Abstract
Derivatives of eigenvalues and eigenvectors of parameter-dependent matrix eigenproblems play a key role in the optimum design of structures in engineering, and in the solution of inverse problems, such as the problem of model updating, which arises, for example, when information on the normal modes of vibration of a structure is used to detect structural damage. Both these applications often involve quadratic eigenvalue problems. Most existing methods for the computation of derivatives of quadratic eigenvalue problems are based on the assumption that repeated eigenvalues have well separated first order derivatives. In this paper we propose new algorithms for computing derivatives of eigenvalues and eigenvectors for quadratic eigenvalue problems under much more general conditions than existing methods, whose effectiveness for repeated or tightly clustered eigenvalues are confirmed by some numerical examples.
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References
Andrew, A.L., Tan, R.C.E.: Computation of derivatives of repeated eigenvalues and the corresponding eigenvectors of symmetric matrix pencils. SIAM. J. Matrix Anal. Appl. 20, 78–100 (1998)
Andrew, A.L., Tan, R.C.E.: Iterative computation of derivatives of repeated eigenvalues and the corresponding eigenvectors. Numer. Linear Algebra Appl. 7, 151–167 (2000)
Backstrom L., Lescovec J.,: Supervised random walks: predicting and recommending links in social networks. In: WSDM’11: Proceedings of Fourth ACM Conferences Web Search Data Mining (2011). doi:10.1145/1935826.1935914
Choi, K.M., Jo, H.K., Kim, W.H., Lee, I.W.: Sensitivity analysis of non-conservative eigensystems. J. Sound Vib. 274, 997–1011 (2004)
Chouchane, M., Guedria, N., Smaoui, H.: Eigensensitivity computation of asymmetric damped systems using an algebraic approach. Mech. Syst. Sig. Process. 21, 2761–2776 (2007)
Eldred, M.S., Venkayya, V.B., Anderson, W.J.: Mode tracking issues in structural optimization. AIAA J. 33, 1926–1933 (1995)
Friswell, M.I., Adhikari, S.: Derivatives of complex eigenvectors using Nelson’s method. AIAA J. 38, 2355–2357 (2000)
Fox, R.L., Kapoor, M.P.: Rates of change of eigenvalues and eigenvectors. AIAA J. 6, 2426–2429 (1968)
Guedria, N., Chouchane, M., Smaoui, H.: Second-order eigensensitivity analysis of asymmetric damped systems using Nelson’s method. J. Sound Vib. 300, 974–992 (2007)
Guedria, N., Smaoui, H., Chouchane, M.: A direct algebraic method for eigensolution sensitivity computation of damped asymmetric systems. Int. J. Comput. Methods Eng. 68, 674–689 (2006)
Haug, E.J., Choi, K.K., Komkov, V.: Design Sensitivity Analysis of Structural Systems. Academic Press, New York (1986)
Karlsson, J., Ericsson, A., Åström, K.: Shape modelling by optimising description length using gradients and parameterisation invariance. Springer Proc. Math. 6, 51–90 (2012)
Kuprov, I., Rodgers, C.T.: Derivatives of spin dynamics simulations. J. Chem. Phys. (2009). doi:10.1063/1.3267086
Lee, I.W., Kim, D.O., Jung, G.H.: Natural frequency and mode shape sensitivities of damped systems: part I, distinct natural frequencies. J. Sound Vib. 223, 399–412 (1999)
Mottershead, J.E., Friswell, M.I.: Model updating in structural dynamics: A survey. J. Sound Vib. 167, 347–375 (1993)
Nelson, R.B.: Simplified calculation of eigenvector derivatives. AIAA J. 14, 1201–1205 (1976)
Qian, J., Andrew, A.L., Chu, D.L., Tan, R.C.E.: Computing derivatives of repeated eigenvalues and corresponding eigenvectors of quadratic eigenvalue problems. SIAM J. Matrix Anal. Appl. 34, 1089–1111 (2013)
Tan, R.C.E., Andrew, A.L.: Computing derivatives of eigenvalues and eigenvectors by simultaneous iteration. IMA J. Numer. Anal. 9, 111–122 (1989)
Tang, J., Ni, W.M., Wang, W.L.: Eigensolutions sensitivity for quadratic eigenproblems. J. Sound Vib. 196, 179–188 (1996)
Tisseur, F., Meerbergen, K.: The quadratic eigenvalue problem. SIAM Rev. 43, 235–286 (2001)
Van Der Aa, N.P.: Sensitivity analysis for grating reconstruction. Eindhoven University of Technology, Thesis (Proefschrift) (2007)
Weber, B., Paultre, P., Proulx, J.: Consistent regularization of nonlinear model updating for damage identification. Mech. Syst. Sig. Proc. 23, 1965–1985 (2009)
Weber, B., Paultre, P.: Damage identification in a truss tower by regularized model updating. ASCE J. Struct. Eng. 136, 307–316 (2010)
Xie, H.Q.: An iterative method for partial derivatives of eigenvectors of quadratric eigenvalue problems. BIT 52, 525–536 (2012)
Xie, H.Q., Dai, H.: Derivatives of repeated eigenvalues and corresponding eigenvectors of damped systems. Appl. Math. Mech. (English Edition) 28, 837–845 (2007)
Xie, H.Q., Dai, H.: Calculation of derivatives of multiple eigenpairs of unsymmetrical quadratic eigenvalue problems. Int. J. Comput. Math. 85, 1815–1831 (2008)
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Chu, D., Qian, J., Tan, R.C.E. (2016). Sensitivity Analysis and Its Numerical Methods for Derivatives of Quadratic Eigenvalue Problems. In: Anderssen, R., et al. Applications + Practical Conceptualization + Mathematics = fruitful Innovation. Mathematics for Industry, vol 11. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55342-7_20
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DOI: https://doi.org/10.1007/978-4-431-55342-7_20
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