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

Abstract

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 Stability of POD Basis Interpolation on Grassmann Manifolds for pMOR 3 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.

A new method to interpolate POD reduced bases–Application to the parametric model order reduction of a gas bearings supported rotor

“…Relation (11) provides the principal angles $$$ \boldsymbol{\Theta} $$$ and the principal vectors $$$ {\mathbf{X}}_0{\mathbf{P}}_1 $$$ and $$$ {\mathbf{X}}_1{\mathbf{Q}}_1 $$$ between the subspaces and , respectively. The matrix $$$ \mathbf{Z} $$$ in relation (5) is then defined if, and only if, the highest principal angle between and is strictly lower than $$$ \pi /2 $$$, see Reference 41 for more details. Noticing that $$$ \left(\mathbf{I}-{\mathbf{X}}_0{\mathbf{X}}_0^T\right){\mathbf{X}}_1 $$$ represents the orthogonal projection of $$$ {\mathbf{X}}_1 $$$ onto , namely $$$ {\mathbf{X}}_{0\perp }{\mathbf{X}}_{0\perp}^T{\mathbf{X}}_1 $$$, we get from (11) to (…”

“…* the component-wise product and where the functions u i , v i are component-wise evaluated. It can be appreciated that formulation (41) distinguishes terms that are linear and quadratic in Ψ, which will be exploited in the reduced order model.…”

Reduced order modeling of modular parameter dependent structures based on proper orthogonal decomposition and mesh tying

“…The parameter-dependent projection matrices are approximated with an interpolation technique using the Grassmann manifold and its tangent spaces, which was first developed in [11]. In solid mechanics, the stability of this interpolation was investigated in the context of hyperelastic transverse isotropic materials in [12]. The review paper [13], gives an overview of MOR of parametric dynamical systems.…”

Parametric Curves Metamodelling Based on Data Clustering, Data Alignment, POD-Based Modes Extraction and PGD-Based Nonlinear Regressions

“…For the sake of completeness, it shall be noticed that the use of POD-based interpolations-PODI-has several limitations and drawbacks, particularly when dealing with non-linear solution manifolds. To alleviate such issues, several works have been conducted in the framework of interpolations on Grassmann manifolds and its tangent space, improving the model robustness over the parametric space (Amsallem and Farhat, 2008;Mosquera et al, 2018Mosquera et al, , 2021Friderikos et al, 2020Friderikos et al, , 2022.…”