  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
systems.
