Based on the hybrid numerical method (HNM) combining with a reduced-basis method (RBM), the real-time transient response of a functionally graded material (FGM) plates is obtained. The large eigenvalue problem in wavenumber domain has been solved through real-time off-line/on-line calculation. At off-line stage, a reduced-basis space is constructed in sample wavenumbers according to the solved eigenvalue problems. The matrices independent of parameters are projected onto the reduced-basis spaces. At on-line stage, the reduced eigenvalue problems of the arbitrary wavenumbers are built. Subsequently, the responses in wavenumber domain are obtained by the approximated eigen-pairs. Because of the application of RBM, the computational cost of transient displacement analysis of FGM plate is decreased significantly, while the accuracy of the solution and the physics of the structure are still retained. The efficiency and validity of the proposed method are demonstrated through a numerical example.
Moore, B.C. (1981) Principal Component Analysis in Linear Systems: Controllability, Observability, and Model Reduction. IEEE Transactions on Automatic Control, 26, 17-32. http://dx.doi.org/10.1109/TAC.1981.1102568
Lall, S., Marsden, J.E. and Glavaski, S. (2002) A Subspace Approach to Balanced Truncation for Model Reduction of Nonlinear Control Systems. International Journal of Robust and Nonlinear Control, 12, 519-535. http://dx.doi.org/10.1002/rnc.657
Ito, K. and Ravindran, S.S. (1997) Reduced Order Methods for Nonlinear Infinite Dimensional Control Systems. Proceedings of the 36th Conference on Decision & Control, San Diego, 10-12 December 1997, 2213-2218. http://dx.doi.org/10.1109/CDC.1997.657101
Maday, Y., Patera, A.T. and Peraire, J. (1999) A General Formulation for a Posteriori Bounds for Output Functionals of Partial Differential Equations; Application to the Eigenvalue Problem. Comptes Rendus de l’Académie des Sciences—Series I, 328, 823-828.
Liu, G.R., Han, X., Xu, Y.G. and Lam, K.Y. (2001) Material Characterization of FGM Plates Using Elastic Waves and an Inverse Procedure. Journal of Composite Materials, 11, 954-971. http://dx.doi.org/10.1106/86AQ-JY72-5VKT-K1NV
Han, X., Liu, G.R., Xi, Z.C. and Lam, K.Y. (2001) Transient Waves in Plates of Functionally Graded Materials. International Journal for Numerical Methods in Engineering, 52, 851-865. http://dx.doi.org/10.1002/nme.237
Veroy, K., Prud’homme, C. and Patera, A.T. (2003) Reduced-Basis Approximation of the Viscous Burgers Equation: Rigorous a Posteriori Error Bounds. Comptes Rendus de l’Académie des Sciences—Series I, 337, 619-624.
Maday, Y., Patera, A.T. and Turinici, G. (2002) A Prior Convergence Theory for Reduced-Basis Approximations of Sing-Parameter Elliptic Partial Differential Equations. Journal of Scientific Computing, 17, 437-446. http://dx.doi.org/10.1023/A:1015145924517