结构可靠度分析代理模型方法研究
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摘要
在结构可靠度分析中，代理模型方法由于能够大幅度减少工程分析中繁重计算量，而得到广泛应用，主要包括传统响应面法以及最近发展起来的kriging法和人工神经网络等。这些方法的基本思想是通过建立近似模型得到变量和响应的函数关系，从而代替可靠度分析中的隐式极限状态函数。然而当面对实际工程问题，特别是多维变量的非线性算例进行可靠度分析时仍存在不足，比如精度不高，效率低下等。本文在现有理论的基础上，对结构可靠性分析代理模型方法进行了深入研究，建立了一些计算精度和效率优于传统方法的结构可靠性分析方法。主要研究工作如下：
(1)对应用统计学中的实验设计理论做了简要介绍，通过对可靠度分析中拟合响应面的常用若干实验设计方法的比较，为传统多项式响应面法提供了实验设计的选取建议。针对多维变量均匀设计的多项式响应面法，分析了由于数据间存在多重相关性，采用普通最小二乘法拟合回归模型时存在的问题，并在已有方法的基础上提出了基于多项式的结构可靠度分析拟线性偏最小二乘响应面法。该方法将均匀设计与偏最小二乘回归技术相结合来近似响应面模型，进行可靠度分析，有效的解决了数据间多重相关性及小样本条件下建立回归模型误差较大的问题。通过算例验证了该方法的适用性，尤其对于多维变量的可靠度问题，与用普通最小二乘拟合的响应面相比，计算结果更加精确。
(2)针对传统响应面法局限于多项式函数，当面对复杂程度和非线性程度较高的问题时，难以得到满意精度。本文提出基于B样条函数变换和核函数变换的非线性偏最小二乘响应面法进行可靠度分析，其主要思想是将原始输入数据通过非线性的核函数或B样条函数映射到高维特征空间，然后在高维空间内实施偏最小二乘回归建立代理模型，该方法能够充分利用样本空间信息，有效捕捉输入输出间的非线性关系，避免了假定极限状态函数形式对可靠度分析结果产生的影响，并且有效处理了高维空间内数据间存在的多重相关性。算例结果表明，提出的两种方法在精度和效率上均优于传统多项式响应面法。
(3)针对可靠度分析中广泛应用的kriging方法，在多维问题中精度和效率较低的问题，通过引入函数的梯度信息，提出了相比与kriging方法精度更高的cokriging方法，并通过算例验证其可靠度分析有效性。同时，将可靠度计算中的重要抽样方法与cokriging模拟技术相结合，提出了一种基于cokriging模拟的重要抽样方法。数值算例表明，该方法在与重要抽样法计算精度相当的情况下，能够大幅度减少模拟的样本数，提高计算效率。
(4)针对神经网络代理模型在求解结构可靠度问题时存在的网络结构难以确定以及参数值选取无理论依据等缺陷，通过引入偏最小二乘技术分别与BP网络和RBF网络相结合，建立基于偏最小二乘和BP网络的混合模型以及偏最小二乘和RBF网络的混合模型，并分别提出以此为代理模型的结构可靠度分析方法。数值算例分析结果表明，本文提出的方法在计算精度和效率方面均优于传统神经网络方法。
Surrogate model methods are widely used in structural reliability to alleviatethe computational burden of engineering analysis at present. Existing methodsinclude the traditional Response Surface Method (RSM) and recently devolopedArtificial Neural Network (ANN) and kriging et al. The basic idea of these methodsis to create approximate models and provide functional relationships between theresponses and variables to replace the implicit limit state function. However, thesemethods are often confronted with kinds of difficulties such as the low accuracy andthe inefficiency for realistic engineering problems, especially in high-dimensionalsystems. Based on the existing methods, this study throughly investigates thesurrogate model methods for structural reliability analysis. Several new methodsthat have the higher accuracy and the improved efficiency when compared with thetraditional methods are proposed. The main work of this study is as follows:
(1) The experiment design methods in applied statistics are briefly introduced.Suggestions for the selection of experiment design methods in traditional responsesurface method are provided through the comparison of some experiment designmethods for response surface fitting in reliability analysis. For the response surfacemethods of multidimensional variables with the uniform design, the limitation of th eoriginal Least Squares (LS) regression induced by multidimensional correlation ofdata in model fitting is analysized. To deal with the limitation, a new approachcalled quasi-linearal Partial Least Squares (PLS) response surface method based onthe traditional polynomial has been proposed. This method combines the uniformdesign and Partial Least Squares technique to estimate the response surface andperform reliability analysis. It can effectively cope with the multidimensionalcorrelation of data and high error when building the regression model under thecondition of small samples. The results of several examples show that the proposedmethod based on the polynomial functions is suitable for structural reliabilityanalysis and has higher accuracy, especially in the high dimensional problem.
(2) Due to the restriction of polynomial functions, the traditional responsesurface method cannot attain satisfactory accuracy for multidimensional variablesand high non-linearity problems. This paper presents the partial least squaresnon-linear regression methods based on the B-spline transform and kernel transformsubstitute for the surrogate model for the calculation of reliability, and the main ideais to first map the input space into a high-dimensional feature space via a nonlinearkernel function or a B-spline function, then to apply partial least squares regressionin the feature space, which can make full use of the sample space information and effectively capture the nonlinear relationship between input variables and outputvariables compared to linear partial least square. Thus the methods can not onlyavoid the influence of assuming the type of the limit state function but also handlethe correlation among variables after space transform. Numerical examples indicatethat the accuracy and efficiency of the two proposed methods are both superior tothe traditional response surface method.
(3) To improve the accuracy and efficiency of the widely used kriging method,especially in multidimensional systems, this paper explores the use of the cokrigingmethod by incorporating the secondary information such as the gradients of thefunction. The improvement of the new method is verified by numerical examples.Then, a simulated importance sampling approach, which combines the importancesampling technique with the cokriging method, is presented for structural reliabilityevaluation. Numerical examples illustrate that the proposed method can greatlydecrease the sample number of Monte Carlo method and improve the efficiency withcomparable accuracy.
(4) To overcome the flaws of artificial neural network (ANN) such as thedifficulty to determine the network struture and the lack of theoretical basis forparameter seletion, two kinds of hybrid artificial neural networks (HANN)integrating Radial Basis Function (RBF) networks and Back Propagation (BP)networks with partial least square technique respectively are proposed. Then thecorresponding structural reliability analysis methods utilizing the hybrid model asthe surrogate model are presented. The results of the numerical examplesdemonstrate that the proposed methods are superior to the traditional artificialneural networks in terms of accuracy and efficiency.
(1)对应用统计学中的实验设计理论做了简要介绍，通过对可靠度分析中拟合响应面的常用若干实验设计方法的比较，为传统多项式响应面法提供了实验设计的选取建议。针对多维变量均匀设计的多项式响应面法，分析了由于数据间存在多重相关性，采用普通最小二乘法拟合回归模型时存在的问题，并在已有方法的基础上提出了基于多项式的结构可靠度分析拟线性偏最小二乘响应面法。该方法将均匀设计与偏最小二乘回归技术相结合来近似响应面模型，进行可靠度分析，有效的解决了数据间多重相关性及小样本条件下建立回归模型误差较大的问题。通过算例验证了该方法的适用性，尤其对于多维变量的可靠度问题，与用普通最小二乘拟合的响应面相比，计算结果更加精确。
(2)针对传统响应面法局限于多项式函数，当面对复杂程度和非线性程度较高的问题时，难以得到满意精度。本文提出基于B样条函数变换和核函数变换的非线性偏最小二乘响应面法进行可靠度分析，其主要思想是将原始输入数据通过非线性的核函数或B样条函数映射到高维特征空间，然后在高维空间内实施偏最小二乘回归建立代理模型，该方法能够充分利用样本空间信息，有效捕捉输入输出间的非线性关系，避免了假定极限状态函数形式对可靠度分析结果产生的影响，并且有效处理了高维空间内数据间存在的多重相关性。算例结果表明，提出的两种方法在精度和效率上均优于传统多项式响应面法。
(3)针对可靠度分析中广泛应用的kriging方法，在多维问题中精度和效率较低的问题，通过引入函数的梯度信息，提出了相比与kriging方法精度更高的cokriging方法，并通过算例验证其可靠度分析有效性。同时，将可靠度计算中的重要抽样方法与cokriging模拟技术相结合，提出了一种基于cokriging模拟的重要抽样方法。数值算例表明，该方法在与重要抽样法计算精度相当的情况下，能够大幅度减少模拟的样本数，提高计算效率。
(4)针对神经网络代理模型在求解结构可靠度问题时存在的网络结构难以确定以及参数值选取无理论依据等缺陷，通过引入偏最小二乘技术分别与BP网络和RBF网络相结合，建立基于偏最小二乘和BP网络的混合模型以及偏最小二乘和RBF网络的混合模型，并分别提出以此为代理模型的结构可靠度分析方法。数值算例分析结果表明，本文提出的方法在计算精度和效率方面均优于传统神经网络方法。
Surrogate model methods are widely used in structural reliability to alleviatethe computational burden of engineering analysis at present. Existing methodsinclude the traditional Response Surface Method (RSM) and recently devolopedArtificial Neural Network (ANN) and kriging et al. The basic idea of these methodsis to create approximate models and provide functional relationships between theresponses and variables to replace the implicit limit state function. However, thesemethods are often confronted with kinds of difficulties such as the low accuracy andthe inefficiency for realistic engineering problems, especially in high-dimensionalsystems. Based on the existing methods, this study throughly investigates thesurrogate model methods for structural reliability analysis. Several new methodsthat have the higher accuracy and the improved efficiency when compared with thetraditional methods are proposed. The main work of this study is as follows:
(1) The experiment design methods in applied statistics are briefly introduced.Suggestions for the selection of experiment design methods in traditional responsesurface method are provided through the comparison of some experiment designmethods for response surface fitting in reliability analysis. For the response surfacemethods of multidimensional variables with the uniform design, the limitation of th eoriginal Least Squares (LS) regression induced by multidimensional correlation ofdata in model fitting is analysized. To deal with the limitation, a new approachcalled quasi-linearal Partial Least Squares (PLS) response surface method based onthe traditional polynomial has been proposed. This method combines the uniformdesign and Partial Least Squares technique to estimate the response surface andperform reliability analysis. It can effectively cope with the multidimensionalcorrelation of data and high error when building the regression model under thecondition of small samples. The results of several examples show that the proposedmethod based on the polynomial functions is suitable for structural reliabilityanalysis and has higher accuracy, especially in the high dimensional problem.
(2) Due to the restriction of polynomial functions, the traditional responsesurface method cannot attain satisfactory accuracy for multidimensional variablesand high non-linearity problems. This paper presents the partial least squaresnon-linear regression methods based on the B-spline transform and kernel transformsubstitute for the surrogate model for the calculation of reliability, and the main ideais to first map the input space into a high-dimensional feature space via a nonlinearkernel function or a B-spline function, then to apply partial least squares regressionin the feature space, which can make full use of the sample space information and effectively capture the nonlinear relationship between input variables and outputvariables compared to linear partial least square. Thus the methods can not onlyavoid the influence of assuming the type of the limit state function but also handlethe correlation among variables after space transform. Numerical examples indicatethat the accuracy and efficiency of the two proposed methods are both superior tothe traditional response surface method.
(3) To improve the accuracy and efficiency of the widely used kriging method,especially in multidimensional systems, this paper explores the use of the cokrigingmethod by incorporating the secondary information such as the gradients of thefunction. The improvement of the new method is verified by numerical examples.Then, a simulated importance sampling approach, which combines the importancesampling technique with the cokriging method, is presented for structural reliabilityevaluation. Numerical examples illustrate that the proposed method can greatlydecrease the sample number of Monte Carlo method and improve the efficiency withcomparable accuracy.
(4) To overcome the flaws of artificial neural network (ANN) such as thedifficulty to determine the network struture and the lack of theoretical basis forparameter seletion, two kinds of hybrid artificial neural networks (HANN)integrating Radial Basis Function (RBF) networks and Back Propagation (BP)networks with partial least square technique respectively are proposed. Then thecorresponding structural reliability analysis methods utilizing the hybrid model asthe surrogate model are presented. The results of the numerical examplesdemonstrate that the proposed methods are superior to the traditional artificialneural networks in terms of accuracy and efficiency.
引文
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