4.3 Conclusion

We have investigated the behaviour of several numerical methods adapted to mechanical systems with impacts. Two major categories of numerical schemes were considered: The first one with no explicit computation of impact times, and the second one consisting of classical methods of Newmark or Runge-Kutta type to which we include an impact approximation procedure. Partial theoretical results have been proved in the latter case, showing the difficulties to establish general results in particular when infinite sequences of impact times occur. All the numerical methods defined have been thoroughly numerically tested, and their order has been computed. The different sources of numerical errors have been identified in the case of accumulations of impact times and has led to the definition of a new algorithm iterating the procedure of impact approximation as long as impacts occur on every time step.

From the mathematical point of view, the Paoli-Schatzman scheme is the best choice because it has been proved to be convergent, provided the equations of the physical model that have to be solved can be written under a form that satisfies the mathematical conditions required to define the scheme. Its main numerical interest is that it does not explicitly compute impact times, but the low order of this method makes it very slow when a relatively high precision is needed. On the other hand, the classical methods associated with impact detection have shown good numerical properties, provided that the algorithms are designed to compute as many impacts as possible. In...