How to Compute Invariant Manifolds and their Reduced Dynamics in High-Dimensional Finite-Element Models

TL;DR

This paper develops computational methods to efficiently calculate invariant manifolds and their reduced dynamics in high-dimensional finite-element models, enabling analysis of complex engineering structures.

Contribution

It introduces novel algorithms for computing invariant manifolds in very high-dimensional nonlinear systems from finite-element discretizations.

Findings

Algorithms successfully applied to models with up to 100,000 degrees of freedom.

Methods enable analysis of nonlinear phenomena in realistic engineering structures.

Computational challenges in high-dimensional systems are effectively addressed.

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.