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.     

Robust optimal design using a second-order tolerance model

In this paper we present a general, quantitative method for developing designs that are robust to variation in design variables and parameters. Variation is defined in terms of tolerances which bracket the expected deviation of uncertain quantities about nominal values. We specifically address the case where input variations are assumed to be random variables that are normally distributed. The method incorporates a second-order tolerance model as part of a nonlinear optimization process. The second-order tolerance model makes it possible to estimate the skewness of function distributions, which are modeled with a three-parameter gamma distribution. We apply the method to determine robust designs for 11 test cases that span a variety of problems; robustness is verified with Monte Carlo simulation. The method enables a designer to understand and account for the effects of tolerances, making it possible to build robustness into an engineering design.     

Sample regeneration algorithm for structural failure probability function estimation

An efficient strategy to approximate the failure probability function in structural reliability problems is proposed. The failure probability function (FPF) is defined as the failure probability of the structure expressed as a function of the design parameters, which in this study are considered to be distribution parameters of random variables representing uncertain model quantities. The task of determining the FPF is commonly numerically demanding since repeated reliability analyses are required. The proposed strategy is based on the concept of augmented reliability analysis, which only requires a single run of a simulation-based reliability method. This paper introduces a new sample regeneration algorithm that allows to generate the required failure samples of design parameters without any additional evaluation of the structural response. In this way, efficiency is further improved while ensuring high accuracy in the estimation of the FPF. To illustrate the efficiency and effectiveness of the method, case studies involving a turbine disk and an aircraft inner flap are included in this study.     

Probabilistic analysis of a pull-out test

This paper presents a sensitivity analysis of the pull-out strength of reinforcement embedded in concrete. Considering both European and French design codes, this failure strength depends on the variability of uncertain parameters such as Young’s modulus of concrete and yield stresses of materials (concrete and steel); moreover, two failure modes can be observed in the studied experimental test. A methodology allowing the characterization of the sensitivity of the pull-out strength to these uncertain parameters is derived. These parameters are modeled by Lognormal random variables. Results show the evolution of the pull-out strength for different anchorage lengths. Probability density functions of the random variable modeling the failure strength are computed using probabilistic methods. A finite element model is also built to quantify uncertainties concerning failure modes, computing 95% confidence intervals.     

复合材料支撑机翼撑杆位置与结构综合优化设计

针对复合材料支撑机翼,发展了一种撑杆位置和结构综合优化设计的方法。在两种严重设计载荷状态下,考虑气动弹性效应和复合材料铺层结构的不确定性,以结构质量最小化为目标,以翼尖垂直变形、翼尖扭角、撑杆屈曲稳定性、颤振速度和强度要求为约束,在一个优化过程中实现了撑杆位置和结构参数的同步优化设计和鲁棒优化设计。结果表明,翼尖垂直变形和颤振速度要求对于撑杆位置影响明显,最优的撑杆展向位置都靠近翼根一侧,同时撑杆的总体稳定性成为同步优化设计的关键约束。鲁棒优化设计得到的撑杆位置和结构参数的最优组合对铺层结构的不确定性摄动具有良好的抗干扰性,鲁棒优化得到的最优撑杆位置会随着设计变量摄动范围而变化,翼尖垂直变形成为鲁棒优化设计的关键约束。     

A comparison between different robust optimum design approaches: Application to tuned mass dampers

This paper is focused on the comparison between different approaches in structural optimization. More precisely, the conventional deterministic optimum design, based on the assumption that the only source of uncertainty concerns the forcing input, is compared to robust single-objective and multi-objective optimum design methods.The analysis is developed by considering as case of study a single-degree-of-freedom system with uncertain parameters, subject to random vibrations and equipped with a tuned mass damper device (TMD). The optimization problem concerns the selection of TMD mechanical characteristics able to enlarge the efficiency of the strategy of vibration reduction.Results demonstrate the importance of performing a robust optimum design and show that the multi-objective robust design methodology provides a significant improvement in performance stability, giving a better control of the design solution choice.     

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.     

Multiresponse optimization using multivariate process capability index

Robust design is an efficient method for product and process improvement which combines experimentation with optimization to create a system that is less sensitive to uncontrollable variation. In this article, a simple and integrated modeling methodology for robust design is proposed. This methodology achieves the robustness objective function and input variables constraints simultaneously. The objective function is written in terms of the multivariate process capability vector (MC_pm) of several competing features of the system under study. The proposed methodology is applicable to general functions of the system performance with random variables. The effectiveness of the methodology is verified using two real‐world examples which are compared with those of other robust design methods. Copyright © 2010 John Wiley & Sons, Ltd.     

Robust optimization of structures subjected to stochastic earthquake with limited information on system parameter uncertainty

The optimization of structures subjected to stochastic earthquake and characterized by uncertain parameters is usually posed in the form of non-linear programming with stochastic performance measures where the uncertain parameters are modelled as random variables. Such an approach, however, cannot be adopted in many real life situations when the limited information about uncertainty can be only modelled as of the uncertain but bounded (UBB) type. A robust optimization strategy for stochastic dynamic systems characterized by UBB parameters is proposed in the present study in the framework of the response surface method (RSM). In evaluating the stochastic constraints, repeated computations of the dynamic responses are avoided by applying an adaptive RSM based on the moving least squares method. Numerical results are presented to highlight the effectiveness of the proposed procedure. The effect of parameter uncertainty is also studied by comparing the results obtained from the proposed optimization approach with the conventional stochastic optimization results.     

Polymorphic uncertain nonlinear programming approach for maximizing the capacity of V-belt driving

In this paper, a polymorphic uncertain nonlinear programming (PUNP) approach is developed to formulate the problem of maximizing the capacity in a system of V-belt driving with uncertainties. The constructed optimization model is found to consist of a nonlinear objective function and some nonlinear constraints with some parameters which are of uncertain nature. These uncertain parameters are interval parameters, random interval parameters, fuzzy parameters or fuzzy interval parameters. To find a robust solution of the problem, a deterministic equivalent formulation (DEF) is established for the polymorphic uncertain nonlinear programming model. For a given satisfaction level, this DEF turns out to be a nonlinear programming involving only interval parameters. A solution method, called a sampling based interactive method, is developed such that a robust solution of the original model with polymorphic uncertainties is obtained by using standard smooth optimization techniques. The proposed method is applied into a real-world design of V-belt driving, and the results indicate that both the PUNP approach and the developed algorithm are useful to the optimization problem with polymorphic uncertainty.     

An uncertain multidisciplinary design optimization method using interval convex models

This article proposes an uncertain multi-objective multidisciplinary design optimization methodology, which employs the interval model to represent the uncertainties of uncertain-but-bounded parameters. The interval number programming method is applied to transform each uncertain objective function into two deterministic objective functions, and a satisfaction degree of intervals is used to convert both the uncertain inequality and equality constraints to deterministic inequality constraints. In doing so, an unconstrained deterministic optimization problem will be constructed in association with the penalty function method. The design will be finally formulated as a nested three-loop optimization, a class of highly challenging problems in the area of engineering design optimization. An advanced hierarchical optimization scheme is developed to solve the proposed optimization problem based on the multidisciplinary feasible strategy, which is a well-studied method able to reduce the dimensions of multidisciplinary design optimization problems by using the design variables as independent optimization variables. In the hierarchical optimization system, the non-dominated sorting genetic algorithm II, sequential quadratic programming method and Gauss–Seidel iterative approach are applied to the outer, middle and inner loops of the optimization problem, respectively. Typical numerical examples are used to demonstrate the effectiveness of the proposed methodology.     

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.     

Consistent and robust ranking in imprecise data envelopment analysis under perturbations of random subsets of data

Data envelopment analysis (DEA) is a non-parametric method for measuring the relative efficiency of a set of decision making units using multiple precise inputs to produce multiple precise outputs. Several extensions to DEA have been made for the case of imprecise data, as well as to improve the robustness of the assessment for these cases. Prevailing robust DEA (RDEA) models are based on mirrored interval DEA models, including two distinct production possibility sets (PPS). However, this approach renders the distance measures incommensurate and violates the standard assumptions for the interpretation of distance measures as efficiency scores. We propose a modified RDEA (MRDEA) model with a unified PPS to overcome the present problem in RDEA. Based on a flexible formulation for the number of variables perturbed, MRDEA calculates the empirical distribution for the interval efficiency for the case of a random number of variables affected. The MRDEA approach also decreases the computational complexity of the RDEA model, as well as significantly increases the discriminatory power of the model without additional information requirements. The properties of the method are demonstrated for four different numerical instances.     

Probabilistic evaluation of fuselage-type composite structures

A methodology is developed to simulate computationally the uncertain behavior of composite structures. The uncertain behavior includes buckling loads, natural frequencies, displacements, stress/strain, etc., which are the consequences of the random variation (scatter) of the primitive (independent random) variables in the constituent, ply, laminate and structural levels. This methodology is implemented in a computer code IPACS (integrated probabilistic assessment of composite structures). A fuselage-type composite structure is analyzed to demonstrate the code's capability. The probability distribution functions of the buckling loads, natural frequency, displacement, strain and stress are computed. The sensitivity of each primitive (independent random) variable to a given structural response is also identified from the analyses.     

Assessing multivariate process/product yield via discrete point approximation

The assessment of multivariate yield is central to the robust design of products/processes. Currently, yield is evaluated via Monte Carlo simulation. However, it requires thousands of replications per simulation to achieve an acceptably precise estimate of yield, this is often tedious and time consuming, thereby rendering it unattractive as an evaluation tool. We propose a discrete point approximation on each design variable, using general Beta distributions, for assessing reasonably precise multivariate yield estimates, which require only a minute fraction of the Monte Carlo replications/simulations required to estimate yield (e.g., 3 and 5 design variables would require only 3³ = 27 and 3⁵ = 243 replications, respectively). The Beta distribution has the desirable property of being able to characterize a wide variety of processes that may or may not be symmetric and which may or may not have a finite operating range. Using an approach that computes the roots of a polynomial, whose degree is determined by the number of discrete points, discrete three point approximations are obtained and tabulated for twenty-five different Beta distributions. Based on several test examples, where design parameters are modeled as independent Beta random variates, our approach appears to be highly accurate, achieving virtually the same multivariate yield estimate as that obtained via Monte Carlo simulation. The substantial reduction in the number of replications and associated computational time required to assess yield makes the iterative adjustment of design parameters a more practical design strategy.     

Variable screening and ranking using sampling-based sensitivity measures

This paper presents a methodology for screening insignificant random variables and ranking significant important random variables using sensitivity measures including two cumulative distribution function (CDF)-based and two mean-response based measures. The methodology features (1) using random samples to compute sensitivities and (2) using acceptance limits, derived from the test-of-hypothesis, to classify significant and insignificant random variables. Because no approximation is needed in either the form of the performance functions or the type of continuous distribution functions representing input variables, the sampling-based approach can handle highly nonlinear functions with non-normal variables. The main characteristics and effectiveness of the sampling-based sensitivity measures are investigated using both simple and complex examples. Because the number of samples needed does not depend on the number of variables, the methodology appears to be particularly suitable for problems with large, complex models that have large numbers of random variables but relatively few numbers of significant random variables.     

Robust observer based control for plasma glucose regulation in type 1 diabetes patient using attractive ellipsoid method

This paper deals with the design of robust observer based output feedback control law for the stabilisation of an uncertain nonlinear system and subsequently apply the developed method for the regulation of plasma glucose concentration in Type 1 diabetes (T1D) patients. The principal objective behind the proposed design is to deal with the issues of intra‐patient parametric variation and non‐availability of all state variables for measurement. The proposed control technique for the T1D patient model is based on the attractive ellipsoid method (AEM). The observer and controller conditions are obtained in terms of linear matrix inequality (LMI), thus allowing to compute easily both the observer and controller gains. The closed‐loop response obtained using the designed controller avoids adverse situations of hypoglycemia and post‐prandial hyperglycemia under uncertain conditions. Further to validate the robustness of the design, closed‐loop simulations of random 200 virtual T1D patients considering parameters within the considered ranges are presented. The results indicate that hypoglycemia and post‐prandial hyperglycemia are significantly reduced in the presence of bounded (±30% ) parametric variability and uncertain exogenous meal disturbance.Inspec keywords: medical control systems, observers, uncertain systems, nonlinear control systems, robust control, control system synthesis, linear matrix inequalities, feedback, sugar, closed loop systems, diseasesOther keywords: virtual T1D patients, type 1 diabetes patients, closed‐loop simulations, uncertain conditions, post‐prandial hyperglycemia, designed controller, closed‐loop response, controller gains, linear matrix inequality, controller conditions, T1D patient model, control technique, intra‐patient parametric variation, principal objective, plasma glucose concentration, uncertain nonlinear system, robust observer based output feedback control law, attractive ellipsoid method, plasma glucose regulation     

Comparison of evidence theory and Bayesian theory for uncertainty modeling

This paper compares Evidence Theory (ET) and Bayesian Theory (BT) for uncertainty modeling and decision under uncertainty, when the evidence about uncertainty is imprecise. The basic concepts of ET and BT are introduced and the ways these theories model uncertainties, propagate them through systems and assess the safety of these systems are presented. ET and BT approaches are demonstrated and compared on challenge problems involving an algebraic function whose input variables are uncertain. The evidence about the input variables consists of intervals provided by experts. It is recommended that a decision-maker compute both the Bayesian probabilities of the outcomes of alternative actions and their plausibility and belief measures when evidence about uncertainty is imprecise, because this helps assess the importance of imprecision and the value of additional information. Finally, the paper presents and demonstrates a method for testing approaches for decision under uncertainty in terms of their effectiveness in making decisions.     

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。     

Robust topology optimization of continuum structures with loading uncertainty using a perturbation method

An approach for the robust topology optimization (RTO) of continuum structures with loading uncertainty is investigated. The loading uncertainties are quantified using the second order Taylor series expansion of uncertain loading magnitudes and directions, and then the response statistic mean and standard deviation of compliance are calculated using the uncertain perturbation propagation method. A robust design Lagrange function considering the compliance objective and finite element constraints is developed, and a sensitivity analysis is performed to calculate the Lagrange coefficients. The Lagrange objective function is optimized using the modified solid isotropic material with penalization (SIMP) algorithm; thus, the optimum material distribution under loading uncertainty is acquired. The proposed methodology is used for the RTO of two examples, revealing its efficiency under both concentrated and distributed uncertain loadings. The accuracy of the results is verified by comparison with similar cases found in the literature where a different modelling approach was used.