Finite element approximation of invariant manifolds by the parameterization method

Gonzalez, Jorge; James, J. D. Mireles; Tuncer, Necibe

doi:10.1007/s42985-022-00214-y

Cited by 6 publications

(4 citation statements)

References 53 publications

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“…Instead, we will implement a fairly recent technique called the parameterization methoddeveloped in the early 2000s [6,7,8]. We merely utilize it here for its benefits as a computational tool (see [26] for a recent review detailing its numerical aspects), but the parameterization method has otherwise created a blooming subject area in dynamical systems, fueling much of today's research [26,41,53,23]. Crucially, the method envisions the manifold as the image of an embedding -rather than the above detailed graph approach -circumventing the issue regarding folds, and hence possibly enlarging its (initial) 4 domain of existence.…”

Section: Parameterizationmentioning

confidence: 99%

“…There are a few recently-emerged techniques that attempt to mitigate these costs via reducing the number of coefficients needed to be stored. For instance, an alternative form of discretization is encouraged in [23] for parabolic PDEs, the finite element method. While this seems advantageous for most applications due to the sparsity of coefficient matrices, finite elements generally require significantly higher resolutions when compared to Chebyshev-Fourier methods in the specific context of fluid flows (especially wall-bounded flows), which would likely diminish the advantages of sparsity.…”

Section: Power Series Expansionsmentioning

confidence: 99%

“…For the case of ODEs, the local theory of invariant manifolds around fixed points (and invariant tori) has now reached a fully-developed state with the emergence of the parameterization method [6,7,8] (and [28,27] for tori). The technique also benefits from numerous computational advantages (over traditional graph transform methods) [26,41,53,23], and hence has been implemented in our numerical procedure. Perhaps among its most appeasing qualities, the resulting algorithm is constrained to converge to a unique smoothest submanifold within the discretized setting (see Theorem 1.1 [6]), thereby eliminating the divergence/ambiguity issues present in e.g.…”

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confidence: 99%

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Spectral Submanifolds of the Navier-Stokes Equations

Buza¹

2023

Preprint

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Spectral subspaces of a linear dynamical system identify a large class of invariant structures that highlight/isolate the dynamics associated to select subsets of the spectrum. The corresponding notion for nonlinear systems is that of spectral submanifolds -manifolds invariant under the full nonlinear dynamics that are determined by their tangency to spectral subspaces of the linearized system. In light of the recently-emerged interest in their use as tools in model reduction, we propose an extension of the relevant theory to the realm of fluid dynamics. We show the existence of a large (and the most pertinent) subclass of spectral submanifolds and foliations -describing the behaviour of nearby trajectories -about fixed points and periodic orbits of the Navier-Stokes equations. Their uniqueness and smoothness properties are discussed in detail, due to their significance from the perspective of model reduction. The machinery is then put to work via a numerical algorithm developed along the lines of the parameterization method, that computes the desired manifolds as power series expansions. Results are shown within the context of 2D channel flows.

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Section: Parameterizationmentioning

confidence: 99%

Section: Power Series Expansionsmentioning

confidence: 99%

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confidence: 99%

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Spectral Submanifolds of the Navier-Stokes Equations

Buza¹

2023

Preprint

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“…A review of model reduction for geometrically nonlinear structures can be found in [21] with more recent references in [32,38]. Some works also considers application to nonlinear PDE solved with the finite element method [39]. .…”

Section: Introductionmentioning

confidence: 99%

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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High order direct parametrisation of invariant manifolds for model order reduction of finite element structures: application to large amplitude vibrations and uncovering of a folding point

et al. 2022

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This paper investigates model-order reduction methods for geometrically nonlinear structures. The parametrisation method of invariant manifolds is used and adapted to the case of mechanical systems in oscillatory form expressed in the physical basis, so that the technique is directly applicable to mechanical problems discretised by the finite element method. Two nonlinear mappings, respectively related to displacement and velocity, are introduced, and the link between the two is made explicit at arbitrary order of expansion, under the assumption that the damping matrix is diagonalised by the conservative linear eigenvectors. The same development is performed on the reduced-order dynamics which is computed at generic order following different styles of parametrisation. More specifically, three different styles are introduced and commented: the graph style, the complex normal form style and the real normal form style. These developments allow making better connections with earlier works using these parametrisation methods. The technique is then applied to three different examples. A clamped-clamped arch with increasing curvature is first used to show an example of a system with a softening behaviour turning to hardening at larger amplitudes, which can be replicated with a single mode reduction. Secondly, the case of a cantilever beam is investigated. It is shown that invariant manifold of the first mode shows a folding point at large amplitudes. This exemplifies the failure of the graph style due to the folding point on a real structure, whereas the normal form style is able to pass over the folding. Finally, a MEMS (Micro Electro Mechanical System) micromirror undergoing large rotations is used to show the importance of using high-order expansions on an industrial example.

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Finite element approximation of invariant manifolds by the parameterization method

Cited by 6 publications

References 53 publications

Spectral Submanifolds of the Navier-Stokes Equations

Spectral Submanifolds of the Navier-Stokes Equations

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

High order direct parametrisation of invariant manifolds for model order reduction of finite element structures: application to large amplitude vibrations and uncovering of a folding point

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