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Projection-based nonlinear model order reduction methods can be used toreduce simulation times for the solution of many PDE-constrained problems. Ithas been observed in literature that such nonlinear reduced-order models (ROMs)based on Galerkin projection sometimes exhibit much larger stable time stepsizes than their unreduced counterparts. This work provides a detailedtheoretical analysis of this phenomenon for structural mechanics. We first showthat many desirable system matrix properties are preserved by the Galerkinprojection. Next, we prove that the eigenvalues of the linearized Galerkinreduced-order system separate the eigenvalues of the linearized originalsystem. Assuming non-negative Rayleigh damping and a time integration using thepopular central difference method, we further prove that the theoretical linearstability time step of the ROM is in fact always larger than or equal to thecritical time step of its corresponding full-order model. We also givemathematical expressions for computing the stable time step size. Finally, weshow that under certain conditions this increase in the stability time stepeven extends to some hyper-reduction methods. The findings can be used tocompute numerical stability time step sizes for the integration of nonlinearROMs in structural mechanics, and to speed up simulations by permitting the useof larger time steps.

Stability conditions for the explicit integration of projection based  nonlinear reduced-order and hyper reduced structural mechanics finite element  models