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随机样本数据的概率表征方法

高等数学研究 2024年第1期27卷 51-57页

作者：宋述芳王家辉吕震宙员婉莹西北工业大学航空学院陕西西安710072

本文通过算例分析了参数概率表征方法的适用性和有效性.对于非参数概率密度估计,介绍了几种拟合变量概率密度函数的方法,通过算例对比了不同方法的拟合效果.

关键词：随机试验样本概率表征参数估计假设检验概率密度函数

Robust Design Optimization and Improvement by Metamodel

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Computer Modeling in Engineering & Sciences 2020年第10期125卷 383-399页

作者： shufang song Lu Wang Yuhua Yan School of Aeronautics Northwestern Polytechnical UniversityXi’an710072China

The robust design optimization(RDO)is an effective method to improve product performance with uncertainty *** robust optimal solution should be not only satisfied the probabilistic constraints but also less sensitive to the variation of design *** are some important issues in RDO,such as how to judge robustness,deal with multi-objective problem and black-box *** this paper,two criteria are proposed to judge the deterministic optimal solution whether satisfies robustness *** robustness measure based on maximum entropy is *** sum method is improved to deal with the objective function,and the basic framework of metamodel assisted robust optimization is also provided for improving the ***,several engineering examples are used to verify the advantages.

关键词： Robust design optimization(RDO) metamodel maximum entropy robustness measure global sensitivity analysis

Sparse grid-based polynomial chaos expansion for aerodynamics of an airfoil with uncertainties

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Chinese Journal of Aeronautics 2018年第5期31卷 997-1011页

作者： Xiaojing WU Weiwei ZHANG shufang song Zhengyin YE School of Aeronautics Northwestern Polytechnical University

The uncertainties can generate fluctuations with aerodynamic characteristics. Uncertainty Quantification（UQ） is applied to compute its impact on the aerodynamic *** addition, the contribution of each uncertainty to aerodynamic characteristics should be computed by uncertainty sensitivity analysis. Non-Intrusive Polynomial Chaos（NIPC） has been successfully applied to uncertainty quantification and uncertainty sensitivity analysis. However, the non-intrusive polynomial chaos method becomes inefficient as the number of random variables adopted to describe uncertainties increases. This deficiency becomes significant in stochastic aerodynamic analysis considering the geometric uncertainty because the description of geometric uncertainty generally needs many parameters. To solve the deficiency, a Sparse Grid-based Polynomial Chaos（SGPC） expansion is used to do uncertainty quantification and sensitivity analysis for stochastic aerodynamic analysis considering geometric and operational uncertainties. It is proved that the method is more efficient than non-intrusive polynomial chaos and Monte Carlo Simulation（MSC） method for the stochastic aerodynamic analysis. By uncertainty quantification, it can be learnt that the flow characteristics of shock wave and boundary layer separation are sensitive to the geometric uncertainty in transonic region. The uncertainty sensitivity analysis reveals the individual and coupled effects among the uncertainty parameters.

关键词： Non-intrusive polynomial chaos sparse grid Stochastic aerodynamic analysis analysis Unceriainty quantification

Reliability Sensitivity Algorithm Based on Stratified Importance Sampling Method for Multiple Failure Modes Systems

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Chinese Journal of Aeronautics 2010年第6期23卷 660-669页

作者： Zhang Feng Lu Zhenzhou Cui Lijie song shufang School of Mechanics and Civil and Architecture Northwestern Polytechnical University Xi 'an 710072 China School of Aeronautics Northwestern Polyteehnical University. Xi 'an 710072 China

Combining the advantages of the stratified sampling and the importance sampling, a stratified importance sampling method （SISM） is presented to analyze the reliability sensitivity for structure with multiple failure modes. In the presented method, the variable space is divided into several disjoint subspace by n-dimensional coordinate planes at the mean point of the random vec- tor, and the importance sampling functions in the subspaces are constructed by keeping the sampling center at the mean point and augmenting the standard deviation by a factor of 2. The sample size generated from the importance sampling function in each subspace is determined by the contribution of the subspace to the reliability sensitivity, which can be estimated by iterative simulation in the sampling process. The formulae of the reliability sensitivity estimation, the variance and the coefficient of variation are derived for the presented SISM. Comparing with the Monte Carlo method, the stratified sampling method and the importance sampling method, the presented SISM has wider applicability and higher calculation efficiency, which is demonstrated by numerical examples. Finally, the reliability sensitivity analysis of flap structure is illustrated that the SISM can be applied to engineering structure.

关键词： multiple failure modes reliability sensitivity Monte Carlo simulation stratified sampling method importance sam-piing method stratified importance sampling method (SISM)

Reliability and Sensitivity Analysis of Transonic Flutter Using Improved Line Sampling Technique

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Chinese Journal of Aeronautics 2009年第5期22卷 513-519页

作者： song shufang Lu Zhenzhou Zhang Weiwei Ye Zhengyin School of Aeronautics Northwestern Polytechnical University Xi'an 710072 China National Key Laboratory of Aerodynamic Design and Research Northwestern Polytechnical University Xi'an 710072 China

The improved line sampling （LS） technique, an effective numerical simulation method, is employed to analyze the probabilistic characteristics and reliability sensitivity of flutter with random structural parameter in transonic flow. The improved LS technique is a novel methodology for reliability and sensitivity analysis of high dimensionality and low probability problem with implicit limit state function, and it does not require any approximating surrogate of the implicit limit state equation. The improved LS is used to estimate the flutter reliability and the sensitivity of a two-dimensional wing, in which some structural properties, such as frequency, parameters of gravity center and mass ratio, are considered as random variables. Computational fluid dynamics （CFD） based unsteady aerodynamic reduced order model （ROM） method is used to construct the aerodynamic state equations. Coupling structural state equations with aerodynamic state equations, the safety margin of flutter is founded by using the critical velocity of flutter. The results show that the improved LS technique can effectively decrease the computational cost in the random uncertainty analysis of flutter. The reliability sensitivity, defined by the partial derivative of the failure probability with respect to the distribution parameter of random variable, can help to identify the important parameters and guide the structural optimization design.

关键词： flutter aeroelastic line sampling technique Monte Carlo simulation uncertainty reduced order model

Variable Importance Measure System Based on Advanced Random Forest

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Computer Modeling in Engineering & Sciences 2021年第7期128卷 65-85页

作者： shufang song Ruyang He Zhaoyin Shi Weiya Zhang School of Aeronautics Northwestern Polytechnical UniversityXi’an710072China AECC Sichuan Gas Turbine Establishment Mianyang621700China

The variable importance measure(VIM)can be implemented to rank or select important variables,which can effectively reduce the variable dimension and shorten the computational *** forest(RF)is an ensemble learning method by constructing multiple decision *** order to improve the prediction accuracy of random forest,advanced random forest is presented by using Kriging models as the models of leaf nodes in all the decision *** to the Mean Decrease Accuracy(MDA)index based on Out-of-Bag(OOB)data,the single variable,group variables and correlated variables importance measures are proposed to establish a complete VIM system on the basis of advanced random *** link of MDA and variance-based sensitivity total index is explored,and then the corresponding relationship of proposed VIM indices and variance-based global sensitivity indices are constructed,which gives a novel way to solve variance-based global ***,several numerical and engineering examples are given to verify the effectiveness of proposed VIM system and the validity of the established relationship.

关键词： Variable importance measure random forest variance-based global sensitivity Kriging model

Machine learning for adjoint vector in aerodynamic shape optimization

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Acta Mechanica Sinica 2021年第9期37卷 1416-1432,I0003页

作者： Mengfei Xu shufang song Xuxiang Sun Wengang Chen Weiwei Zhang Northwestern Polytechnical University Xi’an710072China

Adjoint method is widely used in aerodynamic design because only once solution of flow field is required for it to obtain the gradients of all design variables. However, the computational cost of adjoint vector is approximately equal to that of flow computation. In order to accelerate the solution of adjoint vector and improve the efficiency of adjoint-based optimization, machine learning for adjoint vector modeling is presented. Deep neural network (DNN) is employed to construct the mapping between the adjoint vector and the local flow variables. DNN can efficiently predict adjoint vector and its generalization is examined by a transonic drag reduction of NACA0012 airfoil. The results indicate that with negligible computational cost of the adjoint vector, the proposed DNN-based adjoint method can achieve the same optimization results as the traditional adjoint method.

关键词： Machine learning Deep neural network Adjoint vector modelling Aerodynamic shape optimization Adjoint method

The CDF and its sensitivity analysis of stochastic structure with stochastic excitation by advanced stratified line sampling

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Science China(Physics,Mechanics & Astronomy) 2013年第8期56卷 1559-1567页

作者： song shufang LU ZhenZhou ZHANG WeiWeiNational Key Laboratory of Aerodynamic Design and Research Northwestern Polytechnical University YE ZhengYin School of Aeronautics Northwestern Polytechnical University National Key Laboratory of Aerodynamic Design and Research Northwestern Polytechnical University

For the stochastic structure with stochastic excitation, an advanced stratified line sampling (SLS) method is presented to obtain the cumulative distribution function (CDF) of the structural response and its sensitivity. The advanced stratified line sampling method introduces a set of middle failure subsets firstly. And for each subset, the conventional line sampling can be used to obtain the corresponding value of the response's CDF. At the same time, the sensitivity estimations of each failure subset can also be computed by modifying the important direction and corresponding reliability coefficients. The properties of CDF sensitivity are proved while the performance function is linear with normal random variables. After two simple examples are used to demonstrate the properties of CDF sensitivity and the feasibility of the presented method, the method employed to analyze the CDF and corresponding sensitivity of root bending moment (RBM) responses for the stochastic BAH is wing with gust excitation to a square-edged gust and to a Dryden gust. The results show that the parameters of the second and the fifth order modals exert more influence on the CDF of response than the other ones, and the presented SLS method can more significantly reduce the computational cost compared with Monte Carlo simulation (MCS).

关键词：随机激励随机结构 CDF 敏感性分析灵敏度估计采样累积分布函数随机变量函数

Saddlepoint approximation based line sampling method for uncertainty propagation in fuzzy and random reliability analysis

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Science China(Technological Sciences) 2010年第8期53卷 2252-2260页

作者： LI LuYi,LU ZhenZhou &song shufang School of Aeronautics,Northwestern Polytechnical University,Xi’an 710072,China School of Aeronautics Northwestern Polytechnical University Xi’an China

For structural system with random basic variables as well as fuzzy basic variables,uncertain propagation from two kinds of basic variables to the response of the structure is investigated.A novel algorithm for obtaining membership function of fuzzy reliability is presented with saddlepoint approximation(SA)based line sampling *** the presented method,the value domain of the fuzzy basic variables under the given membership level is firstly obtained according to their membership *** the value domain of the fuzzy basic variables corresponding to the given membership level,bounds of reliability of the structure response satisfying safety requirement are obtained by employing the SA based line sampling method in the reduced space of the random *** this way the uncertainty of the basic variables is propagated to the safety measurement of the structure,and the fuzzy membership function of the reliability is *** to the direct Monte Carlo method for propagating the uncertainties of the fuzzy and random basic variables,the presented method can considerably improve computational efficiency with acceptable *** presented method has wider applicability compared to the transformation method,because it doesn't limit the distribution of the variable and the explicit expression of performance function, and no approximation is made for the performance function during the computing ***,the presented method can easily treat the performance function with cross items of the fuzzy variable and the random variable,which isn't suitably approximated by the existing transformation *** examples are provided to illustrate the advantages of the presented method.

关键词： membership function fuzzy reliability fuzzy variable random variable saddlepoint approximation based line sampling method