Section: Research Program
Axis 1: Modeling and analysis
This axis is dedicated to the modeling and the mathematical analysis of nonsmooth dynamical systems. It consists of four main directions. Two directions are in the continuation of BIPOP activities: 1) multibody vibroimpact systems (Sect. 3.2.1) and 2) excitable systems (Sect. 3.2.2). Two directions are completely new with respect to BIPOP: 3) Nonsmooth geomechanics and natural hazards assessment (Sect. 3.2.3) and 4) Cyberphysical systems (hybrid systems) (Sect. 3.2.4).
Multibody vibroimpact systems
Participants: B. Brogliato, F. Bourrier, G. James, V. Acary

Multiple impacts with or without friction : there are many different approaches to model collisions, especially simultaneous impacts (socalled multiple impacts) [84]. One of our objectives is on one hand to determine the range of application of the models (for instance, when can one use “simplified” rigid contact models relying on kinematic, kinetic or energetic coefficients of restitution?) on typical benchmark examples (chains of aligned beads, rocking block systems). On the other hand, try to take advantage of the new results on nonlinear waves phenomena, to better understand multiple impacts in 2D and 3D granular systems. The study of multiple impacts with (unilateral) nonlinear viscoelastic models (SimonHuntCrossley, KuwabaraKono), or viscoelastoplastic models (assemblies of springs, dashpots and dry friction elements), is also a topic of interest, since these models are widely used.

Artificial or manufactured or ordered granular crystals, metamaterials : Granular metamaterials (or more general nonlinear mechanical metamaterials) offer many perspectives for the passive control of waves originating from impacts or vibrations. The analysis of waves in such systems is delicate due to spatial discreteness, nonlinearity and nonsmoothness of contact laws [86], [72], [73], [79]. We will use a variety of approaches, both theoretical (e.g. bifurcation theory, modulation equations) and numerical, in order to describe nonlinear waves in such systems, with special emphasis on energy localization phenomena (excitation of solitary waves, fronts, breathers).

Systems with clearances, modeling of friction : joint clearances in kinematic chains deserve specific analysis, especially concerning friction modeling [40]. Indeed contacts in joints are often conformal, which involve large contact surfaces between bodies. Lubrication models should also be investigated.

Painlevé paradoxes : the goal is to extend the results in [65], which deal with singlecontact systems, to multicontact systems. One central difficulty here is the understanding and the analysis of singularities that may occur in sliding regimes of motion.
As a continuation of the work in the BIPOP team, our software code, Siconos (see Sect. 5.1) will be our favorite software platform for the integration of these new modeling results.
Excitable systems
Participants: A. Tonnelier, G. James
An excitable system elicits a strong response when the applied perturbation is greater than a threshold [81], [82], [45], [91]. This property has been clearly identified in numerous natural and physical systems. In mechanical systems, nonmonotonic friction law (of spinodaltype) leads to excitability. Similar behavior may be found in electrical systems such as active compounds of neuristor type. Models of excitable systems incorporate strong nonlinearities that can be captured by nonsmooth dynamical systems. Two properties are deeply associated with excitable systems: oscillations and propagation of nonlinear waves (autowaves in coupled excitable systems). We aim at understanding these two dynamical states in excitable systems through theoretical analysis and numerical simulations. Specifically we plan to study:

Thresholdlike models in biology: spiking neurons, gene networks.

Frictional contact oscillators (slider block, BurridgeKnopoff model).

Dynamics of active electrical devices : memristors, neuristors.
Nonsmooth geomechanics and natural hazards assessment
Participants: F. Bourrier, B. Brogliato, G. James, V. Acary

Rockfall impact modeling : Trajectory analysis of falling rocks during rockfall events is limited by a rough modeling of the impact phase [47], [46], [77]. The goal of this work is to better understand the link between local impact laws at contact with refined geometries and the efficient impact laws written for a point mass with a full reset map. A continuum of models in terms of accuracy and complexity will be also developed for the trajectory studies. In particular, nonsmooth models of rolling friction, or rolling resistance will be developed and formulated using optimization problems.

Experimental validation : The participation of IRSTEA with F. Bourrier makes possible the experimental validation of models and simulations through comparisons with real data. IRSTEA has a large experience of lab and insitu experiments for rockfall trajectories modeling [47], [46]. It is a unique opportunity to vstrengthen our model and to prove that nonsmooth modeling of impacts is reliable for such experiments and forecast of natural hazards.

Rock fracturing : When a rock falls from a steep cliff, it stores a large amount of kinetic energy that is partly dissipated though the impact with the ground. If the ground is composed of rocks and the kinetic energy is sufficiently high, the probability of the fracture of the rock is high and yields an extra amount of dissipated energy but also an increase of the number of blocks that fall. In this item, we want to use the capability of the nonsmooth dynamical framework for modeling cohesion and fracture [74], [38] to propose new impact models.

Rock/forest interaction : To prevent damages and incidents to infrastructures, a smart use of the forest is one of the ways to control trajectories (decrease of the runout distance, jump heights and the energy) of the rocks that fall under gravity [58], [59]. From the modeling point of view and to be able to improve the protective function of the forest, an accurate modeling of impacts between rocks and trees is required. Due to the aspect ratio of the trees, they must be considered as flexible bodies that may be damaged by the impact. This new aspect offers interesting modeling research perspectives.
More generally, our collaboration with IRSTEA opens new long term perspectives on granular flows applications such as debris and mud flows, granular avalanches and the design of structural protections. The numerical methods that go with these new modeling approaches will be implemented in our software code, Siconos (see Sect. 5.1)
Cyberphysical systems (hybrid systems)
Participants: V. Acary, B. Brogliato, C. Prieur, A. Tonnelier
Nonsmooth systems have a nonempty intersection with hybrid systems and cyber–physical systems. However, nonsmooth systems enjoy strong mathematical properties (concept of solutions, existence and uniqueness) and efficient numerical tools. This is often the result of the fact that nonsmooth dynamical systems are models of physical systems, and then, take advantage of their intrinsic property (conservation or dissipation of energy, passivity, stability). A standard example is a circuit with $n$ ideal diodes. From the hybrid point of view, this circuit is a piecewise smooth dynamical system with ${2}^{n}$ modes, that can be quite cumbersome to enumerate in order to determinate the current mode. As a nonsmooth system, this circuit can be formulated as a complementarity system for which there exist efficient time–stepping schemes and polynomial time algorithms for the computation of the current mode. The key idea of this research action is to take benefit of this observation to improve the hybrid system modeling tools.
Research actions: There are two main actions in this research direction that will be implemented in the framework of the Inria Project Lab (IPL “ Modeliscale”, see https://team.inria.fr/modeliscale/ for partners and details of the research program):
$\phantom{\rule{1.em}{0ex}}\u2022$Structural analysis of multimode DAE : When a hybrid system is described by a Differential Algebraic Equation (DAE) with different differential indices in each continuous mode, the structural analysis has to be completely rethought. In particular, the reinitialization rule, when a switching occurs from a mode to another one, has to be consistently designed. We propose in this action to use our knowledge in complementarity and (distribution) differential inclusions [33] to design consistent reinitialization rule for systems with nonuniform relative degree vector $({r}_{1},{r}_{2},...,{r}_{m})$ and ${r}_{i}\ne {r}_{j},i\ne j$.
$\phantom{\rule{1.em}{0ex}}\u2022$Cyber–physical in hybrid systems modeling languages : Nowadays, some hybrid modeling languages and tools are widely used to describe and to simulate hybrid systems (modelica , simulink , and see [55] for references therein). Nevertheless, the compilers and the simulation engines behind these languages and tools suffer from several serious weaknesses (failure, weird output or huge sensitivity to simulation parameters), especially when some components, that are standard in nonsmooth dynamics, are introduced (piecewise smooth characteristic, unilateral constraints and complementarity condition, relay characteristic, saturation, dead zone, ...). One of the main reasons is the fact that most of the compilers reduce the hybrid system to a set of smooth modes modeled by differential algebraic equations and some guards and reinitialization rules between these modes. Sliding mode and Zeno–behaviour are really harsh for hybrid systems and relatively simple for nonsmooth systems. With B. Caillaud (Inria HYCOMES) and M. Pouzet (Inria PARKAS), we propose to improve this situation by implementing a module able to identify/describe nonsmooth elements and to efficiently handle them with siconos as the simulation engine. They have already carried out a first implementation [53] in Zelus, a synchronous language for hybrid systems http://zelus.di.ens.fr. Removing the weaknesses related to the nonsmoothness of solutions should improve hybrid systems towards robustness and certification.
$\phantom{\rule{1.em}{0ex}}\u2022$A general solver for piecewise smooth systems This direction is the continuation of the promising result on modeling and the simulation of piecewise smooth systems [37]. As for general hybrid automata, the notion or concept of solutions is not rigorously defined from the mathematical point of view. For piecewise smooth systems, multiplicity of solutions can happen and sliding solutions are common. The objective is to recast general piecewise smooth systems in the framework of differential inclusions with Aizerman–Pyatnitskii extension [37], [61]. This operation provides a precise meaning to the concept of solutions. Starting from this point, the goal is to design and study an efficient numerical solver (time–integration scheme and optimization solver) based on an equivalent formulation as mixed complementarity systems of differential variational inequalities. We are currently discussing the issues in the mathematical analysis. The goal is to prove the convergence of the time–stepping scheme to get an existence theorem. With this work, we should also be able to discuss the general Lyapunov stability of stationary points of piecewise smooth systems.