2020/07/18 by Orestis Friderikos, Friderikos, Orestis, Emmanuel Baranger +5
Engineering · Physics and Astronomy · #Advanced Numerical Methods in Computational Mathematics #Differential Geometry (math.DG) #Elasticity and Material Modeling #FOS: Mathematics #Model Reduction and Neural Networks #Vehicle Dynamics and Control Systems
paper · pdf · doi:10.48550/arxiv.2012.08851
openalex publication_date 2020/07/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This work considers the stability of Proper Orthogonal Decomposition (POD)\nbasis interpolation on Grassmann manifolds for parametric Model Order Reduction\n(pMOR) in hyperelasticity. The article contribution is mainly about stability\nconditions, all defined from strong mathematical background. We show how the\nstability of interpolation can be lost if certain geometrical requirements are\nnot satisfied by making a concrete elucidation of the local character of\nlinearization. To this effect, we draw special attention to the Grassmannian\nExponential map and optimal injectivity condition of this map, related to the\ncut--locus of Grassmann manifolds. From this, explicit stability conditions are\nestablished and can be directly used to determine the loss of injectivity in\npractical pMOR applications. Another stability condition is formulated when\nincreasing the number p of mode, deduced from principal angles of subspaces of\ndifferent dimensions p. This stability condition helps to explain the\nnon-monotonic oscillatory behavior of the error-norm with respect to the number\nof POD modes, and on the contrary, the monotonic decrease of the error-norm in\nthe two benchmark numerical examples considered herein. Under this study, pMOR\nis applied in hyperelastic structures using a non-intrusive approach for\ninserting the interpolated spatial POD ROM basis in a commercial FEM code. The\naccuracy is assessed by \a posteriori error norms defined using the ROM\nFEM solution and its high fidelity counterpart simulation. Numerical studies\nsuccessfully ascertained and highlighted the implication of stability\nconditions. The various stability conditions can be applied to a variety of\nother relevant problems involving parametrized ROMs generation based on POD\nbasis interpolation via Grassmann manifolds.\n