Half-explicit timestepping schemes on velocity level based on time-discontinuous Galerkin methods

Abstract

This paper presents a time-discretization scheme for the simulation of nonsmooth mechanical systems. These consist of rigid and flexible bodies, joints as well as contacts and impacts with dry friction. The benefit of the proposed formalism is both the consistent treatment of velocity jumps, e.g. due to impacts, and the automatic local order elevation in non-impulsive intervals at the same time. For an appropriate treatment of constraints in impulsive and non-impulsive intervals, constraints are implicitly formulated on velocity level in terms of an augmented Lagrangian technique [5]. They are satisfied exactly without any penetration. For efficiency reasons, all other evaluations are explicit which yields a half-explicit method [9, 10, 43, 44, 7, 29, 30].The numerical scheme is an extended timestepping scheme for nonsmooth dynamics according to Moreau [42]. It is based on time-discontinuous Galerkin methods to carry over higher order trial functions of event-driven integration schemes to consistent timestepping schemes for nonsmooth dynamical 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, i.e., with first-order accuracy. Non-impulsive contact forces are integrated with respect to the local order of the trial functions. In order to satisfy the constraints, a set of nonsmooth equations has to be solved in each time step depending on the number of stages; the solution of the velocity jump together with the corresponding impulse yields another nonsmooth equation. All nonsmooth equations are treated separately by semi-smooth Newton methods.The integration scheme on acceleration level was first introduced in [51] labeled forecasting trapezoidal rule. It was analyzed and applied to a decoupled bouncing ball example concerning principal suitability without taking friction into account. In this work, the approach is algorithmically specified, improved and applied to nonlinear multi-contact examples with friction. It is compared to other numerical schemes and it is shown that the newly proposed integration scheme yields a unified behavior for the description of contact mechanical problems.