An augmented Lagrangian and reduced index formulation for the analysis of multibody systems with impacts

This paper presents a formalism for the simulation of nonsmooth dynamic systems with unilateral constraints, which integrates the contributions of impacts with first order accuracy in a fully implicit way, while the other contributions are integrated with second-order accuracy. Here, the smooth contribution of the reaction forces associated with bilateral and unilateral constraints is also integrated with second-order accuracy. Therefore, in free flight phases, the algorithm is strictly equivalent to a classical second-order scheme.
Moreover, following the idea of Gear, Gupta and Leimkuhler, bilateral and unilateral constraints are imposed both at position and velocity levels. This means that the constraints are enforced exactly and no penetration is allowed. In the absence of unilateral constraints, the method leads to a formulation of the equation of motion as an index-2 differential-algebraic equation.
Finally, the method is based on a reformulation of the LCP as an augmented Lagrangian equation with a suitable activation / deactivation criterion for the unilateral constraints. The resulting system of nonsmooth equations is solved using a monolithic semi-smooth Newton algorithm.