Relevant bibliographies by topics / Eigenvalue formulation

Academic literature on the topic 'Eigenvalue formulation'

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Published: 14 March 2023

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Journal articles on the topic "Eigenvalue formulation"

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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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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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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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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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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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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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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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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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Dissertations / Theses on the topic "Eigenvalue formulation"

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Ndow, G. L. "Euclidean-time formulation of the eigenvalue moment method for finite dimensional systems." DigitalCommons@Robert W. Woodruff Library, Atlanta University Center, 1992. http://digitalcommons.auctr.edu/dissertations/3767.

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The eigenvalue moment method (EMM), developed by Handy and Bessis is examined from a Euclidean time reformulation. This alternative approach offers a more elegant and rigorous analysis than the conventional EMM theory. We will look at finite matrix analogues for the Euclidean time dependent problem H'F(x,t) = dt^x.t), analyzed from a moments problem perspective. This will enable the generation of converging upper and lower bounds to the "ground state" eigenvalue without the necessity of a discretization ansatz as is the case in conventional EMM theory.
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Eve, Robin Andrew. "Formulation and implementation of conforming finite element approximations to static and eigenvalue problems for thin elastic shells." Master's thesis, University of Cape Town, 1987. http://hdl.handle.net/11427/22509.

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Bibliography: pages 132-135.
In deriving asymptotic error estimates for a conforming finite element analyses of static thin elastic shell problems, the French mathematician Ciarlet (1976) proposed an approach to the formulation of such problems. The formulation he uses is based on classical shell theory making use of Kirchhoff-Koiter assumptions. The shell problem is posed in two-dimensional space to which the real problem, in three-dimensional space, is related by a mapping of the domain of the problem to the shell mid-surface. The finite element approximation is formulated in terms of the covariant components of the shell mid-surface displacement field. In this study, Ciarlet's formulation is extended to include the eigenvalue problem for the shell. In addition to this, the aim of the study is to obtain some indication of how well this approach might be expected to work in practice. The conforming finite element approximation of both the static and eigenvalue problems are implemented. Particular attention is paid to allowing generality of the shell surface geometry through the use of an approximate mapping. The use of different integration rules, in-plane displacement component interpolation schemes and approximate geometry schemes are investigated. Results are presented for shells of different geometries for both static and eigenvalue analyses; these are compared with independently obtained results.
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Alici, Haydar. "A General Pseudospectral Formulation Of A Class Of Sturm-liouville Systems." Phd thesis, METU, 2010. http://etd.lib.metu.edu.tr/upload/12612435/index.pdf.

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In this thesis, a general pseudospectral formulation for a class of Sturm-Liouville eigenvalue problems is consructed. It is shown that almost all, regular or singular, Sturm-Liouville eigenvalue problems in the Schrö
dinger form may be transformed into a more tractable form. This tractable form will be called here a weighted equation of hypergeometric type with a perturbation (WEHTP) since the non-weighted and unperturbed part of it is known as the equation of hypergeometric type (EHT). It is well known that the EHT has polynomial solutions which form a basis for the Hilbert space of square integrable functions. Pseudospectral methods based on this natural expansion basis are constructed to approximate the eigenvalues of WEHTP, and hence the energy eigenvalues of the Schrö
dinger equation. Exemplary computations are performed to support the convergence numerically.
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Zhao, Sihong. "Dynamic Characterization and Active Modification of Viscoelastic Materials." Miami University Honors Theses / OhioLINK, 2011. http://rave.ohiolink.edu/etdc/view?acc_num=muhonors1303742497.

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Williams, Robert Morton. "The moments formulation for determining eigenvalues of physically important systems." DigitalCommons@Robert W. Woodruff Library, Atlanta University Center, 1986. http://digitalcommons.auctr.edu/dissertations/565.

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A fundamentally new method for determininc’ eigenvalues of linear differential operators is presented. The method involves the application of moments analysis and offers a fast and precise numerical alqorithm for eigenvalue computation, particularly in the strong and intermediate coupling reqimes. The most remarkable feature of this approach is that it provides exponentially converging lower and upper bounds to the eigenvalues. The effectiveness of this method is demonstrated by applyinq it to three important problems: the simple harmonic oscillator potential problem, the quantum potential x2 + ƛ2 / 1 + gx2(studied by Lai and Lin in 1982), and the simplified ideal magnetohydrodynamic (MHD) problem recently studied by Paris et el in 1986. Through the very precise lower and upper bounds obtained, this method of approach gives full support to the analysis of the authors mentioned above.
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Jara-Almonte, J. "Extraction of eigen-pairs from beam structures using an exact element based on a continuum formulation and the finite element method." Diss., Virginia Polytechnic Institute and State University, 1985. http://hdl.handle.net/10919/54300.

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Studies of numerical methods to decouple structure and fluid interaction have reported the need for more precise approximations of higher structure eigenvalues and eigenvectors than are currently available from standard finite elements. The purpose of this study is to investigate hybrid finite element models composed of standard finite elements and exact-elements for the prediction of higher structure eigenvalues and eigenvectors. An exact beam-element dynamic-stiffness formulation is presented for a plane Timoshenko beam with rotatory inertia. This formulation is based on a converted continuum transfer matrix and is incorporated into a typical finite element program for eigenvalue/vector problems. Hybrid models using the exact-beam element generate transcendental, nonlinear eigenvalue problems. An eigenvalue extraction technique for this problem is also implemented. Also presented is a post-processing capability to reconstruct the mode shape each of exact element at as many discrete locations along the element as desired. The resulting code has advantages over both the standard transfer matrix method and the standard finite element method. The advantage over the transfer matrix method is that complicated structures may be modeled with the converted continuum transfer matrix without having to use branching techniques. The advantage over the finite element method is that fewer degrees of freedom are necessary to obtain good approximations for the higher eigenvalues. The reduction is achieved because the incorporation of an exact-beam-element is tantamount to the dynamic condensation of an infinity of degrees of freedom. Numerical examples are used to illustrate the advantages of this method. First, the eigenvalues of a fixed-fixed beam are found with purely finite element models, purely exact-element models, and a closed-form solution. Comparisons show that purely exact-element models give, for all practical purposes, the same eigenvalues as a closed-form solution. Next, a Portal Arch and a Verdeel Truss structure are modeled with hybrid models, purely finite element, and purely exact-element models. The hybrid models do provide precise higher eigenvalues with fewer degrees of freedom than the purely finite element models. The purely exact-element models were the most economical for obtaining higher structure eigenvalues. The hybrid models were more costly than the purely exact-element models, but not as costly as the purely finite element models.
Ph. D.
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Nguyen, Minh Tuan. "Contribution à la formulation symétrique du couplage équations intégrales - éléments finis : application à la géotechnique." Phd thesis, Université Paris-Est, 2010. http://tel.archives-ouvertes.fr/tel-00607258.

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Un des outils numériques les plus utilisés en ingénierie est la méthode des éléments finis, qui peut être mise en o euvre grâce à l'utilisation de nombreux codes de calcul. Toutefois, une difficulté apparaît lors de l'utilisation de la méthode des éléments finis, spécialement en géotechnique, lorsque la structure étudiée est en interaction avec un domaine de dimensions infinies. L'usage courant en ingénierie est alors de réaliser les calculs sur des domaines bornés, mais la définition de la frontière de tels domaines bornés pose de sérieux problèmes. Pour traiter convenablement les problèmes comportant des frontières à l'infini, l'utilisation d'éléments discrets "infinis" est maintenant souvent délaissée au profit de la méthode des équations intégrales ou "méthode des éléments de frontière" qui permet de résoudre un système d'équations aux dérivées partielles linéaire dans un domaine infini en ne maillant que la frontière du domaine à distance finie. La mise en oeuvre du couplage entre la méthode des éléments finis et la méthode des éléments de frontière apparaît donc comme particulièrement intéressante car elle permet de bénéficier de la flexibilité des codes de calcul par éléments finis tout en permettant de représenter les domaines infinis à l'aide de la méthode des éléments de frontière. La méthode est basée sur la construction de la "matrice de raideur" du domaine infini grâce à l'utilisation de la méthode des équations intégrales. Il suffit alors d'assembler la matrice de raideur du domaine infini avec la matrice de raideur du domaine fini représenté par éléments finis. L'utilisation de la méthode la plus simple de traitement des équations intégrales, dite méthode de " collocation " conduit à une matrice de raideur non-symétrique. Par ailleurs, la méthode dite "Singular Galerkin" conduit à une formulation symétrique, mais au prix du calcul d'intégrales hypersingulières. La thèse porte sur une nouvelle formulation permettant d'obtenir une matrice de raideur symétrique sans intégrales hypersingulières, dans le cas de problèmes plans. Quelques applications numériques sont abordées pour des problèmes courants rencontrés en géotechnique
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PIACENTINI, MAURO. "Nonlinear formulation of semidefinite programming and eigenvalue optimization - application to integer quadratic problems." Doctoral thesis, 2012. http://hdl.handle.net/11573/917476.

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Books on the topic "Eigenvalue formulation"

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Friedland, S. The formulation and analysis of numerical methods for inverse eigenvalue problems. New York: Courant Institute of Mathematical Sciences, New York University, 1985.

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Vernon, Thomas A. Finite element formulations for coupled fluid/structure eigenvalue analysis / Thomas A. Vernon. Dartmouth, N.S: Defence Research Establishment Atlantic, 1989.

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Belgacem, Fethi. Elliptic boundary value problems with indefinite weights: Variational formulations of the principal eigenvalue and applications. Harlow: Longman, 1997.

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The Formulation and Analysis of Numerical Methods for Inverse Eigenvalue Problems. Franklin Classics, 2018.

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The Formulation and Analysis of Numerical Methods for Inverse Eigenvalue Problems. Franklin Classics, 2018.

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Belgacem, Fethi. Elliptic Boundary Value Problems with Indefinite Weights, Variational Formulations of the Principal Eigenvalue, and Applications (Research Notes in Mathematics Series). Chapman & Hall/CRC, 1997.

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Book chapters on the topic "Eigenvalue formulation"

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Cozzo, Emanuele, Guilherme Ferraz de Arruda, Francisco Aparecido Rodrigues, and Yamir Moreno. "Polynomial Eigenvalue Formulation." In SpringerBriefs in Complexity, 73–85. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-92255-3_6.

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Seaborn, James B. "Matrix Formulation of the Eigenvalue Problem." In Mathematics for the Physical Sciences, 179–205. New York, NY: Springer New York, 2002. http://dx.doi.org/10.1007/978-1-4684-9279-8_9.

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Kobeissi, Hussein, Youssef Nasser, Amor Nafkha, Oussama Bazzi, and Yves Louët. "A Simple Formulation for the Distribution of the Scaled Largest Eigenvalue and Application to Spectrum Sensing." In Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering, 284–93. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-40352-6_23.

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"Basics of Algebraic Eigenvalue Problem Formulation." In The Boundary Element Method, 89–98. CRC Press, 2004. http://dx.doi.org/10.1201/b17005-12.

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Barrett, Jeffrey A. "The Standard Formulation of Quantum Mechanics." In The Conceptual Foundations of Quantum Mechanics, 42–65. Oxford University Press, 2019. http://dx.doi.org/10.1093/oso/9780198844686.003.0004.

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The standard von Neumann-Dirac formulation of quantum mechanics is presented as a set of five basic rules. We discuss each rule is discussed in turn paying particular attention to the conceptual history of the theory. Of central importance is the standard interpretation of states (the eigenvalue-eigenstate link) and the dynamical laws of the theory (the random collapse dynamics and the deterministic linear dynamics) and how the interpretation and dynamics work together to predict and explain the results of basic quantum experiments. While the focus is on the behavior of electrons, we also briefly consider how the theory uses the same mathematical formalism to treat other phenomena like the behavior of neutral K mesons and qbits in a quantum computer.
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Barnett, Stephen. "Generalized measurements." In Quantum Information. Oxford University Press, 2009. http://dx.doi.org/10.1093/oso/9780198527626.003.0007.

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Extracting information from a quantum system inevitably requires the performance of a measurement, and it is no surprise that the theory of measurement plays a central role in our subject. The physical nature of the measurement process remains one of the great philosophical problems in the formulation of quantum theory. Fortunately, however, it is sufficient for us to take a pragmatic view by asking what measurements are possible and how the theory describes them, without addressing the physical mechanism of the measurement process. This is the approach we shall adopt. We shall find that it leads us to a powerful and general description of both the probabilities associated with measurement outcomes and the manner in which the observation transforms the quantum state of the measured system. The simplest form of measurement was given a mathematical formulation by von Neumann, and we shall refer to measurements of this type as von Neumann measurements or projective measurements. It is this description of measurements that is usually introduced in elementary quantum theory courses. We start with an observable quantity A represented by a Hermitian operator Â, the eigenvalues of which are the possible results of the measurement of A. The relationship between the operator, its eigenstates {|λnñ}, and its (real) eigenvalues {λn} is expressed by the eigenvalue equation . . . Â |λn_ = λn|λn_. (4.1) . . .
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"Formulation of Eigenvalue Problems by the Boundary Integral Equations." In Boundary Integral Equation Methods in Eigenvalue Problems of Elastodynamics and Thin Plates, 39–55. Elsevier, 1985. http://dx.doi.org/10.1016/b978-0-444-42447-1.50008-2.

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"Formulation of Boundary Integral Equations for Steady-State Elastodynamics." In Boundary Integral Equation Methods in Eigenvalue Problems of Elastodynamics and Thin Plates, 29–38. Elsevier, 1985. http://dx.doi.org/10.1016/b978-0-444-42447-1.50007-0.

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"Formulation of Boundary Integral Equations for Thin Plates and Eigenvalue Problems." In Boundary Integral Equation Methods in Eigenvalue Problems of Elastodynamics and Thin Plates, 218–31. Elsevier, 1985. http://dx.doi.org/10.1016/b978-0-444-42447-1.50014-8.

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Roach, G. F., I. G. Stratis, and A. N. Yannacopoulos. "Well Posedness." In Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media Electromagnetics. Princeton University Press, 2012. http://dx.doi.org/10.23943/princeton/9780691142173.003.0004.

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This chapter presents rigorous mathematical results concerning the solvability and well posedness of time-harmonic problems for complex electromagnetic media, with a special emphasis on chiral media. It also presents some results concerning eigenvalue problems in cavities filled with complex electromagnetic materials. The chapter also studies the behaviour of the interior domain problem for a chiral medium in the limit of low chirality. Next, it presents some comments related to the well posedness and solvability of exterior problems. Finally, using an appropriate finite-dimensional space and the variational formulation of the discretised version of the original boundary value problem, this chapter obtains numerical methods for the solution of the Maxwell equations for chiral media.
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Conference papers on the topic "Eigenvalue formulation"

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Yeigh, B. W., and J. A. Hoffman. "Moment formulation for random eigenvalue problems in beams." In ERES 2011. Southampton, UK: WIT Press, 2011. http://dx.doi.org/10.2495/eres110161.

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Feeny, B. F., and U. Farooq. "A State-Variable Decomposition Method for Estimating Modal Parameters." In ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/detc2007-35651.

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A nonsymmetric, generalized eigenvalue problem is constructed from state-variable ensembles. The data-based eigenvalue problem is related to the state-variable formulation of linear multi-degree-of-freedom systems. The inverse-transpose of the eigenvector matrix from this eigenvalue problem converges to the state-variable modal eigenvectors, and the eigenvalues lead to estimates of frequencies and modal damping. The interpretation holds whether damping is modal or nonmodal, and without the need of input data. The method is illustrated on an eight degree-of-freedom mass-spring-dashpot example.
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Chevallier, G., F. Renaud, J. L. Dion, and S. Thouviot. "Complex Eigenvalue Analysis for Structures With Viscoelastic Behavior." In ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2011. http://dx.doi.org/10.1115/detc2011-48897.

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This document deals with a method for eigenvalue extraction for the analysis of structures with viscoelastic materials. A generalized Maxwell model is used to model linear viscoelasticity. Such kind of model necessitates a state-space formulation to perform eigenvalue analysis with standard solvers. This formulation is very close to ADF formulation. The use of several materials on the same structure and during the same analysis may lead to a large number of internal states. This article purpose is to identify simultaneously all the viscoelastic materials and to constrain them to have the same time-constants. As it is usually possible, the size of the state-space problem is therefore widely reduced. Moreover, an accurate method for reducing mass and stiffness operators is proposed; The enhancement of the modal basis allows to obtain good results with large reduction. As the length of the paper is limited, only theoretical development are presented in the present paper while numerical results will be presented in the conference.
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Men, Han, Robert M. Freund, Ngoc C. Nguyen, Joel Saa-Seoane, and Jaime Peraire. "Designing Phononic Crystals With Convex Optimization." In ASME 2013 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2013. http://dx.doi.org/10.1115/imece2013-64694.

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Designing phononic crystals by creating frequency bandgaps is of particular interest in the engineering of elastic and acoustic microstructured materials. Mathematically, the problem of optimizing the frequency bandgaps is often nonconvex, as it requires the maximization of the higher indexed eigenfrequency and the minimization of the lower indexed eigenfrequency. A novel algorithm [1] has been previously developed to reformulate the original nonlinear, nonconvex optimization problem to an iteration-specific semidefinite program (SDP). This algorithm separates two consecutive eigenvalues — effectively maximizing bandgap (or bandwidth) — by separating the gap between two orthogonal subspaces, which are comprised columnwise of “important” eigenvectors associated with the eigenvalues being bounded. By doing so, we avoid the need of computation of eigenvalue gradient by computing the gradient of affine matrices with respect to the decision variables. In this work, we propose an even more efficient algorithm based on linear programming (LP). The new formulation is obtained via approximation of the semidefinite cones by judiciously chosen linear bases, coupled with “delayed constraint generation”. We apply the two convex conic formulations, namely, the semidefinite program and the linear program, to solve the bandgap optimization problems. By comparing the two methods, we demonstrate the efficacy and efficiency of the LP-based algorithm in solving the category of eigenvalue bandgap optimization problems.
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Geyi, Wen. "On the Spurious Solutions in Boundary Integral Formulation for Waveguide Eigenvalue Problems." In 20th European Microwave Conference, 1990. IEEE, 1990. http://dx.doi.org/10.1109/euma.1990.336248.

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Yang, B. "Eigenvalue Inclusion Principles for Distributed Gyroscopic Systems." In ASME 1991 Design Technical Conferences. American Society of Mechanical Engineers, 1991. http://dx.doi.org/10.1115/detc1991-0312.

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Abstract 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 non-dissipative constraints and their effects on the system natural frequencies are studied It is shown that the natural frequencies of the constrained gyroscopic system alternate with those of the unconstrained system.
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Singh, Kumar V., and Yitshak M. Ram. "A Mathematical Model to Overcome the Discrepancies Between Continuous Systems and Their Discrete Approximation." In ASME 2002 Engineering Technology Conference on Energy. ASMEDC, 2002. http://dx.doi.org/10.1115/etce2002/ot-29157.

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The classical finite element and finite difference formulation in structural dynamics leads to an algebraic eigenvalue problem whereas the continuous model however leads to a transcendental eigenvalue problem. This paper demonstrates the discrepancies between continuous systems and their discrete approximations and, introduces a finite dimensional transcendental eigenvalue method, which approximates the spectrum of the continuous system accurately. Illustration of the effectiveness and applicability of such a model has been shown with an example of an axially vibrating tapered rod.
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Silva, Camilo F., Thomas Runte, Wolfgang Polifke, and Luca Magri. "Uncertainty Quantification of Growth Rates of Thermoacoustic Instability by an Adjoint Helmholtz Solver." In ASME Turbo Expo 2016: Turbomachinery Technical Conference and Exposition. American Society of Mechanical Engineers, 2016. http://dx.doi.org/10.1115/gt2016-57659.

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The objective of this paper is to quantify uncertainties in thermoacoustic stability analysis with a Helmholtz solver and its adjoint. Thermoacoustic combustion instability may be described by the Helmholtz equation combined with a model for the flame dynamics. Typically, such a formulation leads to an eigenvalue problem in which the eigenvalue appears under nonlinear terms, such as exponentials related to time delays that result from the flame model. Consequently, the standard adjoint sensitivity formulation should be augmented by first- and second-order correction terms that account for the nonlinearities. Such a formulation is developed in the present paper, and applied to the model of a combustion test rig with a premix swirl burner. The uncertainties considered concern plenum geometry, outlet acoustic reflection coefficient, as well as gain and phase of the flame response. The nonlinear eigenvalue problem and its adjoint are solved by an in-house adjoint Helmholtz solver, based on an axisymmetric finite volume approach. In addition to first-order correction terms of the adjoint formulation, which are often used in literature, second-order terms are also taken into account. It is found that one particular second-order term has significant impact on the accuracy of results. Finally, the Probability Density Function of the growth rate in the presence of uncertainties in input paramters is calculated with Monte Carlo simulations. It is found that the second-order adjoint method, while giving quantitative agreement, requires far less compute resources than Monte Carlo sampling for the full nonlinear eigenvalue problem.
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Sun, Liang, Shuiwang Ji, and Jieping Ye. "A least squares formulation for a class of generalized eigenvalue problems in machine learning." In the 26th Annual International Conference. New York, New York, USA: ACM Press, 2009. http://dx.doi.org/10.1145/1553374.1553499.

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Molzahn, Daniel K., and Bernard C. Lesieutre. "An eigenvalue formulation for determining initial conditions of induction machines in dynamic power system simulations." In 2010 IEEE International Symposium on Circuits and Systems - ISCAS 2010. IEEE, 2010. http://dx.doi.org/10.1109/iscas.2010.5537071.

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Reports on the topic "Eigenvalue formulation"

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Whirl Analysis of an Overhung Disk Shaft System Mounted on Non-rigid Bearings. SAE International, March 2022. http://dx.doi.org/10.4271/2022-01-0607.

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Eigenvalues of a simple rotating flexible disk-shaft system are obtained using different methods. The shaft is supported radially by non-rigid bearings, while the disk is situated at one end of the shaft. Eigenvalues from a finite element and a multi-body dynamic tool are compared against an established analytical formulation. The Campbell diagram based on natural frequencies obtained from the tools differ from the analytical values because of oversimplification in the analytical model. Later, detailed whirl analysis is performed using AVL Excite multi-body tool that includes understanding forward and reverse whirls in absolute and relative coordinate systems and their relationships. Responses to periodic force and base excitations at a constant rotational speed of the shaft are obtained and a modified Campbell diagram based on this is developed. Whirl of the center of the disk is plotted as an orbital or phase plot and its rotational direction noted. Finally, based on the above plots, forward and reverse whirl zones for the two excitation types are established.
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