Working group Nonlinear Waves in Discrete Mechanical Systems


    Inria Grenoble Rhône-Alpes, BIPOP Project Team

    Laboratoire Jean Kuntzmann (UMR CNRS 5224)


Vincent Acary (researcher, Inria), Bernard Brogliato (research director, Inria), Guillaume James (professor, Grenoble INP Ensimag, Inria, LJK), Arnaud Tonnelier (researcher, Inria), Jose Eduardo Morales Morales (PhD candidate, Inria), Franck Pérignon (research engineer, LJK, Inria), Ngoc-Son Nguyen (postdoc Inria 2010-13), Marguerite Gisclon (professor, Chambéry University, LJK, LAMA), Brigitte Bidégaray-Fesquet (researcher, CNRS, LJK)

External collaborators :

Jesus Cuevas (Nonlinear Physics, Sevilla Univ.), Panayotis Kevrekidis (Mathematics and Statistics, UMass), Anna Vainchtein and Lifeng Liu (Mathematics, Univ. of Pittsburgh), Yuli Starosvetsky (Mechanical Engineering, Technion IIT), Caishan Liu (State Key Lab. for Turbulence and Complex Systems, Peking University), Michael Herrmann (Mathematics and Computer Sciences, Münster University)

Overview :

Our group focuses on mathematical analysis, simulation and control of nonlinear wave phenomena in discrete mechanical systems involving many degrees of freedom. Typical examples include :

-impact propagation in granular metamaterials, leading to the formation of solitary waves

-vibrational energy localization (breathers) in oscillator chains

For physical systems with discrete spatial structures, naive continuum limits provide only a poor account of the dynamics, which is better captured by nonlinear lattice differential equations involving typically an infinite number of particles. For example, solitary waves and breathers in granular chains under precompression can be modeled using the celebrated Fermi-Pasta-Ulam lattice. In such spatially discrete models, the analysis of traveling wave solutions leads to challenging mathematical questions related to advance-delay differential equations. Classical phenomena induced by spatial discreteness include robust energy trapping in the form of structurally stable breathers, propagation failure for localized structures (fronts or pulses) and lattice-induced anisotropy.

Our research primarily focuses on two classes of nonlinear systems, either nonsmooth or strongly nonlinear :

-Nonsmooth discrete systems can be characterized by several types of nonsmooth interaction laws, leading to differential inclusions (as in the case of set-valued Coulomb friction law), differential equations with impulsive forces (rigid impacts), or differential equations with piecewise-smooth nonlinearities (Hertz contact law). Such nonlinear systems can be reformulated as complementarity systems.

-Strongly (or essentially) nonlinear discrete systems are characterized by interaction potentials with a vanishing second derivative at the ground state. Classical examples concern the Hertzian contact force between two touching beads (or more general smooth non-conforming surfaces) and geometric nonlinearities implemented with mechanical springs.

In such systems, classical linear and weakly nonlinear wave theory do not apply and unusual physical phenomena can show up. This is one of the essential ideas behind strongly nonlinear metamaterials, in which the main mode of energy propagation corresponds to (essentially) compactly supported solitary waves. The formation of waves localized on a few discrete elements opens up the possibility of controlling stress wave propagation much more efficiently than in classical linear (or weakly-nonlinear) systems dominated by dispersion. For example, experimental applications of granular metamaterials include the non-destructive testing of elastic media interfaced with granular chains, the use of two-dimensional granular crystals for localized wave redirection or the design of acoustic lenses.

Research results:

1 - Analysis of traveling waves in chains of touching beads, logarithmic KdV limit

[JP14] G. James and D. Pelinovsky, Gaussian solitary waves and compactons in Fermi-Pasta-Ulam lattices with Hertzian potentials, Proc. R. Soc. A. 470 (2165), 2014.

Summary : We consider a class of fully-nonlinear Fermi-Pasta-Ulam (FPU) lattices, consisting of a chain of particles coupled by fractional power nonlinearities of order alpha > 1. This class of systems incorporates a classical Hertzian model describing acoustic wave propagation in chains of touching beads in the absence of precompression. We analyze the propagation of localized waves when alpha is close to unity. Solutions varying slowly in space and time are searched with an appropriate scaling, and two asymptotic models of the chain of particles are derived consistently. The first one is a logarithmic KdV equation, and possesses linearly orbitally stable Gaussian solitary wave solutions. The second model consists of a generalized KdV equation with Hölder-continuous fractional power nonlinearity and admits compacton solutions, i.e. solitary waves with compact support. When alpha --> 1+, we numerically establish the asymptotically Gaussian shape of exact FPU solitary waves with near-sonic speed, and analytically check the pointwise convergence of compactons towards the limiting Gaussian profile.

Original publication available at Preprint available on arXiv: pdf file.

[J12] G. James, Periodic travelling waves and compactons in granular chains, J. Nonlinear Sci. 22 (2012), 813-848.

Summary : We study the propagation of an unusual type of periodic travelling waves in chains of identical beads interacting via Hertz's contact forces. Each bead periodically undergoes a compression phase followed by a free flight, due to special properties of Hertzian interactions (fully nonlinear under compression and vanishing in the absence of contact). We prove the existence of such waves close to binary oscillations, and numerically continue these solutions when their wavelength is increased. In the long wave limit, we observe their convergence towards shock profiles consisting of small compression regions close to solitary waves, alternating with large domains of free flight where bead velocities are small. We give formal arguments to justify this asymptotic behaviour, using a matching technique and previous results concerning solitary wave solutions. The numerical finding of such waves implies the existence of compactons, i.e. compactly supported compression waves propagating at a constant velocity, depending on the amplitude and width of the wave. The beads are stationary and separated by equal gaps outside the wave, and each bead reached by the wave is shifted by a finite distance during a finite time interval. Below a critical wavenumber, we observe fast instabilities of the periodic travelling waves leading to a disordered regime.

Original publication available at Preprint available on arXiv: pdf file.

2 - Newton's cradle and discrete p-Schrödinger equation :

[J11] G. James, Nonlinear waves in Newton's cradle and the discrete p-Schrödinger equation, Math. Models Meth. Appl. Sci. 21 (2011), 2335-2377.

Summary : We study nonlinear waves in Newton’s cradle, a classical mechanical system consisting of a chain of beads attached to linear pendula and interacting nonlinearly via Hertz’s contact forces. We formally derive a spatially discrete modulation equation, for small amplitude nonlinear waves consisting of slow modulations of time-periodic linear oscillations. The fully-nonlinear and unilateral interactions between beads yield a nonstandard modulation equation that we call the discrete p-Schrödinger (DpS) equation. It consists of a spatial discretization of a generalized Schrödinger equation with p-Laplacian, with fractional p > 2 depending on the exponent of Hertz’s contact force. We show that the DpS equation admits explicit periodic travelling wave solutions, and numerically find a plethora of standing wave solutions given by the orbits of a discrete map, in particular spatially localized breather solutions. Using a modified Lyapunov-Schmidt technique, we prove the existence of exact periodic travelling waves in the chain of beads, close to the small amplitude modulated waves given by the DpS equation. Using numerical simulations, we show that the DpS equation captures several other important features of the dynamics in the weakly nonlinear regime, namely modulational instabilities, the existence of static and travelling breathers, and repulsive or attractive interactions of these localized structures.

Original publication available at Preprint available on arXiv: pdf file

[JS14] G. James and Y. Starosvetsky, Breather solutions of the discrete p-Schrödinger equation, in Localized Excitations in Nonlinear Complex Systems, Eds. R. Carretero-Gonzalez, J. Cuevas-Maraver, D. Frantzeskakis, N. Karachalios, P. Kevrekidis, F. Palmero-Acebedo, Nonlinear Systems and Complexity 7 (2014), 77-115, Springer.

Summary : We consider the discrete p-Schrödinger (DpS) equation, which approximates small amplitude oscillations in chains of oscillators with fully-nonlinear nearest-neighbors interactions of order alpha = p-1 >1. Using a mapping approach, we prove the existence of breather solutions of the DpS equation with even- or odd-parity reflectional symmetries. We derive in addition analytical approximations for the breather profiles and the corresponding intersecting stable and unstable manifolds, valid on a whole range of nonlinearity orders alpha. In the limit of weak nonlinearity (alpha --> 1^+), we introduce a continuum limit connecting the stationary DpS and logarithmic nonlinear Schrödinger equations. In this limit, breathers correspond asymptotically to Gaussian homoclinic solutions. We numerically analyze the stability properties of breather solutions depending on their even- or odd-parity symmetry. A perturbation of an unstable breather generally results in a translational motion (traveling breather) when alpha is close to unity, whereas pinning becomes predominant for larger values of alpha.

Original publication available at Preprint available on arXiv: pdf file.

[BDJ13] B. Bidégaray-Fesquet, E. Dumas and G. James, From Newton's cradle to the discrete p-Schrödinger equation, SIAM J. Math. Anal. 45 (2013), 3404-3430.

Summary : We investigate the dynamics of a chain of oscillators coupled by fully-nonlinear interaction potentials. This class of models includes Newton's cradle with Hertzian contact interactions between neighbors. By means of multiple-scale analysis, we give a rigorous asymptotic description of small amplitude solutions over large times. The envelope equation leading to approximate solutions is a discrete p-Schrödinger equation formally derived in [J11]. We provide error bounds for the discrete p-Schrödinger approximation valid over large times. Using these error bounds and the breather existence theorem proved in [JS14] for the discrete p-Schrödinger equation, we show the existence of long-lived breather solutions to the original model. For a large class of localized initial conditions, we also estimate the maximal decay of small amplitude solutions over long times.

Original publication available at Preprint available on arXiv: pdf file.

3 - Vibrational energy localization in Newton's cradle : breathers, surface modes and boomerons

[JKC13] G. James, P.G. Kevrekidis and J. Cuevas, Breathers in oscillator chains with Hertzian interactions, Physica D 251 (2013), 39-59.

Summary : We prove nonexistence of breathers (spatially localized and time-periodic oscillations) for a class of Fermi-Pasta-Ulam lattices representing an uncompressed chain of beads interacting via Hertz's contact forces. We then consider the setting in which an additional on-site potential is present, motivated by the Newton's cradle under the effect of gravity. We show the existence of breathers in such systems, using both direct numerical computations and a simplified asymptotic model of the oscillator chain, the so-called discrete p-Schrödinger (DpS) equation. From a spectral analysis, we determine breather stability and explain their translational motion under very weak perturbations. Numerical simulations demonstrate the excitation of traveling breathers from simple initial conditions corresponding to small perturbations at the first site of the chain. This regime is well described by the DpS equation, and is found to occur for physical parameter values in granular chains with stiff local oscillators. In addition, traveling breather propagation can be hindered or even suppressed in other parameter regimes. For soft on-site potentials, a part of the energy remains trapped near the boundary and forms a surface mode. For hard on-site potentials and large to moderate initial excitations, one observes a ``boomeron'', i.e. a traveling breather displaying spontaneous direction-reversing motion. In addition, dispersion is significantly enhanced when a precompression is applied to the chain. Depending on parameters, this results either in the absence of traveling breather excitation on long time scales, or in the formation of a ``nanopteron'' characterized by a sizeable wave train lying at both sides of the localized excitation.

Original publication available at pdf file.

[JCK12] G. James, J. Cuevas and P.G. Kevrekidis, Breathers and surface modes in oscillator chains with Hertzian interactions, Proceedings of the 2012 International Symposium on Nonlinear Theory and its Applications (NOLTA 2012), Palma, Majorca, Spain, 22-26 Oct. 2012, p. 470-473.

Summary : We study localized waves in chains of oscillators coupled by Hertzian interactions and trapped in local potentials. This problem is originally motivated by Newton's cradle, a mechanical system consisting of a chain of touching beads subject to gravity and attached to inelastic strings. We consider an unusual setting with local oscillations and collisions acting on similar time scales, a situation corresponding e.g. to a modified Newton's cradle with beads mounted on stiff cantilevers. Such systems support static and traveling breathers with unusual properties, including double exponential spatial decay, almost vanishing Peierls-Nabarro barrier and spontaneous direction-reversing motion. We prove analytically the double exponential spatial decay of all nonresonant static breather solutions, and the existence of surface modes and static breathers for anharmonic on-site potentials and weak Hertzian interactions.

Preprint available on arXiv: pdf file.

4 - Simulation of dissipative impacts in granular media and more general multibody systems, LZB model

[LZB09] C. Liu, Z. Zhao, B. Brogliato, Frictionless multiple impacts in multibody systems: Part I. Theoretical framework, Proceedings of the Royal Society A (Mathematical, Physical and Engineering Sciences), vol.464, no 2100, pp.3193-3211, December 2008. Part II. Numerical algorithm and simulation results, Proceedings of the Royal Society A (Mathematical, Physical and Engineering Sciences), vol.465, no 2101, pp.1-23, January 2009.

Summary : A new method is proposed that can deal with multi-impact problems and produce energetically consistent and unique post-impact velocities. A distributing law related to the energy dispersion is discovered by mapping the time scale into the impulsive scale for bodies composed of rate-independent materials. It indicates that the evolution of the kinetic energy during the impacts is closely associated with the relative contact stiffness and the relative potential energy stored at the contact points. This distributing law is combined with the Darboux–Keller method of taking the normal impulse as an independent ‘time-like’ variable, which obeys a guideline for the selection of an independent normal impulse. Local energy losses are modelled with energetic coefficients of restitution at each contact point. Theoretical developments are presented in the first part in this paper. The second part is dedicated to numerical simulations where numerous and accurate results prove the validity of the approach.

Original publication available at : (part I), (part II). Preliminary version

[LZB08] C. Liu, Z. Zhao, B. Brogliato, Energy dissipation and dispersion effects in a granular media, Physical Review E, vol.78, no 3, 031307, September 2008.

Summary : The strong interactions between particles will make the energy within the granular materials propagate through the network of contacts and be partly dissipated. Establishing a model that can clearly classify the dissipation and dispersion effects is crucial for the understanding of the global behaviors in the granular materials. For particles with rate-independent material, the dissipation effects come from the local plastic deformation and can be constrained at the energy level by using energetic restitution coefficients. On the other hand, the dispersion effects should depend on the intrinsic nature of the interaction law between two particles. In terms of a bistiffness compliant contact model that obeys the energetical constraint defined by the energetic coefficients, our recent work [LZB09] related to the issue of multiple impacts indicates that the propagation of energy during collisions can be represented by a distributing law. In particular, this law shows that the dispersion effects are dominated by the relative contact stiffness and the relative potential energy stored at the contact points. In this paper, we will apply our theory to the investigation of the wave behavior in granular chain systems. The comparisons between our numerical results and the experimental ones by Falcon et al. Eur. Phys. J. B 5 111 (1998)] for a column of beads colliding against a wall show very good agreement and confirm some conclusions proposed by Falcon et al. Other numerical results associated with the case of several particles impacting a chain, and the collisions between two so-called solitary waves in a Hertzian type chain are also presented.

Original publication available at pdf file

[NB12] N.-S. Nguyen, Bernard Brogliato. Shock dynamics in granular chains: numerical simulations and comparison with experimental tests, Granular Matter, May 2012, Volume 14, Issue 3, pp 341-362.

Summary : The aim of this paper is to simulate the nonlinear wave propagation in granular chains of beads using a recently introduced multiple impact model (LZB model) and to compare numerical results to experimental ones. Different kinds of granular chains are investigated: monodisperse chains, tapered chains and stepped chains. Particular attention is paid to the dispersion effect, and the wave propagation in tapered chains, the interaction of two solitary waves in monodisperse chains, and the formation of solitary wave trains in stepped chains. We show that the main features of the wave propagation observed experimentally in these granular chains are very well reproduced. This proves that the considered multiple impact model and numerical scheme are able to encapsulate the main physical effects that occur in such multibody systems.

Original publication available at Preprint available on HAL.

[NB12] N.S. Nguyen, B. Brogliato, 2014. Multiple Impacts in Dissipative Granular Chains, Springer Verlag, Berlin, Lecture Notes in Applied and Computational Mechanics, vol.72; ISBN 978-3-642-39298-6.

Summary : The extension of collision models for single impacts between two bodies, to the case of multiple impacts (which take place when several collisions occur at the same time in a multibody system) is a challenge in Solid Mechanics, due to the complexity of such phenomena, even in the frictionless case. This monograph aims at presenting the main multiple collision rules proposed in the literature. Such collisions typically occur in granular materials, the simplest of which are made of chains of aligned balls. These chains are used throughout the book to analyze various multiple impact rules which extend the classical Newton (kinematic restitution), Poisson (kinetic restitution) and Darboux-Keller (energetic or kinetic restitution) approaches for impact modelling. The shock dynamics in various types of chains of aligned balls (monodisperse, tapered, decorated, stepped chains) is carefully studied and shown to depend on several parameters: restitution coefficients, contact stiffness ratios, elasticity coefficients (linear or nonlinear force/ indentation relation), and kinetic angles (that depend on the mass ratios). The dissipation and the dispersion of kinetic energy during a multiple impact are mandatory modelling, and are quantified with suitable indices. Particular attention is paid to the ability of the presented laws to correctly predict the wave effects in the chains. Comparisons between many numerical and experimental results are shown, as well as comparisons between four different impact laws in terms of their respective abilities to correctly model dissipation and dispersion of energy.

Monography available at


Modeling and control of waves in strongly nonlinear discrete media.
ITN proposal submitted to H2020 program (2014).

Ondes non linéaires dans les réseaux granulaires et systčmes mécaniques spatialement discrets (2013-14).
IXXI project (Rhône-Alpes Institute for Complex Systems) accepted in November 2012.
This project concerns the study of nonsmooth mechanical systems with a particular focus on granular chains, nonlinear waves and nonlinear modes.

Modelling and Simulation of Multiple Impacts in Multibody Systems (2008-11).
Project funded by the French National Research Agency (ANR) and National Natural Science Foundation of China (NSFC).
Summary and Project description.

Some lecture slides:

B. Brogliato, Multiple impacts and Painlevé paradox, Aussois summer school (Sept 2012). Slides

G. James, Nonlinear Waves in Granular Chains, GdR 3437 DYNOLIN, ENSAM Lille (Oct 2013). Slides

Forthcoming talks:

V. Acary, G. James and F. Pérignon, Periodic motions of coupled impact oscillators, minisymposium Non-Smooth Dynamics, 8th European Nonlinear Dynamics Conference (ENOC), Vienna, July 6-11, 2014.

B. Bidégaray-Fesquet, E. Dumas and G. James, An asymptotic model for small amplitude solutions to Newton's cradle, SIAM Conference on Nonlinear Waves and Coherent Structures, Cambridge, UK, August 11-14, 2014.

G. James, Localized waves in fully nonlinear media, SIAM Conference on Nonlinear Waves and Coherent Structures, Cambridge, UK, August 11-14, 2014.