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

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

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

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

基于蒙特卡罗模拟的高效边坡可靠度修正方法

传统的蒙特卡罗模拟方法在分析由于参数不确定性修正而引起的可靠度修正问题时效率较低。为此,提出了一种基于蒙特卡罗模拟的高效边坡可靠度修正方法,该方法主要包括2个关键步骤:1)根据参数初始分布利用蒙特卡罗模拟方法计算边坡的失效概率,并输出蒙特卡罗模拟的失效样本;2)利用参数统计特征值修正后的联合概率密度函数和蒙特卡罗模拟失效样本计算修正后边坡的失效概率。以两个边坡问题为例说明了所提方法的有效性。结果表明:所提出的方法在计算修正的失效概率过程中无需重新执行蒙特卡罗模拟,计算过程简单、计算效率高。此外,所提方法能够适用于隐式表达功能函数的边坡可靠度修正问题,并能够有效地解决单变量和多变量修正的边坡可靠度修正问题。     

Level-set topology optimization for robust design of structures under hybrid uncertainties

This paper will develop a new robust topology optimization (RTO) method based on level sets for structures subject to hybrid uncertainties, with a more efficient Karhunen-Loève hyperbolic Polynomial Chaos–Chebyshev Interval method to conduct the hybrid uncertain analysis. The loadings and material properties are considered hybrid uncertainties in structures. The parameters with sufficient information are regarded as random fields, while the parameters without sufficient information are treated as intervals. The Karhunen-Loève expansion is applied to discretize random fields into a finite number of random variables, and then, the original hybrid uncertainty analysis is transformed into a new process with random and interval parameters, to which the hyperbolic Polynomial Chaos–Chebyshev Interval is employed for the uncertainty analysis. RTO is formulated to minimize a weighted sum of the mean and standard variance of the structural objective function under the worst-case scenario. Several numerical examples are employed to demonstrate the effectiveness of the proposed RTO, and Monte Carlo simulation is used to validate the numerical accuracy of our proposed method.     

Computation of probabilistic stability measures for a controlled distributed parameter system

Parameter uncertainty can degrade the performance of an otherwise well-designed control system, sometimes leading to system instability. In the context of structural control, performance degradation and instability imply excessive vibration and even structural failure. The ability of a controller to maintain the stability of a system in spite of parameter uncertainty is measured by its robustness, which can be viewed as a probability measure, wherein the joint distribution is of dimension equal to the number of uncertain parameters and the failure hypersurface is defined by the onset of instability in the eigenspace. This observation has led to some recent analyses employing FORM/SORM methods and Monte Carlo simulation.The extension of these concepts to distributed parameter systems is, unfortunately, not immediate. The mere fact that these systems are infinite dimensional precludes the use of much of the machinery available for discrete systems, unless the distributed system is first discretized, which itself introduces error into the analysis, or is represented by an eigenfunction expansion, which requires truncation after some finite number of modes, also a potential source of error. In fact, the system will behave as one with an infinite number of subsystems with highly dependent failure modes in series.In Bergman and Hall, Effect of controller uncertainty on the stability of a distributed parameter system, Structural Safety and Reliability, eds. Schuëller, Shinozuka and Yao, Balkema, Rotterdam, 1993, pp. 210–220, root locus analysis was employed to assess the reliability of the system, requiring the repetitive solution of a transcendental characteristic equation over a range of the parameter under investigation. The loci then provide a mapping from the probability distribution of the random parameter to the probability distribution of the system eigenvalues. This approach was utilized over 30 years ago by Boyce, Random vibration of strings and bars, Proc. of the Fourth US National Congress of Applied Mechanics, Berkeley, 18–21 June, 1962, pp. 77–85, who examined eigenvalue distributions for undamped taut strings and Euler-Bernoulli beams, each subjected to the action of a single point actuator. He demonstrated that, for the case of uncertainty in the actuator gain alone, a simple, closed form mapping leading to the distributions of the eigenvalues of the system could be determined directly from the distribution of the actuator gain, and for uncertainty in the remaining parameters, approximate distributions could be obtained through the application of perturbation methods.In the current paper, the FORM/SORM approach is applied to the taut string problem, where the distributed nature of the system is maintained throughout the analysis. Uncertain parameters, in this case the proportional gain and time delay, are characterized by probability distributions with known mean and variance. Each is transformed to a standard normal variate via Rosenblatt transformations, and the most likely failure point in the parameter space is found using a constrained optimization procedure. The effect of distribution is shown through parameter studies, and verification is provided by Monte Carlo simulation. As expected, time delay is shown to have a pronounced effect upon system robustness.     

在役腐蚀管道动态非概率可靠性分析简

 腐蚀失效是压力管道失效的主要形式之一,研究腐蚀管道的可靠性具有重要理论意义和应用价值.在对腐蚀管道可靠性分析时,概率可靠性模型和模糊可靠性模型对于数据信息的要求较高.而在掌握不确定性信息很少情况下,为了充分利用管道的不确定性信息弥补原始数据的不足,可将腐蚀管道可靠性分析中的材料屈服强度、管道直径、缺陷深度和操作压力等不确定参数视为区间变量,基于区间模型建立一种在役腐蚀管道动态非概率可靠性模型,给出了腐蚀管道剩余寿命预测的简便方法.结合工程实例计算与分析,表明了文中所提出方法的可行性和合理性,并在此基础上,分析了管道的壁厚、缺陷深度、实际压力和腐蚀速率这些区间变量的不同变异系数对非概率可靠性指标的影响,分析结果表明非概率可靠性指标对管道壁厚的变异系数最为敏感.     

Efficient structural reliability methods considering incomplete knowledge of random variable distributions

The determination of an exact distribution function of a random phenomena is not possible using a limited number of observations. Therefore, in the present paper the stochastic properties of a random variable are assumed as uncertain quantities and instead of predefined distribution types the maximum entropy distribution is used. Efficient methods for a reliability analysis considering these uncertain stochastic parameters are presented. Based on approximation strategies this extended analysis requires no additional limit state function evaluations. Later, variance based sensitivity measures are used to evaluate the contribution of the uncertainty of each stochastic parameter to the total variation of the failure probability.     

A probability model of system downtime with implications for optimal warranty design

Traditional approaches toward modeling the availability of a system often do not formally take into account uncertainty over the parameter values of the model. Such models are then frequently criticized because the observed reliability of a system does not match that predicted by the model. This paper extends a recently published segregated failures model so that, rather than providing a single figure for the availability of a system, uncertainty over model parameter values is incorporated and a predictive probability distribution is given. This predictive distribution is generated in a practical way by displaying the uncertainties and dependencies of the parameters of the model through a Bayesian network (BN). Permitting uncertainty in the reliability model then allows the user to determine whether the predicted reliability was incorrect due to inherent variability in the system under study, or due to the use of an inappropriate model. Furthermore, it is demonstrated how the predictive distribution can be used when reliability predictions are employed within a formal decision‐theoretic framework. Use of the model is illustrated with the example of a high‐availability computer system with multiple recovery procedures. An BN is produced to display the relations between parameters of the model in this case and to generate a predictive probability distribution of the system's availability. This predictive distribution is then used to make two decisions under uncertainty concerning the offered warranty policies on the system: a qualitative decision and an optimization over a continuous decision space. Copyright © 2009 John Wiley & Sons, Ltd.     

Integration of computation and testing for reliability estimation

This paper develops a methodology to integrate reliability testing and computational reliability analysis for product development. The presence of information uncertainty such as statistical uncertainty and modeling error is incorporated. The integration of testing and computation leads to a more cost-efficient estimation of failure probability and life distribution than the tests-only approach currently followed by the industry. A Bayesian procedure is proposed to quantify the modeling uncertainty using random parameters, including the uncertainty in mechanical and statistical model selection and the uncertainty in distribution parameters. An adaptive method is developed to determine the number of tests needed to achieve a desired confidence level in the reliability estimates, by combining prior computational prediction and test data. Two kinds of tests — failure probability estimation and life estimation — are considered. The prior distribution and confidence interval of failure probability in both cases are estimated using computational reliability methods, and are updated using the results of tests performed during the product development phase.     

Parametric uncertainty quantification using proper generalized decomposition applied to neutron diffusion

In this paper, a proper generalized decomposition (PGD) approach is employed for uncertainty quantification purposes. The neutron diffusion equation with external sources, a diffusion-reaction problem, is used as the parametric model. The uncertainty parameters include the zone-wise constant material diffusion and reaction coefficients as well as the source strengths, yielding a large uncertain space in highly heterogeneous geometries. The PGD solution, parameterized in all uncertain variables, can then be used to compute mean, variance, and more generally probability distributions of various quantities of interest. In addition to parameterized properties, parameterized geometrical variations of three-dimensional models are also considered in this paper. To achieve and analyze a parametric PGD solution, algorithms are developed to decompose the model's parametric space and semianalytically integrate solutions for evaluating statistical moments. Varying dimensional problems are evaluated to showcase PGD's ability to solve high-dimensional problems and analyze its convergence.     

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.     

Comparison of robust,reliability-based and non-probabilistic topology optimization under uncertain loads and stress constraints

It is nowadays widely acknowledged that optimal structural design should be robust with respect to the uncertainties in loads and material parameters. However, there are several alternatives to consider such uncertainties in structural optimization problems. This paper presents a comprehensive comparison between the results of three different approaches to topology optimization under uncertain loading, considering stress constraints: (1) the robust formulation, which requires only the mean and standard deviation of stresses at each element; (2) the reliability-based formulation, which imposes a reliability constraint on computed stresses; (3) the non-probabilistic formulation, which considers a worst-case scenario for the stresses caused by uncertain loads. The information required by each method, regarding the uncertain loads, and the uncertainty propagation approach used in each case is quite different. The robust formulation requires only mean and standard deviation of uncertain loads; stresses are computed via a first-order perturbation approach. The reliability-based formulation requires full probability distributions of random loads, reliability constraints are computed via a first-order performance measure approach. The non-probabilistic formulation is applicable for bounded uncertain loads; only lower and upper bounds are used, and worst-case stresses are computed via a nested optimization with anti-optimization. The three approaches are quite different in the handling of uncertainties; however, the basic topology optimization framework is the same: the traditional density approach is employed for material parameterization, while the augmented Lagrangian method is employed to solve the resulting problem, in order to handle the large number of stress constraints. Results are computed for two reference problems: similarities and differences between optimized topologies obtained with the three formulations are exploited and discussed.     

多源不确定性条件下气动弹性系统颤振可靠性分析方法

传统的气动弹性系统颤振分析模型大多是在确定性参数条件下建立的,当系统中存在不确定因素时,按确定性方法设计的气动弹性系统存在颤振失效风险.以概率和非概率区间模型为基础,建立了单源不确定性条件下颤振可靠性分析模型;在此基础上,针对含随机和区间多源不确定参数的气动弹性系统颤振可靠性分析问题,提出一种基于分步求解策略的新型混合...     

Reliability–sensitivity analysis using dimension reduction methods and saddlepoint approximations

Reliability–sensitivity, which is considered as an essential component in engineering design under uncertainty, is often of critical importance toward understanding the physical systems underlying failure and modifying the design to mitigate and manage risk. This paper presents a new computational tool for predicting reliability (failure probability) and reliability–sensitivity of mechanical or structural systems subject to random uncertainties in loads, material properties, and geometry. The dimension reduction method is applied to compute response moments and their sensitivities with respect to the distribution parameters (e.g., shape and scale parameters, mean, and standard deviation) of basic random variables. Saddlepoint approximations with truncated cumulant generating functions are employed to estimate failure probability, probability density functions, and cumulative distribution functions. The rigorous analytic derivation of the parameter sensitivities of the failure probability with respect to the distribution parameters of basic random variables is derived. Results of six numerical examples involving hypothetical mathematical functions and solid mechanics problems indicate that the proposed approach provides accurate, convergent, and computationally efficient estimates of the failure probability and reliability–sensitivity. Copyright © 2012 John Wiley & Sons, Ltd.     

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。     

A non-probabilistic structural reliability analysis method based on a multidimensional parallelepiped convex model

Compared with a probability model, a non-probabilistic convex model only requires a small number of experimental samples to discern the uncertainty parameter bounds instead of the exact probability distribution. Therefore, it can be used for uncertainty analysis of many complex structures lacking experimental samples. Based on the multidimensional parallelepiped convex model, we propose a new method for non-probabilistic structural reliability analysis in which marginal intervals are used to express scattering levels for the parameters, and relevant angles are used to express the correlations between uncertain variables. Using an affine coordinate transformation, the multidimensional parallelepiped uncertainty domain and the limit-state function are transformed to a standard parameter space, and a non-probabilistic reliability index is used to measure the structural reliability. Finally, the method proposed herein was applied to several numerical examples.     

Hybrid reliability analysis with uncertain statistical variables,sparse variables and interval variables

There are differences among sampling data and representation types of uncertain statistical variables, sparse variables and interval variables, which increase the complexity of structure reliability analysis. Therefore, a hybrid first order reliability analysis method considering the three types of uncertain variables is demonstrated in this article. First, distribution types and distribution parameters of sparse variables are identified and probabilistically estimated. Secondly, interval variables are transformed into probabilistic types using a uniformity approach. Thirdly, a unified hybrid reliability calculation method considering these uncertain variables simultaneously is demonstrated. The most probable point (MPP) is searched for using the first order reliability method, and then a linear approximation function of performance function is constructed in the neighbourhood of the MPP. Finally, the belief and plausibility measures of the reliability index are efficiently calculated using the theoretical analytical method. Three examples are investigated to demonstrate the effectiveness of the proposed method.     

Fast Fault Tree Analysis for Hybrid Uncertainties Using Stochastic Logic Implemented on Field‐Programmable Gate Arrays: An Application in Quantitative Assessment and mitigation of Welding Defects Risk

This paper presents a stochastic logic‐based method for quantitative risk assessment using fault tree analysis (FTA) that can take into account both types of uncertainties including objective and subjective uncertainties. In the proposed method, each fault tree gate is translated to its corresponding stochastic logic template and then is implemented on a field programmable gate array (FPGA). Because the analysis does not utilize any transformation methods, the results of analysis are more accurate than those methods which are based on transformation from possibility to probability distributions or vice versa. Experimental results for a benchmark fault tree show that this method accelerates analysis time compared to conventional hybrid uncertainty analysis method and transformation methods. The efficiency of the proposed method is demonstrated by implementation in a real steel structure project. The quantitative risk assessment is performed for the incomplete penetration as one of the defects encountered in arc welding process, and its results are compared with transformation methods. The comparison results show the proposed hybrid uncertainty analysis method is also more accurate in comparison to the transformation‐based approaches. Copyright © 2016 John Wiley & Sons, Ltd.     

Non-Parametric Sensitivity Analysis in Linear Programming

A linear programming problem subject to uncertainty in the requirements vector is solved deterministically, then sensitivity analysis is performed to determine the effect of the random variation on this solution. Since there could be an appreciable cost of modifying a solution once implemented, the probability of the random components perturbing a solution is considered. Unlike existing methods of linear programming under uncertainty, this article assumes no knowledge of the distributions of the random variables. Rather, the notion of non-parametric tolerance limits is employed to establish a criterion for changing basic solutions.     

Multidimensional parallelepiped model—a new type of non-probabilistic convex model for structural uncertainty analysis

Non-probabilistic convex models need to be provided only the changing boundary of parameters rather than their exact probability distributions; thus, such models can be applied to uncertainty analysis of complex structures when experimental information is lacking. The interval and the ellipsoidal models are the two most commonly used modeling methods in the field of non-probabilistic convex modeling. However, the former can only deal with independent variables, while the latter can only deal with dependent variables. This paper presents a more general non-probabilistic convex model, the multidimensional parallelepiped model. This model can include the independent and dependent uncertain variables in a unified framework and can effectively deal with complex ‘multi-source uncertainty’ problems in which dependent variables and independent variables coexist. For any two parameters, the concepts of the correlation angle and the correlation coefficient are defined. Through the marginal intervals of all the parameters and also their correlation coefficients, a multidimensional parallelepiped can easily be built as the uncertainty domain for parameters. Through the introduction of affine coordinates, the parallelepiped model in the original parameter space is converted to an interval model in the affine space, thus greatly facilitating subsequent structural uncertainty analysis. The parallelepiped model is applied to structural uncertainty propagation analysis, and the response interval of the structure is obtained in the case of uncertain initial parameters. Finally, the method described in this paper was applied to several numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.