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.     

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.     

对区间不确定性问题的可靠性度量的探讨

实际工程中大量存在不确定性因素,处理不确定性因素的可靠性逐渐成为科学和工程中一个非常重要的概念。区间不确定性是继随机性和模糊性之后被人们研究的又一种不确定性。区间不确定性一般可由区间变量或凸集合模型来描述。近年来,有些文献针对区间不确定性提出了计算非概率可靠性的方法。本文对这些方法进行比较和讨论,并和假定各区间不确定参量在允许取值区间内为具有熵最大的矩形分布,采用概率可靠度的理论来处理问题得到的结果进行了比较。     

精密及缺陷信息条件下的结构可靠性设计

概率方法一直以来被认为是处理不确定现象的最有效的方法.传统的概率可靠性模型建立在随机变量的分布函数基础上,需要较完整的数据信息.在结构设计问题中,有关载荷、抗力等影响因素的统计数据一般是稀少的或缺乏的,信息往往以非完整的形式出现.此外,结构的安全设计还需考虑结构和环境的相互作用,结构设计、分析、评价以及制作过程中可能存在的误差等,有些不确定因素不能归于随机性.因此,需要发展新的结构可靠性设计模型和方法,以克服传统的概率可靠性方法的局限和不足.本文对精密及缺陷信息条件下的可靠性设计模型和方法进行综述,内容包括随机可靠性设计模型与方法、非完整信息下的可靠性分析和设计、模型误差和主观不确定的影响、复杂系统优化求解策略、以及今后的研究展望.      

A parametric study on nonlinear flow-induced dynamics of a fluid-conveying cantilevered pipe in post-flutter region from macro to micro scale

A mathematical formulation is proposed to investigate the nonlinear flow-induced dynamic characteristics of a cantilevered pipe conveying fluid from macro to micro scale. The model is developed by using the extended Hamilton's principle in conjunction with the inextensibility condition and laminar and turbulent flow profiles as well as modified couple stress theory. The current model is capable of recovering the classical model of cantilevered pipe conveying fluid by neglecting the couple stress effect. The governing equation of motion is presented in dimensionless form in a convenient and usable manner. To solve the problem at hand, the integro-partial-differential equation of motion is discretized into a set of ordinary differential equations via Galerkin method. Afterward, a Runge–Kutta's finite difference scheme is employed to evaluate the nonlinear dynamic response of the cantilevered pipe conveying fluid. A parametric study is carried out to examine the influences of mass parameter and dimensionless mean flow velocity on the nonlinear dynamic characteristics of the cantilevered pipe conveying fluid in post-flutter region. The role of size-dependency in the nonlinear behavior of pipe is explored by converting the new set of dimensionless parameters into the conventional one. Eventually, some convergence studies are performed to indicate the reliability of present results.     

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.     

Rigorous derivation of the effective model describing a non-isothermal fluid flow in a vertical pipe filled with porous medium

This paper reports an analytical investigation of non-isothermal fluid flow in a thin (or long) vertical pipe filled with porous medium via asymptotic analysis. We assume that the fluid inside the pipe is cooled (or heated) by the surrounding medium and that the flow is governed by the prescribed pressure drop between pipe’s ends. Starting from the dimensionless Darcy–Brinkman–Boussinesq system, we formally derive a macroscopic model describing the effective flow at small Brinkman–Darcy number. The asymptotic approximation is given by the explicit formulae for the velocity, pressure and temperature clearly acknowledging the effects of the cooling (heating) and porous structure. The theoretical error analysis is carried out to indicate the order of accuracy and to provide a rigorous justification of the effective model.     

Reliability analysis for the uncertainties in vehicle and high-speed railway bridge system based on an improved response surface method for nonlinear limit states

The paper deals with the reliability analysis for the high-speed railway bridge systems. Although the bridge–vehicle interactive system has much more uncertainties in the resistance and loads of trains moving at very high speed compared with static structural analysis, little concern has been engaged to identify which random variable has to be considered in the probabilistic analysis, or what criteria should be selected to determine the probabilistic safety or serviceability. The considered design parameters thus involve uncertainties in stiffness, moment of inertia, damping ratio of primary suspension in terms of load, geometry of girders and slabs, and the mechanical properties of girders in terms of resistance. The considered limit states embrace the safety of trains and comfort of passengers, and the acceptability criteria are based on UIC code. For evaluating the reliability of the time-dependent nonlinear behavior of complex structures, an improved Response Surface Method (RSM) is developed. An adaptive technique and a weight matrix are utilized as an optimizing technique that accelerates the convergence in the reliability analysis. The results of improved RSM, compared with the basic and adaptive RSM, are verified with the improved convergence to the exact solution. The bridge response is analyzed using a new three-dimensional finite element model of high-speed train–bridge interactions. The track structures are idealized using beam elements with the offset of beam nodes and beams on a two-parameter elastic foundation. The vehicle model developed for a 300 km/h train is employed. The calculated reliabilities for performance of the considered bridges and the passenger comfort on board of high-speed trains are compared to the conventional safety indices. The results of this study allow identifying the quantification of uncertainties that can control quality of the high-speed train service.     

Random uncertainty modeling and vibration analysis of a straight pipe conveying fluid

A new procedure on random uncertainty modeling is presented for vibration analysis of a straight pipe conveying fluid when the pipe is fixed at both ends. Taking real conveying condition into account, several randomly uncertain loads and a motion constraint are imposed on the pipe and its corresponding equations of motion, which are established from the Euler–Bernoulli beam theory and the nonlinear Lagrange strain theory previously. Based on the stochastically nonlinear dynamic theory and the Galerkin method, the equations of motion are reduced to the finite discretized ones with randomly uncertain excitations, from which the vibration characteristics of the pipe are investigated in more detail by some previously developed numerical methods and a specific Poincaré map. It is shown that, the vibration modes change not only with the frequency of the harmonic excitation but also with the strength and spectrum width of the randomly uncertain excitations, quasi-periodic-dominant responses can be observed clearly from the point sets in the Poincaré’s cross-section. Moreover, the nonlinear elastic coefficient and location of the motion constraint can be adjusted properly to reduce the transverse vibration amplitude of the pipe.     

Bending-torsional stability analysis of aerodynamically covered pipes with inclined terminal nozzle and concurrent internal and external flows

Stability analysis of a cantilevered pipe with an inclined terminal nozzle as well as simultaneous internal and external fluid flows is investigated in this study. The pipe is embedded in an aerodynamic cover with negligible mass and stiffness simply to streamline the external flow and avoid vortex induced vibrations. The structure of pipe is modeled as an Euler–Bernoulli beam and effects of internal fluid flow including flow-induced inertia, Coriolis and centrifugal forces and the follower force induced by the exhausting jet are taken into account. In addition, neglecting the compressibility effect and using the unsteady Wagner model, aerodynamic loading is determined as a distributed lateral load for any generic structural state. The integral form of coupled equations of motion are obtained using the Hamilton’s principle. Solution to the coupled flexural–torsional equations of motion is realized via the extended Galerkin method. After discretization of the equations of motion, an eigenvalue representation of the problem is obtained. Several parameter studies are then conducted to examine the effects of concurrent fluid flows and other related parameters on the stability margins of the system.     

结构的非概率可靠性方法和概率可靠性方法的比较

对文[8]中提出的非概率可靠性方法和广泛使用的传统的概率可靠性方法，在建模思想、模型结构和基于可靠性的结构优化设计等方面进行了比较研究。进一步阐释了有关概念。得到了一些有益的结论。说明了非概率可靠性方法的有效性和实用性。由于非概率可靠性模型对已知数据的要求较低，计算过程较为简便，从而可使结构可靠性分析和设计中获取数据的难度大大降低。并有效降低计算工作量。在所掌握的原始数据较少的情况下，非概率可靠性方法为结构的可靠性计算提供了一种较好的选择。     

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.      

三参量固体模型粘弹性输流管道的动力特性分析

推导了三参量固体模型粘弹性输流管道的振动微分方程 ,计算了在不同无量纲松弛系数和弹性常数比下管道的无量纲临界流速和无量纲自振复频率 ,并给出了前三阶复频率与流速的关系 .计算结果表明 ,质量比、无量纲松弛系数及无量纲弹性常数比对输流管道的动力特性均有影响 .     

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.     

Modeling of Solute Transport in a 3D Rough-Walled Fracture–Matrix System

Fluid flow and solute transport in a 3D rough-walled fracture–matrix system were simulated by directly solving the Navier–Stokes equations for fracture flow and solving the transport equation for the whole domain of fracture and matrix with considering matrix diffusion. The rough-walled fracture–matrix model was built from laser-scanned surface tomography of a real rock sample, by considering realistic features of surfaces roughness and asperity contacts. The numerical modeling results were compared with both analytical solutions based on simplified fracture surface geometry and numerical results by particle tracking based on the Reynolds equation. The aim is to investigate impacts of surface roughness on solute transport in natural fracture–matrix systems and to quantify the uncertainties in application of simplified models. The results show that fracture surface roughness significantly increases heterogeneity of velocity field in the rough-walled fractures, which consequently cause complex transport behavior, especially the dispersive distributions of solute concentration in the fracture and complex concentration profiles in the matrix. Such complex transport behaviors caused by surface roughness are important sources of uncertainty that needs to be considered for modeling of solute transport processes in fractured rocks. The presented direct numerical simulations of fluid flow and solute transport serve as efficient numerical experiments that provide reliable results for the analysis of effective transmissivity as well as effective dispersion coefficient in rough-walled fracture–matrix systems. Such analysis is helpful in model verifications, uncertainty quantifications and design of laboratorial experiments.     

Uncertainty quantification and global sensitivity analysis of mechanistic one-dimensional models and flow pattern transition boundaries predictions for two-phase pipe flows

The prediction of uncertainties is a growing interest in flow assurance industrial applications, but only few works have been presented on this topic. In this work, an uncertainty quantification and a global sensitivity analysis are performed to quantify the level of confidence in predictions of one-dimensional mechanistic models considering different two-phase flow regimes. A method is proposed for this purpose accounting for the effect of several variables on pressure drop and hold-up predictions by the well-known one-dimensional two-fluid model, such as fluid flow rates, geometry (the inclination angle and the pipe diameter), and fluid properties (density and viscosity); the case of a non-Newtonian shear-thinning fluid behaviour is also considered. Flow pattern transition boundaries, including the stability of the stratified flow regime, are included in this analysis. Monte Carlo simulations were used for the uncertainty quantification while different approaches for the sensitivity analysis (scatter plot, linear regression, the Morris’s method, and the Sobol’s Method) were used and compared to identify the best tool for this family of models. The Sobol’s method appears to be the most convenient approach and a discussion is provided considering different practical cases for gas/liquid and liquid/liquid systems. The most critical input parameters in terms of uncertainty are rigorously identified case by case. A way to reduce the output uncertainty is indicated by the interpretation of the results of the global sensitivity analysis. The conclusions of this analysis gives new insights regarding the degree of uncertainties in predictions of one-dimensional mechanistic models.     

Stability analysis of viscoelastic curved pipes conveying fluid

Based on the Hamilton' s principle for elastic systems of changing mass, a differential equation of motion for viscoelastic curved pipes conveying fluid was derived using variational method, and the complex characteristic equation for the viscoelastic circular pipe conveying fluid was obtained by normalized power series method. The effects of dimensionless delay time on the variation relationship between dimensionless complex frequency of the clamped-clamped viscoelastic circular pipe conveying fluid with the Kelvin-Voigt model and dimensionless flow velocity were analyzed. For greater dimensionless delay time, the behavior of the viscoelastic pipe is that the first, second and third mode does not couple, while the pipe behaves divergent instability in the first and second order mode, then single-mode flutter takes place in the first order mode.     

Uncertainty quantification of a graphite nitridation experiment using a Bayesian approach

In this paper, a stochastic system based Bayesian approach is applied to estimate different model parameters and hence quantify the uncertainty of a graphite nitridation experiment. The Bayesian approach is robust due to its ability to characterize modeling uncertainties associated with the underlying system and is rigorous due to its exclusive foundation on the axioms of probability theory. We choose an experiment by Zhang et al. [1] whose main objective is to measure the reaction efficiency for the active nitridation of graphite by atomic nitrogen. To obtain the primary physical quantity of interest, we need to model and estimate the uncertainty of a number of other physical processes associated with the experimental setup. We use the Bayesian method to obtain posterior probability distributions of all the parameters relevant to the experiment while taking into account uncertainties in the inputs and the modeling errors. We use a recently developed stochastic simulation algorithm which allows for efficient sampling in the high-dimensional parameter space. We show that the predicted reaction efficiency of the graphite nitridation and its uncertainty is ∼3.1 ± 1.0 × 10⁻³ that is slightly larger than the ones deterministically obtained by Zhang et al. [1].     

Deformations of an elastic pipe submitted to gravity and internal fluid flow

This paper describes the deformation of an elastic pipe submitted to gravity and to an internal fluid flow. The pipe is clamped horizontally at one end and free at the other end. As the fluid velocity increases, the shape changes from an elastic beam deflected by its own weight towards an horizontal position. The shape of the pipe is characterized experimentally and is compared with a theoretical model based on the Euler–Bernoulli approximation and the conservation of the fluid momentum. We study how the determination of the pipe deformation provides an estimation of the conveyed fluid flow. Finally, the vertical force produced by the conveyed fluid to lift off a mass is deduced.     

基于高斯过程分类的结构贝叶斯可靠性分析

贝叶斯可靠性方法是处理不完备信息条件下结构可靠性问题的有效途径之一。在实际应用中,由于可靠性分析的计算量较大,常须采用各种近似替代模型以提高计算效率。传统的替代模型方法是对结构的功能函数予以近似建模。这种方法不易定量考虑模型误差对可靠性分析的影响,且难以应用于诸如功能函数不连续和失效域不连通等情况。为此,本文提出一种基于高斯过程分类的替代模型,直接辨识结构的极限状态曲面,并将其应用于结构贝叶斯可靠性分析之中。分析了替代模型不确定性对可靠性预测结果的影响,给出了失效概率分布参数的方差算式,进而提出了改善模型精度的补充采样准则。通过算例验证了方法的适用性和有被性．