A weighted finite element mass redistribution method for dynamic contact problems

TL;DR

This paper introduces a finite element mass redistribution method for 1D wave equations with unilateral boundary conditions, improving stability and reducing oscillations in contact problems.

Contribution

It proposes a novel mass redistribution approach combined with finite elements and introduces a new stable, lightly dissipative scheme for better energy behavior.

Findings

Convergence and error estimates are established.

The method effectively reduces post-impact oscillations.

The new scheme enhances stability and energy dissipation.

Abstract

This paper deals with a one-dimensional wave equation being subjected to a unilateral boundary condition. An approximation of this problem combining the finite element and mass redistribution methods is proposed. The mass redistribution method is based on a redistribution of the body mass such that there is no inertia at the contact node and the mass of the contact node is redistributed on the other nodes. The convergence as well as an error estimate in time are proved. The analytical solution associated with a benchmark problem is introduced and it is compared to approximate solutions for different choices of mass redistribution. However some oscillations for the energy associated with approximate solutions obtained for the second order schemes can be observed after the impact. To overcome this difficulty, an new unconditionally stable and a very lightly dissipative scheme is proposed.