Eigenvalue Problem Research Articles

Dark solitons on elliptic function background for the defocusing Hirota equation

Abstract We investigate dark solitons lying on elliptic function background in the defocusing Hirota equation with third-order dispersion and self-steepening terms. By means of the modified squared wavefunction method, we obtain the Jacobi’s elliptic solution of the defocusing Hirota equation, and solve the related linear matrix eigenvalue problem on elliptic function background. The elliptic N-dark soliton solution in terms of theta functions is constructed by the Darboux transformation and limit technique. The asymptotic dynamical
behaviors for the elliptic N-dark soliton solution as t → ±∞ are studied. Through numerical plots of the elliptic one-, two- and three-dark solitons, the amplification effect on the velocity of elliptic dark solitons, and the compression effect on the soliton spatiotemporal distributions produced by the third-order dispersion and self-steepening terms are discussed.

Mixing between flavor singlets in lattice gauge theories coupled to matter fields in multiple representations

We provide the first extensive, numerical study of the nontrivial problem of mixing between flavor-singlet composite states emerging in strongly coupled lattice field theories with matter field content consisting of fermions transforming in different representations of the gauge group. The theory of interest is the minimal candidate for a composite Higgs model that also accommodates a mechanism for top partial compositeness: the Sp(4) gauge theory coupled to two (Dirac) fermions transforming as the fundamental and three as the two-index antisymmetric representation of the gauge group, respectively. We apply an admixture of APE smearing and Wuppertal smearings, as well as the generalized eigenvalue problem approach, to two-point functions involving flavor-singlet mesons, for ensembles having time extent longer than the space extent. We demonstrate that, in the region of lattice parameter space accessible to this study, both masses and mixing angles can be measured effectively, despite the presence of (numerically noisy) contributions from disconnected diagrams. Published by the American Physical Society 2024

Waveguide finite element modelling for broadband vibration analysis of tyres including rotation and prestress

In the framework of tyre/road noise prediction, our aim is to introduce a waveguide finite element (WFE) method for broadband vibration analysis of rotating inflated structures of arbitrary cross-section. Approximating the geometry as perfectly circular, the finite element discretization is restricted to the two-dimensional cross-section, enabling fast and high-frequency computations. Tyres are particularly challenging to model due to various complicating effects, often approximated or neglected in WFE models : rotation, inflation, structural anisotropy, heterogenetity, damping among others. Our formulation uses Lagrangian perturbations of displacement in Eulerian coordinates, providing a concise way to express equilibrium equations in vibrating media initially moving, prestressed and subjected to fixed point forces (typically, contact excitations in rotating tyres). The method comprises three steps: computing the steady rolling state (including rotation, inflation and centrifugal force), determining the undamped vibration modes superimposed onto the steady state (solving an eigenvalue problem of quadratic type owing to Coriolis acceleration), and obtaining the forced response of the damped structure through specific orthogonality relationships (considering a specific form of damping). The WFE method is compared with a rotating shell analytical model from existing literature and is subsequently applied to a rotating tyre with heterogeneous cross-section up to 4kHz.

Towards estimating acoustic surface impedances at low frequencies in a reverberation chamber

To accurately simulate interior acoustical fields, reliable descriptions of the physical properties of the acoustical space are required. While geometry, propagation medium properties, and source characteristics are, in general, easily accessible, the absorption characteristics of the bounding surfaces of a space can be difficult to obtain. For commonly used materials, databases of absorption coefficients could be used. However, absorption coefficient databases are of limited use at low frequencies, especially below 100 Hz. Furthermore, measurement-based estimations tend to be more reliable. Thus, there is incentive to perform measurement-based estimations of the absorption characteristics of a room's bounding surfaces at low frequencies. This work considers the estimation of sound absorption, given in terms of locally-reacting surface impedance, at modal frequencies. This is made feasible by solving an inverse eigenvalue problem. The spatially-uniform, frequency-dependent impedance of the bounding surfaces of a reverberation chamber is estimated. The method is validated by comparing measured transfer functions to simulated transfer functions, which are generated using the estimated impedance. It is shown that the proposed method can be used to estimate locally-reacting surface impedance from measured room impulse responses at low frequencies.

Efficient modeling of atmospheric infrasound propagation with the Semi-Analytical-Finite-Element (SAFE) method

Downward refraction in the lower atmosphere, attributed to winds and temperature inversion, can enable infrasound to efficiently propagate over long distances inside acoustic waveguides. In this study, we use the semi-analytic finite element (SAFE) method to predict this effect in stratified, inhomogeneous, moving air for range-independent noise propagation over reflective half-planes. We present solutions for different temperature and velocity profiles and compare them to direct numerical solutions of the two-dimensional linearized Euler equations. The SAFE method separates the exact two-dimensional wave equation for a stratified, inhomogeneous, moving medium by expressing the pressure field as a sum of vertical eigenmodes propagating in the range direction. This simplifies the acoustic problem into a one-dimensional eigenvalue problem. As this problem is addressed with the finite-element method, the method can accommodate arbitrary wind and temperature profiles. For large, range-independent domains, the proposed procedure proves to be much more efficient than the tested direct numerical computations. Additionally, it converges against the exact solution of the acoustic problem if enough modes are included in the calculation. Therefore, the method offers a viable procedure for benchmarking other pressure field prediction techniques.

An extended AKNS eigenvalue problem and its affiliated integrable Hamiltonian hierarchies

This paper aims to study an extended 4 × 4 AKNS eigenvalue problem and construct its affiliated integrable hierarchies of bi-Hamiltonian models over the real field. The Lax pair framework serves as the fundamental tool, ensuring integrability through bi-Hamiltonian structures with hereditary recursion operators. We compute illustrative examples of lower-order equations to demonstrate the affiliated integrable hierarchies.

Heat equation for Sturm–Liouville operator with singular propagation and potential

Abstract This article considers the initial boundary value problem for the heat equation with the time-dependent Sturm–Liouville operator with singular potentials. To obtain a solution by the method of separation of variables, the problem is reduced to the problem of eigenvalues of the Sturm–Liouville operator. Further on, the solution to the initial boundary value problem is constructed in the form of a Fourier series expansion. A heterogeneous case is also considered. Finally, we establish the well-posedness of the equation in the case when the potential and initial data are distributions, also for singular time-dependent coefficients.

Finite Element Formulations for Maxwell’s Eigenvalue Problem Using Continuous Lagrangian Interpolations

Abstract We consider nodal-based Lagrangian interpolations for the finite element approximation of the Maxwell eigenvalue problem. The first approach introduced is a standard Galerkin method on Powell–Sabin meshes, which has recently been shown to yield convergent approximations in two dimensions, whereas the other two are stabilized formulations that can be motivated by a variational multiscale approach. For the latter, a mixed formulation equivalent to the original problem is used, in which the operator has a saddle point structure. The Lagrange multiplier introduced to enforce the divergence constraint vanishes in an appropriate functional setting. The first stabilized method consists of an augmented formulation including a mesh dependent term that can be regarded as the Laplacian of the divergence constraint multiplier. The second formulation is based on orthogonal projections, which can be recast as a residual based stabilization technique. We rely on the classical spectral theory to analyze the approximating methods for the eigenproblem. The stability and convergence aspects are inherited from the associated source problems together with an assumption which is discussed numerically. We investigate the performance of the proposed formulations and provide some convergence results validating the theoretical ones for several benchmark tests, including ones with smooth and singular solutions.

On the convergence of the fixed point method for solving neutron transport alpha eigenvalue problems

It was shown that the Fixed Point Method (also known as the Rayleigh Quotient Method) is several times faster than the Critical Search Method for solving neutron transport alpha eigenvalue problems. It was also shown that the Fixed Point Method is able to determine the alpha eigenvalues of sub-critical systems that are beyond the reach of the Critical Search Method. Despite these significant advances, the Fixed Point Method remains an unproven algorithm. This report provides a proof.

Interannual Magneto–Coriolis modes and their sensitivity on the magnetic field within the Earth’s core

Linear modes for which the Coriolis acceleration is almost entirely in balance with the Lorentz force are called Magneto–Coriolis (MC) modes. These MC modes are thought to exist in Earth’s liquid outer core and could therefore contribute to the variations observed in Earth’s magnetic field. The background state on which these waves ride is assumed here to be static and defined by a prescribed magnetic field and zero flow. We introduce a new computational tool to efficiently compute solutions to the related eigenvalue problem, and study the effect of a range of both axisymmetric and non-axisymmetric background magnetic fields on the MC modes. We focus on a hierarchy of conditions that sequentially partition the numerous computed modes into those which are: (i) in principle observable, (ii) those which match a proxy for interannual geomagnetic signal over 1999–2023, and (iii) those which align with core-flows based on recent geomagnetic data. We found that the background field plays a crucial role in determining the structure of the modes. In particular, we found no examples of axisymmetric background fields that support modes consistent with recent geomagnetic changes, but that some non-axisymmetric background fields do support geomagnetically consistent modes.

Geodesic Convexity of the Symmetric Eigenvalue Problem and Convergence of Steepest Descent

We study the convergence of the Riemannian steepest descent algorithm on the Grassmann manifold for minimizing the block version of the Rayleigh quotient of a symmetric matrix. Even though this problem is non-convex in the Euclidean sense and only very locally convex in the Riemannian sense, we discover a structure for this problem that is similar to geodesic strong convexity, namely, weak-strong convexity. This allows us to apply similar arguments from convex optimization when studying the convergence of the steepest descent algorithm but with initialization conditions that do not depend on the eigengap δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document}. When δ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta >0$$\\end{document}, we prove exponential convergence rates, while otherwise the convergence is algebraic. Additionally, we prove that this problem is geodesically convex in a neighbourhood of the global minimizer of radius O(δ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {O}}(\\sqrt{\\delta })$$\\end{document}.

Accurate computation of scattering poles of acoustic obstacles with impedance boundary conditions

We propose a computation method for scattering poles of impedance obstacles. Boundary integral equations are used to formulate the problem. It is shown that the scattering poles are the eigenvalues of some integral operator. Then we employ the Nyström method to discretize the integral operator and obtain a nonlinear matrix eigenvalue problem. The eigenvalues are computed using a multistep parallel spectral indicator method. Numerical examples demonstrate the high accuracy of the proposed method and can serve as the benchmarks. Our study provides a practical approach and can be extended to other scattering problems. This paper continues our previous study on the computation method for scattering poles of sound-soft obstacles.

Dynamic characteristics of a horizontal rectangular vessel partially or fully filled with a fluid

A theoretical framework is presented for free vibration of a flexible horizontal rectangular vessel, whether it is partially or fully filled with a fluid. The vibrational modal patterns of the vessel, whether dry or wet, are classified into symmetric and antisymmetric modes. The theoretical model conceptualizes the vessel as an unfolded plate with line supports simulating the vessel's corners. The motion of contained fluid is described using displacement potentials that adhere to the Laplace equation and fluid boundary conditions. Importantly, the proposed analytical approach accounts for the dynamic interaction between the vessel and the fluid, ensuring compatibility along their contacting surfaces. The Rayleigh-Ritz method is employed to formulate an eigenvalue problem, considering entire kinetic and strain energies of the fluid-filled vessel. The accuracy of the theoretical approach is checked by conducting finite element analyses. Remarkably, the natural frequencies obtained through commercial software align closely with the theoretical predictions for both dry and fluid-filled vessels. In instances of fully fluid-filled vessels, we can discern the presence of fluid contact at the vessel's upper plate by examining the natural frequencies and mode shapes. The proposed method offers applicability in dynamically analyzing water-filled spent fuel casks, enhancing safety during transportation.

Onset of double-diffusive convection in a Poiseuille flow with a uniform internal heat source

The linear stability analysis of the onset of double-diffusive convection in a Poiseuille flow system is investigated. In addition, a volumetric uniform internal heat source is taken into account. In this problem, the horizontal fluid channel is bounded by two plates which are isothermal and isosolutal. The governing parameters are thermal Rayleigh number RaT, solutal Rayleigh number Ras, internal heat source parameter RaI, Prandtl number Pr, and Reynolds number Re. The eigenvalue problem arising from the linear perturbed system of equations is solved numerically using the Chebyshev–Tau method coupled with the QZ algorithm. It is found that the positive solutal Rayleigh number Ras destabilizes the system. Furthermore, it is observed that an increase in the Prandtl number Pr stabilizes the system. Additionally, at Ras = −60, the critical values of the thermal Rayleigh number Rac decreases with R=Re cos ϕ up 2; and increases with R beyond R=2.

Linear and nonlinear stability of double diffusive convection in a micropolar fluid saturated porous layer with magnetic field and throughflow effects

The linear and nonlinear stability of double-diffusive convection in a porous layer saturated with micropolar fluid is examined. A transverse magnetic field is applied to the flow together with vertical throughflow. The normal mode technique is employed for linear stability analysis, whereas the energy method is used for nonlinear stability analysis. The resulting eigenvalue problems corresponding to linear and nonlinear stability theories are solved numerically by employing the bvp4c routine in MATLAB 2022(b). The critical thermal Rayleigh numbers for both linear and nonlinear analyses are computed for the different values of the governing parameters and presented graphically. A comparison is made between linear and nonlinear stability results. It is observed that the flow is more stable whenever a magnetic field is added to the flow, although the subcritical instability region also slightly increases. Increasing the Darcy number, Lewis number, coupling number, and absolute value of the throughflow parameter destabilizes the flow. On the other hand, raising the porosity of the medium and micropolar parameters stabilizes the flow. Furthermore, there is no subcritical gap in the absence of the throughflow effect, which is a good agreement between the linear and nonlinear thresholds.

Acceleration effect on stress singularity and edge strength in piezoelectric/Piezomagnetic heterostructures

The acceleration effect on edge stress singularity and edge strength in piezoelectric/piezomagnetic heterostructures is investigated in this work. For the stress singularity analysis, we propose a stress function based iterative approach based on the Lekhnitskii stress functions with body forces and harmonic assumption of initial stress fields. The acceleration effect is introduced by adding body force terms to the equilibrium equations. The governing equations are obtained by applying variational principle in each process and solved by general eigenvalue problems to obtain homogeneous solutions, as well as to obtain particular solutions based on the forms of load and acceleration conditions. During the iterations, the stress oscillations can be gradually eliminated and the stress concentration can be predicted exactly located at the interfaces. Finally, an example of symmetrically layered heterostructure is presented under both in-plane acceleration and out-of-plane acceleration. It is found that both accelerations have significant effect on the edge normal and shear stresses which may further cause failure. The edge strength is also evaluated by calculating the edge average stress. This work may help to understand the acceleration effect on edge stresses and edge strength for heterostructures, as well as the trend of stress magnitude change caused by external accelerations.

Multiview Tensor Spectral Clustering via Co-Regularization.

Graph-based multi-view clustering encodes multi-view data into sample affinities to find consensus representation, effectively overcoming heterogeneity across different views. However, traditional affinity measures tend to collapse as the feature dimension expands, posing challenges in estimating a unified alignment that reveals both cross-view and inner relationships. To tackle this challenge, we propose to achieve multi-view uniform clustering via consensus representation co-regularization. First, the sample affinities are encoded by both popular dyadic affinity and recent high-order affinities to comprehensively characterize spatial distributions of the HDLSS data. Second, a fused consensus representation is learned through aligning the multi-view low-dimensional representation by co-regularization. The learning of the fused representation is modeled by a high-order eigenvalue problem within manifold space to preserve the intrinsic connections and complementary correlations of original data. A numerical scheme via manifold minimization is designed to solve the high-order eigenvalue problem efficaciously. Experiments on eight HDLSS datasets demonstrate the effectiveness of our proposed method in comparison with the recent thirteen benchmark methods.

Exact closed-form solution for buckling and free vibration of pipes conveying fluid with intermediate elastic supports

In this paper, the exact closed-form solution is given to investigate the influence of the intermediate elastic support on the buckling and free vibration of an elastically supported pipe. According to the Euler–Bernoulli beam theory, the mechanical model of the pipe is established. The exact equilibrium configuration is derived using the generalised function method without enforcing continuity conditions. A simple solution to the eigenvalue problem is formulated using the methods of complex mode superposition and Laplace transformation. The comparative study shows the differences in the supercritical vibration characteristics and highlights the limitations of previous studies. Parametric studies are carried out to investigate the influence of elastic support and intermediate support conditions on the equilibrium configuration, critical flow velocity, and natural frequency. The results demonstrate that the proposed closed-form solution can determine the support conditions that lead to the maximum critical flow velocity and natural frequency of a pipe with multiple intermediate supports. The maximum values are required to adjust the support conditions leading to the nodes of higher-order equilibrium configurations and complex modes. Furthermore, the natural frequencies of the pipe conveying supercritical fluid no longer satisfy the monotonicity for the support stiffness, the symmetry for the support position, and the ‘zero-point’ property for the support number.

Study on the buckling behavior of aluminum alloy sheets - before and after repaired with composite patches

In this paper, the compressive buckling behavior of aluminum alloy plates with an elliptical hole of various sizes is investigated. In order to improve the stability of these thin plates, T700/QY8911 composite laminate is used as a patch to repair the hole. The study included an analysis of the critical and post-critical behaviour using experimental and numerical methods. Experiments focus on buckling loads, post-buckling behavior and the relationship between sizes of holes and buckling load. Meanwhile, the buckling load and buckling mode are determined by finite element analysis, using linear analysis of eigenvalue problems modes, and then, the nonlinear analysis of structures with initiated geometrically imperfection is carried out, studying its post-buckling behavior, damage behavior and transfer of load. The results show that the buckling load of the open-hole specimen is related to size of opening. The existence of patch has a significant influence on stress distribution, and the buckling capability of repaired specimens is noticeably improved to the plate without a hole. And the compression experimental results are consistent with the numerical results, revealing that the developed finite element model of the structure is correct.

A contour integral-based method for nonlinear eigenvalue problems for semi-infinite photonic crystals

In this study, we introduce an efficient method for determining isolated singular points of two-dimensional semi-infinite and bi-infinite photonic crystals, equipped with perfect electric conductor and quasi-periodic mixed boundary conditions. This specific problem can be modeled by a Helmholtz equation and is recast as a generalized eigenvalue problem involving an infinite-dimensional block quasi-Toeplitz matrix. Through an intelligent implementation of cyclic structure-preserving matrix transformations, the contour integral method is elegantly employed to calculate the isolated eigenvalue and to extract a component of the associated eigenvector. Moreover, a propagation formula for electromagnetic fields is derived. This formulation enables rapid computation of field distributions across the expansive semi-infinite and bi-infinite domains, thus highlighting the attributes of edge states. The preliminary MATLAB implementation is available at https://github.com/FAME-GPU/2D_Semi-infinite_PhC.