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.     

Interval spectral stochastic finite element analysis of structures with aggregation of random field and bounded parameters

This paper presents the study on non‐deterministic problems of structures with a mixture of random field and interval material properties under uncertain‐but‐bounded forces. Probabilistic framework is extended to handle the mixed uncertainties from structural parameters and loads by incorporating interval algorithms into spectral stochastic finite element method. Random interval formulations are developed based on K–L expansion and polynomial chaos accommodating the random field Young's modulus, interval Poisson's ratios and bounded applied forces. Numerical characteristics including mean value and standard deviation of the interval random structural responses are consequently obtained as intervals rather than deterministic values. The randomised low‐discrepancy sequences initialized particles and high‐order nonlinear inertia weight with multi‐dimensional parameters are employed to determine the change ranges of statistical moments of the random interval structural responses. The bounded probability density and cumulative distribution of the interval random response are then visualised. The feasibility, efficiency and usefulness of the proposed interval spectral stochastic finite element method are illustrated by three numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Interval Finite Element Analysis using Interval Factor Method

A new method called the interval factor method for the finite element analysis of truss structures with interval parameters is presented in this paper. The structural parameters and applied forces can be considered as interval variables by using the interval factor method, the structural stiffness matrix can then be divided into the product of two parts corresponding to the interval factors and the deterministic value. From the static governing equations of interval finite element method of structures, the structural displacement and stress responses are expressed as the functions of the interval factors. The computational expressions for lower and upper bounds, mean value and interval change ratio of structural static responses are derived by means of the interval operations. The effect of the uncertainty of the structural parameters and applied forces on the structural displacement and stress responses is demonstrated by truss structures.     

区间桁架结构有限元分析的区间因子法

研究了具有区间参数的桁架结构在区间力作用下的有限元分析方法。利用区间因子法,桁架结构材料物理参数、几何尺寸和外荷载均可表达为其区间因子和其确定性量的乘积,进而结构的位移和应力响应也可表达成区间因子们的函数。利用区间算法,推导出了结构位移和应力响应的上、下限和均值的计算表达式。通过算例,分析了结构参数和外荷载的不确定性对结构响应的影响,并验证了模型和方法的合理性与可行性。该方法的优点是能够反映结构某一参数的不确定性对结构响应的影响。     

Weak ternary interval evaluation of fatigue life and its application to mining dump truck

A method of weak ternary interval (TI) evaluation of fatigue life is proposed in this study to reasonably evaluate the nonprobabilistic reliability considering the interval uncertainty. The conventional interval is extended into a TI by introducing an expected value, that is, a value without uncertainty, as the third parameter of the interval to deal with the deviation between the expected value and the median value of the interval. Next, the TI of fatigue life is evaluated by introducing an index of attitude in the exponential function to weaken the unimportant upper bound and deal with the overconservative lower bound. Therefore, the evaluation is reflected in the form of an equivalent fatigue life as an indicator of the nonprobabilistic reliability. Next, the TIs of the fatigue parameters are obtained with limited fatigue test data from a practical engineering application. Finally, the proposed method is applied to a mining dump truck frame to demonstrate its significance and validity.     

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。     

结构区间有限元方程组的一种解法

针对结构静力区间有限元方程组的求解提出了一种简易解法。该法将含区间变量的整体刚度矩阵在区间变量的中值处进行一阶泰勒式展开。在对刚度矩阵展开式进行近似处理之后,将刚度矩阵的逆矩阵用一系列的Neumann展开级数来表示。为减小区间运算的扩张,利用区间乘法运算的次分配律和相关运算规则,导出不确定结构响应量上界、下界的计算式。几个算例结果分析表明:该方法具有较好的精度,是可行和有效的,且易于编程实施。     

Uncertain static plane stress analysis with interval fields

Uncertain static plane stress analysis of continuous structure involving interval fields is investigated in this study. Unlike traditional interval analysis of discrete structure, the interval field is adopted to model the uncertainty, as well as the dependency between the physical locations and degrees of variability, of all interval system parameters presented in the continuous structures. By implementing the flexibility properties of some common structural elements, a new computational scheme is proposed to reformulate the uncertain static plane stress analysis with interval fields into standard mathematical programming problems. Consequently, feasible upper and lower bounds of structural responses can be effectively yet efficiently determined. In addition, the proposed method is adequate to deal with situations involving one‐dimensional and two‐dimensional interval fields, which enhances the pertinence of the proposed approach by incorporating both discrete and continuous structures. In addition, the proposed computational scheme is able to establish the realizations of the uncertain parameters causing the extreme structural responses at zero computational cost. The applicability and credibility of the established computational framework are rigorously justified by various numerical investigations. Copyright © 2016 John Wiley & Sons, Ltd.     

Discussion of “Reliability‐based design optimization with dependent interval variables” by Xiaoping Du,International Journal for Numerical Methods in Engineering 2012; 91:218–228

In the subject paper, a reliability‐based design optimization (RBDO) model with both random and dependent interval uncertainties was proposed based on the First Order Reliability Method. The lower bound of reliability defined in Equation (9) of the subject paper was utilized as the constraint in this RBDO model. The author claimed that it is the minimum reliability with both random and interval variables. However, we prove that it is not the minimum value. It is therefore suggested that the minimum reliability should be used in the RBDO model. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

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.     

Reliability Bounds based on Universal Generating Function and Discrete Stress-strength Interference Model

A method for estimating the component reliability is proposed when the probability density functions of stress and strength can not be exactly determined. For two groups of finite experimental data about the stress and strength,an interval statistics method is introduced. The processed results are formulated as two interval-valued random variables and are graphically represented by using two histograms. The lower and upper bounds of component reliability are proposed based on the universal generating function method and are calculated by solving two discrete stress-strength interference models. The graphical calculations of the proposed reliability bounds are presented through a numerical example and the confidence of the proposed reliability bounds is discussed to demonstrate the validity of the proposed method. It is showed that the proposed reliability bounds can undoubtedly bracket the real reliability value. The proposed method extends the exciting universal generating function method and can give an interval estimation of component reliability in the case of lake of sufficient experimental data. An application example is given to illustrate the proposed method.     

Dynamic reliability analysis for non-stationary non-Gaussian response based on the bivariate vector translation process

Calculation of probability of exceedance for nonstationary non-Gaussian responses remains a great challenge to researchers in the field of structural reliability. In this paper, an analytical solution is proposed for calculating the mean upcrossing rate (MCR) of the non-stationary non-Gaussian responses by approximating the displacement and velocity responses with the bivariate vector translation process, in which the unified Hermite polynomial model (UHPM) is selected as the mapping function. The first four moments (i.e., mean value, standard deviation, skewness, and kurtosis) and cross-correlation function of the displacement and velocity responses needed in UHPM are estimated from some representative samples generated by random function-spectral representation method (RFSRM) and time-domain analysis. Under the Poisson assumption of the upcrossing events, the calculation of extreme value distribution or probability of exceedance for structural response can be determined with the proposed method. The proposed method is applicable to a wide range of structural responses, including asymmetric and hardening or softening responses. Three numerical examples are provided to demonstrate the efficiency and accuracy of the proposed method. It can be concluded that the proposed method provides an accurate and useful tool for dynamic reliability assessment in engineering applications.     

随机参数平面连续体结构的频率拓扑优化设计

摘 要 研究几何和物理参数均为随机变量的平面连续体结构在结构基频约束下的拓扑优化设计问题。以结构总质量均值极小化为目标函数,以结构的形状拓扑信息为设计变量,以结构基频概率可靠性指标为约束条件,构建了随机结构拓扑优化设计数学模型。利用代数综合法,导出了随机参数结构动力响应的均值和均方差的计算表达式。采用渐进结构优化的求解策略与方法,通过两个算例验证了文中模型及求解方法的合理性和可行性。     

Truss layout design under nonprobabilistic reliability-based topology optimization framework with interval uncertainties

A nonprobabilistic reliability-based topology optimization (NRBTO) method for truss structures with interval uncertainties (or unknown-but-bounded uncertainties) is proposed in this paper. The cross-sectional areas of levers are defined as design variables, while the material properties and external loads are regard as interval parameters. A modified perturbation method is applied to calculate structural response bounds, which are the prerequisite to obtain structural reliability. A deviation distance between the current limit state plane and the objective limit state plane, of which the expression is explicit, is defined as the nonprobabilistic reliability index, which serves as a constraint function in the optimization model. Compared with the deterministic topology optimization problem, the proposed NRBTO formulation is still a single-loop optimization problem, as the reliability index is explicit. The sensitivity results are obtained from an analytical approach as well as a direct difference method. Eventually, the NRBTO problem is solved by a sequential quadratic programming method. Two numerical examples are used to testify the validity and effectiveness of the proposed method. The results show significant effects of uncertainties to the topology configuration of truss structures.     

An orthogonal normal transformation of correlated non-normal random variables for structural reliability

In this paper, an efficient and explicit technique is proposed for transforming correlated non-normal random variables into independent standard normal variables based on the three-parameter (3P) lognormal distribution. In contrast with the classic Nataf transformation, the derived equivalent correlation coefficient in non-orthogonal standard normal space of the proposed transformation is expressed as an explicit formula, thereby avoiding tedious iteration algorithm or multifarious empirical formulas. Meanwhile, the applicable range of the original correlation coefficient is determined based on fundamental properties of the proposed expression of correlation distortion and the definition of correlation coefficient. The proposed transformation requires only the first three moments (i.e., mean, standard deviation, and skewness) of basic random variables, as well as their correlation matrix. Therefore, the proposed transformation can also be applied even when the joint distribution or marginal distributions of the basic random variables are unknown. Several numerical examples are presented to demonstrate the user-friendliness, efficiency, and accuracy of the proposed transformation applied in structural reliability analysis involving correlated non-normal random variables.     

考虑载荷作用次数的机械零部件可靠性灵敏度分析方法

 为了研究载荷多次作用时,机械零部件的可靠度及可靠性灵敏度变化规律,从灵敏度角度修改零件的设计参数,降低制造成本,建立一种可靠性模型结合了随机摄动法、Edgeworth级数技术,并考虑了载荷的作用次数．摄动法和Edgeworth级数可以在基本随机参数的前4阶矩已知的情况下,研究具有任意分布参数的机械零件的可靠性灵敏度设计问题,顺序统计量理论考虑了载荷作用次数在可靠度和灵敏度计算中的影响．使用这种模型计算出的可靠度会随着载荷作用次数而变化,这与静态的可靠度计算方法存在差别．以某一型号的螺栓为算例,应用此模型计算了其可靠度、随机变量均值和方差的可靠性灵敏度．由提出的方法得到了可靠度和可靠性灵敏度值及其随载荷作用次数变化的曲线．可靠度及可靠性灵敏度随载荷作用次数变化的规律是：载荷作用次数增加,可靠度值降低,变化趋势单调;载荷作用次数达到最大时,可靠度达到最小;随机变量均值和方差的灵敏度随载荷作用次数变化出现不同的变化趋势,其中螺栓截面直径的均值和方差灵敏度随载荷作用次数的变化最大,随作用次数的增加,螺栓截面直径的参数将对螺栓的可靠性起主要的决定作用．     

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–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.     

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.