Numerical comparison of reduced order models for non-linear vibrations of damped plates
DOI:
https://doi.org/10.13052/17797179.2012.719317Keywords:
non-linear vibration, damping, asymptotic numerical method, harmonic balance method, plates, reduced order model, proper orthogonal decompositionAbstract
This work deals with the computation of the non-linear solutions of the vibration of damped plates by coupling a harmonic balance method and the asymptotic numerical method. These computations can lead to lengthy central processing unit (CPU) times if the solution sought contains an important number of harmonics. In this study, we propose two reduced order models which can be applied to solve this type of problem. Both reduced methods are based on a first computation carried out with a small number of harmonics (here two). Numerical examples of plate vibration show that these algorithms help save a great deal of computational time and can be applied to problems involving numerous harmonics.
Downloads
References
Abdoun, F., Azrar, L., Daya, E.M., & Potier-Ferry, M. (2009). Forced harmonic response of viscoelastic
structures by an asymptotic numerical method. Computers & Structures, 87, 91–100.
Amabili, M., Sarkar, A., & Paıdoussis, M.P. (2003). Reduced-order models for nonlinear vibrations of
cylindrical shells via the proper orthogonal decomposition method. Journal of Fluids and Structures,
, 227–250.
Antoulas, A.C., & Sorensen, D.C. (2001). Approximation of large-scale dynamical systems: An overview.
International Journal of Applied Mathematics and Computer Science, 11(5), 1093–1121.
Batoz, J.-L., & Dhatt, G. (1992). Modeling of structures by the finite element method (Vol. 3: shells).
Paris: Hermès.
Boumediene, F., Duigou, L., Boutyour, E.H., Miloudi, A., & Cadou, J.M. (2011). Nonlinear forced
vibration of damped plates coupling asymptotic numerical method and reduction models. Computational
Mechanics, 47(4), 359–377.
Boumediene, F., Miloudi, A., Cadou, J.M., Duigou, L., & Boutyour, E.H. (2009). Nonlinear forced
vibration of damped plates by an asymptotic numerical method. Computers & Structures, 87, 1508–
Brezillon, A., Girault, G., & Cadou, J.M. (2010). A numerical algorithm coupling a bifurcating indicator
and a direct method for the computation of Hopf bifurcation points in fluid mechanics. Computers
& Fluids, 39(7), 1226–1240.
Cadou, J.M., Cochelin, B., Damil, N., & Potier-Ferry, M. (2001). Asymptotic numerical method for stationary
Navier-Stokes equations and with Petrov-Galerkin formulation. International Journal of
Numerical Methods in Engineering, 50, 825–845.
Cadou, J.M., & Potier-Ferry, M. (2010). A solver combining reduced basis and convergence acceleration
with applications in non-linear elasticity. International Journal for Numerical Methods in Biomedical,
, 1604–1617.
Cochelin, B., Damil, N., & Potier-Ferry, M. (1994). Asymptotic numerical methods and Padé approximants
for non-linear elastic structures. International Journal for Numerical methods in Engineering,
, 1187–1213.
Cochelin, B., & Vergez, C. (2009). A high order purely frequency-based harmonic balance formulation
for continuation of periodic solutions. Journal of Sound and Vibration, 324, 243–262.
Daridon, L., Wattrisse, B., Chrysochoos, A., & Potier-Ferry, M. (2011). Solving fracture problems using
an asymptotic numerical method. Computers & Structures, 89(5–6), 476–484.
Elhage-Hussein, A., Potier-Ferry, M., & Damil, N. (2000). A numerical continuation method based on
Padé approximants. International Journal of Solids and Structures, 37, 6981–7001.
Goncalves, P.B., Silva, F.M.A., & Del Prado, Z.J.G.N. (2008). Low-dimensional models for the nonlinear
vibration analysis of cylindrical shells based on a perturbation procedure and proper orthogonal
decomposition. Journal of Sound and Vibration, 315, 641–663.
Holmes, P., Lumley, J.L., & Berkooz, G. (1996). Turbulence, coherent structures, dynamical systems
and symmetry. Cambridge: Cambridge University Press.
LaBryer, A., & Attar, P.J. (2010). A harmonic balance approach for large-scale problems in nonlinear
structural dynamics. Computers & Structures, 88, 1002–1014.
Lall, S., Krysl, P., & Marsden, J.E. (2003). Structure-preserving model reduction for mechanical systems.
Physica D: Nonlinear Phenomena, 184, 304–318.
Médale, M., & Cochelin, B. (2009). A parallel computer implementation of the asymptotic numerical
method to study thermal convection instabilities. Journal of Computational Physics, 228(22), 8249–
Mickens R.E. (2010). Truly nonlinear oscillations. Harmonic Balance, parameter expansions, iteration,
and averaging methods. USA: World Scientific Publishing Co. Pte. Ltd.
Mottaqui, H., Braikat, B., & Damil, N. (2010). Discussion about parametrization in the asymptotic
numerical method: Appmlication to nonlinear elastic shell. Computer Methods in Applied Mechanics
and Engineering, 199(25–28), 1701–1709.
Potier-Ferry, M., Damil, N., Braikat, B., Descamps, J., Cadou, J.-M., Lei Cao, H., & Elhage Hussein,
A. (1997). Treatment of strong non-linearities by the asymptotic numerical method. Comptes Rendus
de l’Académie des Sciences - Series II, b324(3), 171–177.
Wriggers, P. (2008). Nonlinear finite element methods. Berlin: Springer-Verlag.
Yvonnet, J., & He, Q.C. (2007). The reduced model multiscale method (R3M) for the non-linear homogenization
of hyperelastic media at finite strains. Journal of Computational Physics, 223, 341–368.
