An Anisotropic Sparse Grid Numerical Integration-Based Statistic Moment Estimation Method

Statistic moment estimation is a paramount task in design optimization under uncertainty. In the existing isotropic sparse grid-based statistic moment estimation method, each dimension is considered to be of equal importance and collocated with the same number of integration points. Therefore, “curse of dimensionality” is still very serious. To address this issue, a new statistic moment estimation approach is developed by employing the anisotropic sparse grid technique from the numerical integration point of view in this paper. A new rule allowing different accuracy levels to each dimension according to their varying importance is proposed to determine the eligible multi-index combinations. Integration points can be collocated more reasonably, and more points can be removed from unimportant dimensions to the important ones. The effectiveness of the proposed approach is demonstrated through comparative studies on several mathematical examples and a practical engineering example. It is observed that compared with the existing method, the proposed approach can further mitigate “curse of dimensionality” and significantly improve the accuracy of statistic moment estimation with even less computational cost.     

Algebraic PGD for tensor separation and compression: An algorithmic approach

Proper Generalized Decomposition (PGD) is devised as a computational method to solve high-dimensional boundary value problems (where many dimensions are associated with the space of parameters defining the problem). The PGD philosophy consists in providing a separated representation of the multidimensional solution using a greedy approach combined with an alternated directions scheme to obtain the successive rank-one terms. This paper presents an algorithmic approach to high-dimensional tensor separation based on solving the Least Squares approximation in a separable format of multidimensional tensor using PGD. This strategy is usually embedded in a standard PGD code in order to compress the solution (reduce the number of terms and optimize the available storage capacity), but it stands also as an alternative and highly competitive method for tensor separation.     

A reduced-order model manifold technique for automated structural defects judging using the PGD with analytical validation

Automation is conquering new fields on a daily basis. Aiming for faster and more reliable products, industrials as well as researchers are oriented into automation. Non-destructive testing as well as defect quantification is not an exception. In fact, decisions with minimum allowable error are sought in real-time when facing any potential defect. In this work, we suggest a comprehensive method based on model order reduction techniques to judge if a structure shall be salvaged. The real-time decision is based on multidimensional parametric simulation, performed offline, using the Proper Generalized Decomposition (PGD). The PGD is a model order reduction technique that allows circumventing the curse of dimensionality by using domain decomposition. Therefore, the 6D simulation illustrated in this paper is performed within a few minutes on a standard laptop. Later on, a stress concentration manifold is built and used online for decision-making. The manifold is validated on a few selected solutions solved analytically using an analytical procedure. The aforementioned procedure is developed, in this paper, to calculate the tangential stress around circular holes of different sizes, in an infinite isotropic plate containing any number of holes and subjected to in-plane pressure loading at the tip of the infinite plate. The procedure is based on determining two Muskhelishvili complex potentials in terms of complex Fourier series, and applying the Schwartz alternating method repeatedly until the boundary conditions on the contour of every hole are satisfied.     

Solving parametric complex fluids models in rheometric flows

Inverse identification of complex fluid behaviors is a tricky task because sometimes there are several rheological parameters and the identification procedure itself is quite expensive from the computational time viewpoint. Standard inverse identification procedures solve the model for a choice of the model parameters, and then parameters are updated trying to minimize the gap between the model predictions and some available experimental measures. Thus, the model has to be evaluated for each trial set of the model parameters. When models involve a great number of degrees of freedom the identification procedure becomes a computationally expensive task. In this paper we propose a new procedure able to solve once the model for any value of the model parameters. For this purpose, all the model parameters are considered as extra-coordinates of the model. Thus, the model results finally defined in a multidimensional space including the physical space x, the time t and a number of extra-coordinates related to the model parameters. The solution of such model needs for circumventing the curse of dimensionality illness that suffer multidimensional models.     

Deterministic solution of the kinetic theory model of colloidal suspensions of structureless particles

A direct modeling of colloidal suspensions consists of calculating trajectories of all suspended objects. Due to the large time computing and the large cost involved in such calculations, we consider in this paper another route. Colloidal suspensions are described on a mesoscopic level by a distribution function whose time evolution is governed by a Fokker–Planck-like equation. The difficulty encountered on this route is the high dimensionality of the space in which the distribution function is defined. A novel strategy is used to solve numerically the Fokker–Planck equation circumventing the curse of dimensionality issue. Rheological and morphological predictions of the model that includes both direct and hydrodynamic interactions are presented in different flows.     

A reduced simulation applied to the viscoelastic fatigue of polymers

The paper extends the use of the PGD method to viscoelastic evolution problems described by a large number of internal variables and with a large spectrum of relaxation times. The internal variables evolution is described by a set of linear differential equations that involve many time scales. The feasibility and the robustness of the method are discussed in the case of a polymer in a non-equilibrium state under creep and cyclic loading. The relationships between different time scales (loading and internal variables) are also discussed.     

Global sensitivity analysis based on high-dimensional sparse surrogate construction

Surrogate models are usually used to perform global sensitivity analysis(GSA) by avoiding a large ensemble of deterministic simulations of the Monte Carlo method to provide a reliable estimate of GSA indices. However, most surrogate models such as polynomial chaos(PC) expansions suffer from the curse of dimensionality due to the high-dimensional input space. Thus, sparse surrogate models have been proposed to alleviate the curse of dimensionality. In this paper, three techniques of sparse reconstruction are used to construct sparse PC expansions that are easily applicable to computing variance-based sensitivity indices(Sobol indices). These are orthogonal matching pursuit(OMP), spectral projected gradient for L_1 minimization(SPGL1), and Bayesian compressive sensing with Laplace priors. By computing Sobol indices for several benchmark response models including the Sobol function, the Morris function, and the Sod shock tube problem, effective implementations of high-dimensional sparse surrogate construction are exhibited for GSA.     

On the relation between the global modes and the spectra of drag and lift in periodic wake flowsSur la relation entre les modes globaux et les spectres des forces hydrodynamiques dans un sillage périodique

In this Note we are interested in the relation between the symmetry properties of the global mode envelopes in wake flows and the spectra of the drag and lift forces. We consider the “impulse” formula for the hydrodynamic force and show that the drag force consists of contributions from the even harmonics, and the lift force of contributions from the odd harmonics, only. Our argument explains this well-known empirical fact and is also supported by the computational evidence we provide. Finally, we identify the unsteady wake flows, both controlled and uncontrolled, as belonging to a broader family of “streaming flows”. To cite this article: B. Protas, J.E. Wesfreid, C. R. Mecanique 331 (2003).     

Yang and Yin parameters in the Lorenz system

The history of the Lorenz system is firstly discussed in this paper. In Chinese philosophy, Yin is the negative, historical, or feminine principle in nature, while Yang is the positive, contemporary, or masculine principle in nature. Yin and Yan are two fundamental opposites in Chinese philosophy (therefore, in this paper, these words “Yin parameter,” “Yang parameter,” “historical system,” and “contemporary system” are used to represent the “positive parameter,” “negative parameter,” “time reversed (?t) system,” and “time forward (t) system,” respectively). Simulation results show that chaos of historical Lorenz system can be generated when using “Yin” parameters. To our best knowledge, most characters of contemporary Lorenz system are studied in detail, but there are no articles in making a thorough inquiry about the history of Lorenz system. As a result, the chaos of historical Lorenz system with “Yin parameters” is introduced in this paper and various kinds of phenomena in the historical Lorenz system are investigated by Lyapunov exponents, phase portraits, and bifurcation diagrams.     

Turbulence from 1870 to 1920: The birth of a noun and of a concept

We consider here the works of French, British, and German researchers in fluid mechanics from 1870 to the beginning of the twentieth century. Our aim is to understand how the term “turbulence” introduced by William Thomson in 1887, which was not used by the main researchers of the time, including Joseph Boussinesq, Osborne Reynolds, Lord Rayleigh, Horace Lamb in the first editions of his book, became classical in the 1920s. We trace the first introductions of the terms “turbulence”, “turbulent flow” in the works of relatively unknown researchers between 1889 and 1903, until it reaches the vocabulary of mainstream researchers in fluid mechanics and physics. Our result is that the shift was in 1906–1908, when the term was used in the 1906 edition of the book of Horace Lamb, and in Lanchester's book, followed by a series of papers of German researchers before the First World War.The use of the word “turbulence”, a word used for a long time for crowds or for children, in a scientific context, corresponds to the introduction of a new concept, a new understanding of a scientific phenomenon clearly identified as being different from laminar motion. The study of the use of this term is also the study of the diffusion of a new concept among researchers of the time.     

不确定车轨耦合系统辛随机振动分析

建立了轨道不平顺作用下具有不确定参数车轨耦合系统随机振动评估方法. 车辆系统采用物理坐标下多刚体系统模型,并应用高斯随机变量模拟车体、转向架和轮对一系、二系连接系统中动力学参数具有的不确定性. 采用无穷周期结构进行弹性轨道模拟,在哈密顿状态空间下建立了典型轨道子结构的状态运动方程,通过轮轨耦合关系建立了混合 物理坐标及辛模态坐标车轨耦合系统运动方程. 应用Hermite正交多项式展开得到了耦合系统动力响应相对于不确定性参数的控制方程. 由于利用轨道周期特性建模,所获得的控制方程有效地降低了方程维度. 轮轨接触处轨道不平顺载荷模拟为完全相干多分量平稳随机过程,推广和发展虚拟激励法建立了耦合系统随机振动受不确定动力学 参数影响的量化评估方法. 通过Monte Carlo数值模拟,验证了该方法在不确定参数变异很大时也能够保持较好的精度,具有一定的工程实用性.     

General framework for the identification of constitutive parameters from full-field measurements in linear elasticity

In this paper, several approaches available in the literature for identifying the constitutive parameters of linear elastic materials from full-field measurements are presented and their sensitivity to a white noise added to the data is compared. The first investigated approach is the virtual fields method (VFM). It is shown that the uncertainty of the parameters identified with the VFM when a white noise is added to the data depends on the choice of a relevant set of virtual fields. Optimal virtual fields exist, thus minimizing the uncertainty and providing the “maximum likelihood solution”. The other approaches investigated in this paper are based on finite element model updating (FEMU). It is proved that FEMU approaches actually yield equations similar to the ones derived from the VFM, but with nonoptimal sets of virtual fields. Therefore, the FEMU approaches do not provide the “maximum likelihood solution”. However, the uncertainty of FEMU approaches varies dramatically with the cost function to minimize. On one hand, the FEMU approach based on the “displacement gap” minimization yields equations which are very close to the ones of the VFM approach and therefore, its uncertainty is almost the same as the VFM one. On the other hand, it is shown that other approaches based on the “constitutive equation gap” minimization or the “equilibrium gap” minimization provide biased solutions. For all the approaches, very fast algorithms, converging in only two iterations, have been devised. They are finally applied to real experimental data obtained on an orthotropic composite material. Results confirm the success of two methods: the VFM approach which provides the “maximum likelihood solution” and the FEMU approach based on the “displacement gap” minimization.     

Yield criteria for porous media in plane strain: second-order estimates versus numerical resultsCritère de plasticité des matériaux poreux en déformation plane : Estimations du second ordre et résultats numériques

This Note presents a comparison of some recently developed “second-order” homogenization estimates for two-dimensional, ideally plastic porous media subjected to plane strain conditions with corresponding yield analysis results using a new linearization technique and systematically optimized finite elements meshes. Good qualitative agreement is found between the second-order theory and the yield analysis results for the shape of the yield surfaces, which exhibit a corner on the hydrostatic axis, as well as for the dependence of the effective flow stress in shear on the porosity, which is found to be non-analytic in the dilute limit. Both of these features are inconsistent with the predictions of the standard Gurson model. To cite this article: J. Pastor, P. Ponte Castañeda, C. R. Mecanique 330 (2002) 741–747.     

A new type of non-linear approximation with application to the duffing equation

A new type of trial solution which differs from the usual linear combination of approximating functions is considered. It involves modifying the approximating functions with “form functions;” functions containing undetermined parameters appearing non-linearly, the proper choice of which provide a closer approximation to the large local curvatures which appear in some non-linear problems. In this paper the “form function” approximation is demonstrated for steady-state solutions of the Duffing equation. This equation arises in the problem of non-linear vibration of buckled beams and plates. It is shown that the stability behavior of these steady-state solutions is governed by a Hill equation. It is found that the “form function” approximation gives noticeably better numerical results than, for example, those given by the harmonic balance method. The method also provides additional insight into the non-linear behavior, particularly in the low frequency response region.     

A hybrid phenomenological model for ferroelectroelastic ceramics. Part I: Single phased materials

In this part I of a two part series, a rate-independent hybrid phenomenological constitutive model applicable for single phased polycrystalline ferroelectroelastic ceramics is presented. The term “hybrid” refers to the fact that features from macroscopic phenomenological models and micro-electromechanical phenomenological models are combined. In particular, functional forms for a switching function and the Helmholtz free energy are assumed as in many macroscopic phenomenological models; and the volume fractions of domain variants are used to describe the internal material state, which is a key feature of micro-electromechanical phenomenological models. The approach described in this paper is an attempt to combine the advantages of macroscopic and micro-electromechanical material models. Its potential is demonstrated by comparison with experimental data for barium titanate. Finally, it is shown that the model for single phased materials cannot reproduce the material behavior of morphotropic PZT ceramics based on a realistic choice for the material parameters. This serves as a motivation for part II of the series, which deals with the modeling of morphotropic PZT ceramics taking into account the micro-structural specifics of these materials.     

Sur la réponse transitoire d'une couche solidaire d'un milieu semi-infini ; application à un impact sur le thoraxOn the transient response of a layer welded to a half-space; an insvestigation of impact on the thorax

The theory of generalized rays is dedicated to the investigation of the transient response of elastic layered media. It yields exact solutions, whatever the duration of the applied impulse (i.e., for any frequency content of the excitation). This Note is part of a research project on the modeling of non-penetrating impact on thorax. We investigate a simplified model of the thoracic wall – a layer welded to a semi-infinite medium – submitted to an impulse of variable duration. Among other things, it is found that the response computed for a “long” impulse is consistent with a result of the classical plate theory. To cite this article: Q. Grimal et al., C. R. Mecanique 331 (2003).     

Remarks on the notion of “pressure”

In addition to understanding the various meanings attached to the word “pressure” one also has to comprehend the meanings of the phrases in which the term “pressure” appears. For instance one comes across the following combinations: “static-fluid pressure”, “thermodynamic pressure”, “mechanical pressure”, “contact pressure”, “stagnation pressure”, “vapor pressure”, “electro-osmotic pressure”, etc., One also often comes across the comment that “pressure is the Lagrange multiplier that enforces the constraint of incompressibility” and that “pressure is the mean normal stress”. In general the word “pressure” with different significations, is used with gay abandon without paying proper attention to its usage¹. The distinction in the meanings of the above terms assumes paramount significance when discussing properties of materials, which could possibly depend on “pressure”. In this short note we discuss the distinction between various significations of the word “pressure”, and their implications with regard to response relations for bodies.     

Reconstruction of viscoelastic tissue properties from MR elastography-type measurements

This study is concerned with an optimization-based approach to the identification of “background” viscoelastic properties of soft tissues from magnetic resonance (MR) elastography-type measurements. In this approach, the triaxial tissue displacements, captured by the MR scanner over a suitable subdomain that is free of major heterogeneities, are split into (i) a boundary subset that is used to formulate the forward (Dirichlet) problem, and (ii) an internal subset, employed as “the data” for the inverse (material characterization) problem. For an elevated performance of the minimization scheme, material sensitivities of the featured cost functional are computed semi-analytically via a boundary-integral formulation, resulting in alternative “direct” and adjoint-field sensitivity formulas. The numerical results, obtained assuming input parameters that are relevant to MR elastography, indicate that the proposed approach may provide an effective means for comprehensive multi-frequency characterization of the “background” viscoelasticity of soft tissues.     

Two-Way Coupled Statement of the Problem of Loss of Stability due to Inverse Thermoelastic Phase Transition in a Shape Memory Alloy

A two-way coupled statement of stability problem for shape memory alloy elements is given in the framework of the “fixed load” and “variable load” concepts. It is shown that the largest values of the critical parameters are obtained when solving the problem in the two-way coupled statement in the framework of the “fixed load” concept and the least values are obtained in the oneway coupled statement in the framework of the “variable load” concept.