New Advancements in the Nonsmooth Generalized-α Time Integration Method


The numerical simulation of mechanical systems composed of rigid and flexible bodies interconnected by kinematic joints, both in the frictionless and frictional cases, is intended for the analysis of the dynamic interactions and vibrations in industrial applications such as in automotive, wind turbines and robotic systems.

Non-penetration conditions at the contact points are modelled as unilateral constraints which may cause impact phenomena so that the dynamic response becomes nonsmooth involving velocity jumps and impulsive reaction forces. The case of multiple simultaneous impacts between rigid bodies further increases the complexity of the response.

In many practical situations, the nonsmooth behaviours are localized in space and/or in time. After spatial and time discretization, this implies that velocity jumps and impulsive forces are only observed for a limited number of coordinates and/or during a limited number of time steps. Even though these velocity jumps and impulsive forces are correctly described, the quality of the results within the smooth parts of the motion is also essential.

The most popular time-stepping methods for nonsmooth systems, such as the Moreau-Jean scheme or the Schatzman-Paoli scheme, are robust with respect to the treatment of nonsmooth phenomena. However, they lead to rather poor first-order approximations of the smooth parts of the motion and to high levels of numerical dissipation. Event-driven techniques, which adapt their time steps to the impact instants, cannot be used if accumulation phenomena, involving an infinite series of impacts in a finite time interval, are present.

These observations motivated the recent development of more sophisticated time-stepping algorithms for nonsmooth systems which involve improved approximations of the smoother parts of the motion. A revised version of the nonsmooth generalized-α method introduced in [1] is here presented. It relies on a splitting of the motion into smooth (non-impulsive) and nonsmooth (impulsive) contributions. The smooth contributions are integrated using the second-order generalized-α method whereas the nonsmooth contributions are integrated using a first-order backward Euler scheme. Multiple simultaneous impacts are also considered.

Several examples of nonsmooth dynamic systems are presented. These examples intend to reveal the good properties of the algorithm for systems with bilateral constraints, impacts, accumulation phenomena, flexible bodies, finite contact duration, dynamic activation and deactivation of unilateral constraints, both in the single impact as well as in the multiple impacts case.

  1. Bruls, O., Acary, V., Cardona, A., Simultaneous enforcement of constraints at position and velocity levels in the nonsmooth generalized-α scheme. Computer Methods in Applied Mechanics and Engineering 281, 131–161 (2014)