Bifurcation points and bifurcated branches by an asymptotic numerical method and Padé approximants

Abstract

SUMMARYThe aim of this work is to develop a reliable and fast algorithm to compute bifurcation points and bifurcated branches. It is based upon the asymptotic numerical method (ANM) and Padé approximants. The bifurcation point is detected by analysing the poles of Padé approximants or by evaluating, along the computed solution branch, a bifurcation indicator well adapted to ANM. Several examples are presented to assess the effectiveness of the proposed method, that emanate from buckling problems of thin elastic shells. Especially problems involving large rotations are discussed.

An improved assumed strain solid–shell element formulation with physical stabilization for geometric non‐linear applications and elastic–plastic stability analysis

“…The beam undergoes large rotations, and after a linear pre-buckling regime along the fundamental path, a bifurcation point is detected, resulting in a bifurcated branch with symmetric stable bifurcation. A regular 10×2 mesh, with only a single layer of elements along the thickness, was used as in Reference [107], in which an EAS shell formulation was employed in conjunction with an efficient and accurate technique to pinpoint bifurcation points and to track the associated bifurcation branches. For this reason, those results were taken as reference solutions for the sake of comparison.…”

Modélisation numérique du flambage des plaques minces et applications au laminage

“…L'indicateur de bifurcation [7] [8] est obtenu en introduisant une perturbation fictive dans le problème. Il ne s'agit pas de modifier le problème initial mais de construire un problème auxiliaire qui permet d'évaluer l'indicateur sur toute la branche d'équilibre et de déterminer tous les points singuliers et leurs modes correspondants.…”

On the buckling and post-buckling of core-shell cylinders under thermal loading

“…Introduction of such small perturbation forces is a common technique for solving bifurcation problems by continuation techniques ( Doedel, 1981;Allgower and Georg, 1990 ), even when using commercial finite element codes. This artifice could be avoided by applying a specific procedure to compute the bifurcation branch as in Boutyour et al (2004) ;Vannucci et al (1998) . In this paper, the perturbation force f z allows computing the whole bifurcated branch with a single continuation algorithm.…”

“…It is incorporated via the Enhanced Assumed Strain (EAS) concept to improve the element performance and to avoid locking phenomena such as Poisson thickness locking, shear locking or volume locking. This hybrid shell formulation can describe large rotations and large displacements, and has been successively applied to nonlinear elastic thin-walled structures such as cantilever beam, square plate, cylindrical roof and circular deep arch ( Zahrouni et al, 1999;Boutyour et al, 2004 ). Formulations of geometry and kinematics of the shell element can be found in Xu et al (2014) ; 2015a ); Xu and Potier-Ferry (2016b ).…”