A framework of time integration methods for nonsmooth systems with unilateral constraints

A framework of time integration methods for nonsmooth systems with unilateral constraints is established in this paper, which solves non-impulsive dynamic equations at every step, and deals with impulsive dynamic equations at the end of steps that may involve impacts. The computational procedures of the Generalized-α method and the Bathe method are provided to show this framework, and how to select auxiliary parameters in the procedure is discussed for the benefit of applications. In addition, to avoid high penetration depth and improve the accuracy of detection, an adjustable step size strategy is developed. The provided procedures possess second-order accuracy for smooth motion, and errors caused by the approximate treatment of impacts are controlled by setting an allowable penetration depth. Besides, the observed oscillations of contact forces at contact boundary can be effectively filtered out by the Bathe method. Compared with the classical Moreau-Jean time-stepping scheme, the numerical experiment of the slider-crank mechanism validates that the proposed framework enjoys advantages on accuracy and efficiency.