Numerical analysis for a nonlinear model of elastic strings with moving ends

• Published in: Applied numerical mathematics Vol. 135; pp. 146 - 164
• Main Authors: Rincon, M.A., Liu, I.-S., Huarcaya, W.R., Carmo, B.A.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.01.2019
• Subjects: Error estimates; Iterative method; Moving boundary; Newmark's method; Newton's method; Numerical simulation; Moving boundary; Newton's method; Iterative method; Numerical simulation; Error estimates; Newmark's method
• ISSN: 0168-9274; 1873-5460
• DOI: 10.1016/j.apnum.2018.08.014

In this article, the error estimates for semi-discrete and totally discrete problems of a nonlinear model of elastic strings with moving boundary are established. We consider an extension of the Kirchhoff model, that takes into account the change of length of the string during vibration. The existence and uniqueness theorems of the problem, already known in the literature, will be stated for reference. The present numerical analysis is based on finite element method in spatial variable and finite difference method in time with the Newmark's approximation. Since the problem is nonlinear the resulting algebraic system is nonlinear to be solved by Newton's method. Numerical examples are presented for different kinds of moving boundary to verify the efficiency and feasibility of the method and check the coherence with the theoretical analysis. From the numerical results, the rate of convergence are shown to be consistent with the order of convergence expected from the theoretical ones.