Reduced order modelling for shell finite element structures using the direct parametrisation of invariant manifolds: Hardening/softening transition, resonant dynamics and mode selection

The accurate simulation of the nonlinear dynamics of thin-walled structures is a critical but computationally demanding task. In this contribution, a 7-parameter solid-shell finite element formulation is combined with the direct parametrisation method for invariant manifolds (DPIM), in order to derive accurate and efficient reduced-order models (ROM) accounting for geometric nonlinearity. The method is illustrated in its ability to be used with different yet complementary purposes. On the one hand, low-order tractable models can be obtained, providing simple ROMs that are amenable to giving physical insights and understanding. On the other hand, higher-order solutions are available within the same framework, hence providing accurate and converged solutions. This scheme is carried out on examples with increasing complexity. First, the transition from hardening to softening behaviour for thin shells with shape imperfections is investigated. The 1:2 resonance as a driver of the change of type of nonlinearity is analysed, and a full understanding of the smooth transition is illustrated. Then, shells with varying thicknesses are investigated, and the case of 1:2 internal resonance is further investigated, showing the emergence of isolated solution branches (isola). In the course of the numerical simulations, it is shown how the reduced basis needs to be enlarged to take into account more and more complex resonance scenarios, and some guidelines are provided in order to help the analyst in selecting the master modes. The numerical results highlight the ability of the reduced-order models to provide a fully comprehensive and integrated framework for the understanding and accurate prediction of thin shells’ nonlinear dynamics.