Integral equations and model reduction for fast computation of nonlinear periodic response

Buza, Gergely; Haller, George; Jain, Shobhit

doi:10.1002/nme.6740

Cited by 3 publications

(8 citation statements)

References 19 publications

(66 reference statements)

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“…( 105)). The FRC obtained via local computations of SSM at O (5) agrees with those obtained using global continuation methods involving the harmonic balance method (NLvib [46]) and collocation (coco [13]); the plot shows the displacement amplitude for in the x 3 direction at the tip of the beam (see Table 4 for computation times) relaxing these tolerances further has no effect on the continuation speed. Once again, the computation times in Table 4 indicate orders-of-magnitude higher speed in reliably approximating FRC via local SSMs computations in comparison with global techniques that involve collocation or spectral approximations.…”

Section: Von Kármán Beamsupporting

confidence: 58%

“…On a macrolevel, this wing example resembles a cantilevered beam and we expect a hardening type response. Indeed, the three FRCs at O(3), O (5), and O(7) converge toward a hardening-type response, as shown in Fig. 14b.…”

Section: Aircraft Wingmentioning

confidence: 75%

“…12b. Note that we expect a softening behavior in the FRC of shallow arches (see, e.g., Buza et al [4,5]). Due to excessive memory requirements, this FRC could not be computed using collocation approximations via coco [13] or using harmonic balance approximations via NLvib [46].…”

Section: Shallow-arch Structurementioning

confidence: 76%

“…Hence, we note that the invariance computation automatically accounts for the important physical effects arising due to nonlinearities in the form of the parametrizations W and R of the manifold and its reduced dynamics. These effects, otherwise, are typically captured by a heuristic projection of the governing equation onto carefully selected modes (see Jain et al [41], Buza et al [5] for a discussion).…”

Section: Von Kármán Beammentioning

confidence: 99%

“…We then compute the near-resonant FRC around the first natural frequency via O(3), O (5), and O(7) SSM computations using Lemma 2. Once again, Fig.…”

Section: Shallow-arch Structurementioning

confidence: 99%

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How to compute invariant manifolds and their reduced dynamics in high-dimensional finite element models

2021

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Invariant manifolds are important constructs for the quantitative and qualitative understanding of nonlinear phenomena in dynamical systems. In nonlinear damped mechanical systems, for instance, spectral submanifolds have emerged as useful tools for the computation of forced response curves, backbone curves, detached resonance curves (isolas) via exact reduced-order models. For conservative nonlinear mechanical systems, Lyapunov subcenter manifolds and their reduced dynamics provide a way to identify nonlinear amplitude–frequency relationships in the form of conservative backbone curves. Despite these powerful predictions offered by invariant manifolds, their use has largely been limited to low-dimensional academic examples. This is because several challenges render their computation unfeasible for realistic engineering structures described by finite element models. In this work, we address these computational challenges and develop methods for computing invariant manifolds and their reduced dynamics in very high-dimensional nonlinear systems arising from spatial discretization of the governing partial differential equations. We illustrate our computational algorithms on finite element models of mechanical structures that range from a simple beam containing tens of degrees of freedom to an aircraft wing containing more than a hundred–thousand degrees of freedom.

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Section: Von Kármán Beamsupporting

confidence: 58%

Section: Aircraft Wingmentioning

confidence: 75%

Section: Shallow-arch Structurementioning

confidence: 76%

Section: Von Kármán Beammentioning

confidence: 99%

“…We then compute the near-resonant FRC around the first natural frequency via O(3), O (5), and O(7) SSM computations using Lemma 2. Once again, Fig.…”

Section: Shallow-arch Structurementioning

confidence: 99%

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How to compute invariant manifolds and their reduced dynamics in high-dimensional finite element models

2021

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Collocation‐based harmonic balance framework for highly accurate periodic solution of nonlinear dynamical system

Dai

Zipu

Wang

et al. 2022

Numerical Meth Engineering

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Periodic dynamical systems ubiquitously exist in science and engineering. The harmonic balance (HB) method and its variants have been the most widely‐used approaches for such systems, but are either confined to low‐order approximations or impaired by aliasing and improper‐sampling problems. Here we propose a collocation‐based harmonic balance framework to successfully unify and reconstruct the HB‐like methods. Under this framework a new conditional identity, which exactly bridges the gap between frequency‐domain and time‐domain harmonic analyses, is discovered by introducing a novel aliasing matrix. Upon enforcing the aliasing matrix to vanish, we propose a powerful reconstruction harmonic balance (RHB) method that obtains extremely high‐order (>100) nonaliasing solutions, previously deemed out‐of‐reach, for a range of complex nonlinear systems including the cavitation bubble dynamics, the three‐body problem and the two dimensional airfoil dynamics. We show that the present method is 2–3 orders of magnitude more accurate and simultaneously much faster than the state‐of‐the‐art method. Hence, it has immediate applications in multidisciplinary problems where highly accurate periodic solutions are sought.

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How to Compute Invariant Manifolds and their Reduced Dynamics in High-Dimensional Finite-Element Models

Jain,

Haller

2021

Preprint

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Invariant manifolds are important constructs for the quantitative and qualitative understanding of nonlinear phenomena in dynamical systems. In nonlinear damped mechanical systems, for instance, spectral submanifolds have emerged as useful tools for the computation of forced response curves, backbone curves, detached resonance curves (isolas) via exact reduced-order models. For conservative nonlinear mechanical systems, Lyapunov subcenter manifolds and their reduced dynamics provide a way to identify nonlinear amplitude-frequency relationships in the form of conservative backbone curves. Despite these powerful predictions offered by invariant manifolds, their use has largely been limited to low-dimensional academic examples. This is because several challenges render their computation unfeasible for realistic engineering structures described by finite-element models. In this work, we address these computational challenges and develop methods for computing invariant manifolds and their reduced dynamics in very high-dimensional nonlinear systems arising from spatial discretization of the governing partial differential equations. We illustrate our computational algorithms on finite-element models of mechanical systems.

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Integral equations and model reduction for fast computation of nonlinear periodic response

Cited by 3 publications

References 19 publications

How to compute invariant manifolds and their reduced dynamics in high-dimensional finite element models

How to compute invariant manifolds and their reduced dynamics in high-dimensional finite element models

Collocation‐based harmonic balance framework for highly accurate periodic solution of nonlinear dynamical system

How to Compute Invariant Manifolds and their Reduced Dynamics in High-Dimensional Finite-Element Models

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