偏最小二乘回归在响应面法可靠度分析中的应用

针对目前多维变量可靠度问题中广泛应用的均匀设计响应面法,分析了采用最小二乘法拟合样本数据回归模型时存在的局限性,并在已有方法的基础上提出了一种改进的方法。该方法将均匀设计与偏最小二乘回归技术相结合来回归响应面模型,从而计算结构的失效概率,有效的解决了变量间多重相关性及小样本条件下建立回归模型的问题。通过算例验证了该方法的适用性,尤其对于高维变量的可靠度问题,与最小二乘拟合响应面相比,计算结果更加精确。     

An efficient response surface method using moving least squares approximation for structural reliability analysis

The response surface method (RSM) is widely adopted for structural reliability analysis because of its numerical efficiency. However, the RSM is time consuming for large-scale applications and sometimes shows large errors in the calculation of the sensitivity of the reliability index with respect to random variables. In order to overcome these problems, this study proposes an efficient RSM applying a moving least squares (MLS) approximation instead of the traditional least squares approximation generally used in the RSM. The MLS approximation gives higher weight to the experimental points closer to the most probable failure point (MPFP), which allows the response surface function (RSF) to be closer to the limit state function at the MPFP. In the proposed method, a linear RSF is constructed at first and a quadratic RSF is formed using the axial experimental points selected from the reduced region where the MPFP is likely to exist. The RSF is updated successively by adding one new experimental point to the previous set of experimental points. Numerical examples are presented to demonstrate the improved accuracy and computational efficiency of the proposed method compared to the conventional RSM.     

Adaptive response surface method based on a double weighted regression technique

In structural reliability analysis where the structural response is computed from the finite element method, the response surface method is frequently used. Typically, the response surface is built from polynomials whereof unknown coefficients are estimated from an implicit limit state function numerically defined at fitting points. The locations of these points must be selected in a judicious way to reduce the computational time without deteriorating the quality of the polynomial approximation. To contribute to the development of this method, we propose some improvements. The response surface is successively formed in a cumulative manner. An adaptive construction of the numerical design is proposed. The response surface is fitted by the weighted regression technique, which allows the fitting points to be weighted according to (i) their distance from the true failure surface and (ii) their distance from the estimated design point. This method aims to minimize computational time while producing satisfactory results. The efficiency and the accuracy of the proposed method can be evaluated from examples taken from the literature.     

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.     

结构可靠度分析中的改进响应面法及其应用

在以响应面法分析结构可靠度中,提出了一种区别于通常以插值点为中心展开生成样本点组的新方法:在求解过程中,用插值点逐步替代初始样本点组中距离验算点较远的点,目的是使所选取的样本点集中于真实极限状态曲面上的验算点附近,重新构成下一轮迭代所需的一组样本点,直至满足收敛条件。算例表明,采用新方法可使结构的分析次数显著减少,并改善了对非线性程度较高的极限功能函数求解可靠指标的收敛性。同时,将该方法应用于一座拟建自锚式斜拉-悬吊体系桥正常使用极限状态下的可靠度分析中。     

Cumulative formation of response surface and its use in reliability analysis

An improved response surface method is proposed and applied to the reliability analysis of a stiffened plated structure. The response surface function is formed in a cumulative manner in order to properly account for the second order effects in the response surface with acceptable computational effort involved in the evaluation of the state function. First, a linear response surface is formed by searching for the design point by the first order reliability method. The vector projection technique is used to allocate the sampling points close to the response surface. Then, the linear response surface is improved by adding square terms, and the second order reliability method is employed to search for the design point. All the available sampling points except those generated in the very initial stage are used to obtain a well-conditioned system matrix for regression. Lastly, the obtained response surface is checked for the selected sampling points. If the response surface function is not satisfactory, it is adaptively improved by adding cross terms and removing some of the second order terms. In this way an appropriate incomplete second order response surface can be obtained and used for reliability analysis. Examples are given to demonstrate the features of the proposed method.     

Using simulation techniques to determine optimal operational region for multi-responses problems

Most industrial problems involve multiple-quality characteristics. The methods used for optimising such a multiple-response problem are those normally used for dealing with a single response problem, such as the experimental design, response surface methods (RSM) and regression techniques. There is a major difficulty with this traditional approach on the point estimation of the optimal solution, thus the obtained solution may not be practically feasible. In this research, a simulated regression approach via Monte Carlo simulations and the nonlinear optimisation via the trust-region methods were used to determine an optimal operational region (OOR). In practice, an engineer can refer to this region for adjusting manufacturing parameters in the production environment to minimise the response variations. An example with three controllable variables is used to demonstrate the application of the proposed methodology.     

利用矩点估计法简化响应面可靠度指标的计算

针对响应面可靠度指标计算方法的缺陷,将罗森布鲁斯统计矩点估计法引入到经典响应面方法中对其进行改进。改进后响应面方法,考虑基本随机变量偏态情况,改变了模拟试验中随机变量抽样值的计算方法;直接采用了统计矩计算可靠度指标,使可靠度指标的计算在真实极限状态曲面的某些特殊点上进行;在计算过程中不需拟合近似极限状态曲面和对非线性方程进行线性化处理,不产生迭代误差和累积误差,计算过程简洁明了。最后分别利用改进的响应面方法和经典响应面方法分析了某一矿山大型工程的稳定可靠性,并以蒙特卡洛法计算结果作为准精确解进行了比较,计算结果满足工程要求。     

ASRSM: A Sequential Experimental Design for Response Surface Optimization

Most preset response surface methodology (RSM) designs offer ease of implementation and good performance over a wide range of process and design optimization applications. These designs often lack the ability to adapt the design on the basis of the characteristics of application and experimental space so as to reduce the number of experiments necessary. Hence, they are not cost‐effective for applications where the cost of experimentation is high or when the experimentation resources are limited. In this paper, we present an adaptive sequential response surface methodology (ASRSM) for industrial experiments with high experimentation cost, limited experimental resources, and high design optimization performance requirement. The proposed approach is a sequential adaptive experimentation approach that combines concepts from nonlinear optimization, design of experiments, and response surface optimization. The ASRSM uses the information gained from the previous experiments to design the subsequent experiment by simultaneously reducing the region of interest and identifying factor combinations for new experiments. Its major advantage is the experimentation efficiency such that for a given response target, it identifies the input factor combination (or containing region) in less number of experiments than the classical single‐shot RSM designs. Through extensive simulated experiments and real‐world case studies, we show that the proposed ASRSM method outperforms the popular central composite design method and compares favorably with optimal designs. Copyright © 2012 John Wiley & Sons, Ltd.     

基于支持向量机的小样本响应曲面法研究

当影响因素和响应输出的关系较为复杂时,应用传统响应曲面法(RSM)、非参数响应曲面法(NPRSM)和人工神经网络(ANN)难以拟合真实的响应曲面,不仅需要大的样本量,而且泛化风险大,不易达到全局最优.将RSM归结为可有限制地主动获取样本的小样本机器学习问题,提出了一种基于支持向量机(SVM)的RSM.以大间隔网格取样,利用SVM拟合过程,对拟合方程寻优确定极值大致区域,再逐步缩小间隔求精.算例研究表明,该方法的拟合与泛化性能优于NPRSM和基于ANN的RSM,能在小样本条件下建立全局性数值模型,寻优可以得到多个极值.     

An Efficient Adaptive Sequential Methodology for Expensive Response Surface Optimization

The preset response surface methodology (RSM) designs are commonly used in a wide range of process and design optimization applications. Although they offer ease of implementation and good performance, they are not sufficiently adaptive to reduce the required number of experiments and thus are not cost effective for applications with high cost of experimentation. We propose an efficient adaptive sequential methodology based on optimal design and experiments ranking for response surface optimization (O‐ASRSM) for industrial experiments with high experimentation cost, limited experimental resources, and requiring high design optimization performance. The proposed approach combines the concepts from optimal design of experiments, nonlinear optimization, and RSM. By using the information gained from the previous experiments, O‐ASRSM designs the subsequent experiment by simultaneously reducing the region of interest and by identifying factor combinations for new experiments. Given a given response target, O‐ASRSM identifies the input factor combination in less number of experiments than the classical single‐shot RSM designs. We conducted extensive simulated experiments involving quadratic and nonlinear response functions. The results show that the O‐ASRSM method outperforms the popular central composite design, the Box–Behnken design, and the optimal designs and is competitive with other sequential response surface methods in the literature. Furthermore, results indicate that O‐ASRSM's performance is robust with respect to the increasing number of factors. Copyright © 2012 John Wiley & Sons, Ltd.     

非线性隐式极限状态方程失效概率计算的组合响应面法

提出组合响应面的新方法,用以计算设计点附近非线性程度较大的隐式极限状态方程的失效概率。该方法用主响应面和多个次响应面近似对失效概率贡献较大的区域,其响应面函数形式为不含交叉的二次多项式。主响应面依据传统响应面法通过选择适当的插值点和迭代运算获得,其设计点为主设计点。延坐标轴正负方向偏移主设计点得到拟均值点。以拟均值点为基础得到一组次响应面和次设计点。通过主次响应面在各自设计点处的切平面建立组合响应面近似原隐式极限状态方程,并计算其失效概率。算例结果说明所提方法具有较高精度。     

引入不确定参数的汽车盘式制动器振动稳定性分析

针对汽车制动器的噪声抑制问题,基于可靠性分析理论,将蒙特卡洛法与响应面法相结合,提出了一种汽车盘式制动器系统振动稳定性的可靠性分析方法。该方法针对制动噪声产生具有不确定性的特点,引入随机和区间不确定性参数对制动器系统进行描述,建立包含随机参数和区间参数的制动器不稳定特征值的响应面近似模型,进而采用Sobol′全局灵敏度分析法和蒙特卡洛法分别对不确定参数的全局灵敏度和系统稳定性的可靠度进行分析。用该方法对某车的浮钳盘式制动器系统进行研究,分析了系统稳定性的可靠度和不确定参数的全局灵敏度,甄别了不确定性参数对系统稳定性的影响,并从可靠性角度提出了改善制动器系统振动稳定性的工程措施。     

An Integrated Approach for Enhancing the Quality of the Product by Combining Robust Design and Customer Requirements

Enhancing the quality of the product has always been one considerable concern of production process management, and this subject gave way to implementing so many methods including robust design. In this paper, robust design utilizes response surface methodology (RSM) considering the mean and variance of the response variable regarding system design, parameter design, and tolerance design. In this paper, customer requirements and robust design are regarded simultaneously to achieve enriched quality. Subsequently, with a non‐linear programming, a novel method for integrating RSM and quality function deployment has been proposed to achieve robustness in design. The customer requirements are regarded in every stage of product development process meaning system design, parameter design and tolerance design. To validate the applicability of the proposed approach, it has been implemented in a real case of chemical industry. Research findings show that the proposed method is much better than other existing methods including MSE and dual response methods. According to this method, the resulted mean is better than MSE method, and more importantly, the variance of the process is approximately 14% and 10% lesser than dual response and MSE method. Copyright © 2013 John Wiley & Sons, Ltd.     

On efficient computational schemes to calculate structural failure probabilities

Various methods to calculate structural failure probability are compared in the light of their efficiency and accuracy. The investigation is confined to procedures which provide error estimates. Hence it concentrates on approaches based on numerical integration and simulation where particular attention is focused on variance reduction techniques. Importance sampling procedures are recommended as they provide accurate results with acceptable computational efforts. It is shown that the range of simulation can be identified either by utilizing the design point or, even more effectively, adaptive sampling i.e. an Iterative Fast Monte Carlo (IFM) procedure. Finally, the response surface method (RSM) is recommended for larger structures where each calculation of the limit state function requires a complete Finite Element analysis.     

基于逆响应面法的有限元模型修正

基于响应面法（RSM）的有限元模型修正是以若干设计参数（如密度、弹性模量等）为自变量,以若干特征参数（如固有频率、振型等）为因变量,通过回归分析方法来拟合特征参数关于修正参数的显式表达式。提出的逆响应面法（IRSM）则是以特征参数作为自变量,设计参数作为因变量。利用此法的有限元模型修正可直接根据特征参数的目标值得到设计参数的修正量,而不需要经过迭代计算,有效地提高计算速度和精度。介绍逆响应面法及其应用,讨论使用响应面法和逆响应面法进行有限元模型修正的优缺点和适用范围,分析适合于逆响应面法的逆响应面函数、实验设计方案和回归精度检验的方法。利用逆响应面法对一简支梁进行有限元模型修正的结果表明,逆响应面法能高效准确地修正设计参数,对于输出变量少于输入变量的情况更能显著减少有限元计算次数,适用于复杂的工程结构。     

Robust parameter design using a supersaturated design for a response surface model

Recently, the application of response surface methodology (RSM) to robust parameter design has attracted a great deal of attention. In some cases, experiments are very expensive and may require a great deal of time to perform. Central composite designs (CCDs) and Box and Behnken designs (BBDs), which are commonly used for RSM, may lead to an unacceptably large number of experimental runs. In this paper, a supersaturated design for RSM is constructed and its application to robust parameter design is proposed. A response surface model is fitted using data from the designed experiment and a stepwise variable selection. An illustrative example is presented to show that the proposed method considerably reduces the number of experimental runs, as compared with CCDs and BBDs. Numerical experiments are also conducted in which type I and II error rates are evaluated. The results imply that the proposed method may be effective for finding the effects (i.e. main effects, two‐factor interactions, and pure quadratic effects) of active factors under the ‘effect sparsity’ assumption. Copyright © 2010 John Wiley & Sons, Ltd.     

基于响应面法和Morgenstern-Price法土坡可靠度计算方法

基于响应面法,建立了一种高效的边坡可靠度指标和失效概率近似计算方法。该法在构造响应面函数时,抽样点计算采用Morgenstern-Price法取代传统费时的有限单元法,大大降低了计算工作量。利用Monte-Carlo随机抽样原理,提出了一种能同时确定边坡最危险非圆弧滑动面和最小可靠度指标的随机搜索新算法。该文给出的两个算例验证了方法的实用性和可靠性,其计算结果同时表明:当分别以最小可靠度指标和最小中值安全系数为目标函数时,搜索到的边坡最危险滑动面相差较大。最后,探讨了土性指标(c,φ)的分布概型及相关性对边坡可靠度计算结果的影响。     

遗传算法在响应曲面拟合中的应用

响应曲面方法（RSM）是一种重要的实验设计优化方法,但是当影响因素和响应输出之间的关系异常复杂时,RSM难以拟合影响因素和响应输出之间的响应曲面.遗传算法（GAs）是一种借鉴生物界自然选择和进化发展起来的高度并行、随机、自适应搜索算法,具有强大的搜索能力.本文将遗传算法引入RSM,通过对遗传算法各参数的设计和改进,克服了传统遗传算法易局部收敛的问题.     

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.