A new surrogate modeling method combining polynomial chaos expansion and Gaussian kernel in a sparse Bayesian learning framework

Surrogate modeling techniques have been increasingly developed for optimization and uncertainty quantification problems in many engineering fields. The development of surrogates requires modeling high-dimensional and nonsmooth functions with limited information. To this end, the hybrid surrogate modeling method, where different surrogate models are combined, offers an effective solution. In this paper, a new hybrid modeling technique is proposed by combining polynomial chaos expansion and kernel function in a sparse Bayesian learning framework. The proposed hybrid model possesses both the global characteristic advantage of polynomial chaos expansion and the local characteristic advantage of the Gaussian kernel. The parameterized priors are utilized to encourage the sparsity of the model. Moreover, an optimization algorithm aiming at maximizing Bayesian evidence is proposed for parameter optimization. To assess the performance of the proposed method, a detailed comparison is made with the well-established PC-Kriging technique. The results show that the proposed method is superior in terms of accuracy and robustness.     

Polynomial chaos‐based extended Padé expansion in structural dynamics

The response of a random dynamical system is totally characterized by its probability density function (pdf). However, determining a pdf by a direct approach requires a high numerical cost; similarly, surrogate models such as direct polynomial chaos expansions are not generally efficient, especially around the eigenfrequencies of the dynamical system. In the present study, a new approach based on Padé approximants to obtain moments and pdf of the dynamic response in the frequency domain is proposed. A key difference between the direct polynomial chaos representation and the Padé representation is that the Padé approach has polynomials in both numerator and denominator. For frequency response functions, the denominator plays a vital role as it contains the information related to resonance frequencies, which are uncertain. A Galerkin approach in conjunction with polynomial chaos is proposed for the Padé approximation. Another physics‐based approach, utilizing polynomial chaos expansions of the random eigenmodes, is proposed and compared with the proposed Padé approach. It is shown that both methods give accurate results even if a very low degree of the polynomial expansion is used. The methods are demonstrated for two degree‐of‐freedom system with one and two uncertain parameters. Copyright © 2016 John Wiley & Sons, Ltd.     

Efficient uncertainty propagation for network multiphysics systems

We consider a multiphysics system with multiple component PDE models coupled together through network coupling interfaces, that is, a handful of scalars. If each component model contains uncertainties represented by a set of parameters, a straightforward uncertainty quantification study would collect all uncertainties into a single set and treat the multiphysics model as a black box. Such an approach ignores the rich structure of the multiphysics system, and the combined space of uncertainties can have a large dimension that prohibits the use of polynomial surrogate models. We propose an intrusive methodology that exploits the structure of the network coupled multiphysics system to efficiently construct a polynomial surrogate of the model output as a function of uncertain inputs. Using a nonlinear elimination strategy, we treat the solution as a composite function: the model outputs are functions of the coupling terms, which are functions of the uncertain parameters. The composite structure allows us to construct and employ a reduced polynomial basis that depends on the coupling terms. The basis can be constructed with many fewer PDE solves than the naive approach, which results in substantial computational savings. We demonstrate the method on an idealized model of a nuclear reactor. Copyright © 2014 John Wiley & Sons, Ltd.     

Multi-objective robust design optimization of fatigue life for a welded box girder

To reduce the scatter of fatigue life for welded structures, a robust optimization method is presented in this study based on a dual surrogate modelling and multi-objective particle swam optimization algorithm. Considering the perturbations of material parameters and environment variables, the mean and standard deviation of fatigue life are fitted using dual surrogate modelling and selected as the objective function to be minimized. As an example, a welded box girder is presented to reduce the standard deviation of fatigue life. A set of non-dominated solutions is produced through a multi-objective particle swam optimization algorithm. A cognitive approach is used to select the optimum solution from the Pareto sets. As a comparative study, traditional single objective optimizations are also presented in this study. The results reduced the standard deviation of the fatigue life by about 16.5%, which indicated that the procedure improved the robustness of the fatigue life.     

基于代理模型的复合材料加筋壁板分层损伤定量监测方法

通过建立损伤指数与分层损伤参数间的定量关系,对复合材料加筋壁板分层损伤进行准确定量监测,提出了一种基于代理模型的分层损伤定量监测方法。该方法包括代理模型的建立和逆求解两个过程,通过插值拟合方法建立表征分层损伤参数与损伤指数间定量关系的多项式代理模型;采用损伤概率成像算法获得相对距离和损伤指数,将其代入代理模型中,得到所对应的分层损伤面积;实现对分层损伤进行定量评估的目的。通过一个复合材料加筋壁板分层损伤的定量监测实验,对所提方法的有效性进行了验证。结果表明:基于代理模型的复合材料分层损伤定量监测方法可以实现对分层损伤位置的准确定位,定位误差低于6%;且可实现对分层面积的准确定量评估,定量误差不超过5%。      

Sequential Designs Based on Bayesian Uncertainty Quantification in Sparse Representation Surrogate Modeling

A numerical method, called overcomplete basis surrogate method (OBSM), was recently proposed, which employs overcomplete basis functions to achieve sparse representations. While the method can handle nonstationary response without the need of inverting large covariance matrices, it lacks the capability to quantify uncertainty in predictions. We address this issue by proposing a Bayesian approach that first imposes a normal prior on the large space of linear coefficients, then applies the Markov chain Monte Carlo (MCMC) algorithm to generate posterior samples for predictions. From these samples, Bayesian credible intervals can then be obtained to assess prediction uncertainty. A key application for the proposed method is the efficient construction of sequential designs. Several sequential design procedures with different infill criteria are proposed based on the generated posterior samples. Numerical studies show that the proposed schemes are capable of solving problems of positive point identification, optimization, and surrogate fitting.     

Error modeling for surrogates of dynamical systems using machine learning

A machine learning–based framework for modeling the error introduced by surrogate models of parameterized dynamical systems is proposed. The framework entails the use of high‐dimensional regression techniques (eg, random forests, and LASSO) to map a large set of inexpensively computed “error indicators” (ie, features) produced by the surrogate model at a given time instance to a prediction of the surrogate‐model error in a quantity of interest (QoI). This eliminates the need for the user to hand‐select a small number of informative features. The methodology requires a training set of parameter instances at which the time‐dependent surrogate‐model error is computed by simulating both the high‐fidelity and surrogate models. Using these training data, the method first determines regression‐model locality (via classification or clustering) and subsequently constructs a “local” regression model to predict the time‐instantaneous error within each identified region of feature space. We consider 2 uses for the resulting error model: (1) as a correction to the surrogate‐model QoI prediction at each time instance and (2) as a way to statistically model arbitrary functions of the time‐dependent surrogate‐model error (eg, time‐integrated errors). We apply the proposed framework to model errors in reduced‐order models of nonlinear oil‐water subsurface flow simulations, with time‐varying well‐control (bottom‐hole pressure) parameters. The reduced‐order models used in this work entail application of trajectory piecewise linearization in conjunction with proper orthogonal decomposition. When the first use of the method is considered, numerical experiments demonstrate consistent improvement in accuracy in the time‐instantaneous QoI prediction relative to the original surrogate model, across a large number of test cases. When the second use is considered, results show that the proposed method provides accurate statistical predictions of the time‐ and well‐averaged errors.     

Locally weighted regression models for surrogate-assisted design optimization

We consider engineering design optimization problems where the objective and/or constraint functions are evaluated by means of computationally expensive blackboxes. Our practical optimization strategy consists of solving surrogate optimization problems in the search step of the mesh adaptive direct search algorithm. In this paper, we consider locally weighted regression models to build the necessary surrogates, and present three ideas for appropriate and effective use of locally weighted scatterplot smoothing (LOWESS) models for surrogate optimization. First, a method is proposed to reduce the computational cost of LOWESS models. Second, a local scaling coefficient is introduced to adapt LOWESS models to the density of neighboring points while retaining smoothness. Finally, an appropriate order error metric is used to select the optimal shape coefficient of the LOWESS model. Our surrogate-assisted optimization approach utilizes LOWESS models to both generate and rank promising candidates found in the search and poll steps. The “real” blackbox functions that govern the original optimization problem are then evaluated at these ranked candidates with an opportunistic strategy, reducing CPU time significantly. Computational results are reported for four engineering design problems with up to six variables and six constraints. The results demonstrate the effectiveness of the LOWESS models as well as the order error metric for surrogate optimization.     

Data-driven prediction of the probability of creep–fatigue crack initiation in 316H stainless steel

Stainless steel components in advanced gas-cooled reactors (AGRs) are susceptible to creep–fatigue cracking at high temperatures. Quantifying the probability of creep–fatigue crack initiation requires probabilistic numerical simulations; these are complex and computationally intensive. Here, we present a data-driven approach to develop fast probabilistic surrogate models of creep–fatigue crack initiation in 316H stainless steel. We perform a set of Monte Carlo simulations based on the R5V2/3 high temperature assessment procedure and determine the sensitivity of the probability of crack initiation to loads and operating conditions. The data are used to train different supervised machine learning models considering Bayesian hyperparameter optimization. We discuss the relative performance of such models and show that a gradient tree boosting algorithm results in surrogate models with the highest accuracy.     

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.     

Hybrid approach for remaining useful life prediction of ball bearings

Remaining useful life (RUL) prediction plays an important role in predictive maintenance systems to support decision‐makers for arranging maintenance tasks and related resources. We propose a hybrid approach that is combined an exponential weighted moving average (EWMA) control chart for anomaly detection and machine learning models such as support vector regression (SVR) and random forest regression (RFR) with differential evolution (DE) algorithm to predict the RULs of ball bearings. Here, DE algorithm is used to find the optimal hyperparameters of SVR model. The datasets of ball bearings from the Prognostics Data Repository of NASA are used to compare the prediction performance of different methods. The degradation behavior of training data from the anomaly time to the end of life is used to transfer learning for the testing data in the SVR and RFR models. The results indicate that the proposed methods outperform the other four existing methods in terms of score. Therefore, the proposed hybrid approach is a reliable tool for the RUL prediction of ball bearings.     

Customer demand prediction of service-oriented manufacturing using the least square support vector machine optimized by particle swarm optimization algorithm

Many nonlinear customer satisfaction-related factors significantly influence the future customer demand for service-oriented manufacturing (SOM). To address this issue and enhance the prediction accuracy, this article develops a novel customer demand prediction approach for SOM. The approach combines the phase space reconstruction (PSR) technique with the optimized least square support vector machine (LSSVM). First, the prediction sample space is reconstructed by the PSR to enrich the time-series dynamics of the limited data sample. Then, the generalization and learning ability of the LSSVM are improved by the hybrid polynomial and radial basis function kernel. Finally, the key parameters of the LSSVM are optimized by the particle swarm optimization algorithm. In a real case study, the customer demand prediction of an air conditioner compressor is implemented. Furthermore, the effectiveness and validity of the proposed approach are demonstrated by comparison with other classical predication approaches.     

基于相空间重构与高斯过程的卫星钟差预报

提出了一种基于相空间重构与高斯过程预报卫星钟差的新方法。首先根据星载原子钟的物理特性用多项式进行拟合以提取钟差趋势项,并对拟合后的残差进行经验模态分解,作降噪处理;然后以降噪后的残差时间序列的混沌特性为基础,对其进行相空间重构;最后以重构的相空间为基础,运用高斯过程对残差时间序列进行建模预报,再将预报结果加上趋势项,获得最终的钟差预报值。采用IGS提供的GPS超快速观测钟差建模进行短期预报实验,结果表明,该方法能实时有效地对卫星钟差进行预报,且精度优于超快速预报钟差。     

Efficient global optimization for high-dimensional constrained problems by using the Kriging models combined with the partial least squares method

In many engineering optimization problems, the number of function evaluations is often very limited because of the computational cost to run one high-fidelity numerical simulation. Using a classic optimization algorithm, such as a derivative-based algorithm or an evolutionary algorithm, directly on a computational model is not suitable in this case. A common approach to addressing this challenge is to use black-box surrogate modelling techniques. The most popular surrogate-based optimization algorithm is the efficient global optimization (EGO) algorithm, which is an iterative sampling algorithm that adds one (or many) point(s) per iteration. This algorithm is often based on an infill sampling criterion, called expected improvement, which represents a trade-off between promising and uncertain areas. Many studies have shown the efficiency of EGO, particularly when the number of input variables is relatively low. However, its performance on high-dimensional problems is still poor since the Kriging models used are time-consuming to build. To deal with this issue, this article introduces a surrogate-based optimization method that is suited to high-dimensional problems. The method first uses the ‘locating the regional extreme’ criterion, which incorporates minimizing the surrogate model while also maximizing the expected improvement criterion. Then, it replaces the Kriging models by the KPLS(+K) models (Kriging combined with the partial least squares method), which are more suitable for high-dimensional problems. Finally, the proposed approach is validated by a comparison with alternative methods existing in the literature on some analytical functions and on 12-dimensional and 50-dimensional instances of the benchmark automotive problem ‘MOPTA08’.     

一种基于自适应集成学习代理模型的结构可靠性分析方法

结构可靠性分析需要精确计算结构或系统的失效概率,当结构失效概率低时,运算量大且操作困难。可采用代理模型替代原始性能函数,结合自适应实验设计,在保证准确率的同时大幅减少原始模型的总运行次数。该文提出了基于自适应集成学习代理模型的结构可靠性分析方法,将适应性较广的Kriging与最近发展的PC-Kriging代理模型集成;利用代理模型提供预测点的方差特征,提出新的集成学习函数,识别高预测误差区域,实现高效拟合失效边界;通过主动学习算法在预测误差大和接近极限状态的区域添加采样,迭代更新集成代理模型。通过3个算例,验证了该文方法与单一代理模型结构可靠性分析方法的优势,与AK-MCS+U和AK-MCS+EFF相比,所提方法计算成本低、准确度高。     

基于响应面法的结构动力学模型修正

为了获得精确的结构动力学模型,提出了响应面和优化相结合的方法。利用参数化模型和优化拉丁方试验设计获取样本点构造多项式响应面模型,最小二乘法确定多项式系数并检验响应面的拟合精度。用响应面计算结果与实验结果的误差构造目标函数,自适应模拟退火算法来优化修正响应面参数,将修正后的参数值带入有限元模型得到修正模型。以欧洲航空科技组织的基准模型GARTEUR飞机模型为算例,对比修正前后模态频率,结果表明修正后的模型在测试频段和预测频段具有良好的复现和预测能力,进而验证了基于响应面法与优化方法相结合的结构动力学有限元模型修正的有效性。     

Seismic fragility analysis using stochastic polynomial chaos expansions

Within the performance-based earthquake engineering (PBEE) framework, the fragility model plays a pivotal role. Such a model represents the probability that the engineering demand parameter (EDP) exceeds a certain safety threshold given a set of selected intensity measures (IMs) that characterize the earthquake load. The-state-of-the art methods for fragility computation rely on full non-linear time–history analyses. Within this perimeter, there are two main approaches: the first relies on the selection and scaling of recorded ground motions; the second, based on random vibration theory, characterizes the seismic input with a parametric stochastic ground motion model (SGMM). The latter case has the great advantage that the problem of seismic risk analysis is framed as a forward uncertainty quantification problem. However, running classical full-scale Monte Carlo simulations is intractable because of the prohibitive computational cost of typical finite element models. Therefore, it is of great interest to define fragility models that link an EDP of interest with the SGMM parameters — which are regarded as IMs in this context. The computation of such fragility models is a challenge on its own and, despite a few recent studies, there is still an important research gap in this domain. This comes with no surprise as classical surrogate modeling techniques cannot be applied due to the stochastic nature of SGMM. This study tackles this computational challenge by using stochastic polynomial chaos expansions to represent the statistical dependence of EDP on IMs. More precisely, this surrogate model estimates the full conditional probability distribution of EDP conditioned on IMs. We compare the proposed approach with some state-of-the-art methods in two case studies. The numerical results show that the new method prevails over its competitors in estimating both the conditional distribution and the fragility functions.     

Materials characterization by instrumented indentation using two different approaches

Experimental applications of two different approaches for interpretation of instrumented indentation experiments are proposed in this paper. The first approach is the application of methodology developed for spherical indentation based on models. The second approach is the application of inverse algorithm based on the minimization between experiments and simulated data. It was shown that inverse approach lead to a good prediction of elastic modulus for metals with a face-centred cubic crystal structure. However, for metals with a body-centred cubic crystal structure identified elastic modulus was 10% under values obtained by ultra-sound method. It was then shown that models developed for spherical indentation give more accurate results than inverse approach. Moreover, a sensitivity study of mesh density in inverse identification shows that the more the mesh density the more accurate the results.     

Sparse polynomial chaos expansions of frequency response functions using stochastic frequency transformation

Frequency response functions (FRFs) are important for assessing the behavior of stochastic linear dynamic systems. For large systems, their evaluations are time-consuming even for a single simulation. In such cases, uncertainty quantification by crude Monte-Carlo simulation is not feasible. In this paper, we propose the use of sparse adaptive polynomial chaos expansions (PCE) as a surrogate of the full model. To overcome known limitations of PCE when applied to FRF simulation, we propose a frequency transformation strategy that maximizes the similarity between FRFs prior to the calculation of the PCE surrogate. This strategy results in lower-order PCEs for each frequency. Principal component analysis is then employed to reduce the number of random outputs. The proposed approach is applied to two case studies: a simple 2-DOF system and a 6-DOF system with 16 random inputs. The accuracy assessment of the results indicates that the proposed approach can predict single FRFs accurately. Besides, it is shown that the first two moments of the FRFs obtained by the PCE converge to the reference results faster than with the Monte-Carlo (MC) methods.     

考虑参数不确定性的列车-桥梁垂向耦合振动的PC-ARMAX代理模型研究

车-桥耦合系统不可避免的受到系统参数不确定性的影响,为了研究车-桥耦合系统参数随机性的影响,提出了可考虑动态时变系统参数不确定性的PC-ARMAX (Polynomial Chaos expansions and AutoRegressive Moving-Average with eXogenous inputs) 模型。该模型采用ARMAX模型建立了不同系统参数条件下的代理模型,针对不同系统参数条件下代理模型的参数进行混沌多项式展开。在不考虑随机轨道不平顺影响的条件下,分析了车体质量、二系刚度和阻尼等参数随机性对车-桥响应的影响。研究了轨道不平顺随机性和参数不确定性共同作用的影响。结果表明:该模型的预测结果和蒙特卡洛模拟(MCS)的结果吻合,最大误差仅为2%,计算效率较MCS提高了2个~3个数量级;车体质量参数随机对车辆响应的影响最大,系统参数随机性的影响在车-桥耦合振动分析中是不可忽略,且同时考虑参数不确定性和激励随机性的影响是必要的。