Simultaneous normal form transformation and model-order reduction for systems of coupled nonlinear oscillators

Liu, X.; Wagg, DJ

doi:10.1098/rspa.2019.0042

Cited by 12 publications

(10 citation statements)

References 43 publications

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“…Also, only the first term in the normal form expansion was taken into account, leading to an incorrect prediction of the type of nonlinearity for systems with quadratic and cubic nonlinearity, as underlined in [19]. The problem has then been corrected and the link to reduced-order models underlined in [152]. Other contributions also tackled the problem of systems with periodic coefficients and/or periodic forcing, combining the Lyapunov-Floquet with a normal transform, see, e.g.…”

Section: Applicationsmentioning

confidence: 99%

Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques

2021

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This paper aims at reviewing nonlinear methods for model order reduction in structures with geometric nonlinearity, with a special emphasis on the techniques based on invariant manifold theory. Nonlinear methods differ from linear-based techniques by their use of a nonlinear mapping instead of adding new vectors to enlarge the projection basis. Invariant manifolds have been first introduced in vibration theory within the context of nonlinear normal modes and have been initially computed from the modal basis, using either a graph representation or a normal form approach to compute mappings and reduced dynamics. These developments are first recalled following a historical perspective, where the main applications were first oriented toward structural models that can be expressed thanks to partial differential equations. They are then replaced in the more general context of the parametrisation of invariant manifold that allows unifying the approaches. Then, the specific case of structures dis-C. Touzé (B)

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Section: Applicationsmentioning

confidence: 99%

Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques

2021

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“…The derived expressions are then identical to the one found by first assuming a parametrisation, then solving the resulting tangent and normal homological problems using the normal form style, as stated in [27]. In the framework of mechanical systems, the parametrisation of invariant manifolds has been extensively treated in [28,29]; the application of the normal form approach to model order reduction is instead adopted in the developments led in [9,21,77].…”

Section: Reduction and Normal Form Style Parametrisationmentioning

confidence: 99%

Model Order Reduction based on Direct Normal Form: Application to Large Finite Element MEMS Structures Featuring Internal Resonance

Opreni

Vizzaccaro

Frangi

et al. 2021

Preprint

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Dimensionality reduction in mechanical vibratory systems poses challenges for distributed structures including geometric nonlinearities, mainly because of the lack of invariance of the linear subspaces. A reduction method based on direct normal form computation for large finite element (FE) models is here detailed. The main advantage resides in operating directly from the physical space, hence avoiding the computation of the complete eigenfunctions spectrum. Explicit solutions are given, thus enabling a fully non-intrusive version of the reduction method. The reduced dynamics is obtained from the normal form of the geometrically nonlinear mechanical problem, free of non-resonant monomials, and truncated to the selected master coordinates, thus making a direct link with the parametrisation of invariant manifolds. The method is fully expressed with a complex-valued formalism by detailing the homological equations in a systematic manner, and the link with real-valued expressions is established. A special emphasis is put on the treatment of second-order internal resonances and the specific case of a 1:2 resonance is made explicit. Finally, applications to large-scale models of Micro-Electro-Mechanical structures featuring 1:2 and 1:3 resonances are reported, along with considerations on computational efficiency.

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“…In order to illustrate the previous results, the system composed of a mass connected to two nonlinear springs, is selected. Note that this system has been used in a number of studies so that numerous results are already present in the literature on this example [27,28,59,61,62].…”

Section: Example Systemmentioning

confidence: 99%

Reduced order models for geometrically nonlinear structures: Assessment of implicit condensation in comparison with invariant manifold approach

Shen

Béreux

Frangi

et al. 2021

European Journal of Mechanics - A/Solids

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A comparison between two methods to derive reduced-order models (ROM) for geometrically nonlinear structures is proposed. The implicit condensation and expansion (ICE) method relies on a series of applied static loadings. From this set, a stress manifold is constructed for building the ROM. On the other hand, nonlinear normal modes rely on invariant manifold theory in order to keep the key property of invariance for the reduced subspaces. When the model coefficients are fully known, the ICE method reduces to a static condensation. However, in the framework of finite element discretisation, getting all these coefficients is generally too computationally expensive. The stress manifold is shown to tend to the invariant manifold only when a slow/fast decomposition between master and slave coordinates can be assumed. Another key problem in using the ICE method is related to the fitting procedure when a large number of modes need to be taken into account. A simplified procedure, relying on normal form theory and identification of only resonant monomial terms in the nonlinear stiffness, is proposed and contrasted with the current method. All the findings are illustrated on beams and plates examples.

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Simultaneous normal form transformation and model-order reduction for systems of coupled nonlinear oscillators

Cited by 12 publications

References 43 publications

Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques

Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques

Model Order Reduction based on Direct Normal Form: Application to Large Finite Element MEMS Structures Featuring Internal Resonance

Reduced order models for geometrically nonlinear structures: Assessment of implicit condensation in comparison with invariant manifold approach

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