## Taiwanese Journal of Mathematics

### A NEW EIGENVALUE EMBEDDING APPROACH FOR FINITE ELEMENT MODEL UPDATING

#### Abstract

This paper concerns the eigenvalue embedding problem (EEP) of updating a symmetric finite-element model so that a few troublesome eigenvalues are replaced by some chosen ones, while the remaining large number of eigenvalues and eigenvectors of the original model do not change. Based on the theory established in [2], by sufficiently utilizing the inherent freedom of the EEP, an expression of the parameterized solution to the EEP is derived. This expression is then used to develop a novel numerical method for solving the EEP, in which the parameters in the solutions are optimized in some sense. This method not only utilizes the freedom of the EEP but also removes the limitation of the method proposed in [6]. The results of our numerical experiments show that the present algorithm is feasible and efficient, and can outperform the iterative method in [3] and the method in [6].

#### Article information

Source
Taiwanese J. Math., Volume 14, Number 3A (2010), 911-932.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500405874

Digital Object Identifier
doi:10.11650/twjm/1500405874

Mathematical Reviews number (MathSciNet)
MR2667724

Zentralblatt MATH identifier
1210.15010

#### Citation

Cai, Yunfeng; Xu, Shufang. A NEW EIGENVALUE EMBEDDING APPROACH FOR FINITE ELEMENT MODEL UPDATING. Taiwanese J. Math. 14 (2010), no. 3A, 911--932. doi:10.11650/twjm/1500405874. https://projecteuclid.org/euclid.twjm/1500405874

#### References

• Y. F. Cai, Y. C. Kuo, W. W. Lin and S. F. Xu, Solutions to a quadratic inverse eigenvalue problem, Linear Algebra and its Applications, 430 (2009), 1590-1606.
• Y. F. Cai and S. F. Xu, On a quadratic inverse eigenvalue problem, Inverse Problems, 25 (2009), 085004.
• J. Carvalho, B. N. Datta, W. W. Lin and C. S. Wang, Symmetry preserving eigenvalue embedding in finite-element model updating of vibrating structures, J. Sound and Vibration, 290 (2006), 839-864.
• M. T. Chu, Y. C. Kuo and W. W. Lin, On inverse quadratic eigenvalue problems with partially prescribed eigenstructure, SIAM J. Matrix Analysis and Application, 25 (2004), 995-1020.
• M. T. Chu, W. W. Lin and S. F. Xu, Updating quadratic models with no spill-over effect on unmeasured spectral data, Inverse Problems, 23 (2007), 243-256.
• M. T. Chu and S. F. Xu, Spectral decomposition of real symmetric quadratic $\lambda$-matrices and its applications, Mathematics of Computation, 78 (2009), 293-313.
• B. N. Datta, S. Elhay and Y. M. Ram, Orthogonality and partial pole assignment for the symmetric definite quadratic pencil, Linear Algebra and its Applications, 257 (1997), 29-48.
• B. N. Datta, S. Elhay, Y. M. Ram and D. R. Sarkissian, Partial eigenstructure assignment for the quadratic pencil, Journal of Sound and Vibration, 230 (2000), 101-110.
• B. N. Datta and D. R. Sarkissian, Multi-input partial eigenvalue assignment for the symmetric quadratic pencil, Proceedings of the American Control Conference, San Diego, California, June 2-4, 1999, pp. 2244-2247.
• B. N. Datta and D. R. Sarkissian, Theory and computations of some inverse eigenvalue problems for the quadratic pencil, Contemporary Mathematics, 280 (2001), 221-240.
• B. N. Datta, Finite element model updating and partial eigenvalue assignment in structral dynamics: recent developments on computional methods, Mathematical Modelling and Analysis, Proceedings of the 10th International Conference MMA2005 &CMAM2, Trakai, 2005, pp. 15-27.
• P. Lancaster, Model-updating for symmetric quadratic eigenvalue problems, MIMS EPrint: 2006.407, 2006.
• A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics, Springer-Verlag, New York, 2000.
• Y. M. Ram and S. Elhay, Pole assignment in vibrating systems by multi-input control, Journal of Sound and Vibration, 230 (2001), 101-119.
• Y. M. Ram, Pole-zero assignment of vibratory systems by state feedback control, Journal of Vibration and Control, 4 (1998), 165-185.
• F. Tisseur and K. Meerbergen, The quadratic eigenvalue problem, SIAM Rev., 43 (2001), 235-286.
• http://math.nist.gov/MatrixMarket/data/Harwell-Boeing/bcsstruc1/bcsstk02.html.