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Remarks on the efficiency of POD for model reduction in non‐linear dynamics of continuous elastic systems
Author(s) -
Sampaio Rubens,
Soize Christian
Publication year - 2007
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1991
Subject(s) - basis (linear algebra) , point of delivery , mathematics , basis function , galerkin method , eigenvalues and eigenvectors , reduction (mathematics) , projection (relational algebra) , model order reduction , convergence (economics) , dimension (graph theory) , mathematical optimization , mathematical analysis , algorithm , finite element method , geometry , physics , quantum mechanics , pure mathematics , agronomy , economics , biology , economic growth , thermodynamics
The first objective of this paper is to analyse the efficiency of the reduced models constructed using the proper orthogonal decomposition (POD)‐basis and the LIN‐basis in non‐linear dynamics for continuous elastic systems. The POD‐basis is the Hilbertian basis constructed with the POD method while the LIN‐basis is the Hilbertian basis derived from the generalized continuous eigenvalue problem associated with the underlying conservative part of the continuous elastic system and usually called the eigenmodes of vibration. The efficiency of the POD‐basis or the LIN‐basis is related to the rate of convergence in the frequency domain of the solution constructed with the reduced model with respect to its dimension. A basis will be more efficient than another if the reduced‐order solution of the Galerkin projection converges to the solution of the dynamical system more rapidly than the reduced‐order solution of the other. As a second objective of this paper, we present the usual results concerning the POD method using a continuous formulation, with respect to both time and space variables, and then deriving the numerical approximations. Such a presentation allows convergence discussions to be treated. Six examples in non‐linear elastodynamics problems are presented in order to analyse the efficiency of the POD‐basis and the LIN‐basis. It is concluded that the POD‐basis is not more efficient than the LIN‐basis for the examples treated in non‐linear elastodynamics. Copyright © 2007 John Wiley & Sons, Ltd.