An overview of numerical methods for discrete dynamical systems with frictionless unilateral constraints: mathematical issues and convergence results

The dynamics of rigid multibody systems subjected to frictionless non-penetration conditions leads to vibrations with impacts and velocity jumps, and classical good properties of dynamical systems, such as uniqueness or continuous dependence on data, are not always satisfied. As a consequence reliable simulations of such systems can not be performed easily and the use of any numerical algorithm has to be substantiated by a convergence result. Starting from the basic description of the mechanical problem, we derive its mathematical formulation as a measure differential inclusion. The main difficulties in the approximation of solutions will be discussed and an overview of specially designed algorithms will be proposed together with the corresponding convergence results.