Discussion of the Gear–Gupta–Leimkuhler method for impacting mechanical systems

Abstract

In multibody simulation, the Gear–Gupta–Leimkuhler method or stabilized index 2 formulation for persistent contacts enforces constraints on position and velocity level at the same time. It yields a robust numerical discretization of differential algebraic equations avoiding the drift-off effect and is often more effective than decreasing the time step size to preserve geometric characteristics. In this work, we carry over these benefits to impacting mechanical systems with unilateral constraints. For this kind of a mechanical system, adding the position level constraint to an (event-capturing) timestepping scheme on velocity level even maintains physical consistency of the impulsive discretization. Hence, we propose a timestepping scheme based on Moreau’s midpoint rule, which enables to achieve not only compliance of the impact law, but also of the nonpenetration constraint. The choice of a decoupled and consecutive evaluation of the respective constraints can be interpreted as a not energy-consistent coordinate projection to the nonpenetration constraint at the end of each time step. It is the implicit coupling of position and velocity level, which yields satisfactory results. An implicit evaluation of the right hand side improves stability properties without additional cost. With the prox function formulation, the overall set of nonsmooth equations is solved by a nonsmooth Newton method. Results from simulations of a slider-crank mechanism with unilateral constraints demonstrate the capability of our approach.