Journal
COMPUTATIONAL MECHANICS
Volume 68, Issue 4, Pages 861-883Publisher
SPRINGER
DOI: 10.1007/s00466-021-02050-0
Keywords
Proper Orthogonal Decomposition (POD); Model Order Reduction (MOR); Grassmann manifolds; Stiefel manifolds; Space-Time interpolation; Rigid-viscoplastic FEM
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This work introduces a novel non-intrusive space-time POD basis interpolation scheme, defining ROM spatial and temporal basis curves on compact Stiefel manifolds and incorporating oriented SVD to avoid the bottleneck associated with standard POD interpolation. By applying the proposed method to a specific metal forming process, correlations between the ST POD models and high-fidelity FEM counterpart simulations are demonstrated, showcasing the potential for near real-time parametric simulations using offline computed ROM POD databases.
This work aims to interpolate parametrized reduced order model (ROM) basis constructed via the proper orthogonal decomposition (POD) to derive a robust ROM of the system's dynamics for an unseen target parameter value. A novel non-intrusive space-time (ST) POD basis interpolation scheme is proposed, for which we define ROM spatial and temporal basis curves on compact Stiefel manifolds. An interpolation is finally defined on a mixed part encoded in a square matrix directly deduced using the spacial part, the singular values and the temporal part, to obtain an interpolated snapshot matrix, keeping track of accurate space and temporal eigenvectors. Moreover, in order to establish a well-defined curve on the compact Stiefel manifold, we introduce a new procedure, the so-called oriented SVD. Such an oriented SVD produces unique right and left eigenvectors for generic matrices, for which all singular values are distinct. It is important to notice that the ST POD basis interpolation does not require the construction and the subsequent solution of a reduced-order FEM model as classically is done. Hence it is avoiding the bottleneck of standard POD interpolation which is associated with the evaluation of the nonlinear terms of the Galerkin projection on the governing equations. As a proof of concept, the proposed method is demonstrated with the adaptation of rigid-thermoviscoplastic finite element ROMs applied to a typical nonlinear open forging metal forming process. Strong correlations of the ST POD models with respect to their associated high-fidelity FEM counterpart simulations are reported, highlighting its potential use for near real-time parametric simulations using off-line computed ROM POD databases.
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