Document Zbl 1339.65098

Calculation of eigenpair derivatives for symmetric quadratic eigenvalue problem with repeated eigenvalues. (English) Zbl 1339.65098

Comput. Appl. Math. 35, No. 1, 17-28 (2016).

Summary: In this paper, we consider computing the derivatives of the semisimple eigenvalues and corresponding eigenvectors of symmetric quadratic eigenvalue problem. In the proposed method, the eigenvector derivatives of the symmetric quadratic eigenvalue problem are divided into a particular solution and a homogeneous solution; a simplified method is given to calculate the particular solution by solving a linear system with nonsingular coefficient matrix, the method is numerically stable and efficient. Two numerical examples are included to illustrate the validity of the proposed method.

Cited in 1 Document

MSC:

65L15	Numerical solution of eigenvalue problems involving ordinary differential equations
34L16	Numerical approximation of eigenvalues and of other parts of the spectrum of ordinary differential operators
65F15	Numerical computation of eigenvalues and eigenvectors of matrices

Keywords:

sensitivity analysis; quadratic eigenvalue problem; eigenvalue derivatives; eigenvector derivatives; numerical example

Cite Review PDF

Full Text: DOI

References:

[1]	Adelman HM, Haftka RT (1986) Sensitivity analysis of discrete structural system. AIAA J 24:823-832 · Zbl 0598.15012 · doi:10.2514/3.48671
[2]	Adhikari S (2013a) Structural dynamics with generalized damping models: analysis. Wiley-ISTE · Zbl 0342.65021
[3]	Adhikari S (2013b) Structural dynamics with generalized damping models: identification. Wiley-ISTE
[4]	Adhikari S (1999) Rates of change of eigenvalues and eigenvectors in damped dynamic systems. AIAA J 37:1452-1458 · doi:10.2514/2.622
[5]	Adhikari S, Friswell MI (2001) Eigenderivative analysis of asymmetric non-conservative systems. Int J Numer Method Eng 51:709-733 · Zbl 0992.70016 · doi:10.1002/nme.186.abs
[6]	Andrew AL, Chu K-WE, Lancaster P (1993) Derivatives of eigenvalues and eigenvectors of matrix functions. SIAM J Matrix Anal Appl 14:903-926 · Zbl 0786.15011 · doi:10.1137/0614061
[7]	Andrew AL, Tan RCE (1998) Computation of derivatives of repeated eigenvalues and the corresponding eigenvectors of symmetric matrix pencils. SIAM J Matrix Anal Appl 20:78-100 · Zbl 0919.65026 · doi:10.1137/S0895479896304332
[8]	Andrew AL, Tan RCE (2000) Iterative computation of derivatives of repeated eigenvalues and the corresponding eigenvectors. Numer Linear Algebra Appl 7:151-167 · Zbl 1051.65039 · doi:10.1002/1099-1506(200005)7:4<151::AID-NLA191>3.0.CO;2-M
[9]	Benzi M, Golub GH, Liesen J (2005) Numerical solution of saddle point problems. Acta Numerica 14:1-137 · Zbl 1115.65034 · doi:10.1017/S0962492904000212
[10]	Brandon JA (1991) Second-order design sensitivities to assess the applicability of sensitivity analysis. AIAA J 29:135-139 · doi:10.2514/3.10555
[11]	Choi KM, Jo HK, Kim WH, Lee IW (2004) Sensitivity analysis of non-conservative eigensystems. J Sound Vib 274:997-1011 · doi:10.1016/S0022-460X(03)00660-6
[12]	Fox RL, Kapoor MP (1968) Rates of changes of eigenvalues and eigenvectors. AIAA J 6:2426-2429 · Zbl 0181.53003 · doi:10.2514/3.5008
[13]	Friswell MI, Mottershead JE (1995) Finite element model updating in structural dynamics. Kluwer Academic, Norwell · Zbl 0855.73002 · doi:10.1007/978-94-015-8508-8
[14]	Friswell MI, Adhikari S (2000) Derivatives of complex eigenvectors using nelson’s method. AIAA J 38:2355-2357 · doi:10.2514/2.907
[15]	Garg S (1973) Derivatives of eigensolutions for a general matrix. AIAA J 11:1191-1194 · doi:10.2514/3.6892
[16]	Guedria N, Smaoui H, Chouchane M (2006) A direct algebraic method for eigensolution sensitivity computation of damped asymmetric systems. Int J Numer Method Eng 68:674-689 · Zbl 1128.74015 · doi:10.1002/nme.1732
[17]	Haftka RT, Adelman HM (1989) Recent development in structural sensitivity analysis. Struct Optim 1:137-151 · doi:10.1007/BF01637334
[18]	Haug EJ, Choi KK, Komkov V (1986) Design sensitivity analysis of structural systems. Academic Press, New York · Zbl 0618.73106
[19]	Lee IW, Jung JH (1997) An efficient algebraic method for the computation of natural frequency and mode shape sensitivities: part I, distinct natural frequencies. Comput Struct 62:429-435 · Zbl 0912.73080 · doi:10.1016/S0045-7949(96)00206-4
[20]	Lee IW, Jung JH (1997) An efficient algebraic method for the computation of natural frequency and mode shape sensitivities: part II, multiple natural frequencies. Comput Struct 62:437-443 · Zbl 0912.73080 · doi:10.1016/S0045-7949(96)00207-6
[21]	Lee IW, Kim DO, Jung GH (1999) Natural frequency and mode shape sensitivities of damped systems: part I, distinct natural frequencies. J Sound Vib 223:399-412 · doi:10.1006/jsvi.1998.2129
[22]	Lee IW, Kim DO, Jung GH (1999) Natural frequency and mode shape sensitivities of damped systems: part II, multiple natural frequencies. J Sound Vib 223:413-424 · doi:10.1006/jsvi.1998.2130
[23]	Liu XB (2013) A new method for calculating derivatives of eigenvalues and eigenvectors for discrete structural systems. J Sound Vib 332:1859-1867 · Zbl 1286.05115 · doi:10.1016/j.jsv.2012.11.017
[24]	Messina A, Williams EJ, Contursi T (1998) Structural damage detection by a sensitivity and statistical-based method. J Sound Vib 216:791-808 · doi:10.1006/jsvi.1998.1728
[25]	Mills-Curran WC (1988) Calculation of eigenvector derivatives for structures with repeated eigenvalues. AIAA J 26:867-871 · Zbl 0665.73069 · doi:10.2514/3.9980
[26]	Murthy DV, Haftka RT (1988) Derivatives of eigenvalues and eigenvectors of a general complex matrix. Int J Numer Method Eng 26:293-311 · Zbl 0637.65030 · doi:10.1002/nme.1620260202
[27]	Nelson RB (1976) Simplified calculation of eigenvector derivatives. AIAA J 14:1201-1205 · Zbl 0342.65021 · doi:10.2514/3.7211
[28]	Plaut RH, Huseyin K (1973) Derivatives of eigenvalues and eigenvectors in non-self-adjoint systems. AIAA J 11:250-251 · doi:10.2514/3.6740
[29]	Qian J, Andrew AL, Chu D, Tan RCE (2013) Computing derivatives of repeated eigenvalues and corresponding eigenvectors of quadratic eigenvalue problems. SIAM J Matrix Anal Appl 34:1089-1111 · Zbl 1286.65049 · doi:10.1137/120879841
[30]	Rogers LC (1970) Derivatives of eigenvalues and eigenvectors. AIAA J 8:943-944 · doi:10.2514/3.5795
[31]	Rudisill CS (1974) Derivatives of eigenvalues and eigenvectors for a general matrix. AIAA J 12:721-722 · Zbl 0285.15011 · doi:10.2514/3.49330
[32]	Rudisill CS, Chu Y (1975) Numerical methods for evaluating the derivatives of eigenvalues and eigenvectors. AIAA J 13:834-837 · doi:10.2514/3.60449
[33]	Tang J, Ni WM, Wang WL (1996) Eigensolutions sensitivity for quadratic eigenproblems. J Sound Vib 196:179-188 · Zbl 1235.70010 · doi:10.1006/jsvi.1996.0475
[34]	Xie HQ, Dai H (2007) Derivatives of repeated eigenvalues and corresponding eigenvectors of damped systems. Appl Math Mech (English Ed) 28:837-845 · Zbl 1231.65068 · doi:10.1007/s10483-007-0614-4
[35]	Xie HQ, Dai H (2008) Calculation of derivatives of multiple eigenpairs of unsymmetrical quadratic eigenvalue problems. Int J Comput Math 85:1815-1831 · Zbl 1158.65026 · doi:10.1080/00207160701581525
[36]	Xie HQ (2011) IRAM-based method for eigenpairs and their derivatives of large matrix-valued functions. Numer Linear Algebra Appl 18:513-538 · Zbl 1245.65046 · doi:10.1002/nla.740
[37]	Xu ZH, Wu BS (2008) Derivatives of complex eigenvectors with distinct and repeated eigenvalues. Int J Numer Method Eng 75:945-963 · Zbl 1195.74069 · doi:10.1002/nme.2280
[38]	Zeng QH (1994) Highly accurate modal method for calculating eigenvector derivative in viscous damping systems. AIAA J 33:746-751 · Zbl 0826.73037 · doi:10.2514/3.12453

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.