On practical integration of semi-discretized nonlinear equations of motion. Part 1

Reasons for probable instability and improper convergence

authored by: Aram Soroushian, Peter Wriggers, Jamshid Farjoodi
Abstract: Time integration is the most versatile method for analyzing the general case of nonlinear semi-discretized equations of motion. However, the approximate responses of such analyses generally do not converge properly, and might even display numerical instability. This is a very significant shortcoming especially in practical time integration. Herein, after illustrating that this phenomenon is viable even for very simple nonlinear dynamic models, sources of the shortcoming are discussed in detail. The conclusion is that in time integration of nonlinear dynamic mathematical models of physically stable structural systems, responses may converge improperly for three major reasons. These reasons are: (1) inadequate number of iterations before terminating nonlinearity solutions; (2) deficiencies in the formulation of some time integration methods; and (3) the inherent behaviour of the models of some special dynamic systems. In addition, limitations on computational facilities and improper consideration of these limitations may impair the numerical stability and convergence of the computed responses. The differences between static and dynamic analyses are also discussed from the viewpoint of the numerical errors induced by nonlinearity.
Organisation(s): Institute of Mechanics and Computational Mechanics
External Organisation(s): University of Tehran
Tehranpars
Type: Article
Journal: Journal of sound and vibration
Volume: 284
Pages: 705-731
No. of pages: 27
ISSN: 0022-460X
Publication date: 21.06.2005
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Condensed Matter Physics, Mechanics of Materials, Acoustics and Ultrasonics, Mechanical Engineering
Electronic version(s): https://doi.org/10.1016/j.jsv.2004.07.008 (Access: Unknown)
: Details in the research portal "Research@Leibniz University"

BibTeX

@article{b43e5c29f5d840e0a0f1dd718e7dc534,
title = "On practical integration of semi-discretized nonlinear equations of motion. Part 1: Reasons for probable instability and improper convergence",
abstract = "Time integration is the most versatile method for analyzing the general case of nonlinear semi-discretized equations of motion. However, the approximate responses of such analyses generally do not converge properly, and might even display numerical instability. This is a very significant shortcoming especially in practical time integration. Herein, after illustrating that this phenomenon is viable even for very simple nonlinear dynamic models, sources of the shortcoming are discussed in detail. The conclusion is that in time integration of nonlinear dynamic mathematical models of physically stable structural systems, responses may converge improperly for three major reasons. These reasons are: (1) inadequate number of iterations before terminating nonlinearity solutions; (2) deficiencies in the formulation of some time integration methods; and (3) the inherent behaviour of the models of some special dynamic systems. In addition, limitations on computational facilities and improper consideration of these limitations may impair the numerical stability and convergence of the computed responses. The differences between static and dynamic analyses are also discussed from the viewpoint of the numerical errors induced by nonlinearity.",
author = "Aram Soroushian and Peter Wriggers and Jamshid Farjoodi",
note = "Funding information: The authors gratefully acknowledge the helpful discussions they had on different parts of this paper with Prof. Arthur R. Robinson, Dr. K. Ghayour, and Mr. K. Arzhangi. The first and third authors also express their sincere appreciation to the financial assistance of the Research Council of the University of Tehran for the research report in correspondence to this paper. In addition, the last but by no means the least thanks are given to Prof. A. Der Kiureghian and Prof. J. Retief who caused the need to such a study come into the view of the first author.",
year = "2005",
month = jun,
day = "21",
doi = "10.1016/j.jsv.2004.07.008",
language = "English",
volume = "284",
pages = "705--731",
journal = "Journal of sound and vibration",
issn = "0022-460X",
publisher = "Academic Press Inc.",
number = "3-5",
}