An Efficient Numerical Method for an Approximate Solution of the Beam Equation

TL;DR

This paper introduces a numerical scheme combining the method of lines and finite element methods with Hermite cubic basis functions to efficiently approximate solutions of the unsteady Euler-Bernoulli beam equation, including optimization of mesh parameters.

Contribution

It presents a novel horizontal method of lines approach that reformulates the unsteady beam problem as steady problems solved iteratively with finite elements, enhancing computational efficiency.

Findings

The proposed method accurately approximates the unsteady beam solution.

Mesh parameter optimization improves solution accuracy and computational performance.

Comparisons with exact solutions validate the method's effectiveness.

Abstract

In this paper, we propose a horizontal type method of lines numerical scheme for the unsteady Euler-Bernoulli beam equation. The problem is initially reformulated as a first order system of initial value problems and a suitable one-step difference scheme is used for the highest order temporal derivative which leads to a system of steady beam equations. Then resulted family of steady problems is solved iteratively by the finite element method with Hermite cubic basis functions. This iterative procedure leads to approximations for both the solution of the unsteady problem and its derivatives. All these approximations are compared with the exact ones to illustrate the performance of the proposed method. Moreover, the optimization of the mesh parameters is discussed for both steady and unsteady problems by logarithmic scale plot.