Collision of an Obstacle by an Elastic Bar in a Gravity Field: Solution with Discontinuous Velocity and Space-Time Primal-Dual Active Set Algorithm

A class of one-dimensional dynamic impact models is investigated with respect to non-smooth velocities using variational inequalities and space-time finite element approximation. For the problem of collision of a rigid obstacle by an elastic bar in the gravitational field, a benchmark based on particular solutions to the wave equation is constructed on a partition of rectangle domains. The full discretization of the collision problem is carried out over a uniform space-time triangulation and extended to distorted meshes. For the solution of the corresponding variational inequality, a semi-smooth Newton-based primal-dual active set algorithm is applied. Numerical experiments demonstrate advantages over time-step approximation: a high-precision numerical solution is computed in a few iterations without any spurious oscillations.