Coupling FEM With Parameter Continuation for Analysis of Bifurcations of Periodic Responses in Nonlinear Structures

Abstract

A computational framework is proposed to perform parameter continuation of periodic solutions of nonlinear, distributedparameter systems represented by partial differential equations with timedependent coefficients and excitations. The pathfollowing procedure, encoded in the generalpurpose Matlabbased 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 finiteelement discretization or through reliance on a blackbox discretization. An original contribution of this paper is a systematic treatment of the coupling of coco with Comsolmultiphysics, demonstrating the great flexibility afforded by this computational framework. Comsolmultiphysics 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 primaryresonance, frequencyresponse curves together with their stability and twoparameter 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 inhouse constructed finiteelement discretization scheme, the convergence of which is assessed for increasing number of finite elements.