Determination of Natural Frequencies and Mode Shapes of Structures Using Subspace Iteration Method with Accelerated Starting Vectors
Publication: Journal of Structural Engineering
Volume 131, Issue 7
Abstract
Natural frequencies and mode shapes of structures are determined from eigenvalue analysis. This paper proposes a subspace iteration method with an accelerated Lanczos starting subspace for the efficient eigenvalue analysis of structures. The proposed method uses accelerated Lanczos vectors as starting vectors in order to reduce the number of subspace iterations. Accelerated Lanczos starting vectors are generated by employing the repeated forward reduction and back substitution. The proposed method has less computing time than the subspace iteration method with a conventional Lanczos starting subspace when the number of required eigenpairs is relatively small. The efficiency of the proposed method is verified through numerical examples.
Get full access to this article
View all available purchase options and get full access to this article.
References
Akl, F. A., Dilger, W. H., and Irons, B. M. (1979). “Over-relaxation and subspace iteration.” Int. J. Numer. Methods Eng., 14, 629–630.
Bathe, K. J., and Wilson, E. L. (1972). “Large eigenvalue problems in dynamic analysis.” J. Eng. Mech., 98(6), 1471–1485.
Bathe, K. J., and Ramaswamy, S. (1980). “An accelerated subspace iteration method.” Comput. Methods Appl. Mech. Eng., 23, 313–331.
Bevilacqua, G., Martinelli, L., and Parravicini, G. P. (1996). “Jahn-Teller effect in revisited with a modified Lanczos-type algorithm.” Phys. Rev. B, 54, 7626–7629.
Chen, H. C., and Taylor, R. L. (1988). “Solution of eigenproblems for damped structural systems by the Lanczos algorithm.” Comput. Struct., 30, 151–161.
Cordelli, A. (1994). “Application of accelerated-convergence technique to modified Lanczos calculations.” Nuovo Cimento D, 16, 45–53.
Ericsson, T., and Ruhe, A. (1980). “The spectral transformation Lanczos method for the numerical solution of large sparse generalized symmetric eigenvalue problems.” Math. Comput., 35, 1251–1268.
Grosso, G., Martinelli, L., and Parravicini, G. P. (1993). “A new method for determining excited states of quantum systems.” Nuovo Cimento D, 15, 269–277.
Lam, Y. C., and Bertolini, A. F. (1994). “Acceleration of the subspace iteration method by selective repeated inverse iteration.” Finite Elem. Anal. Design, 18, 309–317.
Lanczos, C. (1950). “An iteration method for the solution of the eigenvalue problem of linear differential and integral operators.” J. Res. Natl. Bur. Stand., 45, 255–282.
Matthies, H. G. (1985). “A subspace Lanczos method for the generalized symmetric eigenproblem.” Comput. Struct., 21, 319–325.
Nour-Omid, B., Parlett, B. N., and Taylor, R. L. (1983). “Lanczos versus subspace iteration for solution of eigenproblems.” Int. J. Numer. Methods Eng., 19, 859–871.
Paige, C. C. (1972). “Computational variants of the Lanczos method for the eigenproblem.” J. Inst. Math. Appl., 10, 373–381.
Parlett, B. N., and Scott, D. S. (1979). “The Lanczos algorithm with selective orthogonalization.” Math. Comput., 33, 217–238.
Qian, Y. Y., and Dhatt, G. (1995). “An accelerated subspace method for generalized eigenproblems.” Comput. Struct., 54, 1127–1134.
Rajakumar, C. (1993). “Lanczos algorithm for the quadratic eigenvalue problem in engineering applications.” Comput. Methods Appl. Mech. Eng., 105, 1–22.
Simon, H. D. (1984). “The Lanczos algorithm with partial reorthogonalization.” Math. Comput., 42, 115–142.
Smith, H. A., Sorensen, D. C., and Singh, R. K. (1993). “A Lanczos-based technique for exact vibration analysis of skeletal structures.” Int. J. Numer. Methods Eng., 36, 1987–2000.
Wang, X., and Zhou, J. (1999). “An accelerated subspace iteration method for generalized eigenproblems.” Comput. Struct., 71, 293–301.
Wilson, E. L., and Itoh, T. (1983). “An eigensolution strategy for large systems.” Comput. Struct., 16, 259–265.
Wilson, E. L., Yuan, M. W., and Dickens, J. M. (1982). “Dynamic analysis by direct superposition of Ritz vectors.” Earthquake Eng. Struct. Dyn., 10, 813–821.
Information & Authors
Information
Published In
Journal of Structural Engineering
Volume 131 • Issue 7 • July 2005
Pages: 1146 - 1149
Copyright
© 2005 ASCE.
History
Received: Feb 2, 2004
Accepted: Nov 29, 2004
Published online: Jul 1, 2005
Published in print: Jul 2005
Notes
Note. Associate Editor: Barry Thomas Rosson
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.