动力可靠度约束下基于概率测度变换-全局收敛移动渐近线法的结构优化设计EI北大核心CSCD

基于动力可靠度的结构优化是实现随机动力系统优化设计的重要途径。针对设计变量为系统中部分随机变量分布均值的情形,提出了一种基于动力可靠度的结构优化设计方法。在该方法中,通过概率密度演化理论实现了结构动力可靠度的高效分析。在此基础上,结合概率测度变换,可以在不增加任何确定性结构分析的前提下,实现动力可靠度对设计变量的灵敏度分析。进而,通过将上述概率密度演化-测度变换方法嵌入全局收敛移动渐近线法,实现了基于动力可靠度的结构优化设计问题的高效求解。数值算例的结果表明,所提方法可以显著降低结构分析次数,具有较高的效率与稳健性。     

一种考虑概率分布的鲁棒优化模型

文章以随机规划中的机会约束思想为指导,根据随机参数的概率分布情况,提出了两种鲁棒性条件约束,并在此基础上建立了一种新的鲁棒优化模型,使模型的可行解控制在一定的鲁棒性指标的范围内。该模型不但可处理约束两端同时含有随机参数的情况,还可以方便地推广到非线性模型中。仿真实例说明了模型的有效性。     

Generating complete disassembly sequences by utilising two-dimensional views

This research work proposes a novel method to generate the complete disassembly sequences for mechanical products by utilising the part interference matrix which contains the removal directions of the parts and the part connection graph which indicates the contact among the parts in the assembly. Contrary to the earlier methods, the proposed method considers the two-dimensional views generated from the computer-aided design assembly model for automatically identifying the part removal directions and for generating the part connection graph. Rules are formulated in the proposed method to identify the part removal directions effectively and also to reduce the size of part connection graph, since the method of identifying the part removal directions and the size of the formulated graph plays a vital role in the generation of disassembly sequence. Finally, a heuristic method is developed to generate the best feasible disassembly sequences. The effectiveness of the approach is illustrated with three examples.     

Solutions of the FPK equation for time-delayed dynamical systems with the continuous time approximation method

This paper presents a method of finite-dimensional Markov process (FDMP) approximation for stochastic dynamical systems with time delay and numerical solutions of probability density functions of the systems. Solutions of probability density functions of time-delayed systems are rare in the literature. The FDMP method preserves the standard state space format of the system, and allows us to apply all the existing methods and theories for analysis and control of stochastic dynamical systems and to compute the probability density functions efficiently. The solutions of the FPK equation for a linear time-delayed stochastic system are presented. The effects of different spectral differentiation schemes for the FDMP method on the probability density functions are compared.     

Risk-averse two-stage stochastic programs in furniture plants

We present two-stage stochastic mixed 0–1 optimization models to hedge against uncertainty in production planning of typical small-scale Brazilian furniture plants under stochastic demands and setup times. The proposed models consider cutting and drilling operations as the most limiting production activities, and synchronize them to avoid intermediate work-in-process. To design solutions less sensitive to changes in scenarios, we propose four models that perceive the risk reductions over the scenarios differently. The first model is based on the minimax regret criteria and optimizes a worst-case scenario perspective without needing the probability of the scenarios. The second formulation uses the conditional value-at-risk as the risk measure to avoid solutions influenced by a bad scenario with a low probability. The third strategy is a mean-risk model based on the upper partial mean that aggregates a risk term in the objective function. The last approach is a restricted recourse approach, in which the risk preferences are directly considered in the constraints. Numerical results indicate that it is possible to achieve significant risk reductions using the risk-averse strategies, without overly sacrificing average costs.     

Dynamic response and reliability analysis of structures with uncertain parameters

An original approach for dynamic response and reliability analysis of stochastic structures is proposed. The probability density evolution equation is established which implies that incremental rate of the probability density function is related to the structural response velocity. Therefore, the response analysis of stochastic structures becomes an initial‐value partial differential equation problem. For the dynamic reliability problem, the solution can be derived through solving the probability density evolution equation with an initial value condition and an absorbing boundary condition corresponding to specified failure criterion. The numerical algorithm for the proposed method is suggested by combining the precise time integration method and the finite difference method with TVD scheme. To verify and validate the proposed method, a SDOF system and an 8‐storey frame with random parameters are investigated in detail. In the SDOF system, the response obtained by the proposed method is compared with the counterparts by the exact solution. The responses and the reliabilities of a frame with random stiffness, subject to deterministic excitation or random excitation, are evaluated by the proposed method as well. The mean, the standard deviation and the reliabilities are compared, respectively, with the Monte Carlo simulation. The numerical examples verify that the proposed method is of high accuracy and efficiency. Moreover, it is found that the probability transition of structural responses is like water flowing in a river with many whirlpools, showing complexity of probability transition process of the stochastic dynamic responses. Copyright © 2004 John Wiley & Sons, Ltd.     

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.     

Comparison of PDEM and MCS: Accuracy and efficiency

The accuracy and efficiency of two methods for stochastic analysis, the probability density evolution method (PDEM) and the Monte Carlo simulation (MCS) method, are compared in terms of how well they reflect the physical properties of stochastic systems. The basic principle and the numerical implementation details of PDEM and MCS are revisited. The analytical solutions of generalized probability density evolution equation (GDEE) for three typical stochastic systems are given and are to be used as the basis for comparing the two methods. It is verified that, with the rational partition of the probability space, the PDEM provides a continuous and complete reflection of physical properties over the whole probability space. Meanwhile, with the help of the numerical solution of GDEE, PDEM is efficient and accurate to describe the process of the probability density evolution of stochastic systems. In contrast, the random samples in the MCS may not reflect the physical properties of a stochastic system adequately, and the local cluster of sample points may cause redundant calculation, which leads to lower computational efficiency. Through three typical numerical examples, the paper compares the accuracy and efficiency of PDEM and MCS specifically. It is shown that, as the numerical approaches for the stochastic response of a system, the PDEM could get much higher numerical accuracy than MCS with the same number of samples. To achieve the same level of calculation accuracy, MCS needs a much higher number of samples than PDEM.     

Novel orthogonal simulated annealing with fractional factorial analysis to solve global optimization problems

A classical simulated annealing (SA) method is a generic probabilistic and heuristic approach to solving global optimization problems. It uses a stochastic process based on probability, rather than a deterministic procedure, to seek the minima or maxima in the solution space. Although the classical SA method can find the optimal solution to most linear and nonlinear optimization problems, the algorithm always requires numerous numerical iterations to yield a good solution. The method also usually fails to achieve optimal solutions to large parameter optimization problems. This study incorporates well-known fractional factorial analysis, which involves several factorial experiments based on orthogonal tables to extract intelligently the best combination of factors, with the classical SA to enhance the numerical convergence and optimal solution. The novel combination of the classical SA and fractional factorial analysis is termed the orthogonal SA herein. This study also introduces a dynamic penalty function to handle constrained optimization problems. The performance of the proposed orthogonal SA method is evaluated by computing several representative global optimization problems such as multi-modal functions, noise-corrupted data fitting, nonlinear dynamic control, and large parameter optimization problems. The numerical results show that the proposed orthogonal SA method markedly outperforms the classical SA in solving global optimization problems with linear or nonlinear objective functions. Additionally, this study addressed two widely used nonlinear functions, proposed by Keane and Himmelblau to examine the effectiveness of the orthogonal SA method and the presented penalty function when applied to the constrained problems. Moreover, the orthogonal SA method is applied to two engineering optimization design problems, including the designs of a welded beam and a coil compression spring, to evaluate the capacity of the method for practical engineering design. The computational results show that the proposed orthogonal SA method is effective in determining the optimal design variables and the value of objective function.     

Probabilistic crack trajectory analysis by a dimension reduction method

This paper proposes a novel analysis method of stochastic crack trajectory based on a dimension reduction approach. The developed method allows efficiently estimating the statistical moments, probability density function and cumulative distribution function of the crack trajectory for cracked elastic structures considering the randomness of the loads, material properties and crack geometries. First, the traditional dimension reduction method is extended to calculate the first four moments of the crack trajectory, in which the responses are eigenvectors rather than scalars. Then the probability density function and cumulative distribution function of the crack trajectory can be obtained using the maximum entropy principle constrained by the calculated moments. Finally, the simulation of the crack propagation paths is realized by using the scaled boundary finite element method. The proposed method is well validated by four numerical examples performed on varied cracked structures. It is demonstrated that this method outperforms the Monte Carlo simulation in terms of computational efficiency, and in the meanwhile, it has an acceptable computational accuracy.     

A multi-objective distribution-free model and method for stochastic disassembly line balancing problem

End-of-life product recycling is a hot research topic in recent years, which can reduce the waste and protect the environment. To disassemble products, the disassembly line balancing is a principal problem that selects tasks and assigns them to a number of workstations under stochastic task processing times. In existing works, stochastic task processing times are usually estimated by probability distributions or fuzzy numbers. However, in real-life applications, only their partial information is accessible. This paper studies a bi-objective stochastic disassembly line balancing problem to minimise the line design cost and the cycle time, with only the knowledge of the mean, standard deviation and upper bound of stochastic task processing times. For the problem, a bi-objective chance-constrained model is developed, which is further approximated into a bi-objective distribution-free one. Based on the problem analysis, two versions of the ?-constraint method are proposed to solve the transformed model. Finally, a fuzzy-logic technique is adapted to propose a preferable solution for decision makers according to their preferences. A case study is presented to illustrate the validity of the proposed models and algorithms. Experimental results on 277 benchmark-based and randomly generated instances show the efficiency of the proposed methods.     

A reduced polynomial chaos expansion method for the stochastic finite element analysis

The stochastic finite element analysis of elliptic type partial differential equations is considered. A reduced method of the spectral stochastic finite element method using polynomial chaos is proposed. The method is based on the spectral decomposition of the deterministic system matrix. The reduction is achieved by retaining only the dominant eigenvalues and eigenvectors. The response of the reduced system is expanded as a series of Hermite polynomials, and a Galerkin error minimization approach is applied to obtain the deterministic coefficients of the expansion. The moments and probability density function of the solution are obtained by a process similar to the classical spectral stochastic finite element method. The method is illustrated using three carefully selected numerical examples, namely, bending of a stochastic beam, flow through porous media with stochastic permeability and transverse bending of a plate with stochastic properties. The results obtained from the proposed method are compared with classical polynomial chaos and direct Monte Carlo simulation results.     

脉动风速连续随机场的降维模拟

基于标准正交随机变量的波数谱表示,通过定义标准正交随机变量集的随机函数形式,建立了连续时空随机场模拟的波数谱-随机函数方法。同时,引入快速傅里叶变换（FFT）的算法,极大地提高了波数谱-随机函数方法的模拟效率。在波数谱-随机函数模拟方法中,仅需两个基本随机变量即可在概率密度层次上描述时空随机场的概率特性,并利用数论方法选取基本随机变量的代表性点集,实现对连续时空随机场模拟的降维表达。数值算例表明,当模拟相同数量的样本时,综合考虑模拟的效率和精度两方面,该文方法与传统的波数谱表示方法不分伯仲,但该文方法所需的基本随机变量最少,生成的代表性样本数量少且构成一个完备的概率集,从而可结合概率密度演化理论实现结构随机动力反应及动力可靠度的精细化分析。最后,结合Kaimal风速谱及Davenport空间相干函数模型,模拟了水平向脉动风速连续随机场,验证了该文方法的有效性和优越性。     

Computing the survival probability density function in jump-diffusion models: A new approach based on radial basis functions

We propose a numerical method to compute the survival (first-passage) probability density function in jump-diffusion models. This function is obtained by numerical approximation of the associated Fokker–Planck partial integro-differential equation, with suitable boundary conditions and delta initial condition. In order to obtain an accurate numerical solution, the singularity of the Dirac delta function is removed using a change of variables based on the fundamental solution of the pure diffusion model. This approach allows to transform the original problem to a regular problem, which is solved using a radial basis functions (RBFs) meshless collocation method. In particular the RBFs approximation is carried out in conjunction with a suitable change of variables, which allows to use radial basis functions with equally spaced centers and at the same time to obtain a sharp resolution of the gradients of the survival probability density function near the barrier. Numerical experiments are presented in which several different kinds of radial basis functions are employed. The results obtained reveal that the numerical method proposed is extremely accurate and fast, and performs significantly better than a conventional finite difference approach.     

Value recovery from end-of-use products facilitated by automated dismantling planning

Circular economy is a promising business model that promotes sustainable development by closing material loops. Making progress toward a circular economy requires the recovery of valuable materials and components from end-of-use products and subsequent reuse of them in some form, thus maximizing the utility of components and materials. Currently, end-of-use products value recovery is carried out without a rational planning, causing the loss of the recoverable value embedded in material and components. To address this problem, dismantling planning and appropriate technologies should be employed to improve the economic performance of end-of-use products value recovery. In this paper, a two-stage dismantling planning method is proposed to find a profitable end-of-use strategy. In the first stage of this method, disassembly optimization model is constructed and can be executed to obtain the optimal disassembly plan allowing maximum preservation of component function value, in a preservative disassembly scenario. To speed up the modeling, a method for automatic generation of AND/OR graph—a structure of incorporating all possible disassembly operations and associated subassemblies, is presented. In the second stage of the method, in order to increase profitability, Pareto analysis is employed to identify bottlenecks to disassembly and automated/destructive technologies are considered to remove the bottlenecks. A hard disk drive serves as a case study to illustrate the suggested method.     

Integrated assembly and disassembly sequence planning using a GA approach

In a product life cycle, an assembly sequence is required to produce a new product at the start, whereas a disassembly sequence is needed at the end. In typical assembly and disassembly sequence planning approaches, the two are performed as two independent tasks. In this way, a good assembly sequence may contradict the cost considerations in the disassembly sequence, and vice versa. In this research, an integrated assembly and disassembly sequence planning model is presented. First, an assembly precedence graph (APG) and a disassembly precedence graph (DPG) are modelled. The two graphs are transformed into an assembly precedence matrix (APM) and a disassembly precedence matrix (DPM). Second, a two-loop genetic algorithm (GA) method is applied to generate and evaluate the solutions. The outer loop of the GA method performs assembly sequence planning. In the inner loop, the reverse order of the assembly sequence solution is used as the initial solution for disassembly sequence planning. A cost objective by integrating the assembly costs and disassembly costs is formulated as the fitness function. The test results show that the developed method using the GA approach is suitable and efficient for the integrated assembly and disassembly sequence planning. Example products are demonstrated and discussed.     

Expert Enhanced Coloured Stochastic Petri Net and its application in assembly/disassembly

This article comprises of an Expert Enhanced Coloured Stochastic Petri Net (rule base system, or RBS) for modelling and analysing assembly/disassembly systems. RBSs are an Enhanced High-level Petri net extended with Close-World-Assumption (CWA). Traditional Petri nets can be used to model RBSs containing explicitly described knowledge. The main focus was is to facilitate and analyse the process planning activities of assembly/disassembly. The advantages of the new modelling approach were: (1) consideration of the non-desirable events, (2) occurrence of assembly/disassembly tasks with regard to colour of the tokens and utilization of probability concept to determine feasible steps, (3) establishing a relationship among components by means of arc labels, and (4) deeper insight into the assembly/disassembly process using high- and low-level petri nets.     

A tractable approximation of non-convex chance constrained optimization with non-Gaussian uncertainties

Chance constrained optimization problems in engineering applications possess highly nonlinear process models and non-convex structures. As a result, solving a nonlinear non-convex chance constrained optimization (CCOPT) problem remains as a challenging task. The major difficulty lies in the evaluation of probability values and gradients of inequality constraints which are nonlinear functions of stochastic variables. This article proposes a novel analytic approximation to improve the tractability of smooth non-convex chance constraints. The approximation uses a smooth parametric function to define a sequence of smooth nonlinear programs (NLPs). The sequence of optimal solutions of these NLPs remains always feasible and converges to the solution set of the CCOPT problem. Furthermore, Karush–Kuhn–Tucker (KKT) points of the approximating problems converge to a subset of KKT points of the CCOPT problem. Another feature of this approach is that it can handle uncertainties with both Gaussian and/or non-Gaussian distributions.     

The probability density evolution method for dynamic response analysis of non‐linear stochastic structures

The probability density evolution method (PDEM) for dynamic responses analysis of non‐linear stochastic structures is proposed. In the method, the dynamic response of non‐linear stochastic structures is firstly expressed in a formal solution, which is a function of the random parameters. In this sense, the dynamic responses are mutually uncoupled. A state equation is then constructed in the augmented state space. Based on the principle of preservation of probability, a one‐dimensional partial differential equation in terms of the joint probability density function is set up. The numerical solving algorithm, where the Newmark‐Beta time‐integration algorithm and the finite difference method with Lax–Wendroff difference scheme are brought together, is studied. In the numerical examples, free vibration of a single‐degree‐of‐freedom non‐linear conservative system and dynamic responses of an 8‐storey shear structure with bilinear hysteretic restoring forces, subjected to harmonic excitation and seismic excitation, respectively, are investigated. The investigations indicate that the probability density functions of dynamic responses of non‐linear stochastic structures are usually irregular and far from the well‐known distribution types. They exhibit obvious evolution characteristics. The comparisons with the analytical solution and Monte Carlo simulation method demonstrate that the proposed PDEM is of fair accuracy and efficiency. Copyright © 2005 John Wiley & Sons, Ltd.