Analysis of dynamic nonlinear thermoviscoelastic beam problems

In this paper, we study a dynamic contact problem between a nonlinear viscoelastic beam
and two rigid obstacles. Thermal effects are also taken into account and the contact is
modeled using the classical Signorini condition. The existence of solutions is proved by
considering approximate problems from a penalization procedure, proving some a priori
estimates and passing to the limit. An exponential decay property is also obtained. Then,
fully discrete approximations of the approximate problem are provided using the finite
element method for the spatial approximation and the implicit Euler scheme for the
discretization of the time derivatives. A stability property is obtained and a priori error
estimates are proved from which, under some additional regularity conditions, the linear
convergence of the algorithm is deduced. Fully discrete approximations of the Signorini
problem are then introduced proceeding in a similar way, for which a stability property is
obtained. Finally, some numerical simulations are performed to demonstrate the accuracy
of the approximation and the behavior of the solution