Abstract
This work concerns the computation of the nonlinear solutions of forced vibration of damped plates. In a recent work (Boumediene et al. in Comput Struct 87:1508–1515, 2009), a numerical method coupling an asymptotic numerical method (ANM), harmonic balance method and Finite Element method was proposed to resolve this type of problem. The harmonic balance method transforms the dynamic equations to equivalent static ones which are solved by using a perturbation method (ANM) and the finite element method. The numerical results presented in reference (Boumediene et al. in Comput Struct 87:1508–1515, 2009) show that the ANM is very efficient and permits one to obtain the nonlinear solutions with few matrix triangulation numbers compared to a classical incremental iterative method. However, putting a great number of harmonics (6 or greater) into the load vector leads to tangent matrices with a great size. The computational time necessary for the triangulation of such matrices can then be large. In this paper, reduced order models are proposed to decrease the size of these matrices and consequently the computational time. We consider two reduced bases. In the first one, the reduced basis is obtained by the resolution of a classical eigenvalue problem. The second one is obtained by using the nonlinear solutions computed during the first step of the calculus which is realized with the ANM. Several classical benchmarks of nonlinear damped plates are presented to show the efficiency of the proposed numerical methods.
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Boumediene F, Miloudi A, Cadou JM, Duigou L, Boutyour EH (2009) Nonlinear forced vibration of damped plates by an asymptotic numerical method. Comput Struct 87: 1508–1515
Azrar L, Boutyour EH, Potier-Ferry M (2002) Non-linear forced vibrations of plates by an asymptotic numerical method. J Sound Vib 252(4): 657–674
Cadou JM, Potier-Ferry M, Cochelin B (2006) A numerical method for the computation of bifurcation points in fluid mechanics. Eur J Mech B Fluids 25(2): 234–254
Boutyour EH, Zahrouni H, Potier-Ferry M, Boudi M (2004) Asymptotic-numerical method for buckling analysis of shell structures with large rotations. J Comput Appl Math 168(1–2): 77–85
Abdoun F, Azrar L, Daya EM, Potier-Ferry M (2009) Forced harmonic response of viscoelastic structures by an asymptotic numerical method. Comput Struct 87(1–2): 91–100
Potier-Ferry M, Cadou JM (2004) Basic ANM algorithms for path following problems. Rev Eur Eléments Finis 13(1–2): 9–32
Médale M, Cochelin B (2009) A parallel computer implementation of the asymptotic numerical method to study thermal convection instabilities. J Comput Phys 228(22): 8249–8262
Cadou JM, Duigou L, Damil N, Potier-Ferry M (2009) Convergence acceleration of iterative algorithms. Applications to thin shell analysis and Navier–Stokes equations. Comput Mech 43(2): 253–264
Kapania RK, Byun C (1993) Reduction methods based on eigenvectors and Ritz vectors for nonlinear transient analysis. Comput Mech 11: 65–82
Pesheck E, Pierre C, Shaw SW (2001) Accurate reduced-order models for a simple rotor blade model using nonlinear normal modes. Math Comput Model 33: 1085–1097
Amabili M, Touze C (2007) Reduced-order models for nonlinear vibrations of fluid-filled circular cylindrical shells: comparison of POD and asymptotic nonlinear normal modes methods. J Fluids Struct 23: 885–903
Touze C, Amabili M (2006) Nonlinear normal modes for damped geometrically nonlinear systems: application to reduced-order modeling of harmonically forced structures. J Sound Vib 298: 958–981
Meyer M, Matthies HG (2003) Efficient model reduction in non-linear dynamics using the Karhunen-Ločve expansion and dual-weighted-residual methods. Comput Mech 31: 179–191
Rizzi SA, Przekop A (2008) System identification-guided basis selection for reduced-order nonlinear response analysis. J Sound Vib 315: 467–485
Potier-Ferry M, Damil N, Braikat B, Descamps J, Cadou J-M, Lei Cao H, Elhage Hussein A (1997) Traitement des fortes non linéarités par la méthode asymptotique numérique. C R Acad Sci Ser IIB Mech Phys Chem Astron 324(3): 171–177
Cochelin B, Vergez C (2009) A high order purely frequency-based harmonic balance formulation for continuation of periodic solutions. J Sound Vib 324(1–2): 243–262
Azrar L, Cochelin B, Damil N, Potier-Ferry M (1993) An asymptotic numerical method to compute the post-bucling behavior of elastic plates and shells. Int J Numer Methods Eng 36: 1251–1277
Cochelin B, Damil N, Potier-Ferry M (1994) Asymptotic numerical methods and Padé approximants for non-linear elastic structures. Int J Numer Methods Eng 37: 1187–1213
Elhage-Hussein A, Potier-Ferry M, Damil N (2000) A numerical continuation method based on Padé approximants. Int J Solids Struct 37: 6981–7001
Hollkamp JJ, Gordon RW, Spottswood SM (2005) Nonlinear modal models for sonic fatigue response prediction: a comparison of methods. J Sound Vib 284: 1145–1163
Amabili M (2004) Nonlinear vibrations of rectangular plates with different boundary conditions: theory and experiments. Comput Struct 82: 2587–2605
Almroth BO, Brogan FA, Stern P (1978) Automatic choice of global shape functions in structural analysis. AIAA J 16: 525–528
Noor AK, Peters JM (1983) Recent advances in reduction methods for instability analysis of structures. Comput Struct 16: 67–80
Noor AK, Peters JM (1980) Reduced basis technique for nonlinear analysis of structures. AIAA J 18(4): 79–0747
Noor AK (1981) Recent advances in reduction methods for nonlinear problems. Comput Struct 13: 31–44
Batoz J-L, Dhatt G (1992) Modélisation des structures par éléments finis, vol 3: coques. Hermčs, Paris
Ribeiro P (2005) Nonlinear vibrations of simply-supported plates by the p-version finite element method. Finite Elements Anal Design 41: 911–924
Bathe KJ (1982) Finite element procedures in engineering analysis. Prentice-Hall, NJ
Cadou JM, Damil N, Potier-Ferry M, Braikat B (2004) Projection technique to improve high order iterative correctors. Finite Elements Anal Design 41(3): 285–309
Cadou JM, Potier-Ferry M (2009) A solver combining reduced basis and convergence acceleration with applications to non-linear elasticity. Commun Numer Methods Eng. doi:10.1002/cnm.1246
LaBryer A, Attar PJ (2010) A harmonic balance approach for large-scale problems in nonlinear structural dynamics. Comput Struct 88: 1002–1014
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Boumediene, F., Duigou, L., Boutyour, E.H. et al. Nonlinear forced vibration of damped plates coupling asymptotic numerical method and reduction models. Comput Mech 47, 359–377 (2011). https://doi.org/10.1007/s00466-010-0549-2
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DOI: https://doi.org/10.1007/s00466-010-0549-2