Éditeur(s) : HAL CCSD
Résumé : International audience
We consider a second-order differential system with Hessian-driven damping . An interesting property of this system is that, after introduction of an auxiliary variable y , it can be equivalently written as a first-order system in time and space. This allows us to extend the analysis to the case of a convex lower semicontinuous potential and so to introduce constraints in the model. When considering the indicator function of a closed convex set, the subdifferential operator takes account of the contact forces. In this setting, by playing with the geometrical damping parameter, we can describe nonelastic shock laws with restitution coefficient. Taking advantage of the infinite dimensional framework, we introduce a nonlinear hyperbolic PDE describing a damped oscillating system with obstacle. The system is dissipative; in the convex case each trajectory weakly converges to a minimizer of the global potential energy function. Exponential stabilization is obtained under strong convexity assumptions.
Differential Equations & Applications

hal-00803214
https://hal.archives-ouvertes.fr/hal-00803214
https://hal.archives-ouvertes.fr/hal-00803214/document
https://hal.archives-ouvertes.fr/hal-00803214/file/Attouch-Mainge-Redont-30August2011.pdf

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
The full numerical simulation of turbulent flows in the context of industrial applications remains a very challenging problem. One of the difficulties is the presence of a large number of interacting scales of different orders of magnitude, ranging from the macroscopic scale to the Kolmogorov dissipation scale. In order to better understand and simulate these interactions, new algorithms of incremental type have recently been introduced, stemming from the theory of infinite-dimensional dynamical systems (see, e.g., the algorithms of Foias, Jolly et al. (1988), Foias, Manley and Temam (1988), Laminie et al. (1993, 1994), Marion and Temam (1989, 1990), and Temam (1990)). These algorithms are based on decompositions of the unknown functions into a large- and a small-scale component, one of the underlying ideas being to approximate the corresponding attractor. In the context of finite element methods, the decomposition of a solution into small- and large-scale components appears when we consider hierarchical bases (Yserentant, 1986; Zienkiewicz et al., 1982). In the present article we derive several estimates of the size of the structures for the linear and nonlinear terms which correspond to interactions of different hierarchical components of the velocity field, and also their time variation. The one-step time variation of these terms can be smaller than the expected accuracy of the computation. Using this remark, we implement an adaptive spatial and temporal multilevel algorithm which treats differently the small- and large-scale components of the flow. We derive several a priori estimates in order to study the perturbation introduced into the approximated equations. All the interaction terms between small- and large-scale structures are frozen during several time steps. Finally, we implement the multilevel method in order to simulate a bidimensional flow described by the Burgers equations. We perform a parametric study of the procedure and its efficiency. The gain in CPU time is significant and the trajectories computed by our multi-scale method remain close to the trajectories obtained with the classical Galerkin method.
ISSN: 0168-9274

hal-00730710
https://hal.archives-ouvertes.fr/hal-00730710

Éditeur(s) : HAL CCSD
Résumé : International audience
In this paper, we compare the effect of rolling resistance at the contacts in granular systems composed of disks with the effect of angularity in granular systems composed of regular polygonal particles. For this purpose, we use contact dynamics simulations. By means of a simple shear numerical device, we investigate the mechanical behavior of these materials in the steady state in terms of shear strength, solid fraction, force and fabric anisotropies, and probability distribution of contact forces. We find that, based on the energy dissipation associated with relative rotation between two particles in contact, the effect of rolling resistance can explicitly be identified with that of the number of sides in a regular polygonal particle. This finding supports the use of rolling resistance as a shape parameter accounting for particle angularity and shows unambiguously that one of the main influencing factors behind the mechanical behavior of granular systems composed of noncircular particles is the partial hindrance of rotations as a result of angular particle shape.
AIP Conference Proceedings
Sydney, Australia

hal-00842799
https://hal.archives-ouvertes.fr/hal-00842799
https://hal.archives-ouvertes.fr/hal-00842799/document
https://hal.archives-ouvertes.fr/hal-00842799/file/Comparison_effects_rolling_Estrada_al.pdf

Éditeur(s) : HAL CCSD Society for Industrial and Applied Mathematics
Résumé : International audience
We present iterative methods for finding the critical points and/or the minima of extended real valued functions of the form $phi = psi+ g-h$, where $psi$ is a differentiable function and g and h are convex, proper, and lower semicontinuous. The underlying idea relies upon the discretization of a first order dissipative dynamical system which allows us to preserve the local feature and to obtain some convergence results. The main theorems not only recover known convergence results in this field but also provide a theoretical basis for the development of new iterative methods.
ISSN: 1052-6234

hal-00779987
https://hal.univ-antilles.fr/hal-00779987
DOI : 10.1137/060655183

Éditeur(s) : HAL CCSD EDP Sciences
Résumé : International audience
In the setting of a real Hilbert space H, we investigate the asymptotic behavior, as time t goes to infinity, of trajectories of second-order damped evolution equations, which are governed by the sum of the gradient operator of a convex differentiable potential function , and a maximal monotone operator which is assumed to be cocoercive. Under a sharp condition involving the viscous and the cocoercive parameters, it is proved that each trajectory weakly converges to an equilibrium. Passing from weak to strong convergence is obtained by introducing an asymptotically vanishing Tikhonov-like regularizing term. As special cases, we recover the asymptotic analysis of the heavy ball with friction dynamic attached to a convex potential, the second-order gradient-projection dynamic, and the second-order dynamic governed by the Yosida approximation of a general maximal monotone operator. The breadth and flexibility of the proposed framework is illustrated through applications in the areas of constrained optimization, dynamical approach to Nash equilibria for noncooperative games, and asymptotic stabilization in the case of a continuum of equilibria.
ISSN: 1292-8119

hal-00803774
https://hal.archives-ouvertes.fr/hal-00803774
https://hal.archives-ouvertes.fr/hal-00803774/document
https://hal.archives-ouvertes.fr/hal-00803774/file/ATTOUCH-MAINGE-09-04-2010-COCV.pdf
DOI : 10.1051/cocv/2010024

Éditeur(s) : HAL CCSD EDP Sciences
Résumé : International audience
Modelling environmental problems leads to mathematical systems with missing data. For instance, weather problems have generally missing initial conditions. The paper is concerned with identifying pollution terms arising in the state equation of some dissipative system with incomplete initial condition. To this aim the so-called sentinel method is used. Here, the problem of determining a sentinel is equivalent to a null-controllability problem for which Carleman inequalities are revisited.
ISSN: 1270-900X

hal-00720818
https://hal.univ-antilles.fr/hal-00720818
https://hal.univ-antilles.fr/hal-00720818/document
https://hal.univ-antilles.fr/hal-00720818/file/esaimpr2.pdf

Éditeur(s) : HAL CCSD American Physical Society
Résumé : International audience
Using contact dynamics simulations, we compare the effect of rolling resistance at the contacts in granular systems composed of disks with the effect of angularity in granular systems composed of regular polygonal particles. In simple shear conditions, we consider four aspects of the mechanical behavior of these systems in the steady state: shear strength, solid fraction, force and fabric anisotropies, and probability distribution of contact forces. Our main finding is that, based on the energy dissipation associated with relative rotation between two particles in contact, the effect of rolling resistance can explicitly be identified with that of the number of sides in a regular polygonal particle. This finding supports the use of rolling resistance as a shape parameter accounting for particle angularity and shows unambiguously that one of the main influencing factors behind the mechanical behavior of granular systems composed of noncircular particles is the partial hindrance of rotations as a result of angular particle shape.
ISSN: 1539-3755

hal-00594891
https://hal.archives-ouvertes.fr/hal-00594891
https://hal.archives-ouvertes.fr/hal-00594891/document
https://hal.archives-ouvertes.fr/hal-00594891/file/Estrada2010ReSubmitV2.pdf
ARXIV : 1105.4418

Éditeur(s) : HAL CCSD Heldermann
Résumé : International audience
We present an iterative method for finding zeroes of maximal monotone operators in a real Hilbert space. The underlying idea relies upon the discretization of a first order dissipative dynamical system which allows us to preserve the local feature, as well as to obtain convergence results. The main theorems do not only recover known convergence results of standard and inertial proximal methods, but also provide a theoretical basis for the application of new iterative methods.
Journal of Convex Analysis

hal-00780210
https://hal.univ-antilles.fr/hal-00780210