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

TL;DR

This paper develops a high-order invariant manifold parametrisation method for nonlinear finite element structures, enabling accurate model reduction and analysis of large amplitude vibrations, including complex phenomena like folding points.

Contribution

It introduces a novel high-order parametrisation approach adapted for physical basis systems, with three styles, and demonstrates its effectiveness on complex nonlinear structural examples.

Findings

Successfully models softening and hardening behaviors in nonlinear structures.

Identifies folding points in invariant manifolds of large amplitude vibrations.

Shows high-order expansions are crucial for industrial-scale MEMS applications.

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 clamped-clamped 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.