Implicit Euler numerical simulations of sliding mode systems

In this report it is shown that the implicit Euler time-discretization of some classes of switching systems with sliding modes, yields a very good stabilization of the trajectory and of its derivative on the sliding surface. Therefore the spurious oscillations which are pointed out elsewhere when an explicit method is used, are avoided. Moreover the method (an {\em event-capturing}, or {\em time-stepping} algorithm) allows for accumulation of events (Zeno phenomena) and for multiple switching surfaces (i.e., a sliding surface of codimension $\geq 2$). The details of the implementation are given, and numerical examples illustrate the developments. This method may be an alternative method for chattering suppression, keeping the intrinsic discontinuous nature of the dynamics on the sliding surfaces. Links with discrete-time sliding mode controllers are studied

Energy conservation and dissipation properties of time-integration methods for the nonsmooth elastodynamics with contact

This research report is devoted to the study of the conservation and the dissipation properties of the mechanical energy of several time-integration methods dedicated to the elasto- dynamics with unilateral contact. Given that the direct application of the standard schemes as the Newmark schemes or the generalized-α schemes leads to energy blow-up, we study two schemes dedicated to the time-integration of nonsmooth systems with contact: the Moreau-Jean scheme and the nonsmooth generalized-α scheme. The energy conservation and dissipation properties of the Moreau-Jean is firstly shown. In a second step, the nonsmooth generalized-α scheme is studied by adapting the previous works of Krenk and Høgsberg in the context of unilateral contact. Finally, the known properties of the Newmark and the Hilber-Hughes-Taylor (HHT) scheme in the unconstrained case are extended without any further assumptions to the case with contact.Ce rapport de recherche propose une étude des propriétés de conservation et de dissipation de l'énergie mécanique pour différents schémas d'intégration en temps de la dynamique élastique avec du contact unilatéral. Sachant que l'application directe des schémas standards de type Newmark et des schémas α-généralisés conduisent à des explosions de l'énergie mécanique, on étudie deux schémas dédiés à l'intégration en temps des systèmes non réguliers avec contact : le schéma de Moreau-Jean et le schéma α-généralisé non-régulier. La conservation de l'énergie et les propriétés de dissipation du schéma de Moreau-Jean sont d'abord démontrées. Dans un second temps, le schéma α-généralisé non-régulier est étudié en adaptant les travaux précurseurs de Krenk et Høgsberg dans le contexte du contact unilatéral. Finalement, les propriétés connues du schéma de Newmark et du schéma Hilber-Hughes-Taylor (HHT) dans le cas régulier sont étendues dans le cas avec contact sans hypothèses supplémentaires

Energy conservation and dissipation properties of time-integration methods for the nonsmooth elastodynamics with contact

This research report is devoted to the study of the conservation and the dissipation properties of the mechanical energy of several time--integration methods dedicated to the elasto--dynamics with unilateral contact. Given that the direct application of the standard schemes as the Newmark schemes or the generalized--$\alpha$ schemes leads to energy blow-up, we study two schemes dedicated to the time--integration of nonsmooth systems with contact: the Moreau--Jean scheme and the nonsmooth generalized--$\alpha$ scheme. The energy conservation and dissipation properties of the Moreau--Jean is firstly shown. In a second step, the nonsmooth generalized--$\alpha$ scheme is studied by adapting the previous works of Krenk and H{\o}gsberg in the context of unilateral contact. Finally, the known properties of the Newmark and the Hilber--Hughes--Taylor (HHT) scheme in the unconstrained case are extended without any further assumptions to the case with contact.Comment: 29 page

Analysis of explicit and implicit discrete-time equivalent-control based sliding mode controllers

Different time-discretization methods for equivalent-control based sliding mode control (ECB-SMC) are presented. A new discrete-time sliding mode control scheme is proposed for linear time-invariant (LTI) systems. It is error-free in the discretization of the equivalent part of the control input. Results from simulations using the various discretized SMC schemes are shown, with and without perturbations. They illustrate the different behaviours that can be observed. Stability results for the proposed scheme are derived

Numerical simulation of monuments by the contact dynamics method

International audienceThe Non Smooth Contact dynamics Method (NSCD or CD) is presented in this paper. The purpose of this method is to deal with large collections of rigid or deformable bodies in contact with unilateral constraints and large friction. The method is applied to monuments made of blocks. The relevance of the modelling is discussed. Several examples of buildings statically an dynamically loaded are presented

An open question : How to solve efficiently 3D frictional contact problem ?

International audienceIn this talk, we want to discuss possible numerical solution procedures for the following discrete frictional contact problem. We will recall a result for the problem in (1) which ensures that a solution exists [3]. In this framework, we will list several algorithms that have been previously developed for solving the SOCCP (1) mainly based variational inequality and nonsmooth equations reformulations. On one hand, we will show that algorithms based on Newton methods for nonsmooth equations solve quickly the problem when they succeed, but suffer from robustness issues mainly if the matrix H has not full rank. On the other hand, the iterative methods dedicated to solving variational inequalities are quite robust but with an extremely slow rate of convergence. To sum up, as far as we know there is no option that combines time efficiency and robustness. To try to answer to this question, we develop an open collection of discrete frictional contact problems called FCLIB http://fclib.gforge.inria.fr in order to offer a large library of problems to compare algorithms on a fair basis. In this work, this collection is solved with the software Siconos and its component Siconos/Numerics http://siconos.gforge.inria.fr

Numerical modeling of three dimensional divided structures by the Non Smooth Contact dynamics method: Application to masonry structures

International audienceThis paper outlines a computational method for modelling 3D divided structures by means of interface models, characterized by unilateral properties. The theoretical framework belongs to the field of non-smooth mechanics which aims at solving problems where severe time and space discontinuities are encountered. Multi-valued and stiff interfaces laws, e.g., Signorini's condition and Coulomb's friction, are solved using tools and formalisms provided by convex analysis. This general framework is adapted to micro-modelling approach of masonry structures, specifying interfaces models to mortar joints behaviour. The various stages in the development and implementation of an algorithm are delineated. Reaching a quasistatic equilibrium of floating structure is discussed and some numerical applications are presented on didactic tests

An overview of Non Smooth Dynamical Systems. Moreau's Sweeping Process, Higher order systems and links with optimization

International audienceThe non smooth approach is based on the Moreau's Sweeping process and its variants. The key idea is to write a right approximation of measures on a finite in- terval, and this leads efficient and robust numerical schemes, called Time–stepping schemes. The first algorithm was the “Catching up algorithm”. In the framework of multi-body dynamics, the derived algorithm is the ”Non Smooth Contact Dy- namics” method[13, 14, 7] which is able to treat several thousands of 3D frictional contact conditions. The time step is no longer of events but only fixed by a a priori error criterion. Therefore, accumulations of events or a large number of events in finite time are handled without difficulties. Furthermore, the convergence analy- sis of this family of schemes leads to existence of solutions for rather complicate systems [8, 16]

Energy conservation and dissipation properties of time-integration methods for nonsmooth elastodynamics with contact

International audienceThis article is devoted to the study of the conservation and the dissi-pation properties of the mechanical energy of several time–integration methods dedicated to the elasto–dynamics with unilateral contact. Given that the direct application of the standard schemes as the Newmark schemes or the generalized– α schemes leads to energy blow-up, we study two schemes dedicated to the time–integration of nonsmooth systems with contact: the Moreau–Jean scheme and the nonsmooth generalized–α scheme. The energy conservation and dissi-pation properties of the Moreau–Jean is firstly shown. In a second step, the nonsmooth generalized–α scheme is studied by adapting the previous works of Krenk and Høgsberg in the context of unilateral contact. Finally, the known properties of the Newmark and the Hilber–Hughes–Taylor (HHT) scheme in the unconstrained case are extended without any further assumptions to the case with contact

Periodic motions of coupled impact oscillators

International audienceWe study existence of time-periodic oscillations in a chain of coupled impact oscillators, for rigid impacts without energy dissipation. We formulate the search of periodic solutions as a boundary value problem incorporating unilateral constraints. This problem is solved numerically and different solution branches corresponding to nonlinear localized modes (breathers) and normal modes are computed