Discussion about parameterization in the asymptotic numerical method: Application to nonlinear elastic shells

The Asymptotic Numerical Method (ANM) is a family of algorithms for path following problems based on the computation of truncated vectorial series with respect to a path parameter “ a” [B. Cochelin, N. Damil, M. Potier-Ferry, Méthode Asymptotique Numérique, Hermès-Lavoisier, Paris, 2007]. In this pa...

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Bibliographic Details
Published in:Computer methods in applied mechanics and engineering Vol. 199; no. 25; pp. 1701 - 1709
Main Authors: Mottaqui, H., Braikat, B., Damil, N.
Format: Journal Article
Language:English
Published: Kidlington Elsevier B.V 01.01.2010
Elsevier
Subjects:
Asymptotic numerical method
Asymptotic properties
Computation
Computer simulation
Continuation methods
Elastic shells
Exact sciences and technology
Fundamental areas of phenomenology (including applications)
Mathematical models
Nonlinear elastic shells
Nonlinearity
Numerical analysis
Parameterization
Parametrization
Physics
Solid mechanics
Static elasticity (thermoelasticity...)
Structural and continuum mechanics
Parameterization
Asymptotic numerical method
Continuation methods
Nonlinear elastic shells
Path following method
Minimization
Continuation method
Modeling
Shell
Non linear acoustics
Truncated shape
Asymptotic approximation
Non linear effect
Arc length
Structural analysis
ISSN:0045-7825, 1879-2138
Online Access:Get full text
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Summary:The Asymptotic Numerical Method (ANM) is a family of algorithms for path following problems based on the computation of truncated vectorial series with respect to a path parameter “ a” [B. Cochelin, N. Damil, M. Potier-Ferry, Méthode Asymptotique Numérique, Hermès-Lavoisier, Paris, 2007]. In this paper, we discuss and compare three concepts of parameterizations of the ANM curves i.e. the definition of the path parameter “ a”. The first concept is based on the classical arc-length parameterization [E. Riks, Some computational aspects of the stability analysis of nonlinear structures, Computer Methods in Applied Mechanics and Engineering, 47 (1984) 219–259], the second is based on the so-called local parameterization [W. C. Rheinboldt, J. V. Burkadt, A Locally parameterized continuation, Acm Transaction on mathematical Software, 9 (1983) 215–235; R. Seydel, A Tracing Branches, World of Bifurcation, Online Collection and Tutorials of Nonlinear Phenomena ( http://www.bifurcation.de), 1999; J. J. Gervais, H. Sadiky, A new steplength control for continuation with the asymptotic numerical method, IAM, J. Numer. Anal. 22, No. 2, (2000) 207–229] and the third is based on a minimization condition of a rest [S. Lopez, An effective parametrization for asymptotic extrapolation, Computer Methods in Applied Mechanics and Engineering, 189 (2000) 297–311]. We demonstrate that the third concept is equivalent to a maximization condition of the ANM step lengths. To illustrate the performance of these proposed parameterizations, we give some numerical comparisons on nonlinear elastic shell problems.
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ISSN:0045-7825
1879-2138
DOI:10.1016/j.cma.2010.01.020