributions. The proposed GPM method is applied to a collection of examples. And its accuracy and efficiency is compared with Monte Carlo simulation, which is used as benchmark solution. Results show that the proposed GPM method is feasible and reliable for efficient uncertainty quantification of complex systems in terms of the computational accuracy and efficiency. Copyright © 2016 John Wiley & Sons, Ltd.     

Data-driven modal surrogate model for frequency response uncertainty propagation

A method is developed for propagation of model parameter uncertainties into frequency response functions based on a modal representation of the equations of motion. Individual local surrogate models of the eigenfrequencies and residue matrix elements for each mode are trained to build a global surrogate model. The computational cost of the global surrogate model is reduced in three steps. First, modes outside the range of interest, necessary to describe the in-band frequency response, are approximated with few residual modes. Secondly, the dimension of the residue matrices for each mode is reduced using principal component analysis. Lastly, multiple surrogate model structures are employed in a mixture. Cheap second-order multivariate polynomial models and more expensive Gaussian process models with different kernels are used to model the modal data. Leave-one-out cross-validation is used for model selection of the local surrogate models. The approximations introduced allow the method to be used for modally dense models at a small computational cost, without sacrificing the global surrogate model’s ability to capture mode veering and crossing phenomena. The method is compared to a Monte Carlo based approach and verified on one industrial-sized component and on one assembly of two car components.     

A reliability-based robust optimization design for the drum brake using adaptive Kriging surrogate model

To decrease random parameters’ influence on the drum brake reliability, the reliability-based robust optimization design (RBROD) of the electric vehicle brake is proposed. Based on the assumption that the maximum temperature of the brake cannot exceed the allowable temperature, a performance function model of thermal–mechanical coupling reliability of drum brakes is established by the adaptive Kriging method, and the analysis of reliability sensitivity and RBROD are conducted. The accuracy of the proposed model is verified by temperature measurement experiment under emergency braking condition. The robust optimization design improves the drum brake reliability to 0.99998 and reduce the influence of the design parameters on the reliability, with the absolute values of the reliability sensitivity and the weight of the drum brake are significantly smaller. Therefore, the objectives of reliability design, robustness design, and optimization design are simultaneously achieved by the proposed methods. Besides, the relative error of the proposed method is 0.373%, the number of function evaluations is 39, and the comparison with four meta-model methods show that the proposed method holds high-accuracy and high-efficiency. This study provides a high-precision theoretical explanation for the robust optimization design of drum brake.     

Gradient projection for reliability-based design optimization using evidence theory

Uncertainty quantification and risk assessment in the optimal design of structural systems has always been a critical consideration for engineers. When new technologies are developed or implemented and budgets are limited for full-scale testing, the result is insufficient datasets for construction of probability distributions. Making assumptions about these probability distributions can potentially introduce more uncertainty to the system than it quantifies. Evidence theory represents a method to handle epistemic uncertainty that represents a lack of knowledge or information in the numerical optimization process. Therefore, it is a natural tool to use for uncertainty quantification and risk assessment especially in the optimization design cycle for future aerospace structures where new technologies are being applied. For evidence theory to be recognized as a useful tool, it must be efficiently applied in a robust design optimization scheme. This article demonstrates a new method for projecting the reliability gradient, based on the measures of belief and plausibility, without gathering any excess information other than what is required to determine these measures. This represents a huge saving in computational time over other methods available in the current literature. The technique developed in this article is demonstrated with three optimization examples.     

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.     

Adaptive improved response surface method for reliability-based design optimization

Reliability-based design optimization (RBDO) requires the evaluation of probabilistic constraints (or reliability), which can be very time consuming. Therefore, a practical solution for efficient reliability analysis is needed. The response surface method (RSM) and dimension reduction (DR) are two well-known approximation methods that construct the probabilistic limit state functions for reliability analysis. This article proposes a new RSM-based approximation approach, named the adaptive improved response surface method (AIRSM), which uses the moving least-squares method in conjunction with a new weight function. AIRSM is tested with two simplified designs of experiments: saturated design and central composite design. Its performance on reliability analysis is compared with DR in terms of efficiency and accuracy in multiple RBDO test problems.     

A robust optimization approach for the capacitated vehicle routing problem with demand uncertainty

In this paper we introduce a robust optimization approach to solve the Vehicle Routing Problem (VRP) with demand uncertainty. This approach yields routes that minimize transportation costs while satisfying all demands in a given bounded uncertainty set. We show that for the Miller-Tucker-Zemlin formulation of the VRP and specific uncertainty sets, solving for the robust solution is no more difficult than solving a single deterministic VRP. Our computational results on benchmark instances and on families of clustered instances show that the robust solution can protect from unmet demand while incurring a small additional cost over deterministic optimal routes. This is most pronounced for clustered instances under moderate uncertainty, where remaining vehicle capacity is used to protect against variations within each cluster at a small additional cost. We compare the robust optimization model with classic stochastic VRP models for this problem to illustrate the differences and similarities between them. We also observe that the robust solution amounts to a clever management of the remaining vehicle capacity compared to uniformly and non-uniformly distributing this slack over the vehicles.     

Enhanced modified reliability index approach for efficient and robust reliability-based design optimization

The reliability index approach (RIA) is one of the effective tools for solving the reliability-based design optimization (RBDO) probabilistic model, which models the uncertainties with probability constraints. However, its wide application in engineering is limited due to low efficiency and convergence problems. The RIA-based modified reliability index approach (MRIA) appears to be very robust and accurate than RIA but yields inefficient for the most probable point (MPP) search with highly nonlinear probabilistic constraints. In this study, an enhanced modified reliability index approach (EMRIA) is developed to improve the efficiency and robustness of searching for MPP and is utilized for RBDO. In the EMRIA, an innovative active set using rigorous inequality is applied to construct the region of exploring for MPP, where the unnecessary probabilistic constraint could be eliminated adaptively during the iterative process. Moreover, the double loop strategy (DLS) is integrated into the EMRIA to strengthen the efficiency and robustness of large-scale RBDO problems. Two numerical examples demonstrated that the EMRIA is an efficient and robust method for MPP search in comparison with current first-order reliability methods. Six RBDO problems quoted also indicate that DLS-based EMRIA has good performance to solve complex RBDO problems.     

An optimization model for a crop deficit irrigation system under uncertainty

In this study, an inexact nonlinear programming model under uncertainty is developed by incorporating a water production function into the crop irrigation system optimization framework. By introducing a time parameter, this model can address the uncertainty associated with the irrigation schedule for different crops and their planting stages. The developed model was applied to a case study of an agricultural water resources management problem to demonstrate its applicability. Through scenario analysis under different precipitation levels, the key planting stage of crops and the amount of water for the irrigation schedule that could significantly affect system benefits were identified. By using intervals to represent uncertain parameters, more reliable and practical decision alternatives were generated through the presented model in typical hydrological years (i.e. wet, normal and dry years).     

基于公理设计和相容决策支持问题法的稳健优化设计方法

 针对多目标稳健优化问题,建立了多目标稳健优化的损失函数,利用灵敏度分析方法确定各设计变量对各设计目标的影响程度,确定主要的设计参数,便于调整和控制设计参数的公差.根据信息公理与损失函数的一致性关系,建立以最小化各目标的总损失函数为目标函数.并在相容决策支持问题法框架基础上,提出一种基于公理设计和相容决策支持问题法的多目标稳健优化设计模型.实例分析表明,提出的方法是可行的.     

Cutting the double loop: Theory and algorithms for reliability-based design optimization with parametric uncertainty

Parametric uncertainties complicate engineering design—confounding regulated design approaches and degrading the performance of reliability efforts. The simplest means to tackle this uncertainty is double-loop simulation , a nested Monte Carlo method that, for practical problems, is intractable. In this work, we introduce a flexible, general approximation technique that obviates the double loop. This approximation is constructed in the context of a novel theory of reliability design under parametric uncertainty: we introduce metrics for measuring the efficacy of reliability-based design optimization strategies ( epistemic design gap and effective reliability ), minimal conditions for controlling uncertain reliability ( precision margin ), and stricter conditions that guarantee the desired reliability at a designed confidence level. We provide a number of examples with open-source code to demonstrate our approaches in a reproducible fashion.     

An interval approach to robust design with parameter uncertainty

In robust design, it is common to estimate empirical models that relate an output response variable to controllable input variables and uncontrollable noise variables from experimental data. However, when determining the optimal input settings that minimise output variability, parameter uncertainties in noise factors and response models are typically neglected. This article presents an interval robust design approach that takes parameter uncertainties into account through the confidence regions for these unknown parameters. To avoid obtaining an overly conservative design, the worst and best cases of mean squared error are both adopted to build an optimisation approach. The midpoint and radius of the interval are used to measure the location and dispersion performances, respectively. Meanwhile, a data-driven method is applied to obtain the relative weights of the location and dispersion performances in the optimisation approach. A simulation example and a case study using automobile manufacturing data from the dimensional tolerance design process are used to demonstrate the effectiveness of the proposed approach. The proposed approach of considering both uncertainties is shown to perform better than other approaches.     

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.     

Enhanced similarity-based metamodel updating strategy for reliability-based design optimization

Metamodels are becoming increasingly popular for handling large-scale optimization problems in product development. Metamodel-based reliability-based design optimization (RBDO) helps to improve the computational efficiency and reliability of optimal design. However, a metamodel in engineering applications is an approximation of a high-fidelity computer-aided engineering model and it frequently suffers from a significant loss of predictive accuracy. This issue must be appropriately addressed before the metamodels are ready to be applied in RBDO. In this article, an enhanced strategy with metamodel selection and bias correction is proposed to improve the predictive capability of metamodels. A similarity-based assessment for metamodel selection (SAMS) is derived from the cross-validation and similarity theories. The selected metamodel is then improved by Bayesian inference-based bias correction. The proposed strategy is illustrated through an analytical example and further demonstrated with a lightweight vehicle design problem. The results show its potential in handling real-world engineering problems.     

Simulated variance optimization for robust design

A method to aid robust design in the presence of design parameter uncertainty is described. For a given relationship between a performance measure (or output parameter) and the uncertain design parameters a probabilistic simulation is used to obtain the variance of the performance measure as a function of the nominal design parameter values. The optimum values of the latter are then obtained as those corresponding to a minimum of the computed variance, determined by means of a particular non-linear optimization algorithm in the presence of constraints. The latter are in the form of limits on the nominal values of the design parameters and a specified value for the performance measure at the nominal design point, i.e. the deterministic design target. Some problems inherent in this type of procedure are discussed and methods of solution are described. A specific example is studied and the results from the present method are compared with those previously obtained by use of another procedure. © 1998 John Wiley & Sons, Ltd.