The Contact Dynamics method: A nonsmooth story

Frédéric Dubois; Vincent Acary; Michel Jean

doi:10.1016/j.crme.2017.12.009

The legacy of Jean-Jacques Moreau in mechanics

The Contact Dynamics method: A nonsmooth story

Frédéric Dubois ^1,² ; Vincent Acary ³; Michel Jean ⁴

¹ LMGC, Univ. Montpellier, CNRS, Montpellier, France
² MIST, Univ. Montpellier, CNRS, IRSN, Montpellier, France
³ LJK, INRIA, Université de Grenoble Alpes, Grenoble, France
⁴ Aix-Marseille Université, CNRS, Centrale Marseille, Marseille, France

Comptes Rendus. Mécanique, Volume 346 (2018) no. 3, pp. 247-262.

Abstract
Résumé

When velocity jumps are occurring, the dynamics is said to be nonsmooth. For instance, in collections of contacting rigid bodies, jumps are caused by shocks and dry friction. Without compliance at the interface, contact laws are not only non-differentiable in the usual sense but also multi-valued. Modeling contacting bodies is of interest in order to understand the behavior of numerous mechanical systems such as flexible multi-body systems, granular materials or masonry. These granular materials behave puzzlingly either like a solid or a fluid and a description in the frame of classical continuous mechanics would be welcome though far to be satisfactory nowadays. Jean-Jacques Moreau greatly contributed to convex analysis, functions of bounded variations, differential measure theory, sweeping process theory, definitive mathematical tools to deal with nonsmooth dynamics. He converted all these underlying theoretical ideas into an original nonsmooth implicit numerical method called Contact Dynamics (CD); a robust and efficient method to simulate large collections of bodies with frictional contacts and impacts. The CD method offers a very interesting complementary alternative to the family of smoothed explicit numerical methods, often called Distinct Elements Method (DEM). In this paper developments and improvements of the CD method are presented together with a critical comparative review of advantages and drawbacks of both approaches.

Lorsque des sauts de vitesse se produisent, la dynamique est dite non régulière. Par exemple, dans les collections de solides supposés rigides rentrant en contact, les sauts sont causés par les chocs et le frottement sec. L'absence de déformabilité fait que les lois de contact sont, non seulement non différentiables au sens usuel, mais aussi multi-valuées. Élaborer des modèles de solides en contact est un moyen de comprendre le comportement de nombreux systèmes mécaniques tels que les systèmes multi-corps flexibles, les matériaux granulaires ou les maçonneries. Les matériaux granulaires se comportent de manière étrange, soit comme des solides, soit comme des fluides, et une description dans le cadre de la mécanique classique des milieux continus, qui serait souhaitable, est loin d'être encore satisfaisante. Jean-Jacques Moreau a contribué, de façon fondamentale, à l'analyse convexe, à la théorie des fonctions à variations bornées et des mesures différentielles ainsi qu'au processus de rafle, outils mathématiques décisifs pour traiter la dynamique non régulière. Il a converti ces idées théoriques sous-jacentes en une méthode numérique originale appelée Contact Dynamics (CD), qui est une méthode non régulière implicite et aussi une méthode robuste et efficace pour simuler de larges collections de solides avec du contact frottant et des impacts. Le méthode CD offre une alternative très intéressante à la famille de méthodes usuelles régularisées explicites, comme la méthode des éléments distincts (DEM). Dans cet article, des développements et des perfectionnements de la méthode CD sont présentés ainsi qu'une étude critique comparative des avantages et inconvénients des deux approches.

Received: 2017-05-20
Accepted: 2017-10-11
Published online: 2017-12-21

DOI: 10.1016/j.crme.2017.12.009

Keywords: Nonsmooth dynamics, Shock, Coulomb law, Contact Dynamics, Discrete element method
Mot clés : Dynamique non régulière, Chocs, Loi de Coulomb, Dynamique des contacts, Méthode par éléments discrets

Author's affiliations:

Frédéric Dubois ^{1, 2}; Vincent Acary ³; Michel Jean ⁴

¹ LMGC, Univ. Montpellier, CNRS, Montpellier, France
² MIST, Univ. Montpellier, CNRS, IRSN, Montpellier, France
³ LJK, INRIA, Université de Grenoble Alpes, Grenoble, France
⁴ Aix-Marseille Université, CNRS, Centrale Marseille, Marseille, France

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Frédéric Dubois; Vincent Acary; Michel Jean. The Contact Dynamics method: A nonsmooth story. Comptes Rendus. Mécanique, Volume 346 (2018) no. 3, pp. 247-262. doi : 10.1016/j.crme.2017.12.009. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2017.12.009/

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