Analysis of dynamic nonlinear thermoviscoelastic beam problems

• The existence and uniqueness of weak solutions are proved for two contact problems.
• An energy decay property is shown for the solutions to both problems.
• A priori error estimates are obtained by using adequate techniques.
• The linear convergence is obtained assuming additional regularity.
• The numerical results show the accuracy and the performance of the approximation.

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.