A new class of conserving algorithms for dynamic contact problems

This contribution introduces a new class of time-stepping algorithms for frictionless contact of solids that exhibit the same conservation laws as the underlying continuum dynamical system. To this end, a new second-order penalty finite element method has been developed that conserves the total energy of the system (including the regularization penalty potential) during persistent contact, and that restores the original energy upon release. Both linear and angular momenta are conserved by the scheme. Furthermore, a new strategy is presented for the enforcement of the associated constraints in the velocity besides the impenetrability constraint in the displacements, while preserving the conservation/restoration properties of the final numerical scheme. In addition, a modification of the previously developed conserving schemes is proposed that assures the correct energy dissipation if desired (even in the highly nonlinear setting of contact/impact problems), leading to contact schemes with high frequency energy dissipation. Representative numerical simulations are presented illustrating the performance of the proposed numerical schemes.