On the stability of POD basis interpolation on Grassmann manifolds for parametric model order reduction

Proper Orthogonal Decomposition (POD) basis interpolation on Grassmann manifolds has been successfully applied to problems of parametric model order reduction (pMOR). In this work we address the necessary stability conditions for the interpolation, all defined from strong mathematical background. A first condition concerns the domain of definition of the logarithm map. Second, 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 the optimal injectivity condition of this map, related to the cut–locus of Grassmann manifolds. From this, an explicit stability condition is established and can be directly used to determine the loss of injectivity in practical pMOR applications. A third stability condition is formulated when increasing the number p of POD modes, deduced from the principal angles of subspaces of different dimensions p. Definition of this condition leads to an understanding of the non-monotonic oscillatory behavior of the Reduced Order Model (ROM) error-norm with respect to the number of POD modes, and on the contrary, the well-behaved monotonic decrease of the error-norm in the two numerical examples presented herein. We have chosen to perform pMOR 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 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 which are general and can be applied to a variety of other linear or nonlinear problems involving parametrized ROMs generation based on POD basis interpolation on Grassmann manifolds.