In this talk we will give an overview of mathematical issues for
vibro-impact problems: existence, (non-)uniqueness and convergence of
approximation techniques.

More precisely we will consider systems of rigid bodies sujected to frictionless unilateral constraints and we will assume that the transmission of the velocity at impacts is governed by Newton's law. Starting from the basic description of the dynamics, we will introduce the mathematical formulation of the problem in terms of measure differential inclusions.

The main difficulties in the study of existence of solutions will be briefly discussed and a counter-example to uniqueness will be given.

Then we will focus on the approximation of the solutions. We will present first the penalty (or normal compliance) method, then two families of time-stepping schemes, at the position or at the velocity level, will be derived from the measure-differential inclusions describing the dynamics. The convergence results of the corresponding approximate solutions will be stated and the main steps of the proofs will be explained.

Lecture slides 1

Lecture slides 2

References:

More precisely we will consider systems of rigid bodies sujected to frictionless unilateral constraints and we will assume that the transmission of the velocity at impacts is governed by Newton's law. Starting from the basic description of the dynamics, we will introduce the mathematical formulation of the problem in terms of measure differential inclusions.

The main difficulties in the study of existence of solutions will be briefly discussed and a counter-example to uniqueness will be given.

Then we will focus on the approximation of the solutions. We will present first the penalty (or normal compliance) method, then two families of time-stepping schemes, at the position or at the velocity level, will be derived from the measure-differential inclusions describing the dynamics. The convergence results of the corresponding approximate solutions will be stated and the main steps of the proofs will be explained.

Lecture slides 1

Lecture slides 2

References:

- M.Schatzman, A class of nonlinear differential equations of second order in time, Nonlinear Anal.,Theory, Methods and Applications, 2(1978)355-373.
- L.Paoli, M.Schatzman, Mouvement à un nombre fini de degrés de liberté avec contraintes unilatérales : cas avec perte d'énergie, Modèl. Math. Anal. Numér. (M2AN), 27-6(1993)673-717.
- M.P.D.Monteiro-Marques, Differential inclusions in non-smooth mechanical problems: shocks and dry friction, Birkhauser, Boston PNLDE 9, 1993.
- R.Dzonou, M.Monteiro-Marques, L.Paoli, A convergence result for a vibro-impact problem with a general inertia operator, Nonlinear Dynamics, 58-1/2(2009) 361-384.
- L.Paoli, Time-stepping approximation of rigid-body dynamics with perfect unilateral constraints I and II, Archive for Rational Mechanics and Analysis, 198(2010)457-503 and 198(2010)505-568.
- L.Paoli, A proximal-like algorithm for vibro-impact problems with a non smooth set of constraints, Journal of Differential Equations, 250(2011)476-514.