Parallel Harmonic Balance Method for Analysis of Nonlinear Dynamical Systems

Blahoš, Jiří; Vizzaccaro, Alessandra; Salles, Loïc; Haddad, Fadi El

doi:10.1115/gt2020-15392

Cited by 9 publications

(3 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Numerical computation of full-order models with the finite element (FE) method for such problems leads to prohibitive computational times that are generally not compatible with the constraints at the design stage, or asks for specific developments using e.g. parallel computing (Blahoš et al 2020;Blahoš 2022). Consequently, the development of efficient and accurate reduced-order models (ROM) that are able to tackle the geometric nonlinearity is a key feature.…”

Section: Introductionmentioning

confidence: 99%

Reduced-order modeling of geometrically nonlinear rotating structures using the direct parametrisation of invariant manifolds

Martin

Opreni

Vizzaccaro

et al. 2023

Journal of Theoretical, Computational and Applied Mechanics

View full text Add to dashboard Cite

The direct parametrisation method for invariant manifolds is a nonlinear reduction technique which derives nonlinear mappings and reduced-order dynamics that describe the evolution of dynamical systems along a low-dimensional invariant-based span of the phase space. It can be directly applied to finite element problems. When the development is performed using an arbitrary order asymptotic expansion, it provides an efficient reduced-order modeling strategy for geometrically nonlinear structures. It is here applied to the case of rotating structures featuring centrifugal effect. A rotating cantilever beam with large amplitude vibrations is first selected in order to highlight the main features of the method. Numerical results show that the method provides accurate reduced-order models (ROMs) for any rotation speed and vibration amplitude of interest with a single master mode, thus offering remarkable reduction in the computational burden. The hardening/softening transition of the fundamental flexural mode with increasing rotation speed is then investigated in detail and a ROM parametrised with respect to rotation speed and forcing frequencies is detailed. The method is then applied to a twisted plate model representative of a fan blade, showing how the technique can handle more complex structures. Hardening/softening transition is also investigated as well as interpolation of ROMs, highlighting the efficacy of the method.

show abstract

Section: Introductionmentioning

confidence: 99%

Reduced-order modeling of geometrically nonlinear rotating structures using the direct parametrisation of invariant manifolds

Martin

Opreni

Vizzaccaro

et al. 2023

Journal of Theoretical, Computational and Applied Mechanics

View full text Add to dashboard Cite

show abstract

“…Then, a MEMS beam res-onator that features a 1:3 internal resonance [68], and a MEMS arch resonator showing 1:2 internal resonance are selected in order to show the ability of the method to deal with systems that feature internally resonant modes. All numerical simulations in the present work are compared with full-order harmonic balance finite element simulations of the systems [69][70][71].…”

Section: Introductionmentioning

confidence: 99%

Model order reduction based on direct normal form: application to large finite element MEMS structures featuring internal resonance

et al. 2021

View full text Add to dashboard Cite

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.

show abstract

“…Analysis of inexact versus exact Newton solvers is motivated by their use in applications (e.g. [5,8,22]) and necessitated by the fact that the HB method requires evaluating a residual defined by Fourier integrals. The Hilbert space formulation makes our analysis applicable in a variety of contexts where HB is used: integral equations (e.g.…”

mentioning

confidence: 99%

Convergence of the harmonic balance method for smooth Hilbert space valued differential-algebraic equations

Steyer¹,

Kuether²

2021

Preprint

View full text Add to dashboard Cite

We analyze the convergence of the harmonic balance method for computing isolated periodic solutions of a large class of continuously differentiable Hilbert space valued differentialalgebraic equations (DAEs). We establish asymptotic convergence estimates for (i) the approximate periodic solution in terms of the number of approximated harmonics and (ii) the inexact Newton method used to compute the approximate Fourier coefficients. The convergence estimates are determined by the rate of convergence of the Fourier series of the exact solution and the structure of the DAE. Both the case that the period is known and unknown are analyzed, where in the latter case we require enforcing an appropriately defined phase condition. The theoretical results are illustrated with several numerical experiments from circuit modeling and structural dynamics.

show abstract

Parallel Harmonic Balance Method for Analysis of Nonlinear Dynamical Systems

Cited by 9 publications

References 12 publications

Reduced-order modeling of geometrically nonlinear rotating structures using the direct parametrisation of invariant manifolds

Reduced-order modeling of geometrically nonlinear rotating structures using the direct parametrisation of invariant manifolds

Model order reduction based on direct normal form: application to large finite element MEMS structures featuring internal resonance

Convergence of the harmonic balance method for smooth Hilbert space valued differential-algebraic equations

Contact Info

Product

Resources

About