  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.
