An extension of the Cayley transform method for a parameterized generalized inverse eigenvalue problem

[EN] Since recent studies have shown that the Cayley transform method can be an effective iterative method for solving the inverse eigenvalue problem, in this work, we consider using an extension of it for solving a type of parameterized generalized inverse eigenvalue problem and prove its locally quadratic convergence. This type of inverse eigenvalue problem, which includes multiplicative and additive inverse eigenvalue problems, appears in many applications. Also, we consider the case where the given eigenvalues are multiple. In this case, we describe a modified problem that is not overdetermined and discuss the extension of the Cayley transform method for this modified problem. Finally, to demonstrate the effectiveness of these algorithms, we present some numerical examples to show that the proposed methods are practical and efficient.The authors would like to express their heartfelt thanks to the editor and anonymous referees for their useful comments and constructive suggestions that substantially improved the quality and presentation of this article. This research was developed during a visit of Z.D. to Universitat Politecnica de Valencia. Z.D. would like to thank the hospitality shown by D. Sistemes Informatics i Computacio, Universitat Politecnica de Valencia. J.E.R. was partially supported by the Spanish Agencia Estatal de Investigacion (AEI) under grant TIN2016-75985-P, which includes European Commission ERDF funds. The authors thank Carmen Campos for useful comments on an initial draft of the article.Dalvand, Z.; Hajarian, M.; Román Moltó, JE. (2020). An extension of the Cayley transform method for a parameterized generalized inverse eigenvalue problem. Numerical Linear Algebra with Applications. 27(6):1-24. https://doi.org/10.1002/nla.2327S124276Chu, M., & Golub, G. (2005). Inverse Eigenvalue Problems. doi:10.1093/acprof:oso/9780198566649.001.0001Hajarian, M., & Abbas, H. (2018). Least squares solutions of quadratic inverse eigenvalue problem with partially bisymmetric matrices under prescribed submatrix constraints. Computers & Mathematics with Applications, 76(6), 1458-1475. doi:10.1016/j.camwa.2018.06.038Hajarian, M. (2019). An efficient algorithm based on Lanczos type of BCR to solve constrained quadratic inverse eigenvalue problems. Journal of Computational and Applied Mathematics, 346, 418-431. doi:10.1016/j.cam.2018.07.025Hajarian, M. (2018). Solving constrained quadratic inverse eigenvalue problem via conjugate direction method. Computers & Mathematics with Applications, 76(10), 2384-2401. doi:10.1016/j.camwa.2018.08.034Chu, M. T., & Golub, G. H. (2002). Structured inverse eigenvalue problems. Acta Numerica, 11, 1-71. doi:10.1017/s0962492902000016Ghanbari, K., & Parvizpour, F. (2012). Generalized inverse eigenvalue problem with mixed eigendata. Linear Algebra and its Applications, 437(8), 2056-2063. doi:10.1016/j.laa.2012.05.020Yuan, Y.-X., & Dai, H. (2009). A generalized inverse eigenvalue problem in structural dynamic model updating. Journal of Computational and Applied Mathematics, 226(1), 42-49. doi:10.1016/j.cam.2008.05.015Yuan, S.-F., Wang, Q.-W., & Xiong, Z.-P. (2013). Linear parameterized inverse eigenvalue problem of bisymmetric matrices. Linear Algebra and its Applications, 439(7), 1990-2007. doi:10.1016/j.laa.2013.05.026Dai, H., Bai, Z.-Z., & Wei, Y. (2015). On the Solvability Condition and Numerical Algorithm for the Parameterized Generalized Inverse Eigenvalue Problem. SIAM Journal on Matrix Analysis and Applications, 36(2), 707-726. doi:10.1137/140972494Gladwell, G. M. L. (1986). Inverse Problems in Vibration. Applied Mechanics Reviews, 39(7), 1013-1018. doi:10.1115/1.3149517Friedland, S., Nocedal, J., & Overton, M. L. (1987). The Formulation and Analysis of Numerical Methods for Inverse Eigenvalue Problems. SIAM Journal on Numerical Analysis, 24(3), 634-667. doi:10.1137/0724043Chan, R. H. (2003). BIT Numerical Mathematics, 43(1), 7-20. doi:10.1023/a:1023611931016Bai, Z.-J., Chan, R. H., & Morini, B. (2004). An inexact Cayley transform method for inverse eigenvalue problems. Inverse Problems, 20(5), 1675-1689. doi:10.1088/0266-5611/20/5/022Shen, W. P., Li, C., & Jin, X. Q. (2011). A Ulm-like method for inverse eigenvalue problems. Applied Numerical Mathematics, 61(3), 356-367. doi:10.1016/j.apnum.2010.11.001Shen, W., & Li, C. (2012). AN ULM-LIKE CAYLEY TRANSFORM METHOD FOR INVERSE EIGENVALUE PROBLEMS. Taiwanese Journal of Mathematics, 16(1). doi:10.11650/twjm/1500406546Aishima, K. (2018). A quadratically convergent algorithm based on matrix equations for inverse eigenvalue problems. Linear Algebra and its Applications, 542, 310-333. doi:10.1016/j.laa.2017.05.019Shen, W. P., Li, C., & Jin, X. Q. (2015). An inexact Cayley transform method for inverse eigenvalue problems with multiple eigenvalues. Inverse Problems, 31(8), 085007. doi:10.1088/0266-5611/31/8/085007Shen, W., Li, C., & Jin, X. (2016). An Ulm-like Cayley Transform Method for Inverse Eigenvalue Problems with Multiple Eigenvalues. Numerical Mathematics: Theory, Methods and Applications, 9(4), 664-685. doi:10.4208/nmtma.2016.y15030Aishima, K. (2018). A quadratically convergent algorithm for inverse eigenvalue problems with multiple eigenvalues. Linear Algebra and its Applications, 549, 30-52. doi:10.1016/j.laa.2018.03.022Li, L. (1995). Sufficient conditions for the solvability of an algebraic inverse eigenvalue problem. Linear Algebra and its Applications, 221, 117-129. doi:10.1016/0024-3795(93)00225-oBiegler-König, F. W. (1981). Sufficient conditions for the solubility of inverse eigenvalue problems. Linear Algebra and its Applications, 40, 89-100. doi:10.1016/0024-3795(81)90142-7Alexander, J. C. (1978). The additive inverse eigenvalue problem and topological degree. Proceedings of the American Mathematical Society, 70(1), 5-5. doi:10.1090/s0002-9939-1978-0487546-3Byrnes, C. I., & Wang, X. (1993). The Additive Inverse Eigenvalue Problem for Lie Perturbations. SIAM Journal on Matrix Analysis and Applications, 14(1), 113-117. doi:10.1137/0614009Wang, Z., & Vong, S. (2013). A Guass–Newton-like method for inverse eigenvalue problems. International Journal of Computer Mathematics, 90(7), 1435-1447. doi:10.1080/00207160.2012.750721Jiang, J., Dai, H., & Yuan, Y. (2013). A symmetric generalized inverse eigenvalue problem in structural dynamics model updating. Linear Algebra and its Applications, 439(5), 1350-1363. doi:10.1016/j.laa.2013.04.021Cox, S. J., Embree, M., & Hokanson, J. M. (2012). One Can Hear the Composition of a String: Experiments with an Inverse Eigenvalue Problem. SIAM Review, 54(1), 157-178. doi:10.1137/080731037Ji, X. (1998). On matrix inverse eigenvalue problems. Inverse Problems, 14(2), 275-285. doi:10.1088/0266-5611/14/2/004Dai, H., & Lancaster, P. (1997). Newton’s Method for a Generalized Inverse Eigenvalue Problem. Numerical Linear Algebra with Applications, 4(1), 1-21. doi:10.1002/(sici)1099-1506(199701/02)4:13.0.co;2-dShu, L., Bo, W., & Ji-zhong, H. (2004). Homotopy solution of the inverse generalized eigenvalue problems in structural dynamics. Applied Mathematics and Mechanics, 25(5), 580-586. doi:10.1007/bf02437606Dai, H. (1999). An algorithm for symmetric generalized inverse eigenvalue problems. Linear Algebra and its Applications, 296(1-3), 79-98. doi:10.1016/s0024-3795(99)00109-3Lancaster, P. (1964). Algorithms for lambda-matrices. Numerische Mathematik, 6(1), 388-394. doi:10.1007/bf01386088Biegler-K�nig, F. W. (1981). A Newton iteration process for inverse eigenvalue problems. Numerische Mathematik, 37(3), 349-354. doi:10.1007/bf01400314Parlett, B. N. (1998). The Symmetric Eigenvalue Problem. doi:10.1137/1.9781611971163Higham, N. J. (2008). Functions of Matrices. doi:10.1137/1.978089871777

Fuzzy finite element model updating of a laboratory wind turbine blade for structural modification detection

Peer reviewedPublisher PD

Solving large-scale dynamic systems using band Lanczos method in Rockwell NASTRAN on CRAY X-MP

The improved cost effectiveness using better models, more accurate and faster algorithms and large scale computing offers more representative dynamic analyses. The band Lanczos eigen-solution method was implemented in Rockwell's version of 1984 COSMIC-released NASTRAN finite element structural analysis computer program to effectively solve for structural vibration modes including those of large complex systems exceeding 10,000 degrees of freedom. The Lanczos vectors were re-orthogonalized locally using the Lanczos Method and globally using the modified Gram-Schmidt method for sweeping rigid-body modes and previously generated modes and Lanczos vectors. The truncated band matrix was solved for vibration frequencies and mode shapes using Givens rotations. Numerical examples are included to demonstrate the cost effectiveness and accuracy of the method as implemented in ROCKWELL NASTRAN. The CRAY version is based on RPK's COSMIC/NASTRAN. The band Lanczos method was more reliable and accurate and converged faster than the single vector Lanczos Method. The band Lanczos method was comparable to the subspace iteration method which was a block version of the inverse power method. However, the subspace matrix tended to be fully populated in the case of subspace iteration and not as sparse as a band matrix

Model correlation and damage location for large space truss structures: Secant method development and evaluation

On-orbit testing of a large space structure will be required to complete the certification of any mathematical model for the structure dynamic response. The process of establishing a mathematical model that matches measured structure response is referred to as model correlation. Most model correlation approaches have an identification technique to determine structural characteristics from the measurements of the structure response. This problem is approached with one particular class of identification techniques - matrix adjustment methods - which use measured data to produce an optimal update of the structure property matrix, often the stiffness matrix. New methods were developed for identification to handle problems of the size and complexity expected for large space structures. Further development and refinement of these secant-method identification algorithms were undertaken. Also, evaluation of these techniques is an approach for model correlation and damage location was initiated

Minimum mass sizing of a large low-aspect ratio airframe for flutter-free performance

A procedure for sizing an airframe for flutter-free performance is demonstrated on a large, flexible supersonic transport aircraft. The procedure is based on using a two level reduced basis or modal technique for reducing the computational cost of performing the repetitive flutter analyses. The supersonic transport aircraft exhibits complex dynamic behavior, has a well-known flutter problem and requires a large finite element model to predict the vibratory and flutter response. Flutter-free designs were produced with small mass increases relative to the wing structural weight and aircraft payload

Iterative method for frequency updating of simple vibrating system

Iterative methods for modification of vibratory characteristics of dynamic systems have attracted a lot of attention as a convenient and more economical way when compared to the traditional and costly structural dynamic optimization processes. Many complicated structures, such as telecommunication towers, chimneys and tall buildings, may be modeled as simple spring-mass systems. This paper presents an iterative method for modification of the frequencies of simple vibrating system consisting of springs and masses. The proposed algorithm may be used to adjust any of the vibration frequencies of a simple vibrating system to the target values within the desired level of accuracy. The method based on the variation of mass and/or stiffness properties of the system is simple yet efficient and needs less computational effort. The efficiency of the method is demonstrated using a numerical example. It is demonstrated that there is a faster convergence for adjustment of the lower frequencies and for the case with stiffness variation of the system rather than mass variation

Eigenstructure assignment in vibrating systems through active and passive approaches

The dynamic behaviour of a vibrating system depends on its eigenstructure, which consists of the eigenvalues and the eigenvectors. In fact, eigenvalues define natural frequencies, damping and settling time, while eigenvectors define the spatial distribution of vibrations, i.e. the mode shape, and also affect the sensitivity of eigenvalues with respect to the system parameters. Therefore, eigenstructure assignment, which is aimed at modifying the system in such a way that it features the desired set of eigenvalues and eigenvectors, is of fundamental importance in mechanical design. However, similarly to several other inverse problems, eigenstructure assignment is inherently challenging, due to its ill-posed nature. Despite the recent advancements of the state of the art in eigenstructure assignment, in fact, there are still important open issues. The available methods for eigenstructure assignment can be grouped into two classes: passive approaches, which consist in modifying the physical parameters of the system, and active approaches, which consist in employing actuators and sensors to exert suitable control forces as determined by a specified control law. Since both these approaches have advantages and drawbacks, it is important to choose the most appropriate strategy for the application of interest. In the present thesis, in fact, are collected passive, active, and even hybrid methods, in which active and passive techniques are concurrently employed. All the methods proposed in the thesis are aimed at solving open issues that emerged from the literature and which have applicative relevance, as well as theoretical. In contrast to several state-of-the-art methods, in fact, the proposed ones implement strategies that enable to ensure that the computed solutions are meaningful and feasible. Moreover, given that in modern mechanical design large-scale systems are increasingly common, computational issues have become a major concern and thus have been adequately addressed in the thesis. The proposed methods have been developed to be general and broadly applicable. In order to demonstrate the versatility of the methods, in the thesis it is provided an extensive numerical assessment, hence diverse test-cases have been used for validation purposes. In order to evaluate without bias the performances of the proposed methods, it has been chosen to employ well-established benchmarks from the literature. Moreover, selected experimental applications are presented in the thesis, in order to determine the capabilities of the developed methods when critically challenged. Given the focus on these issues, it is expected that the methods here proposed can constitute effective tools to improve the dynamic behaviour of vibrating systems and it is hoped that the present work could contribute to spread the use of eigenstructure assignment in the solution of engineering design problems