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.
References
[1]
Guyan, R.J. (1965) Reduction of Stiffness and Mass Matrices. AIAA Journal, 3, 380-381. http://dx.doi.org/10.2514/3.2874
[2]
Wilson, E.L. and Bayo, E.P. (1967) Use of Special Ritz Vectors in Dynamic Substructure Analysis. AIAA Journal, 5, 1944-1954.
[3]
Sirovich, L. and Kirby, M. (1987) Low-Dimensional Procedure for the Characterization of Human Faces. Journal of the Optical Society of America A, 4, 519-524. http://dx.doi.org/10.1364/JOSAA.4.000519
[4]
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
[5]
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
[6]
Willcox, K. and Peraire, J. (2002) Balanced Model Reduction via the Proper Orthogonal Decomposition. AIAA Journal, 40, 2323-2330. http://dx.doi.org/10.2514/2.1570
[7]
McGowan, D.M. and Bostic, S.W. (1993) Comparison of Advanced Reduced-Basis Methods for Transient Structural Analysis. AIAA Journal, 31, 1712-1719. http://dx.doi.org/10.2514/3.11834
[8]
Machiels, L., Maday, Y., Oliveira, I.B., Patera, A.T. and Rovas, D.V. (2000) Output Bounds for Reduced-Basis Approximations of Symmetric Positive Definite Eigenvalue Problems. Comptes Rendus de l’Académie des Sciences—Series I, 331, 152-158.
[9]
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
[10]
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.
[11]
Liu, G.R. and Xi, Z.C. (2002) Elastic Waves in Anisotropic Laminates. CRC Press, Boca Raton.
[12]
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
[13]
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
[14]
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.
[15]
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
[16]
Touloukian, Y.S. (1967) Thermo-Physical Properties of High Temperature Solid Materials. Macmillan, New York.
[17]
Liu, G.R. (1998) A Step-by-Step Method of Rule-of-Mixture of Fiber- and Particle-Reinforced Composite Materials. Composite Structures, 40, 313-322. http://dx.doi.org/10.1016/S0263-8223(98)00033-6
