• Title, Summary, Keyword: Eigenvalue Problem

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On lower bounds of eigenvalues for self adjoint operators

  • Lee, Gyou-Bong
    • Journal of the Korean Mathematical Society
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    • v.31 no.3
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    • pp.477-492
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    • 1994
  • For the eigenvalue problem of $Au = \lambda u$ where A is considered as a semi-bounded self-adjoint operator on a Hilbert space, we are used to apply two complentary methods finding upper bounds and lower bounds to the eigenvalues. The most popular method for finding upper bounds may be the Rayleigh-Ritz method which was developed in the 19th century while a method for computing lower bounds may be the method of intermediate eigenvalue problems which has been developed since 1950's. In the method of intermediate eigenvalue problems (IEP), we consider the original operator eigenvalue problem as a perturbation of a simpler, resolvable, self-adjoint eigenvalue problem, called a base problem, that gives rough lower bounds.

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THE SOLVABILITY CONDITIONS FOR A CLASS OF CONSTRAINED INVERSE EIGENVALUE PROBLEM OF ANTISYMMETRIC MATRICES

  • PAN XIAO-PING;HU XI-YAN;ZHANG LEI
    • Journal of the Korean Mathematical Society
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    • v.43 no.1
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    • pp.87-98
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    • 2006
  • In this paper, a class of constrained inverse eigenvalue problem for antisymmetric matrices and their optimal approximation problem are considered. Some sufficient and necessary conditions of the solvability for the inverse eigenvalue problem are given. A general representation of the solution is presented for a solvable case. Furthermore, an expression of the solution for the optimal approximation problem is given.

POSTPROCESSING FOR THE RAVIART-THOMAS MIXED FINITE ELEMENT APPROXIMATION OF THE EIGENVALUE PROBLEM

  • Kim, Kwang-Yeon
    • Korean Journal of Mathematics
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    • v.26 no.3
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    • pp.467-481
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    • 2018
  • In this paper we present a postprocessing scheme for the Raviart-Thomas mixed finite element approximation of the second order elliptic eigenvalue problem. This scheme is carried out by solving a primal source problem on a higher order space, and thereby can improve the convergence rate of the eigenfunction and eigenvalue approximations. It is also used to compute a posteriori error estimates which are asymptotically exact for the $L^2$ errors of the eigenfunctions. Some numerical results are provided to confirm the theoretical results.

On eigenvalue problem of bar structures with stochastic spatial stiffness variations

  • Rozycki, B.;Zembaty, Z.
    • Structural Engineering and Mechanics
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    • v.39 no.4
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    • pp.541-558
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    • 2011
  • This paper presents an analysis of stochastic eigenvalue problem of plane bar structures. Particular attention is paid to the effect of spatial variations of the flexural properties of the structure on the first four eigenvalues. The problem of spatial variations of the structure properties and their effect on the first four eigenvalues is analyzed in detail. The stochastic eigenvalue problem was solved independently by stochastic finite element method (stochastic FEM) and Monte Carlo techniques. It was revealed that the spatial variations of the structural parameters along the structure may substantially affect the eigenvalues with quite wide gap between the two extreme cases of zero- and full-correlation. This is particularly evident for the multi-segment structures for which technology may dictate natural bounds of zero- and full-correlation cases.

An Eigenvalue Sensitivity Analysis of the Iterative Eigenvalue Calculation Algorithm (반복계산에 의한 고유치 계산 알고리즘에서의 고유치 감도해석)

  • Kim, Deok-Young
    • Proceedings of the KIEE Conference
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    • pp.217-219
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    • 2001
  • This paper presents a new eigenvalue sensitivity analysis method based on AESOPS algorithm. The additional calculation steps are derived from the original AESOPS algorithm. The additional calculation steps are performed directly from the AESOPS algorithm after iteratively calculating electro-mechanical oscillation modes in small signal stability problems. Owing to the structural characteristics of partitioned sub-matrix of state space equations, the partial differentiation terms of system state matrix for obtaining eigenvalue sensitivity indices can be calculated very simply. By the method presented in this paper, the AESOPS algorithm can be used in controller design problem as well as analysis of small signal stability problem.

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A study on Stress Singularities for V-notched Cracks in Anisotropic and/or Pseudo-isotropic Dissimilar Materials

  • Cho, Sang-Bong;Kim, Jin-kwang
    • International Journal of Precision Engineering and Manufacturing
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    • v.3 no.2
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    • pp.22-32
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    • 2002
  • V-notched crack problems can be formulated as eigenvalue problems. The problem ova v-notched crack in anisotropic and/or pseudo-isotropic dissimilar materials was formulated as an eigenvalue problem to discuss stress singularities. The eigenvalue problem was served by the commercial numerical program; MATHEMATICA. The specific data of eigenvalues possessing the stress singularity were obtained. Stress singularities fur v-notched cracks in anisotropic and/or pseudo-isotropic dissimilar materials were discussed according to the relation between wedge angle and material property. It was shown that there are three cases of eigenvalues possessing the stress singularity; one real, two real and one complex.

Alternative approach for the derivation of an eigenvalue problem for a Bernoulli-Euler beam carrying a single in-span elastic rod with a tip-mounted mass

  • Gurgoze, Metin;Zeren, Serkan
    • Structural Engineering and Mechanics
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    • v.53 no.6
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    • pp.1105-1126
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    • 2015
  • Many vibrating mechanical systems from the real life are modeled as combined dynamical systems consisting of beams to which spring-mass secondary systems are attached. In most of the publications on this topic, masses of the helical springs are neglected. In a paper (Cha et al. 2008) published recently, the eigencharacteristics of an arbitrary supported Bernoulli-Euler beam with multiple in-span helical spring-mass systems were determined via the solution of the established eigenvalue problem, where the springs were modeled as axially vibrating rods. In the present article, the authors used the assumed modes method in the usual sense and obtained the equations of motion from Lagrange Equations and arrived at a generalized eigenvalue problem after applying a Galerkin procedure. The aim of the present paper is simply to show that one can arrive at the corresponding generalized eigenvalue problem by following a quite different way, namely, by using the so-called "characteristic force" method. Further, parametric investigations are carried out for two representative types of supporting conditions of the bending beam.

An Application of the Multigrid Method to Eigenvalue problems (복합마디방법의 고유치문제에 응용)

  • Lee, Gyou-Bong;Kim, Sung-Soo;Sung, Soo-Hak
    • The Journal of Natural Sciences
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    • v.8 no.2
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    • pp.9-11
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    • 1996
  • We apply a full mutigrid scheme to computing eigenvalues of the Laplace eigenvalue problem with Dirichlet boundary condition. We use finite difference method to get an algebraic equation and apply inverse power method to estimating the smallest eigenvalue. Our result shows that combined method of inverse power method and full multigrid scheme is very effective in calculating eigenvalue of the eigenvalue problem.

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Probabilistic Finite Element Analysis of Eigenvalue Problem(Buckling Reliability Analysis of Frame Structure) (고유치 문제의 확률 유한요소 해석(Frame 구조물의 좌굴 신뢰성 해석))

  • 양영순;김지호
    • Proceedings of the Computational Structural Engineering Institute Conference
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    • pp.22-27
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    • 1990
  • Since an eigenvalue problem in structural analysis has been recognized as an important process for the assessment of structural strength, it is usually to be carried out the eigenvalue analysis or buckling analysis of structures when the compression behabiour of the member is dorminant. In general, various variables involved in the eigenvalue problem have also shown their variability. So it is natural to apply the probabilistic analysis into such problem. Since the limit state equation for the eigenvalue analysis or buckling reliability analysis is expressed implicitly in terms of random variables involved, the probabilistic finite element method is combined with the conventional reliability method such as MVFOSM and AFOSM for the determination of probability of failure due to buckling. The accuracy of the results obtained by this method is compared with results from the Monte Carlo simulations. Importance sampling method is specially chosen for overcomming the difficulty in a large simulation number needed for appropriate accurate result. From the results of the case study, it is found that the method developed here has shown good performance for the calculation of probability of buckling failure and could be used for checking the safety of the calculation of probability of buckling failure and could be used for checking the safely of frame structure which might be collapsed by either yielding or buckling.

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