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

TL;DR

This paper introduces a direct normal form-based model order reduction method for large finite element models of mechanical systems, effectively handling internal resonances and nonlinearities without requiring full eigenfunction spectra.

Contribution

It presents a non-intrusive, physically-based reduction technique using direct normal form computation, explicitly addressing internal resonances in large-scale FE models.

Findings

Effective reduction of large FE models with internal resonances

Explicit handling of 1:2 and 1:3 internal resonances

Demonstrated computational efficiency in MEMS structures

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.