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.     

Effects of uncertain suspension parameters on dynamic responses of the railway vehicle system

Due to the manufacture error, design tolerance and time-varying factors, the suspension parameters of railway vehicles are always uncertain. This paper investigates the stochastic vibration of the railway vehicle system with uncertain suspension parameters. The energy method and Hamilton’s principle are adopted to derive the governing equations of the deterministic railway vehicle system, in which the rigid and flexible modes of the railway car body can be considered. Based on the deterministic model, the polynomial chaos expansion (PCE) method is further employed to perform the uncertain analysis of the railway vehicle system. The global sensitivity analysis of the stochastic response of the railway vehicle with uncertain parameters is further carried out based on the PCE method and Sobol indices. The accuracy of the proposed method is validated by comparing the obtained random results with those from the published literature and satisfactory agreements can be observed between them. Furthermore, the effects of uncertain suspension parameters on the stochastic vibration characteristics of the railway vehicle system are discussed, which can be used as the reference for the dynamic design of the railway vehicle system. The numerical results show that the computational efficiency of the PCE method is significantly improved compared with the Monte Carlo method.     

A modal correction method for non-stationary random vibrations of linear systems

The computational effort in determining the dynamic response of linear systems is usually reduced by adopting the well-known modal analysis along with modal truncation of higher modes. However, in the case in which the contribution of higher modes is not negligible, modal correction methods have been introduced to improve the accuracy of the dynamic response, for both deterministic and stochastic input. In the latter case the random response is usually corrected via various methods determined as rough extensions of methods originally proposed for deterministic input. Consequently the efficiency of the correction methods is not suitable, from both theoretical and computational points of view. In this paper, a new approach to cope with the non-stationary response of linear systems is presented. The proposed modal correction method provides a correction term determined as a pseudo-stationary contribution of the equation governing either first-order or second-order statistics. Owing to the fact that no truncation criteria are well established for random vibration study, the proposed modal correction method offers a suitable vehicle for determining very accurately the stochastic response of MDOF linear systems under Gaussian stationary and non stationary excitation as evidenced in the numerical applications.     

Time response of structures with uncertain parameters under stochastic inputs based on coupled polynomial chaos expansion and component mode synthesis methods

This article presents a numerical procedure to compute the stochastic dynamic response of large finite element models with uncertain parameters based on polynomial chaos and component mode synthesis methods. Polynomial chaos expansions at higher orders are used to derive the statistical solution of the dynamic response as well as the Monte Carlo simulation procedure. Based on various component mode synthesis methods, the size of the model is reduced. These methods are coupled with polynomial chaos expansion and the explicit mathematical formulations are given. Numerical results illustrating the accuracy and efficiency of the proposed coupled methodological procedures are presented.     

Compromise design of stochastic dynamical systems: A reliability-based approach

This paper presents a procedure for obtaining compromise designs of structural systems under stochastic excitation. In particular, an effective strategy for determining specific Pareto optimal solutions is implemented. The design goals are defined in terms of deterministic performance functions and/or performance functions involving reliability measures. The associated reliability problems are characterized by means of a large number of uncertain parameters (hundreds or thousands). The designs are obtained by formulating a compromise programming problem which is solved by a first-order interior point algorithm. The sensitivity information required by the proposed solution strategy is estimated by an approach that combines an advanced simulation technique with local approximations of some of the quantities associated with structural performance. An efficient Pareto sensitivity analysis with respect to the design variables becomes possible with the proposed formulation. Such information is used for decision making and tradeoff analysis. Numerical validations show that only a moderate number of stochastic analyses (reliability estimations) has to be performed in order to find compromise designs. Two example problems are presented to illustrate the effectiveness of the proposed approach.     

A two-step method for analysis of linear systems with uncertain parameters driven by Gaussian noise

A two-step method is proposed to find state properties for linear dynamic systems driven by Gaussian noise with uncertain parameters modeled as a random vector with known probability distribution. First, equations of linear random vibration are used to find the probability law of the state of a system with uncertain parameters conditional on this vector. Second, stochastic reduced order models (SROMs) are employed to calculate properties of the unconditional system state. Bayesian methods are applied to extend the proposed approach to the case when the probability law of the random vector is not available. Various examples are provided to demonstrate the usefulness of the method, including the random vibration response of a spacecraft with uncertain damping model.     

A generalized stochastic perturbation technique for plasticity problems

The main aim of this paper is to present an algorithm and the solution to the nonlinear plasticity problems with random parameters. This methodology is based on the finite element method covering physical and geometrical nonlinearities and, on the other hand, on the generalized nth order stochastic perturbation method. The perturbation approach resulting from the Taylor series expansion with uncertain parameters is provided in two different ways: (i) via the straightforward differentiation of the initial incremental equation and (ii) using the modified response surface method. This methodology is illustrated with the analysis of the elasto-plastic plane truss with random Young’s modulus leading to the determination of the probabilistic moments by the hybrid stochastic symbolic-finite element method computations.     

Non-stationary random response analysis of structures with uncertain parameters

This paper proposes a non-stationary random response analysis method of structures with uncertain parameters. The structural physical parameters and the input parameters are considered as random variables or interval variables. By using the pseudo-excitation method and the direct differentiation method (DDM), the analytical expression of the time-varying power spectrum and the time-varying variance of the structure response can be obtained in the framework of first order perturbation approaches. In addition, the analytical expression of the first-order and second-order partial derivative (e.g., time-varying sensitivity coefficient) for the time-varying power spectrum and the time-varying variance of the structure response expressed via the uncertainty parameters can also be determined. Based on this and the perturbation technique, the probabilistic and non-probabilistic analysis methods to calculate the upper and lower bounds of the time-varying variance of the structure response are proposed. Finally the effectiveness of the proposed method is demonstrated by numerical examples compared with the Monte Carlo solutions and the vertex solutions.     

Random fatigue crack growth in mixed mode by stochastic collocation method

Fatigue crack growth is uncertain, either for cracking rate or direction. The stochastic models proposed in the literature suffer from limited applicability or lack of physical meaning. In this paper, a new stochastic collocation method is proposed to solve mixed mode fatigue crack growth problems with uncertain parameters. This approach has the advantage of non-intrusive nature methods, such as Monte-Carlo simulations, since it allows us to decouple the stochastic and the mechanical computations. The proposed numerical implementation is very simple, as it requires only repetitive runs of deterministic finite element analysis at some specific points in the random space. The method describes a precise approximation of the mechanical response corresponding to the fatigue life, in order to assess the stochastic properties, namely the statistical moments and the probability density function of fatigue life. The performance of the stochastic collocation method for dealing with this kind of problems has been evaluated through two numerical examples, showing the high performance for practical applications. Moreover, the proposed method is extended in the last example to the failure probability assessment, with respect to the target service life.     

Application of the generalized perturbation-based stochastic boundary element method to the elastostatics

This paper is entirely devoted to the demonstration of a solution for some boundary value problems of isotropic linear elastostatics with random parameters using the boundary element method. The stochastic perturbation technique in its general nth-order Taylor series expansion version is used to express all the random parameters and the state functions of the problem. These expansions inserted in the classical deterministic equilibrium statement return up to the nth-order (both PDEs and matrix) equations. Contrary to the previous implementations of the stochastic perturbation technique, any order partial derivatives with respect to the random input are derived from the deterministic structural response function (SRF) at a given point. This function is approximated using polynomials by the least-squares method from the multiple solution of the initial deterministic problem solved for the expectations of random structural parameters. First two probabilistic moments have been computed symbolically here using the computational MAPLE environment, also as the polynomial expressions including perturbation parameter ε. It should be mentioned that such a generalized perturbation approach makes it possible to analyze all types of random variables (not only Gaussian) and to compute even higher probabilistic moments with a priori given accuracy. The entire methodology can be implemented after minor modifications to analyze nonlinear phenomena for both statics and dynamics of even heterogeneous domains.     

基于同伦分析方法的随机结构静力响应求解

该文提出了一种基于同伦分析方法的求解含随机参数结构的静力响应的新方法。该方法将随机静力平衡方程重新进行同伦构造,利用含随机变量和趋近函数的同伦级数展式来表示结构的随机静力位移响应,该同伦级数的各阶确定性系数和趋近函数可通过对一系列的变形方程求解得到。由于趋近函数的引入,该同伦级数解相较于传统的摄动法有更大的收敛范围,对于含较大变异性随机参数的结构也能获得不错的求解精度。同时,该文提出了一种降维策略来提高该方法的计算效率。数值算例表明,与目前广泛应用的广义正交多项式展开法（GPC）相比,从计算精度上看,该文方法的3阶展开与GPC 2阶展开相当,该文方法的6阶展开与GPC 4阶展开相当,而计算时间上前者均明显少于后者。此外,该文方法也可以方便地应用到随机结构的几何非线性分析当中,并具有较好的计算精度和计算效率。     

Efficient spectral response of locally uncertain linear systems

It is difficult to model any real dynamical system with fully deterministic characteristics and yet, capture its behavior reasonably. Randomness arises from many sources, such as uncertain material properties, assumptions involved in structural modeling, and the stochastic nature of input forces. Thus, the random vibration analysis of systems with uncertain parameters is a crucial component of structural design and optimization procedures. In this paper, a new method is presented for fast spectral analysis of locally uncertain systems subjected to random inputs based on the response of one such system (called the nominal system). Unlike other methods, such as modal expansion methods, the proposed method is applicable to general uncertainties in the damping and stiffness matrices with the sole restriction that the system remains stable with probability one. Moreover, the proposed method yields exact responses for the perturbed systems and its accuracy is not affected by the size or magnitude of the uncertainties. However, the degree of locality of the uncertainty dictates the observed gains in computational efficiency when using the proposed method. When the uncertainty is extremely localized, one can expect gains in computational efficiency of two to three orders of magnitude while only modest gains of 2–3 times are observed when half the model is uncertain. Two numerical examples are presented to illustrate the accuracy and gains in computational efficiency of the proposed method.     

Uncertainty quantification in non-linear dynamic response of functionally graded materials plate

This study deals with the stochastic non-linear dynamic response of functionally graded materials (FGMs) plate with uncertain system properties subjected to time-dependent uniformly distributed transverse load in thermal environments. System properties, such as material properties of each constituent's material, volume fraction index, and transverse load, are taken as uncorrelated random input variables. Material properties are assumed as temperature dependent (TD). The formulation is based on higher-order shear deformation theory (HSDT) with von-Karman non-linear strain kinematics using modified C° continuity. A Newton–Raphson-based non-linear finite element method along with a first-order perturbation technique (FOPT) and Monte Carlo sampling (MCS) is outlined to examine the second-order statistics (mean, standard deviation (SD), and probability density function (PDF)) of the non-linear dynamic response of the FGM plate. The governing dynamic equation is solved by Newmark's time integration scheme. The effects of volume fraction index, load parameters, plate thickness ratios, and temperature changes with random system properties are examined through parametric studies. The present outlined approach is validated with the results available in the literature and by MCS.     

Interval spectral stochastic finite element analysis of structures with aggregation of random field and bounded parameters

This paper presents the study on non‐deterministic problems of structures with a mixture of random field and interval material properties under uncertain‐but‐bounded forces. Probabilistic framework is extended to handle the mixed uncertainties from structural parameters and loads by incorporating interval algorithms into spectral stochastic finite element method. Random interval formulations are developed based on K–L expansion and polynomial chaos accommodating the random field Young's modulus, interval Poisson's ratios and bounded applied forces. Numerical characteristics including mean value and standard deviation of the interval random structural responses are consequently obtained as intervals rather than deterministic values. The randomised low‐discrepancy sequences initialized particles and high‐order nonlinear inertia weight with multi‐dimensional parameters are employed to determine the change ranges of statistical moments of the random interval structural responses. The bounded probability density and cumulative distribution of the interval random response are then visualised. The feasibility, efficiency and usefulness of the proposed interval spectral stochastic finite element method are illustrated by three numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Probability density evolution method for dynamic response analysis of structures with uncertain parameters

Probability density evolution method is proposed for dynamic response analysis of structures with random parameters. In the present paper, a probability density evolution equation (PDEE) is derived according to the principle of preservation of probability. With the state equation expression, the PDEE is further reduced to a one-dimensional partial differential equation. The numerical algorithm is studied through combining the precise time integration method and the finite difference method with TVD schemes. The proposed method can provide the probability density function (PDF) and its evolution, rather than the second-order statistical quantities, of the stochastic responses. Numerical examples, including a SDOF system and an 8-story frame, are investigated. The results demonstrate that the proposed method is of high accuracy and efficiency. Some characteristics of the PDF and its evolution of the stochastic responses are observed. The PDFs evidence heavy variance against time. Usually, they are much irregular and far from well-known regular distribution types. Additionally, the coefficients of variation of the random parameters have significant influence on PDF and second-order statistical quantities of responses of the stochastic structure.The support of the Natural Science Funds for Distinguished Young Scholars of China (Grant No.59825105) and the Natural Science Funds for Innovative Research Groups of China (Grant No.50321803) are gratefully appreciated.     

Single machine scheduling problem with stochastic sequence-dependent setup times

In this study, we consider stochastic single machine scheduling problem. We assume that setup times are both sequence dependent and uncertain while processing times and due dates are deterministic. In the literature, most of the studies consider the uncertainty on processing times or due dates. However, in the real-world applications (i.e. plastic moulding industry, appliance assembly, etc.), it is common to see varying setup times due to labour or setup tools availability. In order to cover this fact in machine scheduling, we set our objective as to minimise the total expected tardiness under uncertain sequence-dependent setup times. For the solution of this NP-hard problem, several heuristics and some dynamic programming algorithms have been developed. However, none of these approaches provide an exact solution for the problem. In this study, a two-stage stochastic-programming method is utilised for the optimal solution of the problem. In addition, a Genetic Algorithm approach is proposed to solve the large-size problems approximately. Finally, the results of the stochastic approach are compared with the deterministic one to demonstrate the value of the stochastic solution.     

Comparison of finite element reliability methods

The spectral stochastic finite element method (SSFEM) aims at constructing a probabilistic representation of the response of a mechanical system, whose material properties are random fields. The response quantities, e.g. the nodal displacements, are represented by a polynomial series expansion in terms of standard normal random variables. This expansion is usually post-processed to obtain the second-order statistical moments of the response quantities. However, in the literature, the SSFEM has also been suggested as a method for reliability analysis. No careful examination of this potential has been made yet. In this paper, the SSFEM is considered in conjunction with the first-order reliability method (FORM) and with importance sampling for finite element reliability analysis. This approach is compared with the direct coupling of a FORM reliability code and a finite element code. The two procedures are applied to the reliability analysis of the settlement of a foundation lying on a randomly heterogeneous soil layer. The results are used to make a comprehensive comparison of the two methods in terms of their relative accuracies and efficiencies.     

Variability response functions for beams with nonlinear constitutive laws

The ability to determine probabilistic information of response quantities in structural mechanics (e.g. displacements, stresses) is restricted due to lack of information on the probabilistic characteristics of uncertain system parameters. The concept of the Variability Response Function (VRF) has been proposed as a means to systematically capture the effect of the stochastic spectral characteristics of uncertain system parameters modeled by homogeneous stochastic fields on the uncertain structural response. The key property of the VRF in its classical sense is its independence from the marginal probability distribution function (PDF) and the spectral density function (SDF) of the uncertain system parameters (it depends only on the deterministic structural configuration and boundary conditions). In this paper, the existence, the uniqueness, and the SDF- and PDF-independence of a variability response function is formally proven for the first time for statically determinate beam structures following a specific class of nonlinear constitutive laws (power laws). For statically indeterminate nonlinear structures, the generalized variability response function (GVRF) methodology is shown to produce GVRFs for statically indeterminate nonlinear beams with a square-root constitutive law that are almost SDF-independent and only mildly dependent on the marginal PDF. This PDF-dependence is not significant and all GVRFs computed in this study have very similar shapes. This is important as it implies that conclusions related to the effect of correlation length scales on the response uncertainty can be inferred in general. However, the GVRF methodology for nonlinear statically indeterminate structures is only suitable when a closed-form expression is known to exist for the VRF of statically determinate structures having the same constitutive law.     

Fast numerical methods for robust optimal design

A fast numerical approach for robust design optimization is presented. The core of the method is based on the state-of-the-art fast numerical methods for stochastic computations with parametric uncertainty. These methods employ generalized polynomial chaos (gPC) as a high-order representation for random quantities and a stochastic Galerkin (SG) or stochastic collocation (SC) approach to transform the original stochastic governing equations to a set of deterministic equations. The gPC-based SG and SC algorithms are able to produce highly accurate stochastic solutions with (much) reduced computational cost. It is demonstrated that they can serve as efficient forward problem solvers in robust design problems. Possible alternative definitions for robustness are also discussed. Traditional robust optimization seeks to minimize the variance (or standard deviation) of the response function while optimizing its mean. It can be shown that although variance can be used as a measure of uncertainty, it is a weak measure and may not fully reflect the output variability. Subsequently a strong measure in terms of the sensitivity derivatives of the response function is proposed as an alternative robust optimization definition. Numerical examples are provided to demonstrate the efficiency of the gPC-based algorithms, in both the traditional weak measure and the newly proposed strong measure.     

A novel shape function approach of dynamic load identification for the structures with interval uncertainty

Aiming at uncertain structures, a computational inverse approach is proposed to identify the dynamic load on the basis of the shape function method and interval analysis. The forward model for an uncertain structure is established through the relationship between the uncertain load vector and the assembly matrix of the uncertain responses of the shape function loads in each discrete element in time domain. The uncertainty is characterized by the interval with a closed bounded set of uncertain parameters. On the basis of interval analysis method, the load identification for uncertain structures can be transformed into two kinds of deterministic inverse problems, namely the deterministic dynamic load identification and the first order derivatives of the unknown load to each parameter both at the midpoints of the uncertain parameters. In order to eliminate the ill-posedness of inversion, the regularization method is adopted to solve the deterministic equations. Two numerical examples demonstrates the effectiveness of the proposed method, and example one also gives the identified result using Monte Carlo method to compare with that using the proposed method.