NON-UNIQUE SOLUTION WITH DISCONTINUOUS VELOCITY FOR COLLISION OF ELASTIC BAR WITH HIGH INITIAL SPEED BY ST-PDAS RESTORING ENERGY

Abstract Motivated by inequality constraints appearing in optimization context, we study non-smooth variational problems and methods appropriate for their accurate solution. The use of Lagrange multipliers and merit functions leads to semi-smooth Newton methods and equivalent primal-dual active-set iterative algorithms. For application in dynamic contact mechanics, we investigate the collision of a rigid obstacle by a one-dimensional elastic bar. The problem is described by the wave equation subjected to complementarity conditions, which weak solution is characterized by a discontinuous velocity. The collision problem may have a non-unique solution for a high initial speed exceeding the propagation speed of elastic waves. Multiple solutions are constructed analytically. For the unique solution that restores the energy of the bar after rebound, the primal-dual active-set method is implemented within space-time finite elements.