Dynamic contact of a beam–rod system with Signorini typed contact conditions and thermal effects

This paper provides mathematical and numerical analyses for a beam–rod system with thermal effects. Its motion is described by a partial differential equation system with Signorini typed contact conditions. These conditions that cause a nonlinear model are interpreted as complementarity conditions (CCs) with a convolution. In particular, the convolution plays a role in incorporating a thermal effect of the surface of a rigid obstacle, which inspires us to investigate possibilities of proving conservation of energy under some assumptions. We employ the one-step- θ schemes to show that numerical trajectories for a variational formulation and the CCs are convergent. Additionally, an alternative approach supports the convergence results which are proved through the time discretizations. Finite element methods are combined with time discretization techniques to propose the fully discrete numerical schemes. Numerical stability is validated and simulations with selected data are presented as well.