On Nonlocal Computation of Eigenfrequencies of Beams Using Finite Difference and Finite Element Methods

Authors

• Noël Challamel, Université Européenne de Bretagne
• Vincent Picandet, Université Européenne de Bretagne
• Isaac Elishakoff, Florida Atlantic University
• Chien Ming Wang, National University of Singapore
• Bernard Collet, Sorbonne Université
• Thomas Michelitsch, Sorbonne Université

Author Type

Outside Researcher

Co-Author Type 1

Outside Researcher

Co-Author Type 2

Faculty

Co-Author Type 3

Outside Researcher

College

Engineering and Computer Science

Department

Ocean and Mechanical Engineering

Document Type

Article

Publication/Event/Conference Title

International Journal of Structural Stability and Dynamics

Publication Status

Version of Record

Abstract

In this paper, we show that two numerical methods, specifically the finite difference method and the finite element method applied to continuous beam dynamics problems, can be asymptotically investigated by some kind of enriched continuum approach (gradient elasticity or nonlocal elasticity). The analysis is restricted to the vibrations of elastic beams, and more specifically the computation of the natural frequencies for each numerical method. The analogy between the finite numerical approaches and the equivalent enriched continuum is demonstrated, using a continualization procedure, which converts the discrete numerical problem into a continuous one. It is shown that the finite element problem can be transformed into a system of finite difference equations. The convergence rate of the finite numerical approaches is quantified by an equivalent Rayleigh's quotient. We also present some exact analytical solutions for a first-order finite difference method, a higher-order finite difference method or a cubic Hermitian finite element, valid for arbitrary number of nodes or segments. The comparison between the exact numerical solution and the approximated nonlocal approaches shows the efficiency of the continualization methodology. These analogies between enriched continuum and finite numerical schemes provide a new attractive framework for potential applications of enriched continua in computational mechanics.

DOI

10.1142/S0219455415400088

Publication Date

10-3-2015

Recommended Citation

Challamel, Noël; Picandet, Vincent; Elishakoff, Isaac; Wang, Chien Ming; Collet, Bernard; and Michelitsch, Thomas, "On Nonlocal Computation of Eigenfrequencies of Beams Using Finite Difference and Finite Element Methods" (2015). Faculty Scholarship. 305.
https://digitalcommons.fau.edu/faculty_papers/305