Eigenvalue Problem Research Articles

An inexact Matrix-Newton method for solving NEPv

In this paper, an inexact Newton method for solving real-valued nonlinear eigenvalue problems with eigenvector dependency (NEPv) is introduced that is able to solve the problem on a matrix level. Our main contribution is to derive a variant of Newton's method that uses global Krylov methods such as global GMRES to solve the linear operator equation necessary to compute the Newton correction in a matrix-free way. The advantages that this second order method has over the well-established SCF algorithm are explained and visualized by a variety of numerical experiments.

Superinertial Internal Tides Propagating along the Coast: Dynamics and Energetics Revealed through Topographic Modes

Abstract Internal tides are an important energy source for diapycnal mixing in the ocean interior. Subinertial internal tides are known to have a cross-sectional structure that is baroclinic in deep areas but more barotropic in shallower areas, making the conventional barotropic–baroclinic decomposition ineffective to separate internal tides from surface tides. To address this issue, Tanaka introduced a “topographic mode” which represents internal motions with a barotropic vertical structure and proposed a new energy diagram enabling us to quantify the generation of subinertial internal tides. Here, the applicability of this new energy diagram to superinertial internal tides at various latitudes is investigated by idealized numerical experiments with uniform surface tides passing over a continental slope with a bump in the coastline. The calculated results show that both the topographic and baroclinic modes are generated over the bump and propagate downstream along the coast, inseparably forming a topographically modified internal Kelvin wave whose cross-sectional structure closely resembles that obtained by solving a two-dimensional eigenvalue problem for a superinertial frequency. The energy conversion rate from the surface mode to the topographic mode is about half of the energy conversion rate from the surface mode to the baroclinic mode when the tidal frequency is slightly superinertial and maintains a nonnegligible contribution even when the tidal frequency is substantially superinertial. The findings of this study indicate that the new energy diagram can be applied seamlessly across the critical latitudes and possibly provides a global distribution of the internal tide generation significantly larger than the previous estimates. Significance Statement Quantifying the generation of internal tides is essential for accurate climate modeling, since they are a major energy source of vertical mixing in the ocean. It is known that diurnal internal tides change their characteristics poleward of 30° latitude, where their motion is dominated by horizontal rotation rather than vertical oscillation. This study demonstrates that a recently proposed energy diagram for quantifying the generation of diurnal internal tides poleward of 30° latitude is also effective equatorward of 30° latitude. This is because the rotational motion remains significant even equatorward of 30° latitude, being a fundamental component of the internal tides over a sloping seafloor. The application of this energy diagram will improve the global estimate of internal tide generation.

Analytical solutions for free vibrations of rectangular cuboid elastic lattices and their continuous approximations

This paper presents analytical solutions for the free vibration of an elastic cuboid lattice (rectangular parallelepiped) with sliding supports along its planar boundaries. The lattice model includes both central and angular interactions. The free vibration problem is solved by formulating a 3D difference eigenvalue problem, and exact solutions for the eigenfrequencies and eigenmodes are derived analytically using a trigonometric discrete displacement field. A cubic equation for the eigenfrequency squares is obtained, with solutions determined using Cardano's formula. The derived exact solutions for the finite cuboid lattice are validated by comparison with solutions obtained from a discrete algebraic method, calibrated for accurate stiffness and mass properties both within the lattice and at its boundaries. Furthermore, these exact solutions for the 3D lattice are benchmarked against analytical solutions for the corresponding 3D continuum, based on classical elasticity, lattice-based gradient elasticity, and lattice-based nonlocal elasticity theories, demonstrating their accuracy and reliability for free vibration analysis.

[formula omitted]-conforming triangular spectral element method for quad-curl problems

In this paper, we consider the H(curl2)-conforming triangular spectral element method to solve the quad-curl problems. We first explicitly construct the H(curl2)-conforming elements on triangles through the contravariant transform and the affine mapping from the reference element to physical elements. These constructed elements possess a hierarchical structure and can be categorized into the kernel space and non-kernel space of the curl operator. We then establish H(curl2)-conforming triangular spectral element spaces and the corresponding mixed formulated spectral element approximation scheme for the quad-curl problems and related eigenvalue problems. Subsequently, we present the best spectral element approximation theory in H(curl2;Ω)-seminorms. Notably, the degrees of polynomials in the kernel space solely impact the convergence rate of the (L2(Ω))2-norm of uh, without affecting the semi-norm of H(curl;Ω) and H(curl2;Ω). This observation enables us to derive eigenvalue approximations from either the upper or lower side by selecting different degrees of polynomials for the kernel space and non-kernel space of the curl operator. Finally, numerical results demonstrate the effectiveness and efficiency of our method.

Monotonicity, asymptotics and level sets for principal eigenvalues of some elliptic operators with shear flow

We investigate the joint effects of diffusion and advection on principal eigenvalues of some elliptic operators with shear flow. Some monotonicity and asymptotic behaviors of principal eigenvalues, with respect to diffusion rate and flow amplitude, are established. These analyses lead to a classification of topological structures of level sets for principal eigenvalues, as a function of diffusion rate and flow amplitude. Our analytical results provide a unifying viewpoint to understand mixing enhancement and dispersal-induced growth, which are apparently two unrelated phenomena, one in fluid mechanics and the other in population dynamics. In our analysis, some limiting Hamilton-Jacobi equations, blowup argument and limiting generalized principal eigenvalue problems play critical roles.

Pulses in singularly perturbed reaction-diffusion systems with slowly mixed nonlinearity.

This article is concerned with the existence and spectral stability of pulses in singularly perturbed two-component reaction-diffusion systems with slowly mixed nonlinearity. In this paper, the slow nonlinearity is referred to be "mixed" in the sense that it is generated by a trigonometric function multiplied by a power function. We demonstrate via geometric singular perturbation theory that this model can support both the single-pulse and the double-hump solutions. The presence of the slowly mixed nonlinearity complicates the stability analysis on pulses, since the conditions that govern their stability can no longer be explicitly computed. We remove this difficulty by introducing the hypergeometric functions followed by a comparison theorem. By doing so, the "slow-fast" eigenvalues can be determined via the nonlocal eigenvalue problem method. We prove that the double-hump solution is always unstable, while the single-pulse solution can be stable under certain parameter conditions.

Exact solution to the Wheeler-DeWitt equation: Early and current Universe

In this paper, we present exact solutions to the Wheeler-DeWitt equation in two different scenarios: the early Universe, where the ordering parameter of kinetic energy is important, and the current Universe, where the ordering parameter effect is negligible. To make the exact solutions as general as possible, we incorporate as many different types of energy density as possible into the Hamiltonian, including baryonic and non-baryonic matter (dark matter), radiation, vacuum, and quintessence (dark energy). In the early Universe scenario, we obtain exact solutions in terms of the Biconfluent Heun functions, whereas in the current Universe, the exact solutions are given in terms of the Triconfluent Heun functions. Furthermore, by applying the polynomial conditions to each case, we obtain a constraint equation that supports the notion that the Wheeler-DeWitt equation can be viewed as an eigenvalue problem for the cosmological constant.

Randomized methods for computing joint eigenvalues, with applications to multiparameter eigenvalue problems and root finding

AbstractIt is well known that a family of $${n}\times {n}$$ n × n commuting matrices can be simultaneously triangularized by a unitary similarity transformation. The diagonal entries of the triangular matrices define the n joint eigenvalues of the family. In this work, we consider the task of numerically computing approximations to such joint eigenvalues for a family of (nearly) commuting matrices. This task arises, for example, in solvers for multiparameter eigenvalue problems and systems of multivariate polynomials, which are our main motivations. We propose and analyze a simple approach that computes eigenvalues as one-sided or two-sided Rayleigh quotients from eigenvectors of a random linear combination of the matrices in the family. We provide some analysis and numerous numerical examples, showing that such randomized approaches can compute semisimple joint eigenvalues accurately and lead to improved performance of existing solvers.

Динамічна нестійкість при взаємодії композитної оболонки, виготовленої за адитивними технологіями, з газовим потоком

The dynamic instability of a composite cylindrical-conical thin-walled structure interacting with a supersonic gas flow is analyzed. This structure consists of three layers. The middle layer is manufactured by FDM additive technologies from ULTEM material. The top and bottom layers are manufactured from carbon-filled plastic. Free linear vibrations of the thin-walled structure are studied by Rayleigh?Ritz semi-analytical method to obtain a model of dynamic instability. The free linear vibrations are analyzed numerically. The obtained eigenfrequencies and eigenmodes are in close agreement with the data obtained by the ANSYS commercial software. The calculated eigenmodes were used to construct a model of composite cylindrical-conical shell instability. This model of instability is a system of ordinary linear differential equations in the generalized coordinates of the thin-walled structure. The supersonic gas flow is described by a piston theory, which accounts for the angle of attack. The study of the dynamic instability of the composite cylindrical-conical shell reduces to analyzing the trivial equilibrium instability of the system of ordinary differential equations. Characteristic exponents are calculated to analyze the stability of the trivial solution. These characteristic exponents are calculated from an eigenvalue problem. If the angle of attack is 12° and the Mach number is small, the minimal value of the critical pressure is observed for three circular waves. If the Mach number is increased, the minimal critical pressure is observed for four and five circular waves. If the angle of attack is 6° and the Mach number is small, the minimal critical pressure is observed for two circular waves. If the Mach number is increased, the minimal critical pressure is observed for three and four circular waves. The dynamic stability is lost for eigenmodes with a small number of circular waves

Co-designing ab initio electronic structure methods on a RISC-V vector architecture

Ab initio electronic structure applications are among the most widely used in High-Performance Computing (HPC), and the eigenvalue problem is often their main computational bottleneck. This article presents our initial efforts in porting these codes to a RISC-V prototype platform leveraging a wide Vector Processing Unit (VPU). Our software tester is based on a mini-app extracted from the ELPA eigensolver library. The user-space emulator Vehave and a RISC-V vector architecture implemented on an FPGA were tested. Metrics from both systems and different vectorisation strategies were extracted, ranging from the simplest and most portable one (using autovectorisation and assisting this by fusing loops in the code) to the more complex one (using intrinsics). We observed a progressive reduction in the number of vectorised instructions, executed instructions and computing cycles with the different methodologies, which will lead to a substantial speed-up in the calculations. The obtained outcomes are crucial in advancing the porting of computational materials and molecular science codes to (post)-exascale architectures using RISC-V-based technologies fully developed within the EU. Our evaluation also provides valuable feedback for hardware designers, engineers and compiler developers, making this use case pivotal for co-design efforts.

Extraction of Spectra in the Shell Model MonteCarlo Method Using Imaginary-Time Correlation Matrices.

Nuclear energy levels are usually calculated using conventional diagonalization methods in the framework of the configuration-interaction (CI) shell model but these methods are prohibited in very large model spaces. The shell model MonteCarlo (SMMC) method is a powerful technique for calculating thermal and ground-state observables of nuclei in very large model spaces, but it is challenging to extract nuclear spectra in this approach. We present a novel method to extract within SMMC low-lying energy levels for given values of a set of good quantum numbers such as spin and parity. The method is based on imaginary-time correlation matrices (ITCMs) of one-body densities that satisfy asymptotically a generalized eigenvalue problem. We validate the method in a light nucleus that allows comparison with exact diagonalization results of the CI shell-model Hamiltonian. The method is broadly applicable to quantum many-body systems in other disciplines.

A Finite Integral Transform-Based Generalized Eigenvalue Formulation for Buckling of Orthotropic Rectangular Plates with Rotational Restraints

Although the finite integral transform (FIT) method has been developed for buckling analysis of rectangular thin plates, the existing formulation involves solving complex nonlinear determinantal equations, leading to potentially questionable numerical results. To address this challenge, a new FIT-based formulation is established that transforms the solution of a complex nonlinear equation into that of a straightforward generalized eigenvalue problem. New analytical solutions for buckling of orthotropic rectangular thin plates with rotational restraints are obtained. The solution procedure is implemented via the following four steps: imposing the two-dimensional FIT on the governing equation; inputting deflection conditions of four edges, by which the relations between the transformed deflections and specific unknowns are provided; imposing the one-dimensional FIT on the elastic conditions to replace the unknowns with the transformed deflections, a generalized eigenvalue problem is constructed; determining the final analytical solutions by solving the generalized eigenvalue problem. Comprehensive highly accurate buckling load/mode solutions for typical rotationally restrained orthotropic plates are presented as benchmarks. The effects of boundary conditions, rotational fixity factors, and loading ratios on the buckling behaviors of orthotropic plates are expediently investigated adopting the present solutions.

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.

A-priori and a-posteriori error estimates for discontinuous Galerkin method of the Maxwell eigenvalue problem

This paper is devoted to a-priori and a-posteriori error analysis of discontinuous Galerkin (DG) method for the Maxwell eigenvalue problem. The discrete compactness of DG space is proved so that the Babuška and Osborn spectral approximation theory can be applicable in the a-priori error analysis. Then we prove the optimal error estimates for DG eigenfunctions in mesh-dependent norm and DG eigenvalues. A special contribution of this work is to prove that the error in L2-norm for smooth eigenfunctions is of higher order than that in mesh-dependent norm, so that the DG eigenvalues can approximate the true eigenvalues from upper. Another contribution of this work is to provide a-posteriori error analysis for the DG method. A reliable a-posteriori error estimator is analyzed. The upper bound property of DG eigenvalues and the robustness of adaptive methods are verified through numerical experiments.

Optimal weighting strategies for maximizing contrast-to-noise ratio in photon counting CT images.

Photon counting detectors (PCDs) with energy discrimination capabilities have the potential to generate grayscale CT images with improved contrast-to-noise ratio (CNR) through optimal weighting of their spectral measurements. This study evaluates the CNR performance of grayscale CT projections and images generated from spectral measurements of PCDs using three energy-weighting strategies: pre-log weighting, post-log weighting, and material decomposition (MD) weighting. This study provides the expressions of optimal weights and maximum achievable CNR of these energy-weighting strategies, which only require the knowledge of detected bin counts and do not require information of PCD energy responses or imaging techniques. We defined and solved a generalized eigenvalue problem to obtain the maximum achievable CNR in the projection domain for low-contrast tasks using three energy-weighting strategies: pre-log weighting (weighted sum of energy bin counts), post-log weighting (weighted sum of line integrals), and MD weighting (weighted sum of basis material thicknesses, which is equivalent to virtual monoenergetic images [VMIs]). These expressions only contain energy bin counts from PCD measurements. We used a realistic PCD energy response model to simulate the detected bin counts and conducted Monte Carlo simulations of different contrast tasks and phantoms to evaluate the projection- and image-domain CNR performance of these energy-weighting strategies. Additionally, the total counts method (a special case of pre-log weighting with unity weights) was included for comparison. We also conducted Gammex head and body phantom scans on an edge-on-irradiated silicon PCCT prototype to evaluate the image-domain CNR performance of these energy-weighting strategies. The results show that pre-log, post-log, and MD weighting strategies generate approximately equal projection-domain maximum achievable CNR, with a difference of less than 2%, and outperform the total counts method. These three energy-weighting strategies also generate approximately equal image-domain maximum CNR when the contrast task is located at the center of a homogeneous phantom. Pre-log weighting generates the highest image-domain CNR for an off-center contrast task location or inhomogeneous phantoms while also outperforming the total counts method. We derived the expression of projection-domain maximum achievable CNR using three energy-weighting strategies. Our results suggest that using pre-log weighting strategies enables fast grayscale CT image generation with high CNR from spectral PCD measurements for inhomogeneous phantoms and off-center region of interests (ROIs).

Nonlinear Model for LTB Prediction of RHS Beams Under Combined Axial and Bending Loads With Distortional Deformation Allowed

ABSTRACTThe aim of the present study is to develop an efficient and accurate analytical model for the lateral torsional buckling (LTB) prediction of RHS beams. To this end, a coherent nonlinear geometric theory is formulated assuming large torsion angles. This theory is based on a new kinematic model that incorporates the distortion deformations in addition to the bending and torsion ones. Ritz's method is employed to discretize the governing equilibrium equations basing on trigonometric shape functions, and then the eigenvalue problem is defined according to the fundamental state. The Buckling loads are obtained by imposing the singularity condition of the tangential stiffness matrix. The numerical investigation is carried out to prove the accuracy of the proposed model in LTB load evaluation, when compared with the nonlinear shell finite element simulation using Abaqus software. The numerical results reveal that the classical linear models associated with moderate torsion angles, such as those provided by Generalized Beam Theory (GBT) and the classical eigenvalue shell finite element method, tend to underestimate the correct value of LTB loads of simply supported RHS beams. In accordance with linear and non‐linear models, the numerical study is focused on the impact of load height, intensity of introduced compressive load and section ratio b/h in the LTB loads prediction.

An Improved Two‐Step Method for Inverse Eigenvalue Problems

ABSTRACTIn this article, we investigate the numerical solutions of the inverse eigenvalue problem (IEP). Motivated by the techniques of Aishima in 2018, we propose here an improved two‐step method for the IEP. Compared with other existing methods, the proposed method has higher convergence order and/or requires less operations. Under the assumption that the relative generalized Jacobian matrices at a solution are nonsingular, the proposed method is proved to be convergent with cubic root‐convergence rate. Numerical examples demonstrate the effectiveness of the improved method.

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.

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.

An Energy Approach to the Modal Identification of a Variable Thickness Quartz Crystal Plate

The primary objective of modal identification for variable thickness quartz plates is to ascertain their dominant operating mode, which is essential for examining the vibration of beveled quartz resonators. These beveled resonators are plate structures with varying thicknesses. While the beveling process mitigates some spurious modes, it still presents challenges for modal identification. In this work, we introduce a modal identification technique based on the energy method. When a plate with variable thickness is in a resonant state of thickness–shear vibration, the proportions of strain energy and kinetic energy associated with the thickness–shear mode in the total energy reach their peak values. Near this frequency, their proportions are the highest, aiding in identifying the dominant mode. Our research was based on the Mindlin plate theory, and appropriate modal truncation were conducted by retaining three modes for the coupled vibration analysis. The governing equation of the coupled vibration was solved for eigenvalue problem, and the modal energy proportions were calculated based on the determined modal displacement and frequency. Finally, we computed the eigenvalue problems at different beveling time, as well as the modal energies associated with each mode. By calculating the energy proportions, we could clearly identify the dominant mode at each frequency. Our proposed method can effectively assist engineers in identifying vibration modes, facilitating the design and optimization of variable thickness quartz resonators for sensing applications.