Improved dynamic analysis of structures with mechanical uncertainties under deterministic input

This paper addresses the dynamic analysis of linear systems with uncertain parameters subjected to deterministic excitation. The conventional methods dealing with stochastic structures are based on series expansion of stochastic quantities with respect to uncertain parameters, by means of either Taylor expansion, perturbation technique or Neumann expansion and evaluate the first- and second-order moments of the response by solving deterministic equations. Unfortunately, these methods lead to significant error when the coefficients of variation of uncertainties are relatively large. Herein, an improved first-order perturbation approach is proposed, which considers the stochastic quantities as the sum of their mean and deviation. Comparisons with conventional second-order perturbation approach and Monte Carlo simulations illustrate the efficiency of the proposed method. Applications are discussed in order to investigate the influence of mass, damping and stiffness uncertainty on the dynamic response of the system.     

A multi-stage stochastic programming approach for production planning with uncertainty in the quality of raw materials and demand

Motivated by the challenges encountered in sawmill production planning, we study a multi-product, multi-period production planning problem with uncertainty in the quality of raw materials and consequently in processes yields, as well as uncertainty in products demands. As the demand and yield own different uncertain natures, they are modelled separately and then integrated. Demand uncertainty is considered as a dynamic stochastic data process during the planning horizon, which is modelled as a scenario tree. Each stage in the demand scenario tree corresponds to a cluster of time periods, for which the demand has a stationary behaviour. The uncertain yield is modelled as scenarios with stationary probability distributions during the planning horizon. Yield scenarios are then integrated in each node of the demand scenario tree, constituting a hybrid scenario tree. Based on the hybrid scenario tree for the uncertain yield and demand, a multi-stage stochastic programming (MSP) model is proposed which is full recourse for demand scenarios and simple recourse for yield scenarios. We conduct a case study with respect to a realistic scale sawmill. Numerical results indicate that the solution to the multi-stage stochastic model is far superior to the optimal solution to the mean-value deterministic and the two-stage stochastic models.     

Optimization of aluminium sheet hot stamping process using a multi-objective stochastic approach

This article aims to investigate the means to obtain optimal hot stamping process parameters and the influence of the stochastic variability of these parameters on forming quality. A multi-objective stochastic approach, integrating response surface methodology (RSM), multi-objective genetic algorithm optimization non-dominated sorting genetic algorithm II (NSGA-II) and the Monte Carlo simulation (MCS) method is proposed in this article to achieve this goal. RSM was used to establish the relationship between the process parameters and forming quality indices. NSGA-II was utilized to obtain a Pareto frontier, which consists of a series of optimal process parameters. The MCS method was employed to study and reduce the influence of a stochastic property of these process parameters on forming quality. The results confirmed the efficiency of the proposed multi-objective stochastic approach during optimization of the hot stamping process. Robust optimal process parameters guaranteeing good forming quality were also obtained using this approach.     

An efficient framework for the reliability-based design optimization of large-scale uncertain and stochastic linear systems

This paper is focused on the development of an efficient reliability-based design optimization algorithm for solving problems posed on uncertain linear dynamic systems characterized by large design variable vectors and driven by non-stationary stochastic excitation. The interest in such problems lies in the desire to define a new generation of tools that can efficiently solve practical problems, such as the design of high-rise buildings in seismic zones, characterized by numerous free parameters in a rigorously probabilistic setting. To this end a novel decoupling approach is developed based on defining and solving a limited sequence of deterministic optimization sub-problems. In particular, each sub-problem is formulated from information pertaining to a single simulation carried out exclusively in the current design point. This characteristic drastically limits the number of simulations necessary to find a solution to the original problem while making the proposed approach practically insensitive to the size of the design variable vector. To demonstrate the efficiency and strong convergence properties of the proposed approach, the structural system of a high-rise building defined by over three hundred free parameters is optimized under non-stationary stochastic earthquake excitation.     

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。     

Stochastic analysis and robust optimization for a deck lid inner panel stamping

FE-simulation and optimization are widely used in the stamping process to improve design quality and shorten development cycle. However, the current simulation and optimization may lead to non-robust results due to not considering the variation of material and process parameters. In this study, a novel stochastic analysis and robust optimization approach is proposed to improve the stamping robustness, where the uncertainties are involved to reflect manufacturing reality. A meta-model based stochastic analysis method is developed, where FE-simulation, uniform design and response surface methodology (RSM) are used to construct meta-model, based on which Monte-Carlo simulation is performed to predict the influence of input parameters variation on the final product quality. By applying the stochastic analysis, uniform design and RSM, the mean and the standard deviation (SD) of product quality are calculated as functions of the controllable process parameters. The robust optimization model composed of mean and SD is constructed and solved, the result of which is compared with the deterministic one to show its advantages. It is demonstrated that the product quality variations are reduced significantly, and quality targets (reject rate) are achieved under the robust optimal solution. The developed approach offers rapid and reliable results for engineers to deal with potential stamping problems during the early phase of product and tooling design, saving more time and resources.     

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.     

Polymorphic uncertain nonlinear programming approach for maximizing the capacity of V-belt driving

In this paper, a polymorphic uncertain nonlinear programming (PUNP) approach is developed to formulate the problem of maximizing the capacity in a system of V-belt driving with uncertainties. The constructed optimization model is found to consist of a nonlinear objective function and some nonlinear constraints with some parameters which are of uncertain nature. These uncertain parameters are interval parameters, random interval parameters, fuzzy parameters or fuzzy interval parameters. To find a robust solution of the problem, a deterministic equivalent formulation (DEF) is established for the polymorphic uncertain nonlinear programming model. For a given satisfaction level, this DEF turns out to be a nonlinear programming involving only interval parameters. A solution method, called a sampling based interactive method, is developed such that a robust solution of the original model with polymorphic uncertainties is obtained by using standard smooth optimization techniques. The proposed method is applied into a real-world design of V-belt driving, and the results indicate that both the PUNP approach and the developed algorithm are useful to the optimization problem with polymorphic uncertainty.     

Identification of stochastic loads applied to a non-linear dynamical system using an uncertain computational model and experimental responses

The paper is devoted to the identification of stochastic loads applied to a non-linear dynamical system for which experimental dynamical responses are available. The identification of the stochastic load is performed using a simplified computational non-linear dynamical model containing both model uncertainties and data uncertainties. Uncertainties are taken into account in the context of the probability theory. The stochastic load which has to be identified is modelled by a stationary non-Gaussian stochastic process for which the matrix-valued spectral density function is uncertain and is then modelled by a matrix-valued random function. The parameters to be identified are the mean value of the random matrix-valued spectral density function and its dispersion parameter. The identification problem is formulated as two optimization problems using the computational stochastic model and experimental responses. A validation of the theory proposed is presented in the context of tubes bundles in Pressurized Water Reactors.     

2.5维C/SiC复合材料弹性参数不确定性识别方法研究

提出了一种2.5维C/SiC编织复合材料弹性参数不确定性识别方法。采用刚度平均法获得复合材料等效弹性参数理论预测值。选取对结构动态特性影响较大的3个弹性参数E11,E22和G12作为待识别参数;在确定性识别结果基础上,采用拉丁超立方体采样构造随机试验样本,开展不确定性参数识别方法仿真研究。仿真结果表明,针对考虑弹性参数不确定性的2.5维C/SiC复合材料,采用所提出的方法能够准确识别材料弹性参数的均值与标准差,建立反映实际结构动态特性统计意义的精确动力学模型。     

Evaluation model and algorithm of product disassembly process with stochastic feature

Disassembly planning is considered as the optimization of disassembly sequences with the target of the shortest disassembly time, the lowest disassembly cost, and the minimum disassembly energy consumption. However, obsolete products suffer from the influence of a variety of uncertainties, the disassembly process of products has the strong uncertain feature. Traditionally, to account for this uncertainty, each removal operation or removal task is assumed to be an activity or event with certain probability, and the determination of the optimal path of a disassembly process is merely a probabilistic planning problem based on this assumption. In this article, based on the established stochastic disassembly network graph, combined with different disassembly decision-making criterion, typical stochastic models for disassembly time analysis are developed. In addition, a two-phase approach is proposed to solve the typical stochastic models. Initially, according to different removal probability density functions, disassembly probability density functions of feasible disassembly paths are determined by a time-domain method or frequency-domain method, and additionally, after the disassembly probability density functions have been obtained, the quantitative evaluation of a product disassembly process and stochastic optimization of feasible disassembly paths are realized by a numerical solution method. Finally, a numerical example is illustrated to test the proposed concepts and the effectiveness of the proposed approach.     

Uncertainty quantification using evidence theory in multidisciplinary design optimization

Advances in computational performance have led to the development of large-scale simulation tools for design. Systems generated using such simulation tools can fail in service if the uncertainty of the simulation tool's performance predictions is not accounted for. In this research an investigation of how uncertainty can be quantified in multidisciplinary systems analysis subject to epistemic uncertainty associated with the disciplinary design tools and input parameters is undertaken. Evidence theory is used to quantify uncertainty in terms of the uncertain measures of belief and plausibility. To illustrate the methodology, multidisciplinary analysis problems are introduced as an extension to the epistemic uncertainty challenge problems identified by Sandia National Laboratories.After uncertainty has been characterized mathematically the designer seeks the optimum design under uncertainty. The measures of uncertainty provided by evidence theory are discontinuous functions. Such non-smooth functions cannot be used in traditional gradient-based optimizers because the sensitivities of the uncertain measures are not properly defined. In this research surrogate models are used to represent the uncertain measures as continuous functions. A sequential approximate optimization approach is used to drive the optimization process. The methodology is illustrated in application to multidisciplinary example problems.     

Optimization of structures under material parameter uncertainty using evidence theory

An evidence-based approach is developed for optimization of structural components under material parameter uncertainty. The approach is applied to evidence-based design optimization (EBDO) of externally stiffened circular tubes under axial impact load using an isotropic–elastic–plastic plasticity model to simulate dynamic material behaviour. Uncertainty modelling considers the changes in material parameters that are caused by variability in material properties as well as incertitude and errors in experimental data and procedure to determine the material parameters. Spatial variation of material parameters across the structural component is modelled using a field joint belief structure and propagated for the calculation of evidence-based objective function and design constraints. Surrogate models are used in both uncertainty propagation and solution of the optimization problem. The methodology and the solution to the EBDO example problem are presented and discussed.     

Analysis and optimization of clutch judder based on a hybrid uncertain model with random and interval variables

Clutch judder has serious impacts on the noise, vibration and harshness performance. In this article, a simplified dynamic model with nonlinear friction torque is developed to simulate clutch judder, and the stability and dynamic response of the clutch are analysed. The real part of the judder modal eigenvalue, the moment when the clutch enters the stick state and the fluctuation level of the driving part of the clutch are treated as the evaluation indices. An uncertain hybrid model with random and interval variables is used to describe the uncertainty of parameters and a hybrid perturbation vertex method is formulated to compute the uncertainty. Furthermore, parameters with high sensitivities are used as design variables and uncertainty-based optimization is conducted to reduce clutch judder. The optimization results strongly validate that the proposed method is very effective in improving the robustness of the clutch judder performance.     

引入不确定参数的汽车盘式制动器振动稳定性分析

针对汽车制动器的噪声抑制问题,基于可靠性分析理论,将蒙特卡洛法与响应面法相结合,提出了一种汽车盘式制动器系统振动稳定性的可靠性分析方法。该方法针对制动噪声产生具有不确定性的特点,引入随机和区间不确定性参数对制动器系统进行描述,建立包含随机参数和区间参数的制动器不稳定特征值的响应面近似模型,进而采用Sobol′全局灵敏度分析法和蒙特卡洛法分别对不确定参数的全局灵敏度和系统稳定性的可靠度进行分析。用该方法对某车的浮钳盘式制动器系统进行研究,分析了系统稳定性的可靠度和不确定参数的全局灵敏度,甄别了不确定性参数对系统稳定性的影响,并从可靠性角度提出了改善制动器系统振动稳定性的工程措施。     

Probabilistic nonconvex constrained optimization with fixed number of function evaluations

A methodology is proposed for the efficient solution of probabilistic nonconvex constrained optimization problems with uncertain. Statistical properties of the underlying stochastic generator are characterized from an initial statistical sample of function evaluations. A diffusion manifold over the initial set of data points is first identified and an associated basis computed. The joint probability density function of this initial set is estimated using a kernel density model and an Itô stochastic differential equation (ISDE) constructed with this model as its invariant measure. This ISDE is adapted to fluctuate around the manifold yielding additional joint realizations of the uncertain parameters, design variables, and function values, which are obtained as solutions of the ISDE. The expectations in the objective function and constraints are then accurately evaluated without performing additional function evaluations. The methodology brings together novel ideas from manifold learning and stochastic Hamiltonian dynamics to tackle an outstanding challenge in stochastic optimization. Three examples are presented to highlight different aspects of the proposed methodology.     

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.     

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.     

Robust design with imprecise random variables and its application in hydrokinetic turbine optimization

In robust design, uncertainty is commonly modelled with precise probability distributions. In reality, the distribution types and distribution parameters may not always be available owing to limited data. This research develops a robust design methodology to accommodate the mixture of both precise and imprecise random variables. By incorporating the Taguchi quality loss function and the minimax regret criterion, the methodology mitigates the effects of not only uncertain parameters but also uncertainties in the models of the uncertain parameters. Hydrokinetic turbine systems are a relatively new alternative energy technology, and both precise and imprecise random variables exist in the design of such systems. The developed methodology is applied to the robust design optimization of a hydrokinetic turbine system. The results demonstrate the effectiveness of the proposed methodology.     

基于薄层单元的螺栓连接结构接触面不确定性参数识别

提出了一种螺栓连接接触面不确定性参数识别方法,首先采用薄层单元对接触面进行参数化,然后根据不确定性识别方法识别薄层单元参数。以四螺栓搭接结构试验模型为研究对象,开展接触面不确定性参数识别方法仿真研究。采用Monte-Carlo方法构造待识别参数真实值样本,代入基准有限元模型中计算获得具有统计意义的仿真试验数据;采用不确定性参数识别方法预测薄层单元参数均值与标准差,仿真结果表明:该方法能够较为准确的模拟接触面法向和切向接触刚度,并显著提高连接结构的建模效率,建立反映真实结构动态性能统计特征的有限元模型。