A new hybrid reliability-based design optimization method under random and interval uncertainties

This article proposes a new method for hybrid reliability-based design optimization under random and interval uncertainties (HRBDO-RI). In this method, Monte Carlo simulation (MCS) is employed to estimate the upper bound of failure probability, and stochastic sensitivity analysis (SSA) is extended to calculate the sensitivity information of failure probability in HRBDO-RI. Due to a large number of samples involved in MCS and SSA, Kriging metamodels are constructed to substitute true constraints. To avoid unnecessary computational cost on Kriging metamodel construction, a new screening criterion based on the coefficient of variation of failure probability is developed to judge active constraints in HRBDO-RI. Then a projection-outline-based active learning Kriging is achieved by sequentially select update points around the projection outlines on the limit-state surfaces of active constraints. Furthermore, the prediction uncertainty of Kriging metamodel is quantified and considered in the termination of Kriging update. Several examples, including a piezoelectric energy harvester design, are presented to test the accuracy and efficiency of the proposed method for HRBDO-RI.     

Linearly constrained minimum-'normalised variance' beamforming against heavy-tailed impulsive noise of unknown statistics

A new beamforming approach to combat the arbitrary unknown heavy-tailed impulsive noises including all a-stable noises with infinite variance or infinite mean is presented. The new approach, termed as linearly constrained minimum-`normalized variance? beamformer (LCMNV), is formulated as one to minimise the normalised variance of the beamformer?s output, subject to a pre-specified set of linear constraints. The normalised variance is defined as a pseudo-correlation function of the instantaneously adaptive, infinity-norm snapshot-normalised data, as an alternative to the customary `fractional lower-order moments? (FLOM) for heavy-tailed impulsive noise environments. The proposed beamformer is in essence second-order statistics based, and produces an instantaneously scaled beamformer output. The LCMNV beamformer outperforms the FLOM beamformer with the following advantages: (i) computationally simpler with a closed-form solution, (ii) requiring no prior information or estimation of the effective characteristic exponents of the impulsive noises, (iii) applicable to a wider class of heavy-tailed impulsive noises and (iv) offering better interferencerejection ability.     

Active learning polynomial chaos expansion for reliability analysis by maximizing expected indicator function prediction error

Assessing the failure probability of complex aeronautical structure is a difficult task in presence of uncertainties. In this paper, active learning polynomial chaos expansion (PCE) is developed for reliability analysis. The proposed method firstly assigns a Gaussian Process (GP) prior to the model response, and the covariance function of this GP is defined by the inner product of PCE basis function. Then, we show that a PCE model can be derived by the posterior mean of the GP, and the posterior variance is obtained to measure the local prediction error as Kriging model. Also, the expectation of the prediction variance is derived to measure the overall accuracy of the obtained PCE model. Then, a learning function, named expected indicator function prediction error (EIFPE), is proposed to update the design of experiment of PCE model for reliability analysis. This learning function is developed under the framework of the variance-bias decomposition. It selects new points sequentially by maximizing the EIFPE that considers both the variance and bias information, and it provides a dynamic balance between global exploration and local exploitation. Finally, several test functions and engineering applications are investigated, and the results are compared with the widely used Kriging model combined with U and expected feasibility function learning function. Results show that the proposed method is efficient and accurate for complex engineering applications.     

基于Stochastic Kriging模型的不确定性序贯试验设计方法

不确定性研究中需要计算大量重复样本,这无疑对计算量较大的数值模拟提出了巨大的挑战.通过试验设计方法可以有效地减少不确定性研究中的计算量,然而,目前考虑不确定性的试验设计方法研究大多仍专注于传统试验设计方法.针对这一问题,为了通过更为合理的计算资源分配得到更精准的不确定性评估,基于有限样本的Stochastic Kriging模型提出了针对不确定性问题的三阶段序贯试验设计方法.首先,通过特定位置的采样对IMSE进行简化,构建了预选步进信息选取策略,通过预选增量样本总个数以及各取样位置处的分布信息,达到随机代理模型目标精度要求;同时,基于IMSE构建了基于步进信息的单轮选点试验设计准则,以同时考虑设计变量的取样位置及其分布信息.由算例与传统方法的对比分析可知,所建立方法通过等量的采样得到了精度更高的随机代理模型,验证了其在不确定性问题中的可行性和优势.     

Non-parametric estimation of conditional moments for sensitivity analysis

In this paper, we consider the non-parametric estimation of conditional moments, which is useful for applications in global sensitivity analysis (GSA) and in the more general emulation framework. The estimation is based on the state-dependent parameter (SDP) estimation approach and allows for the estimation of conditional moments of order larger than unity. This allows one to identify a wider spectrum of parameter sensitivities with respect to the variance-based main effects, like shifts in the variance, skewness or kurtosis of the model output, so adding valuable information for the analyst, at a small computational cost.     

A parallel adaptive sampling strategy to accelerate the sampling process during the modeling of a Kriging metamodel

ABSTRACT

Higher modeling efficiency is an important goal for the modeling of a Kriging (KG) metamodel, and the sampling approach affects the modeling efficiency directly. Considering the effect of the employed correlation model on prediction accuracy of a KG model, a multiple KG models based parallel adaptive sampling strategy (MKPAS) is proposed using the combination forecasting method, in which the added new points in the sampling process are determined using multiple KG models with different correlation models. The effectiveness of the proposed approach is verified by two low dimensional benchmark functions as well as a high dimensional one. And an engineering application is also used to demonstrate the effectiveness of the proposed MKPAS approach. The results show that the proposed approach can improve the modeling efficiency of a KG model significantly compared with other ordinary sampling approaches.     

Stochastic Polynomial Interpolation for Uncertainty Quantification With Computer Experiments

Multivariate polynomials are increasingly being used to construct emulators of computer models for uncertainty quantification. For deterministic computer codes, interpolating polynomial metamodels should be used instead of noninterpolating ones for logical consistency and prediction accuracy. However, available methods for constructing interpolating polynomials only provide point predictions. There is no known method that can provide probabilistic statements about the interpolation error. Furthermore, there are few alternatives to grid designs and sparse grids for constructing multivariate interpolating polynomials. A significant disadvantage of these designs is the large gaps between allowable design sizes. This article proposes a stochastic interpolating polynomial (SIP) that seeks to overcome the problems discussed above. A Bayesian approach in which interpolation uncertainty is quantified probabilistically through the posterior distribution of the output is employed. This allows assessment of the effect of interpolation uncertainty on estimation of quantities of interest based on the metamodel. A class of transformed space-filling design and a sequential design approach are proposed to efficiently construct the SIP with any desired number of runs. Simulations demonstrate that the SIP can outperform Gaussian process (GP) emulators. This article has supplementary material online.     

Bayesian Metamodeling for Computer Experiments Using the Gaussian Kriging Models

In the past two decades, more and more quality and reliability activities have been moving into the design of product and process. The design and analysis of computer experiments, as a new frontier of the design of experiments, has become increasingly popular among modern companies for optimizing product and process conditions and producing high‐quality yet low‐cost products and processes. This article mainly focuses on the issue of constructing cheap metamodels as alternatives to the expensive computer simulators and proposes a new metamodeling method on the basis of the Gaussian stochastic process model or Gaussian Kriging. Rather than a constant mean as in ordinary Kriging or a fixed mean function as in universal Kriging, the new method captures the overall trend of the performance characteristics of products and processes through a more accurate mean, by efficiently incorporating a scheme of sparseness prior–based Bayesian inference into Kriging. Meanwhile, the mean model is able to adaptively exclude the unimportant effects that deteriorate the prediction performance. The results of an experiment on empirical applications demonstrate that, compared with several benchmark methods in the literature, the proposed Bayesian method is not only much more effective in approximation but also very efficient in implementation, hence more appropriate than the widely used ordinary Kriging to empirical applications in the real world. Copyright © 2011 John Wiley & Sons, Ltd.     

A frequency domain metamodeling approach to transient sensitivity analysis

This paper develops a metamodel approach to approximate the transient relationship between a univariate output response and one or more continuous-valued input factors in a computer simulation. The approach is based on a variant of Frequency Domain Methodology (FDM) in which a metamodel is hypothesized in the time domain; the analysis and parameter estimation are performed in the frequency domain; and finally prediction and inference are made back in the time domain. The switching of domains for analysis and estimation is advantageous because it permits the simultaneous consideration of multiple input factor changes. The metamodel is then used to estimate the mean transient function of the output response after a discontinuous change has been made to each individual input factor. The methodology is illustrated on an M/M/1 queue and a three-station tandem queue.     

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.     

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.     

Integration of computation and testing for reliability estimation

This paper develops a methodology to integrate reliability testing and computational reliability analysis for product development. The presence of information uncertainty such as statistical uncertainty and modeling error is incorporated. The integration of testing and computation leads to a more cost-efficient estimation of failure probability and life distribution than the tests-only approach currently followed by the industry. A Bayesian procedure is proposed to quantify the modeling uncertainty using random parameters, including the uncertainty in mechanical and statistical model selection and the uncertainty in distribution parameters. An adaptive method is developed to determine the number of tests needed to achieve a desired confidence level in the reliability estimates, by combining prior computational prediction and test data. Two kinds of tests — failure probability estimation and life estimation — are considered. The prior distribution and confidence interval of failure probability in both cases are estimated using computational reliability methods, and are updated using the results of tests performed during the product development phase.     

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.     

Metamodel-based inverse method for parameter identification: elastic–plastic damage model

This article proposed a metamodel-based inverse method for material parameter identification and applies it to elastic–plastic damage model parameter identification. An elastic–plastic damage model is presented and implemented in numerical simulation. The metamodel-based inverse method is proposed in order to overcome the disadvantage in computational cost of the inverse method. In the metamodel-based inverse method, a Kriging metamodel is constructed based on the experimental design in order to model the relationship between material parameters and the objective function values in the inverse problem, and then the optimization procedure is executed by the use of a metamodel. The applications of the presented material model and proposed parameter identification method in the standard A 2017-T4 tensile test prove that the presented elastic–plastic damage model is adequate to describe the material's mechanical behaviour and that the proposed metamodel-based inverse method not only enhances the efficiency of parameter identification but also gives reliable results.     

基于序贯层次Kriging模型的微型飞行器机身结构设计优化

在基于仿真模型的工程设计优化中,采用高精度、高成本的分析模型会导致计算量大,采用低精度、低成本的分析模型会导致设计优化结果的可信度低,难以满足实际工程的要求。为了有效平衡高精度与低成本之间的矛盾关系,通过建立序贯层次Kriging模型融合高/低精度数据,采用大量低成本、低精度的样本点反映高精度分析模型的变化趋势,并采用少量高成本、高精度的样本点对低精度分析模型进行校正,以实现对优化目标的高精度预测。为了避免层次Kriging模型误差对优化结果的影响,将层次Kriging模型与遗传算法相结合,根据6σ设计准则计算每一代最优解的预测区间,具有较大预测区间的当前最优解即为新的高精度样本点。同时,在优化过程中序贯更新层次Kriging模型,提高最优解附近的层次Kriging模型的预测精度,从而保证设计结果的可靠性。将所提出的方法应用于微型飞行器机身结构的设计优化中,以验证该方法的有效性和优越性。采用具有不同单元数的网格模型分别作为低精度分析模型和高精度分析模型,利用最优拉丁超立方设计分别选取60个低精度样本点和20个高精度样本点建立初始层次Kriging模型,采用本文方法求解并与直接采用高精度仿真模型求解的结果进行比较。结果表明,所提出的方法能够有效利用高/低精度样本点处的信息,建立高精度的层次Kriging模型;本文方法仅需要少量的计算成本就能求得近似最优解,有效提高了设计效率,为类似的结构设计优化问题提供了参考。     

Expected improvement based infill sampling for global robust optimization of constrained problems

A novel adaptive sampling scheme for efficient global robust optimization of constrained problems is proposed. The method addresses expensive to simulate black-box constrained problems affected by uncertainties for which only the bounds are known, while the probability distribution is not available. An iterative strategy for global robust optimization that adaptively samples the Kriging metamodel of the computationally expensive problem is proposed. The presented approach is tested on several benchmark problems and the average performance based on 100 runs is evaluated. The applicability of the method to engineering problems is also illustrated by applying robust optimization on an integrated photonic device affected by manufacturing uncertainties. The numerical results show consistent convergence to the global robust optimum using a limited number of expensive simulations.     

Evolutionary reliable regional Kriging surrogate for expensive optimization

This work proposes a sequential optimization algorithm, EORKS, combining a Kriging surrogate from an adaptive sampling and an iterative constrained search in the dynamic reliable regions to reduce the sampling size in expensive optimization. A surrogate established from small samples is liable to limited generality, which leads to a false prediction of optimum. EORKS applies Kriging variance to establish the reliable region neighbouring the learning samples to constrain the evolutionary searches of the surrogate. The verified quasi-optimum is used as an additional sample to dynamically update the regional model according to the prediction accuracy. A hybrid infilling strategy switches between the iterative quasi-optima and the maximum expected improvement from Kriging to prevent early convergence of local optimum. EORKS provides superior optima in several benchmark functions and an engineering design problem, using much smaller samples compared with the literature results, which demonstrates the sampling efficiency and searching robustness.     

A weighted three-point-based methodology for variance estimation

It is widely accepted that variations in manufacturing processes are inevitable and should be taken into account during analysis and design processes. However, estimating uncertainty propagation in an end-product caused by these variations is a very challenging task, especially when a computationally expensive effort is already needed in deterministic models, such as simulations of sheet metal forming. The focus of this article is on the variance estimation of a system response using sensitivity-based methods. A weighted three-point-based strategy for efficiently and effectively estimating the variance of a system response is proposed. Three first-order derivatives of each variable are used to describe the non-linear behaviour and estimate the variance of a system. A methodology for determining the optimal locations and weights of the three points along each axis is proposed and illustrated for the cases where each variable follows either a normal distribution or a uniform distribution. An extension of the weighted three-point-based strategy is introduced to take into account the interaction between parameters. In addition, an extension is given for mean estimation of the system response without requiring more data. The considerable improvement in accuracy compared with the traditional first-order approximation is demonstrated in a number of test problems. The proposed method requires significantly less computational effort than the Monte Carlo method.     

An efficient approach for statistical moments estimation of structural response based on a novel adaptive hybrid dimension-reduction method

Statistical moments estimation is one of the main topics for the analysis of a stochastic system, but the balance among the accuracy, efficiency, and versatility for different methods of statistical moments estimation still remains a challenge. In this paper, a novel point estimate method (PEM) based on a new adaptive hybrid dimension-reduction method (AH-DRM) is proposed. Firstly, the adaptive cut-high-dimensional model representation (cut-HDMR) is briefly reviewed, and a novel AH-DRM is developed, where the high-order component functions of the adaptive cut-HDMR are further approximated by multiplicative forms of the low-order component functions. Secondly, a new point estimation method (PEM) based on the AH-DRM is proposed for statistical moments estimation. Finally, several examples are investigated to demonstrate the performance of the proposed PEM. The results show the proposed PEM has fairly high accuracy and good versatility for statistical moments estimation.     

Controlling first four moments for robust optimization

The paper addresses the solution of robust moment-based optimization problems after a multipoint reformulation. The first four moments are considered (i.e. mean, variance, skewness and kurtosis) going beyond classical engineering optimization based on the control of the mean and variance . In particular, the impact on the design of a control of the third and fourth moments are discussed. The multipoint formulation leads to discrete expressions for the moments. linking moment-based and multipoint optimizations. The linearity of the sums in the discrete moments permits an easy evaluation of their gradients with respect to the design variables. Optimal sampling issues are analyzed and a procedure is proposed to quantify the confidence level on the robustness of the design. The proposed formulation is fully parallel and the time-to-solution is comparable to single-point situations. It is applied to three problems: an analytical least-square minimization problem, a shape optimization problem with a reduced-order model, and a full aircraft shape optimization robust over a range of transverse winds.