A Nonsmooth Modified Symplectic Integration Scheme for Frictional Contact Dynamics of Rigid--Flexible Multibody Systems

Constructing an efficient high-order nonsmooth time-stepping integration scheme for rigid–flexible multibody systems with frictional contacts and impacts is a challenging task. This paper presents a nonsmooth modified symplectic integration scheme that can automatically achieve second-order convergence if possible. This scheme is obtained in three main steps. Firstly, the position coordinates are used to interpolate midpoint approximated integrals, which include velocity jumps implicitly; Secondly, a two-timestep central difference scheme is formulated to approximate the measure differential inclusions; Thirdly, composing two half-condensed two-timestep schemes leads to a one-timestep scheme, which retains the structure of a second-order scheme. Furthermore, while bilateral constraints are satisfied at the position level, unilateral constraints at the velocity level are subject to constraint drifts. To alleviate this issue, a classified cone complementarity problem is formulated to restrain penetration from deteriorating and sticking contacts from fictitious slipping. After time discretization, a set of nonlinear equations with cone complementarity conditions at each timestep is solved by a two-layer iterative computational strategy, where the outer layer is Newton iteration and the inner layer is a primal–dual interior point method. Several numerical examples demonstrate the high accuracy attained by the proposed method. Nonsmooth dynamics of a rigid–flexible robot involving high-speed and high-friction impacts can be simulated faster than real-time.