AJCM  Vol.5 No.3 , September 2015
Adaptive Reduced Basis Methods Applied to Structural Dynamic Analysis
Abstract: The reduced basis methods (RBM) have been demonstrated as a promising numerical technique for statics problems and are extended to structural dynamic problems in this paper. Direct step-by-step integration and mode superposition are the most widely used methods in the field of the finite element analysis of structural dynamic response and solid mechanics. Herein these two methods are both transformed into reduced forms according to the proposed reduced basis methods. To generate a reduced surrogate model with small size, a greedy algorithm is suggested to construct sample set and reduced basis space adaptively in a prescribed training parameter space. For mode superposition method, the reduced basis space comprises the truncated eigenvectors from generalized eigenvalue problem associated with selected sample parameters. The reduced generalized eigenvalue problem is obtained by the projection of original generalized eigenvalue problem onto the reduced basis space. In the situation of direct integration, the solutions of the original increment formulation corresponding to the sample set are extracted to construct the reduced basis space. The reduced increment formulation is formed by the same method as mode superposition method. Numerical example is given in Section 5 to validate the efficiency of the presented reduced basis methods for structural dynamic problems.
Cite this paper: Huang, Y. and Huang, Y. (2015) Adaptive Reduced Basis Methods Applied to Structural Dynamic Analysis. American Journal of Computational Mathematics, 5, 317-328. doi: 10.4236/ajcm.2015.53029.

[1]   Guyan, R.J. (1965) Reduction of Stiffness and Mass Matrices. AIAA Journal, 3, 380-381.

[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.

[4]   Moore, B.C. (1981) Principal Component Analysis in Linear Systems: Controllability, Observability, and Model Reduction. IEEE Transactions on Automatic Control, 26, 17-32.

[5]   Friswell, M.I., Garvey, S.D. and Penny, J.E.T. (1995) Model Reduction Using Dynamic and Iterative IRS Technique. Journal of Sound and Vibration, 186, 311-323.

[6]   Kammer, D.C. (1987) Test-Analysis Model Development Using an Exact Modal Reduction. The International Journal of Analytical and Experimental Modal Analysis, 2, 174-179.

[7]   Leung, A.Y.T. (1978) An Accurate Method of Dynamic Condensation in Structural Analysis. International Journal for Numerical Methods in Engineering, 12, 1705-1715.

[8]   O’Callahan, J.C. (1989) A Procedure for an Improved Reduced System (IRS) Model. Proceedings of the 7th International Modal Analysis Conference, Las Vegas, 17-21.

[9]   Singh, M.P. and Suarez, L.E. (1992) Dynamic Condensation with Synthesis of Substructure Eigenproperties. Journal of Sound and Vibration, 159, 581-606.

[10]   Van Woerkom, P.Th.L.M. (1990) Mathematical Models of Flexible Spacecraft Dynamics: A Survey of Order Reduction Approaches. Control Theory and Advanced Technology, 4, 609-632.

[11]   Qu, Z.Q. and Chang, W.J. (1997) An Extended Order Method (EOM) for Nonlinear Eigenvalue Problems. Mechanics and Practice, 19, 25-26.

[12]   Noor, A.K. (1994) Recent Advances and Applications of Reduction Methods. Applied Mechanics Reviews, 47, 125-146.

[13]   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.

[14]   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.

[15]   Willcox, K. and Peraire, J. (2002) Balanced Model Reduction via the Proper Orthogonal Decomposition. AIAA Journal, 40, 2323-2330.

[16]   Qu, Z.Q. (2002) Model Reduction for Dynamical Systems with Local Nonlinearities. AIAA Journal, 40, 327-333.

[17]   Almroth, B.O., Stern, P. and Brogan, F.A. (1978) Automatic Choice of Global Shape Functions in Structural Analysis. AIAA Journal, 16, 525-528.

[18]   Noor, A.K. and Peters, J.M. (1980) Reduced Basis Technique for Nonlinear Analysis of Structures. AIAA Journal, 18, 455-462.

[19]   Machiels, L., Maday, Y., Oliveira, I.B., Patera, A.T. and Rovas, D.V. (2003) Output Bounds for Reduced-Basis Approximations of Symmetric Positive Definite Eigenvalue Problems. Comptes Rendus de l’Académie des Sciences - Series I - Mathematics, 331, 152-158.

[20]   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—Mathematics, 328, 823-828.

[21]   Akcelik, V., Bielak, J., Biros, G., Epanomeritakis, I., Fernandez, A., Ghattas, O., et al. (2003) High Resolution Forward and Inverse Earthquake Modeling on Terasacale Computers. Proceedings of SC2003, Phoenix, 15-21 November 2003.