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

Abstract

Invariant manifolds are important constructs for the quantitative and qualitative understanding of nonlinear phenomena in dynamical systems. In nonlinear damped mechanical systems, for instance, spectral submanifolds have emerged as useful tools for the computation of forced response curves, backbone curves, detached resonance curves (isolas) via exact reduced-order models. For conservative nonlinear mechanical systems, Lyapunov subcenter manifolds and their reduced dynamics provide a way to identify nonlinear amplitude–frequency relationships in the form of conservative backbone curves. Despite these powerful predictions offered by invariant manifolds, their use has largely been limited to low-dimensional academic examples. This is because several challenges render their computation unfeasible for realistic engineering structures described by finite element models. In this work, we address these computational challenges and develop methods for computing invariant manifolds and their reduced dynamics in very high-dimensional nonlinear systems arising from spatial discretization of the governing partial differential equations. We illustrate our computational algorithms on finite element models of mechanical structures that range from a simple beam containing tens of degrees of freedom to an aircraft wing containing more than a hundred–thousand degrees of freedom.

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

“…Consequently, prior developments led in [17,18,21] with damping included were low-order approximations of the SSM, either with a graph style or a normal form style. Along the same lines, and as remarked in [47], the computations proposed in this contribution as well as those shown for example in [44,45,46], are approximations of the unique SSM, which is reached at a very high order only. Importantly, all lower order approximations of the SSM share the invariance property, up to the selected order, and can be thus used safely to provide accurate ROMs.…”

“…In short, whereas normal form expansion, as proposed in [20,21,42,43], first computes the complete nonlinear mapping and then reduces by selecting a few master normal coordinates, the parametrisation method first reduces by selecting the master coordinates, and then computes the expansions, with the added value that different solutions are possible, thus offering the possibility of using either a graph style or a normal form style. With this initial choice, the developments are thus closer to those already reported in [44,45,46], where arbitrary order expansions have already been shown, together with the possibility of using either graph or normal form style. The main differences can be listed as follows: (i) the focus here is on large FE models of mechanical systems for which the damping matrix is diagonalised by the eigenvectors of the conservative system; (ii) thanks to this assumption, displacement and velocity mappings can be treated separately allowing to show the relationship between the two at generic order and to retrieve homological equations in the sole displacement mapping; (iii) a number of implementation details on the treatment of the direct computation are reported in order to decrease the computational burden (e.g.…”

“…As a direct consequence, the size of the problem will double and become 2N . As shown for example in [54,44], numerous different formulations can be used to write Eq. ( 1), leading to different properties in terms of the symmetry of the resulting matrices.…”

“…Elaborating on previous results on normal forms, a direct approach has been proposed in [42] and further developed in [43,27], allowing one to express the reduced dynamics with normal coordinates in an invariant-based span of the phase space. Leveraging on SSM, a direct approach has also been proposed in [44,45,46], taking into account the damping and proposing arbitrary order approximations in an automated framework.…”

Preface to the special issue NODYCON 2021, Second International Nonlinear Dynamics Conference, Feb. 16–19, 2021

“…The first topical area titled Concepts and methods in nonlinear dynamics includes articles concerning the computation of invariant manifolds in high-dimensional finite element models [ 1 ], the computation of the basins of attraction of multi-stable dynamical systems [ 2 ], reduced-order models for vertical sloshing employing neural networks [ 3 ], the problem of continuous representations of piecewise-smooth nonlinear systems [ 4 ], and analytical approaches to nonlinear singular traveling waves in compressible thermo-hyperelastic cylindrical shells [ 5 ].…”