High order direct parametrisation of invariant manifolds for model order reduction of finite element structures: application to large amplitude vibrations and uncovering of a folding point

Abstract

This paper investigates model-order reduction methods for geometrically nonlinear structures. The parametrisation method of invariant manifolds is used and adapted to the case of mechanical systems expressed in the physical basis, so that the technique is directly applicable to problems discretised by the finite element method. Two nonlinear mappings, respectively related to displacement and velocity, are introduced, and the link between the two is made explicit at arbitrary order of expansion. The same development is performed on the reduced-order dynamics which is computed at generic order following the different styles of parametrisation. More specifically, three different styles are introduced and commented: the graph style, the complex normal form style and the real normal form style. These developments allow making better connections with earlier works using these parametrisation methods. The technique is then applied to three different examples. A clampedclamped arch with increasing curvature is first used to show an example of a system with a softening behaviour turning to hardening at larger amplitudes, which can be replicated with a single mode reduction. Secondly, the case of a cantilever beam is investigated. It is shown that the invariant manifold of the first mode shows a folding point at large amplitudes which is not connected to an internal resonance. This exemplifies the failure of the graph style due to the folding point, whereas the normal form style is able to pass over the folding. Finally, A MEMS micromirror undergoing large rotations is used to show the importance of using high-order expansions on an industrial example.

Deep learning-based reduced order models for the real-time simulation of the nonlinear dynamics of microstructures

“…It is worth stressing that the generation of a suitable ROM is a tough challenge even for the most advanced and recent techniques, like the Direct Normal Form approach, mainly because the torsional mode is not the lowest-frequency one (it is the third) and is not well separated from the other modes. Indeed, the quadratic formulation of the Direct Normal Form approach implemented by Opreni et al [51] fails and a high order expansion is required, as remarked by Vizzaccaro et al [67]. Similar difficulties have been experienced with the Implicit Condensation approach [19].…”

“…Compared to other surrogate models exploiting machine/deep learning algorithms, a distinguishing feature of POD-G DL-ROMs is their capability to compute the whole solution field, for any new parameter instance and time instant, at testing time, thus enabling the extremely efficient evaluation of any output quantity of interest depending on the solution field. The technique proposed in this work can be considered as the Data Driven counterpart of the Direct Normal Form method for invariant manifolds that has been recently applied to large scale finite element systems of mechanical structures [68,51,67].…”

Virtual twins of nonlinear vibrating multiphysics microstructures: physics-based versus deep learning-based approaches

“…For dissipative systems, the picture gets more complicated, as the whole phase space is foliated by invariant manifolds tangent at the origin to the linear subspace [10]. 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 [11,12,13,14] in which direct approaches, called Direct Parametrization of Invariant Manifolds (DPIM), bypass the requirement of computing the whole modal basis. Applications to complex structures with millions of degrees of freedom (DOFs) and featuring also internal resonances and parametric excitation have been recently demonstrated in [13,15].…”

“…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 [11,12,13,14] in which direct approaches, called Direct Parametrization of Invariant Manifolds (DPIM), bypass the requirement of computing the whole modal basis. Applications to complex structures with millions of degrees of freedom (DOFs) and featuring also internal resonances and parametric excitation have been recently demonstrated in [13,15]. However, their extension to coupled problems and nonlinearities of generic type is still an open issue, and requires dedicated developments.…”

Finite Element Approximation of Invariant Manifolds by the Parameterization Method

“…More recently the parameterization method has been used to develop a mathematically rigorous approach to optimal mode selection in nonlinear model reduction by projecting onto spectral submanifolds [31,4,7]. This research direction has been further developed and combined with large finite element systems demonstrating its potential for industrial applications [52,46].…”