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1. Comparison of Reduction Methods for Finite Element Geometrically Nonlinear Beam Structures

Author

Yichang Shen, Alessandra Vizzaccaro, Nassim Kesmia, Ting Yu, Loïc Salles, Olivier Thomas, and Cyril Touzé

Subjects
reduced-order model, direct normal form, geometric nonlinearity, modal derivatives, implicit condensation and expansion, Physics, QC1-999
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).

Published
2021
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2. Comparison of Reduction Methods for Finite Element Geometrically Nonlinear Beam Structures

Author

Loic Salles, Olivier Thomas, Yichang Shen, Alessandra Vizzaccaro, Nassim Kesmia, Ting Yu, Cyril Touzé, Institut des Sciences de la mécanique et Applications industrielles (IMSIA - UMR 9219), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS)-Université Paris-Saclay-EDF R&D (EDF R&D), EDF (EDF)-EDF (EDF), Imperial College London, Laboratoire d’Ingénierie des Systèmes Physiques et Numériques (LISPEN), Arts et Métiers Sciences et Technologies, HESAM Université (HESAM)-HESAM Université (HESAM), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-EDF R&D (EDF R&D), and HESAM Université - Communauté d'universités et d'établissements Hautes écoles Sorbonne Arts et métiers université (HESAM)-HESAM Université - Communauté d'universités et d'établissements Hautes écoles Sorbonne Arts et métiers université (HESAM)

Subjects
Technology, reduced-order model, SPECTRAL SUBMANIFOLDS, DIMENSION REDUCTION, 02 engineering and technology, Mechanics, COMPUTATION, Curvature, 01 natural sciences, CIRCULAR CYLINDRICAL-SHELLS, Modal derivatives, Implicit condensation and expansion, Mécanique: Vibrations [Sciences de l'ingénieur], Engineering, Quadratic equation, Geometric nonlinearity, 0203 mechanical engineering, SYSTEMS, SLOW-FAST DECOMPOSITION, 0103 physical sciences, Mécanique: Mécanique des structures [Sciences de l'ingénieur], Invariant (mathematics), 010301 acoustics, direct normal form, Physics, REDUCED-ORDER MODELS, Science & Technology, Mathematical analysis, [SPI.MECA.VIBR]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Vibrations [physics.class-ph], implicit condensation and expansion, geometric nonlinearity, modal derivatives, [SPI.MECA]Engineering Sciences [physics]/Mechanics [physics.med-ph], Linear subspace, Finite element method, lcsh:QC1-999, RESPONSE PREDICTION, Engineering, Mechanical, Reduced-order model, Nonlinear system, 020303 mechanical engineering & transports, model order reduction, [SPI.MECA.STRU]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Structural mechanics [physics.class-ph], LARGE-AMPLITUDE VIBRATIONS, SPHERICAL-SHELLS, [SPI.GCIV.DV]Engineering Sciences [physics]/Civil Engineering/Dynamique, vibrations, Reduction (mathematics), Beam (structure), lcsh:Physics
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).

Published
2021

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