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Summary: This study aims to analyze the buckling and post‐buckling behaviors of multilayer nanocomposite beams reinforced with graphene oxide powders (GOPs) at low concentrations. The GOPs are randomly oriented and evenly distributed throughout the composite layers, with their weight fraction varying in the thickness direction. The Halpin‐Tsai model is employed to estimate each layer's effective material properties. The nonlinear governing equations for the FG‐GOPRC beam are derived based on the first‐order shear deformation beam theory, and a nonlinear algebraic system is formulated using the principle of potential energy. In this research, a comprehensive parametric analysis is conducted to examine the influence of various factors on the buckling and post‐buckling behaviors of the composite beams. These factors include the distribution pattern and weight fraction of graphene platelets (GPLs) nanofillers, as well as the geometry, size, slenderness ratio, and boundary conditions of the beams. Through the analysis, the study highlights the significant strengthening effect of these factors on the buckling and post‐buckling performance of the composite beams. The findings from this investigation contribute to a deeper understanding of the behavior of multilayer nanocomposite beams reinforced with GOPs. This knowledge can be valuable in designing and optimizing of composite structures for enhanced buckling resistance and post‐buckling performance. [ABSTRACT FROM AUTHOR]

Conscious efforts on reduction of greenhouse gas emissions have led to an energy transition to renewable energy, however uncertainties of renewable energy production have resulted in higher thermal cycling demands from conventional power plants. Thermal load cycling at high temperature regions of steam turbine components leads to enhanced creep-fatigue damage accumulation. It is well established that such damage mechanism is numerically best predicted by unified constitutive modeling including damage as a variable as per the formalism of continuum damage mechanics at an expense of considerable computational efforts using finite element analysis. In this paper, the non-iterative Asymptotic Numerical Method (ANM), currently limited to partial cycle analysis with linear hardening plasticity model, is proposed for the first time to address cyclic viscoplasticity problems including damage capable of handling multiple cycles and most generalized loading conditions. Regularization techniques additionally necessary to implement loading-unloading-reloading criteria and advanced constitutive models etc. are presented. The constitutive model chosen for the formulation includes the non-linear multiple back stress variable modified Chaboche model to include damage combined with modified Chaboche-Rousselier isotropic hardening model to include damage, power law for viscoplasticity, Lemaitre's damage potential and Kachanov-Rabotnov's creep damage law. The method is verified with defined error measures and then applied to two high pressure steam turbine rotors, one with and another without thermal stress relief groove (TSRG) at the inlet under service type loading conditions to study the beneficial effect of the TSRG on creep-fatigue damage evolution. The accumulated errors of the proposed ANM and computational time are compared to a conventional Newton-Raphson solution. [ABSTRACT FROM AUTHOR]

The literature about the asymptotic numerical method (ANM) is reviewed in this paper as well as its application to hyperelasticity and elastoplasticity. ANM is a generic continuation method based on the computation of Taylor series for solving nonlinear partial differential equations. Modern techniques of high-order differentiation provide simple tools for computing these power series, the corresponding algorithms for finite strain elasticity and elastoplasticity being summarized here. Taylor series is not only a computation tool, but it contains also useful information about the structure of the considered solution curve. The paper ends with a short historical account about the development of this numerical method. [ABSTRACT FROM AUTHOR]

Introduction: The impact of vibrations excited by incident sound fields has become a major concern today, due to its influence on the performance of systems and installations. Vibrations have the potential to cause considerable dynamic disturbances and instabilities, which can lead to significant structural and functional damage. Consequently, it is crucial to control vibration phenomena right from the system design phase. To solve the problem of vibration, it is sometimes possible to increase the damping level of the structure by incorporating a damping treatment. Objective: The aim of this paper is to present a simplified numerical approach to study the vibro-acoustic responses of structures with PCLD "Passive Constrained Layer Damping" treatment in the thermal environment, taking into account the frequency and temperature dependence of the different viscoelastic behavior laws. Material and Methods: The modal stability procedure MSP is based on the finite element method in order to discretize and formulate the equation of motion. The asymptotic numerical method "ANM" is applied to approximate the solution of complex eigenvalue problems and construct the modal basis. The variability of the frequency responses is evaluated by a Monte Carlo simulation (MCS) combined with MSP and ANM to evaluate the stochastic behavior of a sandwich beam with random properties. Results: The comparison with the direct frequency responses (DFR) demonstrates that the results are highly satisfactory in terms of the validity of the present MSP approach. A comparative study of viscoelastic behavior models was carried out to evaluate their damping properties provided to the structure. The viscoelastic materials provide significant damping particularly for amplitudes corresponding to the high frequencies. This is in contrast to the responses obtained without the viscoelastic layer. Conclusion: The obtained results show the importance of viscoelastic damping, which has a significant effect on the vibro-acoustic behavior, implying the improvement of the damping of the structure, especially for large frequencies and high temperatures. [ABSTRACT FROM AUTHOR]

An accurate numerical tool is presented in this work to investigate the stationary incompressible Navier–Stokes equations. The proposed approach is based on a pseudo‐spectral method for discretizing the differential equations and the asymptotic numerical method to convert nonlinear systems into linear algebraic equations. The coupling of the spectral method with the asymptotic numerical method is considered as an efficient algorithm to solve any nonlinear differential equations. Their efficiency and robustness are examined here on the flow fluid in different canal with different geometries. These computational efficiency and performance have been analysed via several numerical and benchmark examples of incompressible fluid flow in lid‐driven cavity and vortex shedding over L‐Shaped cavity and fluid flow around a square obstacle. The validation of the proposed approach is made by comparison between the obtained results and those calculated using a finite element method or Ansys commercial code. This validation asserts that the presented numerical tool can be promise for solving fluid flow problems with high accuracy. [ABSTRACT FROM AUTHOR]

Summary: Numerical simulation of stretch‐induced wrinkling in thin elastic sheets is a challenging problem due to a vanishing bending stiffness and the coexistence of superabundant equilibrium solutions. In this work, we present a computational framework to capture the morphological evolution of stretch‐induced wrinkles. The application of modified Föppl‐von Kármán plate model results in a fourth‐order partial differential equation. The convergence of finite‐element solutions necessitates C1$$ {C}^1 $$‐continuous approximations. Herein, Powell‐Sabin B‐splines, which are based on triangles, are utilized for both the approximation of the field variables and the description of the geometry. To trace the wrinkling behavior in thin sheets, a path‐following technique using asymptotic numerical method is considered. The advantage of this method is an adaptive step length, which works incredibly well near the bifurcation points and allows for the computation of the post‐bifurcation diagrams with a quite small perturbation. The versatility and accuracy of the developed computational approach are assessed in three case studies, featuring wrinkling in highly stretched rectangular and annular thin sheets. [ABSTRACT FROM AUTHOR]

This paper presents mathematical modelling and simulation founded on an Asymptotic Numerical Method (ANM) for the rubber-like material hyperelastic behaviour. The basic models such as Mooney-Rivlin, and the Gent partial models and the Hybrid Integral Approach (HIA) model are considered. The HIA is a strong nonlinear model developed from the inverse Langevin function approximation is considered here for its efficiency. Explicit formulations of each model are presented as well as the associated asymptotic developments in the frame of the ANM. Various types of boundary conditions effects are analysed for each considered hyperelastic model. The effectiveness of the ANM compared to the Newton-Raphson method is demonstrated and particularly for the HIA model. We present as results stress-strain curves for the implementation of these different models where we specify the influence of the Imposed Displacement for tensile and compressive stresses when the structure is clamped and simply supported. In term of this study, we observe that the complete HIA model present the curves with the difference hardly perceptible when we vary results in term of stress-strain curves with different imposed Displacement. But when strain is high than 5%, the Mooney-Rivlin and the Gent partial models, presents the high disparities with different imposed Displacement. This observation justifying the limitations of partial models in term of characterisation of the behaviour law of hyperplastic materials. [ABSTRACT FROM AUTHOR]

The non-linear quasi-static buckling of thin-walled beams with arbitrary cross-sections is studied by developing coupling strategies between the composing plates (web, flanges). The element adopted for modeling the plates is a 7-parameter shell element using the Enhanced Assumed Strain concept (Büchter et al., 1994). Different nodal shell directors exist at the interface between plates for thin-walled beams with arbitrary cross-sections, requiring appropriate strategies. The coupling between plates first considers Lagrange multipliers at the interface, following the mortar method on the surface between the plates. Second a simplified pre-processing method is proposed by modifying the shell directors of the nodes close to the interface. In the case of the quasi-static buckling of thin-walled beams with L-shaped and I-shaped cross-sections, the two coupling strategies are thoroughly assessed by using three different solution procedures: Newton–Raphson, Newton–Riks and Asymptotic Numerical Method (ANM). The pre-processing method is simple and turns out to be robust and efficient. • Non-linear buckling analysis of thin-walled beams with arbitrary cross-section. • Mortar approach for coupling 7-parameter shell elements (with Enhanced Assumed Strain). • Pre-processing approach for dealing with discontinuity of shell directors. • Comparisons between solving strategies: Newton–Raphson with EAS, Riks, Asymptotic Numerical Method. • Reference computation with hexahedral 20-node elements. [ABSTRACT FROM AUTHOR]

The paper deals with the numerical computation of difficult problems requiring many steps and many degrees of freedom such as the finite element analysis of wrinkling of film–substrate systems. The asymptotic numerical method (ANM) is well adapted to such computations but with a progressive loss of accuracy during step chaining. Thus, correction phases are necessary, which are rarely carried out within ANM. A convergence acceleration algorithm and a step-length adaptation have been included to limit the growth of computation time and to strengthen the reliability of the procedure. This modified version of the ANM is assessed by simulating the appearance and evolution of sinusoidal wrinkles under uniaxial compression.

Composite materials and structures are inherently in-homogeneous across multiple scales. Multi-scale modelling offers opportunities to apprehend the coupling of material behaviour and characteristics from the micro- to meso- and macro-scales. In this paper, a multi-scale finite element method (FE2) is proposed to compute the modal properties of visco-elastic heterogeneous composite materials in terms of damping frequencies and modal loss factors. In the proposed FE2-based vibration analysis, two finite elements (FE) calculations are carried out in a nested manner, one at the macro-scale and the other at the micro-scale. Unlike conventional analysis, the developed analysis does not require homogenized constitutive properties because these are derived from the micro-scale FE simulations at the representative volume element (RVE) level. The non-linearity at the micro-scale is accounted by using a frequency dependent Young’s modulus. The Asymptotic Numerical Method (ANM) and its automatic differentiation is used to solve the non-linear numerical problem. ANM consists of solving an analytical non-linear problem with a path-following (or continuation) method associated with a high-order perturbation technique. Compared with existing methods, the originality of the proposed approach lies in its ability to account for the frequency dependence of Young’s modulus in visco-elastic microstructure. Using the automatic differentiation makes the proposed approach enough flexible and generic to deal with damped and undamped vibration analyses of composite materials structures.

This paper couples the Carrera's Unified Formulation (CUF) with the Asymptotic Numerical Method (ANM) for investigating geometrically nonlinear behaviors of beam structures. On the basis of CUF, a family of advanced one-dimensional beam models is firstly established by deriving a fundamental nucleus. The ANM that is an efficient and robust nonlinear solver is then used to solve the resulted nonlinear systems. Several typical nonlinear problems in thin/thick beam structures are carried out, and numerical results demonstrate the validity and efficiency of the proposed framework. Besides, the importance of using high-order kinematics for beam structures undergoing large deformations is emphasized. [ABSTRACT FROM AUTHOR]

In this paper, a mathematical procedure is elaborated based on large deformation, finite element and perturbation methods for thermo-large amplitude analysis of anisotropic laminated and sandwich FGM plates with temperature-dependent properties. Nonlinear thermal dependence distribution following a power law is adapted. Various types of multilayered composite plates with position and temperature-dependent material properties are considered. Mathematical formulations and explicit relationships are presented in the frame of finite element methods. The structure is subjected to linear and uniform temperature variations. The temperature and the displacement are expanded in power series with unknown terms, and the numerical solution is obtained by the finite element method. The homotopy method is used for nonlinear thermal buckling, and a continuation procedure is elaborated for nonlinear thermal–displacement responses. The nonlinear thermal critical buckling load is first computed and the postbuckling equilibrium path of FGM and laminated plates under thermal loading is investigated. Temperature-dependent properties and geometry effects as well as boundary conditions on the nonlinear thermal buckling and postbuckling behaviors are analyzed. [ABSTRACT FROM AUTHOR]

Solving nonlinear systems is an important problem. Numerical continuation methods efficiently solve certain nonlinear systems. The Asymptotic Numerical Method (ANM) is a powerful continuation method that usually converges faster than Newtonian methods. ANM explores the landscape of the function by following a parameterized solution curve approximated with a high-order power series. Although ANM has successfully solved a few graphics and engineering problems, prior to our work, applying ANM to new problems required significant effort because the standard ANM assumes quadratic functions, while manually deriving the power series expansion for nonquadratic systems is a tedious and challenging task. This paper presents a novel solver, SANM, that applies ANM to solve symbolically represented nonlinear systems. SANM solves such systems in a fully automated manner. SANM also extends ANM to support many nonquadratic operators, including intricate ones such as singular value decomposition. Furthermore, SANM generalizes ANM to support the implicit homotopy form. Moreover, SANM achieves high computing performance via optimized system design and implementation. We deploy SANM to solve forward and inverse elastic force equilibrium problems and controlled mesh deformation problems with a few constitutive models. Our results show that SANM converges faster than Newtonian solvers, requires little programming effort for new problems, and delivers comparable or better performance than a hand-coded, specialized ANM solver. While we demonstrate on mesh deformation problems, SANM is generic and potentially applicable to many tasks. [ABSTRACT FROM AUTHOR]

This study aims to link critical Reynolds numbers associated with either steady-state or temporal bifurcations to fluid flow regimes described by macroscopic laws (Darcy, Forchheimer) in a periodic 3D Kelvin foam, using direct numerical simulations. We first identify the permeability and inertial coefficient of the Darcy's law and the Forchheimer's one. We explicit the different flow regimes that are accounted for in the Forchheimer's framework (Darcian, weak inertia and strong inertia regimes, respectively). We present an original systematic way to determine the critical Reynolds number associated with both regime transitions. We calculate them over a wide range of porosities and present two power-law correlations that locate these regime transitions for engineering purposes. We have performed pore-scale resolved calculations for various porosities with the Asymptotic Numerical Method (ANM) to find steady-state bifurcations, if any, along with a Linear Stability Analysis (LSA) to find temporal (Hopf) bifurcations from steady-state base flows. All the computed bifurcations occur at Reynolds numbers in the vicinity of the transition from weak to strong inertia regimes, where a change of behavior takes place. On the other hand, no bifurcation has been found in the transition between Darcian and weak inertia regimes. [Display omitted] • Identification of macroscopic properties of a periodic Kelvin cell. • Flow regimes Transition identification. • 3D pore scale numerical simulations. • First time that the ANM and LSA method were used to automatically detect bifurcations in such system. • To correlate critical Reynolds of found bifurcations to the flow regimes. [ABSTRACT FROM AUTHOR]

In this article, a modal analysis of rotating viscoelastic sandwich beams with uniform cross sections and with various viscoelastic stiffness laws is conducted. The governing equations of motion are derived using the virtual work principle and discretized by the finite element procedure. The obtained nonlinear eigenvalue problems are solved by coupling the homotopy and asymptotic numerical techniques leading to complex solutions composed of natural frequencies and loss factors. Further, the effects of the rotating speed, thickness and core layer position on the dynamic characteristics are investigated. Furthermore, the effects of the frequency dependence of the shear modulus is investigated for a beam subject to various boundary conditions. Then, numerical tests are performed and the obtained results coincide very well with the literature and those provided by the commercial software Abaqus. [ABSTRACT FROM AUTHOR]

A dynamic analysis of the behavior of sandwich beams with a viscoelastic core under the action of a moving load is performed considering their geometrical asymmetry. The use of viscoelastic materials integrated into structures in front of the moving load is considered as a new suggestion to enhance their stability. A high-order theory, taking into consideration longitudinal and rotational inertias was used to examine the viscoelastic damping properties composite sandwich beams with account of their geometrical asymmetry and the frequency-dependent behavior of their viscoelastic core. [ABSTRACT FROM AUTHOR]

The paper deals with the numerical computation of difficult problems requiring many steps and many degrees of freedom such as the finite element analysis of wrinkling of film--substrate systems. The asymptotic numerical method (ANM) is well adapted to such computations but with a progressive loss of accuracy during step chaining. Thus, correction phases are necessary, which are rarely carried out within ANM. A convergence acceleration algorithm and a step-length adaptation have been included to limit the growth of computation time and to strengthen the reliability of the procedure. This modified version of the ANM is assessed by simulating the appearance and evolution of sinusoidal wrinkles under uniaxial compression. [ABSTRACT FROM AUTHOR]

This paper describes a generic Taylor series-based continuation method, the so-called asymptotic numerical method, to compute the bifurcation diagrams of nonlinear systems. The key point of this approach is the quadratic recast of the equations as it allows to treat in the same way a wide range of dynamical systems and their solutions. Implicit differential-algebraic equations, forced or autonomous, possibly with time-delay or fractional-order derivatives are handled in the same framework. The static, periodic and quasi-periodic solutions can be continued and also transient solutions. [ABSTRACT FROM AUTHOR]

New algorithms, combining asymptotic numerical method (ANM) and method of fundamental solutions, are proposed to compute bifurcation points on branch solutions of a nonlinear bi‐harmonic problem. Three methods, mainly based on asymptotic developments framework, are then proposed. The first one consists in exploiting the ANM step accumulation close to the bifurcation points on a solution branch, the second method allows the introduction of an indicator that vanishes at the bifurcation points, and finally the first real root of the Padé approximant denominator represents the third bifurcation indicator. Two numerical examples are considered to analyze the robustness of these algorithms. [ABSTRACT FROM AUTHOR]

Flexible operation of the conventional power plants is necessary for the integration of renewables in the energy mix. Steam turbine rotors are one of the most affected components due to the daily thermal load cycling requirements. In order to accurately study the effect of cycling on fatigue, it is necessary to solve cyclic 3D elasto-plasticity problems for a large number of cycles. The established methods are iterative in nature to arrive at the current yield surface for each elasto-plastic step and are computationally extensive. In this paper, we propose the non-iterative Asymptotic Numerical Method (ANM) based solution technique for 3D cyclic elasto-plasticity problems for the first time by a novel regularized Kuhn-Tucker condition. Regularization techniques used in previous works take care of the partial cycle only, i.e., elastic loading, elastic-plastic transition, elasto-plastic flow and elastic unloading for simple linear work hardening models of plasticity. In the present work, the proposed method extends the applicability of ANM by incorporating elastic-plastic transition and elasto-plasticity in reverse loading, second unloading and repeated loading for any number of cycles so that the computationally efficient fatigue analysis can be carried out. In addition, the extension also includes the application of ANM to non-linear kinematic hardening through the Chaboche plasticity model combined with isotropic hardening. For these advancements, additional regularizations are proposed and implemented in a Finite Element Code. The obtained results are then compared to the solutions obtained from the conventional Newton Raphson (NR) solution technique. The proposed ANM is applied to a steam turbine rotor undergoing a typical thermal load cycling and the cyclic elasto-plastic response is compared with that obtained using the conventional Newton Raphson solution technique. [Display omitted] • A computationally efficient Asymptotic Numerical Method for multi-cycle 3D elastoplastic analysis with nonlinear hardening is proposed. • A new regularization of Kuhn-Tucker conditions is proposed and a regularized expression for the plastic strain rate is derived. • ANM depicts 20%–50% reduction in computational time with an accuracy >99% compared to an iterative Newton Raphson (NR) method. [ABSTRACT FROM AUTHOR]

In this work, we have investigated numerically the disappearance of wrinkles from a tended membrane by the Asymptotic Numerical Method (ANM) using the finite-element DKT18. The ANM is a path-following technique that has been used to solve bifurcation problems. We show numerically the influence of the terms corresponding to the membrane displacement gradient in the Föppl–von Kármán (FvK) theory on the bifurcation curves in the case of a stretched elastic membrane. We will also study numerically, by using the ANM algorithm, the influence of the thickness and of the aspect ratio on the re-stabilization of a rectangular elastic membrane during stretching. The results obtained by our model are compared with those obtained using the industrial code ABAQUS. [ABSTRACT FROM AUTHOR]

• A multi-scale beam model is derived by coupling CUF and ANM under the FE2 framework. • It benefits from both the efficiency of 1D CUF beam model and multi scale framework. • The introduction of CUF accelerates the resolution without compromising accuracy. In this paper, a new paradigm for Carrera's Unified Formulation (CUF) based on multiscale structural modelling is accomplished by bridging micromechanics and the advanced CUF one-dimensional/beam structural theories by means of the Multilevel Finite Element (also known as FE2) framework. Under the framework of the FE2 method, the analysis is divided into a macroscopic/structural problem and a microscopic/material problem. At the macroscopic level, several higher-order refined beam elements can be easily implemented via CUF by deriving a fundamental nucleus that is independent of the approximation order over the thickness and the number of nodes per element (they are free parameters of the formulation). The unknown macroscopic constitutive law is obtained by numerical homogenisation of a Representative Volume Element (RVE) at the microscopic level. Vice versa, the microscopic deformation gradient is calculated from the macroscopic model. Information is passed between the two scales in a FE2 sense. The resulting nonlinear problem is solved through the Asymptotic Numerical Method (ANM) that is more reliable and less Newton-Raphson's one. The developed models are used as a first attempt to investigate the microstructure effect on the macrostructure geometrically nonlinear response. The proposed paradigm is used for investigating the effect of microscale imperfections (not straight carbon fibres) on the macroscale response (instability). Results are assessed in terms of accuracy and computational costs towards full FEM solutions. Three factors have been considered for an imperfection sensitivity parametric analysis: the defect wavelength as well as the amplitude and the size of RVE. [ABSTRACT FROM AUTHOR]

Summary: This paper is concerned with a Taylor series–based continuation algorithm, the so‐called Asymptotic Numerical Method (ANM). It describes a generic continuation procedure to apply the ANM principle at best, in other words, that presents a high level of genericity without paying the price of this genericity by low computing performances. The way to quadratically recast a system of equation is now part of the method itself, and the way to handle elementary transcendental function is detailed with great attention. A sparse tensorial formalism is introduced for the internal representation of the system, which, when combined with a block condensation technique, provides a good computational efficiency of the ANM. Three examples are developed to show the performance and the versatility of the implementation of the continuation tool. Its robustness and its accuracy are explored. Finally, the potentiality of this method for complex nonlinear finite element analysis is enlightened by treating 2D elasticity problems with geometrical nonlinearities. [ABSTRACT FROM AUTHOR]

This article presents an extension of the asymptotic numerical method combined with the harmonic balance method to the continuation of periodic orbits of delay differential equations. The equations can be forced or autonomous and possibly of neutral type. The approach developed in this paper requires the system of equations to be written in a quadratic formalism which is detailed. The method is applied to various systems, from Van der Pol and Duffing oscillators to toy models of clarinet and saxophone. The harmonic balance method is ascertained from a comparison to standards time-integration solvers. Bifurcation diagrams are drawn which are sometimes intricate, showing the robustness of this method. [ABSTRACT FROM AUTHOR]

For compliant adhesive contact problems based on the Lennard-Jones potential, the non-convexity of the latter leads to jump-in and jump-off instabilities which can hardly be traced by using classical algorithms. In this work, we combine an adapted Asymptotic Numerical Method (ANM) and the multiscale Arlequin method to trace efficiently these instabilities. The ANM is used to trace the entire unstable solution path in a branch-by-branch manner. The Arlequin method is used to achieve a refined resolution in the vicinity of the contact surface and to reduce possible spurious numerical oscillations due to coarse surface discretizations. Numerical results, validated by comparison with available ones, reveal the accuracy, efficiency and robustness of the proposed global methodology. [ABSTRACT FROM AUTHOR]

Abstract This work aims to develop a computationally efficient multiscale model to accurately simulate and analyze the instability phenomena of long fiber reinforced composites. The multilevel finite element method (FE2) based on the homogenization theory is adopted to realize the real-time interaction between the microscopic and the macroscopic levels. To increase the computational efficiency in the microscopic level and control the nonlinear calculations, the Fourier-related analysis, is carried out on the representative volume elements (RVEs), where all fast varying unknowns are replaced by slowly varying unknowns. The established nonlinear multiscale system is solved by the Asymptotic Numerical Method (ANM), which is more reliable and less time consuming than other classical iterative methods. This model is used to study the structural instability and the fiber micro-buckling of long fiber reinforced composites. The effects of fiber wavelength on the buckling and post-buckling of macro structure are discussed. [ABSTRACT FROM AUTHOR]

Abstract As a soft material with an almost negligible bending stiffness, a membrane may easily lose its mechanical stability. To capture its entire instability process, intensive computation is required, especially in the case of short wave length. The objective of this paper is to construct an efficient model to simulate and study the instability phenomena of circular membranes. By using the method of Fourier series with slowly variable coefficients in the circumferential direction, a new family of one-dimensional reduced finite elements are developed to study the three-dimensional problems. The nonlinear system is solved by the Asymptotic Numerical Method(ANM), which is reliable and efficient for tracing the bifurcation points and post-buckling equilibrium path compared with other classical non-linear solution algorithms. The accuracy and efficiency of the reduced model is verified by simulating the instability phenomena in stretched annular membranes and a compressed circular plate. The relation between critical loads and bifurcation patterns and the evolution of stress components in the entire wrinkling process are discussed. This study provides new simulation schemes to explore the instability in circular membranes under complex loadings and boundary conditions. [ABSTRACT FROM AUTHOR]

Abstract In the present work, we propose a numerical analysis of instability and bifurcations for geometrically nonlinear elasticity problems. These latter are solved by using the Asymptotic Numerical Method (ANM) associated with the Method of Fundamental Solutions (MFS). To compute bifurcation points and to determine the critical loads, we propose three techniques. The first one is based on a geometrical indicator obtained by analyzing the Taylor series. The second one exploits the properties of the Padé approximants, and the last technique uses an analytical bifurcation indicator. Numerical examples are studied to show the efficiency and the reliability of the proposed algorithms. [ABSTRACT FROM AUTHOR]

Abstract In this paper, the vibration behaviour and damping performances of Flax Fiber Reinforced Epoxy (FFRE) are investigated. Static tests using FFRE composite beams are carried out leading to the identification of elastic properties of each layer. Then, three FFRE composite plates are elaborated and used in experimental vibration tests to identify their eigenfrequencies and their modal damping. In numerical part, a constant complex representation of the stiffness is assumed and the loss factors are supposed constants and identified from the first experimental vibration mode. The homogenization technique and the finite element method are applied to describe their damped dynamic behaviour. The resolution of the resulting non-linear problem is carried out using the Asymptotic Numerical Method (ANM). Experimental results show that the modal damping is greater when the flax fibers are oriented at 90 °. Comparing numerical and experimental results, the proposed finite element modelling enables to estimate the damped eigenfrequencies with high accuracy. However, the predicted modal loss factors are over-estimated compared to experimental ones. It is concluded that the present modelling should be improved considering the frequency dependence of damping. [ABSTRACT FROM AUTHOR]

This paper addresses the numerical computation of periodic solutions of nonlinear differential systems involving fractional derivatives. For this purpose, the Harmonic Balance Method and the Asymptotic Numerical Method are combined, generalizing an approach largely followed in non-fractional systems. This enables to perform the continuation of periodic solutions of fractional systems with respect to a system parameter or to the fractional order. In the particular case of a constant fractional order, the results are validated by a successful comparison with an alternative formulation based on the diffusive representation of fractional operators. The new numerical strategy presented here allows to simulate phenomena still lacking from theoretical foundations. For example, the numerical experiments proposed here lead to a bifurcation similar to the Hopf bifurcation, well known in the case of non-fractional systems. Throughout this article, the Weyl derivative is used; its link with the classic Caputo derivative is elucidated. [ABSTRACT FROM AUTHOR]

In strip rolling flatness of the final product is controlled inline with a roll sensor placed at the outgoing of the tandem mill to measure the longitudinal (in the rolling direction) internal stress profile throughout the coil. This profile involves resultant force and sometimes resultant moment (due to preset anomaly for example) equilibrated by the coiler. When strip is relaxed these two quantities are eliminated and there remain residual stresses in the strip. Since negative residual stresses are the main origin of defects, operators subtract the resultant force from measurements to visualize the residual stress profile in real time and detect negative values, without considering the resultant moment. In this paper we show that to predict flatness defects correctly we have to subtract resultant moment too. Otherwise, unpredictable defects manifest in the delivered product. [ABSTRACT FROM AUTHOR]

The design of high-damping composite viscoelastic sandwich structures (CVSS) requires continuous study of damping properties according to some parameters. The common design parameters that directly affect the damping properties of CVSS are the fibers'orientation and the thicknesses of the composite layers. However, until nowthere has been no method for continuously studying the influences of any design parameter on the damping properties of CVSS. Also, to study the influence of a design parameter on the damping properties of CVSS, computations are made for various discrete values of this parameter. These calculations are qualified incrementally, and are very expensive in computation time and do not allowto followcontinuously the effects of the parameter on the damping properties of CVSS. In order to tackle these problems, we propose in this article a numerical method for studying the damping properties variation of CVSS according to a chosen modeling parameter, through the resolution of a generic, residual, complex nonlinear eigenvalues problems having frequency dependence and a modeling parameter that describes a study interval. This method is based on the asymptotic numericalmethod, automatic differentiation, homotopy technique and continuation. An application is performed to study the variation of the damping properties of a threelayered symmetric viscoelastic sandwich plates according to the fibers orientation of their orthotropic faces layers. [ABSTRACT FROM AUTHOR]

Data-driven computational homogenization has been proposed recently for the analyses of composite structures. Its basic idea is to construct an equivalent stress–strain database of composites via offline homogenization on the representative volume element and conduct online macroscopic simulation through distance-minimizing data-driven computing. Thanks to the scale separation of concurrent multiscale systems, this framework allows for improving online computational efficiency. However, high-density database construction in the offline stage remains a burdensome and time-consuming task. To this end, this work proposed an efficient approach that associates computational homogenization with the Asymptotic Numerical Method (ANM) to construct a high-density database. Being a reliable and efficient perturbation technique, the ANM allows for accurate tracking of the displacement–load paths and easily generates abundant equivalent stress–strain data on the paths. A fiber reinforced composite material with fiber buckling has been considered to demonstrate the accuracy and efficiency of the proposed method for the database construction of composites. [ABSTRACT FROM AUTHOR]

In this paper, we propose an efficient and accurate approach to investigate the post-buckling behavior of sandwich structures. In this framework, a novel one-dimensional layer-wise model using Euler-Bernoulli beam theory in the skins and higher-order kinematics in the core is proposed. The resulting nonlinear governing equations are then solved by the Asymptotic Numerical Method (ANM) with a bifurcation indicator, which is more reliable and efficient than the classical iterative methods, e.g., Newton-Raphson method, in terms of detecting critical points and computing bifurcated branches. Several numerical tests, i.e., global buckling, local wrinkling and global-local-coupling instability phenomena of sandwich beams, are performed and the results show that the proposed approach is able to efficiently and precisely characterize the critical loads and the post-buckling behaviors of sandwich structures. Finally, the effect of three aspects, i.e., kinematics, strain-displacement relationships and interpolation functions on the computational accuracy of predicting these instability phenomena are investigated. [ABSTRACT FROM AUTHOR]

Highlights • A new numerical method is proposed for bifurcation analysis. • Taylor series with respect to loading are combined with Taylor series with respect to space variables. • The adaptive step length allows to compute post-bifurcation paths with very small perturbations. Abstract A new numerical technique for post-buckling analysis is presented by combining the asymptotic numerical method (ANM) and the Taylor meshless method (TMM). These two methods are based on Taylor series, with respect to a scalar load parameter for ANM, and with respect to the space variables for TMM. The advantage of ANM is an adaptive step length, which is very efficient near bifurcation points. The specificity of TMM is a quasi-exact solution of the partial differential equations inside the domain, which leads to a strong reduction of the number of degrees of freedom. [ABSTRACT FROM AUTHOR]

We present an efficient numerical scheme based on asymptotic numerical method for continuation of spatial equilibria of special Cosserat rods. Using quaternions to represent rotation, the equations of static equilibria of special Cosserat rods are posed as a system of thirteen first order ordinary differential equations having cubic nonlinearity. The derivatives in these equations are further discretized to yield a system of cubic polynomial equations. As asymptotic-numerical methods are typically applied to polynomial systems having quadratic nonlinearity, a modified version of this method is presented in order to apply it directly to our cubic nonlinear system. We then use our method for continuation of equilibria of the follower load problem and demonstrate our method to be highly efficient when compared to conventional solvers based on the finite element method. Finally, we demonstrate how our method can be used for computing the buckling load as well as for continuation of postbuckled equilibria of hemitropic rods. [ABSTRACT FROM AUTHOR]

In this paper, a method for the quasi-static simulation of flexible cables assembly in the context of automotive industry is presented. The cables geometry and behavior encourage to employ a geometrically exact beam model. The 3D kinematics is then based on the position of the centerline and on the orientation of the cross-sections, which is here represented by rotational quaternions. Their algebraic nature leads to a polynomial form of equilibrium equations. The continuous equations obtained are then discretized by the finite element method and easily recast under quadratic form by introducing additional slave variables. The asymptotic numerical method, a powerful solver for systems of quadratic equations, is then employed for the continuation of the branches of solution. The originality of this paper stands in the combination of all these methods which leads to a fast and accurate tool for the assembly process of cables. This is proved by running several classical validation tests and an industry-like example. [ABSTRACT FROM AUTHOR]

In this paper, we present a numerical analysis of the buckling of elastic thin structures by using Asymptotic Numerical Method (ANM) and Padé approximants. We consider the case of cylindrical shells submitted to external pressure and we show how the perturbation technique can be applied for conservative and non-conservative loadings. A shell formulation well adapted to thin structures and which uses the Enhanced Assumed Strain (EAS) concept and a three-dimensional constitutive law without condensation is considered in this study. Numerical examples are presented with comparison between conservative and non-conservative loading in term of critical pressure. • Numerical analysis of the buckling of elastic thin structure submitted to external pressure. • Asymptotic Numerical Method, and Padé approximants are successfully used. • The external pressure is considered as a non-conservative, follower loading on the thin structure. • Büchter-Ramm finite elements and EAS assumption have been used. [ABSTRACT FROM AUTHOR]