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Journal articles on the topic 'Eigenvalue formulation'

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Author: Grafiati

Published: 14 March 2023

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1

Gürgöze, M. "On Various Eigenvalue Problem Formulations for Viscously Damped Linear Mechanical Systems." International Journal of Mechanical Engineering Education 33, no. 3 (July 2005): 235–43. http://dx.doi.org/10.7227/ijmee.33.3.5.

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The state-space method is frequently used to obtain the eigenvalues of a viscously damped linear mechanical system. Differences in the definition of the state vector and auxiliary matrices found in the literature lead to differences in the formulation of the eigenvalue problems and this in turn can cause difficulties for students on mechanical vibration courses. In this study, various eigenvalue problem formulations in different textbooks have been examined, relationships between them have been established and results have been applied to a numerical example of a system with two degrees of freedom.

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2

Renshaw, A. A., and C. D. Mote. "Local Stability of Gyroscopic Systems Near Vanishing Eigenvalues." Journal of Applied Mechanics 63, no. 1 (March 1, 1996): 116–20. http://dx.doi.org/10.1115/1.2787185.

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Vanishing eigenvalues of a gyroscopic system are always repeated and, as a result of this degeneracy, their eigenfunctions represent a combination of constant displacements with zero velocity and the displacements derived from constant, nonzero velocity. In a second-order formulation of the equations of motion, the assumption of harmonic vibration is not sufficiently general to represent this motion as the displacements derived from constant, nonzero velocity are not included. In a first order formulation, however, the assumption of harmonic vibration is sufficient. Solvability criteria are required to determine the complete form of such eigenfunctions and in particular whether or not their velocities are identically zero. A conjecture for gyroscopic systems is proposed that predicts whether the eigenvalue locus is imaginary or complex in the neighborhood of a vanishing eigenvalue. If the velocities of all eigenfunctions with vanishing eigenvalues are identically zero, the eigenvalues are imaginary; if any eigenfunction exists whose eigenvalue is zero but whose velocity is nonzero, the corresponding eigenvalue locus is complex. The conjecture is shown to be true for many commonly studied gyroscopic systems; no counter examples have yet been found. The conjecture can be used to predict divergence instability in many cases without extensive computation.

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3

Gardini, Francesca, Gianmarco Manzini, and Giuseppe Vacca. "The nonconforming Virtual Element Method for eigenvalue problems." ESAIM: Mathematical Modelling and Numerical Analysis 53, no. 3 (May 2019): 749–74. http://dx.doi.org/10.1051/m2an/2018074.

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We analyse the nonconforming Virtual Element Method (VEM) for the approximation of elliptic eigenvalue problems. The nonconforming VEM allows to treat in the same formulation the two- and three-dimensional case. We present two possible formulations of the discrete problem, derived respectively by the nonstabilized and stabilized approximation of theL2-inner product, and we study the convergence properties of the corresponding discrete eigenvalue problem. The proposed schemes provide a correct approximation of the spectrum, in particular we prove optimal-order error estimates for the eigenfunctions and the usual double order of convergence of the eigenvalues. Finally we show a large set of numerical tests supporting the theoretical results, including a comparison with the conforming Virtual Element choice.

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4

Wanxie, Zhong, F. W. Williams, and P. N. Bennett. "Extension of the Wittrick-Williams Algorithm to Mixed Variable Systems." Journal of Vibration and Acoustics 119, no. 3 (July 1, 1997): 334–40. http://dx.doi.org/10.1115/1.2889728.

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A precise integration algorithm has recently been proposed by Zhong (1994) for dynamic stiffness matrix computations, but he did not give a corresponding eigenvalue count method. The Wittrick-Williams algorithm gives an eigenvalue count method for pure displacement formulations, but the precise integration method uses a mixed variable formulation. Therefore the Wittrick-Williams method is extended in this paper to give the eigenvalue count needed by the precise integration method and by other methods involving mixed variable formulations. A simple Timoshenko beam example is included.

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Parker, R. G., and C. J. Mote. "Vibration and Coupling Phenomena in Asymmetric Disk-Spindle Systems." Journal of Applied Mechanics 63, no. 4 (December 1, 1996): 953–61. http://dx.doi.org/10.1115/1.2787252.

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This paper analytically treats the free vibration of coupled, asymmetric disk-spindle systems in which both the disk and spindle are continuous and flexible. The disk and spindle are coupled by a rigid clamping collar. The asymmetries derive from geometric shape imperfections and nonuniform clamping stiffness at the disk boundaries. They appear as small perturbations in the disk boundary conditions. The coupled system eigenvalue problem is cast in terms of “extended” eigenfunctions that are vectors of the disk, spindle, and clamp displacements. With this formulation, the eigenvalue problem is self-adjoint and the eigenfunctions are orthogonal. The conciseness and clarity of this formulation are exploited in an eigensolution perturbation analysis. The amplitude of the disk boundary condition asymmetry is the perturbation parameter. Exact eigensolution perturbations are derived through second order. For general boundary asymmetry distributions, simple rules emerge showing how asymmetry couples the eigenfunctions of the axisymmetric system and how the degenerate pairs of axisymmetric system eigenvalues split into distinct eigenvalues. Additionally, properties of the formulation are ideal for use in modal analyses, Ritz-Galerkin discretizations, and extensions to gyroscopic or nonlinear analyses.

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6

Ashokkumar, Chimpalthradi R., George WP York, and Scott F. Gruber. "Proportional–integral–derivative controller family for pole placement." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 231, no. 20 (May 20, 2016): 3791–97. http://dx.doi.org/10.1177/0954406216651893.

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In this paper, linear time-invariant square systems are considered. A procedure to design infinitely many proportional–integral–derivative controllers, all of them assigning closed-loop poles (or closed-loop eigenvalues), at desired locations fixed in the open left half plane of the complex plane is presented. The formulation accommodates partial pole placement features. The state-space realization of the linear system incorporated with a proportional–integral–derivative controller boils down to the generalized eigenvalue problem. The generalized eigenvalue-eigenvector constraint is transformed into a system of underdetermined linear homogenous set of equations whose unknowns include proportional–integral–derivative parameters. Hence, the proportional–integral–derivative solution sets are infinitely many for the chosen closed-loop eigenvalues in the eigenvalue-eigenvector constraint. The solution set is also useful to reduce the tracking errors and improve the performance. Three examples are illustrated.

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7

Sandi, Horea, and Ioan Sorin Borcia (†). "An Approach to Some Non-Classical Eigenvalue Problems of Structural Dynamics." Mathematical Modelling in Civil Engineering 11, no. 4 (December 1, 2015): 21–32. http://dx.doi.org/10.1515/mmce-2015-0017.

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Abstract Two main shortcomings of common formulations, encountered in the literature concerning the linear problems of structural dynamics are revealed: the implicit, not discussed, postulation, of the use of Kelvin – Voigt constitutive laws (which is often infirmed by experience) and the calculation difficulties involved by the attempts to use other constitutive laws. In order to overcome these two categories of shortcomings, the use of the bilateral Laplace – Carson transformation is adopted. Instead of the dependence on time, t, of a certain function f (t), the dependence of its image f# (p) on the complex parameter p = χ + iω (ω: circular frequency) will occur. This leads to the formulation of associated non-classical eigenvalue problems. The basic relations satisfied by the eigenvalues λr#(p) and the eigenvectors vr#(p) of dynamic systems are examined (among other, the property of orthogonality of eigenvectors is replaced by the property of pseudo-orthogonality). The case of points p = p’, where multiple eigenvalues occur and where, as a rule, chains of principal vectors are to be considered, is discussed. An illustrative case, concerning a non-classical eigenvalue problem, is presented. Plots of variation along the ω axis, for the real and imaginary components of eigenvalues and eigenvectors, are presented. A brief final discussion closes the paper.

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8

Mora, David, and Iván Velásquez. "A virtual element method for the transmission eigenvalue problem." Mathematical Models and Methods in Applied Sciences 28, no. 14 (December 30, 2018): 2803–31. http://dx.doi.org/10.1142/s0218202518500616.

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In this paper, we analyze a Virtual Element Method (VEM) for solving a non-self-adjoint fourth-order eigenvalue problem derived from the transmission eigenvalue problem. We write a variational formulation and propose a [Formula: see text]-conforming discretization by means of the VEM. We use the classical approximation theory for compact non-self-adjoint operators to obtain optimal order error estimates for the eigenfunctions and a double order for the eigenvalues. Finally, we present some numerical experiments illustrating the behavior of the virtual scheme on different families of meshes.

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9

Cotta, Renato M., Carolina Palma Naveira-Cotta, and Diego C. Knupp. "Nonlinear eigenvalue problem in the integral transforms solution of convection-diffusion with nonlinear boundary conditions." International Journal of Numerical Methods for Heat & Fluid Flow 26, no. 3/4 (May 3, 2016): 767–89. http://dx.doi.org/10.1108/hff-08-2015-0309.

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Purpose – The purpose of this paper is to propose the generalized integral transform technique (GITT) to the solution of convection-diffusion problems with nonlinear boundary conditions by employing the corresponding nonlinear eigenvalue problem in the construction of the expansion basis. Design/methodology/approach – The original nonlinear boundary condition coefficients in the problem formulation are all incorporated into the adopted eigenvalue problem, which may be itself integral transformed through a representative linear auxiliary problem, yielding a nonlinear algebraic eigenvalue problem for the associated eigenvalues and eigenvectors, to be solved along with the transformed ordinary differential system. The nonlinear eigenvalues computation may also be accomplished by rewriting the corresponding transcendental equation as an ordinary differential system for the eigenvalues, which is then simultaneously solved with the transformed potentials. Findings – An application on one-dimensional transient diffusion with nonlinear boundary condition coefficients is selected for illustrating some important computational aspects and the convergence behavior of the proposed eigenfunction expansions. For comparison purposes, an alternative solution with a linear eigenvalue problem basis is also presented and implemented. Originality/value – This novel approach can be further extended to various classes of nonlinear convection-diffusion problems, either already solved by the GITT with a linear coefficients basis, or new challenging applications with more involved nonlinearities.

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Manolis, George D., and Georgios I. Dadoulis. "On the Numerical Treatment of the Temporal Discontinuity Arising from a Time-Varying Point Mass Attachment on a Waveguide." Algorithms 16, no. 1 (January 3, 2023): 26. http://dx.doi.org/10.3390/a16010026.

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A vibrating pylon, modeled as a waveguide, with an attached point mass that is time-varying poses a numerically challenging problem regarding the most efficient way for eigenvalue extraction. The reason is three-fold, starting with a heavy mass attachment that modifies the original eigenvalue problem for the stand-alone pylon, plus the fact that the point attachment results in a Dirac delta function in the mixed-type boundary conditions, and finally the eigenvalue problem becomes time-dependent and must be solved for a sequence of time steps until the time interval of interests is covered. An additional complication is that the eigenvalues are now complex quantities. Following the formulation of the eigenvalue problem as a system of first-order, time-dependent matrix differential equations, two eigenvalue extraction methods are implemented and critically examined, namely the Laguerre and the QR algorithms. The aim of the analysis is to identify the most efficient technique for interpreting time signals registered at a given pylon as a means for detecting damage, a procedure which finds application in structural health monitoring of civil engineering infrastructure.

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11

FUß, STEFANIE, STUART C. HAWKINS, and STEFFEN MARBURG. "AN EIGENVALUE SEARCH ALGORITHM FOR THE MODAL ANALYSIS OF A RESONATOR IN FREE SPACE." Journal of Computational Acoustics 19, no. 01 (March 2011): 95–109. http://dx.doi.org/10.1142/s0218396x11004304.

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In this article we present an algorithm for the three-dimensional numerical simulation of the sound spectrum and the propagation of acoustic radiation inside and around long slender hollow objects. The fluid inside and close to the object is meshed by Lagrangian tetrahedral finite elements. To obtain results in the far field of the object, complex conjugated Astley-Leis infinite elements are used. To apply these infinite elements the finite element domain is meshed either in a spherical or an ellipsoidal shape. Advantages and disadvantages of both shapes regarding the form of the object are discussed in this article. The formulation leads to a quadratic eigenvalue problem with real, large and nonsymmetric matrices. An eigenvalue search algorithm is implemented to concentrate on the computation of the interior eigenmodes. This algorithm is based on a linearization of the quadratic problem in a state space formulation. The search algorithm uses a complex shift to efficiently extract the relevant eigenvalues only.

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12

Yang, Yidu, Wei Jiang, Yu Zhang, Wenjun Wang, and Hai Bi. "A Two-Scale Discretization Scheme for Mixed Variational Formulation of Eigenvalue Problems." Abstract and Applied Analysis 2012 (2012): 1–29. http://dx.doi.org/10.1155/2012/812914.

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This paper discusses highly efficient discretization schemes for mixed variational formulation of eigenvalue problems. A new finite element two-scale discretization scheme is proposed by combining the mixed finite element method with the shifted-inverse power method for solving matrix eigenvalue problems. With this scheme, the solution of an eigenvalue problem on a fine gridKhis reduced to the solution of an eigenvalue problem on a much coarser gridKHand the solution of a linear algebraic system on the fine gridKh. Theoretical analysis shows that the scheme has high efficiency. For instance, when using the Mini element to solve Stokes eigenvalue problem, the resulting solution can maintain an asymptotically optimal accuracy by takingH=O(h4), and when using thePk+1-Pkelement to solve eigenvalue problems of electric field, the calculation results can maintain an asymptotically optimal accuracy by takingH=O(h3). Finally, numerical experiments are presented to support the theoretical analysis.

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13

Miller, U., S. Bograd, A. Schmidt, and L. Gaul. "Eigenpath Following for Systems with Symmetric Complex-Valued Stiffness Matrices." Shock and Vibration 17, no. 4-5 (2010): 397–405. http://dx.doi.org/10.1155/2010/849840.

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A vibration analysis of a structure with joints is performed. The simulation is conducted with finite element software capable of performing a numeric modal analysis with hysteretic damping assumption. The joints are modeled with thin layer elements, representing dissipation and stiffness of the joints. The matrices describing the system consist of the mass, as well as real and complex-valued stiffness matrices. If the eigenvalues of this system are found in one step, due to the mode crossing occurring for the closely spaced modes, it is difficult and time consuming to assign calculated modal damping factors to the corresponding undamped eigenvalues. In order to avoid this problem, an eigenvalue following method is used. The outcome of the solution is the graphical presentation of continuous eigenvalue paths, showing the change in the eigenvalues from the undamped to the fully damped case. For every undamped eigenvalue exists its equivalent eigenfrequency and damping factor that can be used for further numerical analysis.In scope of this article a Predictor-Corrector and a Rayleigh-Quotient Iteration algorithms are applied to the problem. The algorithms are tested specifically on the type of matrices resulting from the weakly damped hysteretic formulation arising from the simulation of metallic structures with joints.

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14

Parker, R. G. "On the Eigenvalues and Critical Speed Stability of Gyroscopic Continua." Journal of Applied Mechanics 65, no. 1 (March 1, 1998): 134–40. http://dx.doi.org/10.1115/1.2789016.

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In order to provide analytical eigenvalue estimates for general continuous gyroscopic systems, this paper presents a perturbation analysis to determine approximate eigenvalue loci and stability conclusions in the vicinity of critical speeds and zero speed. The perturbation analysis relies on a formulation of the general continuous gyroscopic system eigenvalue problem in terms of matrix differential operators and vector eigenfunctions. The eigenvalue λ appears only as λ2 in the formulation, and the smoothness of λ2 at the critical speeds and zero speed is the essential feature. First-order eigenvalue perturbations are determined at the critical speeds and zero speed. The derived eigenvalue perturbations are simple expressions in terms of the original mass, gyroscopic, and stiffness operators and the critical-speed/zero-speed eigenfunctions. Prediction of whether an eigenvalue passes to or from a region of divergence instability at the critical speed is determined by the sign of the eigenvalue perturbation. Additionally, eigenvalue perturbation at the critical speeds and zero speed yields approximations for the eigenvalue loci over a range of speeds. The results are limited to systems having one independent eigenfunction associated with each critical speed and each stationary system eigenvalue. Examples are presented for an axially moving tensioned beam, an axially moving string on an elastic foundation, and a rotating rigid body. The eigenvalue perturbations agree identically with exact solutions for the moving string and rotating rigid body.

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15

CAMPOS, RAFAEL G., J. L. LÓPEZ-LÓPEZ, and R. VERA. "LATTICE CALCULATIONS ON THE SPECTRUM OF DIRAC AND DIRAC–KÄHLER OPERATORS." International Journal of Modern Physics A 23, no. 07 (March 20, 2008): 1029–38. http://dx.doi.org/10.1142/s0217751x08038470.

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We use a lattice formulation to study the spectra of the Dirac and the Dirac–Kähler operators on the 2-sphere. This lattice formulation uses differentiation matrices which yield exact values for the derivatives of polynomials, preserving the Leibniz rule in subspaces of polynomials of low degree and therefore, this formulation can be used to study the fermion–boson symmetry on the lattice. In this context, we find that the free Dirac and Dirac–Kähler operators on the 2-sphere exhibit fermionic as well as bosonic-like eigensolutions for which the corresponding eigenvalues and the number of states are computed. In the Dirac case these solutions appear in doublets, except for the bosonic mode with zero eigenvalue, indicating the possible existence of a supersymmetry of the square of the Dirac operator.

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16

Coyette, J. P., and K. R. Fyfe. "An Improved Formulation for Acoustic Eigenmode Extraction from Boundary Element Models." Journal of Vibration and Acoustics 112, no. 3 (July 1, 1990): 392–97. http://dx.doi.org/10.1115/1.2930521.

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The acoustic boundary element method has been utilized primarily as a direct response analysis technique. Recently, methodologies have been published that enable the formulation of an eigenvalue problem from a boundary element model. These techniques, however, have been very slow due to the inefficient synthesis techniques used in solving the algebraic problems. In this paper, an alternative procedure is obtained which indirectly calculates the desired acoustic variables. This technique greatly enhances the calculation speed in setting up the eigenvalue problem and in determining the field pressures. A subspace iteration technique is outlined to solve the generalized unsymmetric eigenvalue problem. Examples are presented to show the accuracy of the method.

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17

Handy, C. R., B. G. Giraud, and D. Bessis. "Dynamical-system formulation of the eigenvalue moment method." Physical Review A 44, no. 3 (August 1, 1991): 1505–15. http://dx.doi.org/10.1103/physreva.44.1505.

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18

Liu, Qing, and Ayato Mitsuishi. "Principal eigenvalue problem for infinity Laplacian in metric spaces." Advanced Nonlinear Studies 22, no. 1 (January 1, 2022): 548–73. http://dx.doi.org/10.1515/ans-2022-0028.

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Abstract This article is concerned with the Dirichlet eigenvalue problem associated with the ∞ \infty -Laplacian in metric spaces. We establish a direct partial differential equation approach to find the principal eigenvalue and eigenfunctions in a proper geodesic space without assuming any measure structure. We provide an appropriate notion of solutions to the ∞ \infty -eigenvalue problem and show the existence of solutions by adapting Perron’s method. Our method is different from the standard limit process via the variational eigenvalue formulation for p p -Laplacian in the Euclidean space.

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19

Kobeissi, Hafez, Majida Kobeissi, and Chafia H. Trad. "On nonintegral E corrections in perturbation theory: application to the perturbed Morse oscillator." Canadian Journal of Physics 72, no. 1-2 (January 1, 1994): 80–85. http://dx.doi.org/10.1139/p94-013.

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A new formulation of the Rayleigh–Schrödinger perturbation theory is applied to the derivation of the vibrational eigenvalues of the perturbed Morse oscillator (PMO). This formulation avoids the conventional projection of the Ψ corrections on the basis of unperturbed eigenfunctions [Formula: see text], or the projection of the nonhomogeneous Schrödinger equations on [Formula: see text], it gives simple expressions for each E correction [Formula: see text] free of summations and integrals. When the PMO is characterized by the potential U = UM + UP (where UM is the unperturbed Morse potential), the eigenvalue of a vibrational level ν is given by: [Formula: see text]. According to the new formulation the correction £, [Formula: see text] is given by [Formula: see text], where σp(r) is a particular solution of the nonhomogeneous differential equation y″ + f y = sp; here [Formula: see text], sp is well known for each p: for p = 0, [Formula: see text]; for [Formula: see text]. For the numerical application one single routine is used, that of integrating y″ + f y = s, where the coefficients are known as well as the initial values. An example is presented for the Huffaker PMO of the (carbon monoxide) CO-X1Σ+ state. The vibrational eigenvalues Eν are obtained to a good accuracy (with p = 4) even for high levels. This result confirms the validity of this new formulation and gives a semianalytic expression for the PMO eigenvalues.

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20

Adylina, E. M., S. A. Igonin, and L. V. Stepanova. "ABOUT A NON-LINEAR TASK ON EIGENVALUES INCURRING FROM THE ANALYSIS OF TENSIONS AT THE FATIGUE CRACK TIP." Vestnik of Samara University. Natural Science Series 18, no. 3.1 (June 7, 2017): 83–102. http://dx.doi.org/10.18287/2541-7525-2012-18-3.1-83-102.

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An analytical solution of the nonlinear eigenvalue problem arising from the fatigue crack growth problem in a damaged medium in coupled formulation is obtained. The perturbation technique is used. The method allows to ﬁnd the analytical dependence of eigenvalue on parameters of the kinetic equation of the damage evolution law.

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21

Jung, Jaesoon, Seongyeol Goo, and Junghwan Kook. "Predicting anti-resonance frequencies using a novel eigenvalue formulation." Finite Elements in Analysis and Design 191 (September 2021): 103525. http://dx.doi.org/10.1016/j.finel.2021.103525.

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22

Ren, Shixian, Yu Zhang, and Ziqiang Wang. "An efficient spectral-Galerkin method for a new Steklov eigenvalue problem in inverse scattering." AIMS Mathematics 7, no. 5 (2022): 7528–51. http://dx.doi.org/10.3934/math.2022423.

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<abstract><p>An efficient spectral method is proposed for a new Steklov eigenvalue problem in inverse scattering. Firstly, we establish the weak form and the associated discrete scheme by introducing an appropriate Sobolev space and a corresponding approximation space. Then, according to the Fredholm Alternative, the corresponding operator forms of weak formulation and discrete formulation are derived. After that, the error estimates of approximated eigenvalues and eigenfunctions are proved by using the spectral approximation results of completely continuous operators and the approximation properties of orthogonal projection operators. We also construct an appropriate set of basis functions in the approximation space and derive the matrix form of the discrete scheme based on the tensor product. In addition, we extend the algorithm to the circular domain. Finally, we present plenty of numerical experiments and compare them with some existing numerical methods, which validate that our algorithm is effective and high accuracy.</p></abstract>

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23

Wang, Ying-xiao, and Shou-qiang Du. "A Kind of Stochastic Eigenvalue Complementarity Problems." Mathematical Problems in Engineering 2018 (2018): 1–9. http://dx.doi.org/10.1155/2018/7397592.

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With the development of computer science, computational electromagnetics have also been widely used. Electromagnetic phenomena are closely related to eigenvalue problems. On the other hand, in order to solve the uncertainty of input data, the stochastic eigenvalue complementarity problem, which is a general formulation for the eigenvalue complementarity problem, has aroused interest in research. So, in this paper, we propose a new kind of stochastic eigenvalue complementarity problem. We reformulate the given stochastic eigenvalue complementarity problem as a system of nonsmooth equations with nonnegative constraints. Then, a projected smoothing Newton method is presented to solve it. The global and local convergence properties of the given method for solving the proposed stochastic eigenvalue complementarity problem are also given. Finally, the related numerical results show that the proposed method is efficient.

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24

Korek, M., and K. Fakhreddine. "A canonical approach for computing the eigenvalues of the Schrödinger equation for double-well potentials." Canadian Journal of Physics 78, no. 11 (November 1, 2000): 969–76. http://dx.doi.org/10.1139/p00-072.

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The problem of obtaining the eigenvalues of the Schrödinger equation for a double-well potential function is considered. By replacing the differential Schrödinger equation by a Volterra integral equation the wave function will be given by [Formula: see text] where the coefficients ai are obtained from the boundary conditions and the fi are two well-defined canonical functions. Using these canonical functions, we define an eigenvalue function F(E) = 0; its roots E1, E2, ... are the eigenvalues of the corresponding double-well potential. The numerical application to analytical potentials (either symmetric or asymmetric) and to a numerical potential of the (2)1 [Formula: see text] state of the molecule Na2 shows the validity and the high accuracy of the present formulation. PACS No.: 03.65Ge

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25

Yang, B. "Eigenvalue Inclusion Principles for Discrete Gyroscopic Systems." Journal of Applied Mechanics 59, no. 2S (June 1, 1992): S278—S283. http://dx.doi.org/10.1115/1.2899501.

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In his famous treatise The Theory of Sound, Rayleigh enunciated an eigenvalue inclusion principle for modified discrete nongyroscopic systems. According to this principle, the natural frequencies of a nongyroscopic system without and with modification are alternatively located along the positive real axis. Although vibration and dynamics of discrete gyroscopic systems have been extensively studied, the problem of inclusion principles for discrete gyroscopic systems has not been addressed. This paper presents several eigenvalue inclusion principles for a class of discrete gyroscopic systems. A transfer function formulation is proposed to describe modified gyroscopic systems. Six types of modifications and their effects on the system natural frequencies are studied. It is shown that the transfer function formulation provides a systematic and convenient way to handle modification problems for discrete gyroscopic systems.

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26

Yang, B. "Eigenvalue Inclusion Principles for Distributed Gyroscopic Systems." Journal of Applied Mechanics 59, no. 3 (September 1, 1992): 650–56. http://dx.doi.org/10.1115/1.2893773.

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In his famous treatise The Theory of Sound, Rayleigh enunciated an eigenvalue inclusion principle for the discrete, self-adjoint vibrating system under a constraint. According to this principle, the natural frequencies of the discrete system without and with the constraint are alternately located along the positive real axis. Although it is commonly believed that the same rule also applied for distributed vibrating systems, no proof has been given for the distributed gyroscopic system. This paper presents several eigenvalue inclusion principles for a class of distributed gyroscopic systems under pointwise constraints. A transfer function formulation is proposed to describe the constrained system. Five types of nondissipative constraints and their effects on the system natural frequencies are studied. It is shown that the transfer function formulation is a systematic and convenient way to handle constraint problems for the distributed gyroscopic system.

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27

Rocha, Rogério Vinícius Matos, Dany Sanchez Dominguez, Susana Marrero Iglesias, and Ricardo Carvalho de Barros. "Finite Element Method with Spectral Green's Function in Slab Geometry for Neutron Diffusion in Multiplying Media and One Energy Group." TEMA (São Carlos) 17, no. 2 (September 4, 2016): 173. http://dx.doi.org/10.5540/tema.2016.017.02.0173.

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The physical phenomenon of neutrons transport associated with eigenvalue problems appears in the criticality calculations of nuclear reactors and can be treated as a diffusion process. This paper presents a new method to solve eigenvalue problems of neutron diffusion in slab geometry and one energy group. This formulation combines the Finite Element Method, considered an intermediate mesh method, with the Spectral Green's Function Method, which is free of truncation errors, and it is considered a coarse mesh method. The novelty of this formulation is to approach the spatial moments of the neutron flux distribution by the first-order polynomials obtained from the spectral analysis of diffusion equation. The approximations provided by the new formulation allow obtaining accurate results in coarse mesh calculations. To validate the method, we compare the results obtained with the methods described in the literature, specifically the Diamond Difference method. The accuracy and the computational performance of the proposed formulation were characterized by solving benchmarks problems with a high degree of heterogeneity.

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Yang, Yidu, Yu Zhang, and Hai Bi. "Multigrid Discretization and Iterative Algorithm for Mixed Variational Formulation of the Eigenvalue Problem of Electric Field." Abstract and Applied Analysis 2012 (2012): 1–25. http://dx.doi.org/10.1155/2012/190768.

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This paper discusses highly finite element algorithms for the eigenvalue problem of electric field. Combining the mixed finite element method with the Rayleigh quotient iteration method, a new multi-grid discretization scheme and an adaptive algorithm are proposed and applied to the eigenvalue problem of electric field. Theoretical analysis and numerical results show that the computational schemes established in the paper have high efficiency.

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29

Polyachenko, E. V. "Instability of the cored barotropic disc: the linear eigenvalue formulation." Monthly Notices of the Royal Astronomical Society 478, no. 3 (May 29, 2018): 4268–75. http://dx.doi.org/10.1093/mnras/sty1402.

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Davey, Kent R., Hsiu Chi Han, and Larry Turner. "3D transient eddy current fields using theu‐vintegral‐eigenvalue formulation." Journal of Applied Physics 63, no. 4 (February 15, 1988): 991–96. http://dx.doi.org/10.1063/1.339998.

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31

Nebot-Gil, I., J. Sanchez-Marin, J. L. Heully, J. P. Malrieu, and D. Maynau. "Eigenvalue problem formulation of coupled-cluster expansions through intermediate Hamiltonians." Chemical Physics Letters 234, no. 1-3 (March 1995): 45–49. http://dx.doi.org/10.1016/0009-2614(95)00026-z.

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32

Zingoni, Alphose. "A group-theoretic finite-difference formulation for plate eigenvalue problems." Computers & Structures 112-113 (December 2012): 266–82. http://dx.doi.org/10.1016/j.compstruc.2012.08.009.

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Ali1, Ashraf, C. Rajakumar, and Shah M. Yunus. "On the formulation of the acoustic boundary element eigenvalue problems." International Journal for Numerical Methods in Engineering 31, no. 7 (May 20, 1991): 1271–82. http://dx.doi.org/10.1002/nme.1620310704.

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Boiko, Andrey V., Kirill V. Demyanko, and Yuri M. Nechepurenko. "Asymptotic boundary conditions for the analysis of hydrodynamic stability of flows in regions with open boundaries." Russian Journal of Numerical Analysis and Mathematical Modelling 34, no. 1 (February 25, 2019): 15–29. http://dx.doi.org/10.1515/rnam-2019-0002.

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Abstract A new approach to formulation of asymptotic boundary conditions for eigenvalue problems arising in numerical analysis of hydrodynamic stability of such shear flows as boundary layers, separations, jets, wakes, characterized by almost constant velocity of the main flow outside the shear layer or layers is proposed and justified. This approach makes it possible to formulate and solve completely the temporal and spatial stability problems in the locally-parallel approximation, reducing them to ordinary algebraic eigenvalue problems.

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Attaporn, Wisessint, and Hideo Koguchi. "FEM Formulation and Analysis of Elasto-Plastic Stress Singularity." Key Engineering Materials 324-325 (November 2006): 915–18. http://dx.doi.org/10.4028/www.scientific.net/kem.324-325.915.

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The present study shows a new idea for investigating characteristics of stress singularity field around a vertex associated with elasto-plastic properties of materials. FEM formulation for elasto-plastic stress singularity analysis is expressed to investigate an eigenvalue and the intensity of singularity.The elasto-plastic stress singularity in a flip chip joint was investigated using the FEM formulation. After that, the possibility of delamination on the flip chip joint was discussed.

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Nogae, Kazuyoshi, Eiji Andoh, and Norio Kamiya. "Boundary Element Eigenvalue Analysis of the Helmholtz Equation by New Complex-Valued Formulation. 1st Report, Formulation." Transactions of the Japan Society of Mechanical Engineers Series C 60, no. 579 (1994): 3854–58. http://dx.doi.org/10.1299/kikaic.60.3854.

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Olver, Sheehan, Raj Rao Nadakuditi, and Thomas Trogdon. "Sampling unitary ensembles." Random Matrices: Theory and Applications 04, no. 01 (January 2015): 1550002. http://dx.doi.org/10.1142/s2010326315500021.

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We develop a computationally efficient algorithm for sampling from a broad class of unitary random matrix ensembles that includes but goes well beyond the straightforward to sample Gaussian unitary ensemble (GUE). The algorithm exploits the fact that the eigenvalues of unitary ensembles (UEs) can be represented as a determinantal point process whose kernel is given in terms of orthogonal polynomials. Consequently, our algorithm can be used to sample from UEs for which the associated orthogonal polynomials can be numerically computed efficiently. By facilitating high accuracy sampling of non-classical UEs, the algorithm can aid in the experimentation-based formulation or refutation of universality conjectures involving eigenvalue statistics that might presently be unamenable to theoretical analysis. Examples of such experiments are included.

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Belgacem, Fethi Bin Muhammad, and Ahmed Abdullatif Karaballi. "Principal eigenvalue characterization connected with stochastic particle motion in a finite interval." Journal of Applied Mathematics and Stochastic Analysis 15, no. 4 (January 1, 2002): 349–56. http://dx.doi.org/10.1155/s104895330200028x.

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In this paper, we show that despite their distinction, both the Statonovich and Îto s calculi lead to the same reactive Fokker-Planck equation: ∂p∂t−∂∂x[D∂p∂x−bp]=λmp, (1) describing stochastic dynamics of a particle moving under the influence of an indefinite potential m(x,t), a drift b(x,t), and a constant diffusion D. We treat the periodic-parabolic eigenvalue problem (1) for finite domains having absorbing barriers. We show that under conditions required by the maximum principle, the positive principal eigenvalue λ* (and the negative principal λ* eigenvalue) is connected to the probability eigendensity function p(x,t) by a Raleigh-Ritz like formulation. In the process, we establish the manner of effect of the drift and any inducing potential on the size of the principal eigenvalue. We show that the degree of convexity of the potential plays a major role in this regard.

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Xi, Yingxia, Xia Ji, and Shuo Zhang. "A multi-level mixed element scheme of the two-dimensional Helmholtz transmission eigenvalue problem." IMA Journal of Numerical Analysis 40, no. 1 (October 29, 2018): 686–707. http://dx.doi.org/10.1093/imanum/dry061.

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Abstract In this paper, we present a multi-level mixed element scheme for the Helmholtz transmission eigenvalue problem on polygonal domains that are not necessarily able to be covered by rectangular grids. We first construct an equivalent linear mixed formulation of the transmission eigenvalue problem and then discretize it with Lagrangian finite elements of low regularities. The proposed scheme admits a natural nested discretization, based on which we construct a multi-level scheme. Optimal convergence rate and optimal computational cost can be obtained with the scheme.

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Kamiya, N., and S. T. Wu. "Generalized eigenvalue formulation of the helmholtz equation by the trefftz method." Engineering Computations 11, no. 2 (February 1994): 177–86. http://dx.doi.org/10.1108/02644409410799218.

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Ma, Z., E. Yamashita, and S. Xu. "Transverse scattering matrix formulation for a class of waveguide eigenvalue problems." IEEE Transactions on Microwave Theory and Techniques 41, no. 6 (1993): 1044–51. http://dx.doi.org/10.1109/22.238522.

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Lee, Min Jae, Han Gyu Joo, Deokjung Lee, and Kord Smith. "Coarse mesh finite difference formulation for accelerated Monte Carlo eigenvalue calculation." Annals of Nuclear Energy 65 (March 2014): 101–13. http://dx.doi.org/10.1016/j.anucene.2013.10.025.

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Betcke, T. "A GSVD formulation of a domain decomposition method forplanar eigenvalue problems." IMA Journal of Numerical Analysis 27, no. 3 (November 2, 2006): 451–78. http://dx.doi.org/10.1093/imanum/drl030.

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Friedland, S., J. Nocedal, and M. L. Overton. "The Formulation and Analysis of Numerical Methods for Inverse Eigenvalue Problems." SIAM Journal on Numerical Analysis 24, no. 3 (June 1987): 634–67. http://dx.doi.org/10.1137/0724043.

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Cai, Yun-Feng, Jiang Qian, and Shu-Fang Xu. "The formulation and numerical method for partial quadratic eigenvalue assignment problems." Numerical Linear Algebra with Applications 18, no. 4 (August 28, 2010): 637–52. http://dx.doi.org/10.1002/nla.745.

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Pereda, José A., Ángel Vegas, Luis F. Velarde, and Oscar González. "An FDFD eigenvalue formulation for computing port solutions in FDTD simulators." Microwave and Optical Technology Letters 45, no. 1 (February 25, 2005): 1–3. http://dx.doi.org/10.1002/mop.20704.

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Glowinski, Roland, Shingyu Leung, Hao Liu, and Jianliang Qian. "On the numerical solution of nonlinear eigenvalue problems for the Monge-Ampère operator." ESAIM: Control, Optimisation and Calculus of Variations 26 (2020): 118. http://dx.doi.org/10.1051/cocv/2020072.

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In this article, we report the results we obtained when investigating the numerical solution of some nonlinear eigenvalue problems for the Monge-Ampère operator v → det D2v. The methodology we employ relies on the following ingredients: (i) a divergence formulation of the eigenvalue problems under consideration. (ii) The time discretization by operator-splitting of an initial value problem (a kind of gradient flow) associated with each eigenvalue problem. (iii) A finite element approximation relying on spaces of continuous piecewise affine functions. To validate the above methodology, we applied it to the solution of problems with known exact solutions: The results we obtained suggest convergence to the exact solution when the space discretization step h → 0. We considered also test problems with no known exact solutions.

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A. Ferreira, L. G., C. C. Pagani Júnior, E. M. Gennaro, and C. De Marqui Junior. "A Numerical Study of a Rotor Induced Flow Based on a Finite-State Dynamic Wake Model." Trends in Computational and Applied Mathematics 22, no. 2 (June 28, 2021): 307–24. http://dx.doi.org/10.5540/tcam.2021.022.02.00307.

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A Helicopter rotor undergoes unsteady aerodynamic loads ruled by the aeroelastic coupling between the elastic blades and the dynamic wake induced by rotary wings. Modeling the dynamic interaction between the structural and aerodynamic fields is a key point to understand aeroelastic phenomena associated with rotor stability, flow induced vibration and noise generation, among others. In this study, we address the Generalized Dynamic Wake Model, which describes the inflow velocity field at the rotor disk as a superposition of a finite number of induced flow states. It is a mature model that has been validated based on experimental data and numerically investigated from an eigenvalue problem formulation, whose eigenvalues and eigenvectors provide a deeper insight on the dynamic wake behavior. The paper extends the results presented in the literature to date in order to support physical interpretation of inflow states drawn from the finite-state wake model for flight conditions varying from hover to edgewise flight. The discussion of the wake model mathematical formulation is also oriented towards practical engineering applications to fill a gap in the literature.

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Szajek, Krzysztof, Wojciech Sumelka, Krzysztof Bekus, and Tomasz Blaszczyk. "Designing of Dynamic Spectrum Shifting in Terms of Non-Local Space-Fractional Mechanics." Energies 14, no. 2 (January 19, 2021): 506. http://dx.doi.org/10.3390/en14020506.

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In this paper, the applicability of the space-fractional non-local formulation (sFCM) to design 1D material bodies with a specific dynamic eigenvalue spectrum is discussed. Such a formulated problem is based on the proper spatial distribution of material length scale, which maps the information about the underlying microstructure (it is important that the material length scale is one of two additional material parameters of sFCM compared to the classical local continuum mechanics—the second one, the order of fractional continua—is treated herein as a scaling parameter only). Technically, the design process for finding adequate length scale distribution is not trivial and requires the use of an inverse optimization procedure. In the analysis, the objective function considers a subset of eigenvalues reduced to a single value based on Kreisselmeier–Steinhauser formula. It is crucial that the total number of eigenvalues considered must be smaller than the limit which comes from the ratio of the sFCM length scale to the length of the material body.

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GRADO-CAFFARO, M. A., and M. GRADO-CAFFARO. "A MODEL FOR QUANTUM SYSTEMS OF TWO-BAND EIGENVALUE SPECTRUM." Modern Physics Letters B 18, no. 28n29 (December 20, 2004): 1449–52. http://dx.doi.org/10.1142/s0217984904007918.

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A matrix model to describe quantitatively certain physical properties of quantum systems of two-band eigenvalue spectrum is proposed. This formulation is a generalization of Wannier-type representations and is applicable to various situations where physical properties are defined in tensor form.

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