Quadratic Eigenvalue Problem Research Articles

Computing Quadratic Eigenvalues and Solvent by a New Minimization Method and a Split-Linearization Technique

To solve quadratic eigenvalue problems (QEPs), especially the gyroscopic systems, two methods are proposed: an iterative direct detection method (DDM) of the complex eigenvalues of the original QEP, and a split-linearization method (SLM) for finding the solvent matrix, which results to a standard linear eigenvalue problem (LEP) being solved to compute all eigenvalues by the symmetry extension. Reducing the dimension to one-half, the LEP is recast in a simpler QEP involving the square of the solvent. We set up two new merit functions which are minimized to detect the complex eigenvalues from the original QEP and a simpler QEP. For each eigen-parameter the merit function consists of the Euclidean norm of each derived eigen-equation, whose vector variable is solved from a derived inhomogeneous linear system. Then, the golden section search algorithm is employed to minimize the merit functions and locate the complex eigenvalue as a local minimal point. The results are compared with that computed by the cyclic-reduction-based solvent (CRS) method.

Stress fields at skin-stringer junctions in composite aircraft fuselages

This paper presents a semi-analytical analysis approach for the determination of stress fields in the vicinity of skin-stringer junctions in stiffened composite panels. The situation considered in this paper is representative for a typical stiffened panel in a modern composite aircraft fuselage. The analysis method employs a two-tier approach employing a global model based on CLPT on the one hand, and a local approach on the other hand in the form of a layerwise displacement formulation. This allows for the detailed computation of the stress concentrations in the vicinity of the skin-stringer junction. The layerwise formulation utilizes a discretization of the laminate layers into mathematical layers. The principle of the minimum of the total elastic potential yields the governing equations of the given problem, and an exponential approach leads to a quadratic eigenvalue problem that can be solved numerically. The analysis method shows excellent accuracy of the stress results in comparison with comparative finite element computations at a fraction of the computational time and effort that is required for numerical analyses.

Plug-and-play analytical paradigm for the scattering of plane waves by ‘layer-cake’ periodic systems

We investigate the scattering of scalar plane waves in two dimensions by a heterogeneous layer that is periodic in the direction parallel to its boundary. On describing the layer as a union of periodic laminae, we develop a solution of the scattering problem by merging the concept of propagator matrices and that of Bloch eigenstates featured by the unit cell of each lamina. The featured Bloch eigenstates are obtained by solving the quadratic eigenvalue problem (QEVP) that seeks a complex-valued wavenumber normal to the layer boundary given (i) the excitation frequency, and (ii) real-valued wavenumber parallel to the boundary—that is preserved throughout the system. Spectral analysis of the QEVP reveals sufficient conditions for discreteness of the eigenvalue spectrum and the fact that all eigenvalues come in complex-conjugate pairs. By deploying the factorization afforded by the propagator matrix approach, we demonstrate that the contribution of individual eigenvalues (and so eigenmodes) to the solution diminishes exponentially with absolute value of their imaginary part, which then forms a rational basis for truncation of the factorized Bloch (FB)-wave solution. The proposed methodology caters for the optimal design of rainbow traps, energy harvesters and metasurfaces, whose potency to manipulate waves is decided not only by the individual dispersion characteristics of the component laminae, but also by ordering and generally fitting of the latter into a composite layer. Using the FB approach, evaluation of trial configurations—as generated by the permutation and window translation/stretching of the component laminae—can be accelerated by decades.

Finite element analysis for free vibration of pipes conveying fluids–physical significance of complex mode shapes

A finite element formulation is presented for the natural vibration analysis of pipes conveying fluids. The solution of the resulting quadratic eigenvalue problem generally yields complex eigenvalues and eigenvectors. The present study then develops a robust mathematical procedure that combines the real and imaginary components of the eigenvectors to form physically attainable (i.e., real) mode shapes. The procedure yields a family of solutions that is more general than previously known solutions. The well-known classical mode shape is shown to be recoverable as a special case from the present solution. The study provides new insights on the effects of viscous damping, axial compressive force, and the flexibility of intermediate pipe supports on the response. Additionally, the study develops a novel algorithm based on Hermitian angles between eigenvectors to automate the tracing of mode evolution in the frequency-velocity plots.

Nonreciprocal propagation of bending waves in spatiotemporally modulated prestressed beam

Breaking time or/and spatial symmetry can lead to unprecedent mechanical responses of a structure, the design of which is usually very intricate. This paper proposes a simple design of a beam structure with spatiotemporal modulation. In fact, a time-varying axial load applied to the ends of an elastic beam will induce space-time dependent axial force propagating in the form of longitudinal wave in the beam, which can modulate spatiotemporally the propagation of transverse bending waves. The governing equation is derived, of which the Floquet solution leads to a quadratic eigenvalue problem that can be solved for the frequency as a function of the wavenumber. Theoretical results show the existence of asymmetric wavenumber bandgaps and the unique dispersion diagram which is not periodic with respect to time or space. The breaking of temporal symmetry enables nonreciprocal propagation of transverse bending waves, which is confirmed by the finite element simulations. This modulation method offers promise for applications in the fields of elastic wave filter, frequency converter and magnitude amplifier.

Three-Dimensional Plate Dynamics in the Framework of Space-Fractional Generalized Thermoelasticity: Theory and Validation

This work aims to study the dynamics of 3D plates under uniform and nonuniform temperature distributions in the framework of the space-fractional generalized thermoelasticity (S-FGT) approach. The quadratic eigenvalue problem is obtained, which means that the thermoelastic damping plays a meaningful role due to the plate’s thermal energy absorption. The plate’s complex frequency spectrum and mode shapes (free ends) under two different temperature distributions are considered for different values of the fractional continua order α and the length scale parameter l. For the first four frequencies, the fractional modes closest to the experimental results and the classical modes are presented with the absolute differences between them. For the nonuniform temperature distribution case, the mode shape analysis is performed assuming that modulus of elasticity, thermal expansion, and specific heat parameters are functions of the temperature. The primary outcomes of the paper can be stated as follows: 1) the S-FGT approach analysis gives more reliable results than the classical (local) theory; 2) the peak point of the out-of-plane mode amplitude is shifted toward the warmed zone; 3) a mode shifting is observed for the uniform temperature distribution in contrast to the nonuniform temperature distribution; 4) the fractional order derivative and length scale parameter depend on temperature, similar to other material properties such as elastic modulus, specific heat, and coefficients of thermal expansion; 5) a decrease in the fractional order is observed, while temperature increases for the fixed length scale parameter. These novelties indicate that the S-FGT approach establishes a new model for analyzing materials under heating, and the results may be beneficial for designing thermal structures.

Lead Green's functions from quadratic eigenvalue problems without mode velocity calculations.

In quantum transport calculations, the proper handling of incoming and outgoing modes for retarded Green's functions is achieved via the lead self-energies. Computationally efficient and accurate methods to calculate the self-energies are thus very important. Here we present an alternative method for calculating lead self-energies which improves on a standard approach to solving quadratic eigenvalue problems that arise in quantum transport modeling. The method is based on a perturbative analysis of the generalized Schur decomposition to determine the relevant set of eigenvalues for transmitting modes. This allows us to circumvent finding the velocities of the modes (left- or right-moving) that are needed in order to calculate the lead Green's function from translationally invariant Green's functions. This saves computational time irrespective of the value of the imaginary part added to the energy. We compare our method with two existing methods-a popular iterative method and a standard eigenvalue method that explicitly calculates the velocities of the propagating modes. Our comparison shows that both eigenvalue methods are more robust than the iterative method. Furthermore, the comparison also shows that above a small threshold of propagating modes, the standard eigenvalue method requires extra computation time over our perturbation method. This excess of computation time grows linearly with the number of propagating modes.

Lateral torsional buckling of unsymmetric beams by FEM with linear elements

As the exact solution of lateral torsional buckling of elastic prismatic beams is practically limited to the simple case of simply supported beams under equal end moments, other loading conditions and boundary conditions require more practical solutions to the problem. As the general solution of lateral torsional buckling of unsymmetric beams is a function of the sign of the bending moment along the axis of the beam, i.e. location of the shear center concerning the compression zone of the cross section, application of the current traditional empirical expressions such as coefficient of the moment, Cb, becomes highly inconsistent and unwarranted to capture the correct critical moment as acknowledged by the American Institute of Steel Construction, AISC. The finite element method, FEM, offers a feasible alternative to overcome these shortcomings. FEM is formulated in its simplest form of linear elements for the general case of lateral torsional buckling of unsymmetric cross sections. Finite element development shows that the characteristic equation is of the nonlinear quadratic eigenvalue problem type. Using the classical polynomial shape functions for beams, FEM proves to be extremely accurate and can overcome the high inconsistencies and discrepancies embedded in the application of the classical methods.

Developing HSS iteration schemes for solving the quadratic matrix equation AX2+BX+C=0$AX^{2}+BX+C=0$

AbstractThe quadratic matrix equation (QME) occurs in the branches such as the quadratic eigenvalue problems and quasi‐birth‐death processes. Also, the numerical solution of QMEs is an essential step in many computational methods for linear‐quadratic and robust control, filtering, controller order reduction, inner‐outer factorization, spectral factorization, and other applications. In this study, schemes are presented to solve the QME based on the Hermitian and skew‐Hermitian splitting (HSS). It is shown that the proposed schemes converge to the solutions of the QME. Finally, some examples are solved to discover the application of the schemes in comparison with Newton's method.

Multiscale Asymptotic Computations for the Elastic Quadratic Eigenvalue Problem in Periodically Composite Structure

A multiscale analysis and computational method based on the Second-Order Two-Scale (SOTS) approach are proposed for the elastic quadratic eigenvalue problems in the periodic composite domain. Two typical quadratic eigenvalue problems with different damping effects are considered, and by the asymptotic expansions of both the eigenfunctions and eigenvalues, the first- and second-order cell functions, the microscale features of this heterogeneous materials are defined successively. Then, the homogenized quadratic eigenvalue problems are derived and the second-order expansions of the eigenfunctions are formed. The eigenvalues are also broadened to the second-order terms by introducing proper auxiliary elastic functions defined in the composite structure, and the nonlinear expressions of the correctors of the eigenvalues are derived. The finite element procedures are established, linearized methods are discussed for solving the quadratic eigenvalue problems and the second-order asymptotic computations are performed. Effectiveness of the asymptotic model is demonstrated by both the qualitative and quantitative comparisons between the computed SOTS approximations and the reference solutions, and the converging behavior of the eigenfunctions are numerically verified. It is also indicated that the second-order correctors are of importance to reconstruct the detailed information of the original eigenfunctions within the micro cells.

Multiscale asymptotic analysis and algorithm for the quadratic eigenvalue problem in composite materials

Multiscale asymptotic analysis and algorithm for the quadratic eigenvalue problem in composite materials

Molecular vibrations in the presence of velocity-dependent forces.

A semiclassical theory of small oscillations is developed for nuclei that are subject to velocity-dependent forces in addition to the usual interatomic forces. When the velocity-dependent forces are due to a strong magnetic field, novel effects arise-for example, the coupling of vibrational, rotational, and translational modes. The theory is first developed using Newtonian mechanics and we provide a simple quantification of the coupling between these types of modes. We also discuss the mathematical structure of the problem, which turns out to be a quadratic eigenvalue problem rather than a standard eigenvalue problem. The theory is then re-derived using the Hamiltonian formalism, which brings additional insight, including a close analogy to the quantum-mechanical treatment of the problem. Finally, we provide numerical examples for the H2, HT, and HCN molecules in a strong magnetic field.

Computation of high-frequency magnetoelastic waves in layered materials

The direct calculation of magnetoelastic wave dispersion in layered media is presented using an efficient, accurate computational technique. The governing, coupled equations for elasticity and magnetism, the Navier and Landau-Lifshitz equations, respectively, are linearized to form a quadratic eigenvalue problem that determines a complex web of wavenumber-frequency dispersion branches and their corresponding mode profiles. Numerical discretization of the eigenvalue problem via a spectral collocation method (SCM) is employed to determine the complete dispersion maps for both a single, finite-thickness magnetic layer and a finite magnetic-nonmagnetic double-layer. The SCM, previously used to study elastic waves in non-magnetic media, is fast, accurate, and adaptable to a variety of sample configurations and geometries. Emphasis is placed on the extremely high frequency regimes being accessed in ultrafast magnetism experiments. The dispersion maps and modes provide insight into how energy propagates through the coupled system, including how energy can be transferred between elastic- and magnetic-dominated waves as well as between different layers. The numerical computations for a single layer are further understood by a simplified analytical calculation in the high-frequency, exchange-dominated regime where the resonance condition required for energy exchange (an anticrossing) between quasi-elastic and quasi-magnetic dispersion branches is determined. Nonresonant interactions are shown to be well approximated by the dispersion of uncoupled elastic and magnetic waves. The methods and results provide fundamental theoretical tools to model and understand current and future magnetic devices powering spintronic innovation.

Time-dependent nuclear-electronic orbital Hartree-Fock theory in a strong uniform magnetic field.

In an ultrastrong magnetic field, with field strength B ≈ B0 = 2.35 × 105 T, molecular structure and dynamics differ strongly from that observed on the Earth. Within the Born-Oppenheimer (BO) approximation, for example, frequent (near) crossings of electronic energy surfaces are induced by the field, suggesting that nonadiabatic phenomena and processes may play a more important role in this mixed-field regime than in the weak-field regime on Earth. To understand the chemistry in the mixed regime, it therefore becomes important to explore non-BO methods. In this work, the nuclear-electronic orbital (NEO) method is employed to study protonic vibrational excitation energies in the presence of a strong magnetic field. The NEO generalized Hartree-Fock theory and time-dependent Hartree-Fock (TDHF) theory are derived and implemented, accounting for all terms that result as a consequence of the nonperturbative treatment of molecular systems in a magnetic field. The NEO results for HCN and FHF- with clamped heavy nuclei are compared against the quadratic eigenvalue problem. Each molecule has three semi-classical modes owing to the hydrogen-two precession modes that are degenerate in the absence of a field and one stretching mode. The NEO-TDHF model is found to perform well; in particular, it automatically captures the screening effects of the electrons on the nuclei, which are quantified through the difference in energy of the precession modes.

Singular quadratic eigenvalue problems: linearization and weak condition numbers

The numerical solution of singular eigenvalue problems is complicated by the fact that small perturbations of the coefficients may have an arbitrarily bad effect on eigenvalue accuracy. However, it has been known for a long time that such perturbations are exceptional and standard eigenvalue solvers, such as the QZ algorithm, tend to yield good accuracy despite the inevitable presence of roundoff error. Recently, Lotz and Noferini quantified this phenomenon by introducing the concept of $$\delta $$ -weak eigenvalue condition numbers. In this work, we consider singular quadratic eigenvalue problems and two popular linearizations. Our results show that a correctly chosen linearization increases $$\delta $$ -weak eigenvalue condition numbers only marginally, justifying the use of these linearizations in numerical solvers also in the singular case. We propose a very simple but often effective algorithm for computing well-conditioned eigenvalues of a singular quadratic eigenvalue problems by adding small random perturbations to the coefficients. We prove that the eigenvalue condition number is, with high probability, a reliable criterion for detecting and excluding spurious eigenvalues created from the singular part.

Interior point methods for solving Pareto eigenvalue complementarity problems

In this paper, we propose to solve Pareto eigenvalue complementarity problems by using interior-point methods. Precisely, we focus the study on an adaptation of the Mehrotra Predictor Corrector Method (MPCM) and a Non-Parametric Interior Point Method (NPIPM). We compare these two methods with two alternative methods, namely the Lattice Projection Method (LPM) and the Soft Max Method (SM). On a set of data generated from the Matrix Market, the performance profiles highlight the efficiency of MPCM and NPIPM for solving eigenvalue complementarity problems. We also consider an application to a concrete and large size situation corresponding to a geomechanical fracture problem. Finally, we discuss the extension of MPCM and NPIPM methods to solve quadratic pencil eigenvalue problems under conic constraints.

New backward error bounds of Rayleigh–Ritz projection methods for quadratic eigenvalue problem

Rayleigh–Ritz projection is one of the efficient procedures to project a large-scale quadratic eigenvalue problem (QEP) into a small-scale QEP by properly choosing a low-dimensional subspace. One common way for solving the projected QEP is to recast it via linearization. In this paper, we establish bounds for backward errors of approximate eigenpairs of QEP relative to those of a linearization. These bounds give useful information to predict the numerical stability of Rayleigh–Ritz projection eigensolvers followed by a linearization. We present results of numerical experiments that support the predictions of the backward error analysis.

Free Vibrations of Multi-Degree Structures: Solving Quadratic Eigenvalue Problems with an Excitation and Fast Iterative Detection Method

For the free vibrations of multi-degree mechanical structures appeared in structural dynamics, we solve the quadratic eigenvalue problem either by linearizing it to a generalized eigenvalue problem or directly treating it by developing the iterative detection methods for the real and complex eigenvalues. To solve the generalized eigenvalue problem, we impose a nonzero exciting vector into the eigen-equation, and solve a nonhomogeneous linear system to obtain a response curve, which consists of the magnitudes of the n-vectors with respect to the eigen-parameters in a range. The n-dimensional eigenvector is supposed to be a superposition of a constant exciting vector and an m-vector, which can be obtained in terms of eigen-parameter by solving the projected eigen-equation. In doing so, we can save computational cost because the response curve is generated from the data acquired in a lower dimensional subspace. We develop a fast iterative detection method by maximizing the magnitude to locate the eigenvalue, which appears as a peak in the response curve. Through zoom-in sequentially, very accurate eigenvalue can be obtained. We reduce the number of eigen-equation to n−1 to find the eigen-mode with its certain component being normalized to the unit. The real and complex eigenvalues and eigen-modes can be determined simultaneously, quickly and accurately by the proposed methods.

Fast optimization of viscosities for frequency‐weighted damping of second‐order systems

AbstractWe consider frequency‐weighted damping optimization for vibrating systems described by a second‐order differential equation. The goal is to determine viscosity values such that eigenvalues are kept away from certain undesirable areas on the imaginary axis. To this end, we present two complementary techniques. First, we propose new frameworks using nonsmooth constrained optimization problems, whose solutions both damp undesirable frequency bands and maintain the stability of the system. These frameworks also allow us to weight, which frequency bands are the most important to damp. Second, we also propose a fast new eigensolver for the structured quadratic eigenvalue problems (QEPs) that appear in such vibrating systems. In order to be efficient, our new eigensolver exploits special properties of diagonal‐plus‐rank‐one (DPR1) complex symmetric matrices, which we leverage by showing how each QEP can be transformed into a short sequence of such linear eigenvalue problems. The result is an eigensolver that is substantially faster than standard techniques. By combining this new solver with our new optimization frameworks, we obtain our overall algorithm for fast computation of optimal viscosities. The efficiency and performance of our new approach are verified and illustrated on several numerical examples.

Weight splitting iteration methods to solve quadratic nonlinear matrix equation [formula omitted

The quadratic nonlinear matrix equationQ(Y)=MY2+NY+P,occurs in many applications such as the Quasi-Birth-Death processes, pseudo-spectra for quadratic eigenvalue problems and the quadratic eigenvalue problems. In this work, we propose efficient parametric iterative methods for finding the solution of this quadratic matrix equation based on weight splitting (WS) on matrices M and N, separately. We show that the proposed methods converge to the solution of the quadratic nonlinear matrix equation under conditions. Every iteration in proposed methods requires the solution of one or two linear matrix equations. Finally, various numerical examples indicate that the obtained methods in terms of CPU time, accuracy and computational cost are superior to the famous methods.