Asymptotic Numerical Method Research Articles

A Fourier-related FE2 multiscale model for instability phenomena of long fiber reinforced materials

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.

Three-dimensional numerical simulation of material mixing observed in FSW using a mesh-free approach

This paper presents a three-dimensional numerical simulation of material mixing observed in the friction stir welding process using a mesh-free approach. This mesh-free approach is based on the coupling of the moving least squares and the techniques of the asymptotic numerical method which is applied on a strong formulation to avoid numerical integration. The strong formulation is obtained from equations of the mass conservation and of the moment in the three-dimensional case. Numerical examples are given to illustrate that this mesh-free approach is accurate, effective and feasible, especially in the material mixing problem. Comparisons with the classical iterative solver are performed.

A modeling and resolution framework for wrinkling in hyperelastic sheets at finite membrane strain

Wrinkles commonly occur in uniaxially stretched rectangular hyperelastic membranes with clamped-clamped boundaries, and can vanish upon excess stretching. Here we develop a modeling and resolution framework to solve this complex instability problem with highly geometric and material nonlinearities. We extend the nonlinear Föppl-von Kármán thin plate model to finite membrane strain regime for various compressible and incompressible hyperelastic materials. Under plane stress condition, 2D hyperelastic constitutive models can be systematically deduced based on general 3D strain energy potentials, e.g., Saint-Venant Kirchhoff, neo-Hookean, Mooney-Rivlin, Gent model and Gent-Gent model. Moreover, we establish a novel and efficient numerical resolution framework combining a path-following continuation technique by Asymptotic Numerical Method (ANM) and a discretization by a spectral method. The main advantages of this framework include the generality for both compressible and incompressible materials, ease of programming, high precision and efficient continuation predictor. Based on the proposed approach, effect of different incompressible constitutive models on the post-buckling response is investigated, which shows that restabilization points and wrinkling amplitudes are quantitatively influenced. However, for compressible materials, Poisson’s ratio plays a critical role in the wrinkling and restabilization behavior. We find that smaller Poisson’s ratio makes later onset of wrinkling, lower amplitude and earlier disappearance of wrinkles. Besides, severe strain-stiffening phenomena are explored by accounting for phenomenological models such as Gent model and Gent-Gent model. Efficiency and accuracy of the proposed modeling and resolution framework were examined by comparing with some benchmarks.

Investigations of the Effects of Geometric Imperfections on the Nonlinear Static and Dynamic Behavior of Capacitive Micomachined Ultrasonic Transducers.

In order to investigate the effects of geometric imperfections on the static and dynamic behavior of capacitive micomachined ultrasonic transducers (CMUTs), the governing equations of motion of a circular microplate with initial defection have been derived using the von Kármán plate theory while taking into account the mechanical and electrostatic nonlinearities. The partial differential equations are discretized using the differential quadrature method (DQM) and the resulting coupled nonlinear ordinary differential equations (ODEs) are solved using the harmonic balance method (HBM) coupled with the asymptotic numerical method (ANM). It is shown that the initial deflection has an impact on the static behavior of the CMUT by increasing its pull-in voltage up to 45%. Moreover, the dynamic behavior is affected by the initial deflection, enabling an increase in the resonance frequencies and the bistability domain and leading to a change of the frequency response from softening to hardening. This model allows MEMS designers to predict the nonlinear behavior of imperfect CMUT and tune its bifurcation topology in order to enhance its performances in terms of bandwidth and generated acoustic power while driving the microplate up to 80% beyond its critical amplitude.

Nonlinear dynamics of micromechanical resonator arrays for mass sensing

This paper investigates the mass sensing capability of an array of a few identical electrostatically actuated microbeams, as a first step toward the implementation of arrays of thousands of such resonant sensors. A reduced-order model is considered, and Taylor series are used to simplify the nonlinear electrostatic force. Then, the harmonic balance method associated with the asymptotic numerical method, as well as time integration or averaging methods, is applied to this model, and its results are compared. In this paper, two- and three-beam arrays are studied. The predicted responses exhibit complex branches of solutions with additional loops due to the influence of adjacent beams. Moreover, depending on the applied voltages, the solutions with and without added mass exhibit large differences in amplitude which can be used for detection. For symmetric configurations, the symmetry breaking induced by an added mass is exploited to improve mass sensing.

An asymptotic numerical method to solve compliant Lennard-Jones-based contact problems involving adhesive instabilities

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.

Experimental and numerical investigation of the damping of flax–epoxy composite plates

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.

The dynamics of disappearing pulses in a singularly perturbed reaction–diffusion system with parameters that vary in time and space

We consider the evolution of multi-pulse patterns in an extended Klausmeier equation with parameters that change in time and/or space. We formally show that the full PDE dynamics of a N-pulse configuration can be reduced to a N-dimensional dynamical system describing the dynamics on a N-dimensional manifold MN. Next, we determine the local stability of MN via the quasi-steady spectrum associated to evolving N-pulse patterns, which provides explicit information on the boundary ∂MN. Following the dynamics on MN, a N-pulse pattern may move through ∂MN and ‘fall off’ MN. A direct nonlinear extrapolation of our linear analysis predicts the subsequent fast PDE dynamics as the pattern ‘jumps’ to another invariant manifold MM, and specifically predicts the number N−M of pulses that disappear. Combining the asymptotic analysis with numerical simulations of the dynamics on the various invariant manifolds yields a hybrid asymptotic–numerical method describing the full process that starts with a N-pulse pattern and typically ends in the trivial homogeneous state without pulses. We extensively test this method against PDE simulations and deduce general conjectures on the nature of pulse interactions with disappearing pulses. We especially consider the differences between the evolution of irregular and regular patterns. In the former case, the disappearing process is gradual: irregular patterns lose their pulses one by one. In contrast, regular, spatially periodic, patterns undergo catastrophic transitions in which either half or all pulses disappear. However, making a precise distinction between these two drastically different processes is quite subtle, since irregular N-pulse patterns that do not cross ∂MN typically evolve towards regularity. hybrid asymptotic-numerical method

Continuation of periodic solutions for systems with fractional derivatives

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.

A Fourier based reduced model for wrinkling analysis of circular membranes

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.

An efficient asymptotic-numerical method to solve nonlinear systems of one-dimensional balance laws

An asymptotic-numerical method to solve the initial-boundary value problems for systems of balance laws in one space dimension, on the half space is developed. Expansions in powers of $$t^{-1/2}$$ are used, in view of the precise asymptotic behavior recently established on theoretical bases. This approach increases considerably the efficiency of a previous one, where just expansions in inverse powers of t were made. Numerical examples and comparisons with the Godunov, the asymptotic high order, and the asymptotic-numerical method earlier developed are presented. Expanding the solution in powers of $$t^{-1/2}$$ instead of $$t^{-1}$$ , a saving of about one-half of the CPU time can be realized, still achieving the same accuracy.

An efficient approach to investigate the post-buckling behaviors of sandwich structures

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.

Optimal Postbuckling Design of Variable Angle Tow Composite Plates

Perturbation-based approximation methods are widely used in preliminary design studies of thin-walled structures. In this paper, postbuckling analysis of a variable-angle-tow composite plate is performed using the perturbation-based asymptotic numerical method, which transforms the nonlinear problem into a set of well-posed recursive linear problems. These linear problems are solved using a novel generalized differential-integral quadrature method, and the postbuckling solutions are sought over a finite load step size around the critical buckling point using asymptotic expansions. The accuracy of the asymptotic numerical method in evaluating the initial postbuckling of variable-angle-tow plates under compression is investigated. Subsequently, a novel postbuckling optimization approach based on asymptotic numerical method results is proposed for the design of variable-angle-tow laminates. The postbuckling features obtained from asymptotic numerical method are used in an efficient two-level optimization framework for the design of variable-angle-tow plates. At the first level, a globally convergent method of moving asymptotes is adopted to determine the optimal lamination parameter distributions that maximize the postbuckling performance of the variable-angle-tow plate. At the second level, a genetic algorithm is used to convert the optimal lamination parameter distributions into realistic variable-angle-tow layups. The optimization studies are performed for square variable-angle-tow plates for axial/biaxial compression under different in-plane boundary conditions.

An asymptotic numerical method for continuation of spatial equilibria of special Cosserat rods

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.

Design of a quasi-periodic vibration energy harvester based on an electromagnetic technique

In the present paper, a quasi-periodic vibration energy harvester with magnetic coupling is proposed using the benefits of the energy localization. The proposed quasi-periodic system consists of moving magnets held by elastic structures and coupled by a magnetic force. The mistuning of the device can be achieved by changing either the linear mechanical stiffness or the mass. The whole system has been modelled by forced Duffing equations for each degree of freedom, which include the magnetic nonlinearity and the mechanical damping. The governing equations have then been solved using the harmonic balance method coupled with the asymptotic numerical method. The obtained numerical results show that the total harvested power was increased by 11 % with a bandwidth of 2.7 % thanks to the energy localization phenomenon compared to the reference case.

An implicit algorithm for the dynamic study of nonlinear vibration of spur gear system with backlash

In this work, we propose some regularization techniques to adapt the implicit high order algorithm based on the coupling of the asymptotic numerical methods (ANM) (Cochelin et al., Méthode Asymptotique Numérique, Hermès-Lavoisier, Paris, 2007; Mottaqui et al., Comput. Methods Appl. Mech. Eng. 199 (2010) 1701–1709; Mottaqui et al., Math. Model. Nat. Phenom. 5 (2010) 16–22) and the implicit Newmark scheme for solving the non-linear problem of dynamic model of a two-stage spur gear system with backlash. The regularization technique is used to overcome the numerical difficulties of singularities existing in the considered problem as in the contact problems (Abichou et al., Comput. Methods Appl. Mech. Eng. 191 (2002) 5795–5810; Aggoune et al., J. Comput. Appl. Math. 168 (2004) 1–9). This algorithm combines a time discretization technique, a homotopy method, Taylor series expansions technique and a continuation method. The performance and effectiveness of this algorithm will be illustrated on two examples of one-stage and two-stage gears with spur teeth. The obtained results are compared with those obtained by the Newton–Raphson method coupled with the implicit Newmark scheme.

Asymptotic Analysis and Numerical Method for Singularly Perturbed Eigenvalue Problems

In this paper, we study the asymptotic analysis and numerical method for singularly perturbed eigenvalue problems (SPEPs). We first make a close asymptotic analysis of the SPEP and prove that, for ...

A numerical method based on Taylor series for bifurcation analyses within Föppl–von Karman plate theory

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.

A finite element/quaternion/asymptotic numerical method for the 3D simulation of flexible cables

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.

On the use of Padé approximant in the Asymptotic Numerical Method ANM to compute the post-buckling of shells

In this paper, we present a new ANM continuation algorithm with a predictor based on a new Padé approximant and without the use of a correction process. The ANM is a numerical method to obtain the solution of a nonlinear problem as a succession of branches [1–7]. Each branch is represented by a vectorial series which is obtained by inverting only one tangent stiffness matrix. The series representation can be replaced by a rational representation which reduces the number of branches necessary to obtain the entire branch of desired solution. In this work, we discuss the use of a new vectorial Padé approximants in the ANM algorithm. In previous works [1, 2, 4, 5], the Padé approximants have been introduced after an orthonormalization of the terms of the vectorial series. In a recent article [8], we have demonstrated that the coefficients biM of the Padé approximants can be chosen in an arbitrary manner. For this purpose, we propose a new choice of vectorial Padé approximants {UpM} at order M which minimizes the relative error between two consecutive vectorial Padé approximants {UpM} at order M and {UpM−1} at order M − 1. This minimization has been made by a judicious choice of the coefficients biM of the Padé {UpM} at order M as a function of coefficients biM−1 of the Padé approximants {UpM−1} at order M − 1 obtained by the classical process of Gram-Schmidt orthonormalization. A comparison of the obtained results with those computed by the use of classical Padé approximants is presented.