Coupling FEM with parameter continuation for analysis of bifurcations of periodic responses in nonlinear structures

Giovanni Formica, Andrea Arena, Walter Lacarbonara, Harry Dankowicz

Research output: Contribution to journalArticlepeer-review

Abstract

A computational framework is proposed to perform parameter continuation of periodic solutions of nonlinear, distributed-parameter systems represented by partial differential equations with time-dependent coefficients and excitations. The path-following procedure, encoded in the general-purpose Matlab-based computational continuation core (referred to below as coco), employs only the evaluation of the vector field of an appropriate spatial discretization; for example as formulated through an explicit finite-element discretization or through reliance on a black-box discretization. An original contribution of this paper is a systematic treatment of the coupling of coco with Comsol multiphysics, demonstrating the great flexibility afforded by this computational framework. Comsol multiphysics provides embedded discretization algorithms capable of accommodating a great variety of mechanical/physical assumptions and multiphysics interactions. Within this framework, it is shown that a concurrent bifurcation analysis may be carried out together with parameter continuation of the corresponding monodromy matrices. As a case study, we consider a nonlinear beam, subject to a harmonic, transverse direct excitation for two different sets of boundary conditions and demonstrate how the proposed approach may be able to generate results for a variety of structural models with great ease. The numerical results include primary-resonance, frequency-response curves together with their stability and two-parameter analysis of multistability regions bounded by the loci of fold bifurcations that occur along the resonance curves. In addition, the results of comsol are validated for the Mettler model of slender beams against an in-house constructed finite-element discretization scheme, the convergence of which is assessed for increasing number of finite elements.

Original languageEnglish (US)
Article number021013
JournalJournal of Computational and Nonlinear Dynamics
Volume8
Issue number2
DOIs
StatePublished - Jan 1 2013

Keywords

  • bifurcations
  • nonlinear PDEs
  • nonlinear finite elements
  • parameter continuation
  • path following
  • space discretization
  • stability

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Mechanical Engineering
  • Applied Mathematics

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