Dynamic analysis of bridge with non-Gaussian uncertainties under a moving vehicle

An analysis method on the bridge-vehicle interaction problem with uncertainties is proposed. The bridge is modeled as a simply supported Euler-Bernoulli beam with non-Gaussian material parameters with a vehicle moving on top modeled by a deterministic four degrees-of-freedom mass-spring system. The non-Gaussian uncertainty in bridge is modeled by the Spectral Stochastic Finite Element Method (Ghanem and Spanos (1991) [17]), and the mathematical model of the coupled bridge-vehicle system, with the road surface roughness assumed as a Gaussian random process, will be solved by the Newmark-β method. The proposed model is verified by the Monte Carlo Simulation with numerical examples. Different levels of uncertainties in both the excitation and system parameters are investigated. Criteria on the selection of both the order of Polynomial Chaos and the threshold for truncation in the Karhunen-Loève expansion are provided. Results show that the proposed algorithm is promising for the dynamic analysis of the bridge-vehicle interaction problem even with a high level of system and excitation uncertainties.     

On Solution Regularity of Linear Hyperbolic Stochastic PDE Using the Method of Characteristics

The generalized Polynomial Chaos (gPC) method is one of the most widely used numerical methods for solving stochastic differential equations. Recently, attempts have been made to extend the the gPC to solve hyperbolic stochastic partial differential equations (SPDE). The convergence rate of the gPC depends on the regularity of the solution. It is shown that the characteristics technique can be used to derive general conditions for regularity of linear hyperbolic PDE, in a detailed case study of a linear wave equation with a random variable coefficient and random initial and boundary data.     

复合随机振动分析的混合PC-PEM方法

由于加工、制造等原因,实际结构系统往往所具有很多不确定性,准确评估随机系统的动力学行为不仅具有实际意义,而且是近年来结构动力学理论的一个研究热点。本文研究了同时考虑结构模型参数与所受外激励载荷具有不确定性的复合随机振动问题。结构模型参数的不确定性采用随机变量模拟,外激励载荷的不确定性采用随机过程模拟,提出了结构随机振动响应评估的混合混沌多项式-虚拟激励(PC-PEM)方法。数值算例研究了参数不确定性在21杆桁架中的传播,讨论了响应的一阶、二阶统计矩,并同蒙特卡洛方法进行对比表明提出方法的正确性和有效性。本文的工作对于考虑不确定的复杂装备与结构系统的随机振动分析具有很好的借鉴意义。     

Reliability analysis of structures based on a probability‐uncertainty hybrid model

The traditional reliability analysis method based on probabilistic method requires probability distributions of all the uncertain parameters. However, in practical applications, the distributions of some parameters may not be precisely known due to the lack of sufficient sample data. The probabilistic theory cannot directly measure the reliability of structures with epistemic uncertainty, ie, subjective randomness and fuzziness. Hence, a hybrid reliability analysis (HRA) problem will be caused when the aleatory and epistemic uncertainties coexist in a structure. In this paper, by combining the probability theory and the uncertainty theory into a chance theory, a probability‐uncertainty hybrid model is established, and a new quantification method based on the uncertain random variables for the structural reliability is presented in order to simultaneously satisfy the duality of random variables and the subadditivity of uncertain variables; then, a reliability index is explored based on the chance expected value and variance. Besides, the formulas of the chance theory‐based reliability and reliability index are derived to uniformly assess the reliability of structures under the hybrid aleatory and epistemic uncertainties. The numerical experiments illustrate the validity of the proposed method, and the results of the proposed method can provide a more accurate assessment of the structural system under the mixed uncertainties than the ones obtained separately from the probability theory and the uncertainty theory.     

A unified stochastic framework for robust topology optimization of continuum and truss-like structures

In this article, a unified framework is introduced for robust structural topology optimization for 2D and 3D continuum and truss problems. The uncertain material parameters are modelled using a spatially correlated random field which is discretized using the Karhunen–Loève expansion. The spectral stochastic finite element method is used, with a polynomial chaos expansion to propagate uncertainties in the material characteristics to the response quantities. In continuum structures, either 2D or 3D random fields are modelled across the structural domain, while representation of the material uncertainties in linear truss elements is achieved by expanding 1D random fields along the length of the elements. Several examples demonstrate the method on both 2D and 3D continuum and truss structures, showing that this common framework provides an interesting insight into robustness versus optimality for the test problems considered.     

A two-pronged approach for springback variability assessment using sparse polynomial chaos expansion and multi-level simulations

In this study, we show that stochastic analysis of metal forming process requires both a high precision and low cost numerical models in order to take into account very small perturbations on inputs (physical as well as process parameters) and to allow for numerous repeated analysis in a reasonable time. To this end, an original semi-analytical model dedicated to plain strain deep drawing based on a Bending-Under-Tension numerical model (B-U-T model) is used to accurately predict the influence of small random perturbations around a nominal solution estimated with a full scale Finite Element Model (FEM). We introduce a custom sparse variant of the Polynomial Chaos Expansion (PCE) to model the propagation of uncertainties through this model at low computational cost. Next, we apply this methodology to the deep drawing process of U-shaped metal sheet considering up to 8 random variables.     

Error propagation for quantile estimation via combining Polynomial Chaos expansions and metalog distributions

Introducing parametric uncertainties to models is a quintessential step of simulation-based risk assessment and rare event simulations. A novel quantile estimation method is proposed based on generalized Polynomial Chaos expansions and metalog distribution estimation. We propose to derive the metalog distribution based on the statistical information of an expansion of the random model. The advantage of latter lies in a reduced number of evaluations of the full model, while deriving the metalog distribution avoids sampling-related challenges of quantile estimation. Error estimates and an algorithm to choose the relevant number of statistical moments are developed and analyzed, giving a framework for assessing the method applicability and the accuracy of the quantile estimation. The proposed method and the analysis are demonstrated on numerical examples. The method is applied to a complex fluid flow problem in transpiration cooling.     

Interval uncertain method for multibody mechanical systems using Chebyshev inclusion functions

This study proposes a new uncertain analysis method for multibody dynamics of mechanical systems based on Chebyshev inclusion functions The interval model accounts for the uncertainties in multibody mechanical systems comprising uncertain‐but‐bounded parameters, which only requires lower and upper bounds of uncertain parameters, without having to know probability distributions. A Chebyshev inclusion function based on the truncated Chebyshev series, rather than the Taylor inclusion function, is proposed to achieve sharper and tighter bounds for meaningful solutions of interval functions, to effectively handle the overestimation caused by the wrapping effect, intrinsic to interval computations. The Mehler integral is used to evaluate the coefficients of Chebyshev polynomials in the numerical implementation. The multibody dynamics of mechanical systems are governed by index‐3 differential algebraic equations (DAEs), including a combination of differential equations and algebraic equations, responsible for the dynamics of the system subject to certain constraints. The proposed interval method with Chebyshev inclusion functions is applied to solve the DAEs in association with appropriate numerical solvers. This study employs HHT‐I3 as the numerical solver to transform the DAEs into a series of nonlinear algebraic equations at each integration time step, which are solved further by using the Newton–Raphson iterative method at the current time step. Two typical multibody dynamic systems with interval parameters, the slider crank and double pendulum mechanisms, are employed to demonstrate the effectiveness of the proposed methodology. The results show that the proposed methodology can supply sufficient numerical accuracy with a reasonable computational cost and is able to effectively handle the wrapping effect, as cosine functions are incorporated to sharpen the range of non‐monotonic interval functions. Copyright © 2013 John Wiley & Sons, Ltd.     

结构随机分析的向量型层递响应面法

结构随机分析的传统响应面法通常用于显化表示某一随机响应量在结构特定点的涨落,属于标量型响应面法。为克服传统响应面法的局限性,提出了结构总体节点位移向量显化表示的层递响应面法,属于向量型响应面法。首先利用Karhunen-Loève级数线性展开随机刚度矩阵和节点荷载向量,并定义随机场均值处的刚度矩阵作为预处理器,进而生成预处理Krylov子空间;然后将结构总体节点位移向量在该空间中层递展开,建立向量型的层递响应面,有效保持了节点位移之间的协调性;最后分析了层递响应面与混沌多项式之间的关系,给出了样本点选取原则,建立了节点位移向量的均值和协方差计算公式。通过算例分析,验证了层递响应面法的高精度、全域性和快速收敛性。     

A meshfree-Galerkin method in modelling and synthesizing spatially varying soil properties

Physical properties of soil vary from point to point in space and exhibit great uncertainty, suggesting random field as a natural approach in modelling and synthesizing these properties. The significance of considering spatial variability and uncertainty of soil properties is greatly manifested in the probabilistic seismic risk analysis of soil–structural system (nonlinear dynamic analysis under earthquake loading), where modelling and synthesis of the spatial variability and uncertainty of soil properties are necessary. This paper introduces a meshfree-Galerkin approach within the Karhunen–Loève (K–L) expansion scheme for representation of spatial soil properties modelled as the random fields. The meshfree shape functions are introduced and employed as a set of basis functions in the Galerkin scheme to obtain the eigen-solutions of integral equation of K–L expansion. An optimization scheme is proposed for the resulting eigenvectors in treating the compatibility between the target and analytical covariance models. Assessments of the meshfree-Galerkin method are conducted for the resulting eigen-solutions and the representation of covariance models for various homogeneous and nonhomogeneous random fields. The accuracy and validity of the proposed approach are demonstrated through the modelling and synthesis of the spatial field models inferred from the field measurements.     

Robust topology optimization for periodic structures by combining sensitivity averaging with a semianalytical method

Uncertainty factors play an important role in the design of periodic structures because structures with small periodic design spaces are extremely sensitive to loading uncertainty. Therefore, for the first time, this paper proposes a framework for robust topology optimization (RTO) of periodic structures assuming that load uncertainties follow a Gaussian distribution. In this framework, the expected value and variance of structural compliance can be easily computed using a semianalytical method combined with probability theory, which is important for RTO when uncertain variables follow probabilistic distributions. To obtain optimal topologies, the bidirectional evolutionary structural optimization method is used. Structural periodicity is calculated using a strategy of sensitivity averaging and consistency constraints. To eliminate the influence of numerical units when comparing the optimal results to deterministic and RTO solutions, a generic coefficient of variation is defined as the robust index, which contains both the expected value and variance. The proposed framework is verified through the optimization of both 2D and 3D structures with periodicity. Computational results demonstrate the feasibility and effectiveness of the proposed framework for designing robust periodic structures under loading uncertainties.     

An isogeometric collocation method for efficient random field discretization

This paper presents an isogeometric collocation method for a computationally expedient random field discretization by means of the Karhunen-Loève expansion. The method involves a collocation projection onto a finite-dimensional subspace of continuous functions over a bounded domain, basis splines (B-splines) and nonuniform rational B-splines (NURBS) spanning the subspace, and standard methods of eigensolutions. Similar to the existing Galerkin isogeometric method, the isogeometric collocation method preserves an exact geometrical representation of many commonly used physical or computational domains and exploits the regularity of isogeometric basis functions delivering globally smooth eigensolutions. However, in the collocation method, the construction of the system matrices for a d-dimensional eigenvalue problem asks for at most d-dimensional domain integrations, as compared with 2d-dimensional integrations required in the Galerkin method. Therefore, the introduction of the collocation method for random field discretization offers a huge computational advantage over the existing Galerkin method. Three numerical examples, including a three-dimensional random field discretization problem, illustrate the accuracy and convergence properties of the collocation method for obtaining eigensolutions.     

A polynomial‐based method for topology optimization of phononic crystals with unknown‐but‐bounded parameters

The design and analysis of phononic crystals (PnCs) are generally based on the deterministic models without considering the effects of uncertainties. However, uncertainties that existed in PnCs may have a nontrivial impact on their band structure characteristics. In this paper, a sparse point sampling–based Chebyshev polynomial expansion (SPSCPE) method is proposed to estimate the extreme bounds of the band structures of PnCs. In the SPSCPE, the interval model is introduced to handle the unknown‐but‐bounded parameters. Then, the sparse point sampling scheme and the finite element method are used to calculate the coefficients of the Chebyshev polynomial expansion. After that, the SPSCPE method is applied for the band structure analysis of PnCs. Meanwhile, the checkerboard and hinge phenomena are eliminated by the hybrid discretization model. In the end, the genetic algorithm is introduced for the topology optimization of PnCs with unknown‐but‐bounded parameters. The specific frequency constraint is considered. Two numerical examples are investigated to demonstrate the effectiveness of the proposed method.     

An iterative polynomial chaos approach toward stochastic elastostatic structural analysis with non-Gaussian randomness

Stochastic analysis of structure with non-Gaussian material property and loading in the framework of polynomial chaos (PC) is considered. A new approach for the solution of stochastic mechanics problem with random coefficient is presented. The major focus of the method is to consider reduced size of expansion in an iterative manner to overcome the problem of large system matrix in conventional PC expansion. The iterative method is based on orthogonal expansion of stochastic responses and generation of an iterative PC based on the responses of the previous iteration. The polynomials are evaluated using Gram-Schmidt orthogonalization process. The numbers of random variables in PC expansion are reduced by considering only the dominant components of the response characteristics, which is evaluated using Karhunen-Loève (KL) expansion. In case of random material field problem, the KL expansion is used to discretize and simulate the non-Gaussian random field. Independent component analysis (ICA) is carried out on the non-Gaussian KL random variables to minimize statistical dependence. The usefulness of the proposed method in terms of accuracy and computational efficiency is examined. From the numerical analysis of three different types of structural mechanics problems, the proposed iterative method is observed to be computationally more efficient and accurate than conventional PC method for solution of linear elastostatic structural mechanics problems.     

Characterization of Cast Iron Microstructure Through Fluctuation and Fractal Analyses of Ultrasonic Backscattered Signals Combined with Classification Techniques

This work aims at evaluating the performance of pattern recognition methods in the identification of different microstructures presented by cast iron, namely, lamellar, vermicular and nodular microstructures, through the statistical fluctuation and fractal analyses of backscattered ultrasonic signals. The signals were obtained with a broad band ultrasonic probe with a central frequency of 5 MHz. The statistical fluctuations of the ultrasonic signals were analyzed by means of Hurst (RSA) and detrended-fluctuation analyses (DFA), and the fractal analyses were carried out by applying the minimal cover and box-counting techniques to the signals. The curves obtained from the statistical fluctuations and fractal analyses, as functions of the time window, were processed by using four pattern classification techniques, namely, principal-component analysis (PCA), Karhunen-Loève transformation (KLT), neural networks and Gaussian classifier. The best results were obtained by Karhunen-Loève expansion and neural networks, where an approximately 100% success rate has been reached for the classification of the different microstructures as well as for the training and the testing sets of events. The results presented correspond to an average taken over 100 randomly chosen sets of events. These results indicate that, within the techniques used, the Karhunen-Loève transformation and neural network associated with the statistical fluctuation analyses (RSA and DFA) are the best tools for the recognition of the different cast iron microstructures. It is worthwhile pointing out that the microstructure classification was made by using backscattering signals acquired during pulse echo ultrasonic nondestructive testing only. Therefore, that approach is a promising method for material characterization.     

Karhunen–Loève decomposition of random fields based on a hierarchical matrix approach

The simulation of the behavior of structures with uncertain properties is a challenging issue, because it requires suitable probabilistic models and adequate numerical tools. Nowadays, it is possible to perform probabilistic investigations of the structural performance, which take into account a space‐variant uncertainty characterization of the structures. Given a structural solver and the probabilistic models, the reliability analysis of the structural response depends on the continuous random fields approximation, which is carried out by means of a finite set of random variables. The paper analyzes the main aspects of discretization in the case of 2D problems. The combination of the well‐known Karhunen–Loève series expansion, the finite element method and the hierarchical matrices approach is proposed in the paper. Copyright © 2013 John Wiley & Sons, Ltd.     

A new reliability analysis method for uncertain structures with random and interval variables

In this paper, a new reliability analysis method is developed for uncertain structures with mixed uncertainty. In our problem, the uncertain parameters with sufficient information are treated by random distributions, while some ones with limited information can only be given variation intervals. A complex nesting optimization will be involved when using the existing methods to compute such a hybrid reliability, which will lead to extremely low efficiency or instable convergence performance. In this paper, an equivalent model is firstly created for the hybrid reliability, which is a conventional reliability analysis problem with only random variables. Thus only through computing the reliability of the equivalent model the original hybrid reliability can be easily evaluated. Based on the above equivalent model, an algorithm with high efficiency and robust convergence performance is then constructed for computation of the above hybrid reliability with both random and interval variables. Two numerical examples are provided to demonstrate the effectiveness of the present method.     

Fuzzy interval perturbation method for uncertain heat conduction problem with interval and fuzzy parameters

This paper proposes a fuzzy interval perturbation method (FIPM) and a modified fuzzy interval perturbation method (MFIPM) for the hybrid uncertain temperature field prediction involving both interval and fuzzy parameters in material properties and boundary conditions. Interval variables are used to quantify the non‐probabilistic uncertainty with limited information, whereas fuzzy variables are used to represent the uncertainty associated with the expert opinions. The level‐cut method is introduced to decompose the fuzzy parameters into interval variables. FIPM approximates the interval matrix inverse by the first‐order Neumann series, while MFIPM improves the accuracy by considering higher‐order terms of the Neumann series. The membership functions of the interval temperature field are eventually derived using the fuzzy decomposition theorem. Three numerical examples are provided to demonstrate the feasibility and effectiveness of the proposed methods for solving heat conduction problems with hybrid uncertain parameters, pure interval parameters, and pure fuzzy parameters, respectively. Copyright © 2015 John Wiley & Sons, Ltd.     

基于区间理论的不确定结构随机疲劳损伤估计方法

将结构体系中不确定参数定义为区间变量,在随机疲劳谱分析方法的基础上,提出一种计算平稳高斯荷载作用下不确定结构疲劳损伤的新方法。该方法采用区间参数模型定义结构的不确定性,应用功率谱密度描述外荷载的随机性;利用有理级数和单位对称区间显式表达结构区间频响函数和不确定结构在平稳高斯荷载作用下的动力响应区间;根据Tovo-Benasciutti疲劳损伤预测模型,计算不确定结构在随机荷载作用下的疲劳损伤区间期望率;并可通过调整相应不确定参数的单位对称区间近似估计该不确定参数不同不确定半径的疲劳损伤区间期望率。通过数值算例,将该文提出的随机疲劳区间分析方法与顶点法进行比较,验证了该方法的准确性和适用性。     

Uncertainty quantification of subcritical bifurcations

Analysing and quantifying parametric uncertainties numerically is a tedious task, even more so when the system exhibits subcritical bifurcations. Here a novel interpolation based approach is presented and applied to two simple models exhibiting subcritical Hopf bifurcation. It is seen that this integrated interpolation scheme is significantly faster than traditional Monte Carlo based simulations. The advantages of using this scheme and the reason for its success compared to other uncertainty quantification schemes like Polynomial Chaos Expansion (PCE) are highlighted. The paper also discusses advantages of using an equi-probable node distribution which is seen to improve the accuracy of the proposed scheme. The probabilities of failure (POF) are defined and plotted for various operating conditions. The possibilities of extending the above scheme to experiments are also discussed.