General Eigenvalue Research Articles

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.

Dynamic and modal analysis of nearly incompressible structures with stabilised displacement-volumetric strain formulations

This paper presents a dynamic formulation for the simulation of nearly incompressible structures using a mixed finite element method with equal-order interpolation pairs. Specifically, the nodal unknowns are the displacement and the volumetric strain component, something that makes possible the reconstruction of the complete stain at the integration point level and thus enables the use of strain-driven constitutive laws. Furthermore, we also discuss the resulting eigenvalue problem and how it can be applied for the modal analysis of linear elastic solids. The article puts special emphasis on the stabilisation technique used, which becomes crucial in the resolution of the generalised eigenvalue problem. In particular, we prove that using a variational multiscale method assuming the sub-grid scales to lie in the finite element space orthogonal to that of the approximation, namely the Orthogonal Sub-Grid Scales (OSGS), results in a convenient linear and symmetric generalised eigenvalue problem. The correctness, convergence and performance of the method are proven by solving a series of two- and three-dimensional examples.

Match-based solution of general parametric eigenvalue problems

We describe a novel algorithm for solving general parametric (nonlinear) eigenvalue problems. Our method has two steps: first, high-accuracy solutions of non-parametric versions of the problem are gathered at some values of the parameters; these are then combined to obtain global approximations of the parametric eigenvalues. To gather the non-parametric data, we use non-intrusive contour-integration-based methods, which, however, cannot track eigenvalues that migrate into/out of the contour as the parameter changes. Special strategies are described for performing the combination-over-parameter step despite having only partial information on such migrating eigenvalues. Moreover, we dedicate a special focus to the approximation of eigenvalues that undergo bifurcations. Finally, we propose an adaptive strategy that allows one to effectively apply our method even without any a priori information on the behavior of the sought-after eigenvalues. Numerical tests are performed, showing that our algorithm can achieve remarkably high approximation accuracy.

Efficient Mixed-Precision Matrix Factorization of the Inverse Overlap Matrix in Electronic Structure Calculations with AI-Hardware and GPUs.

In recent years, a new kind of accelerated hardware has gained popularity in the artificial intelligence (AI) community which enables extremely high-performance tensor contractions in reduced precision for deep neural network calculations. In this article, we exploit Nvidia Tensor cores, a prototypical example of such AI-hardware, to develop a mixed precision approach for computing a dense matrix factorization of the inverse overlap matrix in electronic structure theory, S-1. This factorization of S-1, written as ZZT = S-1, is used to transform the general matrix eigenvalue problem into a standard matrix eigenvalue problem. Here we present a mixed precision iterative refinement algorithm where Z is given recursively using matrix-matrix multiplications and can be computed with high performance on Tensor cores. To understand the performance and accuracy of Tensor cores, comparisons are made to GPU-only implementations in single and double precision. Additionally, we propose a nonparametric stopping criteria which is robust in the face of lower precision floating point operations. The algorithm is particularly useful when we have a good initial guess to Z, for example, from previous time steps in quantum-mechanical molecular dynamics simulations or from a previous iteration in a geometry optimization.

Vibration of embedded restrained composite tube shafts with nonlocal and strain gradient effects

Torsional vibration response of a circular nanoshaft, which is restrained by the means of elastic springs at both ends, is a matter of great concern in the field of nano-/micromechanics. Hence, the complexities arising from the deformable boundary conditions present a formidable obstacle to the attainment of closed-form solutions. In this study, a general method is presented to calculate the torsional vibration frequencies of functionally graded porous tube nanoshafts under both deformable and rigid boundary conditions. Classical continuum theory, upgraded with nonlocal strain gradient elasticity theory, is employed to reformulate the partial differential equation of the nanoshaft. First, torsional vibration equation based on the nonlocal strain gradient theory is derived for functionally graded porous nanoshaft embedded in an elastic media via Hamilton’s principle. The ordinary differential equation is found by discretizing the partial differential equation with the separation of variables method. Then, Fourier sine series is used as the rotation function. The necessary Stokes' transformation is applied to establish the general eigenvalue problem including the different parameters. For the first time in the literature, a solution that can analyze the torsional vibration frequencies of functionally graded porous tube shafts embedded in an elastic media under general (elastic and rigid) boundary conditions on the basis of nonlocal strain gradient theory is presented in this study. The results obtained show that while the increase in the material length scale parameter, elastic media and spring stiffnesses increase the frequencies of nanoshafts, the increase in the nonlocal parameter and functionally grading index values decreases the frequencies of nanoshafts. The detailed effects of these parameters are discussed in the article.

A Power Method for Computing the Dominant Eigenvalue of a Dual Quaternion Hermitian Matrix

In this paper, we first study the projections onto the set of unit dual quaternions, and the set of dual quaternion vectors with unit norms. Then we propose a power method for computing the dominant eigenvalue of a dual quaternion Hermitian matrix. For a strict dominant eigenvalue, we show the sequence generated by the power method converges to the dominant eigenvalue and its corresponding eigenvector linearly. For a general dominant eigenvalue, we establish linear convergence of the standard part of the dominant eigenvalue. Based upon these, we reformulate the simultaneous localization and mapping problem as a rank-one dual quaternion completion problem. A two-block coordinate descent method is proposed to solve this problem. One block has a closed-form solution and the other block is the best rank-one approximation problem of a dual quaternion Hermitian matrix, which can be computed by the power method. Numerical experiments are presented to show the efficiency of our proposed power method.

On the stability analysis of a restrained functionally graded nanobeam in an elastic matrix with neutral axis effects

Abstract In this work, a general eigenvalue solution of an arbitrarily constrained nonlocal strain gradient nanobeam made of functionally graded material is presented for the first time for the stability response by the effect of the Winkler foundation. Elastic springs at the ends of the nanobeam are considered in the formulation, which have not been considered in most studies. In order to analyze deformable boundary conditions, linear equation systems are derived in terms of infinite power series by using the Fourier sine series together with the Stokes’ transform. The higher-order force boundary conditions are used to obtain a coefficient matrix including different end conditions, power-law index, elastic medium, and small-scale parameters. A general eigenvalue problem of technical interest, associated with nonlocal strain gradient theory, is mathematically evaluated and presented in detail. Parametric results are obtained to investigate the effects of material length scale parameter, Winkler stiffness, power-law index, nonlocal parameter, and elastic springs at the ends. In addition, the effects of the other higher-order elasticity theories simplified from nonlocal strain gradient theory are also investigated and some benchmark results are presented.

An application of stability charts to prediction of buckling instability in tapered columns via Galerkin’s method

This work tackles the mathematical modeling of buckling problem to obtain their critical loads in tapered columns subjected to concentrated and axial distributed loads. The governing model is a general eigenvalue problem that has no exact solution due to some new terms included. A semi-analytical technique satisfying the boundary conditions is proposed for the solution procedure. The minimum residual Galerkin’s method is suggested due to its effectiveness as a semi-analytical tool for the buckling problem to obtain the buckling shape modes by using admissible periodic functions. The study investigates the buckling instability and the responses of tapered columns with different periodic trial shape functions as approximations to the exact solutions. Based on the eigenvalue problem, Galerkin’s method is employed to obtain the transition curves to represent the critical loads. The stability charts (Ince–Strutt diagrams) among the parameters of the problem are proposed to explain the elastic stability of different tapered columns subjected to different shapes of cross sections and distributed weights. Consequently, the influences of the included parameters on the critical buckling loads are discussed. Among the different tapered columns presented, some parameters in the proposed distributions have a big influence on the critical buckling load and the creation of the instability regions in the chart for the clamped-clamped boundary conditions. The results are verified using the analytical solutions for some specific known problems.

General soliton solutions for the complex reverse space-time nonlocal mKdV equation on a finite background

Three kinds of Darboux transformations are constructed by means of the loop group method for the complex reverse space-time (RST) nonlocal modified Korteweg–de Vries equation, which are different from that for the PT symmetric (reverse space) and reverse time nonlocal models. The N-periodic, the N-soliton, and the N-breather-like solutions, which are, respectively, associated with real, pure imaginary, and general complex eigenvalues on a finite background are presented in compact determinant forms. Some typical localized wave patterns such as the doubly periodic lattice-like wave, the asymmetric double-peak breather-like wave, and the solitons on singly or doubly periodic waves are graphically shown. The essential differences and links between the complex RST nonlocal equations and their local or PT symmetric nonlocal counterparts are revealed through these explicit solutions and the solving process.

A universal variational quantum eigensolver for non-Hermitian systems

Many quantum algorithms are developed to evaluate eigenvalues for Hermitian matrices. However, few practical approach exists for the eigenanalysis of non-Hermintian ones, such as arising from modern power systems. The main difficulty lies in the fact that, as the eigenvector matrix of a general matrix can be non-unitary, solving a general eigenvalue problem is inherently incompatible with existing unitary-gate-based quantum methods. To fill this gap, this paper introduces a Variational Quantum Universal Eigensolver (VQUE), which is deployable on noisy intermediate scale quantum computers. Our new contributions include: (1) The first universal variational quantum algorithm capable of evaluating the eigenvalues of non-Hermitian matrices—Inspired by Schur’s triangularization theory, VQUE unitarizes the eigenvalue problem to a procedure of searching unitary transformation matrices via quantum devices; (2) A Quantum Process Snapshot technique is devised to make VQUE maintain the potential quantum advantage inherited from the original variational quantum eigensolver—With additional O(log2N)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(log_{2}{N})$$\\end{document} quantum gates, this method efficiently identifies whether a unitary operator is triangular with respect to a given basis; (3) Successful deployment and validation of VQUE on a real noisy quantum computer, which demonstrates the algorithm’s feasibility. We also undertake a comprehensive parametric study to validate VQUE’s scalability, generality, and performance in realistic applications.

Porosity effects on the dynamic response of arbitrary restrained FG nanobeam based on the MCST

Abstract In this study, two different general eigenvalue problems for nanobeams made of functionally graded material with pores in their sections according to Rayleigh beam theory using modified couple stress theory are established. Fourier sine series and Stokes transformation are used for the solution. First, the partial differential equation of motion of the problem is discretized into an ordinary differential equation. Then, the Fourier sine series of infinite series is substituted into this ordinary differential equation to determine the Fourier coefficient. Using the force boundary conditions of the system, Stokes’ transformation is performed at both ends to include elastic spring parameters. The unknown displacement terms are discretized to form two eigenvalue problems. By solving these eigenvalue problems, vibration frequencies for different boundary conditions can be found analytically. The variations of some parameters are discussed in a series of graphs.

On the conditioning of some structured generalized eigenvalue problems

This work continues the analysis of the conditioning of a Hankel structured generalized eigenvalue problem (GEP) started in [1]. The considered generalized eigenvalue problem appears in exponential analysis and sparse interpolation.
 We generalize the proof in [1] and add expressions for the relative condition numbers of two reformulations of the GEP, a reformulation as a Loewner GEP valid for general complex data, and a compression to a Hankel+Toeplitz GEP in the case of real data. Both reformulations are compared to the original Hankel GEP. The analysis is concluded with ample numerical illustrations.

Validation of the Townsend criterion for ignition of volume gas discharges

An eigenvalue problem governing ignition of volume discharges is formulated in the drift approximation with account of photoionization and the presence of multiple ion species with various reactions. The Townsend ignition (self-sustainment) criterion is deduced by partial integration of this eigenvalue problem neglecting the photoionization, and, in the case of electronegative gases, also the detachment. The Townsend criterion is applied to three examples of discharge ignition in high-pressure air, two of them referring to negative coronas in concentric-cylinder and point-to-plane configurations and one to a configuration with weakly nonuniform electric field. The comparison of the results obtained from different forms of the Townsend criterion with those given by the accurate numerical solution of the full eigenvalue problem shows that the neglect of the diffusion of the charged particles produces an error in the ignition voltage of the order of or less. The effect of photoionization for these configurations is also weak as expected. The lower and upper estimates of the effect of negative ions over the ignition voltage are obtained from different forms of the Townsend criterion. A form of the Townsend criterion, which employs an effective attachment coefficient in air and takes into account, in an approximate way, also the detachment, gives a virtually exact value of the inception voltage in all the cases considered. Practical recommendations are given on when one can use the Townsend self-sustainment criterion, and when one should resort to a numerical solution of the general eigenvalue problem governing discharge inception.

Deep Learning Solution of the Eigenvalue Problem for Differential Operators

Solving the eigenvalue problem for differential operators is a common problem in many scientific fields. Classical numerical methods rely on intricate domain discretization and yield nonanalytic or nonsmooth approximations. We introduce a novel neural network-based solver for the eigenvalue problem of differential self-adjoint operators, where the eigenpairs are learned in an unsupervised end-to-end fashion. We propose several training procedures for solving increasingly challenging tasks toward the general eigenvalue problem. The proposed solver is capable of finding the M smallest eigenpairs for a general differential operator. We demonstrate the method on the Laplacian operator, which is of particular interest in image processing, computer vision, and shape analysis among many other applications. In addition, we solve the Legendre differential equation. Our proposed method simultaneously solves several eigenpairs and can be easily used on free-form domains. We exemplify it on L-shape and circular cut domains. A significant contribution of this work is an analysis of the numerical error of this method. In particular an upper bound for the (unknown) solution error is given in terms of the (measured) truncation error of the partial differential equation and the network structure.

A Framework of Hybrid Transceiver Optimizations With Eigenvalue Constraints for Multi-Hop Networks

In this paper, we propose a general framework on the hybrid analog-digital transceiver design for multi-hop communications. For the inclusive purpose, a transceiver model unifying both linear and nonlinear transceivers has been taken into account. Various performance metrics, including the most representative capacity and weighted mean-squared error (MSE), have been investigated in a unified manner. In particular, to meet practical needs for the quality of services (QoS), a general eigenvalue power constraint model is introduced, which contains a sum power constraint and box eigenvalue constraints as special cases. Specifically, by carefully designing the auxiliary analog and digital beamformers, the multi-hop transceiver optimization is decomposed into a series of independent sub-problems, where the analog beamformers for different hops are completely decoupled. Based on that, this framework establishes a majorization-minimization (MM) based analog beamformer design algorithm, which is able to handle the complicated weighted unit-modulus matrix optimizations by finding their semi-closed-form solutions. Furthermore, an efficient waterfilling algorithm is proposed for the digital beamformer designs to deal with the difficulties of optimizations subject to the multiple eigenvalue power constraints. The numerical results are provided to demonstrate the performance advantages of the proposed framework.

Concentration of the number of intersections of random eigenfunctions on flat tori

We show that in two dimensional flat torus the number of intersections between random eigenfunctions of general eigenvalues and a given smooth curve is almost exponentially concentrated around its mean, even when the randomness is not gaussian.

On the General Sum Distance Spectra of Digraphs

Let G be a strongly connected digraph, and dG(vi,vj) denote the distance from the vertex vi to vertex vj and be defined as the length of the shortest directed path from vi to vj in G. The sum distance between vertices vi and vj in G is defined as sdG(vi,vj)=dG(vi,vj)+dG(vj,vi). The sum distance matrix of G is the n×n matrix SD(G)=(sdG(vi,vj))vi,vj∈V(G). For vertex vi∈V(G), the sum transmission of vi in G, denoted by STG(vi) or STi, is the row sum of the sum distance matrix SD(G) corresponding to vertex vi. Let ST(G)=diag(ST1,ST2,…,STn) be the diagonal matrix with the vertex sum transmissions of G in the diagonal and zeroes elsewhere. For any real number 0≤α≤1, the general sum distance matrix of G is defined as SDα(G)=αST(G)+(1−α)SD(G). The eigenvalues of SDα(G) are called the general sum distance eigenvalues of G, the spectral radius of SDα(G), i.e., the largest eigenvalue of SDα(G), is called the general sum distance spectral radius of G, denoted by μα(G). In this paper, we first give some spectral properties of SDα(G). We also characterize the digraph minimizes the general sum distance spectral radius among all strongly connected r-partite digraphs. Moreover, for digraphs that are not sum transmission regular, we give a lower bound on the difference between the maximum vertex sum transmission and the general sum distance spectral radius.

Improved bisection eigenvalue method for band symmetric Toeplitz matrices

We apply a general bisection eigenvalue algorithm, developed forHermitian matrices with quasiseparable representations, to the particular caseof real band symmetric Toeplitz matrices. We show that every band symmetricToeplitz matrix $T_q$ with bandwidth $q$ admits the representation$T_q=A_q+H_q$, where the eigendata of $A_q$ are obtained explicitly andthe matrix $H_q$ has nonzero entries only in two diagonal blocksof size $(q-1)\times (q-1)$. Based on this representation, one obtainsan interlacing property of the eigenvalues of the matrix $T_q$ and the knowneigenvalues of the matrix $A_q$. This allows us to essentially improve the performance of thebisection eigenvalue algorithm. We also present an algorithm tocompute the corresponding eigenvectors.

Size-dependent nonlinear post-buckling analysis of functionally graded porous Timoshenko microbeam with nonlocal integral models

Strain-driven (ɛD) and stress-driven (σD) two-phase local/nonlocal integral models (TPNIM) are applied to study the size-dependent nonlinear post-buckling behaviors of functionally graded porous Timoshenko microbeams. The differential governing equations and boundary conditions of motion are derived on the basis of the principle of minimum potential energy and expressed in nominal form. The integral relation between local and nonlocal stresses is transformed unitedly into equivalent differential form with constitutive constraints for ɛD- and σD-TPNIMs. Bending deflection and cross-sectional rotation are derived explicitly with Laplace transformation for linear buckling. Taking into account boundary conditions and constitutive constraints, a general eigenvalue problem is obtained to determine linear buckling mode shape (LBMS) and nominal buckling load (NBL). Local and nonlocal LBMS based Ritz–Galerkin methods and general differential quadrature method (GDQM) based Newton–Raphson’s method are utilized to obtain the numerical solutions for linear and nonlinear NBLs. Numerical results show that nonlocal LBMS based Ritz–Galerkin method and GDQM would lead to same prediction for linear NBLs as analytical method, and nonlocal LBMSs based Ritz–Galerkin and GDQM based Newton–Raphson’s method would lead to exact same prediction for nonlinear post-buckling loads. However, local LBMS based Ritz–Galerkin method is failed to provide accurate prediction for both linear NBLs and nonlinear post-buckling loads, through the difference between local and nonlocal LBMSs is not significant. The influence of nonlocal parameters, porous distribution patterns and boundary conditions on the linear buckling and nonlinear post-buckling behaviors are investigated numerically.

An advanced general dominant eigenvalue method of accelerating successive substitution during flash calculation for compositional reservoir model

The efficiency and accuracy of phase equilibrium calculations are essential in compositional reservoir models. Usually, a significant part of the computational effort in compositional reservoir simulations is spent on phase equilibrium calculations. The nonlinear nature of phase equilibrium calculations requires an iterative solution procedure. Although the successive substitution method (SSM) is robust and simple to implement, it suffers from slow convergence, especially near the critical point of the mixture. The general dominant eigenvalue method (GDEM) has been widely used to accelerate SSM, but its stability and efficiency deteriorate as the temperature and pressure approach the critical point. This paper proposes a modified form of GDEM to improve its performance in the near-critical region. The modifications have two aspects. First, the liquid phase fraction in the mixture is added as a variable when performing GDEM acceleration, improving both stability and efficiency. The second modification is a post-calibration step imposed to replace the conventional criterion, which is applied before triggering GDEM. With the help of the post-calibration step, the stability of the modified GDEM is ensured, and more importantly, the calculation efficiency can be improved. Numerical tests of three hydrocarbon mixtures, including different numbers of components, show that the stability of the modified GDEM is almost the same as SSM and that its calculation efficiency is much higher than SSM and the conventional GDEM. Cited as: Wang, X., Wei, D., Wang, X., Zhao, X., Li, J., Noetinger, B. An advanced general dominant eigenvalue method of accelerating successive substitution during flash calculation for compositional reservoir model. Advances in Geo-Energy Research, 2022, 6(3): 241-251. https://doi.org/10.46690/ager.2022.03.07