Model reduction to spectral submanifolds and forced-response calculation in high-dimensional mechanical systems

Ponsioen, Sten; Jain, Shobhit; Haller, George

doi:10.1016/j.jsv.2020.115640

Cited by 64 publications

(82 citation statements)

References 28 publications

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“…While the SSMs can also be directly used to approximate the steady-state response [19][20][21] in nonlinear mechanical systems, their computational feasibility for realistic high-dimensional problems is a subject of ongoing research. Our method relies on SSM theory, but still employs the widely applied linear projection to obtain a ROM, which is straightforward to implement and whose computational advantages are well understood.…”

Section: Resultsmentioning

confidence: 99%

“…Several computational applications of the parametrization method have developed algorithms to solve these equations (e.g. [20][21][22]). For the second-order system (2.1), Verasztó et al [23] have already developed matrix equations to determine W k (see eqns (E.7-E.9) in [23]), but their expressions were less amenable to numerical implementation.…”

Section: Spectral Submanifoldsmentioning

confidence: 99%

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Using spectral submanifolds for optimal mode selection in nonlinear model reduction

2021

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Model reduction of large nonlinear systems often involves the projection of the governing equations onto linear subspaces spanned by carefully selected modes. The criteria to select the modes relevant for reduction are usually problem-specific and heuristic. In this work, we propose a rigorous mode-selection criterion based on the recent theory of spectral submanifolds (SSMs), which facilitates a reliable projection of the governing nonlinear equations onto modal subspaces. SSMs are exact invariant manifolds in the phase space that act as nonlinear continuations of linear normal modes. Our criterion identifies critical linear normal modes whose associated SSMs have locally the largest curvature. These modes should then be included in any projection-based model reduction as they are the most sensitive to nonlinearities. To make this mode selection automatic, we develop explicit formulae for the scalar curvature of an SSM and provide an open-source numerical implementation of our mode-selection procedure. We illustrate the power of this procedure by accurately reproducing the forced-response curves on three examples of varying complexity, including high-dimensional finite-element models.

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Section: Resultsmentioning

confidence: 99%

Section: Spectral Submanifoldsmentioning

confidence: 99%

Using spectral submanifolds for optimal mode selection in nonlinear model reduction

2021

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“…A remarkable advancement within the correct and formal definition and properties of nonlinear normal modes has been given by Haller et al in a series of papers [28,29,30,31]. In the conservative framework, existence and uniqueness of the searched invariant structures are given by the Lyapunov center theorem [32], stating that under non-resonance conditions, a two-dimensional manifold densely filled with periodic orbits exists for each couple of imaginary eigenvalues.…”

Section: Introductionmentioning

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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“…A recent development generalizes the notion of invariant manifolds and NNMs in order to propose a mathematically well-justified framework, allowing one to tackle conservative as well as dissipative systems. Spectral submanifolds (SSMs) were introduced in [31], and their use in model reduction was emphasized with different applications [32][33][34][35] to underline the accuracy of the method.…”

Section: Introductionmentioning

confidence: 99%

Comparison of Reduction Methods for Finite Element Geometrically Nonlinear Beam Structures

et al. 2021

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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).

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Model reduction to spectral submanifolds and forced-response calculation in high-dimensional mechanical systems

Cited by 64 publications

References 28 publications

Using spectral submanifolds for optimal mode selection in nonlinear model reduction

Using spectral submanifolds for optimal mode selection in nonlinear model reduction

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

Comparison of Reduction Methods for Finite Element Geometrically Nonlinear Beam Structures

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