Numerical analysis of a non-clamped dynamic thermoviscoelastic contact problem

TL;DR

The paper tackles a non-clamped dynamic thermoviscoelastic contact problem with a non-monotone friction law, coupling displacement and temperature. It adopts a variational formulation based on Clarke subdifferential (hemivariational inequality) and provides a fully discrete finite element scheme. Under standard regularity assumptions, it proves optimal a priori error estimates linear in the spatial mesh size $h$ and time step $k$, with no smallness restrictions on the data, and confirms the results with numerical simulations in 2D and 3D. The work extends prior results to non-clamped bodies and demonstrates reliable, efficient simulation of frictional contact with thermal effects for engineering applications.

Abstract

In this work, we analyze a non-clamped dynamic viscoelastic contact problem involving thermal effect. The friction law is described by a non-monotone relation between the tangential stress and the tangential velocity. This leads to a system of second-order inclusion for displacement and a parabolic equation for temperature. We provide a fully discrete approximation of the problem and find optimal error estimates without any smallness assumption on the data. The theoretical result is illustrated by numerical simulations.

Figures (6)

• Figure 1: Solution at chosen time moments $t$. Plots present the temperature on the left and relative temperature error on the right for $h=2^{-5}$ and $k=2^{-9}$.
• Figure 2: Numerical error estimate for velocity (left) and temperature (right).
• Figure 3: First simulation - base data
• Figure 4: Second simulation - increased friction
• Figure 5: Third simulation - increased heat generated by friction