A new method to interpolate POD reduced bases–Application to the parametric model order reduction of a gas bearings supported rotor

Abstract

SummaryThe proper orthogonal decomposition (POD) is successfully employed in a variety of projection‐based methods for parametric model order reduction (pMOR) of large dynamical systems. It extracts the most energetic modes describing the dynamics of a system from time snapshots of the solution. In practice, these snapshots are computed at user‐defined parameters of the system with a high fidelity model. Then either all the snapshots are concatenated to extract a global POD basis, or a different POD basis is extracted for each parameter value. In the latter case, for unseen target parameters, the POD bases need to be adapted with a dedicated technique. An established method addressing this task is the interpolation on the tangent space to the Grassmann manifold (ITSGM). It interpolates the linear subspaces spanned by the known POD bases, discarding by construction the knowledge on the modes ordering by energy levels. In this paper, we formalize an ordered reduced basis interpolation (ORBI) technique which preserves such ordering. This approach improves the adaptation accuracy of the resulting POD basis as more information is taken into account in the construction of the interpolation operator. Numerical investigations are conducted on the parametric reduced order model of a dynamical system representing a small‐scale gas‐bearings supported rotor. Results show that the proposed method is more accurate than the ITSGM method at a similar computation cost. Trained at only three points of the parameters space, the developed hyper reduced order model (h‐ROM) performs to faster simulations with satisfactory accuracy, even far from the training points.

Plug‐and‐play adaptive surrogate modeling of parametric nonlinear dynamics in frequency domain

“…From this point of view, our proposed hybrid method improves the joint one by increasing the approximation efficiency in one of the parameters, namely, frequency. Notably, in contrast to our observations concerning polynomials in Section 5.1.1, the chosen higher-order approximation strategy in frequency is numerically stable, due to the beneficial properties of the barycentric expansion, see Equation (9). • The greedy MRI method performs adaptive sampling more effectively than LASGs.…”

“…In gMRI, the set of support frequencies is incrementally built, starting from the two endpoints of the interval Ω. Each new support frequency 𝜔 j,S j +1 is selected based on the current surrogate ξj , specifically, based on the denominator of the weight functions 𝜙 j,i , see Equation (9). The termination condition is based on a user-defined tolerance tol 𝜔 , which determines the desired accuracy in frequency of the local surrogate.…”