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

2012 ◽  
Vol 8 (2) ◽  
Author(s):  
Giovanni Formica ◽  
Andrea Arena ◽  
Walter Lacarbonara ◽  
Harry Dankowicz
Keyword(s):  
Finite Element ◽  
Primary Resonance ◽  
Path Following ◽  
Parameter Analysis ◽  
Finite Element Discretization ◽  
Response Curves ◽  
Computational Framework ◽  
Element Discretization ◽  
Explicit Finite Element ◽  
Parameter Continuation

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 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 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.

Nonlinear Finite Element-Based Path Following of Periodic Solutions

2011 ◽  
Author(s):  
Andrea Arena ◽  
Giovanni Formica ◽  
Walter Lacarbonara ◽  
Harry Dankowicz
Keyword(s):  
Finite Element ◽  
Vector Field ◽  
Frequency Response ◽  
Periodic Solutions ◽  
Parametric Resonance ◽  
Primary Resonance ◽  
Path Following ◽  
State Variables ◽  
Response Curves ◽  
Space Dependence

A computational framework is proposed to path follow the periodic solutions of nonlinear spatially continuous systems and more general coupled multiphysics problems represented by systems of partial differential equations with time-dependent excitations. The set of PDEs is cast in first order differential form (in time) u˙ = f(u,s,t;c) where u(s,t) is the vector collecting all state variables including the velocities/time rates, s is a space coordinate (here, one-dimensional systems are considered without lack of generality for the space dependence) and t denotes time. The vector field f depends, in general, not only on the classical state variables (such as positions and velocities) but also on the space gradients of the leading unknowns. The space gradients are introduced as part of the state variables. This is justified by the mathematical and computational requirements on the continuity in space up to the proper differential order of the space gradients associated with the unknown position vector field. The path following procedure employs, for the computation of the periodic solutions, only the evaluation of the vector field f. This part of the path following procedure within the proposed combined scheme was formerly implemented by Dankowicz and coworkers in a MATLAB software package called COCO. The here proposed procedure seeks to discretize the space dependence of the variables using finite elements based on Lagrangian polynomials which leads to a discrete form of the vector field f. A concurrent bifurcation analysis is carried out by calculating the eigenvalues of the monodromy matrix. A hinged-hinged nonlinear beam subject to a primary-resonance harmonic transverse load or to a parametric-resonance horizontal end displacement is considered as a case study. Some primary-resonance frequency-response curves are calculated along with their stability to assess the convergence of the discretization scheme. The frequency-response curves are shown to be in close agreement with those calculated by direct integration of the PDEs through the FE software called COMSOL Multiphysics. Besides primary-resonance direct forcing conditions, also parametric forcing causing the principal parametric resonance of the lowest two bending modes is considered through construction of the associated transition curves. The proposed approach integrates algorithms from the finite element and bifurcation domains thus enabling an accurate and effective unfolding of the bifurcation and post-bifurcation scenarios of nonautonomous PDEs with the underlying structures.

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