An enhanced parametric nonlinear reduced order model for imperfect structures using Neumann expansion

Abstract

We present an enhanced version of the parametric nonlinear reduced order model for shape imperfections in structural dynamics we studied in a previous work [1]. The model is computed intrusively and with no training using information about the nominal geometry of the structure and some user-defined displacement fields representing shape defects, i.e. small deviations from the nominal geometry parametrized by their respective amplitudes. The linear superposition of these artificial displacements describe the defected geometry and can be embedded in the strain formulation in such a way that, in the end, nonlinear internal elastic forces can be expressed as a polynomial function of both these defect fields and the actual displacement field. This way, a tensorial representation of the internal forces can be obtained and, owning the reduction in size of the model given by a Galerkin projection, high simulation speed-ups can be achieved. We show that by adopting a rigorous deformation framework we are able to achieve better accuracy as compared to the previous work. In particular, exploiting Neumann expansion in the definition of the Green-Lagrange strain tensor, we show that our previous model is a lower order approximation with respect to the one we present now. Two numerical examples of a clamped beam and a MEMS gyroscope finally demonstrate the benefits of the method in terms of speed and increased accuracy.

Neural modal ordinary differential equations: Integrating physics-based modeling with neural ordinary differential equations for modeling high-dimensional monitored structures

“…To achieve this, we propose to blend a dynamical version (Girin et al, 2020) of a variational autoencoder (VAE) (Kingma and Welling, 2013), with a projection basis containing the eigenmodes that are derived from the linearization of a physics-based model, termed as neural modal ODEs. We justify these components in the proposed architecture as follows: (a) the majority of the aforementioned projection-based methods, which commonly rely on proper orthogonal decomposition (POD (Liang et al, 2002), have been applied for the reduction of nonlinear models/simulators (Abgrall and Amsallem, 2016;Amsallem et al, 2015;Balajewicz et al, 2016;Peherstorfer and Willcox, 2016;Marconia et al, 2021;. In this case, we rely on the availability of actual measured data but not simulations of full-order models, which may bear with model bias.…”