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

TL;DR

This paper extends the Gear-Gupta-Leimkuhler method to impacting mechanical systems with unilateral constraints, proposing a stable, energy-consistent time-stepping scheme that enforces impact and non-penetration laws simultaneously.

Contribution

It introduces a novel implicit coupling of position and velocity constraints for impact simulation, improving stability and physical consistency in multibody dynamics.

Findings

Stable simulation of impact with unilateral constraints demonstrated

Implicit coupling yields physically consistent impulsive discretization

Newton scheme effectively solves nonsmooth equations in simulations

Abstract

In multibody simulation, the Gear-Gupta-Leimkuhler method for only 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. 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 a 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 non-penetration constraint. The choice of a decoupled and consecutive evaluation of the respective constraints can be interpreted as a not energy-consistent projection to the non-penetration 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 Newton scheme. Results from simulations of a slider-crank mechanism with unilateral constraints demonstrate the capability of our approach.