Discussion about parameterization in the asymptotic numerical method: Application to nonlinear elastic shells - Mottaqui, H.; Braikat, B.; Damil, N. (Computer methods in applied mechanics and engineering, 2010)
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.