vibration Article Comparison of Reduction Methods for Finite Element Geometrically Nonlinear Beam Structures Yichang Shen 1 , Alessandra Vizzaccaro 2 , Nassim Kesmia 1 , Ting Yu 1 , Loïc Salles 2 , Olivier Thomas 3 and Cyril Touzé 1, * Citation: Shen, Y.; Vizzaccaro, A.; Kesmia, N.; Yu, T.; Salles, L.; Thomas, O.; Touzé, C. Comparison of Reduction Methods for Finite Element Geometrically Nonlinear Beam Structures. Vibration 2021, 4, 175–204. https://doi.org/10.3390/ vibration4010014 Academic Editor: Paolo Tiso Received: 13 January 2021 Accepted: 1 March 2021 Published: 4 March 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: c 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Institut of Mechanical Sciences and Industrial Applications (IMSIA), ENSTA Paris, CNRS, EDF, CEA, Institut Polytechnique de Paris, 91762 Palaiseau, France; [email protected] (Y.S.); [email protected] (N.K.);[email protected] (T.Y.) 2 Vibration University Technology Centre, Imperial College London, London SW7 2AZ, UK; [email protected] (A.V.); [email protected] (L.S.) 3 Arts et Métiers Institute of Technology, LISPEN, HESAM Université, 8 boulevard Louis XIV, 59000 Lille, France; [email protected] * Correspondence: [email protected]; Tel.: +33-1-69-31-97-34 Abstract: The aim of this contribution is to present numerical comparisons of model-order reduction methods for geometrically nonlinear structures in the general framework of finite element (FE) procedures. Three different methods are compared: the implicit condensation and expansion (ICE), the quadratic manifold computed from modal derivatives (MD), and the direct normal form (DNF) procedure, the latter expressing the reduced dynamics in an invariant-based span of the phase space. The methods are first presented in order to underline their common points and differences, highlighting in particular that ICE and MD use reduction subspaces that are not invariant. A simple analytical example is then used in order to analyze how the different treatments of quadratic nonlinearities by the three methods can affect the predictions. Finally, three beam examples are used to emphasize the ability of the methods to handle curvature (on a curved beam), 1:1 internal resonance (on a clamped-clamped beam with two polarizations), and inertia nonlinearity (on a cantilever beam). Keywords: reduced-order model; direct normal form; geometric nonlinearity; modal derivatives; implicit condensation and expansion 1. Introduction Model reduction methods have been investigated for a long time for thin structures experiencing large-amplitude vibrations with geometric nonlinearities [1,2]. The two main identified difficulties are that the nonlinearity is distributed, and that the dynamical phenomena displayed by these nonlinear vibrations are numerous, including jump phe- nomena [3], bifurcations of solutions [4,5], internal resonance and modal interactions [6–9], strong couplings [10], transition to chaos [11,12], and wave turbulence [13]. Consequently, deriving accurate and predictive reduced-order models (ROMs) requires tackling these two problems in such a manner that the possible dynamics of the ROM can mimic all of the complexity of the full-order solution. In this contribution, only the nonlinear reduction methods, where a curved subspace is derived as a reduction manifold and/or a nonlinear mapping is used, are considered. All of the linear methods based on optimal orthogonal ba- sis selection, such as Proper Orthogonal Decomposition (POD), are not taken into account in the discussion and comparisons, since they have already been covered in numerous articles. The interested reader is referred, e.g., to [14–19] and references therein for the literature on these linear methods. Focusing on the nonlinear methods, the first steps can be traced back to the pioneering work by Shaw and Pierre, who introduced invariant manifolds—tangent at the origin to the Vibration 2021, 4, 175–204. https://doi.org/10.3390/vibration4010014 https://www.mdpi.com/journal/vibration
Comparison of Reduction Methods for Finite Element Geometrically Nonlinear Beam Structures
Jun 19, 2023
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