Eigenmode computation of cavities with perturbed geometry using matrix perturbation methods applied on generalized eigenvalue problems. (English) Zbl 1392.78017
Summary: Generalized eigenvalue problems are standard problems in computational sciences. They may arise in electromagnetic fields from the discretization of the Helmholtz equation by for example, the finite element method (FEM). Geometrical perturbations of the structure under concern lead to a new generalized eigenvalue problems with different system matrices. Geometrical perturbations may arise by manufacturing tolerances, harsh operating conditions or during shape optimization. Directly solving the eigenvalue problem for each perturbation is computationally costly. The perturbed eigenpairs can be approximated using eigenpair derivatives. Two common approaches for the calculation of eigenpair derivatives, namely modal superposition method and direct algebraic methods, are discussed in this paper. Based on the direct algebraic methods an iterative algorithm is developed for efficiently calculating the eigenvalues and eigenvectors of the perturbed geometry from the eigenvalues and eigenvectors of the unperturbed geometry.
MSC:
| 78M10 | Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory |
| 65F15 | Numerical computation of eigenvalues and eigenvectors of matrices |
Keywords:
generalized eigenvalue problem; matrix perturbation; modal sensitivity analysis; Slater’s theorem; eigenvalue reanalysis; microwave cavityReferences:
| [1] | Ghaboussi, J.; Wu, X. S., Numerical methods in computational mechanics, (2016), CRC Press |
| [2] | Bathe, K. J.; Wilson, E. L., Solution methods for eigenvalue problems in structural mechanics, Int. J. Numer. Methods Eng., 6, 213-226, (1973) · Zbl 0252.73062 |
| [3] | Bittnar, Z.; Šejnoha, J., Numerical methods in structural mechanics, (1996), Thomas Telford |
| [4] | Brackebusch, K.; van Rienen, U., Eigenmode computation for cavities with perturbed geometry based on a series expansion of unperturbed eigenmodes, (Proceedings of 3rd IPAC, New Orleans, USA, (May 2012)), 20-25 |
| [5] | Brackebusch, K.; Glock, H.; van Rienen, U., Calculation of high frequency fields in resonant cavities based on perturbation theory, (Proceedings of 2nd IPAC, San Sebastian, Spain, (September 2011)), 4-9 |
| [6] | Meidlinger, D., A general perturbation theory for cavity mode field patterns, (Proceedings of SRF2009, Berlin, Germany, (2009)), 523-527 |
| [7] | Baldwin, J. F.; Hutton, S. G., Natural modes of modified structures, AIAA J., 23, 1737-1743, (1985) |
| [8] | Lü, Z. H.; Shao, C.; Feng, Z. D., High accuracy step-by-step perturbation method of the generalized eigenvalue problem and its application to parametric history analysis of structural vibration characteristics, J. Sound Vib., 164, 459-469, (1993) · Zbl 0925.73384 |
| [9] | Zhang, T.; Golub, G. H.; Law, K., Eigenvalue perturbation and generalized Krylov subspace method, Appl. Numer. Math., 27, 2, 185-202, (1998) · Zbl 0931.65039 |
| [10] | Chen, S. H.; Yang, X. W., Extended kirsch combined method for eigenvalue reanalysis, AIAA J., 38, 927-930, (2000) |
| [11] | Kirsch, U.; Bogomolni, M., Procedures for approximate eigenproblem reanalysis of structures, Int. J. Numer. Methods Eng., 60, 1969-1986, (2004) · Zbl 1069.74024 |
| [12] | Chen, S. H.; Wu, X. M.; Yang, Z. J., Eigensolution reanalysis of modified structures using epsilon-algorithm, Int. J. Numer. Methods Eng., 66, 2115-2130, (2006) · Zbl 1110.74846 |
| [13] | Günel, S.; Zoral, E. Y., Parametric history analysis of resonance problems via step-by-step eigenvalue perturbation technique, IET Microw. Antennas Propag., 4, 466-476, (2010) |
| [14] | Wang, P. B.; Patadia, C. S., Efficient eigensolution reanalysis using sensitivity data, (2nd International Conference on Engineering Optimization, Lisbon, Portugal, (2010)) |
| [15] | Zou, W.; Xu, T.; Zhang, H.; Xu, T., Fast structural optimization with frequency constraints by genetic algorithm using adaptive eigenvalue reanalysis methods, Struct. Multidiscip. Optim., 43, 799-810, (2011) |
| [16] | Liu, X.; Shao, Q.; Xu, T.; Guo, G., Natural frequency modification using frequency-shift combined approximations algorithm, (Recent Developments in Intelligent Systems and Interactive Applications, IISA 2016, Advances in Intelligent Systems and Computing, vol. 541, (2017)), 86-91 |
| [17] | Kirsch, U., Improved stiffness-based first-order approximations for structural optimization, AIAA J., 33, 143-150, (1995) · Zbl 0824.73041 |
| [18] | Fox, R. L.; Kapoor, M. P., Rates of change of eigenvalues and eigenvectors, AIAA J., 6, 2426-2429, (1968) · Zbl 0181.53003 |
| [19] | Nelson, R. B., Simplified calculation of eigenvector derivatives, AIAA J., 14, 1201-1205, (1976) · Zbl 0342.65021 |
| [20] | Dailey, R. L., Eigenvector derivatives with repeated eigenvalues, AIAA J., 27, 486-491, (1989) |
| [21] | Ojalvo, I. U., Efficient computation of modal sensitivities for systems with repeated frequencies, AIAA J., 26, 361-366, (1988) · Zbl 0682.73043 |
| [22] | Mills-Curran, W. C., Calculation of eigenvector derivatives for structures with repeated eigenvalues, AIAA J., 26, 867-871, (1988) · Zbl 0665.73069 |
| [23] | Friswell, M. I., The derivatives of repeated eigenvalues and their associated eigenvectors, Transactions of American Society of Mechanical Engineers, J. Vib. Acoust., 118, 390-397, (1996) |
| [24] | Shaw, J.; Jayasuriya, S., Modal sensitivities for repeated eigenvalues and eigenvalue derivatives, AIAA J., 30, 850-852, (1992) · Zbl 0774.65019 |
| [25] | Wang, P.; Dai, H., Calculation of eigenpair derivatives for asymmetric damped systems with distinct and repeated eigenvalues, Int. J. Numer. Methods Eng., 103, 501-515, (2015) · Zbl 1352.65122 |
| [26] | Li, L.; Hu, Y.; Wang, X., A parallel way for computing eigenvector sensitivity of asymmetric damped systems with distinct and repeated eigenvalues, Mech. Syst. Signal Process., 30, 61-77, (2012) |
| [27] | Lee, I.-W.; Jung, G.-H., An efficient algebraic method for the computation of natural frequency and mode shape sensitivities—part I. distinct natural frequencies, Comput. Struct., 62, 429-435, (1997) · Zbl 0912.73080 |
| [28] | Lee, I.-W.; Jung, G.-H., An efficient algebraic method for the computation of natural frequency and mode shape sensitivities—part II. multiple natural frequencies, Comput. Struct., 62, 437-443, (1997) · Zbl 0912.73080 |
| [29] | Rylander, T.; Ingelström, P.; Bondeson, A., The finite element method, (Computational Electromagnetics, (2013), Springer), 93-184 · Zbl 1259.78002 |
| [30] | Nakatsukasa, Y., Algorithms and perturbation theory for matrix eigenvalue problems and the singular value decomposition, (2011), University of California Davis, Ph.D. thesis |
| [31] | Golub, G. H.; Van Loan, C. F., Matrix computation, (1996), The Johns Hopkins University Press · Zbl 0865.65009 |
| [32] | Bathe, K. J., Finite element procedures, (2006) |
| [33] | To, W. M.; Ewins, D. J., Structural modification analysis using Rayleigh quotient iteration, Int. J. Mech. Sci., 32, 169-179, (1990) · Zbl 0712.73052 |
| [34] | Aune, B.; Bandelmann, R.; Bloess, D.; Bonin, B.; Bosotti, A.; Champion, M.; Crawford, C.; Deppe, G.; Dwersteg, B.; Edwards, D., Superconducting TESLA cavities, Phys. Rev. Spec. Top., Accel. Beams, 3, (2000) |
| [35] | Galek, T.; Zadeh, S. Gorgi; van Rienen, U.; Brackebusch, K., Study of higher order modes in multi-cell cavities for BESSY-VSR upgrade, (Proceedings of IPAC2014, Dresden, Germany, (2014)), 412-414 |
| [36] | Shemelin, V., Optimal choice of cell geometry for a multicell superconducting cavity, Phys. Rev. Spec. Top., Accel. Beams, 12, 11, (2009) |
| [37] | Belomestnykh, S.; Shemelin, V., High-β cavity design — a tutorial, (SRF International Workshop, Ithaca, New York, (2005)) |
| [38] | Matlab, (2014), The MathWorks Inc. Natick, Massachusetts, version 8.4.0.150421 (R2014b) |
| [39] | R.B. Lehoucq, D.C. Sorensen, C. Yang, ARPACK users’ guide: solution of large scale eigenvalue problems with implicitly restarted Arnoldi methods, 1998.; R.B. Lehoucq, D.C. Sorensen, C. Yang, ARPACK users’ guide: solution of large scale eigenvalue problems with implicitly restarted Arnoldi methods, 1998. · Zbl 0901.65021 |
| [40] | Slater, J. C., Microwave electronics, Rev. Mod. Phys., 18, 4, 441-512, (1946) · Zbl 0063.07079 |
| [41] | Pozar, D. M., Microwave engineering, (2011), John Wiley |
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