Abstract
AbstractThe asymptotic‐numerical method (ANM) is a path following technique which is based on high order power series expansions. In this paper, we analyse its behaviour when it is applied to the continuation of a branch with bifurcation points. We show that when the starting point of the continuation is near a bifurcation, the radius of convergence of the power series is exactly the distance from the starting point to the bifurcation. This leads to an accumulation of small steps around the bifurcation point. This phenomenon is related to the presence of inevitable imperfections in the FE models. We also explain that, depending on the maximal tolerated residual error (out‐of‐balance error), the ANM continuation may continue to follow the fundamental path or it may turn onto the bifurcated path without applying any branch switching technique. Copyright © 2003 John Wiley & Sons, Ltd.
Highlights
The asymptotic-numerical-method (ANM) is an alternative to the classical Newton–Raphson techniques [1] for making the continuation of a non-linear solution with respect to a parameter, typically in structural mechanics, for tracing an equilibrium path with respect to a loading parameter [2,3,4]
We show that when the starting point of the continuation is near a bifurcation, the radius of convergence of the power series is exactly the distance from the starting point to the bifurcation
We have explained the behaviour of the ANM continuation when there is a bifurcation on a path
Summary
The asymptotic-numerical-method (ANM) is an alternative to the classical Newton–Raphson techniques [1] for making the continuation of a non-linear solution with respect to a parameter, typically in structural mechanics, for tracing an equilibrium path with respect to a loading parameter [2,3,4]. The basic principle of the ANM continuation is to determine the path by a succession of high order power series expansions (perturbation method) with respect to a well chosen path parameter It is a high order ‘continuous’ predictor without any correction. The step-length has not to be estimated in advance and, if necessary, to be adjusted according to the convergence behaviour of the corrector. It is always determined after the computation of the power series by analysing their radius of convergence. Using this crucial information, the design of robust continuation algorithms with automatically determined optimal step-length is easy. We begin with a minimal review of the ANM continuation, before embarking on the detail of the continuation of a path with a bifurcation
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