Analysis and numerical simulation of the nonlinear beam equation with moving ends

TL;DR

This paper develops and analyzes a numerical method for simulating small amplitude elastic beam motions with moving ends, demonstrating stability, convergence, and energy decay through finite element and finite difference techniques.

Contribution

It introduces an efficient numerical scheme with quadratic convergence for nonlinear beam equations with moving boundaries, validated by theoretical analysis and numerical experiments.

Findings

The method achieves quadratic convergence in space and time.

Numerical simulations confirm theoretical stability and energy decay.

Approximate solutions closely match exact solutions in test cases.

Abstract

The numerical analysis for the small amplitude motion of an elastic beam with internal damping is investigated in domain with moving ends. An efficient numerical method is constructed to solve this moving boundary problem. The stability and convergence of the method is studied, and the errors of both the semi-discrete and fully-discrete schemes are derived, using Hermite's polynomials as a base function, proving that the method has order of quadratic convergence in space and time. Numerical simulations using the finite element method associated with the finite difference method (Newmark's method) are employed, for one-dimensional and two-dimensional cases. To validate the theoretical results, tables are shown comparing approximate and exact solutions. In addition, numerically the uniform decay rate for energy and the order of convergence of the approximate solution are also shown.