Uncertainty and sensitivity analysis for models with correlated parameters

When conducting sensitivity and uncertainty analysis, most of the global sensitivity techniques assume parameter independence. However, it is common that the parameters are correlated with each other. For models with correlated inputs, we propose that the contribution of uncertainty to model output by an individual parameter be divided into two parts: the correlated contribution (by the correlated variations, i.e. variations of a parameter which are correlated with other parameters) and the uncorrelated contribution (by the uncorrelated variations, i.e. the unique variations of a parameter which cannot be explained by any other parameters). So far, only a few studies have been conducted to obtain the sensitivity index for a model with correlated input. But these studies do not distinguish between the correlated and uncorrelated contribution of a parameter. In this study, we propose a regression-based method to quantitatively decompose the total uncertainty in model output into partial variances contributed by the correlated variations and partial variances contributed by the uncorrelated variations. The proposed regression-based method is then applied in three test cases. Results show that the regression-based method can successfully measure the uncertainty contribution in the case where the relationship between response and parameters is approximately linear.     

A new estimator for sensitivity analysis of model output: An application to the e-business readiness composite indicator

In this paper we propose and test a generalisation of the method originally proposed by Sobol’, and recently extended by Saltelli, to estimate the first-order and total effect sensitivity indices. Exploiting the symmetries and the dualities of the formulas, we obtain additional estimates of first-order and total indices at no extra computational cost. We test the technique on a case study involving the construction of a composite indicator of e-business readiness, which is part of the initiative “e-Readiness of European enterprises” of the European Commission “e-Europe 2005” action plan. The method is used to assess the contribution of uncertainties in (a) the weights of the component indicators and (b) the imputation of missing data on the composite indicator values for several European countries.     

Analytical solutions using a higher-order refined theory for the static analysis of antisymmetric angle-ply composite and sandwich plates

Analytical formulations and solutions to the static analysis of simply supported anti-symmetric angle-ply composite and sandwich plates hitherto not reported in the literature based on a higher-order refined theory already reported in the literature are presented. The theoretical model presented herein incorporates laminate deformations, which account for the effect of transverse shear deformation and a non-linear variation of in-plane displacements with respect to the thickness coordinate. The transverse displacement is assumed to be constant throughout the thickness. The equations of equilibrium are obtained using principle of minimum potential energy. Solutions are obtained in closed form using Navier's technique by solving the boundary value problem. Accuracy of the theoretical formulations and the solution method is first ascertained by comparing the results with that already reported in the literature. After establishing the accuracy of the solutions, numerical results with real properties are presented for the multilayer antisymmetric angle-ply composite and sandwich plates, which will serve as a benchmark for future investigations.     

A general first-order global sensitivity analysis method

Fourier amplitude sensitivity test (FAST) is one of the most popular global sensitivity analysis techniques. The main mechanism of FAST is to assign each parameter with a characteristic frequency through a search function. Then, for a specific parameter, the variance contribution can be singled out of the model output by the characteristic frequency. Although FAST has been widely applied, there are two limitations: (1) the aliasing effect among parameters by using integer characteristic frequencies and (2) the suitability for only models with independent parameters. In this paper, we synthesize the improvement to overcome the aliasing effect limitation [Tarantola S, Gatelli D, Mara TA. Random balance designs for the estimation of first order global sensitivity indices. Reliab Eng Syst Safety 2006; 91(6):717–27] and the improvement to overcome the independence limitation [Xu C, Gertner G. Extending a global sensitivity analysis technique to models with correlated parameters. Comput Stat Data Anal 2007, accepted for publication]. In this way, FAST can be a general first-order global sensitivity analysis method for linear/nonlinear models with as many correlated/uncorrelated parameters as the user specifies. We apply the general FAST to four test cases with correlated parameters. The results show that the sensitivity indices derived by the general FAST are in good agreement with the sensitivity indices derived by the correlation ratio method, which is a non-parametric method for models with correlated parameters.     

基于Morris方法的纤维复合材料结构件固化均匀性的全局灵敏度分析

纤维增强树脂基复合材料结构件的残余应力问题是制约其在航空航天、汽车和建筑领域大规模应用的关键问题。复合材料固化过程中温度场和固化度场的非均匀性是引起残余热应力和固化收缩应力的重要因素。为了探讨纤维复合材料结构件在固化成型过程中固化工艺温度、热传导系数、对流换热系数及结构件厚度对固化均匀性的敏感程度,采用数值模拟分析了这4个关键参数对温度场和固化度场均匀性的影响规律。模拟结果表明:升高固化工艺温度,复合材料温度场的非均匀性增大,固化度场的非均匀性减小;增大对流换热系数和热传导系数,复合材料温度场和固化度场的非均匀性减小;增加复合材料结构件的厚度,复合材料温度场和固化度场的非均匀性增大。在此基础上,应用Morris全局灵敏度分析方法对4个关键参数对复合材料固化均匀性的影响程度进行定量分析,得到固化均匀性的影响因素按灵敏程度由大到小的顺序为:结构件厚度、热传导系数、固化工艺温度、对流换热系数。     

Polynomial chaos expansion for sensitivity analysis

In this paper, the computation of Sobol's sensitivity indices from the polynomial chaos expansion of a model output involving uncertain inputs is investigated. It is shown that when the model output is smooth with regards to the inputs, a spectral convergence of the computed sensitivity indices is achieved. However, even for smooth outputs the method is limited to a moderate number of inputs, say 10-20, as it becomes computationally too demanding to reach the convergence domain. Alternative methods (such as sampling strategies) are then more attractive. The method is also challenged when the output is non-smooth even when the number of inputs is limited.     

Effect of transverse cracks on the elastic properties of high temperature angle-ply laminated composites

The development of transverse cracks can be detrimental to the stiffness and dimensional stability of composite laminates. In this investigation, a modified shear lag analysis, taking into account the concept of stress perturbation function, is employed to evaluate the effect of transverse cracks on the stiffness reduction in high temperature angle-ply laminated composites. The results present well the effect of high temperature and the fibre orientation of the outer layers on the degradation of mechanical properties of the angle-ply polymer composite laminates.     

Higher order refined computational model with 12 degrees of freedom for the stress analysis of antisymmetric angle-ply plates – analytical solutions

Analytical formulations and solutions for the stress analysis of simply supported antisymmetric angle-ply composite and sandwich plates hitherto not reported in the literature based on a higher order refined computational model with twelve degrees of freedom already reported in the literature are presented. The theoretical model presented herein incorporates laminate deformations which account for the effects of transverse shear deformation, transverse normal strain/stress and a nonlinear variation of in-plane displacements with respect to the thickness coordinate thus modelling the warping of transverse cross sections more accurately and eliminating the need for shear correction coefficients. In addition, two higher order computational models, one with nine and the other with five degrees of freedom already available in the literature are also considered for comparison. The equations of equilibrium are obtained using Principle of Minimum Potential Energy (PMPE). Solutions are obtained in closed form using Navier’s technique by solving the boundary value problem. Accuracy of the theoretical formulations and the solution method is first ascertained by comparing the results with that already available in the literature. After establishing the accuracy of the solutions, numerical results with real properties using all the computational models are presented for the stress analysis of multilayer antisymmetric angle-ply composite and sandwich plates, which will serve as a benchmark for future investigations.     

Geometrically nonlinear composite beam structures: design sensitivity analysis

The purpose of this paper is to develop a finite element model for optimal design of composite laminated thin-walled beam structures, with geometrically nonlinear behavior, including post-critical behavior. A continuation paper will be presented with design optimization applications of this model. The structural deformation is described by an updated Lagrangean formulation. The structural response is determined by a displacement controlled continuation method. A two-node Hermitean beam element is used. The beams are made from an assembly of flat-layered laminated composite panels. Beam cross-section mass and stiffness property matrices are presented.

Design sensitivities are imbedded into the finite element modeling and assembled in order to perform the structural design sensitivity analysis. The adjoint structure method is used. The lamina orientation and the laminate thickness are selected as the design variables. Displacement, failure index, critical load and natural frequency are considered as performance measures. The critical load constraint calculated as the limit point of the nonlinear response is also considered, but a new method is proposed, replacing it by a displacement constraint.     

Uncertainty analysis of structural output with closed-form expression based on surrogate model

Uncertainty analysis (UA) is the process that quantitatively identifies and characterizes the output uncertainty and has a crucial implication in engineering applications. The research of efficient estimation of structural output moments in probability space plays an important part in the UA and has great engineering significance. Given this point, a new UA method based on the Kriging surrogate model related to closed-form expressions for the perception of the estimation of mean and variance is proposed in this paper. The new proposed method is proven effective because of its direct reflection on the prediction uncertainty of the output moments of metamodel to quantify the accuracy level. The estimation can be completed by directly using the redefined closed-form expressions of the model’s output mean and variance to avoid excess post-processing computational costs and errors. Furthermore, a novel framework of adaptive Kriging estimating mean (AKEM) is demonstrated for more efficiently reducing uncertainty in the estimation of output moment. In the adaptive strategy of AKEM, a new learning function based on the closed-form expression is proposed. Based on the closed-form expression which modifies the computational error caused by the metamodeling uncertainty, the proposed learning function enables the updating of metamodel to reduce prediction uncertainty efficiently and realize the decrease in computational costs. Several applications are introduced to prove the effectiveness and efficiency of the AKEM compared with a universal adaptive Kriging method. Through the good performance of AKEM, its potential in engineering applications can be spotted.     

An approximate sensitivity analysis of results from complex computer models in the presence of epistemic and aleatory uncertainties

This paper focuses on sensitivity analysis of results from computer models in which both epistemic and aleatory uncertainties are present. Sensitivity is defined in the sense of “uncertainty importance” in order to identify and to rank the principal sources of epistemic uncertainty. A natural and consistent way to arrive at sensitivity results in such cases would be a two-dimensional or double-loop nested Monte Carlo sampling strategy in which the epistemic parameters are sampled in the outer loop and the aleatory variables are sampled in the nested inner loop. However, the computational effort of this procedure may be prohibitive for complex and time-demanding codes. This paper therefore suggests an approximate method for sensitivity analysis based on particular one-dimensional or single-loop sampling procedures, which require substantially less computational effort. From the results of such sampling one can obtain approximate estimates of several standard uncertainty importance measures for the aleatory probability distributions and related probabilistic quantities of the model outcomes of interest. The reliability of the approximate sensitivity results depends on the effect of all epistemic uncertainties on the total joint epistemic and aleatory uncertainty of the outcome. The magnitude of this effect can be expressed quantitatively and estimated from the same single-loop samples. The higher it is the more accurate the approximate sensitivity results will be. A case study, which shows that the results from the proposed approximate method are comparable to those obtained with the full two-dimensional approach, is provided.     

CDF sensitivity analysis technique for ranking influential parameters in the performance assessment of the proposed high-level waste repository at Yucca Mountain, Nevada, USA

A cumulative distribution function (CDF)-based method has been used to perform sensitivity analysis on a computer model that conducts total system performance assessment of the proposed high-level nuclear waste repository at Yucca Mountain, and to identify the most influential input parameters affecting the output of the model. The performance assessment computer model referred to as the TPA code, was recently developed by the US nuclear regulatory commission (NRC) and the center for nuclear waste regulatory analyses (CNWRA), to evaluate the performance assessments conducted by the US department of energy (DOE) in support of their license application. The model uses a probabilistic framework implemented through Monte Carlo or Latin hypercube sampling (LHS) to permit the propagation of uncertainties associated with model parameters, conceptual models, and future system states. The problem involves more than 246 uncertain parameters (also referred to as random variables) of which the ones that have significant influence on the response or the uncertainty of the response must be identified and ranked. The CDF-based approach identifies and ranks important parameters based on the sensitivity of the response CDF to the input parameter distributions. Based on a reliability sensitivity concept [AIAA Journal 32 (1994) 1717], the response CDF is defined as the integral of the joint probability-density-function of the input parameters, with a domain of integration that is defined by a subset of the samples. The sensitivity analysis does not require explicit knowledge of any specific relationship between the response and the input parameters, and the sensitivity is dependent upon the magnitude of the response. The method allows for calculating sensitivity over a wide range of the response and is not limited to the mean value.     

不确定条件下复合材料结构的全局灵敏度分析

基于不确定条件下结构的全局灵敏度分析理论,研究了输入变量的不确定性对复合材料结构输出响应量方差和失效概率的影响。考虑材料力学性能、铺设角、铺层厚度及加载载荷的不确定性,利用基于方差和基于失效概率的全局灵敏度分析方法,对复合材料结构输出位移和强度比的不确定性来源进行分析,得到输入变量的全局灵敏度排序结果。对复合材料工字梁结构进行算例分析,验证了所得排序结果的有效性,为工程实际中复合材料结构稳定性优化设计提供了一定的指导。     

Application of sensitivity analysis to oil refinery emissions

Catalyst emissions from fluidising catalytic cracking units have the potential to impact significantly on the environmental compliance of oil refineries. Traditionally it has been assumed that gas velocity and fine particles significantly impact on emission levels. Through the use of a simple fluidised bed model, sensitivity analysis was conducted to identify the key operating parameters that influence emission rates. It was found that in addition to velocity, density and mid sized particles are the most influential factors for emission rates. Further work is needed to identify how these parameters can be altered during normal operations to reduce catalyst emissions.     

基于剩余应变的复合材料结构寿命预测

为实现对复合材料结构的寿命预测,对已有的复合材料疲劳寿命预测模型进行了研究,确定了基本的刚度降模型,提出了剩余应变的概念,并将其应用到渐进疲劳损伤方法中,以Abaqus为平台,编写UMAT子程序,实现了对复合材料结构的寿命预测及疲劳损伤扩展分析.针对某碳纤维增强复合材料TS800开展相关试验,试验结果与预测结果吻合较好.研究表明,本文所改进的渐进疲劳损伤方法能较好地完成对复合材料结构的寿命预测.     

Implementation and evaluation of nonparametric regression procedures for sensitivity analysis of computationally demanding models

The analysis of many physical and engineering problems involves running complex computational models (simulation models, computer codes). With problems of this type, it is important to understand the relationships between the input variables (whose values are often imprecisely known) and the output. The goal of sensitivity analysis (SA) is to study this relationship and identify the most significant factors or variables affecting the results of the model. In this presentation, an improvement on existing methods for SA of complex computer models is described for use when the model is too computationally expensive for a standard Monte-Carlo analysis. In these situations, a meta-model or surrogate model can be used to estimate the necessary sensitivity index for each input. A sensitivity index is a measure of the variance in the response that is due to the uncertainty in an input. Most existing approaches to this problem either do not work well with a large number of input variables and/or they ignore the error involved in estimating a sensitivity index. Here, a new approach to sensitivity index estimation using meta-models and bootstrap confidence intervals is described that provides solutions to these drawbacks. Further, an efficient yet effective approach to incorporate this methodology into an actual SA is presented. Several simulated and real examples illustrate the utility of this approach. This framework can be extended to uncertainty analysis as well.     

Response surfaces and sensitivity analyses for an environmental model of dose calculations

A parametric sensitivity analysis is carried out on GASCON, a radiological impact software describing the radionuclides transfer to the man following a chronic gas release of a nuclear facility. An effective dose received by age group can thus be calculated according to a specific radionuclide and to the duration of the release. In this study, we are concerned by 18 output variables, each depending of approximately 50 uncertain input parameters. First, the generation of 1000 Monte-Carlo simulations allows us to calculate correlation coefficients between input parameters and output variables, which give a first overview of important factors. Response surfaces are then constructed in polynomial form, and used to predict system responses at reduced computation time cost; this response surface will be very useful for global sensitivity analysis where thousands of runs are required. Using the response surfaces, we calculate the total sensitivity indices of Sobol by the Monte-Carlo method. We demonstrate the application of this method to one site of study and to one reference group near the nuclear research Center of Cadarache (France), for two radionuclides: iodine 129 and uranium 238. It is thus shown that the most influential parameters are all related to the food chain of the goat's milk, in decreasing order of importance: dose coefficient “effective ingestion”, goat's milk ration of the individuals of the reference group, grass ration of the goat, dry deposition velocity and transfer factor to the goat's milk.     

0.01～961.78℃分温区90国际温标的传播不确定度方程

通过直接对内插方程求导,获得了0.01~961.78℃分温区的传播不确定度方程。其灵敏度系数的基本结构与内插方程相同,仍然是组成原内插方程的基础函数的线性组合,但组合系数不同,且仍是一分段函数。     

Thermo-viscoelastic analysis of the integrated T-shaped composite structures

A thermo-viscoelastic finite element analysis is used to investigate the residual stresses and the curing deformation of the integrated T-shaped composite structure. First, a three dimensional (3D) incremental viscoelastic constitutive equation is established and implemented into the finite element software ABAQUS to predict the full field warpage profiles of the integrated T-shaped structures. These results are validated based on the measured data obtained from digital speckle correlation technology. Second, the effects of the cooling rate on the warpage deformation and the residual stresses of the integrated T-shaped composite structure are studied. Finally, the relationships between the different curing strategies and the corresponding residual stresses are studied, and it shows that the Outside-to-Inside curing strategy will develop the smallest residual stresses for the integrated T-shaped composite structures.     

Variance decomposition-based sensitivity analysis via neural networks

This paper illustrates a method for efficiently performing multiparametric sensitivity analyses of the reliability model of a given system. These analyses are of great importance for the identification of critical components in highly hazardous plants, such as the nuclear or chemical ones, thus providing significant insights for their risk-based design and management. The technique used to quantify the importance of a component parameter with respect to the system model is based on a classical decomposition of the variance. When the model of the system is realistically complicated (e.g. by aging, stand-by, maintenance, etc.), its analytical evaluation soon becomes impractical and one is better off resorting to Monte Carlo simulation techniques which, however, could be computationally burdensome. Therefore, since the variance decomposition method requires a large number of system evaluations, each one to be performed by Monte Carlo, the need arises for possibly substituting the Monte Carlo simulation model with a fast, approximated, algorithm. Here we investigate an approach which makes use of neural networks appropriately trained on the results of a Monte Carlo system reliability/availability evaluation to quickly provide with reasonable approximation, the values of the quantities of interest for the sensitivity analyses. The work was a joint effort between the Department of Nuclear Engineering of the Polytechnic of Milan, Italy, and the Institute for Systems, Informatics and Safety, Nuclear Safety Unit of the Joint Research Centre in Ispra, Italy which sponsored the project.