Model Order Reduction based on Direct Normal Form: Application to Large Finite Element MEMS Structures Featuring Internal Resonance

Abstract

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.

Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques

“…A last known drawback of the method relies on its inability to properly take into account cases where iner-tia nonlinearity is important, as underlined for example in the case of the cantilever beam [117,208,257], or a micromirror in [213]. A method to bring a correction for this specific case has been implemented recently in [208].…”

“…The DNF has then been completely rewritten in [213], using complex-valued formalism inherited from using first-order, state-space formulation from the beginning of the calculation. The main advantages of the complete rewriting are to use symmetric formulations throughout the derivation, while keeping clear the mechanical context by always referring, in all the developments, to mass and stiffness matrix M and K, quadratic and cubic terms G and H. The homological equations have then been fully rewritten with these notations, opening the doors to numerous extensions of the method in order to go to higher orders or to add new physical phenomena described by new forces.…”

“…with Ψ , Υ polynomial mappings in R, S. Following the real-valued expressions, and by considering only second-order terms for the sake of brevity, the nonlinear mapping reads [213,297]:…”

“…The unknown vectors of coefficients âi j , bi j , ĉi j , αi j , βi j and γ i j are derived from the second-order homological equations, see [213,279,287] for more details. These vectors of coefficients are the equivalent in physical coordinates of the vectors in Eq.…”

“…( 48). The developments reported in [213,297] settles down the full formulation, unfolds the link with the parametrisation method of invariant manifolds, and opens the door to higher-order developments by giving the recursive general formulations. From the computational point of view, the real-valued formalism can also be fully written in a non-intrusive manner, such that the method as derived in [297] can be used from any FE code, provided the code allows a user to script for performing matrix operations online, such that all outputs can be computed without the need to export the full mass and stiffness matrices.…”