Biased Monte Carlo optimization: the basic approach

It is well-known that the Monte Carlo method is very successful in tackling several kinds of system simulations.It often happens that one has to deal with rare events, and the use of a variance reduction technique is almost mandatory, in order to have Monte Carlo efficient applications. The main issue associated with variance reduction techniques is related to the choice of the value of the biasing parameter. Actually, this task is typically left to the experience of the Monte Carlo user, who has to make many attempts before achieving an advantageous biasing.A valuable result is provided: a methodology and a practical rule addressed to establish an a priori guidance for the choice of the optimal value of the biasing parameter. This result, which has been obtained for a single component system, has the notable property of being valid for any multicomponent system.In particular, in this paper, the exponential and the uniform biases of exponentially distributed phenomena are investigated thoroughly.     

Reliability importance analysis of Markovian systems at steady state using perturbation analysis

Sensitivity analysis has been primarily defined for static systems, i.e. systems described by combinatorial reliability models (fault or event trees). Several structural and probabilistic measures have been proposed to assess the components importance. For dynamic systems including inter-component and functional dependencies (cold spare, shared load, shared resources, etc.), and described by Markov models or, more generally, by discrete events dynamic systems models, the problem of sensitivity analysis remains widely open. In this paper, the perturbation method is used to estimate an importance factor, called multi-directional sensitivity measure, in the framework of Markovian systems. Some numerical examples are introduced to show why this method offers a promising tool for steady-state sensitivity analysis of Markov processes in reliability studies.     

Reliability‐based topology optimization of structures under stress constraints

In this paper, we propose an approach for reliability‐based design optimization where a structure of minimum weight subject to reliability constraints on the effective stresses is sought. The reliability‐based topology optimization problem is formulated by using the performance measure approach, and the sequential optimization and reliability assessment method is employed. This strategy allows for decoupling the reliability‐based topology optimization problem into 2 steps, namely, deterministic topology optimization and reliability analysis. In particular, the deterministic structural optimization problem subject to stress constraints is addressed with an efficient methodology based on the topological derivative concept together with a level‐set domain representation method. The resulting algorithm is applied to some benchmark problems, showing the effectiveness of the proposed approach.     

Product Assembly Cost Estimation Based on Artificial Neural Networks

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Importance sampling for reliability assessment of dynamic systems under general random process excitation

This paper develops a reliability assessment method for dynamic systems subjected to a general random process excitation. Safety assessment using direct Monte Carlo simulation is computationally expensive, particularly when estimating low probabilities of failure. The Girsanov transformation-based reliability assessment method is a computationally efficient approach intended for dynamic systems driven by Gaussian white noise, and this approach can be extended to random process inputs that can be represented as transformations of Gaussian white noise. In practice, dynamic systems may be subjected to inputs that may be better modeled as non-Gaussian and/or non-stationary random processes, which are not easily transformable to Gaussian white noise. We propose a computationally efficient scheme, based on importance sampling, which can be implemented directly on a general class of random processes — both Gaussian and non-Gaussian, and stationary and non-stationary. We demonstrate that this approach is in fact equivalent to Girsanov transformation when the uncertain inputs are Gaussian white noise processes. The proposed approach is demonstrated on a linear dynamic system driven by Gaussian white noise and Brownian bridge processes, a multi-physics aero-thermo-elastic model of a flexible panel subjected to hypersonic flow, and a nonlinear building frame subjected to non-stationary non-Gaussian random process excitation.     

Dynamic reliability analysis of gear transmission system based on sparse grid numerical integration and saddle-point approximation method

Gear systems are widely used in various mechanical transmission systems. This paper aims to develop an effective and practical method for dynamic reliability analysis of gear transmission system. The proposed method can comprehensively evaluate the dynamic reliability of gear transmission system by adopting the fourth-moment SPA method. First, a nonlinear dynamics model of a single-stage spur gear transmission system is established, which simultaneously takes into account the nonlinear backlash, time-varying meshing stiffness, and static transmission error. After that, a dynamic reliability model for the tooth surface contact fatigue failure of gear system is established with the uncertainty of the motion, structure, and material parameters using stress-strength interference (SSI) theory. To be specific, the sparse grid numerical integration (SGNI) method is applied to solve the statistical characteristic parameters of the dynamic reliability of the system. The probability distribution of the performance function is obtained with the fourth-moment SPA method. Test examples show that the results of the proposed method are consistent with the results obtained by the Monte Carlo simulation (MCS) and superior to the maximum entropy with fractional moments (ME-FM) method, which verifies the effectiveness of this approach. Finally, the dynamic reliability of the gear transmission system with respect to load times is evaluated.     

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.     

Robust compliance topology optimization based on the topological derivative concept

In this paper, we present an approach for robust compliance topology optimization under volume constraint. The compliance is evaluated considering a point‐wise worst‐case scenario. Analogously to sequential optimization and reliability assessment, the resulting robust optimization problem can be decoupled into a deterministic topology optimization step and a reliability analysis step. This procedure allows us to use topology optimization algorithms already developed with only small modifications. Here, the deterministic topology optimization problem is addressed with an efficient algorithm based on the topological derivative concept and a level‐set domain representation method. The reliability analysis step is handled as in the performance measure approach. Several numerical examples are presented showing the effectiveness of the proposed approach. Copyright © 2015 John Wiley & Sons, Ltd.     

Geometric insight into the challenges of solving high-dimensional reliability problems

In this paper we adopt a geometric perspective to highlight the challenges associated with solving high-dimensional reliability problems. Adopting a geometric point of view we highlight and explain a range of results concerning the performance of several well-known reliability methods.

We start by investigating geometric properties of the N-dimensional Gaussian space and the distribution of samples in such a space or in a subspace corresponding to a failure domain. Next, we discuss Importance Sampling (IS) in high dimensions. We provide a geometric understanding as to why IS generally does not work in high dimensions [Au SK, Beck JL. Importance sampling in high dimensions. Structural Safety 2003;25(2):139–63]. We furthermore challenge the significance of “design point” when dealing with strongly nonlinear problems. We conclude by showing that for the general high-dimensional nonlinear reliability problems the selection of an appropriate fixed IS density is practically impossible.

Next, we discuss the simulation of samples using Markov Chain Monte Carlo (MCMC) methods. Firstly, we provide a geometric explanation as to why the standard Metropolis–Hastings (MH) algorithm does “not work” in high-dimensions. We then explain why the modified Metropolis–Hastings (MMH) algorithm introduced by Au and Beck [Au SK, Beck JL. Estimation of small failure probabilities in high dimensions by subset simulation. Probabilistic Engineering Mechanics 2001;16(4):263–77] overcomes this problem. A study of the correlation of samples obtained using MMH as a function of different parameters follows. Such study leads to recommendations for fine-tuning the MMH algorithm. Finally, the MMH algorithm is compared with the MCMC algorithm proposed by Katafygiotis and Cheung [Katafygiotis LS, Cheung SH. Application of spherical subset simulation method and auxiliary domain method on a benchmark reliability study, Structural Safety 2006 (in print)] in terms of the correlation of samples they generate.     

Reliability-based evaluation of automotive wind noise quality

This paper proposes an efficient method for the reliability analysis of a vehicle body–door subsystem with respect to an important quality issue—wind noise. A nonlinear seal model is constructed for the automotive wind noise problem and the limit state function is evaluated using finite element analysis. A multi-modal adaptive importance sampling (AIS) method is developed for reliability analysis at system level, to improve the efficiency of the Monte Carlo simulation. The method can easily handle implicit and time-dependent limit-state functions, with variables of any statistical distributions. Existing analytical as well as simulation-based methods are also investigated to solve the car door wind noise problem. It is demonstrated through this industrial application problem that the multi-modal AIS method is superior to existing methods in terms of efficiency and accuracy. A generalized framework for reliability estimation is then established for series system reliability problems with large numbers of random variables and complicated, implicit limit states.     

A bounded optimal control for maximizing the reliability of randomly excited nonlinear oscillators with fractional derivative damping

In this paper, a bounded optimal control for maximizing the reliability of randomly excited nonlinear oscillators with fractional derivative damping is proposed. First, the partially averaged It? equations for the energy processes of individual degree of freedom are derived by using the stochastic averaging method. Second, the dynamical programming equations for the control problems of maximizing the reliability function and maximizing the mean first passage time are established from the partially averaged It? equations by using the dynamical programming principle. The optimal control law is derived from the dynamical programming equation and control constraints. Third, the conditional reliability function and mean first passage time of the optimally controlled system are obtained by solving the backward Kolmogorov equation and Pontryagin equation associated with the fully averaged It? equation, respectively. The application of the proposed procedure and effectiveness of the control strategy are illustrated by using two examples. Besides, the effect of fractional derivative order on the reliability of the optimally controlled system is examined.     

聚合物泡沫座垫的多滞回环型动力学特性的一种建模方法

该文进行了汽车座椅聚合物泡沫座垫多滞回环型动力学特性的一种新的建模方法研究。基于对泡沫座垫的动态特性实验测试结果分析,构造了座垫的非线性动态特性的准静态分量和动态分量子模型,并进行了模型及其参数辨识。采用滞回偏移量的概念,建立了准静态分量的非线性特性模型;基于分数阶导数,建立了简明、精细的动态分量子模型。通过系列实验测试结果,验证了所建立聚合物泡沫座垫的静、动态叠加非线性力学模型,表明该模型具有较高的精度。     

A novel framework for the reliability modelling of repairable multistate complex mechanical systems considering propagation relationships

Due to the propagation, amplification, and concatenation in a failure process, the reliabilities of repairable multistate complex mechanical systems (RMCMSs) may be affected by a significant fluctuation due to a small exception associated with a reliability indicator. Focused on the problems arising from the lack of propagation relationships among fault modes, functional components, and failure causes in conventional reliability models, a novel framework for reliability modelling is proposed to comprehensively analyse the reliabilities of RMCMSs. First, the reliability models are abstracted as weighted and directed networks with five layers. Second, an improved failure mode and effects analysis (IFMEA) method combined with the D‐number method and VIKOR approach is presented to determine the importance of reliability nodes. Third, a cut set of the reliability model is generated by any exception of a reliability indicator by considering the propagation relationships, and the reliability sensibility index is defined to characterize the fluctuations in system reliability. The effectiveness of the proposed framework is demonstrated in an actual reliability modelling application. As an intuitive method, the proposed framework inherits the advantages of conventional models but overcomes the drawbacks of these existing methods. Therefore, this method can be flexibly and efficiently used in the reliability modelling of RMCMSs. Moreover, the approach provides a foundation for comprehensive and dynamic reliability analysis and the failure mechanism mining of RMCMSs, and it can be used in other engineering applications.     

基于镜像激励的结构动力学系统的设计点激励

在随机振动及结构可靠性研究中,动力学系统的设计点激励有着不可替代的作用,但非线性动力学系统设计点激励的计算方法仍是当今研究者的焦点之一。该文利用振子自由振动响应的镜像激励,给出了高斯白噪声激励下非线性系统的设计点激励,并将其应用到首穿失效概率估计问题中,与原始的蒙特卡罗模拟相比较,两者体现了高度的一致性。为进一步说明该文方法的正确性,针对线性系统,利用解析方法获得设计点激励的准确值,利用镜像方法所得近似值,将其均应用到首穿失效概率的计算中,数值例子显示,两种方法所得设计点激励稍有不同,但在计算首穿失效概率时,展现出同样的有效性。     

Error-based stopping criterion for the combined adaptive Kriging and importance sampling method for reliability analysis

Metamodel-based method is a wise reliability analysis technique because it uses the metamodel to substitute the actual limit state function under the predefined accuracy. Adaptive Kriging (AK) is a famous metamodel in reliability analysis for its flexibility and efficiency. AK combined with the importance sampling (IS) method abbreviate as AK–IS can extremely reduce the size of candidate sampling pool in the updating process of Kriging model, which makes the AK-based reliability method more suitable for estimating the small failure probability. In this paper, an error-based stopping criterion of updating the Kriging model in the AK–IS method is constructed and two considerable maximum relative error estimation methods between the failure probability estimated by the current Kriging model and the limit state function are derived. By controlling the maximum relative error, the accuracy of the estimate can be adjusted flexibly. Results in three case studies show that the error-based stopping criterion based AK–IS method can achieve the predefined accuracy level and simultaneously enhance the efficiency of updating the Kriging model.     

An efficient global optimization algorithm for multifactor dynamic systems

Abstract

In this study we present an efficient global optimization method, DIviding RECTangle (DIRECT) algorithm, for parametric analysis of dynamic systems. In a bounded constrained problem the DIRECT algorithm explores multiple potentially optimal subspaces in one search. The algorithm also eliminates the need for derivative calculations which are required in some efficient gradient‐based methods. In this study the first optimization example is to find the dynamic parameters of a tennis racket. The second example is a biomechanical parametric study of a heel‐toe running model governed by six factors. The effectiveness of the DIRECT algorithm is compared with a genetic algorithm in an analysis of heel‐toe running. The result shows that the DIRECT algorithm obtains an improved result in 83% less execution time. It is demonstrated that the straightforward DIRECT algorithm provides a general procedure for solving global optimization problems efficiently and confidently.     

Analytic Critical Flow Method (ACFM): A Reliability Allocation Method Based on Analytic Hierarchy Process

RAMS is an acronym for reliability, availability, maintainability and safety. These four properties concern the application of important methodologies to design and manage complex systems. In the present research, starting from the analysis of several literature reliability allocation techniques, a reliability allocation method has been implemented called analytic critical flow method (ACFM). Critical flow method is a reliability allocation method for series-parallel configurations, based on failure analysis of each unit of the system. The new approach is based on critical flow method, whose results are matched with the analytic hierarchy process multicriteria method. The result is a dynamic model that combines the advantages of the allocation method and the multicriteria approach. The need to develop the ACFM is the outcome of a careful analysis of the current military and commercial approaches. In particular, no literature method takes into account to assign a different level of significance (weight) to the different units of the system, simultaneously to the considered factors. The proposed approach has been applied and compared with other traditional methods on an aerospace prototype (series-parallel configuration), where the reliability allocation process is rigorous. The results demonstrate the effectiveness of the new approach and its ability to overcome the criticalities highlighted in literature.     

Monte carlo methods in reliability engineering

The potential of the Monte Carlo method as a common place tool for a broad range of reliability and logistic problems is discussed. The mathematical features of the method are illustrated and biasing methods aimed at reducing the cost of the calculation are described. These methods combine efficiency with simplicity of implementation in multipurpose computer codes. A multipurpose computer code package developed on the basis of the Monte Carlo method is described.     

Interval dynamic reliability analysis of mechanical components under multistage load based on strength degradation

To further study the law of strength degradation, the residual strength degradation model is established based on the definition of fatigue damage, considering the interaction of various uncertain factors and time factors in service environment. Combined with equivalent damage model, a nonlinear cumulative damage model is proposed, which takes the interaction among loading loads into account and improves the accuracy of calculation. Additionally, the equivalent transformation of multistage load is studied using interval theory. According to the interval dynamic nonprobability reliability prediction model, a dynamic reliability analysis of the interval model is carried out. Dynamic reliability of the component is analyzed under multistage load accumulation damage to verify the effectiveness of the method.     

Reliability updating with equality information

In many instances, information on engineering systems can be obtained through measurements, monitoring or direct observations of system performances and can be used to update the system reliability estimate. In structural reliability analysis, such information is expressed either by inequalities (e.g. for the observation that no defect is present) or by equalities (e.g. for quantitative measurements of system characteristics). When information Z is of the equality type, the a priori probability of Z is zero and most structural reliability methods (SRM) are not directly applicable to the computation of the updated reliability. Hitherto, the computation of the reliability of engineering systems conditional on equality information was performed through first- and second-order approximations. In this paper, it is shown how equality information can be transformed into inequality information, which enables reliability updating by solving a standard structural system reliability problem. This approach enables the use of any SRM, including those based on simulation, for reliability updating with equality information. It is demonstrated on three numerical examples, including an application to fatigue reliability.