Quadratic Eigenvalue Problem Research Articles

Approximation of damped quadratic eigenvalue problem by dimension reduction

This paper presents an approach to the efficient calculation of all or just one important part of the eigenvalues of the parameter dependent quadratic eigenvalue problem (λ2(v)M+λ(v)D(v)+K)x(v)=0, where M, K are positive definite Hermitian n × n matrices and D(v) is an n × n Hermitian semidefinite matrix which depends on a damping parameter vector v=[v1…vk]∈R+k. With the new approach one can efficiently (and accurately enough) calculate all (or just part of the) eigenvalues even for the case when the parameters vi, which in this paper represent damping viscosities, are of the modest magnitude. Moreover, we derive two types of approximations with corresponding error bounds. The quality of error bounds as well as the performance of the achieved eigenvalue tracking are illustrated in several numerical experiments.

Existence and Stability of Equilibrium of DC Microgrid With Constant Power Loads

The problem of existence and stability of equilibria of DC microgirds with constant power loads methods is addressed in this paper. Constant power loads (CPLs) often cause instability due to its negative impedance characteristics. What is the worse, CPLs may cause that the system admits no equilibria for its nonlinearity. The purpose of paper can be summarized as: A) designing a controller to overcome CPLs instability; B) obtaining the sufficient conditions to guarantee the existence of the equilibria; C) Under the conditions of equilibria existing, obtaining the sufficient conditions of local stability. For the first objective, the method based on virtual resistance and virtual inductance is proposed. For the second objective, we transform the problem of solvability of quadratic equations into the existence of the fixed point of a mapping. Thus, the analytical sufficient conditions is obtained based on Tarski fixed point theorem. Moreover, a numerical method is also provided to reduce conservatism. For the third objective, the small-signal model is established to predict the system qualitative behavior around equilibria. The stability conditions are obtained by using quadratic eigenvalue problem (GEP) theorem. These explicit conditions are obtained as a function of system parameters and it lead to build reliable microgrid.

Solving constrained quadratic inverse eigenvalue problem via conjugate direction method

Inverse eigenvalue problems are among the most important problems in numerical linear algebra. This paper is concerned with the problem of designing an iterative method for a quadratic inverse eigenvalue problem of the form MXΛ2+GXΛ+KX=0 where M, G and K should be partially doubly symmetric under a prescribed submatrix constraint. This kind of quadratic inverse eigenvalue problem includes the classical and generalized inverse eigenvalue problems as special cases. We propose the conjugate direction (CD) method for computing the least Frobenius norm solutions of this constrained quadratic inverse eigenvalue problem in a finite number of steps in the absence of round-off errors. The CD method can automatically determine the solvability of this problem. Finally, the overall performance and efficiency of the CD method for computing approximate solutions of the problem are discussed.

Nonlinear interval eigenvalue problems for damped spring-mass system

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

A polynomial eigenvalue approach for multiplex networks

We explore the block nature of the matrix representation of multiplex networks, introducing a new formalism to deal with its spectral properties as a function of the inter-layer coupling parameter. This approach allows us to derive interesting results based on an interpretation of the traditional eigenvalue problem. Specifically, our formalism is based on the reduction of the dimensionality of a matrix of interest but increasing the power of the characteristic polynomial, i.e, a polynomial eigenvalue problem. This approach may sound counterintuitive at first, but it enable us to relate the quadratic eigenvalue problem for a 2-Layer multiplex network with the spectra of its respective aggregated network. Additionally, it also allows us to derive bounds for the spectra, among many other interesting analytical insights. Furthermore, it also permits us to directly obtain analytical and numerical insights on the eigenvalue behavior as a function of the coupling between layers. Our study includes the supra-adjacency, supra-Laplacian and the probability transition matrices, which enables us to put our results under the perspective of structural phases in multiplex networks. We believe that this formalism and the results reported will make it possible to derive new results for multiplex networks in the future.

Normal Modes and Modal Reduction in Exterior Acoustics

The Helmholtz equation for exterior acoustic problems can be solved by the finite element method in combination with conjugated infinite elements. Both provide frequency-independent system matrices, forming a discrete, linear system of equations. The homogenous system can be understood as a quadratic eigenvalue problem of normal modes (NMs). Knowledge about the only relevant NMs, which — when doing modal superposition — still provide a sufficiently accurate solution for the sound pressure and sound power in comparison to the full set of modes, leads to reduced computational effort. Properties of NMs and criteria of modal reduction are discussed in this work.

Resonances of enclosed shallow water basins with slender floating elastic bodies

A model for the hydroelastic response of enclosed shallow water basins with floating slender plates of large span, in variable bathymetry is presented. The eigenproblem for the system’s resonant behaviour is formulated in the strong and variational form. Based on the variational form, and special hydroelastic finite elements, a quadratic eigenvalue problem is derived for the eigenfrequencies and eigenmodes of the basin-floating body system. Benchmark examples, for simplified configurations, are used to study the effects of parameters like the plate’s span, position, draft, mass, stiffness and depth of the basin. It is observed that the presence of the plate can alter significantly the eigenstates of the pond if the depth of the basin is relatively small, compared to the plate draft. In such cases the plate span and position play a crucial role. The hydroelastic analysis can be important even in cases where the depth increases and the upper surface elevation differences, in the presence or absence of the floating plate, appear to be less significant. This is due to the importance in accurately calculating the curvature and hence bending moments inside the plate. The proposed model could be relevant to the seiche formation analysis in reservoirs with floating photovoltaic platforms or ice-covered lakes.

Effect of concrete creep on dynamic stability behavior of slender concrete-filled steel tubular column

An analytical procedure for dynamic stability of CFST column accounting for the creep of concrete core is proposed. The long-term effect of creep of concrete core is formulated based on the creep model by the ACI 209 committee and the age-adjusted effective modulus method (AEMM). The equations of boundary frequencies accounting for the effects of concrete creep are derived by the Bolotin's theory and solved as a quadratic eigenvalue problem. The effectiveness of the proposed method and the characteristics of time-varying distribution of instability regions are numerically surveyed. It is shown that the CFST column becomes dynamically unstable even when the sum of the sustained static load and the amplitude of the dynamic excitation is much lower than the static instability load. It is also found that due to the time effects of concrete creep under the sustained static load, the same excitation, that cannot induce dynamic instability in the early stage of sustained loading, can induce the dynamic instability in a few days later. The critical amplitude and frequency of the dynamic excitation can decrease by 6% and 3% in 5 days, and 11% and 6% in 100 days.

Convergence of a lowest-order finite element method for the transmission eigenvalue problem

The transmission eigenvalue problem arises in scattering theory. The main difficulty in its analysis is the fact that, depending on the chosen formulation, it leads either to a quadratic eigenvalue problem or to a non-classical mixed problem. In this paper we prove the convergence of a mixed finite element approximation. This approach, which is close to the Ciarlet–Raviart discretization of biharmonic problems, is based on Lagrange finite elements and is one of the less expensive methods in terms of the amount of degrees of freedom. The convergence analysis is based on classical abstract spectral approximation result and the theory of mixed finite element methods for solving the stream function–vorticity formulation of the Stokes problem. Numerical experiments are reported in order to assess the efficiency of the method.

An efficient algorithm based on Lanczos type of BCR to solve constrained quadratic inverse eigenvalue problems

This paper proposes the Lanczos type of biconjugate residual (BCR) algorithm for solving the quadratic inverse eigenvalue problem L 2 X Λ 2 + L 1 X Λ + L 0 X = 0 where L 2 , L 1 and L 0 should be partially bisymmetric under a prescribed submatrix constraint. An analysis reveals that the algorithm obtains the solutions of the constrained quadratic inverse eigenvalue problem in finitely many steps in the absence of round-off errors. Finally numerical results are performed to confirm the analysis and to illustrate the efficiency of the proposed algorithm.

An iterative method for obtaining the Least squares solutions of quadratic inverse eigenvalue problems over generalized Hamiltonian matrix with submatrix constraints

In this paper, we consider a class of constrained matrix quadratic inverse eigenvalue problem and its optimal approximation problem. It is proved that the proposed algorithm always converge to the generalized Hamiltonian solutions with a submatrix constraint of Problem 1.1 within finite iterative steps in the absence of roundoff error. In addition, by choosing a special kind of initial matrices, it is shown that the minimum norm solution of Problem 1.1 can be obtained consequently. At last, for a given matrix group in the solution set of Problem 1.1, it is proved that the unique optimal approximation solution of Problem 1.2 can be also obtained. Some numerical results are reported to demonstrate the efficiency of our algorithm.

Least squares solutions of quadratic inverse eigenvalue problem with partially bisymmetric matrices under prescribed submatrix constraints

The inverse eigenvalue problems play an important role in broad application areas such as system identification, Hopfield neural networks, control design, mass–spring system and molecular spectroscopy. This paper proposes an algorithm that yields a new method to efficiently and accurately compute the partially bisymmetric solutions (M,C,K) under prescribed submatrix constraints of the quadratic inverse eigenvalue problem MXΛ2+CXΛ+KX=0. The algorithm is developed based on the conjugate gradient normal equations residual minimizing (CGNR) method. We discuss the convergence properties of the algorithm. Finally, the performance of the algorithm is tested on two numerical examples and compared to the previous algorithm.

Dynamic stability of doubly-curved multilayered shells subjected to arbitrarily oriented angular velocities: Numerical evaluation of the critical speed

The paper is focused on the evaluation of the critical speed of rotating doubly-curved multilayered shell structures. The theoretical framework developed to this aim is based on a general formulation capable to define several Higher-order Shear Deformation Theories (HSDTs) in a unified manner. The current approach can deal easily with angular velocities applied about a generic axis of the structure. This aspect represents a clear advancement with respect to the formulations available in the literature, which are developed mainly to investigate the dynamic behavior of rotating shells of revolution (disks, circular cylinders and conical shells), in which the angular velocity is applied about their revolution axis. It is important to underline that the effects of both Coriolis and centripetal accelerations on the dynamic response of shell structures, characterized by various geometric shapes, are included in the model. The quadratic eigenvalue problem that lies behind the free vibration analysis in hand is solved numerically by means of the well-known Generalized Differential Quadrature (GDQ) method. The critical speed is obtained as a result of many parametric investigations, which are defined for increasing values of the applied angular velocities. Finally, it should be mentioned that the current research falls within the aim of the study of the dynamic stability of rotating structures. For this purpose, several considerations concerning the flutter and divergence phenomena are presented.

Acoustic vibration problem for dissipative fluids

In this paper we analyze a finite element method for solving a quadratic eigenvalue problem derived from the acoustic vibration problem for a heterogeneous dissipative fluid. The problem is shown to be equivalent to the spectral problem for a noncompact operator and athorough spectral characterization is given. The numerical discretization of the problem is based on Raviart-Thomas finite elements. The method is proved to be free of spurious modes and to converge with optimal order. Finally, we report numerical tests which allow us to assess the performance of the method.

Efficient model based on genetic programming and spline functions to find modes of unconventional waveguides

The contribution of this work is twofold: the authors developed an accurate model to solve the vector wave equation of radially-layered inhomogeneous waveguides based on spline function expansions and automated grid construction by genetic programming, and thenemployed this model to analyse the propagation of electromagnetic waves within oil wells. The developed model uses a spline expansion of the fields to convert the wave equation into a quadratic eigenvalue problem where eigenvectors represent the coefficients of the splines and eigenvalues represent the propagation constant of the eigenmode. The present study compared the proposed model using the classical winding number technique. The results obtained for the first eigenmodes of a typical oil well geometry were more accurate than those obtained by the winding number method. Moreover, the authors model could find a larger amount of eigenmodes for a fixed azimuthal parameter than the standard approach.

Massively parallel structural acoustics in Sierra-SD

Sierra-SD is a massively parallel finite element application for structural dynamics and acoustics. Problems on the order of 2 billion degrees of freedom and running on as up to one hundred thousand distributed memory cores have been solved. Sierra-SD offers a wide range of modern acoustic capabilities. Unbounded problems can be solved with either an absorbing boundary condition, infinite elements, or perfectly matched layers. Problems can be solved in the time domain, frequency domain, or as a linear or quadratic eigenvalue problem. Structural acoustics problems can be solved with monolithic strong coupling, or a Multiple Program Multiple Data (MPMD) coupling with the linear and nonlinear structural Sierra applications. One way handoff is available from other applications in the Sandia National Laboratories’ Sierra Mechanics suite, enabling loading through the Lighthill tensor from incompressible fluid codes to calculate far field noise. Real world applications include ship shock loading and vibration of reentry bodies. [Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-NA-0003525.]

On the transmission eigenvalue problem for the acoustic equation with a negative index of refraction and a practical numerical reconstruction method

In this paper, we consider the two-dimensional Maxwell's equations with the TM mode in pseudo-chiral media. The system can be reduced to the acoustic equation with a negative index of refraction. We first study the transmission eigenvalue problem (TEP) for this equation. By the continuous finite element method, we discretize the reduced equation and transform the study of TEP to a quadratic eigenvalue problem by deflating all nonphysical zeros. We then estimate half of the eigenvalues are negative with order of \begin{document}$O(1)$\end{document} and the other half of eigenvalues are positive with order of \begin{document}$O(10^2)$\end{document} . In the second part of the paper, we present a practical numerical method to reconstruct the support of the inhomogeneity by the near-field measurements, i.e., Cauchy data. Based on the linear sampling method, we propose the truncated singular value decomposition to solve the ill-posed near-field integral equation, at one wave number which is not a transmission eigenvalue. By carefully chosen an indicator function, this method produce different jumps for the sampling points inside and outside the support. Numerical results show that our method is able to reconstruct the support reliably.

Stability Analysis and Stabilization Methods of DC Microgrid With Multiple Parallel-Connected DC–DC Converters Loaded by CPLs

Constant power loads may yield instability due to the well-known negative impedance characteristic. This paper analyzes the factors that cause instability of a dc microgrid with multiple dc–dc converters. Two stabilization methods are presented for two operation modes: 1) constant voltage source mode; and 2) droop mode, and sufficient conditions for the stability of the dc microgrid are obtained by identifying the eigenvalues of the Jacobian matrix. The key is to transform the eigenvalue problem to a quadratic eigenvalue problem. When applying the methods in practical engineering, the salient feature is that the stability parameter domains can be estimated by the available constraints, such as the values of capacities, inductances, maximum load power, and distances of the cables. Compared with some classical methods, the proposed methods have wider stability region. The simulation results based on MATLAB/simulink platform verify the feasibility of the methods.

Carrier Frequency Offset Estimation in OFDM systems as a Quadratic Eigenvalue Problem

Carrier Frequency Offset Estimation in OFDM systems as a Quadratic Eigenvalue Problem

Computing the full spectrum of large sparse palindromic quadratic eigenvalue problems arising from surface Green's function calculations

Full spectrum of a large sparse ⊤-palindromic quadratic eigenvalue problem (⊤-PQEP) is considered arguably for the first time in this article. Such a problem is posed by calculation of surface Green's functions (SGFs) of mesoscopic transistors with a tremendous non-periodic cross-section. For this problem, general purpose eigensolvers are not efficient, nor is advisable to resort to the decimation method etc. to obtain the Wiener–Hopf factorization. After reviewing some rigorous understanding of SGF calculation from the perspective of ⊤-PQEP and nonlinear matrix equation, we present our new approach to this problem. In a nutshell, the unit disk where the spectrum of interest lies is broken down adaptively into pieces small enough that they each can be locally tackled by the generalized ⊤-skew-Hamiltonian implicitly restarted shift-and-invert Arnoldi (G⊤SHIRA) algorithm with suitable shifts and other parameters, and the eigenvalues missed by this divide-and-conquer strategy can be recovered thanks to the accurate estimation provided by our newly developed scheme. Notably the novel non-equivalence deflation is proposed to avoid as much as possible duplication of nearby known eigenvalues when a new shift of G⊤SHIRA is determined. We demonstrate our new approach by calculating the SGF of a realistic nanowire whose unit cell is described by a matrix of size 4000×4000 at the density functional tight binding level, corresponding to a 8×8nm2 cross-section. We believe that quantum transport simulation of realistic nano-devices in the mesoscopic regime will greatly benefit from this work.