[en] The simulation of mechanical systems subjected to impacts and friction requires to solve highly non-linear systems of equations stemming from the Signorini’s conditions and the Coulomb friction’s law, which present several practical difficulties, which are not yet completely solved.
For systems involving nonsmooth phenomena the common methods used to study the dynamics of structures with a finite element spatial discretization, such as the Newmark method, the Hilbert-Hughes-Taylor method (HHT) and the standard generalized-α method, can no longer be applied. These implicit integrators assume that the kinematic variables are smooth. However, in the presence of instantaneous changes of the velocity, which occurs due to the impact effects, these methods may produce numerical solutions with notable precision losses, non physical behaviors and the generation of fictitious energy at the contact instant. Thus, nonsmooth time integration methods able to deal with nonsmooth motion equations are needed.
This work presents the development of a robust and accurate time integrator. The integrator is built upon a previously developed nonsmooth generalized-α scheme time integrator which was able to deal well with nonsmooth dynamical problems avoiding any constraint drift phenomena and capturing vibration effects without introducing too much numerical dissipation. However, when dealing with problems involving nonlinear bilateral constraints and/or flexible elements, it is necessary to adopt small time-step sizes to ensure the convergence of the numerical scheme. In order to tackle these problems more efficiently, a fully decoupled version of the nonsmooth generalized-α method is proposed in this work, avoiding these inconveniences.
To account for friction, a new node-to-face contact element compatible with the proposed nonsmooth generalized-α solver has been developed. The node and face can be attached, each one, to either flexible or rigid bodies. For the sake of robustness and numerical performance, the frictional contact problem is treated using an augmented Lagrangian technique inspired by the work of Alart and Curnier for quasi-static problems.
Finally, this methodology has been implemented in the general purpose finite element software Oofelie. The algorithm and the frictional contact element have been coded using the existing data structure and in a non intrusive manner, in order to preserve the compatibility with the existing utilities and the wide element library. Using this implementation several numerical experiments have been done.
The results of theses examples have been compared to analytical solutions or previous numerical solutions obtained by other authors showing good agreement and convergence rate.