Analysis and numerical simulation of dynamic contact problem with friction in thermo–viscoelasticity

M. Bouallala^a, EL.-H. Essoufi^b, Y. Ouafik<sup>c

a</sup> Department of Mathematics and Computer Science, Polydisciplinary faculty, Cadi Ayyad University, B.P. 4162, Safi, Morocco
^b Faculty of Science and Technology, Hassan 1st University Settat Laboratory Mathematics, 26000 Settat, Morocco.
^c National School of Applied Sciences of Safi, Cadi Ayyad University, B.P. 4162, Safi, Morocco

Abstract: The focus of our study is a dynamic frictional contact model that involves a viscoelastic body and a conductive foundation. We use Coulomb's law to describe the frictional behavior, while a normal compliance model is employed to simulate the contact. We formulate a variational formulation for the problem, and we establish the existence of its unique weak solution using the Banach fixed point theorem. We propose a fully discrete scheme, using the finite element method for the spatial approximation and the Euler scheme for the discretization of the time derivatives. The errors on the solutions are derived, and the linear convergence is obtained under suitable regularity hypotheses. Some numerical simulations are included to show the performance of method.

Keywords: Viscoelastic material, heat transfer, frictional contact, Banach fixed point, finite element approximation, error estimates, numerical simulations.

MSC: 74D99, 74M15, 37C25, 65M60, 74S05, 81T80, 74F05

Language: English

 English version: Ufa Mathematical Journal, 2025, 17:4, 156–175