Consistent Time-Integration Schemes for Flexible Multibody Systems with Friction and Impacts

This work deals with nonsmooth mechanical systems and their discretization for numerical simulation. The mechanical systems consist of rigid and flexible bodies, joints as well as contacts and impacts with dry friction. We present a framework, which consistently treats velocity jumps, e.g., due to impacts, and benefits from nice properties of established integration methods. In impulsive as well as nonimpulsive intervals, constraints are implicitly formulated on velocity level in terms of an augmented Lagrangian technique. They are satisfied exactly without any penetration and solved by semi-smooth Newton methods. We present time-discontinuous Galerkin methods, which embed classical timestepping schemes from nonsmooth mechanics and yield consistent higher-order time-integration methods including impulsive forces. They transform higher order trial functions of event-driven integration schemes into consistent timestepping schemes for nonsmooth mechanical systems with friction and impacts. Splitting separates the portion of impulsive contact forces from the portion of non-impulsive contact forces. Impacts are included within the discontinuity of the piecewise continuous trial functions, that is, with first-order accuracy. Non-impulsive contact forces are integrated with respect to the local order of the trial functions. As a consequence, the computing time can be significantly reduced. A further abstraction shows that also time-integration methods from computational mechanics with high-frequency damping profit from impulsive corrections. We compare the generalized-α method, the Bathe method and the ED-α method. These implicit time-integration schemes are only chosen as illustrative examples. Each base integration scheme tailored for a specific application can be embedded in the concept of mixed timestepping schemes, like it is introduced and presented in the present work. Mixed timestepping schemes are formulated on velocity level. For preservation of geometric constraints, we offer the extension to projected timestepping schemes. The Gear-Gupta-Leimkuhler method or stabilized index 2 formulation enforces constraints on position and velocity level at the same time. It yields a robust numerical discretization avoiding the drift-off effect. Adding the position level constraint to a timestepping scheme on velocity level maintains physical consistency of the impulsive discretization. We demonstrate the capability of our formalism with various nonlinear and flexible multi-contact examples with friction and impacts. It is shown that the newly proposed integration schemes yield a unified behavior for the description of contact mechanical problems.