Eigenvalue Solver Research Articles

Damage detection of multi-storeyed shear structure using sparse and noisy modal data

In the present paper, a method for identifying damage in a multi storeyed shear building structure is presented using minimum number of modal parameters of the structure. A damage at any level of the structure may lead to a major failure if the damage is not attended at appropriate time. Hence an early detection of damage is essential. The proposed identification methodology requires experimentally determined sparse modal data of any particular mode as input to detect the location and extent of damage in the structure. Here, the first natural frequency and corresponding partial mode shape values are used as input to the model and results are compared by changing the sensor placement locations at different floors to conclude the best location of sensors for accurate damage identification. Initially experimental data are simulated numerically by solving eigen value problem of the damaged structure with inclusion of random noise on the vibration characteristics. Reliability of the procedure has been demonstrated through a few examples of multi storeyed shear structure with different damage scenarios and various noise levels. Validation of the methodology has also been done using dynamic data obtained through experiment conducted on a laboratory scale steel structure.

Simulation of birefringence effects on the dominant transversal laser resonator mode caused by anisotropic crystals.

Birefringence effects can have a significant influence on the polarization state as well as on the transversal mode structure of laser resonators. This work introduces a flexible, fast and fully vectorial algorithm for the analysis of resonators containing homogeneous, anisotropic optical components. It is based on a generalization of the Fox and Li algorithm by field tracing, enabling the calculation of the dominant transversal resonator eigenmode. For the simulation of light propagation through the anisotropic media, a fast Fourier Transformation (FFT) based angular spectrum of plane waves approach is introduced. Besides birefringence effects, this technique includes intra-crystal diffraction and interface refraction at crystal surfaces. The combination with numerically efficient eigenvalue solvers, namely vector extrapolation methods, ensures the fast convergence of the method. Furthermore a numerical example is presented which is in good agreement to experimental measurements.

Eigenspectra and mode coalescence of temporal instability in two-phase channel flow

The stability of two immiscible fluids with different densities and viscosities is examined for channel flow. A multi-domain Chebyshev collocation spectral method is used for solving the coupled Orr-Sommerfeld stability equations for the entire spectrum of eigenvalues and associated eigenfunctions. Numerical solution of the eigenvalue problem is obtained with the QZ eigenvalue solver and is validated with analytical results derived in the long and short wave limits. A parametric study is carried out to investigate the spectral characteristics and eigenfunction structures related to the shear and interfacial modes of instability. The interactions and mode switching between the two instability modes are investigated. Under certain conditions, the two modes are found to become unstable simultaneously. Mode coalescence can occur in either the stable or the unstable part of the spectrum. In general, the eigenfunction of the most dangerous mode is observed to vary sharply in the neighborhood of the interface and the critical points.

A type of multi-level correction scheme for eigenvalue problems by nonconforming finite element methods

In this paper, a type of multi-level correction scheme is proposed to solve eigenvalue problems by nonconforming finite element methods. With this new scheme, the accuracy of eigenpair approximations can be improved after each correction step which only needs to solve a source problem on finer finite element space and an eigenvalue problem on a coarse finite element space. This correction scheme can improve the efficiency of solving eigenvalue problems by the nonconforming finite element method. Furthermore, the same as the direct eigenvalue solving by the nonconforming finite element method, this multi-level correction method can also produce the lower-bound approximations of the eigenvalues.

A multigrid method for eigenvalue problems based on shifted-inverse power technique

A multigrid method is proposed to solve eigenvalue problems by means of the finite element method based on the shifted-inverse power iteration technique. With this scheme, solving eigenvalue problem is transformed to solving a series of nonsingular boundary value problems on multilevel meshes. As replacing the difficult eigenvalue solving by an easier solving of boundary value problems, the multigrid way can improve the overall efficiency of the eigenvalue problem solving. Some numerical experiments are presented to validate the efficiency of this new method.

Linear Stability of Double Diffusive Convection of Hadley-prats Flow with Viscous Dissipation in a Porous Media

Linear stability analysis of double diffusive convection in a horizontal fluid saturated porous layer has been carried out. Effects of viscous dissipation, horizontal mass flow are taken into account. Combined effects of horizontal mass flow, viscous dissipation may cause instability in the fluid system. To carry out linear stability analysis, disturbances are the form of longitudinal rolls, transverse rolls have been considered. In order to solve eigenvalue problem numerically, Runge-Kutta and shooting methods have been employed for the case of longitudinal rolls. Chebyshev-Tau method has been used for the case of transverse rolls. Critical wave number aC , critical vertical thermal Rayleigh number RaC are evaluated for assigned values of flow governing parameters. For the onset of convection, physical explanation is given.

Neural network iterative diagonalization method to solve eigenvalue problems in quantum mechanics.

We propose a multi-layer feed-forward neural network iterative diagonalization method (NNiDM) to compute some eigenvalues and eigenvectors of large sparse complex symmetric or Hermitian matrices. The NNiDM algorithm is developed by using the complex (or real) guided spectral transform Lanczos (cGSTL) method, thick restart technique, and multi-layered basis contraction scheme. Artificial neurons (or nodes) are defined by a set of formally orthogonal Lanczos polynomials, where the biases and weights are dynamically determined through a series of cGSTL iterations and small matrix diagonalizations. The algorithm starts with one random vector. The last output layer produces wanted eigenvalues and eigenvectors near a given reference value via a linear transform diagonalization approach. Since the algorithm uses the spectral transform technique, it is capable of computing interior eigenstates in dense spectrum regions. The general NNiDM algorithm is applied for calculating energies, widths, and wavefunctions of two typical molecules HO2 and CH4 as examples.

Nonsymmetric Preconditioning for Conjugate Gradient and Steepest Descent Methods 1

We numerically analyze the possibility of turning off post-smoothing (relaxation) in geometric multigrid when used as a preconditioner in conjugate gradient linear and eigenvalue solvers for the 3D Laplacian. The geometric Semicoarsening Multigrid (SMG) method is provided by the hypre parallel software package. We solve linear systems using two variants (standard and flexible) of the preconditioned conjugate gradient (PCG) and preconditioned steepest descent (PSD) methods. The eigenvalue problems are solved using the locally optimal block preconditioned conjugate gradient (LOBPCG) method available in hypre through BLOPEX software. We observe that turning off the post-smoothing in SMG dramatically slows down the standard PCG-SMG. For flexible PCG and LOBPCG, our numerical results show that post-smoothing can be avoided, resulting in overall acceleration, due to the high costs of smoothing and relatively insignificant decrease in convergence speed. We numerically demonstrate for linear systems that PSD-SMG and flexible PCG-SMG converge similarly if SMG post-smoothing is off. We experimentally show that the effect of acceleration is independent of memory interconnection. A theoretical justification is provided.

New Algorithms for Computing the Real Structured Pseudospectral Abscissa and the Real Stability Radius of Large and Sparse Matrices

We present two new algorithms for investigating the stability of large and sparse matrices subject to real perturbations. The first algorithm computes the real structured pseudospectral abscissa and is based on the algorithm for computing the pseudospectral abscissa proposed by Guglielmi and Overton [SIAM J. Matrix Anal. Appl., 32 (2011), pp. 1166--1192]. It entails finding the rightmost eigenvalues for a sequence of large matrices, and we demonstrate that these eigenvalue problems can be solved in a robust manner by an unconventional eigenvalue solver. We also develop an algorithm for computing the real stability radius of a real and stable matrix, which utilizes a recently developed technique for detecting the loss of stability in a large dynamical system. Both algorithms are tested on large and sparse matrices.

Computation of Complex Eigenmodes for Resonators Filled With Gyrotropic Materials

In this thesis, the numerical computation of complex eigenmodes of cavity resonators filled with magnetically biased gyrotropic material is demonstrated. For this purpose, a dedicated solver based on the Finite Integration Technique (FIT) has been developed, efficiently implemented as well as successfully verified. Gyrotropic field problems arise, for instance, for the calculation of the resonance frequencies of ferrite-loaded resonators like the GSI SIS 18 cavity. Ferrites exhibit gyromagnetic properties with an anisotropic permeability, which furthermore depends both on frequency and bias field. Similarly, gyroelectric material such as magnetized plasmas can be described by a frequency- and bias field dependent permittivity tensor. Since these material tensors affect the system matrix of the eigenvalue problem, a dedicated solver is required. In this thesis, the FIT with a hexaedral staircase filling is employed for discretization. In the standard formulation, it is, however, limited to diagonally anisotropic materials. Hence, as one of the goals of this thesis, the FIT has been extended to gyromagnetic as well as gyroelectric materials in frequency domain. The derived expressions for the non-diagonal material matrices are fully consistent with the standard FIT when applied to non-gyrotropic materials. Moreover, their structure is manifestly Hermitian in the lossless case, even for non-equidistant grids. Due to the above-mentioned material requirements, the newly developed solver consists of two components: The first one is a magnetostatic solver based on the H_i-algorithm supporting nonlinear material to calculate the magnetic field excited by the bias current. Having obtained the field distribution, the material properties are evaluated locally in each mesh cell at the specified working point. The second component is a Jacobi-Davidson type eigenvalue solver for the iterative solution of the nonlinear eigenproblem. To be capable of handling material losses, the eigensolver also supports non-Hermitian eigenproblems. What is more, efficient parallel computing on machines with distributed memory is possible. To this end, an ordering of the FIT-DOFs different from the standard scheme is implemented, which results in an increased computation to communication ratio. Furthermore, all DOFs that vanish a priori due to several reasons are completely removed from the vectors and matrices. All in all, gyrotropic eigenproblems discretized with several millions of mesh cells can be solved in a reasonable time by the developed solver. The validity of the numerically obtained results is confirmed by thorough comparisons with (semi-)analytical calculations. As an application example, an eigenmode analysis of the GSI SIS 18 cavity is carried out. Since the required material data are not available in the data sheet of the manufacturer, designated measurements of the magnetic characteristics of the Ferroxcube 8C12m ferrite ring cores, which are installed inside the GSI SIS 18 cavities, were performed. Among these characteristics are the complex permeability as a function of frequency and bias magnetic field strength at low radio-frequency power levels as well as the B-H curve. The measurement methods together with the detailed data analysis including the presentation of evaluated data are supplemented to this thesis. The scalar, isotropic permeability retrieved this way is used for the cavity simulations. The obtained values for the resonance frequency and quality factor for the fundamental mode are in accordance with available measurement data. To demonstrate the further potential of the solver, also higher-order modes are investigated and an outlook on possibly advantageous 2-directional bias schemes is given.

Improving robustness of the FEAST algorithm and solving eigenvalue problems from graphene nanoribbons

AbstractWe consider the FEAST eigensolver, introduced by Polizzi in 2009 [5]. We describe an improvement concerning the reliability of the algorithm and discuss an application in the solution of eigenvalue problems from graphene modeling. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A Linear-Complexity Finite-Element-Based Eigenvalue Solver for Efficient Analysis of 3-D On-Chip Integrated Circuits

This letter presents a linear-complexity finite-element-based eigenvalue solver for efficient analysis of 3-D on-chip integrated circuits. In this solver, from the original 3-D quadratic eigenvalue problem governing the on-chip circuits, we formulate a new generalized eigenvalue problem to efficiently compute the eigenvalues and eigenvectors of physical interest. We also develop an efficient linear-complexity solution for the matrix equation involved in the solution of the generalized eigenvalue problem. Numerical results demonstrate the accuracy and efficiency of the proposed eigenvalue solver.

Stability analysis of thick piezoelectric metal based FGM plate using first order and higher order shear deformation theory

This paper presents the stability analysis of functionally graded plate integrated with piezoelectric actuator and sensor at the top and bottom face, subjected to electrical and mechanical loading. The finite element formulation is based on first order and higher order shear deformation theory, degenerated shell element, von-Karman hypothesis and piezoelectric effect. The equation for static analysis is derived by using the minimum energy principle and solutions for critical buckling load is obtained by solving eigenvalue problem. The material properties of the functionally graded plate are assumed to be graded along the thickness direction according to simple power law function. Two types of boundary conditions are used, such as SSSS (simply supported) and CSCS (simply supported along two opposite side perpendicular to the direction of compression and clamped along the other two sides). Sensor voltage is calculated using present analysis for various power law indices and FG (functionally graded) material gradations. The stability analysis of piezoelectric FG plate is carried out to present the effects of power law index, material variations, applied mechanical pressure and piezo effect on buckling and stability characteristics of FG plate.

Local k-proximal plane clustering

k-Plane clustering (kPC) and k-proximal plane clustering (kPPC) cluster data points to the center plane, instead of clustering data points to cluster center in k-means. However, the cluster center plane constructed by kPC and kPPC is infinitely extending, which will affect the clustering performance. In this paper, we propose a local k-proximal plane clustering (LkPPC) by bringing k-means into kPPC which will force the data points to center around some prototypes and thus localize the representations of the cluster center plane. The contributions of our LkPPC are as follows: (1) LkPPC introduces localized representation of each cluster center plane to avoid the infinitely confusion. (2) Different from kPPC, our LkPPC constructs cluster center plane that makes the data points of the same cluster close to both the same center plane and the prototype, and meanwhile far away from the other clusters to some extent, which leads to solve eigenvalue problems. (3) Instead of randomly selecting the initial data points, a Laplace graph strategy is established to initialize the data points. (4) The experimental results on several artificial datasets and benchmark datasets show the effectiveness of our LkPPC.

Resonant dynamics of arbitrarily shaped meta-atoms

Meta-atoms, nanoantennas, plasmonic particles, and other small scatterers are commonly modeled in terms of their modes. However these modal solutions are seldom determined explicitly, due to the conceptual and numerical difficulties in solving eigenvalue problems for open systems with strong radiative losses. Here these modes are directly calculated from Maxwell's equations expressed in integral operator form, by finding the complex frequencies which yield a homogeneous solution. This gives a clear physical interpretation of the modes, and enables their conduction or polarization current distribution to be calculated numerically for particles of arbitrary shape. By combining the modal current distribution with a scalar impedance function, simple yet accurate models of scatterers are constructed which describe their response to an arbitrary incident field over a broad bandwidth. These models generalize both equivalent-dipole and and equivalent-circuit models to finite-sized structures with multiple modes. They are applied here to explain the frequency-splitting for a pair of coupled split rings, and the accompanying change in radiative losses. The approach presented in this paper is made available in an open-source code.

The 10,240‐member ensemble Kalman filtering with an intermediate AGCM

AbstractThe local ensemble transform Kalman filter (LETKF) with an intermediate atmospheric general circulation model (AGCM) is implemented with the Japanese 10 petaflops (floating point operations per second) “K computer” for large‐ensemble simulations of 10,240 members, 2 orders of magnitude greater than the typical ensemble size of about 100. The computational challenge includes the eigenvalue decomposition of 10,240 × 10,240 dense covariance matrices at each grid point. Using the efficient eigenvalue solver for the K computer, the LETKF computations are accelerated by a factor of 8, allowing a 3 week experiment of 10,240‐member LETKF with an intermediate AGCM for the first time. The flow‐dependent 10,240‐member ensemble revealed meaningful long‐range error correlations at continental scales. The surface pressure error correlation shows teleconnection patterns like the Pacific North American pattern. Specific humidity error correlation shows continental scale wave trains. Investigations with different ensemble sizes suggest that at least several hundred members be necessary to capture these continental scale error correlations.

Determining bound states in a semiconductor device with contacts using a nonlinear eigenvalue solver

We present a nonlinear eigenvalue solver enabling the calculation of bound solutions of the Schrodinger equation in a system with contacts. We discuss how the imposition of contacts leads to a nonlinear eigenvalue problem and discuss the numerics for a one- and two-dimensional potential. We reformulate the problem so that the eigenvalue problem can be efficiently solved by the recently proposal rational Krylov method for nonlinear eigenvalue problems, known as NLEIGS. In order to improve the convergence of the method, we propose a holomorphic extension such that we can easily deal with the branch points introduced by a square root. We use our method to determine the bound states of the one-dimensional Poschl---Teller potential, a two-dimensional potential describing a particle in a canyon and the multi-band Hamiltonian of a topological insulator.

New Pogo Analysis Method Using Rational Fitting and Three-Dimensional Tank Modeling

Considering the flexibility of a liquid rocket propellant tank and the importance of the relationship between oscillatory pressure and outflow near the tank outlet, a three-dimensional modeling technique is developed to provide a more accurate tank simulation. This paper also presents a rational function fitting method that could be used to transform transcendental equations of a feed-line transfer function into a homogeneous second-order matrix form. It is shown that this form of feed-line transfer function can be easily coupled together with governing equations of a liquid rocket structural system, which could be further solved for system stability using mature eigenvalue solvers. By comparison with the telemetry data and traditional methods, this pogo stability method proves to be highly accurate and efficient.

A multigrid method for eigenvalue problem

A multigrid method is proposed to solve the eigenvalue problem by the finite element method based on the combination of the multilevel correction scheme for the eigenvalue problem and the multigrid method for the boundary value problem. In this scheme, solving eigenvalue problem is transformed to a series of solutions of boundary value problems by the multigrid method on multilevel meshes and a series of solutions of eigenvalue problems on the coarsest finite element space. Besides the multigrid scheme, all other efficient iteration methods can also serve as the linear algebraic solver for the associated boundary value problems. The total computational work of this scheme can reach the optimal order as same as solving the corresponding boundary value problem. Therefore, this type of iteration scheme improves the overall efficiency of the eigenvalue problem solving. Some numerical experiments are presented to validate the efficiency of the method.

Dynamics of Two-dimensional Electron Gas in Non-uniform magnetic field

We have theoretically studied dynamics of the two-dimensional electron system (2DES) placed in a strong laterally non-uniform magnetic field, which appears due to ferromagnetic film on the top of heterostructure. It is shown that lateral inhomogeneity of a strong magnetic field allows itself "magnetic gradient" or special magnetic-edge magnetoplasmons. This mechanism is different from usual "density gradient" edge magnetoplasmons. We have solved self-consistently Poisson equation for non-uniform density distribution of the 2DES for realistic heterostructure together with hydrodynamic equation of 2D Fermi liquid. As a result eigen value problem has been obtained that corresponds to the motion of charge density wave perpendicular to magnetic gradient. It is shown that for non-monotonic distribution of magnetic field "magnetic gradient" magnetoplasmons may move in both directions. To solve eigen value problem we have compared two types of numerical approaches: first is grid method that diagonalizes large Hermitian matrix and second is semi-analytical approach that expand each eigen mode on the set of orthogonal functions.