An augmented Lagrangian and reduced index formulation for the analysis of multibody systems with impacts

BRÜLS OLIVIER; ACARY VINCENT; CAVALIERI FEDERICO J.; VIRLEZ GEOFFREY; CARDONA ALBERTO

Zagreb

Congreso; Multibody Dynamics 2013, ECCOMAS Thematic Conference; 2013

University of Zagreb, Faculty of Mechanical Engineering and Naval Architecture

In multibody dynamics, contact conditions can lead to impacts and/or abrupt changes in the velocities and, as a consequence, the design of a consistent and stable time integration scheme requires great care. There exist two main groups of numerical integration schemes for nonsmooth systems, namely, event-driven schemes and time-stepping schemes. Eventdriven schemes are based on an accurate event detection and the time step is adapted such that the end of the step coincides with an event. At this time instant, the event is solved with the help of an impact law. Such schemes are accurate for the free flight smooth motions, but fail to handle frequent transitions in a short time. In practice, they are especially suitable for small multibody systems with a limited number of events. Contrary to the event-driven schemes, time-stepping schemes, e.g. the Moreau-Jean scheme [1, 2] and the Schatzman-Paoli scheme [3, 4], do not adapt their time step size on events but only on some accuracy requirements if needed. Time-stepping methods have been proven to be convergent and robust, and are extensively applied as the solution to the nonsmooth system models. In contrast to event-driven schemes, time-stepping schemes are expected to have order-one accuracy even in the smooth part of the motion. Thus, the accuracy is less satisfactory unless a very small step size is applied. However, it remains robust and efficient even for a large number of events as it is usual in structural dynamics. The present contribution proposes a time-stepping scheme with improved accuracy in the smooth part of motion. A previous work [5, 6] was based on the observation that, if all impulsive terms are treated using an Euler implicit integration scheme to ensure consistency, some terms in the equations of motion, such as the elastic forces in a flexible system, are smooth and can be integrated in the time domain using a second-order scheme, e.g. the generalized- method. This means that different integration formulae are used for the different contributions to the equation of motion. Globally, the order of convergence of the method is still limited to one. Nevertheless, the advantage of this approach is that the numerical dissipation of the generalized- method is significantly smaller than the numerical dissipation of Euler implicit scheme, so that the energy behavior is strongly improved, especially for mechanical systems exhibiting both impacts and structural vibrations. In some sense, in this scheme, the high numerical dissipation of the Euler implicit scheme is only acting locally at the contact region and during the contact time, and not on the full system during the whole trajectory. If the method described in [5, 6] appears quite promising, it still exhibits a few limitative features. Firstly, all reaction forces associated with both bilateral and unilateral constraints are treated as impulses and integrated using a first-order scheme. This is a safe approach which guarantees the stability and the consistency of the numerical solution. However, in many practical situations, these constraint reaction forces may actually be smooth for long periods of time where all active contacts stay closed and no impact occur. In these situations, the algorithm still relies on an order-one integration scheme so that its accuracy is limited. Secondly, bilateral and unilateral constraints are only satisfied at velocity level. One thus expects that the constraints at position level are not exactly satisfied and that some drift-off phenomena can occur which may result in non-physical solutions, unless suitable projection techniques are implemented. Thirdly, the solution algorithm relies on a formulation of the problem as a linear complementarity problem (LCP) for which a specific LCP solver is required. These various limitations are addressed in the present work.