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.     

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.     

Reliability study of fracture mechanics based non‐probabilistic interval analysis model

In this paper, we advanced a new interval reliability analysis model for fracture reliability analysis. Based on the non‐probabilistic stress intensity factor interference model and the ratio of the volume of the safe region to the total volume of the region associated with the variation of the standardized interval variables is suggested as the measure of structural non‐probabilistic reliability. We use this theory to calculate the reliability of structure based on fracture criterion. This model needs less uncertain information, so it has less limitation for analysing an uncertain structure or system. Examples of practical application are given to explain the simplicity and practicability of this model by comparing the interval reliability analysis model with probabilistic reliability analysis model.     

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 single-loop optimization method for reliability analysis with second order uncertainty

Reliability analysis may involve random variables and interval variables. In addition, some of the random variables may have interval distribution parameters owing to limited information. This kind of uncertainty is called second order uncertainty. This article develops an efficient reliability method for problems involving the three aforementioned types of uncertain input variables. The analysis produces the maximum and minimum reliability and is computationally demanding because two loops are needed: a reliability analysis loop with respect to random variables and an interval analysis loop for extreme responses with respect to interval variables. The first order reliability method and nonlinear optimization are used for the two loops, respectively. For computational efficiency, the two loops are combined into a single loop by treating the Karush–Kuhn–Tucker (KKT) optimal conditions of the interval analysis as constraints. Three examples are presented to demonstrate the proposed method.     

The interval estimation of reliability for probabilistic and non-probabilistic hybrid structural system

The aim of this paper is to improve evaluation of the reliability of probabilistic and non-probabilistic hybrid structural system. Based on the probabilistic reliability model and interval arithmetic, a new model of interval estimation for reliability of the hybrid structural system was proposed. Adequately considering all uncertainties affecting the hybrid structural system, the lower and upper bounds of reliability for the hybrid structural system were obtained through the probabilistic and non-probabilistic analysis. In the process of non-probabilistic analysis, the interval truncation method was used. In addition, a recognition method of the main failure modes in the hybrid structural system was presented. A five-bar statically indeterminate truss structure and an intermediate complexity wing structure were used to demonstrate the new model is more suitable for analysis and design of these structural systems in comparison with the probabilistic model. The results also show that the method of recognition of main failure modes is effective. In addition, range obtained through interval estimation is shown to be more credible than certain results of other reliability models.     

热弹耦合梁结构动力响应的区间数值分析

研究了含有区间参数梁结构在温度载荷和力载荷共同作用下的动力响应问题,考虑材料变形与传热的相互影响,建立了梁在热弹耦合下的动力学有限元模型,并给出了对结构瞬态热传导方程与动力学方程进行相互交替迭代求解的计算方法。针对结构响应不确定性问题,以不确定参数作为约束变量,通过寻求结构响应函数的区间范围,将区间问题转化为优化问题,并利用遗传算法给出了结构响应函数的区间界限。通过算例及与概率有限元方法的计算结果比较,表明文中所提出方法的可行性和有效性,并获得在热弹耦合作用下梁结构的固有振动频率有所增加,而振动响应振幅则逐渐减弱的结论。该方法只需已知不确定参数所在范围的界限,而无需其他统计信息,为解决区间参数热弹耦合梁问题提供了一种途径。     

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.     

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.     

基于区间理论的不确定结构随机疲劳损伤估计方法

将结构体系中不确定参数定义为区间变量,在随机疲劳谱分析方法的基础上,提出一种计算平稳高斯荷载作用下不确定结构疲劳损伤的新方法。该方法采用区间参数模型定义结构的不确定性,应用功率谱密度描述外荷载的随机性;利用有理级数和单位对称区间显式表达结构区间频响函数和不确定结构在平稳高斯荷载作用下的动力响应区间;根据Tovo-Benasciutti疲劳损伤预测模型,计算不确定结构在随机荷载作用下的疲劳损伤区间期望率;并可通过调整相应不确定参数的单位对称区间近似估计该不确定参数不同不确定半径的疲劳损伤区间期望率。通过数值算例,将该文提出的随机疲劳区间分析方法与顶点法进行比较,验证了该方法的准确性和适用性。     

梁结构刚度动态非概率可靠性分析

 为了定量分析在疲劳载荷作用下梁在不同寿命期内刚度的可靠性,建立梁结构物理性能退化的精确公式就十分重要.依据疲劳载荷造成的累积损伤对材料极限应力的影响,基于材料剩余强度模型,利用材料强度与弹性模量之间的关系,推导出结构弹性模量的退化表达式,并在此基础上,提出梁弹性模量退化系数的递推表达式,推导出圆截面梁剩余抗弯刚度的表达式.在对结构可靠性分析时,概率可靠性模型和模糊可靠性模型对于原始数据信息要求较高.为了充分利用结构的不确定性信息弥补原始数据的不足,将梁的初始弹性模量及所受的疲劳载荷等看作区间变量,利用区间模型建立基于刚度退化的梁刚度动态非概率可靠性模型.最后,结合工程实例的计算表明了该方法对梁的刚度退化分析及其刚度动态可靠性分析是可行、有效和合理的.     

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.     

Propagation of uncertain structural properties described by imprecise Probability Density Functions via response surface method

The present study addresses the analysis of structures with uncertain properties modelled as random variables characterized by imprecise Probability Density Functions (PDFs), namely PDFs with interval basic parameters (mean-value, variance, etc.). Due to imprecision in the probabilistic model, the statistics of the response and the failure probability are described by interval quantities. An efficient procedure for evaluating the bounds of such quantities is developed. The proposed method stems from the application of a ratio of polynomial response surface (Impollonia and Sofi, 2003; Sofi and Romeo, 2018) in conjunction with the classical probabilistic analysis and the so-called Improved Interval Analysis via Extra Unitary Interval (IIA via EUI) (Muscolino and Sofi, 2012). Interval response statistics are derived as approximate explicit functions of the interval parameters describing imprecise probabilities. The range of the interval failure probability is estimated in terms of the interval reliability index once the bounds of the interval mean-value and variance of the response are evaluated.Numerical results concerning a frame structure and a grid structure with uncertain Young’s moduli characterized by imprecise PDFs are presented. The accuracy of the proposed method along with the influence of randomness and imprecision of the input parameters on response statistics and reliability assessment are investigated.     

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.     

区间参数结构动力优化的改进方法

针对区间参数结构,提出一种改进的动力响应的区间优化方法。由于区间优化问题一般要比确定性优化问题的求解复杂得多,因此,通过优化结构动力响应区间值的上界,将区间优化问题转化为近似的确定性优化问题。为了得到结构动力响应更加准确的区间值,把结构动力响应Taylor展开式中的一阶导数也看成区间的,这样得到的区间值能近似包含精确值。在区间优化方法中,设计变量的中值和半径都被选为优化变量,可以得到比传统确定性优化方法更多的优化信息。把该方法应用于典型刚架结构,优化结果表明,区间优化方法不仅能得到与传统优化方法大致相当的设计变量最优值,还能得到实际问题中当设计变量取不到最优值而有微小变化时,目标函数值的一个变化范围。     

New reliability modeling methods for structural systems with hybrid uncertainty

The present study investigates the hybrid reliability modeling of structures in which the inputs contain both random variables and interval variables. Hybrid uncertainty is divided into three categories, including random variables mixed with random variables, interval variables mixed with interval variable, and random variables mixed with interval variables. In order to perform the reliability analysis of structural systems, first, the Bayes method is proposed in the present study to obtain distribution parameters of random variables. Moreover, the self-sample method is introduced to obtain the interval boundaries based on the least available measuring data. Then, the reliability models are established for three situations and the reliability indices are defined and derived accordingly. The abovementioned three types of reliability indices outline the general situation of structural systems. Finally, the specific calculation process is described in details through different examples. Furthermore, the accuracy and efficiency of the proposed method is discussed by comparing the results obtained from the Monte Carlo simulation and those of other methods. The obtained results indicate that the performance of the proposed model in solving reliability modeling problems is better.     

A probabilistic approach for representation of interval uncertainty

In this paper, we propose a probabilistic approach to represent interval data for input variables in reliability and uncertainty analysis problems, using flexible families of continuous Johnson distributions. Such a probabilistic representation of interval data facilitates a unified framework for handling aleatory and epistemic uncertainty. For fitting probability distributions, methods such as moment matching are commonly used in the literature. However, unlike point data where single estimates for the moments of data can be calculated, moments of interval data can only be computed in terms of upper and lower bounds. Finding bounds on the moments of interval data has been generally considered an NP-hard problem because it includes a search among the combinations of multiple values of the variables, including interval endpoints. In this paper, we present efficient algorithms based on continuous optimization to find the bounds on second and higher moments of interval data. With numerical examples, we show that the proposed bounding algorithms are scalable in polynomial time with respect to increasing number of intervals. Using the bounds on moments computed using the proposed approach, we fit a family of Johnson distributions to interval data. Furthermore, using an optimization approach based on percentiles, we find the bounding envelopes of the family of distributions, termed as a Johnson p-box. The idea of bounding envelopes for the family of Johnson distributions is analogous to the notion of empirical p-box in the literature. Several sets of interval data with different numbers of intervals and type of overlap are presented to demonstrate the proposed methods. As against the computationally expensive nested analysis that is typically required in the presence of interval variables, the proposed probabilistic representation enables inexpensive optimization-based strategies to estimate bounds on an output quantity of interest.     

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...     

汽车车身耐撞性设计参数的区间稳健性优化

基于区间分析,提出了一种考虑公差的汽车车身耐撞性稳健优化设计模型,可在有效降低耐撞性能对设计参数波动敏感性的同时实现公差范围的最大化。该模型首先利用对称公差来描述汽车碰撞模型中车身关键耐撞部件的主要尺寸、位置和形状等设计参数本身的不确定性,然后将参数设计和公差设计相结合,建立了以稳健性评价指标和公差评价指标为优化目标,设计变量名义值和公差同步优化的多目标优化模型。再次,利用区间可能度处理不确定约束,将该优化模型转换为确定性多目标优化模型。最后,将该模型应用于两个汽车耐撞性优化设计问题,并通过序列二次规划法和改进的非支配排序遗传算法进行求解,结果表明该方法及稳健优化设计模型可行且实用。