Exponential stability and numerical analysis of a thermoelastic diffusion beam with rotational inertia and second sound

Abstract

We study the dynamic behavior of a thermoelastic diffusion beam with rotational inertia and second sound, clamped at one end and free to move between two stops at the other. The contact with the stops is modeled with the normal compliance condition. The system, recently derived by Aouadi (2015), describes the behavior of thermoelastic diffusion thin plates under Cattaneo’s law for heat and mass diffusion transmission to remove the physical paradox of infinite propagation speeds of the classical Fourier’s and Fick’s laws. The system of equations is a coupling of a hyperbolic equation with four parabolic equations. It poses some new mathematical and numerical difficulties due to the lack of regularity and the nonlinear boundary conditions. The exponential stability of the solutions to the contact problem is obtained in the presence of rotational inertia thanks to a structural damping term. We propose a finite element approximation and we prove that the associated discrete energy decays to zero. Finally, we give an error estimate assuming extra regularity on the solution and we present some results of our numerical experiments.