An enhanced energy conserving time stepping algorithm for frictionless particle contacts

Abstract

An Enhanced Energy Conserving Algorithm (EECA) formulation for time integration of frictionless contact-impact problems is presented. In it the energy, linear and angular momentum are conserved for every contact using an enhanced Penalty method. Previous formulations for these problems have shown that the total bodies' energy decreases for contact due to an artificial energy transfer between the penalty spring and the contacting bodies. Consequently, they are not able to reproduce a physical response after a single contact, introducing errors in trajectories and velocities. Through the conserving balance equations, EECA computes a physical response by inserting for every contact an additional amount of linear momentum and contact force. The structure of these equations defines the additional linear momentum to restore the energy and the enhanced Penalty method based on a spring and a dashpot. This method approximately enforces the first and second Kuhn-Tucker conditions. The new algorithm has been applied to several frictionless rigid problems using the Discrete Element Method. The first two problems consist of the simulation and analytical comparison of the Newton's Cradle and Carom problems (billiard pool problem). The last two are the hopper filling process and the breaking of a pool ball's triangular arrangement, both of which involve a medium number of contacts. Application of this formulation will be straightforward to elastic and general-shaped bodies using the Finite Element Method.