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.

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

“…Haller and Ponsioen [29]). These results in [60,73] provide loworder approximations to SSMs [29], whose computation up to arbitrarily high orders of accuracy has already been automated in prior work [61,63] for mechanical systems with diagonalized linear part. A major computational advance in the approach of Vizzaccaro et al [73] is the non-intrusive use of finite element software to compute normal form coefficients up to cubic order.…”

“…More recently, Vizzaccaro et al [73] and Opreni et al [60] have computed normal forms on second order, proportionally damped mechanical systems with up to cubic nonlinearities and derived explicit expressions up to cubic-order accuracy (see also Touzé et al [71] for a review). This is a direct application of the parametrization method via a normal form style parametrization to formally compute assumed invariant manifolds whose existence/uniqueness is a priori unclear (cf.…”

Frequency combs in a MEMS resonator featuring 1:2 internal resonance: ab initio reduced order modelling and experimental validation

“…First, only moderate transformations can be addressed, meaning that inertia nonlinearities cannot be included. This implies, for instance, that IC in the proposed form cannot be applied, for example, to micromirrors in large rotations [63], nor to cantilevers with large tip deflections. Secondly, a slow/fast decomposition of the system is required, which means that the active slave coordinates should have eigenfrequencies well above those of the master coordinates.…”

“…A NNM has been firstly defined as a synchronous vibration of the system and then generalized by the notion of invariant manifold [75,[81][82][83] and spectral submanifold [38,66]. Despite the generation of ROMs based on the concept of NNM has been proposed for both small systems with few degrees of freedom (dofs) and complex structures involving inertia and geometrical nonlinearities [63,89], its extension to multiphysic problems (e.g. electromechanics in MEMS) has not been addressed yet and poses important computational challenges.…”

Reduced Order Modeling of Nonlinear Vibrating Multiphysics Microstructures with Deep Learning-Based Approaches

“…For dissipative systems, the picture becomes more complicated as the whole phase space is foliated by an invariant manifolds tangent at the origin to the linear subspace [ 11 ]. The application of these methods to large finite element models (FEMs) has remained sporadic until recently but is currently receiving an impressive boost by contributions [ 12 , 13 , 14 , 15 ] in which a direct approach, called the direct parametrization of invariant manifolds (DPIM), bypasses the requirement of computing the whole modal basis. Applications to complex structures with millions of degrees of freedom (DOFs) and featuring internal resonances and parametric excitation have been recently demonstrated in [ 14 , 16 ].…”