Modelling, Analysis and Numerical Simulation of a Spring-Rods System with Unilateral Constraints

TL;DR

This paper develops a mathematical model for two elastic rods connected by a nonlinear spring, proving its unique solvability, analyzing convergence, and validating results through finite element numerical simulations.

Contribution

It introduces a variational formulation of the elastic rods system with unilateral constraints and provides the first proof of unique weak solvability along with numerical validation.

Findings

Proved unique weak solvability of the model

Established convergence results with mechanical interpretation

Validated numerical simulations confirming theoretical results

Abstract

In this paper we consider a mathematical model which describes the equilibrium of two elastic rods attached to a nonlinear spring. We derive the variational formulation of the model which is in the form of an elliptic quasivariational inequality for the displacement field. We prove the unique weak solvability of the problem, then we state and prove some convergence results, for which we provide the corresponding mechanical interpretation. Next, we turn to the numerical approximation of the problem based on a finite element scheme. We use a relaxation method to solve the discrete problems that we implement on the computer. Using this method, we provide numerical simulations which validate our convergence results.