Interaction and multiscale mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Advances in solution of classical generalized eigenvalue problem

  • Chen, P. (LTCS, Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University) ;
  • Sun, S.L. (LTCS, Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University) ;
  • Zhao, Q.C. (LTCS, Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University) ;
  • Gong, Y.C. (LTCS, Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University) ;
  • Chen, Y.Q. (LTCS, Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University) ;
  • Yuan, M.W. (LTCS, Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University)
  • Received : 2007.07.01
  • Accepted : 2008.05.12
  • Published : 2008.06.25
⟨ Previous Next ⟩

Abstract

Owing to the growing size of the eigenvalue problem and the growing number of eigenvalues desired, solution methods of iterative nature are becoming more popular than ever, which however suffer from low efficiency and lack of proper convergence criteria. In this paper, three efficient iterative eigenvalue algorithms are considered, i.e., subspace iteration method, iterative Ritz vector method and iterative Lanczos method based on the cell sparse fast solver and loop-unrolling. They are examined under the mode error criterion, i.e., the ratio of the out-of-balance nodal forces and the maximum elastic nodal point forces. Averagely speaking, the iterative Ritz vector method is the most efficient one among the three. Based on the mode error convergence criteria, the eigenvalue solvers are shown to be more stable than those based on eigenvalues only. Compared with ANSYS's subspace iteration and block Lanczos approaches, the subspace iteration presented here appears to be more efficient, while the Lanczos approach has roughly equal efficiency. The methods proposed are robust and efficient. Large size tests show that the improvement in terms of CPU time and storage is tremendous. Also reported is an aggressive shifting technique for the subspace iteration method, based on the mode error convergence criteria. A backward technique is introduced when the shift is not located in the right region. The efficiency of such a technique was demonstrated in the numerical tests.

Keywords

References

  1. Akl, F.A., Dilger, W.H. and Irons, B.M. (1979), "Over-relaxation and subspace iteration", Int. J. Numer. Meth. Engg., 14, 629-630. https://doi.org/10.1002/nme.1620140414
  2. Akl, F.A., Dilger, W.H. and Irons, B.M. (1982), "Acceleration of subspace iteration", Int. J. Numer. Meth. Engg., 18, 583-589. https://doi.org/10.1002/nme.1620180408
  3. Bathe, K.J. (1996), Finite Element Procedure, Prentice-Hall, Englewood Cliffs, New Jersey. Bathe, K.J. and Ramaswamy, S. (1980), "An accelerated subspace iteration method", Comput. Meth. Appl. Mech. Engg., 23, 313-331. https://doi.org/10.1016/0045-7825(80)90012-2
  4. Bathe, K.J. and Wilson, E.L. (1972), "Large eigenvalue problems in dynamic analysis", J. Eng. Mech. Div., ASCE, 98, 1471-1485.
  5. Bertolini, A.F. and Lam, Y.C. (1995), "Accelerated reduction of subspace upper bound by multiple inverse iteration", Comput. Systems Eng., 6, 67-72. https://doi.org/10.1016/0956-0521(95)00007-M
  6. Chen, P., Gong, Y.C., Chen, Y.Q., et al. (2007), "On the computable eigenvalue error bound in the subspace iteration method", Submitted to J. of Computational Mechanics (in revision).
  7. Chen, P. and Sun, S.L. (2005), "A new high performance sparse static solver in finite element analyses", Acta Mechanica Solida Sinica, 18, 248-255.
  8. Chen, P., Zheng, D., Sun, S.L. and Yuan, M.W. (2003), "High performance sparse static solver in finite element analyses with loop-unrolling", Advances in Engineering Software, 34, 203-215. https://doi.org/10.1016/S0965-9978(02)00128-X
  9. Clint, M. and Jennings, A. (1970), "The evaluation of eigenvalue and eigenvectors of real symmetric matrices by simultaneous iteration", The Comput. J., 13, 76-80. https://doi.org/10.1093/comjnl/13.1.76
  10. Duff, I.S., Grimes, R. and Lewis, J. (1992), The User's Guide for the Harwell-Boeing Sparse Matrix Collection (Release I), Technical Report TR/PA/92/86, CERFACS, Lyon, France.
  11. Golub, G.H. (1972), "Some uses of the Lanczos algorithm in numerical linear algebra", in Miller J.J.H. eds. Topics in Numerical Analysis, 173-194.
  12. Gong, Y.C., Zhou, H.W., Yuan, M.W. and Chen, P. (2005), "Comparison of subspace iteration, iterative Ritz vector method and iterative Lanczos method", J. Vib. Eng., 18, 227-232 (in Chinese).
  13. Grimes, R.G., Lewis, J.G. and Simon, H.D. (1994), "A shifted block Lanczos algorithm for solving sparse symmetric generalized eigenproblems", SIAM J. Matrix. Anal. Appl., 15(1), 228-272. https://doi.org/10.1137/S0895479888151111
  14. Hughes, T.J.R. (1987), The Finite Element Method-Linear Static and Dynamic Finite Element Analysis,. Prentice-Hall, Englewood Cliffs, New Jersey.
  15. Lee, J.H. and Kim, H.T. (2005), "Hybrid method for natural frequency extraction: performance improvement using Newton-Raphson method", J. Electromagnetic Waves and Applications, 19, 1043-1055. https://doi.org/10.1163/156939305775526061
  16. Matthies, H. (1985), "Computable error bounds for the generalized symmetric eigenproblem", Communications Appl Num Methods, 1, 33-38. https://doi.org/10.1002/cnm.1630010107
  17. Nguyen, D.T., Bunting, C.F., Moeller, M.J., Runesha, H. and Qin, J. (2000), "Subspace and Lanczos sparse eigen-solvers for finite element structural and electromagnetic applications", Advances in Engineering Software, 31, 599-606. https://doi.org/10.1016/S0965-9978(00)00021-1
  18. Press, W.H. et al. (1996), Numerical Recipes in Fortran 90: the Art of Parallel Scientific Computing (2nd ed), Cambridge University Press, London.
  19. Rajendran, S. and Liew, K.M. (2003), "A variant of the subspace iteration algorithm for generalized eigenproblems", Int. J. Numer. Meth. Eng., 57, 2027-2042. https://doi.org/10.1002/nme.753
  20. Underwood, R. (1975), "An iterative block Lanczos method for the solution of large sparse symmetric eigenproblems", Ph.D. Dissertation, Stanford University.
  21. Wilson, E.L. and Itoh, T. (1983), "An eigensolution strategy for large systems", Comput. Struct., 16, 259-265. https://doi.org/10.1016/0045-7949(83)90166-9
  22. 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. https://doi.org/10.1002/eqe.4290100606
  23. Yamamoto, Y. and Ohtsubo, H. (1976), "Subspace iteration accelerated by using Chebyshev polynomials for eigenvalue problems with symmetric matrices", Int. J. Numer. Meth. Engg., 10, 935-944. https://doi.org/10.1002/nme.1620100418
  24. Yang, Y. B., Hung, H. H. and Hsu, L. C. (2008), "Ground vibrations due to underground trains considering soiltunnel interaction", Interaction and Multiscale Mechanics, An Int. J., 1(1), 157-175. https://doi.org/10.12989/imm.2008.1.1.157
  25. Yuan, M.W., et al. (2005), SAP84-A General purpose Program on Microcomputer (version 6.5), Peking University, Beijing, China (in Chinese).
  26. Yuan, M.W., et al. (1989), "The WYD method in large eigenvalue problems", Eng. Comput., 6, 49-57. https://doi.org/10.1108/eb023759
  27. Zhao, Q.C., Chen, P., Peng, W.B., Gong, Y.C. and Yuan M.W. (2007), "Accelerate subspace iteration with aggressive shift", Comput. Struct., 85(19-20), 1562-1578. https://doi.org/10.1016/j.compstruc.2006.11.033

Cited by

  1. Investigation on efficiency and applicability of subspace iteration method with accelerated starting vectors for calculating natural modes of structures vol.37, pp.5, 2008, https://doi.org/10.12989/sem.2011.37.5.561
  2. Free Vibration Analysis of Thin Circular Cylindrical Shell with Closure Using Finite Element Method vol.20, pp.1, 2008, https://doi.org/10.1007/s13296-019-00277-5