基于最小参数区间集的不确定结构响应分析

考虑到实际工程问题中普遍存在不确定性,完成了针对工程结构从定量化到传播的完整不确定性分析过程．通过建立包含全部有限样本点的最小区间/超立方体域来描述不确定参数的变化范围;借助于最小区间参数集,开展了不确定结构传播分析的研究工作以确定其最有利/不利响应．此外,进一步就给出的区间分析方法同经典概率方法的相容性进行了分析和探究．采用2个数值算例很好地论证了所述方法的正确性和可行性．     

Uncertainty quantification of multi-dimensional parameters for composite laminates based on grey mathematical theory

In order to perform uncertainty quantification of elastic mechanical properties for composite laminates with multi-dimensional parameters, this paper is to develop a novel quantification approach based on grey mathematical theory. Here, uncertain parameters are modeled as correlated interval variables by virtue of some limited experimental points. The developed method not only can eliminate big errors in experimental points, but also can estimate uncertain information including nominal values, uncertain intervals, auto and mutual uncertainties of elastic properties. Besides, it can give out feasible domains of mechanical properties when considering mutual uncertainties for uncertainty propagation analysis. The numerical examples are implemented to demonstrate the feasibility and availability of the developed method. The results show that the developed method can become an important and powerful tool for uncertainty quantification of composite laminates with mutual uncertainties.     

Correlation propagation for uncertainty analysis of structures based on a non-probabilistic ellipsoidal model

Traditional non-probabilistic methods for uncertainty propagation problems evaluate only the lower and upper bounds of structural responses, lacking any analysis of the correlations among the structural multi-responses. In this paper, a new non-probabilistic correlation propagation method is proposed to effectively evaluate the intervals and non-probabilistic correlation matrix of the structural responses. The uncertainty propagation process with correlated parameters is first decomposed into an interval propagation problem and a correlation propagation problem. The ellipsoidal model is then utilized to describe the uncertainty domain of the correlated parameters. For the interval propagation problem, a subinterval decomposition analysis method is developed based on the ellipsoidal model to efficiently evaluate the intervals of responses with a low computational cost. More importantly, the non-probabilistic correlation propagation equations are newly derived for theoretically predicting the correlations among the uncertain responses. Finally, the multi-dimensional ellipsoidal model is adopted again to represent both uncertainties and correlations of multi-responses. Three examples are presented to examine the accuracy and effectiveness of the proposed method both numerically and experimentally.     

Sparse regression Chebyshev polynomial interval method for nonlinear dynamic systems under uncertainty

This paper proposes a new higher-efficiency interval method for the response bound estimation of nonlinear dynamic systems, whose uncertain parameters are bounded. This proposed method uses sparse regression and Chebyshev polynomials to help the interval analysis applied on the estimation. It is also a non-intrusive method which needs much fewer evaluations of original nonlinear dynamic systems than the other Chebyshev polynomials based interval methods. By using the proposed method, the response bound estimation of nonlinear dynamic systems can be performed more easily, even if the numerical simulation in nonlinear dynamic systems is costly or the number of uncertain parameters is higher than usual. In our approach, the sparse regression method “elastic net” is adopted to improve the sampling efficiency, but with sufficient accuracy. It alleviates the sample size required in coefficient calculation of the Chebyshev inclusion function in the sampling based methods. Moreover, some mature technologies are adopted to further reduce the sample size and to guarantee the accuracy of the estimation. So that the number of sampling, which solves the certain ordinary differential equations (ODEs), can be reduced significantly in the Chebyshev interval method. Three numerical examples are presented to illustrate the efficiency of proposed interval method. In particular, the last two examples are high dimension uncertain problems, which can further exhibit the ability to reduce the computational cost.     

Exact bounds for the sensitivity analysis of structures with uncertain-but-bounded parameters

Based on interval mathematical theory, the interval analysis method for the sensitivity analysis of the structure is advanced in this paper. The interval analysis method deals with the upper and lower bounds on eigenvalues of structures with uncertain-but-bounded (or interval) parameters. The stiffness matrix and the mass matrix of the structure, whose elements have the initial errors, are unknown except for the fact that they belong to given bounded matrix sets. The set of possible matrices can be described by the interval matrix. In terms of structural parameters, the stiffness matrix and the mass matrix take the non-negative decomposition. By means of interval extension, the generalized interval eigenvalue problem of structures with uncertain-but-bounded parameters can be divided into two generalized eigenvalue problems of a pair of real symmetric matrix pair by the real analysis method. Unlike normal sensitivity analysis method, the interval analysis method obtains informations on the response of structures with structural parameters (or design variables) changing and without any partial differential operation. Low computational effort and wide application rang are the characteristic of the proposed method. Two illustrative numerical examples illustrate the efficiency of the interval analysis.     

An iteration method for predicting static response of nonlinear structural systems with non-deterministic parameters

Confident numerical method is a crucial issue in the field of structural health monitoring. This paper focuses on uncertainty propagation in nonlinear structural systems with non-deterministic parameters. An interval-based iteration method is proposed on the basis of interval analysis and Taylor series expansion. The proposed method aims to improve the bounds of static response calculated by the point-based iteration method. In the proposed method, the iterative interval of static response is updated by revising the lower and upper bounds, respectively, which is the essential difference in comparison with the previous point-based iteration method. In this paper, interval parameters are employed to quantify the non-deterministic parameters instead of random parameters in the case of insufficient sample data. Iterative scheme is established based on the first-order Taylor series expansion. For the implementation of interval-based iteration method, a general procedure is formulated. Moreover, the important source of the limitation of point-based iteration method is revealed profoundly, and the good performance of the proposed method is demonstrated by three numerical comparisons.     

A new uncertainty analysis for the transformation method

In this paper, a new uncertainty analysis for the transformation method (TM) is proposed. As a practical implementation of fuzzy arithmetic, the TM is a convenient tool for the simulation and analysis of systems with uncertain parameters that are expressed by fuzzy numbers. The proposed uncertainty analysis and the sensitivity analysis of the TM complete each other in providing some quantification of the relationship between the uncertainties of the system input and the system output. The computation of gain factors is proposed, which allows the estimation of the absolute and relative measures of uncertainty. These measures allow the quantification of the influence of the uncertainty of the input on the uncertainty of the output.     

基于误差理论的区间主成分分析及其应用

针对区间数样本,传统的主成分分析需进行拓展。首先讨论了区间样本数据的两种主要来源,即观测误差和符号数据分析。然后将区间数看作一个由中点和半径构成的具有一定误差的数,从误差理论出发,研究基于误差传递公式的区间主成分分析方法,并获得以区间数为表达形式的主成分。最后,结合我国2005年第四季度股票市场的数据进行了实证分析。结果表明,面对海量数据,区间PCA较传统PCA更容易从总体上把握样本的属性。     

Fuzzy confidence interval for pH titration curve

In this paper we present a new method of confidence interval identification for Takagi–Sugeno fuzzy models in the case of the data with regionally changeable variance. The method combines a fuzzy identification methodology with some ideas from applied statistics. The idea is to find, on a finite set of measured data, the confidence interval defined by the lower and upper bounds. The confidence interval which defines the band that contains the measurement values with certain confidence. The method can be used when describing a family of uncertain nonlinear functions or when the systems with uncertain physical parameters are observed. In our example the proposed method is applied to model the pH-titration curve.     

Dynamic load identification for interval structures under a presupposition of ‘being included prior to being measured’

Influences of structural uncertainties in the dynamic load identification are always significant and need to be quantified. In case of insufficient information available, intervals are favorable for modelling uncertainties. To perform the interval propagation in an inverse problem, this paper develops a sequential dual-stage interval identification method under a presupposition that each noisy response, which is an accomplished measurement for reconstructing unknown loads, should be included in the corresponding interval response of the structure exerted by interval loads to be identified. The proposed method transforms the interval identification problem into a classical one at the midpoint of interval parameters and an optimization model for minimizing the radius of each interval load. The effectiveness of the proposed method is validated by a spatial truss subjected to multiple forces due to the inclusion of each unknown load in the corresponding load. Besides, regularized solutions without exact knowledge of the accuracy loss are recommended to be used as few as possible in the interval identification of unknown loads.     

Non-probabilistic Bayesian update method for model validation

Model validation is the principal strategy to evaluate the accuracy and reliability of computational simulations. A systematic model validation procedure including uncertainty quantification, model update and prediction is described based on a non-probabilistic interval model. The crucial technical challenge in model validation is limited data, thus the non-probabilistic interval model is adopted to describe uncertain parameters. To establish the model update formula, the concepts of the interval escape rate and interval coverage rate are first described. Then, not only can the possibility of failure be estimated but also the credibility of the possibility of failure based on the proposed model validation method. The data in the validation experiment are used to update the credibility of each interval model, while the data from the accreditation experiment are used to conduct a final check of the validated models. To demonstrate that the proposed method can be applied to model validation problems successfully, a validation benchmark, the static frame challenge problem, is implemented. In addition, a practical aviation structure engineering validation problem is described. The results of these two validation problems show the feasibility and effectiveness of the proposed model validation method. The theoretical framework proposed in this paper is also suitable for model validation of computational simulations in other research fields.     

Interval uncertainty analysis for static response of structures using radial basis functions

This paper proposes a new interval uncertainty analysis method for static response of structures with unknown-but-bounded parameters by using radial basis functions (RBFs). Recently, collocation methods (CM) which apply orthogonal polynomials are proposed to solve interval uncertainty quantification problems with high accuracy. These methods overcome the drawback of Taylor expansion based methods, which are prone to overestimate the response bounds. However, the form of orthogonal basis functions is very complicated in higher dimensions, which may restrict their application when there exist relatively more interval parameters. In contrast to orthogonal basis function, the form of radial basis function (RBF) is simple and stays the same in whatever dimension. This study introduces RBFs into interval analysis of structures and provides a relatively simple approach to solve structural response bounds accurately. A surrogate model of real structural response with respect to interval parameters is constructed with the RBFs. The extrema of the surrogate model can be calculated by some auxiliary methods. The static response bounds can be obtained accordingly. Two numerical examples are used to verify the proposed method. The engineering application of the proposed method is performed by a center wing-box. The results prove the effectiveness of the proposed method.     

An interval uncertainty analysis method for structural response bounds using feedforward neural network differentiation

This paper proposes a new interval uncertainty analysis method for structural response bounds with uncertain‑but-bounded parameters by using feedforward neural network (FNN) differentiation. The information of partial derivative may be unavailable analytically for some complicated engineering problems. To overcome this drawback, the FNNs of real structural responses with respect to structure parameters are first constructed in this work. The first-order and second-order partial derivative formulas of FNN are derived via the backward chain rule of partial differentiation, thus the partial derivatives could be determined directly. Especially, the influences of structures of multilayer FNNs on the accuracy of the first-order and second-order partial derivatives are analyzed. A numerical example shows that an FNN with the appropriate structure parameters is capable of approximating the first-order and second-order partial derivatives of an arbitrary function. Based on the parameter perturbation method using these partial derivatives, the extrema of the FNN can be approximated without requiring much computational time. Moreover, the subinterval method is introduced to obtain more accurate and reliable results of structural response with relatively large interval uncertain parameters. Three specific examples, a cantilever tube, a Belleville spring, and a rigid-flexible coupling dynamic model, are employed to show the effectiveness and feasibility of the proposed interval uncertainty analysis method compared with other methods.     

The reliability analysis of probabilistic and interval hybrid structural system

The aim of this paper is to evaluate the reliability of probabilistic and interval hybrid structural system. The hybrid structural system includes two kinds of uncertain parameters—probabilistic parameters and interval parameters. Based on the interval reliability model and probabilistic operation, a new probabilistic and interval hybrid reliability model is proposed. Firstly, we use the interval reliability model to analyze the performance function, and then sum up reliability of all regions divided by the failure plane. Based on the presented optimal criterion enumerating the main failure modes of hybrid structural system and the relationship of failure modes, the reliability of structure system can be obtained. By means of the numerical examples, the hybrid reliability model and the traditional probabilistic reliability model are critically contrasted. The results indicate the presented reliability model is more suitable for analysis and design of these structural systems and it can ensure the security of system well, and it only needs less uncertain information.     

Novel methods on comparing grey numbers

Uncertain decision-making is an important branch of decision-making theory. It is crucial to describe uncertain information, which determine the decision-making is effective or not. This paper first presents a brief survey of the existing methods on denoting uncertain information, such as fuzzy mathematics, stochastic and interval methods, analyzes the merits and demerits of these methods. Then the paper proposes a novel method grey systems theory to describe uncertain information and gives the novel definition of grey number on the basis of probability distribution. Subsequently a novel probability method on comparing grey numbers, especially discrete grey numbers and interval grey numbers, is studied. When an interval grey number satisfied to continuous uniform distribution, it will be degenerated into an interval number. Finally three numerical examples are investigated to demonstrate the effectiveness of the present method.     

A new nonlinear interval programming method for uncertain problems with dependent interval variables

This paper proposes a new nonlinear interval programming method that can be used to handle uncertain optimization problems when there are dependencies among the interval variables. The uncertain domain is modeled using a multidimensional parallelepiped interval model. The model depicts single-variable uncertainty using a marginal interval and depicts the degree of dependencies among the interval variables using correlation angles and correlation coefficients. Based on the order relation of interval and the possibility degree of interval, the uncertain optimization problem is converted to a deterministic two-layer nesting optimization problem. The affine coordinate is then introduced to convert the uncertain domain of a multidimensional parallelepiped interval model to a standard interval uncertain domain. A highly efficient iterative algorithm is formulated to generate an efficient solution for the multi-layer nesting optimization problem after the conversion. Three computational examples are given to verify the effectiveness of the proposed method.     

A Chebyshev interval method for nonlinear dynamic systems under uncertainty

This paper proposes a new interval analysis method for the dynamic response of nonlinear systems with uncertain-but-bounded parameters using Chebyshev polynomial series. Interval model can be used to describe nonlinear dynamic systems under uncertainty with low-order Taylor series expansions. However, the Taylor series-based interval method can only suit problems with small uncertain levels. To account for larger uncertain levels, this study introduces Chebyshev series expansions into interval model to develop a new uncertain method for dynamic nonlinear systems. In contrast to the Taylor series, the Chebyshev series can offer a higher numerical accuracy in the approximation of solutions. The Chebyshev inclusion function is developed to control the overestimation in interval computations, based on the truncated Chevbyshev series expansion. The Mehler integral is used to calculate the coefficients of Chebyshev polynomials. With the proposed Chebyshev approximation, the set of ordinary differential equations (ODEs) with interval parameters can be transformed to a new set of ODEs with deterministic parameters, to which many numerical solvers for ODEs can be directly applied. Two numerical examples are applied to demonstrate the effectiveness of the proposed method, in particular its ability to effectively control the overestimation as a non-intrusive method.     

An enhanced subinterval analysis method for uncertain structural problems

This paper proposes an enhanced subinterval analysis method to predict the bounds of structural response with interval parameters, which could deal with problems with relatively large uncertainties of the parameters. The intervals are first divided into several subintervals, and two expansion routes are then constructed based on the sensitivity analysis. Two subinterval sets are selected according to the expansion points on the routes, and the first order Taylor expansion method is then adopted to complete the subinterval analysis. Based on the selected subinterval sets, the upper and lower bounds of the structural response are further obtained by employing the interval union operation. An adaptive convergence approach is presented to determine the appropriate number of subintervals. Four numerical examples are investigated to demonstrate the validity of the proposed method.     

区间参数结构振动问题的矩阵摄动法

当结构的参数具有不确定性时，结构的固有频率也将具有某种程度的不确定性．本文讨论了区间参数结构的振动问题，将区间参数结构的特征值问题归结为两个不同的特征值问题来求解．提出了求解区间参数结构振动问题的矩阵摄动方法．数值运算结果表明，本文所提出方法具有运算量小，结果精度高等优点．     

Structural reliability analysis with imprecise random and interval fields

This paper investigates the issue of reliability assessment for engineering structures involving mixture of stochastic and non-stochastic uncertain parameters through the Finite Element Method (FEM). Non-deterministic system inputs modelled by both imprecise random and interval fields have been incorporated, so the applicability of the structural reliability analysis scheme can be further promoted to satisfy the intricate demand of modern engineering application. The concept of robust structural reliability profile for systems involving hybrid uncertainties is discussed, and then a new computational scheme, namely the unified interval stochastic reliability sampling (UISRS) approach, is proposed for assessing the safety of engineering structures. The proposed method provides a robust semi-sampling scheme for assessing the safety of engineering structures involving multiple imprecise random fields with various distribution types and interval fields simultaneously. Various aspects of structural reliability analysis with multiple imprecise random and interval fields are explored, and some theoretically instructive remarks are also reported herein.