Abstract
Parametric simulations of thermomechanical metal forming processes still remain computational costly and difficult due to inherent strong non-linearities. To this end, Reduced Order Models (ROMs) are introduced to decrease the computational time in large scale models and provide near-optimal solutions in acceptable times. ROMs based on the Proper Orthogonal Decomposition (POD) are usually capable of accurately reproducing the dynamics of high-fidelity FEM simulations and offer the potential for near real-time analysis. However, ROMs are not robust with respect to parameter changes and must often be rebuilt for each parameter variation. This work aims to interpolate ROM POD basis associated with a limited number of training points on Grassmann manifolds, so as to obtain a robust ROM corresponding to a target parameter. A novel Space-Time (ST) POD basis interpolation, where the reduced spatial and time basis are separately interpolated on Grassmann manifolds, is proposed. Good correlations of the ROM ST models with respect to their associated high-fidelity FEM counterpart simulations are found. Hence, application of the ROM adaptation method for near real-time metal forming simulations using off-line computed ROM POD databases can be possible.
Highlights
Computational metal forming has been widely used in a variety of applications in academia laboratories and manufacturing industry over the last decades
For a new parameter value, an interpolation method is proposed using the underlying spanned subspaces of the Proper Orthogonal Decomposition (POD) basis [3]. Since these matrices are obtained via the Singular Value Decomposition (SVD), they describe the optimal projection operators, which reduce the problem from the full space to a low dimensional subspace
POD basis interpolation is performed using local maps on Grassmann manifolds by evaluating the geodesic paths between the subspaces on this manifold, all this being done in the framework of Riemannian geometry, which is a specific matter of differential geometry
Summary
Computational metal forming has been widely used in a variety of applications in academia laboratories and manufacturing industry over the last decades. For a new parameter value, an interpolation method is proposed using the underlying spanned subspaces of the POD basis [3] Since these matrices are obtained via the Singular Value Decomposition (SVD), they describe the optimal projection operators, which reduce the problem from the full space to a low dimensional subspace. POD basis interpolation is performed using local maps on Grassmann manifolds by evaluating the geodesic paths between the subspaces on this manifold, all this being done in the framework of Riemannian geometry, which is a specific matter of differential geometry In this context, a novel non-intrusive Space-Time POD interpolation is proposed here allowing near-real time predictions since no new ROM FEM models have to be solved, as required by the standard POD approach.
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