The influence of time integrator on contact/impact problems using the positional finite element method

Resumo

The dynamic structural problems involving contact/impact are strongly nonlinear, and can lead to spu-
rious numerical oscillations, generating unsatisfactory results or even convergence problems depending on the

applied temporal and spatial discretization techniques. The use of Newmark method with traditional parameters is
proven to be inefficient in this case, making necessary the application of specialized time integration techniques.
Some authors propose alternative values for the Newmark parameters to circumvent this problem, introducing a

numerical damping in the system, and reducing artificially the high frequency oscillations. However, this strat-
egy is highly sensitive to the time discretization, decreasing the accuracy of the results when the time steps are

not sufficiently refined. As an alternative, one can employ the alpha-generalized time integration method, which

allows the control of numerical dissipation by using appropriate parameters. In this work, we apply different com-
binations of said parameters, including the ones which reproduce the Newmark method and its variations, in order

to analyze the numerical stability of two-dimensional impact problems. The applied computational framework
is the positional finite element method, which is characterized by using positions as nodal parameters, instead of
displacements, and naturally considering geometrical nonlinearities in its formulation. The applied constitutive
model is the Neo-Hookean, for large strain. For the numerical implementation of structural contact, we make
use of a node-to-segment model with Lagrange multipliers, employing a contact detection algorithm based on the

intersection of trajectories. Finally, a representative numerical example is proposed with different time integra-
tion techniques. The results indicate that the adequate choice of alpha-generalized parameters can lead to quite

significant improvements to the numerical stability when compared to the traditional Newmark method and its
modifications.