Reduced-Order Modelling of nonlinear dynamics in thin structures via invariant manifolds

At Cnam, Paris, October 15th 2026, 1.30 p.m.

Zixu Xia
ATER, Laboratoire MACS, Le Cnam, Paris

Thin-walled structures are widely encountered in mechanical, civil, and aerospace engineering, where their lightweight design often makes them particularly sensitive to geometric nonlinearities. At large vibration amplitudes, nonlinear modal interactions, internal resonances, bifurcations, and changes in hardening or softening behaviour may arise, making high-fidelity finite element simulations computationally demanding.

This seminar presents a reduced-order modelling framework for the nonlinear dynamics of thin structures based on the Direct Parametrisation of Invariant Manifolds (DPIM). The proposed methodology combines a geometrically nonlinear 7-parameter continuum solid-shell finite element formulation with high-order invariant-manifold reduction.

Rather than projecting the dynamics onto a fixed linear modal subspace, DPIM constructs a nonlinear manifold tangent to selected modal subspaces and simultaneously derives the associated reduced dynamics. Periodically forced responses are subsequently analysed in the frequency domain using the harmonic balance method, together with continuation and stability analysis. Applications to flat plates and curved shells illustrate how the resulting reduced-order models can reproduce complex nonlinear phenomena with substantially lower computational cost than the corresponding full-order models.

Particular attention is given to curvature-induced hardening-to-softening transitions, 1:2 internal resonance, geometric and thickness variations, isolated solution branches, and the selection of appropriate master modes. The framework is further extended to thin structures subjected to rotational effects.