Remarks on the efficiency of POD for model reduction in non-linear dynamics of continuous elastic systems

International Journal for Numerical Methods in Engineering

Volumn 72, Issue 1, 2007, Pages 22-45

  Sampaio, Rubens ^a   Soize, Christian ^b  

^a Mechanical Engineering Department   (Brazil)

^b UNIVERSITÉ GUSTAVE EIFFEL   (France)

Author keywords

Karhunen Lo ve basis; Non linear dynamics; POD; Reduced model; Snapshot

Indexed keywords

EIGENVALUES AND EIGENFUNCTIONS; ELASTICITY; FREQUENCY DOMAIN ANALYSIS; GALERKIN METHODS; VIBRATIONS (MECHANICAL);

CONTINUOUS FORMULATION; NON-LINEAR ELASTODYNAMICS; PROPER ORTHOGONAL DECOMPOSITION (POD); REDUCED MODELS;

DYNAMICAL SYSTEMS;

DYNAMICAL SYSTEMS; EIGENVALUES AND EIGENFUNCTIONS; ELASTICITY; FREQUENCY DOMAIN ANALYSIS; GALERKIN METHODS; VIBRATIONS (MECHANICAL);

EID: 34848875725     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.1991     Document Type: Article

References (19)

1. Loéve M. Probability Theory (3rd edn). Van Nostrand: New York, 1963.
2. Holmes P, Lumley JL, Berkooz G. Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press: Cambridge, 1997.
3. Kunisch E, Volkwein S. Galerkin proper orthogonal decomposition methods for parabolic problems. Numerische Mathematik 2001; 90(1):117-148.
4. Kunisch E, Volkwein S. Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM Journal on Numerical Analysis 2002; 40(1):233-253.
5. Azeez MFA, Vakakis AF. Proper orthogonal decomposition (POD) of a class of vibroimpact oscillations. Journal of Sound and Vibration 2001; 240(5):859-889.
6. Meyer M, Matthies HG. Efficient model reduction in non-linear dynamics using the Karhunen-Loéve expansion and dual-weighted-residual methods. Computational Mechanics 2003; 31:179-191.
7. Park HM, Cho DH. The use of the Karhunen-Loéve decomposition for the modeling of distributed parameters systems. Chemical Engineering Science 1996; 51(1):81-98.
8. Rowley CW, Marsden JE. Reconstruction equations and the Karhunen-Loéve expansion for systems with symmetry. Physica D 2000; 142:1-19.
9. Wolter C, Trindade MA, Sampaio R. Reduced-order model for an impacting beam using the Karhunen-Loéve expansion. Tendências em Matemática Aplicada e Computacional (TEMA) 2002; 3(2):217-226.
10. Trindade MA, Wolter C, Sampaio R. Karhunen-Loève decomposition of coupled axial/bending vibrations of beams subject to impacts. Journal of Sound and Vibration 2005; 279:1015-1036.
11. Dautray R, Lions JL. Mathematical Analysis and Numerical Methods for Science and Technology, vols. 1-3, Springer: Berlin, 1992.
12. Ohayon R, Soize C. Structural Acoustics and Vibration: Mechanical Models, Variational Formulations and Discretization. Academic Press: San Diego, 1998.
13. Bellizzi S, Sampaio R. POMs analysis of randornly vibrating systems obtained from Karhunen-Loève expansion. Journal of Sound and Vibration 2006; 297:774-793.
14. Soize C. Reduced models in the medium frequency range for general dissipative structural-dynamics systems. European Journal of Mechanics - A/Solids 1998; 17(4):657-685.
15. Guelfand IM, Vilenkin NY. Les Distributions. Tome 4. Application de l'Analyse Harmonique. Dunod: Paris, 1967.
16. Zienkiewicz OC, Taylor RL. Finite Element Method, vols. 1-3. Butterworth-Heinomann: London, 2000.
17. Chatelin F. Eigenvalues of Matrices. Wiley: New York, 1993.
18. Bathe KJ, Wilson EL. Numerical Methods in Finite Element Analysis. Prentice-Hall: New York, 1976.
19. Sirovich L. Turbulence and the dynamics of coherent structures, Part I: coherent structures. Quarterly of Applied Mathematics 1987; 45(3):561-571.