Posing Multibody Dynamics With Friction and Contact as a Differential Complementarity Problem

Abstract

which was introduced to solve the rigid multibody dynamics problem in the presence of friction and contact. The discretized equations of motion obtained here are identical to the ones in Tasora and Anitescu 2011 ["A Matrix-Free Cone Complementarity Approach for Solving Large-Scale, Nonsmooth, Rigid Body Dynamics," Comput. Methods Appl. Mech. Eng., 200(5-8), pp. [439][440][441][442][443][444][445][446][447][448][449][450][451][452][453]; what is different is the process of obtaining these equations. Instead of using maximum dissipation conditions as the basis for the Coulomb friction model, the approach detailed uses complementarity conditions that combine with contact unilateral constraints to augment the classical index-3 differential algebraic equations of multibody dynamics. The resulting set of differential, algebraic, and complementarity equations is relaxed after time discretization to a cone complementarity problem (CCP) whose solution represents the first-order optimality condition of a quadratic program with conic constraints. The method discussed herein has proven reliable in handling large frictional contact problems. Recently, it has been used with promising results in fluid-solid interaction applications. Alas, this solution is not perfect, and it is hoped that the detailed account provided herein will serve as a starting point for future improvements.

Resolving force indeterminacy in contact dynamics using compatibility conditions

“…From a practical standpoint, we have to evaluate B at the beginning of the time step and we make the assumption that B does not change substantially during the step. This is a good assumption in a sufficiently-resolved simulation and is standard practice in CD formulations [23], [27].…”

“…We consider the methods for computing Φ(q) and its gradient an implementation detail, and will not address them further here. A similar "gap function" formalism is used in [23], where more implementation details are provided. An authoritative reference for the implementation of contact detection systems is [10].…”

“…The time integration scheme above is used to discretize the complementarity condition. Like other formulations of CD [23], we impose the complementarity condition at the end of the timestep in order to ensure that the no-penetration constraint is satisfied at all steps of a simulation.…”

“…The second class of methods, developed by the study of nonsmooth mechanics, makes the simplifying assumption of perfectly rigid bodies. These methods are grouped under the term Contact Dynamics (CD), and were pioneered by J.J. Moreau [20,21,22] with significant contributions to the problem formulation and solution algorithms in more recent years [1,27,30,14,17,23]. Assuming perfect rigidity has two primary impacts on the method, distinguishing it from DEM methods.…”

The contact problem in Lagrangian systems with redundant frictional bilateral and unilateral constraints and singular mass matrix. The all-sticking contacts problem

“…The existence and uniqueness of solutions for this type of problems, has been studied in [3] (the existence of solutions to (46) could also be analysed using results in [50], due to the particular form of the function ϕ(·) in (81)), whose results have been used in [20] for the analysis of an inclusion simpler than (45) (equivalently an MLCP simpler than (43), see [20,Equation (16)]; and the MLCP in [20, Equation (34)] has a structure different than (43) because it involves two different multipliers). To analyse the contact MLCP (43), we will rely on the material in Appendix B.…”