An efficient hybrid reliability analysis method with random and interval variables

Random and interval variables often coexist. Interval variables make reliability analysis much more computationally intensive. This work develops a new hybrid reliability analysis method so that the probability analysis (PA) loop and interval analysis (IA) loop are decomposed into two separate loops. An efficient PA algorithm is employed, and a new efficient IA method is developed. The new IA method consists of two stages. The first stage is for monotonic limit-state functions. If the limit-state function is not monotonic, the second stage is triggered. In the second stage, the limit-state function is sequentially approximated with a second order form, and the gradient projection method is applied to solve the extreme responses of the limit-state function with respect to the interval variables. The efficiency and accuracy of the proposed method are demonstrated by three 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.     

随机-区间型天线结构有限元及可靠性分析

构建了对随机-区间混合型天线结构的有限元及可靠性分析模型,提出了一种新的处理不确定性因素的结构有限元分析方法,给出了结构保精度和保强度两工况的概率描述。同时考虑了结构的物理参数、几何参数的随机性和作用风载荷的区间性。首先将随机变量固定,利用区间因子法求得结构位移和应力响应的区间范围,然后在区间内任意点处利用随机因子法求结构响应的随机分布范围。构造了天线反射面位移响应和结构单元应力响应不确定变量的数字特征计算公式,进而得到结构各响应量的可靠性指标。对一8m口径天线结构进行了分析,分析结果表明文中所提方法具有合理性和可行性。     

An inverse reliability analysis method for reliability-based design optimization with random and dependent interval variables constrained within ellipsoids

In practical design problems, interval variables exist. Many existing methods can handle only independent interval variables. Some interval variables, however, are dependent. In this work, dependent interval variables constrained within a multi-ellipsoid convex set are considered and incorporated into reliability-based design optimization (RBDO). An efficient RBDO method is proposed by employing the sequential single-loop procedure, which separates the coupled reliability analysis procedure from the deterministic optimization procedure. In the reliability analysis procedure, a single-loop optimization for the inverse reliability analysis is performed, and an efficient inverse reliability analysis method for searching for the worst-case most probable point (WMPP) is developed. The search method contains two stages. The first stage deals the situation where the WMPP is on the boundary of the feasible region, while the second stage accommodates the situation where the WMPP is inside the feasible region by interpolation. Three examples are used for a demonstration.     

Reliability sensitivity analysis with random and interval variables

In reliability analysis and reliability‐based design, sensitivity analysis identifies the relationship between the change in reliability and the change in the characteristics of uncertain variables. Sensitivity analysis is also used to identify the most significant uncertain variables that have the highest contributions to reliability. Most of the current sensitivity analysis methods are applicable for only random variables. In many engineering applications, however, some of uncertain variables are intervals. In this work, a sensitivity analysis method is proposed for the mixture of random and interval variables. Six sensitivity indices are defined for the sensitivity of the average reliability and reliability bounds with respect to the averages and widths of intervals, as well as with respect to the distribution parameters of random variables. The equations of these sensitivity indices are derived based on the first‐order reliability method (FORM). The proposed reliability sensitivity analysis is a byproduct of FORM without any extra function calls after reliability is found. Once FORM is performed, the sensitivity information is obtained automatically. Two examples are used for demonstration. Copyright © 2009 John Wiley & Sons, Ltd.     

Dynamic reliability analysis model for structure with both random and interval uncertainties

International Journal of Mechanics and Materials in Design - Aiming at analyzing the safety of the dynamic structure involving both input random variables and the interval ones, a new dynamic...     

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.     

Non-stationary random response analysis of structures with uncertain parameters

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

A new reliability measure based on specified minimum distances before the locations of random variables in a finite interval

A new reliability measure is proposed and equations are derived which determine the probability of existence of a specified set of minimum gaps between random variables following a homogeneous Poisson process in a finite interval. Using the derived equations, a method is proposed for specifying the upper bound of the random variables' number density which guarantees that the probability of clustering of two or more random variables in a finite interval remains below a maximum acceptable level. It is demonstrated that even for moderate number densities the probability of clustering is substantial and should not be neglected in reliability calculations.In the important special case where the random variables are failure times, models have been proposed for determining the upper bound of the hazard rate which guarantees a set of minimum failure-free operating intervals before the random failures, with a specified probability. A model has also been proposed for determining the upper bound of the hazard rate which guarantees a minimum availability target. Using the models proposed, a new strategy, models and reliability tools have been developed for setting quantitative reliability requirements which consist of determining the intersection of the hazard rate envelopes (hazard rate upper bounds) which deliver a minimum failure-free operating period before random failures, a risk of premature failure below a maximum acceptable level and a minimum required availability. It is demonstrated that setting reliability requirements solely based on an availability target does not necessarily mean a low risk of premature failure. Even at a high availability level, the probability of premature failure can be substantial. For industries characterised by a high cost of failure, the reliability requirements should involve a hazard rate envelope limiting the risk of failure below a maximum acceptable level.     

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.     

An interval random perturbation method for structural‐acoustic system with hybrid uncertain parameters

An interval random model is introduced for the response analysis of structural‐acoustic systems that lack sufficient information to construct the precise probability distributions of uncertain parameters. In the interval random model, the uncertain parameters are treated as random variables, whereas some distribution parameters of random variables with limited information are expressed as interval variables instead of precise values. On the basis of the interval random model, the interval random structural‐acoustic finite element equation is constructed, and an interval random perturbation method for solving this interval random equation is proposed. In the proposed method, the interval random matrix and vector are expanded by the first‐order Taylor series, and the response vector of the structural‐acoustic system is calculated by the matrix perturbation method. According to the linear monotonicity of the response vector, the lower and upper bounds of the response vector are calculated by the vertex method. On the basis of the lower and upper bounds, the intervals of expectation and standard variance of the response vector are obtained by the random interval moment method. The numerical results on a shell structural‐acoustic model and an automobile passenger compartment with flexible front panel demonstrate the effectiveness and efficiency of the proposed method. Copyright © 2013 John Wiley & Sons, Ltd.     

随机激励下随机结构动力可靠性灵敏度分析

对于随机激励下随机结构动力可靠性的灵敏度分析问题,在加权非线性响应面法的基础之上建立了随机结构动力可靠性灵敏度分析方法.所提方法从随机结构无条件动力可靠度的表达式出发,首先将随机结构的动力可靠性分析问题转化成传统的静力可靠性分析问题,然后采用基于加权非线性响应面法的Monte-Carlo可靠性灵敏度分析方法求解动力可靠性灵敏度值.算例表明该方法的计算结果是合理的,并且由于加权非线性法具有较高的效率和精度,因而所提方法具有一定的工程意义.     

区间随机桁架结构动力特性分析方法研究

摘 要：针对区间随机桁架结构的动力特性分析,提出了一种区间随机有限元方法。当结构的物理参数和几何尺寸同时具有区间随机性时,利用区间因子法和随机因子法建立了结构的刚度矩阵和质量矩阵;从结构振动的瑞利商表达式出发,利用区间运算推导了结构动力特性区间随机变量的计算式;进而利用随机变量的矩法和代数综合法,推导出了结构特征值的数字特征的计算式。最后通过算例分析了区间随机桁架结构参数的区间随机性对其动力特性的影响,计算结果表明该方法是可行和有效的。
     

A new hybrid reliability-based design optimization method under random and interval uncertainties

This article proposes a new method for hybrid reliability-based design optimization under random and interval uncertainties (HRBDO-RI). In this method, Monte Carlo simulation (MCS) is employed to estimate the upper bound of failure probability, and stochastic sensitivity analysis (SSA) is extended to calculate the sensitivity information of failure probability in HRBDO-RI. Due to a large number of samples involved in MCS and SSA, Kriging metamodels are constructed to substitute true constraints. To avoid unnecessary computational cost on Kriging metamodel construction, a new screening criterion based on the coefficient of variation of failure probability is developed to judge active constraints in HRBDO-RI. Then a projection-outline-based active learning Kriging is achieved by sequentially select update points around the projection outlines on the limit-state surfaces of active constraints. Furthermore, the prediction uncertainty of Kriging metamodel is quantified and considered in the termination of Kriging update. Several examples, including a piezoelectric energy harvester design, are presented to test the accuracy and efficiency of the proposed method for HRBDO-RI.     

A new adaptive response surface method for reliability analysis

Response surface method is a convenient tool to assess reliability for a wide range of structural mechanical problems. More specifically, adaptive schemes which consist in iteratively refine the experimental design close to the limit state have received much attention. However, it is generally difficult to take into account a lot of variables and to well handle approximation error. The method, proposed in this paper, addresses these points using sparse response surface and a relevant criterion for results accuracy. For this purpose, a response surface is built from an initial Latin Hypercube Sampling (LHS) where the most significant terms are chosen from statistical criteria and cross-validation method. At each step, LHS is refined in a region of interest defined with respect to an importance level on probability density in the design point. Two convergence criteria are used in the procedure: The first one concerns localization of the region and the second one the response surface quality. Finally, a bootstrap method is used to determine the influence of the response error on the estimated probability of failure. This method is applied to several examples and results are discussed.     

Fuzzy interval perturbation method for uncertain heat conduction problem with interval and fuzzy parameters

This paper proposes a fuzzy interval perturbation method (FIPM) and a modified fuzzy interval perturbation method (MFIPM) for the hybrid uncertain temperature field prediction involving both interval and fuzzy parameters in material properties and boundary conditions. Interval variables are used to quantify the non‐probabilistic uncertainty with limited information, whereas fuzzy variables are used to represent the uncertainty associated with the expert opinions. The level‐cut method is introduced to decompose the fuzzy parameters into interval variables. FIPM approximates the interval matrix inverse by the first‐order Neumann series, while MFIPM improves the accuracy by considering higher‐order terms of the Neumann series. The membership functions of the interval temperature field are eventually derived using the fuzzy decomposition theorem. Three numerical examples are provided to demonstrate the feasibility and effectiveness of the proposed methods for solving heat conduction problems with hybrid uncertain parameters, pure interval parameters, and pure fuzzy parameters, respectively. Copyright © 2015 John Wiley & Sons, Ltd.     

A practical engineering method for fuzzy reliability analysis of mechanical structures

The fuzzy sets theory in reliability analyses is studied. The structure stress is related to several other variables, such as structure sizes, material properties, external loads; in most cases, it is difficult to be expressed in a mathematical formula, and the related variables are not random variables, but fuzzy variables or other uncertain variables which have not only randomness but also fuzziness. In this paper, a novel approach is presented to use the finite element analysis as a “numerical experiment” tool, and to find directly, by fuzzy linear regression method, the statistical property of the structure stress. Based on the fuzzy stress–random strength interference model proposed in this paper, the fuzzy reliability of the mechanical structure can be evaluated. The compressor blade of a given turbocharger is then introduced as a realistic example to illustrate the approach.     

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

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

基于区间分析的舰船装备可靠性模糊分配方法

针对舰船装备可靠性分配工作中具有多种影响因素且部分影响因素较难定量分析的问题,提出了一种基于AHP、模糊综合评判和区间分析的可靠性模糊综合分配方法.采用模糊综合评判构建了系统的可靠性分配模型,并设计出分配的技术路线;综合分析了8种影响可靠性分配的因素及其量化方法,并采用AHP构造了系统的可靠性分配递阶层次模型;采用区间数代替单一数值来表达模糊信息,并将定量分析与定性分析相结合,合理利用专家经验和相似系统的可靠性数据来确定判断矩阵和各影响因素的权重,从而有效解决了舰船装备可靠性模糊综合分配过程中的不确定性问题.最后,以某型舰电力系统可靠性分配为例,研究表明:该方法可操作性强、分配合理有效,具有较好的工程实践指导意义.