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.     

Stochastic modeling and identification of an uncertain computational dynamical model with random fields properties and model uncertainties

This paper is devoted to the construction and to the identification of a probabilistic model of random fields in the presence of modeling errors, in high stochastic dimension and presented in the context of computational structural dynamics. Due to the high stochastic dimension of the random quantities which have to be identified using statistical inverse methods (challenging problem), a complete methodology is proposed and validated. The parametric–nonparametric (generalized) probabilistic approach of uncertainties is used to perform the prior stochastic models: (1) system-parameters uncertainties induced by the variabilities of the material properties are described by random fields for which their statistical reductions are still in high stochastic dimension and (2) model uncertainties induced by the modeling errors are taken into account with the nonparametric probabilistic approach in high stochastic dimension. For these two sources of uncertainties, the methodology consists in introducing prior stochastic models described with a small number of parameters which are simultaneously identified using the maximum likelihood method and experimental responses. The steps of the methodology are explained and illustrated through an application.     

Modelling of solute transport in a mild heterogeneous porous medium using stochastic finite element method: Effects of random source conditions

Randomness in the source condition other than the heterogeneity in the system parameters can also be a major source of uncertainty in the concentration field. Hence, a more general form of the problem formulation is necessary to consider randomness in both source condition and system parameters. When the source varies with time, the unsteady problem, can be solved using the unit response function. In the case of random system parameters, the response function becomes a random function and depends on the randomness in the system parameters. In the present study, the source is modelled as a random discrete process with either a fixed interval or a random interval (the Poisson process). In this study, an attempt is made to assess the relative effects of various types of source uncertainties on the probabilistic behaviour of the concentration in a porous medium while the system parameters are also modelled as random fields. Analytical expressions of mean and covariance of concentration due to random discrete source are derived in terms of mean and covariance of unit response function. The probabilistic behaviour of the random response function is obtained by using a perturbation‐based stochastic finite element method (SFEM), which performs well for mild heterogeneity. The proposed method is applied for analysing both the 1‐D as well as the 3‐D solute transport problems. The results obtained with SFEM are compared with the Monte Carlo simulation for 1‐D problems. Copyright © 2007 John Wiley & Sons, Ltd.     

Stochastic approach to estimate the arterial pressure

The aim of this paper is to illustrate the application of a stochastic approach to estimate the human common carotid arterial pressure. The analysis took into account the possible random uncertainties of the problem inputs such as geometric information and mechanical model parameters so that it is called a probabilistic parametric approach. Based on the only available information reported in literature, entropy maximum principle was used to develop probabilistic density functions for every random variable. In addition, in vivo human experimental data were considered for the determination of the so-called mean or deterministic model. Furthermore, numerical simulations of Monte Carlo were carried out involving the dispersion of all the uncertain parameters. Results showed that uncertainty of 5% led to error up to 20% in the arterial pressure estimation. Convergence was proved and a region with a confidence probability of 95% was constructed to allow the prediction of the random response of the arterial pressure. Eventually, we managed numerous calculations to analyze the influence of each random variable of the problem inputs over the arterial pressure evolution.     

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。     

A probabilistic method for field solutions in two-phase flows

Solid-particle motion and related transport phenomena in two-phase flow are fluctuating processes in space and time. A deterministic method can describe only partially the intrinsic physics of these processes. In this paper, the fluctuations of the flow parameters are modelled by considering the spatial correlations, and a probabilistic computational method for two-phase flow is presented. The probabilistic governing equations have been discretized in space using a finite volume method, and then solved by applying the Neumann expansion method. This last method is time efficient, and its convergence can be guaranteed even for large fluctuations. A liquid-solid particle mixture flow in a circular pipe is taken as an example. Computational results illustrate the merit of the probabilistic approach for the prediction of two-phase flow phenomena.     

Quantification of stochastically stable representative volumes for random heterogeneous materials

This paper details a procedure to determine lower bounds on the size of representative volume elements (RVEs) by which the size of the RVE can be quantified objectively for random heterogeneous materials. Here, attention is focused on granular materials with various distributions of inclusion size and volume fraction of inclusions. An extensive analysis of the RVE size dependence on the various parameters is performed. Both deterministic and stochastic parameters are analysed. Also, the effects of loading mode and the parameter of interest are studied. As the RVE size is a function of the material, some material properties such as Young's modulus and Poisson's ratio are analysed as factors that influence the RVE size. The lower bound of RVE size is found as a function of the stochastically distributed volume fraction of inclusions; thus the stochastic stability of the obtained results is assessed. To this end a newly defined concept of stochastic stability (DH-stability) is introduced by which stochastic effects can be included in the stability considerations. DH-stability can be seen as an extension of classical Lyapunov stability. As is shown, DH-stability provides an objective tool to establish the lower bound nature of RVEs for fluctuations in stochastic parameters.     

Three-dimensional computational fluid dynamics (volume of fluid) modelling coupled with a stochastic discrete phase model for the performance analysis of an invert trap experimentally validated using field sewer solids

Invert traps are used to trap sewer solids flowing into a sewer drainage system. The performance of the invert trap in an open rectangular channel was experimentally and numerically analysed using field sewer solids collected from a sewer drain. Experiments showed that the free water surface rises over the central opening (slot) of the invert trap, which reduces the velocity near the slot and allows more sediment to be trapped in comparison with the case for the fixed-lid model (assuming closed conduit flow with a shear-free top wall) used by earlier investigators. This phenomenon cannot be modelled using a closed conduit model as no extra space is provided for the fluctuation of the water surface, whereas this space is provided in the volume of fluid (VOF) model in the form of air space in ANSYS Fluent 14.0 software. Additionally, the zero atmospheric pressure at the free water surface cannot be modelled in a fixed-lid model. In the present study, experimental trap efficiencies of the invert trap using field sewer solids were fairly validated using a three-dimensional computational fluid dynamics model (VOF model) coupled with a stochastic discrete phase model. The flow field (i.e., velocities) predicted by the VOF model were compared with experimental velocities obtained employing particle image velocimetry. The water surface profile above the invert trap predicted by the VOF model was found to be in good agreement with the experimentally measured profile. The present study thus showed that the VOF model can be used with the stochastic discrete phase model to well predict the performance of invert traps.     

Uncertainties of unidirectional composite strength under tensile loading and variation of environmental condition

A probabilistic strength model is developed for unidirectional composites with fibers in hexagonal arrays. The model assumes that, a central core of broken fibers surrounded by unbroken fibers which are subjected to unidirectional tensile loading. The proposed approach consists in using a modified shear lag model to calculate the ineffective lengths and stress concentrations around fiber breaks. The main feature in the model lies in incorporating the variation of composite properties due to temperature and moisture, in order to predict degradation of fibers and matrix characteristics. The strength degradation is often seen as a result of changes in ineffective lengths at fiber breaks, leading to stress concentrations in intact neighboring fibers. As fiber breaks are intrinsically random, the variability of input data allows us to describe the probabilistic model by using the Monte-Carlo method. The sensitivities of the mechanical response are evaluated regarding the uncertainties in design variables such as Young’s modulus of fibers and matrix, fiber reference strength, shear yield stress, fiber volume fraction and shear parameter defining the shear stress in the inelastic region.     

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.     

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.      

Water entrainment due to spillway surface jets

Strong flow entrainment has been observed downstream of spillways constructed with flow deflectors. This water entrainment has important environmental and ecological impacts because it improves the mixing of powerhouse and spillway flows, but may negatively impact fish migration or create adverse flow conditions.

Most studies found in the literature attempt to explain this entrainment with turbulent mixing. Both reduced-scale hydraulic models and single-phase, isotropic RANS models grossly under-predict the degree of entrainment observed in prototypes. In this paper, an anisotropic model that accounts for the bubble volume fraction and attenuation of the normal velocity fluctuations at the free surface is presented. The model adequately predicts the main mechanisms causing water entrainment and compares well against experimental data for a round surface jet and for Brownlee Dam at model scale. It is shown that appropriate entrainment can only be captured if the turbulence anisotropy and the two-phase nature of the flow are modelled.     

Rotary motion of the parametric and planar pendulum under stochastic wave excitation

In this paper a rotary motion of a pendulum subjected to a parametric and planar excitation of its pivot mimicking random nature of sea waves has been studied. The vertical motion of the sea surface has been modelled and simulated as a stochastic process, based on the Shinozuka approach and using the spectral representation of the sea state proposed by Pierson–Moskowitz model. It has been investigated how the number of wave frequency components used in the simulation can be reduced without the loss of accuracy and how the model relates to the real data. The generated stochastic wave has been used as an excitation to the pendulum system in numerical and experimental studies. For the first time, the rotary response of a pendulum under stochastic wave excitation has been studied. The rotational number has been used for statistical analysis of the results in the numerical and experimental studies. It has been demonstrated how the forcing arrangement affects the probability of rotation of the parametric pendulum.     

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.     

Computation of particle dispersion in turbulent liquid flows using an efficient Lagrangian trajectory model

The dispersion of solid particles in a turbulent liquid flow impinging on a centrebody through an axisymmetric sudden expansion was investigated numerically using a Eulerian–Lagrangian model. Detailed experimental measurements at the inlet were used to specify the inlet conditions for two-phase flow computations. The anisotropy of liquid turbulence was accounted for using a second-moment Reynold stress transport model. A recently developed stochastic–probabilistic model was used to enhance the computational efficiency of Lagrangian trajectory computations. Numerical results of the stochastic–probabilistic model using 650 particle trajectories were compared with those of the conventional stochastic discrete-delta-function model using 18 000 particle trajectories. In addition, results of the two models were compared with experimental measurements. © 1998 John Wiley & Sons, Ltd.     

Random-field model for the elasticity tensor of anisotropic random media

This Note deals with the construction of a non-Gaussian positive definite matrix-valued random field whose mathematical properties allow the fourth-order elasticity tensor of random non homogeneous anisotropic three dimensional elastic media to be modelled. If the usual parametric probabilistic approach was used, then 21 mutually dependent random fields should be modelled and identified by using experimental data. Such an approach would be very difficult because the systems of the marginal probability distributions of these random fields have to be identified due to the fact that, for a boundary value problem, the displacement field of the random medium is a non-linear mapping of the random elasticity tensor. The theory presented in this paper allows such a probabilistic model of the fourth-order elasticity tensor field to be constructed and depends only of four scalar parameters: three spatial correlation lengths and one parameter allowing the level of the random fluctuations to be controlled. To cite this article: C. Soize, C. R. Mecanique 332 (2004).

Résumé

On présente la construction d'un champ aléatoire à valeurs dans les matrices définies positives dont les propriétés mathématiques permettent de modéliser le tenseur d'élasticité du quatrième ordre des mileux élastiques anisotropes tridimensionnels aléatoires. Si l'approche probabiliste paramétrique usuelle était utilisée, alors il serait nécessaire de modéliser et d'identifier à l'aide de données expérimentales 21 champs aléatoires mutuellement dépendants. Une telle approche serait très difficile de part le fait que le système de lois marginales de ces champs aléatoires doit être identifié parce que, pour un problème aux limites, le champ de déplacement est une transformation non linéaire du tenseur d'élasticité. La théorie présentée dans ce papier permet de construire une modélisation probabiliste du champ de tenseur d'élasticité qui ne dépend que de quatre paramètres scalaires : trois échelles de corrélation spatiale et un paramètre permettant de contrôler le niveau des fluctuations aléatoires. Pour citer cet article : C. Soize, C. R. Mecanique 332 (2004).     

Non-linear dynamics of a drill-string with uncertain model of the bit–rock interaction

The drill-string dynamics is difficult to predict due to the non-linearities and uncertainties involved in the problem. In this paper a stochastic computational model is proposed to model uncertainties in the bit–rock interaction model. To do so, a new strategy that uses the non-parametric probabilistic approach is developed to take into account model uncertainties in the bit–rock non-linear interaction model. The mean model considers the main forces applied to the column such as the bit–rock interaction, the fluid–structure interaction and the impact forces. The non-linear Timoshenko beam theory is used and the non-linear dynamical equations are discretized by means of the finite element method.     

Micromechanical characterizing elastic,thermoelastic and viscoelastic properties of functionally graded carbon nanotube reinforced polymer nanocomposites

In this work, elastic, thermoelastic and viscoelastic properties of functionally graded carbon nanotube reinforced polymer nanocomposites are investigated using a 3-dimensional micromechanics-based approach. The main advantage of the proposed micromechanical model is its ability to give closed-form formulation for predicting the effective properties of nanocomposites. In the micromechanical modeling, the interphase formed due to non-boned van der Waals interaction between the continuous CNT and polymer matrix is considered through employing an individual representative volume element. The validity of the model is examined by comparing its results with other theoretical approaches and experimental data available in the literature. The effects of various types of CNTs arrangement in the matrix, i.e. uniform distribution and different functionally graded distributions on the elastic, thermoelastic and viscoelastic properties of polymer nanocomposites are investigated in detail. Furthermore, random arrangement of CNTs in the matrix is modelled. The influences of CNT/polymer matrix interphase and CNT volume fraction on the effective properties of nanocomposites are also studied. Finally, the viscoelastic response of nanocomposites under multiaxial loading is extracted and interpreted.     

Hysteretic deteriorating model for quasi-brittle materials based on micromechanical damage approach

A damage model, which is based on the stochastic modeling of the microstructures, is developed for the quasi-brittle materials subjected to repeated loading. According to this model, the overall response of the material is represented with a series of micro-elements joined in parallel. A combined model is proposed for the micro-element considering the fracture as well as the hysteretic energy dissipation. To account for the progressive failure, the random fracture strains are assigned to the micro-elements. Therefore the overall parallel bundle is considered as a stationary random field. Then by averaging the microscopic random field, the overall loading, unloading and reloading curves are derived analytically. Two hysteretic rules are derived from the proposed model, and the overall hysteretic deteriorating behaviors could be well reproduced. To demonstrate the validity of the present model, the numerical results are shown against the stochastic simulated curves as well as the experimental data. The present model provides an alternative approach for the efficient modeling of the hysteretic deteriorating behaviors for the quasi-brittle materials.