Eigenvalue and eigenvector approximate analysis for repeated eigenvalue problems

Abstract. An algorithm to perform a higher-order sensitivity analysis for electromagnetic eigenvalue problems is presented. By computing the eigenvalue and eigenvector derivatives, the Brillouin Diagram for periodic structures can be calculated. The discrete model is described using the Finite Integration Technique (FIT) with periodic boundaries, and the sensitivity analysis is performed with respect to the phase shift φ between the periodic boundaries. For validation, a reference solution is calculated by solving multiple eigenvalue problems (EVP). Furthermore, the eigenvalue derivatives are compared to reference values using finite difference (FD) formulas.

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Nonlinear interval eigenvalue problems for damped spring-mass system

Engineering Computations ◽

10.1108/ec-04-2017-0128 ◽

2018 ◽

Vol 35 (6) ◽

pp. 2272-2286 ◽

Cited By ~ 4

Author(s):

Snehashish Chakraverty ◽

Nisha Rani Mahato

Keyword(s):

Eigenvalue Problems ◽

Solution Procedure ◽

Damping Factor ◽

Computationally Efficient ◽

Mass System ◽

Content Type ◽

Nonlinear Eigenvalue Problems ◽

Closed Intervals ◽

Quadratic Eigenvalue ◽

Eigenvalue And Eigenvector

Purpose In structural mechanics, systems with damping factor get converted to nonlinear eigenvalue problems (NEPs), namely, quadratic eigenvalue problems. Generally, the parameters of NEPs are considered as crisp values but because of errors in measurement, observation or maintenance-induced errors, the parameters may have uncertain bounds of values, and such uncertain bounds may be considered in terms of closed intervals. As such, this paper aims to deal with solving nonlinear interval eigenvalue problems (NIEPs) with respect to damped spring-mass systems having interval parameters. Design/methodology/approach Two methods, namely, linear sufficient regularity perturbation (LSRP) and direct sufficient regularity perturbation (DSRP), have been proposed for solving NIEPs based on sufficient regularity perturbation method for intervals. LSRP may be used for solving NIEPs by linearizing the eigenvalue problems into generalized interval eigenvalue problems, and DSRP may be considered as a direct solution procedure for solving NIEPs. Findings LSRP and DSRP methods help in computing the lower and upper eigenvalue and eigenvector bounds for NIEPs which contain the crisp eigenvalues. Further, the DSRP method is computationally efficient compared to LSRP. Originality/value The efficiency of the proposed methods has been validated by example problems of NIEPs. Moreover, the procedures may be extended for other nonlinear interval eigenvalue application problems.

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A Practical Algorithm for the Efficient Computation of Eigenvector Derivatives

Volume 3B: 15th Biennial Conference on Mechanical Vibration and Noise — Acoustics, Vibrations, and Rotating Machines ◽

10.1115/detc1995-0486 ◽

1995 ◽

Author(s):

R. M. Lin ◽

Z. Wang ◽

M. K. Lim

Keyword(s):

Finite Element ◽

Structural Design ◽

Eigenvalue Problems ◽

Computational Cost ◽

Efficient Computation ◽

Eigenvalues And Eigenvectors ◽

Practical Algorithm ◽

Eigenvector Derivatives ◽

Eigenvalue And Eigenvector ◽

Derivatives Of

Abstract Derivatives of eigenvalues and eigenvectors have become increasingly important in the development of modern numerical methods for areas such as structural design optimization, dynamic system identification and dynamic control, and the development of effective and efficient methods for the calculation of such derivatives has remained to be an active research area for several decades. In this paper, a practical algorithm has been developed for efficiently computing eigenvector derivatives of generalized symmetric eigenvalue problems. For eigenvector derivative of a separate mode, the computation only requires the knowledge of eigenvalue and eigenvector of the mode itself and an inverse of system matrix accounts for most computation cost involved. In the case of two close modes, the modal information of both modes is required and the eigenvector derivatives can be accurately determined simultaneously at minor additional computational cost. Further, the method has been extended to the case of practical structural design where structural modifications are made locally and the eigenvalues and eigenvectors and their derivatives are of interest. By combining the proposed algorithm together with the proposed inverse iteration technique and singular value decomposition theory, eigenproperties and their derivatives can be very efficiently computed. Numerical results from a practical finite element model have demonstrated the practicality of the proposed method. The proposed method can be easily incorporated into commercial finite element packages to improve the computational efficiency of eigenderivatives needed for practical applications.

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Refinements in the Approximate Analysis of Web-Stiffened Sandwich Structures

Journal of Applied Mechanics ◽

10.1115/1.3423200 ◽

1973 ◽

Vol 40 (4) ◽

pp. 992-996 ◽

Cited By ~ 2

Author(s):

Y. N. Chen ◽

F. Cicero ◽

J. Kempner

Keyword(s):

Present Method ◽

Eigenvalue Problems ◽

Sandwich Structures ◽

Normal Modes ◽

Free Vibrations ◽

Approximate Analysis ◽

One Dimensional ◽

Elastic Beams ◽

Energy Formulation

Presented in this work is a method of construction of approximate functions in connection with the energy formulation of certain eigenvalue problems of web-stiffened sandwich structures. The construction is based upon the method of Young together with a group of fundamental functions deduced from Timoshenko’s flexural equations for elastic beams. The analysis is exemplified and numerically tested by the eigenvalue problems of free vibrations and buckling of one-dimensional sandwich structures. Results indicate that the present method possesses advantages over similar constructions oriented from the classical flexural normal modes of Bernoulli-Euler.

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