Direct parametrisation of invariant manifolds for non-autonomous forced systems including superharmonic resonances

International audience ; The direct parametrisation method for invariant manifold is a model-order reduction technique that can be applied to nonlinear systems described by PDEs and discretised e.g. witha finite element procedure in order to derive efficient reduced-order models (ROMs). In non-linear vibrations, it has already been applied to autonomous and non-autonomous problemsto propose ROMs that can compute backbone and frequency-response curves of structures with geometric nonlinearity. While previous developments used a first-order expansion tocope with the non-autonomous term, this assumption is here relaxed by proposing a different treatment. The key idea is to enlarge the dimension of the parametrising coordinateswith additional entries related to the forcing. A new algorithm is derived with this starting assumption and, as a key consequence, the resonance relationships appearing throughthe homological equations involve multiple occurrences of the forcing frequency, showing that with this new development, ROMs for systems exhibiting a superharmonic resonance,can be derived. The method is implemented and validated on academic test cases involving beams and arches. It is numerically demonstrated that the method generates efficient ROMsfor problems involving 3:1 and 2:1 superharmonic resonances, as well as converged results for systems where the first-order truncation on the non-autonomous term showed a clear limitation.