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 is 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, a new unconditionally stable and a very lightly dissipative scheme is proposed.
Highlights
The present paper highlights some new numerical results obtained for a one-dimensional elastodynamic contact problem
Dynamical contact problems play a crucial role in structural mechanics as well as in biomechanics and a considerable amount of engineering and mathematical literature has been dedicated to this topic last decades
This manuscript focuses on the weighted mass redistribution method which is well adapted to approximate elastodynamic contact problems
Summary
The present paper highlights some new numerical results obtained for a one-dimensional elastodynamic contact problem. Renard give rise to an undesirable energy blow-up during the time integration, the reader is referred to [KLR08, DEP11, DP*13] as well as to the references therein for further details To overcome these difficulties, some numerical methods based on the Newmark scheme for solving impact problems are proposed in [CTK91]. Numerical experiments presented in this work show that the undesirable phase shift between the approximate and analytical solutions disappears and all the properties of the mass redistribution method mentioned above are preserved. They highlight that the weighted mass redistribution method is well adapted to deal with contact problems.
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