Non-intrusive methods have been used since two decades to derive reduced-order models for geometrically nonlinear structures, with a particular emphasis on the so-called STiffness Evaluation Procedure (STEP), relying on the static application of prescribed displacements in a finite-element context. We show that a particularly slow convergence of the modal expansion is observed when applying the method with 3D elements, because of nonlinear couplings occurring with very high frequency modes involving 3D thickness deformations. Focusing on the case of flat structures, we first show by computing all the modes of the structure that a converged solution can be exhibited by using either static condensation or normal form theory. We then show that static modal derivatives provide the same solution with fewer calculations. Finally, we propose a modified STEP, where the prescribed displacements are imposed solely on specific degrees of freedom of the structure, and show that this adjustment also provides efficiently a converged solution.

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The generation of ultra-low frequency acoustic noise (0.1 to 1 Hz) by the nonlinear interaction of ocean surface gravity waves is well established. More controversial are the quantitative theories that attempt to predict the recorded noise levels and their variability. Here a single theoretical framework is used to predict the noise level associated with propagating pseudo-Rayleigh modes and evanescent acoustic-gravity modes. The latter are dominant only within 200 m from the sea surface, in shallow or deep water. At depths larger than 500 m, the comparison of a numerical noise model with hydrophone records from two open-ocean sites near Hawaii and the Kerguelen islands reveal: (a) Deep ocean acoustic noise at frequencies 0.1 to 1 Hz is consistent with the Rayleigh wave theory, in which the presence of the ocean bottom amplifies the noise by 10 to 20 dB; (b) in agreement with previous results, the local maxima in the noise spectrum support the theoretical prediction for the vertical structure of acoustic modes; and (c) noise level and variability are well predicted for frequencies up to 0.4 Hz. Above 0.6 Hz, the model results are less accurate, probably due to the poor estimation of the directional properties of wind-waves with frequencies higher than 0.3 Hz.