On the stability of POD Basis Interpolation via Grassmann Manifolds for Parametric Model Order Reduction in Hyperelasticity

Orestis Friderikos (1) , Emmanuel Baranger (2) , Marc Olive (2) , David Néron (1)

1 LMT - Laboratoire de Mécanique et Technologie
2 LMT - Laboratoire de Mécanique et Technologie

A rigorous mathematical approach on the Proper Orthogonal Decomposition (POD) basis interpolation via Grassmann manifolds for parametric Model Order Reduction (pMOR) is given. The work focuses on issues related to interpolation on Grassmann manifolds making a concrete elucidation of the local character of linearization. It is shown how the stability of the interpolation can be lost if certain geometrical conditions are met. To this effect, we draw special attention to the injectivity radius and the cut-locus on the Grassmannian which are essential to establish well defined local maps. Here, pMOR is applied in hyperelastic structures using a non-intrusive approach for inserting the interpolated spatial POD ROM basis in a commercial code. The accuracy of the method is assessed by a posteriori error norms defined using the ROM FEM solution and its high fidelity counterpart simulation. High quality correlations of the error norms are found for both numerical examples. One important outcome concerns the accuracy with respect to the number of POD modes in which the error norm shows a non-monotonic decrease with a spurious oscillatory behavior.

Mots clés: Parametric Model Order Reduction (pMOR); Proper Orthogonal Decomposition; Grassmann manifolds; Hyperelasticity * Corresponding author

Domaines: Matériaux et structures en mécanique [physics.class-ph]; Géométrie différentielle [math.DG]

Orestis Friderikos, Emmanuel Baranger, Marc Olive, David Néron. On the stability of POD Basis Interpolation via Grassmann Manifolds for Parametric Model Order Reduction in Hyperelasticity. 2020. ⟨hal-02902214v1⟩