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An algorithm and a method to search bifurcation points in non‐linear problems
Author(s) -
Chambon R.,
Crochepeyre S.,
Charlier R.
Publication year - 2001
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.199
Subject(s) - uniqueness , mathematics , bifurcation , simple (philosophy) , bifurcation theory , algorithm , numerical analysis , mathematical optimization , nonlinear system , mathematical analysis , philosophy , epistemology , quantum mechanics , physics
Abstract This paper is devoted to a numerical method able to help the determination of the bifurcation threshold in non‐linear time‐independent continuum mechanic problems. First, some theoretical results about uniqueness are recalled. In the framework of the large‐strain assumption, the differences between the classical finite‐step problem and the rate problem are presented. An iterative algorithm able to solve the rate problem is given. Using different initializations, it is seen in some numerical experiments that it is possible with this algorithm to get different solutions when the underlying mathematical problem solved does not enjoy a uniqueness property. The constitutive equations used have been chosen to be simple enough to deduce some theoretical knowledge about the corresponding uniqueness problems. Finally, a method is given which is able in some case to give an upper bound of the bifurcation threshold. Copyright © 2001 John Wiley & Sons, Ltd.