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

This work considers the stability of Proper Orthogonal Decomposition (POD) basis interpolation on Grassmann manifolds for parametric Model Order Reduction (pMOR) in hyperelasticity. The article contribution is mainly about stability conditions, all defined from strong mathematical background. We show how the stability of interpolation can be lost if certain geometrical requirements are not satisfied by making a concrete elucidation of the local character of linearization. To this effect, we draw special attention to the Grassmannian Exponential map and optimal injectivity condition of this map, related to the cut--locus of Grassmann manifolds. From this, explicit stability conditions are established and can be directly used to determine the loss of injectivity in practical pMOR applications. Another stability condition is formulated when increasing the number p of mode, deduced from principal angles of subspaces of different dimensions p. This stability condition helps to explain the non-monotonic oscillatory behavior of the error-norm with respect to the number of POD modes, and on the contrary, the monotonic decrease of the error-norm in the two benchmark numerical examples considered herein. Under this study, pMOR is applied in hyperelastic structures using a non-intrusive approach for inserting the interpolated spatial POD ROM basis in a commercial FEM code. The accuracy is assessed by \emph{a posteriori} error norms defined using the ROM FEM solution and its high fidelity counterpart simulation. Numerical studies successfully ascertained and highlighted the implication of stability conditions. The various stability conditions can be applied to a variety of other relevant problems involving parametrized ROMs generation based on POD basis interpolation via Grassmann manifolds.