An eigenanalysis-based bifurcation indicator proposed in the framework of a reduced-order modeling technique for non-linear structural analysis

The Koiter–Newton method is a reduced order modeling technique which allows us to trace efficiently the entire equilibrium path of a non-linear structural analysis. In the framework of buckling the method is capable to handle snap-back and snap-through phenomena but may fail to predict reliably bifurcation branches along the equilibrium path. In this contribution we extend the original Koiter–Newton approach with a reliable and accurate bifurcation indicator which is based on an eigenanalysis of the reduced order tangent stiffness matrix. The proposed indicator has a negligible numerical effort since all computations refer to the reduced order model which is typically of very small dimension. The extension allows the identification of bifurcation points and a tracing of corresponding bifurcation branches in each sector of the equilibrium path. The performance of the method in terms of reliability, accuracy and computational effort is demonstrated with several examples.     

Optimized sixth‐order monotonicity‐preserving scheme by nonlinear spectral analysis

In this paper, sixth‐order monotonicity‐preserving optimized scheme (OMP6) for the numerical solution of conservation laws is developed on the basis of the dispersion and dissipation optimization and monotonicity‐preserving technique. The nonlinear spectral analysis method is developed and is used for the purpose of minimizing the dispersion errors and controlling the dissipation errors. The new scheme (OMP6) is simple in expression and is easy for use in CFD codes. The suitability and accuracy of this new scheme have been tested through a set of one‐dimensional, two‐dimensional, and three‐dimensional tests, including the one‐dimensional Shu–Osher problem, the two‐dimensional double Mach reflection, and the Rayleigh–Taylor instability problem, and the three‐dimensional direct numerical simulation of decaying compressible isotropic turbulence. All numerical tests show that the new scheme has robust shock capturing capability and high resolution for the small‐scale waves due to fewer numerical dispersion and dissipation errors. Moreover, the new scheme has higher computational efficiency than the well‐used WENO schemes. Copyright © 2013 John Wiley & Sons, Ltd.     

Analysis on secondary instability of shear layer based on the concept of phase synchronization

A concept of phase synchronization point is proposed, and then a model is built using this concept to explain secondary instabilities. This model has been used to determine the conditions of K- and H-type secondary instabilities, which are coincident with the conditions published in literatures. It also can be used to analyze other secondary instability phenomena. For example, the numerical results validate the analysis results in the case of 1/3rd subharmonic mode secondary instability. Furthermore, the numerical results indicate that the spanwise wave number of 3D disturbance has significant effect on the secondary instability.     

Lyapunov exponents as a criterion for the dynamic pull-in instability of electrostatically actuated microstructures

The dynamic pull-in instability of double clamped microscale beams actuated by a suddenly applied distributed electrostatic force and subjected to non-linear squeeze film damping is investigated. A reduced order model is built using the Galerkin decomposition with undamped linear modes as base functions and verified through comparison with numerical finite differences solution. The stability analysis of a beam actuated by one and two electrodes symmetrically located at two sides of the beam and operated by a step-input voltage is performed by evaluating the largest Lyapunov exponent, the sign of which defines the character of the response. It is shown that this approach provides an efficient quantitative criterion for the evaluation of dynamic pull-in instability, especially when combined with compact reduced order models. Based on the Lyapunov exponent criterion, the influence of various parameters on the beam dynamic stability is investigated.     

Two scale damage model and related numerical issues for thermo-mechanical High Cycle Fatigue

On the idea that fatigue damage is localized at the microscopic scale, a scale smaller than the mesoscopic one of the Representative Volume Element (RVE), a three-dimensional two scale damage model has been proposed for High Cycle Fatigue applications. It is extended here to anisothermal cases and then to thermo-mechanical fatigue. The modeling consists in the micromechanics analysis of a weak micro-inclusion subjected to plasticity and damage embedded in an elastic meso-element (the RVE of continuum mechanics). The consideration of plasticity coupled with damage equations at microscale, altogether with Eshelby–Kröner localization law, allows to compute the value of microscopic damage up to failure for any kind of loading, 1D or 3D, cyclic or random, isothermal or anisothermal, mechanical, thermal or thermo-mechanical. A robust numerical scheme is proposed in order to make the computations fast. A post-processor for damage and fatigue (DAMAGE_2005) has been developed. It applies to complex thermo-mechanical loadings. Examples of the representation by the two scale damage model of physical phenomena related to High Cycle Fatigue are given such as the mean stress effect, the non-linear accumulation of damage. Examples of thermal and thermo-mechanical fatigue as well as complex applications on real size testing structure subjected to thermo-mechanical fatigue are detailed.     

Viscoelastic flow analysis using the software OpenFOAM and differential constitutive equations

Viscoelastic fluids are of great importance in many industrial sectors, such as in food and synthetic polymers industries. The rheological response of viscoelastic fluids is quite complex, including combination of viscous and elastic effects and non-linear phenomena. This work presents a numerical methodology based on the split-stress tensor approach and the concept of equilibrium stress tensor to treat high Weissenberg number problems using any differential constitutive equations. The proposed methodology was implemented in a new computational fluid dynamics (CFD) tool and consists of a viscoelastic fluid module included in the OpenFOAM, a flexible open source CFD package. Oldroyd-B/UCM, Giesekus, Phan-Thien–Tanner (PTT), Finitely Extensible Nonlinear Elastic (FENE-P and FENE-CR), and Pom–Pom based constitutive equations were implemented, in single and multimode forms. The proposed methodology was evaluated by comparing its predictions with experimental and numerical data from the literature for the analysis of a planar 4:1 contraction flow, showing to be stable and efficient.     

A comparison of numerical methods applied to non-linear adsorption columns

Eight numerical schemes (first-order upstream finite difference, MacCormack, explicit Taylor–Galerkin, random choice, flux-corrected transport, ENO, TVD, and Euler–Lagrange methods) are compared on the basis of their computational efficiency for one-dimensional non-linear convection–diffusion problems. For the ideal chromatographic equation for which an exact solution exists, errors plotted against computational times show that the best methods are the random choice, Euler–Lagrange and flux-corrected MacCormack methods. Even when significant diffusion is added to the model, steep gradients are possible because of non-linearities. In such an instance, the random choice and flux-corrected transport methods give the best performance. One can now tackle more complicated problems and refer to this comparative study in order to choose an adequate numerical method which will provide sufficiently accurate results at a reasonable cost.     

Development and elaboration of numerical method for simulating gas–liquid–solid three-phase flows based on particle method

A numerical method for simulating gas–liquid–solid three-phase flows based on the moving particle semi-implicit (MPS) approach was developed in this study. Computational instability often occurs in multiphase flow simulations if the deformations of the free surfaces between different phases are large, among other reasons. To avoid this instability, this paper proposes an improved coupling procedure between different phases in which the physical quantities of particles in different phases are calculated independently. We performed numerical tests on two illustrative problems: a dam-break problem and a solid-sphere impingement problem. The former problem is a gas–liquid two-phase problem, and the latter is a gas–liquid–solid three-phase problem. The computational results agree reasonably well with the experimental results. Thus, we confirmed that the proposed MPS method reproduces the interaction between different phases without inducing numerical instability.     

Ho's theorem in global-local mode interaction of pin-jointed bar structures

This paper is focused on the interaction phenomena among a global critical mode and some local Eulerian critical modes in pin-jointed structures. These phenomena are framed within Koiter's theory of elastic instability, by an asymptotic reduction into cubic systems. The aim is to present an algorithm for the appraisal of the lowest critical load characterizing the structure under the effect of small imperfections. First of all, the Ho's theorem, concerning the definition of the most dangerous imperfection, is presented and discussed. Then, a FEM code aimed at the determination of the most dangerous shape for the imperfection, and at performing the related sensitivity analysis, is implemented, by superimposing a proper FE beam model (able to model Eulerian instability) to a non-linear FE model for spatial pin-jointed structures. Some numerical results having a practical interest are presented and discussed.     

Three-dimensional arbitrary Lagrangian–Eulerian numerical prediction method for non-linear free surface oscillation

A numerical prediction method has been proposed to predict non-linear free surface oscillation in an arbitrarily-shaped three-dimensional container. The liquid motions are described with Navier–Stokes equations rather than Laplace equations which are derived by assuming the velocity potential. The profile of a liquid surface is precisely represented with the three-dimensional curvilinear co-ordinates which are regenerated in each computational step on the basis of the arbitrary Lagrangian–Eulerian (ALE) formulation. In the transformed space, the governing equations are discretized on a Lagrangian scheme with sufficient numerical accuracy and the boundary conditions near the liquid surface are implemented in a complete manner. In order to confirm the applicability of the present computational technique, numerical simulations are conducted for the free oscillations of viscid and inviscid liquids and for highly non-linear oscillation. In addition, non-linear sloshing motions caused by horizontal and vertical excitations and a transition from non-linear sloshing to swirling are numerically predicted in three-dimensional cylindrical containers. Conclusively, it is shown that these sloshing motions associated with high non-linearity are reasonably predicted with the present numerical technique. © 1998 John Wiley & Sons, Ltd.     

Higher-order impact modeling of sandwich structures with flexible core

In this study, a higher-order impact model is presented to simulate the response of a soft-core sandwich beam subjected to a foreign object impact. A free vibration problem of sandwich beams is first solved, and the results are validated by comparing with numerical finite element modeling results of ABAQUS and the solution by Frostig and Baruch [Frostig, Y., Baruch, M., 1994. Free vibration of sandwich beams with a transversely flexible core: a high order approach. Journal of Sound and Vibration 176(2), 195–208]. Then a foreign object impact process is incorporated in the higher-order model, and the contact force and deflection history as well as the propagation of transverse normal, shear, and axial stresses during the impact are analyzed and discussed. The validity of the model in the impact response predictions is demonstrated by comparing with finite element solutions of LS-DYNA. The calculated stresses caused by a foreign object impact are then used to assess failure locations, failure time, and failure modes in sandwich beams, which are shown to compare well with the available experimental results. The effects of impact mass, initial velocity, core stiffness, and core height on the impact stresses generated in the beams are discussed. The influences of impact mass and initial velocity on the contact force history are close to those by the linearized impact solution, but the proposed higher-order impact model captures the non-linear impact process and different generated stresses. Compared to the fully backed sandwich case, the core height shows a great influence over the impact process of a simply supported sandwich system, in which the global behavior of the sandwich is dominant; while the core stiffness shows minor effect over the impact process. The higher-order impact model of sandwich beams developed in the study provides accurate predictions of the generated stresses and impact process and can be used effectively in design analysis of anti-impact structures made of sandwich materials.     

Estimation of material properties of nanocomposite structures

A method for the numerical modelling of mechanical behaviour of nanocomposite materials reinforced with the carbon nanotubes, based on computational homogenization as a multi-scale method, is presented. Since the carbon nanotube inside of the representative volume element (RVE) is modelled as a space frame structure, theoretical background and a proper way of modelling of carbon nanotubes is given. Novelty in this paper is an incorporation of interactions, based on the weak van der Waals forces and modelled by nonlinear rod elements, into a multiscale model as described below. An algorithm is developed for analysis of those interactions. Since the problem of modelling nanocomposite structures is a three-dimensional multi-scale problem, one part of this work is dedicated to multi-scale modelling methods, especially to the first order computational homogenization. Computational homogenization and representative volume element are the basis of the presented numerical model of the nanocomposites. Nano scale model is based on beam and non-linear rod finite elements. For the purpose of the software verification, examples, i.e. models of the nanocomposite material are presented. Obtained results are compared with the results given by the other authors.     

A strain localization analysis using a viscoplastic softening model for clay

Strain localization has become an attractive subject in geomechanics during the past decade. Shear bands are well known to develop in clay specimens during the straining process. Strain localization is closely related to plastic instability. In the present paper, a non-linear instability condition for the viscoplastic strain softening model during the creep process is firstly obtained. It is found that the proposed viscoplastic model is capable of describing plastic instability. Secondly, a two-dimensional linear instability analysis is performed and the preferred orientation for the growth of fluctuation and the instability condition are derived. It is worth noting that the two instability conditions are equivalent. Finally, the behavior of the clay is numerically analyzed in undrained plane-strain compression tests by the finite element method, considering a transport of pore water in the material at a quasi-static strain rate. The numerical results show that the model can predict strain localization phenomena, such as shear banding. From the numerical calculations, the effects of strain rate and permeability are discussed.     

Evaluation a priori d'un modèle non-linéaire de turbulenceA priori evaluation and improvement of a non-linear model for turbulent flows

The aim of this work is a priori evaluation and improvement of a non-linear model for turbulent flows using the results from direct numerical simulation of Navier–Stokes equations. The algebraic explicit non-linear model recently proposed by Rumsey C.L. et al. [1] is studied. The data base used here comes from a direct numerical simulation of a turbulent flow through a square duct. For this flow, this study shows that the hypothesis of equilibrium state for the anisotropic tensor is correct. The analysis is made using the maps of the second and third invariants of the Reynolds stress tensor. The approach used permits to conclude that the model using a wall function improves the numerical prediction of the anisotropy. To cite this article: O. El Yahyaoui et al., C. R. Mecanique 330 (2002) 27–34     

Discrete particle modeling of granular Rayleigh–Taylor instability

The gravitational air–grain Rayleigh–Taylor (RT) flow instability in a Hele-Shaw cell was studied using a parallel three-dimensional discrete particle model (DPM). The onset of flow instability and the development of fingering flow structures were well captured by the model. Power spectra analysis of solid volume fraction field indicated the non-linear coarsening process of the fingering flow structures. The sensitivity of the flow patterns to the initial porosity, the Atwood number, and the ratio of particle size to the Hele-Shaw cell width was also demonstrated. The excellent agreement of DPM simulation results with the reported experimental observations proved the robustness and reliability of the numerical approach to model complex multiphase flows such as granular RT instability.     

The effect of viscosity,surface tension and non-linearity on Richtmyer–Meshkov instability

A weakly non-linear theoretical model of the Richtmyer–Meshkov instability between two viscous fluids with surface tension is proposed. The model is based on the application of singular perturbations techniques to the incompressible Navier–Stokes equations written for two superposed immiscible fluids. A simple analytical law of interface deformation is obtained, in which the effects of viscosity, surface tension and non-linearities appear under the form of independent terms. The model gives also access to the velocity and pressure distribution in the fluids, which can be of interest for estimating vorticity diffusion in the fluids. A comparison with accurate direct numerical simulations confirms the validity of the proposed theory. The interface deformation law is then applied to typical experimental configurations in order to estimate the relative influence of surface tension, viscosity and non-linearities on the growth of perturbations for each of the chosen cases.     

蜂窝夹芯试件破坏行为分析

通过平面拉伸试验和双悬臂梁试验研究了蜂窝夹芯试件的破坏行为. 在平面拉伸试验中, 发现的破坏模式不是预期的面芯界面脱胶破坏而是面板层间分层破坏;在双悬臂梁试验中, 发现了一种与以往文献报导中不同的破坏模式(面板层间分层,预制裂纹偏转和面板分层扩 展依次出现). 针对试验中所发现的新的破坏模式,结合粘结模型,建立了基于 蔡-希尔破坏准则和能量准则的计算模型. 模拟结果与试验结果对比发现,所建立 的计算模型能够很好地模拟所发现的破坏行为.     

A simple meso-macro model based on SQP for the non-linear analysis of masonry double curvature structures

A 3D model for the evaluation of the non-linear behavior of masonry double curvature structures is presented. In the model, the heterogeneous assemblage of blocks is substituted with a macroscopically equivalent homogeneous non-linear material. At the meso-scale, a curved running bond representative element of volume (REV) constituted by a central block interconnected with its six neighbors is discretized through of a few six-noded rigid wedge elements and rectangular interfaces. Non linearity is concentrated exclusively on joints reduced to interface, exhibiting a frictional behavior with limited tensile and compressive strength with softening. The macroscopic homogenous masonry behavior is then evaluated on the REV imposing separately increasing internal actions (in-plane membrane actions, meridian and parallel bending, torsion and out-of-plane shear). This simplified approach allows to estimate heuristically the macroscopic stress–strain behavior of masonry at the meso-scale. The non-linear behavior so obtained is then implemented at a structural level in a novel FE non-linear code, relying on an assemblage of rigid infinitely resistant six-noded wedge elements and non-linear interfaces, exhibiting deterioration of the mechanical properties.Several numerical examples are analyzed, consisting of two different typologies of masonry arches (a parabolic vault and an arch in a so-called “skew” disposition), a ribbed cross vault, a hemispherical dome and a cloister vault. To fully assess numerical results, additional non-linear FE analyses are presented. In particular, a simplified model is proposed, which relies in performing at a structural level a preliminary limit analysis – which allows to identify the failure mechanism – and subsequently in modeling masonry through elastic elements and non-linear interfaces placed only in correspondence or near the failure mechanism provided by limit analysis. Simulations performed through an equivalent macroscopic material with orthotropic behavior and possible softening are also presented, along with existing experimental evidences (where available), in order to have a full insight into the capabilities and limitations of the approach proposed.     

改进的物理融合神经网络在瑞利?泰勒不稳定性问题中的应用

基于相场法的物理融合神经网络PF-PINNs被成功用于两相流动的建模, 为两相流动的高精度直接数值模拟提供了全新的技术手段. 相场法作为一种新兴的界面捕捉方法, 其引入确保了界面的质量守恒, 显著提高了相界面的捕捉精度; 但是相场法中高阶导数的存在也降低了神经网络的训练速度. 为了提升计算训练过程的效率, 本文在PF-PINNS框架下, 参考深度混合残差方法MIM, 将化学能作为辅助变量以及神经网络的输出之一, 并修改了物理约束项的形式, 使辅助变量与相分数的关系式由硬约束转为了软约束. 上述两点改进显著降低了自动微分过程中计算图的规模, 节约了求导过程中的计算开销. 同时, 为了评估建立的PF-PINNS在雷诺数较高、计算量较大的场景中的建模能力, 本文将瑞利?泰勒RT不稳定性问题作为验证算例. 与高精度谱元法的定性与定量对比结果表明, 改进PF-PINNs有能力捕捉到两相界面的强非线性演化过程, 且计算精度接近传统算法, 计算结果符合物理规律. 改进前后的对比结果表明, 深度混合残差方法能够显著降低PF-PINNS的训练用时. 本文所述方法是进一步提升神经网络训练速度的重要参考资料, 并为探索高精度智能建模方法提供了全新的见解.      

Rotating a pendulum with an electromechanical excitation

The objective of this paper is showing investigation of pendulum rotations via vertical, non-linear electromechanical excitation generated using a RLC-circuit-powered solenoid, which is originally built for an electro-vibro-impact mechanism. Various non-linear phenomena of pendulum dynamics, namely period-1 rotation, period-1 oscillation and period-2 oscillation, have been observed experimentally from the proposed apparatus. A mathematical model has been developed for the experimental rig and the system parameters have also been identified for the mathematical model. Finally, numerical results have been generated using the developed mathematical model and identified parameters, and their correlations with experimental observations have been discussed.