Model reduction for constrained mechanical systems via spectral submanifolds

Li, Mingwu; Jain, Shobhit; Haller, George

doi:10.1007/s11071-023-08300-5

Cited by 21 publications

(4 citation statements)

References 80 publications

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“…Many high-dimensional chaotic dynamical systems can be approximated by a low-dimensional system (35)(36)(37)(38). Although the underlying dynamics of earthquakes and Slow Slip cycles are often chaotic (24)(25)(26)(27)(28), in certain examples, it has been observed that the chaotic attractors are low dimensional (8,39) which mathematically implies that we can approximate the evolution of the sequence of events using parameters in a finite-dimensional space instead of an infinite function space.…”

Section: Model Reduction and Forecast Schemementioning

confidence: 99%

Spatiotemporal forecast of extreme events in a dynamical model of earthquake sequences

Kaveh,

Avouac,

Stuart

2024

Preprint

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Seismic (‘earthquakes’) and aseismic (‘slow earthquakes’) slip events result from episodic slips on faults and are often chaotic due to stress heterogeneity. Their predictability in nature is a widely open question. Here, we forecast extreme events in a numerical model of a single fault governed by rate-and-state friction, which produces realistic sequences of slow events with a wide range of magnitudes and inter-event times. The complex dynamics of this system arise from partial ruptures. As the system self-organizes, prestress is confined to a chaotic attractor of a relatively small dimension. We identify the instability regions (corresponding to particular stress distributions) within this attractor which are precursors of large events. We show that large events can be forecasted in time and space based on the determination of these instability regions in a low-dimensional space and the knowledge of the current slip rate on the fault.

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Section: Model Reduction and Forecast Schemementioning

confidence: 99%

Spatiotemporal forecast of extreme events in a dynamical model of earthquake sequences

Kaveh,

Avouac,

Stuart

2024

Preprint

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“…In its initial formulation, SSM reduction constructed the parametrization of the invariant manifold and the reduced dynamics as a Taylor expansion around a steady state, which proved fruitful in reduced-order modelling for general mechanical systems (e.g. Jain & Haller 2022; Li, Jain & Haller 2023). In addition to its strict mathematical foundation, a noteworthy advantage of SSM-based model reduction over projection-based methods is that the dimension of the slowest non-resonant spectral subspace a priori determines the dimension of the reduced-order model.…”

Section: Introductionmentioning

confidence: 99%

Capturing the edge of chaos as a spectral submanifold in pipe flows

Kaszás,

Haller

2024

J. Fluid Mech.

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An extended turbulent state can coexist with the stable laminar state in pipe flows. We focus here on short pipes with additional discrete symmetries imposed. In this case, the boundary between the coexisting basins of attraction, often called the edge of chaos, is the stable manifold of an edge state, which is a lower-branch travelling wave solution. We show that a low-dimensional submanifold of the edge of chaos can be constructed from velocity data using the recently developed theory of spectral submanifolds (SSMs). These manifolds are the unique smoothest nonlinear continuations of non-resonant spectral subspaces of the linearized system at stationary states. Using very low-dimensional SSM-based reduced-order models, we predict transitions to turbulence or laminarization for velocity fields near the edge of chaos.

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“…DAEs are for example the starting formalism of the Manlab numerical continuation package that uses the asymptotic numerical method on quadratic DAE to compute their solutions as a parameter is varied [43], or can be used for time integration [44]. Considering the computation of reduced-order models for DAE using invariant manifolds has been considered recently in [45] with the spectral submanifold framework. However, in this work the authors transform the DAE into an ODE by taking the derivative of the algebraic equation with relation to time as many times as necessary.…”

Section: Introductionmentioning

confidence: 99%

“…45) Taylor (order 11) NF (order 45) Taylor (order 21) NF (order 45) Taylor (order 31) NF (order 45) ODE for sin NF (order 45) ODE for sin NF (order 45) Chebyshev 1 (cartesian)) NF (order 45) Chebyshev 2 (quaternion-like)) NF (order 45) Chebyshev 3 NF (order 45) Chebyshev 4…”

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High order invariant manifold model reduction for systems with non-polynomial non-linearities: geometrically exact finite-element structures and validity limit

GROLET,

Vizzaccaro,

Debeurre

et al. 2024

Preprint

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This paper considers the computation of reduced-order models for systems of ordinary differential equations that include non-polynomial non-linearities. An targeted example is the case of a geometrically exact model of highly flexible slender structure, that includes, after space discretisation, trigonometric non-linear terms. With a suitable change of variables, this system can be rewritten in an equivalent one with polynomial non-linearities at most quadratic, at the price of introducing additional variables linked to algebraic equations, leading to a differential algebraic set of equations (DAE) to be solved. This DAE is reduced thanks to a normal form parametrisation of its invariant manifolds and selecting a set of master ones. Arbitrary order expansions are detailed for the coefficients of the change of variable and the reduced dynamics, using linear algebra in the space of multivariate polynomials of a given degree. In the case of a single non-linear mode reduction, a criterion to evaluate the quality of the normal form results is also proposed based on an estimation of the convergence radius of the polynomial asymptotic expansion representing truncated series. The method is then applied to compute a single mode reduction of three test cases -- a Duffing oscillator, a simple pendulum and a clamped clamped beam with von~K\'arm\'an model --, in order to investigate the effect of the algebraic part of the DAE on the quality of the model reduction and its validity range. Then, the more involved case of a cantilever beam modelled by geometrically exact finite elements is considered, underlining the ability of the method to produce accurate and converged results in a range of amplitude that can be bounded thanks to a convergence criterion.

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Model reduction for constrained mechanical systems via spectral submanifolds

Cited by 21 publications

References 80 publications

Spatiotemporal forecast of extreme events in a dynamical model of earthquake sequences

Spatiotemporal forecast of extreme events in a dynamical model of earthquake sequences

Capturing the edge of chaos as a spectral submanifold in pipe flows

High order invariant manifold model reduction for systems with non-polynomial non-linearities: geometrically exact finite-element structures and validity limit

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