Numerical time integration methods for nonsmooth systems. Part I. Low relative degree and electrical networks applications Part II. Nonsmooth mechanical systems with impacts

Vincent Acary 1
1 BIPOP - Modelling, Simulation, Control and Optimization of Non-Smooth Dynamical Systems
Inria Grenoble - Rhône-Alpes, LJK - Laboratoire Jean Kuntzmann, INPG - Institut National Polytechnique de Grenoble
Abstract : In the first talk, numerical time integration methods will be presented for nonsmooth dynamical systems with low relative degree. With the consistency of initial conditions, the relative degree r, which plays a similar role of the index for differential algebraic equations, governs the smoothness of the solution and therefore the design of numerical integration schemes. After a brief presentation of the formulation of nonsmooth dynamical systems with low relative r ≥ 1, several event-capturing schemes will be presented according to the smoothness of the expected solution. The talk will be illustrated with applications to electrical networks, where the ideal components (diodes, transistors, Switches) are modeled by nonsmooth components based on the variational inequalities theory and the complementarity theory. In the second talk, the problem of the time-integration of mechanical systems sub- jected to unilateral contact with Coulomb's friction and impacts is addressed. After an introduction on time-stepping schemes for mechanical systems, a study of new high reso- lution and higher order time integration methods for nonsmooth multibody systems will be presented. The term "high resolution" applies to methods that are at least first order methods when non smooth events are encountered and of higher order on smooth solu- tions. We focus our work on event-capturing methods where accurate findings of events are not performed. We propose adaptive time-step strategies for standard time-stepping schemes (Moreau's time stepping scheme) and we derive a new time-stepping scheme based on rough localization of events. The performance of these new schemes will be shown on standard academic examples such as a bounding ball with accumulation of impacts, linear impacting oscillator and double pendulum with impacts. This work paves the way to new schemes for an efficient numerical time integration of non smooth mechanical and raises some open problems about the order of accuracy of standard time-stepping schemes for general non smooth systems.
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Vincent Acary. Numerical time integration methods for nonsmooth systems. Part I. Low relative degree and electrical networks applications Part II. Nonsmooth mechanical systems with impacts. SDS 2010 - 6th Workshop on Structural Dynamical System: Computational Aspects, Dept. of Maths, University of Bari, Jun 2010, Capitolo, Monopoli, Italy. ⟨inria-00522334⟩

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