Publications

2021

• Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques
  
  • Touzé Cyril
  • Vizzaccaro Alessandra
  • Thomas Olivier
  Nonlinear Dynamics, Springer Verlag, 2021, 105. This paper aims at reviewing nonlinear methods for model order reduction of structures with geometric nonlinearity, with a special emphasis on the techniques based on invariant manifold theory. Nonlinear methods differ from linear-based techniques by their use of a nonlinear mapping instead of adding new vectors to enlarge the projection basis. Invariant manifolds have been first introduced in vibration theory within the context of nonlinear normal modes (NNMs) and have been initially computed from the modal basis, using either a graph representation or a normal form approach to compute mappings and reduced dynamics. These developments are first recalled following a historical perspective, where the main applications were first oriented toward structural models that can be expressed thanks to partial differential equations (PDE). They are then replaced in the more general context of the parametrisation of invariant manifold that allows unifying the approaches. Then the specific case of structures discretized with the finite element method is addressed. Implicit condensation, giving rise to a projection onto a stress manifold, and modal derivatives, used in the framework of the quadratic manifold, are first reviewed. Finally, recent developments allowing direct computation of reduced-order models (ROMs) relying on invariant manifolds theory are detailed. Applicative examples are shown and the extension of the methods to deal with further complications are reviewed. Finally, open problems and future directions are highlighted. (10.1007/s11071-021-06693-9)
  DOI : 10.1007/s11071-021-06693-9
• A stable, unified model for resonant Faraday cages
  
  • Delourme Bérangère
  • Lunéville Éric
  • Marigo Jean-Jacques
  • Maurel Agnès
  • Mercier Jean-François
  • Pham Kim
  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Royal Society, 2021, 477 (2245), pp.20200668. We study some effective transmission conditions able to reproduce the effect of a periodic array of Dirichlet wires on wave propagation, in particular when the array delimits an acoustic Faraday cage able to resonate. In the study of Hewett & Hewitt (2016 Proc. R. Soc. A 472 , 20160062 ( doi:10.1098/rspa.2016.0062 )) different transmission conditions emerge from the asymptotic analysis whose validity depends on the frequency, specifically the distance to a resonance frequency of the cage. In practice, dealing with such conditions is difficult, especially if the problem is set in the time domain. In the present study, we demonstrate the validity of a simpler unified model derived in Marigo & Maurel (2016 Proc. R. Soc. A 472 , 20160068 ( doi:10.1098/rspa.2016.0068 )), where unified means valid whatever the distance to the resonance frequencies. The effectiveness of the model is discussed in the harmonic regime owing to explicit solutions. It is also exemplified in the time domain, where a formulation guaranteeing the stability of the numerical scheme has been implemented. (10.1098/rspa.2020.0668)
  DOI : 10.1098/rspa.2020.0668
• Model Order Reduction based on Direct Normal Form: Application to Large Finite Element MEMS Structures Featuring Internal Resonance
  
  • Opreni Andrea
  • Vizzaccaro Alessandra
  • Frangi Attilio
  • Touzé Cyril
  Nonlinear Dynamics, Springer Verlag, 2021. 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. (10.1007/s11071-021-06641-7)
  DOI : 10.1007/s11071-021-06641-7
• Invariant integrals and asymptotic fields near the front of a curved planar crack
  
  • Stolz Claude
  International Journal of Fracture, Springer Verlag, 2021. A plane crack is considered and the influence of local curvature of the crack front on the local mechanical fields is studied. The main goal is to determine the stress intensity factors along a curved planar crack in linear elasticity with accuracy. This is obtained by determination of new test fields and the use of bilinear forms, issued from invariant integrals, which separate the local modes of fracture. (10.1007/s10704-021-00552-9)
  DOI : 10.1007/s10704-021-00552-9