Discrete Energy behavior of a damped Timoshenko system
For researchers studying Timoshenko systems, this work provides a numerical method that faithfully replicates theoretical energy decay behaviors, but it is an incremental application of existing numerical techniques to a specific problem.
This paper develops a numerical scheme for a damped Timoshenko system that reproduces key discrete energy properties (positivity, conservation, decay rates) matching known analytical results. The scheme combines finite element and finite difference methods to handle linear and nonlinear dampings.
In this article, we consider a one-dimensional Timoshenko system subject to different types of dissipation (linear and nonlinear dampings). Based on a combination between the finite element and the finite difference methods, we design a discretization scheme for the different Timoshenko systems under consideration. We first come up with a numerical scheme to the free-undamped Timoshenko system. Then, we adapt this numerical scheme to the corresponding linear and nonlinear damped systems. Interestingly, this scheme reaches to reproduce the most important properties of the discrete energy. Namely, we show for the discrete energy the positivity, the energy conservation property and the different decay rate profiles. We numerically reproduce the known analytical results established on the decay rate of the energy associated with each type of dissipation.