Convergence analysis of semi-smooth Newton method for mixed FEM approximations of dynamic two-body contact and crack problems

A class of elastodynamic problems describing contact between two deformable bodies as well as non-penetrating cracks in a single body is considered in the framework of FEM approximation. For time discretization, the Hilber–Hughes–Taylor (HHT-$\alpha$) method extending Newmark schemes is incorporated. Using mixed variational formulation of the fully discrete contact problem, a semi-smooth Newton method of solution is provided with the locally super-linear convergence. An equivalent primal–dual active set algorithm validates monotone properties of global convergence for the Newton iterates provided by M-matrix property. Numerical solution of the Signorini contact with rigid obstacle is presented for isotropic body in 2D using benchmark and moving load experiment.