Practical Integration of Semidiscretized Nonlinear Equations of Motion

Proper Convergence for Systems with Piecewise Linear Behavior

authored by: Aram Soroushian, Peter Wriggers, Jamshid Farjoodi
Abstract: Time integration is the most versatile tool for analyzing semidiscretized equations of motion. The responses are approximations, with deviations from the exact responses mainly depending on the integration method and the integration step sizes. When repeating the analyses with smaller steps, the responses generally converge to the exact responses. However, the convergence trends are different in linear and nonlinear analyses.Whereas in linear analyses, by decreasing the sizes of integration steps, the errors decrease with a rate, depending on the orders of accuracy, in nonlinear analyses, the change in errors might be unpredictable. The main reason is the inconsistency between the integration steps sizes and the residuals of nonlinearity iterations. In this paper, based on careful selection of nonlinearity tolerances, a methodology and a method to overcome this inconsistency for semidiscretized systems with piecewise linear behavior are introduced. When the responses converge, except for systems with very complex behaviors, the proposed method leads to proper convergence, with tolerable computational costs. In addition, by implementing the proposed method, more reliable error estimations can be expected from convergence-based accuracy controlling methods.
Organisation(s): Institute of Continuum Mechanics
External Organisation(s): International Institute of Earthquake Engineering and Seismology (IIEES)
University of Tehran
Type: Article
Journal: Journal of engineering mechanics
Volume: 139
Pages: 114-145
No. of pages: 32
ISSN: 0733-9399
Publication date: 17.03.2012
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Mechanics of Materials, Mechanical Engineering
Electronic version(s): https://doi.org/10.1061/(ASCE)EM.1943-7889.0000434 (Access: Unknown)
: Details in the research portal "Research@Leibniz University"

BibTeX

@article{f98af306a27347b4ab488cb36f3a174c,
title = "Practical Integration of Semidiscretized Nonlinear Equations of Motion: Proper Convergence for Systems with Piecewise Linear Behavior",
abstract = "Time integration is the most versatile tool for analyzing semidiscretized equations of motion. The responses are approximations, with deviations from the exact responses mainly depending on the integration method and the integration step sizes. When repeating the analyses with smaller steps, the responses generally converge to the exact responses. However, the convergence trends are different in linear and nonlinear analyses.Whereas in linear analyses, by decreasing the sizes of integration steps, the errors decrease with a rate, depending on the orders of accuracy, in nonlinear analyses, the change in errors might be unpredictable. The main reason is the inconsistency between the integration steps sizes and the residuals of nonlinearity iterations. In this paper, based on careful selection of nonlinearity tolerances, a methodology and a method to overcome this inconsistency for semidiscretized systems with piecewise linear behavior are introduced. When the responses converge, except for systems with very complex behaviors, the proposed method leads to proper convergence, with tolerable computational costs. In addition, by implementing the proposed method, more reliable error estimations can be expected from convergence-based accuracy controlling methods.",
keywords = "Accuracy-controlling methods, Nonlinearity residual, Nonlinearity tolerance, Order of accuracy, Piecewise linear, Proper convergence, Pseudoerror, Space of nonlinearity, Time integration, Time step",
author = "Aram Soroushian and Peter Wriggers and Jamshid Farjoodi",
year = "2012",
month = mar,
day = "17",
doi = "10.1061/(ASCE)EM.1943-7889.0000434",
language = "English",
volume = "139",
pages = "114--145",
journal = "Journal of engineering mechanics",
issn = "0733-9399",
publisher = "American Society of Civil Engineers (ASCE)",
number = "2",
}