https://hal.science/hal-03285817Opreni, AndreaAndreaOpreniPOLIMI - Politecnico di Milano [Milan]Vizzaccaro, AlessandraAlessandraVizzaccaroUniversity of Bristol [Bristol]Frangi, AttilioAttilioFrangiPOLIMI - Politecnico di Milano [Milan]Touzé, CyrilCyrilTouzéIMSIA - UMR 9219 - Institut des Sciences de la mécanique et Applications industrielles - CEA - Commissariat à l'énergie atomique et aux énergies alternatives - ENSTA Paris - École Nationale Supérieure de Techniques Avancées - Université Paris-Saclay - CNRS - Centre National de la Recherche Scientifique - EDF R&D - EDF R&D - EDF - EDFModel Order Reduction based on Direct Normal Form: Application to Large Finite Element MEMS Structures Featuring Internal ResonanceHAL CCSD2021invariant manifold parametrisationnormal formnonlinear normal modesharmonic balancemodel order reductionnon-intrusive method[SPI.MECA.VIBR] Engineering Sciences [physics]/Mechanics [physics.med-ph]/Vibrations [physics.class-ph][MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS][SPI.MECA.STRU] Engineering Sciences [physics]/Mechanics [physics.med-ph]/Structural mechanics [physics.class-ph]Touzé, Cyril2021-07-13 16:10:562023-03-24 14:53:222021-07-15 18:42:49enJournal articleshttps://hal.science/hal-03285817/document10.1007/s11071-021-06641-7application/pdf1Dimensionality 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.