An Euler-Bernoulli beam with dynamic contact: Discretization, convergence, and numerical results

In this paper, we formulate a time-discretization using the implicit Euler method for contact conditions and the midpoint rule for the elastic part of the equations. The energy functional is defined, and convergence for the time-discretization is investigated. Our time-discretization leads to energy dissipation. Using this time discretization and the finite element method with B-spline basis functions, we compute numerical solutions. We show that there is a converging subsequence, and the limit of any such converging subsequence is a solution of the dynamic impact problem. In order to solve the linear complementarity problem that arises in the numerical method, we use a smoothed guarded Newton method. We also investigate numerically the question of whether the numerical solutions converge strongly to their limit and if energy is conserved for the limit. Our numerical results give some evidence that this is so.