The finite element method for the elastic body with the Tresca friction and a modified Signorini-type contact interface conditions

We study finite element approximations for the dynamics of a linear elastic body with a frictional contact interface. The tangential motion is governed by the Tresca friction law, while the normal contact is described by a modified Signorini condition involving both displacement and velocity through a parameter $\\delta$. This condition formally includes the usual non-penetration condition when $\\delta=0$ and the Signorini condition on velocity in the limit $\\delta=\\infty$. We first propose a semi-discrete finite element scheme based on a mass-lumping treatment of the contact terms. The scheme is formulated both as a variational inequality and as an equivalent Lagrange multiplier system. Its well-posedness is proved by a regularization argument, and stability and error estimates are derived. We then introduce a fully discrete scheme and establish its well-posedness, stability, and convergence under a suitable time-step restriction. Finally, we develop several iterative methods, including an Uzawa algorithm, an active/inactive set algorithm, and an augmented Lagrangian algorithm. Numerical experiments are carried out to test the convergence rate of the proposed scheme and the efficiency of the iterative algorithms.