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 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.