A stochastic model for elasticity tensors with uncertain material symmetries

In this paper, we consider the probabilistic modeling of media exhibiting uncertainties on material symmetries. More specifically, we address both the construction of a stochastic model and the definition of a methodology allowing the numerical simulation (and consequently, the inverse experimental identification) of random elasticity tensors whose mean distance (in a sense to be defined) to a given class of material symmetry is specified. Following the eigensystem characterization of the material symmetries, the proposed approach relies on the probabilistic model derived in Mignolet and Soize (2008), allowing the variance of selected eigenvalues of the elasticity tensor to be partially prescribed. In this context, a new methodology (regarding in particular the parametrization of the model) is defined and illustrated in the case of transversely isotropic materials. The efficiency of the approach is demonstrated by computing the mean distance of the random elasticity tensor to a given material symmetry class, the distance and projection onto the space of transversely isotropic tensors being defined by considering the Riemmanian metric and the Euclidean projection, respectively. It is shown that the methodology allows the above distance to be (partially) reduced as the overall level of statistical fluctuations increases, no matter the initial distance of the mean model used in the simulations. A comparison between this approach and the initial nonparametric approach introduced in Soize (2008) is finally provided.     

Dynamic stability of a pipe conveying fluid with an uncertain computational model

This paper deals with the problem of a pipe conveying fluid of interest in several engineering applications, such as micro-systems or drill-string dynamics. The deterministic stability analysis developed by Paidoussis and Issid (1974) is extended to the case for which there are model uncertainties induced by modeling errors in the computational model. The aim of this work is twofold: (1) to propose a probabilistic model for the fluid–structure interaction considering modeling errors and (2) to analyze the stability and reliability of the stochastic system. The Euler–Bernoulli beam model is used to model the pipe and the plug flow model is used to take into account the internal flow in the pipe. The resulting differential equation is discretized by means of the finite element method and a reduced-order model is constructed from some eigenmodes of the beam. A probabilistic approach is used to model uncertainties in the fluid–structure interaction. The proposed strategy takes into account global uncertainties related to the noninertial coupled fluid forces (related to damping and stiffness). The resulting random eigenvalue problem is used to analyze flutter and divergence unstable modes of the system for different values of the dimensionless flow speed. The numerical results show the random response of the system for different levels of uncertainty, and the reliability of the system for different dimensionless speeds and levels of uncertainty.     

Assessment of Digital Image Correlation Measurement Accuracy in the Ultimate Error Regime: Improved Models of Systematic and Random Errors

The literature contains many studies on assessment of DIC uncertainties, particularly in the ultimate error regime, when the shape function used to describe the material transformation perfectly matches the actual transformation. For pure sub-pixel translations, bias and random errors obtained for experimental or synthetic images show more complex evolution versus the fractional part of displacement than those predicted by the existing theoretical models. Indeed, small deviations arise, mainly around integer values of imposed displacements for noisy images, and they are interpreted as the unrepresentativeness of the underlying hypotheses of the theoretical models. In a first step, differences between imposed and measured displacements are analysed: random error is independent of fractional displacement, and systematic error does not decrease for values close to integer displacements whatever the noise level. Therefore, new prediction models are proposed based on the analysis of identified phenomena from synthetic speckle-pattern 8-bit images. The statistical approach used in this paper generalizes the methods proposed in the literature and mimics the experimental methodology usually used for displacement measurements performed in different subsets in the same image. Two closed-form expressions for the systematic and random errors and a linear interpolation scheme are developed. These models, depending only on image properties and the imposed displacement, are built with a very limited number of parameters. It is then possible to predict the evolution of bias and random errors from one to four images.     

Theoretical framework and experimental procedure for modelling mesoscopic volume fraction stochastic fluctuations in fiber reinforced composites

Many engineering materials exhibit fluctuations and uncertainties on their macroscopic mechanical properties. This randomness results from random fluctuations observed at a lower scale, especially at the meso-scale where microstructural uncertainties generally occur. In the present paper, we first propose a complete theoretical stochastic framework (that is, a relevant probabilistic model as well as a non-intrusive stochastic solver) in which the volume fraction at the microscale is modelled as a random field whose statistical reduction is performed using a Karhunen–Loeve expansion. Then, an experimental procedure dedicated to the identification of the parameters involved in the probabilistic model is presented and relies on a non-destructive ultrasonic method. The combination of the experimental results with a micromechanical analysis provides realizations of the volume fraction random field. In particular, it is shown that the volume fraction can be modelled by a homogeneous random field whose spatial correlation lengths are determined and may provide conditions on the size of the meso-volumes to be considered.     

A stochastic homogenization approach to estimate bone elastic properties

The mechanical properties of bone tissue depend on its hierarchical structure spanning many length scales, from the organ down to the nanoscale. Multiscale models allow estimating bone mechanical properties at the macroscale based on information on bone organization and composition at the lower scales. However, the reliability of these estimates can be questioned in view of the many uncertainties affecting the information which they are based on. In this paper, a new methodology is proposed, coupling probabilistic modeling and micromechanical homogenization to estimate the material properties of bone while taking into account the uncertainties on the bone micro- and nanostructure. Elastic coefficients of bone solid matrix are computed using a three-scale micromechanical homogenization method. A probabilistic model of the uncertain parameters allows propagating the uncertainties affecting their actual values into the estimated material properties of bone. The probability density functions of the random variables are constructed using the Maximum Entropy principle. Numerical simulations are used to show the relevance of this approach.     

Stochastic framework for modeling the linear apparent behavior of complex materials: Application to random porous materials with interphases

This paper is concerned with the modeling of randomness in multiscale analysis of heterogeneous materials. More specifically, a framework dedicated to the stochastic modeling of random properties is first introduced. A probabilistic model for matrix-valued second-order random fields with symmetry propertries, recently proposed in the literature, is further reviewed. Algorithms adapted to the Monte Carlo simulation of the proposed representation are also provided. The derivations and calibration procedure are finally exemplified through the modeling of the apparent properties associated with an elastic porous microstructure containing stochastic interphases.     

不确定动力系统平稳随机响应分析

由于设计、建造以及测量等诸多不确定因素的影响,通常的有限元力学分析模型只是原型结构的一种均值近似,采用随机结构模型是更为合理的.本文应用随机矩阵模拟不确定线性动力系统有限元模型中质量阵、阻尼阵和刚度阵的随机不确定性,并进一步建立此类非参数概率系统在平稳随机外载作用下动力响应的虚拟激励高效求解算法.数值结果表明,均值有限元模型和随机矩阵模型的动力响应具有很大的差异.对于精细制造,模型的随机性是不能忽略的,本文提出的算法为此类问题求解提供了一条有效途径.     

具有随机参数的含裂纹板弯曲应力强度因子的统计分析

本文首次应用随机有限元法研究了具有随机参数的含裂纹板裂纹尖端弯曲应力强度因子的统计性质。文中首先给出了杂交模式的裂纹尖端奇异单元的刚度矩阵,然后基于随机场的局部平均理论和一阶泰勒展开得到了应力强度因子均值和方差的计算公式。作为数例,详细讨论了杨氏模量、泊松比及板厚度的不确定性对应力强度因子的影响。     

An efficient 3D stochastic finite element method

Real life structural systems are characterized by their inherent or externally induced uncertainties in the design parameters. This study proposes a stochastic finite element tool efficient to take account of these uncertainties. Here uncertain structural parameter is modeled as homogeneous Gaussian stochastic field and commonly used two-dimensional (2D) local averaging technique is extended and generalized for 3D random field. This is followed by Cholesky decomposition of respective covariance matrix for digital simulation. By expanding uncertain stiffness matrix about its reference value, the Neumann expansion method is introduced blended with direct Monte Carlo simulation. This approach involves decomposition of stiffness matrix only once for the entire simulated structure. Thus substantial saving of CPU time and also the scope of tackling several stochastic fields simultaneously are the basic advantages of the proposed algorithm. Accuracy and efficiency of this method with reference to example problem is also studied here and numerical results validate its superiority over direct simulation method or first-order perturbation approach.     

Sensitivity,probabilistic and stochastic analysis of the thermo-piezoelectric phenomena in solids by the stochastic perturbation technique

The main aim here is to present the application of the generalized stochastic perturbation technique to thermo-piezoelectric analysis of solid continua. The general nth order Taylor series representation for all random input parameters and the state functions is employed to formulate the coupled thermo-electro-elasticity equilibrium equations of the additional order; a determination of any probabilistic moments and characteristics is described; the discretization of the problem in terms of the Stochastic perturbation-based Finite Element Method is also provided. Since this expansion includes the lowest order partial derivatives, the structural sensitivity analysis using direct differentiation is performed at the same time with probabilistic modeling contrasted with the Monte-Carlo simulation results. The probabilistic approach is extended here towards an accounting for the stochastic ageing processes, which appear frequently in aggressive external environments and under dynamic excitation. The two parametric stochastic process with Gaussian initial value and ageing velocity is tested for this purpose. The entire procedure is tested on the example of the thermo-electro-elastic pulsation of the beam modeled analytically using the symbolic software MAPLE, where polynomial approximations, design sensitivities, probabilistic moments and their histories are computed and visualized.     

Stochastic non-linear response of a flexible rotor with local non-linearities

The effects of uncertainties on the non-linear dynamics response remain misunderstood and most of the classical stochastic methods used in the linear case fail to deal with a non-linear problem. So we propose to take into account of uncertainties into non-linear models, by coupling the Harmonic Balance Method (HBM) and the Polynomial Chaos Expansion (PCE). The proposed method called the Stochastic Harmonic Balance Method (Stochastic-HBM) is based on a new formulation of the non-linear dynamic problem in which not only the approximated non-linear responses but also the non-linear forces and the excitation pulsation are considered as stochastic parameters. Expansions on the PCE basis are performed by passing via an Alternate Frequency Time method with Probabilistic Collocation (AFTPC) for estimating the stochastic non-linear forces in the stochastic domain and the frequency domain. In the present paper, the Stochastic Harmonic Balance Method (Stochastic-HBM) that is applied to a flexible non-linear rotor system, with random parameters modeled as random fields, is presented. The Stochastic-HBM combined with an Alternate Frequency-Time method with Probabilistic Collocation (AFTPC) allows us to solve dynamical problems with non-regular non-linearities in presence of uncertainties. In this study, the procedure is developed for the estimation of stochastic non-linear responses of the rotor system with different regular and non-regular non-linearities. The finite element rotor system is composed of a shaft with two disks and two flexible bearing supports where the non-linearities are due to a radial clearance or a cubic stiffness. A numerical analysis is performed to analyze the effect of uncertainties on the non-linear behavior of this rotor system by using the Stochastic-HBM. Furthermore, the results are compared with those obtained by applying a classical Monte-Carlo simulation to demonstrate the efficiency of the proposed methodology.     

Uncertainty quantification for dynamics of geometrically nonlinear structures coupled with internal acoustic fluids in presence of sloshing and capillarity

In this paper, we propose an uncertainty quantification analysis, which is the continuation of a recent work performed in a deterministic framework. The fluid–structure system under consideration is the one experimentally studied in the sixties by Abramson, Kana, and Lindholm from the Southwest Research Institute under NASA contract. This coupled system is constituted of a linear acoustic liquid contained in an elastic tank that undergoes finite dynamical displacements, inducing geometrical nonlinear effects in the structure. The liquid has a free surface on which sloshing and capillarity effects are taken into account. The problem is expressed in terms of the acoustic pressure field in the fluid, of the displacement field of the elastic structure, and of the normal elevation field of the free surface. The nonlinear reduced-order model constructed in the recent work evoked above is reused for implementing the nonparametric probabilistic approach of uncertainties. The objective of this paper is to present a sensitivity analysis of this coupled fluid–structure system with respect to uncertainties and to use a classical statistical inverse problem for carrying out the experimental identification of the hyperparameter of the stochastic model. The analysis show a significant sensitivity of the displacement of the structure, of the acoustic pressure in the liquid, and of the free-surface elevation to uncertainties in both linear and geometrically nonlinear simulations.     

随机结构系统的一般实矩阵特征值问题的概率分析

由于工程实际结构的复杂性和所用材料在统计上的离散性以及测量、加工、制造误差的存在，必然导致具有随机参数的随机结构振动系统，按结构参数的性质来划分，随机振动问题包括两方面内容：(1)确定结构问题；(2)随机结构问题。本文以现代数学理论为依托，研究了随机结构系统的一般实矩阵的特征值问题。根据Kronecker代数、向量值和矩阵值函数的灵敏度分析、一般二阶矩法和概率摄动技术给出了计算随机结构系统的一般实矩阵的特征值和特征向量的数值方法，可以有效地得出随机结构系统的一般实矩阵的特征向量的统计量，发展了2D矩阵值函数的随机结构系统的特征值问题概率分析理论。     

剪切型钢筋混凝土柱临界斜裂缝倾角的概率模型

对于剪切型钢筋混凝土(RC)柱,传统的确定性临界斜裂缝倾角模型难以有效考虑其材料参数、几何尺寸和边界约束条件等方面不确定性因素的影响,导致计算精度有限,且离散性较大。鉴于此,本文研究建立了剪切型RC柱临界斜裂缝倾角的概率模型。首先,基于变角桁架模型理论,建立了剪切型RC柱临界裂缝倾角的确定性模型;然后,考虑主观不确定性和客观不确定性因素的影响,结合贝叶斯理论和马尔可夫链蒙特卡洛(MCMC)法,建立了剪切型RC柱临界斜裂缝倾角的概率模型,并推导了均值和方差的解析表达式,从而能够合理描述临界斜裂缝倾角的概率分布特性;最后,利用试验数据对比验证了该模型的有效性,并校准了现有确定性模型的置信水平,进而确定了不同置信水平下剪切型RC柱临界斜裂缝倾角的概率特征值。     

Shells with random geometric imperfections simulation — based approach

The paper concerns the stochastic analysis of the static response of materially and geometrically non-linear models of imperfect shell structures. Geometric imperfections of shells are described using random variables or random fields. The Monte Carlo method combined with a finite element program analysis is employed. The proposed approach is used to analyse the non-linear post-buckling behaviour of a shallow cylindrical shell. The numerically obtained critical load histograms allow to estimate the structure reliability. The accuracy of simulation-based approach is discussed.     

P-CS不确定性量化模型与其性能数据驱动更新方法

在实际工程中, 广泛存在大量的不确定性信息, 直接或间接影响着工程结构形式设计、结构性能评估与预测以及在役结构损伤识别等工作的开展与决策. 这些多源不确定性信息往往需要用多种不同的不确定性量化模型加以描述; 与此同时, 不确定性变量在使用过程中可能随时间变化且难以直接测量, 需要间接根据性能测试信息在使用工程中更新不确定性量化模型. 为兼顾上述两个问题, 本文基于等概率变换原则提出了一种P-CS (probability-convex set) 不确定性量化模型, 该模型将不确定性变量用概率随机变量与非概率凸集变量组合表征, 可统一表达概率模型、非概率模型以及非精确概率模型, 实现多源、多类型不确定性的统一量化. 本文进一步基于贝叶斯理论提出了一种针对该P-CS不确定性量化模型的性能数据驱动更新方法. 该更新方法根据性能测试数据信息更新P-CS不确定性量化模型参数取值的信度分布, 从而根据后验信度分布计算得出当前P-CS不确定性量化模型参数集合. 通过数值算例详述了P-CS不确定性量化模型的构建方法与其概率、非概率特性, 并验证了性能数据驱动更新P-CS模型方法的适用性.      

Evaluating large eddy simulation results based on error analysis

The quantification of the prediction accuracy in large eddy simulations (LES) is very challenging due to various interacting errors associated with this approach. When dealing with errors in LES using implicit filtering, numerical and modeling errors have drawn the interest of many researchers. Little attention has been paid to other sources of discrepancies between LES results and reference data, namely sampling errors, influence of the initial conditions, improper boundary conditions or uncertainties issuing from reference data. A framework of metrics that includes all these issues is addressed in the present paper to study subgrid-scale (SGS) models for LES and to quantify their prediction accuracy and computational costs. The method is applied to a simple wall-bounded turbulent flow at moderate Reynolds number. It turns out from the results obtained with six commonly used SGS models that wall-adapting models (WALE and SIGMA) and localized dynamic models reproduce the physics of the flow field more faithfully, reveal a superior prediction accuracy and have a similar computational cost than models using van Driest wall damping. Especially at the viscous wall region (\(r^+<50\)), wall-adapting and localized dynamic models are more accurate, reflecting the proper near wall behavior of such models. Relying on the analysis of sources of various errors, uncertainties in LES are estimated and systematically assessed, and their influence on simulation results is quantified. Finally, engineering estimations of the required averaging time to obtain basic estimates of statistical quantities with a predetermined degree of accuracy are suggested.