Original Eigenvalue Research Articles

Inverse problem for the two-dimensional discrete Schrödinger equation in a square

The potential of the two-dimensional discrete Schrodinger equation can be reconstructed from a part of its spectrum and prescribed symmetry conditions of the basis eigenfunctions. The discrete potential along with the missing eigenvalues is found by solving a polynomial system of equations, which is derived and solved using the REDUCE computer algebra system. To ensure the convergence of the iterative process implemented in the Numeric package in REDUCE, proper initial data must be specified. The prescribed eigenvalues are perturbed original eigenvalues corresponding to the zero discrete potential. The original eigenvalues provide the natural initial data for the corresponding missing eigenvalues. In the case of a square, there are many multiple eigenvalues among the original eigenvalues. The direct application of the variant of the Newton method implemented in the Numeric package in REDUCE is impossible in the case of multiple initial data. A modification of the method proposed earlier for calculating the discrete potential of the two-dimensional discrete Schrodinger equation in a square is illustrated by an example.

A BEM-BASED MESHLESS METHOD FOR ELASTIC BUCKLING ANALYSIS OF PLATES

In this paper, a BEM-based meshless method is developed for buckling analysis of elastic plates with various boundary conditions that include elastic supports and restraints. The proposed method is based on the concept of the Analog Equation Method (AEM) of Katsikadelis. According to this method, the original eigenvalue problem for a governing differential equation of buckling is replaced by an equivalent plate bending problem subjected to an appropriate fictitious load under the same boundary conditions. The fictitious load is established using a technique based on BEM and approximated by using the radial basis functions. The eigenmodes of the actual problem are obtained from the known integral representation of the solution for the classical plate bending problem, which is derived using the fundamental solution of the biharmonic equation. Thus, the kernels of the boundary integral equations are conveniently established and evaluated. The method has all the advantages of the pure BEM. To validate its effectiveness, accuracy as well as applicability of the proposed method, numerical results of various problems are presented.

Computing eigenvalue bounds of structures with uncertain‐but‐non‐random parameters by a method based on perturbation theory

AbstractIn this paper, an eigenvalue problem that involves uncertain‐but‐non‐random parameters is discussed. A new method is developed to evaluate the reliable upper and lower bounds on frequencies of structures for these problems. In this method the matrix in the deviation amplitude interval is considered to be a perturbation around the nominal value of the interval matrix, and the upper and lower bounds to the maximum and minimum eigenvalues of this perturbation matrix are computed, respectively. Then based on the matrix perturbation theory, the eigenvalue bounds of the original interval eigenvalue problem can be obtained. Finally, two numerical examples are provided and the results show that the proposed method is reliable and efficient. Copyright © 2006 John Wiley & Sons, Ltd.

A Preconditioning Mass Matrix to Avoid the Ill-Posed Two-Fluid Model

Two-fluid models are central to the simulation of transport processes in two-phase homogenized systems. Even though this physical model has been widely accepted, an inherently nonhyperbolic and nonconservative ill-posed problem arises from the mathematical point of view. It has been demonstrated that this drawback occurs even for a very simplified model, i.e., an inviscid model with no interfacial terms. Much effort has been made to remedy this anomaly and in the literature two different types of approaches can be found. On one hand, extra terms with physical origin are added to model the interphase interaction, but even though this methodology seems to be realistic, several extra parameters arise from each added term with the associated difficulty in their estimation. On the other hand, mathematical based-work has been done to find a way to remove the complex eigenvalues obtained with two-fluid model equations. Preconditioned systems, characterized as a projection of the complex eigenvalues over the real axis, may be one of the choices. The aim of this paper is to introduce a simple and novel mathematical strategy based on the application of a preconditioning mass matrix that circumvents the drawback caused by the nonhyperbolic behavior of the original model. Although the mass and momentum conservation equations are modified, the target of this methodology is to present another way to reach a steady-state solution (using a time marching scheme), greatly valued by researchers in industrial process design. Attaining this goal is possible because only the temporal term is affected by the preconditioner. The obtained matrix has two parameters that correct the nonhyperbolic behavior of the model: the first one modifies the eigenvalues removing their imaginary part and the second one recovers the real part of the original eigenvalues. Besides the theoretical development of the preconditioning matrix, several numerical results are presented to show the validity of the method.

Numerical verification of a gap condition for a linearized nonlinear Schrödinger equation

We make a detailed numerical study of the spectrum of two Schrödinger operators L± arising from the linearization of the supercritical nonlinear Schrödinger equation (NLS) about the standing wave, in three dimensions. This study was motivated by a recent result of the second author on the conditional asymptotic stability of solitary waves in the case of a cubic nonlinearity. Underlying the validity of this result is a spectral condition on the operators L±, namely that they have no eigenvalues nor resonances in the gap (a region of the positive real axis between zero and the continuous spectrum), which we call the gap property. The present numerical study verifies this spectral condition and shows further that the gap property holds for NLS exponents of the form 2 β + 1, as long as β* < β ⩽ 1, where Our strategy consists of rewriting the original eigenvalue problem via the Birman–Schwinger method. From a numerical analysis viewpoint, our main contribution is an efficient quadrature rule for the kernel 1/|x − y| in , i.e. proved spectrally accurate. As a result, we are able to give similar accuracy estimates for all our eigenvalue computations. We also propose an improvement in Petviashvili's iteration for the computation of standing wave profiles which automatically chooses the radial solution.All our numerical experiments are reproducible. The Matlab code can be downloaded from http://www.acm.caltech.edu/~demanet/NLS/.

Optical Response of Photonic Crystals Requiring High Precision Band Calculation in the Form ofk(ω) Including Evanescent Waves

We give a general method to calculate photonic band structure in the form of wave number k as a function of frequency ω, which is required whenever we want to calculate signal intensity related with photonic band structure. This method is based on the fact that the elements of the coefficient matrix for the plane wave expansion of the Maxwell equations contain wave number up to the second order, which allows us to rewrite the original eigenvalue equation for ω 2 into that for wave number. This method is much better, especially for complex wave numbers, than the transfer matrix method of Pendry, which gives the eigenvalues in the form of exp [i k d ]. The present method leads to a good improvement in the calculation of reflectivity spectrum by eliminating unphysical part due to insufficient precision of numerical results, showing the cross-section dependence of a surface of given Miller index more precisely.

Pseudospectral approximation of eigenvalues of derivative operators with non-local boundary conditions

By taking as a “prototype problem” a one-delay linear autonomous system of delay differential equations we present the problem of computing the characteristic roots of a retarded functional differential equation as an eigenvalue problem for a derivative operator with non-local boundary conditions given by the particular system considered. This theory can be enlarged to more general classes of functional equations such as neutral delay equations, age-structured population models and mixed-type functional differential equations. It is thus relevant to have a numerical technique to approximate the eigenvalues of derivative operators under non-local boundary conditions. In this paper we propose to discretize such operators by pseudospectral techniques and turn the original eigenvalue problem into a matrix eigenvalue problem. This approach is shown to be particularly efficient due to the well-known “spectral accuracy” convergence of pseudospectral methods. Numerical examples are given.

Functional-discrete Method for Eigenvalue Transmission Problem with Periodic Boundary Conditions

AbstractWe apply a functional-discrete method with convergence rate not worse than that of a geometric series for an eigenvalue transmission problem with periodic boundary conditions. It is shown that the convergence rate increases with an increase in the ordinal number of trial eigenvalue. Based on asymptotic behavior of eigenvalues of the basic problem and the functional-discrete method, qualitative results concerning the arrangement of the eigenvalues of the original eigenvalue transmission problem with periodic boundary conditions are proved. A number of numerical examples are given to support the theory.

Diagonalization of replicated transfer matrices for disordered Ising spin systems

We present an alternative procedure for solving the eigenvalue problem of replicated transfer matrices describing disordered spin systems with (random) 1D nearest neighbor bonds and/or random fields, possibly in combination with (random) long range bonds. Our method is based on transforming the original eigenvalue problem for a $2^n\times 2^n$ matrix (where $n\to 0$) into an eigenvalue problem for integral operators. We first develop our formalism for the Ising chain with random bonds and fields, where we recover known results. We then apply our methods to models of spins which interact simultaneously via a one-dimensional ring and via more complex long-range connectivity structures, e.g. $1+\infty$ dimensional neural networks and `small world' magnets. Numerical simulations confirm our predictions satisfactorily.

An algorithm using co‐ordinate transformation for free vibration analysis of fully incompressible materials

AbstractThis paper presents an algorithm for conducting eigenvalue analysis of fully incompressible isotropic materials (a Poisson's ratio of exactly 0.5). This algorithm is based on the concept of co‐ordinate transformation defined by eigenvectors with positive eigenvalues of the system stiffness matrix. The transformation to the new co‐ordinates of orthogonal bases in the system potential energy can reduce the size of the system equations. The original eigenvalue problem is reduced to a smaller one that can be solved. This algorithm possesses two drawbacks. One is the increase of system degrees of freedom (DOF) due to the mixed method for incompressible problems; the other is due to additional eigenvalue calculation. However, the present algorithm is inherently able to compensate for the above two drawbacks by reducing the DOF. Numerical demonstrations indicate that the exact transformation yields accurate results that converge to analytical results. They also indicate that an approximated transformation retaining only several per cent of all the eigenvectors produces accurate results in the lower frequency range that is generally of interest in engineering analyses. Therefore, the proposed algorithm is judged to be an efficient procedure for eigenvalue problems of fully incompressible materials. This co‐ordinate‐transformation scheme, derived from the system stiffness matrix, will be applicable to static incompressible problems. Copyright © 2003 John Wiley &amp; Sons, Ltd.

Model reduction for parametric instability analysis in shells conveying fluid

Flexible pipes conveying fluid are often subjected to parametric excitation due to time-periodic flow fluctuations. Such systems are known to exhibit complex instability phenomena such as divergence and coupled-mode flutter. Investigators have typically used weighted residual techniques, to reduce the continuous system model into a discrete model, based on approximation functions with global support, for carrying out stability analysis. While this approach is useful for straight pipes, modelling based on FEM is needed for the study of complicated piping systems, where the approximation functions used are local in support. However, the size of the problem is now significantly larger and for computationally efficient stability analysis, model reduction is necessary. In this paper, model reduction techniques are developed for the analysis of parametric instability in flexible pipes conveying fluids under a mean pressure. It is shown that only those linear transformations which leave the original eigenvalues of the linear time invariant system unchanged are admissible. The numerical technique developed by Friedmann and Hammond (Int. J. Numer. Methods Eng. Efficient 11 (1997) 1117) is used for the stability analysis. One of the key research issues is to establish criteria for deciding the basis vectors essential for an accurate stability analysis. This paper examines this issue in detail and proposes new guidelines for their selection.

Separation of Variables in Deformed Cylinders

We study the Laplace operator subject to Dirichlet boundary conditions in a two-dimensional domain that is one-to-one mapped onto a cylinder (rectangle or infinite strip). As a result of this transformation the original eigenvalue problem is reduced to an equivalent problem for an operator with variable coefficients. Taking advantage of the simple geometry we separate variables by means of the Fourier decomposition method. The ODE system obtained in this way is then solved numerically, yielding the eigenvalues of the operator. The same approach allows us to find complex resonances arising in some noncompact domains. We discuss numerical examples related to quantum waveguide problems. The aim of these experiments is to compare the method based on the separation of variables with the standard finite-volume procedure. For the most computationally difficult examples related to domains with narrow throats one can clearly see the advantages of the proposed method.

Augmentation of the Generalised n × n Eigenvalue Equation to a Generalised ( n +1)×( n +1) Eigenvalue Equation

Generalised n×n eigenvalue equation B|Φi〉=λiSb|Φi〉 (i=1,...,n) where B and Sb are n×n Hermitian matrices while Sb is in addition positive definite is considered. This equation is augmented to a generalised (n+1)(n+1) eigenvalue equation H|Ψk〉=ekS|Ψk〉 (k=1,...,n+1) where Hermitian matrices H and S represent matrices B and Sb, respectively, augmented by one additional row and one additional column. It is shown how the eigenvalues ek and the eigenvectors |Ψk〉 of the augmented eigenvalue equation can be expressed in terms of the eigenvalues λi and the eigenvectors |Φi〉 of the original eigenvalue equation. Operation count to obtain by this method all augmented eigenvalues and eigenvectors is of the order O(n2). Unless matrices involved are of some special kind such as sparse matrices or alike, this operation count is one order of magnitude smaller than operation count required by other presently known methods. In many practical cases operation count to obtain a single selected eigenvalue and/or eigenvector by this method is of the order O(n). In the case of the generalised eigenvalue equation, all other methods usually require again O(n3) operations, even if only a single eigenvalue and/or eigenvector is required. Thus in many cases of interest operation count to obtain a selected eigenvalue and/or eigenvector by this method is two orders of magnitude smaller than operation count required by other methods.

Altering structures to clear vibration frequencies from a band

Structures vibrate, and sometimes with frequencies that are not wanted. Structures are commonly required to be resonance-free within certain frequency ranges, and if they do have undesirable frequencies, these can be moved by changing the structure stiffness, or mass, or both. A mixed stiffness/flexibility formulation of the vibration problem presents alternative condensations to stiffness and flexibility eigenvalue equations for an altered structure. The flexibility form gives more compact equations, and this is developed to solve a parent problem where a structure has a single frequency in a nominated band, to be removed by adjusting the stiffness of a brace stressing the structure in a single way. Interestingly, if the original eigenvalue problem has a Sturm sequence, the frequency exclusion problem can be solved without determining any frequency or mode of the original structure.

Parametric solutions of the eigenvalue equation for a convectively cooled slab

The convective cooling of a slab by an ambient fluid under the most general linear and homogeneous boundary conditions is considered. For the roots of the corresponding transcendental eigenvalue equation an explicit formula is written down in a parametric form. The practical consequences of this representation, among them certain “singular solutions” which cannot be obtained by a direct numerical treatment of the original eigenvalue equation, are discussed in detail.

Quantum QSAR and the eigensystems of stochastic quantum similarity matrices

Row or column stochastic quantum similarity matrices (SM) are by construction non-symmetric square arrays. Thus, although representing a source of partial order within quantum object (QO) sets, they are not so easily connected to QO descriptors as the parent symmetric quantum SM do. In this paper, the eigensystems of such stochastic SM are analyzed and connected to the original quantum SM eigenvalues and eigenvectors. Simple relationships are found, providing row or column stochastic quantum SM with the power to be used as QO descriptors in both classical or quantum QSAR frameworks.

Anisotropic surface relief diffraction gratings under arbitrary plane wave incidence

A modal method is used for the analysis under oblique incidence of a diffraction grating made of anisotropic material. The problem is studied viewing the structure as the cascade of junctions between periodic arrays of anisotropic slab waveguides with the same period and different heights. This diffraction problem is formulated in terms of an integral equation that enforces the continuity of the transverse magnetic field at the junction. The unknown is the transverse electric field at the junction. It is possible to use also another formulation, where the role of the two fields is exchanged. The kernels of these equations are the relevant Green's functions, which are expressed in terms of eigenfunction expansions. The determination of the modes of the various regions composed of arrays of anisotropic dielectric slabs has been carried out by the method of spectral elements, whereby the field components are represented in a polynomial basis and the original differential eigenvalue problem is converted into an algebraic one. The integral equation is solved numerically by the method of moments and each junction is characterized by its generalized scattering matrix (GSM). Finally, the diffraction efficiencies of the grating are obtained by combining the various GSM's.

Invariant and Orthonormal Scalar Measures Derived from Magnetic Resonance Diffusion Tensor Imaging

A diffusion tensor is a mathematical construct used to describe water diffusion in complicated biological structures. It describes a process which occurs in all directions simultaneously. It is difficult to comprehend or graphically display the information in the diffusion tensor. This paper describes a coordinate system approach for producing scalar measures which characterize key aspects of the diffusion tensor. The eigenvalues of the diffusion tensor are introduced as the three elements of a point in a Cartesian coordinate system. The Cartesian coordinates are then expressed in cylindrical and spherical coordinates. The orthonormal coordinates of the spherical system are particularly useful scalar measures of attributes of the diffusion tensor: One coordinate contains all the information about the overall magnitude of diffusion. Another contains all of the anisotropy information. The third coordinate contains all of the information about skewness. No information is lost when transforming the original eigenvalues to spherical coordinates.

Solution of a transcendental eigenvalue problem via interval analysis

A knowledge of the complex roots λ of the transcendental eigenvalue equation sin λα=± λsin α is essential in the analysis of the slow viscous fluid flow in the neighbourhood of a sharp corner which subtends an angle α ϵ (0, 2π] to the fluid. Existing methods for finding all roots λ essentially require an a priori knowledge of the solution structure; given that { λ 1, λ 2, …, λ m } are known, gl m+1 is determined via iterations, and/or a solution procedure initiated by λ m . We present herein a general interval analysis method which exhaustively finds all roots λ via only the original eigenvalue equation: no other information is required. The interval-analysis method automatically guarantees root existence and uniqueness while simultaneously providing error bounds.

High-Supersonic/Hypersonic Flutter of Prismatic Composite Plate/Shell Panels

A spline e nite strip modal method is developed for e utter instability analysis of prismatic composite plate/shell panels exposed to high-supersonic/hypersonic e ow. The structural model is formulated within the context of e rstorder shear deformation shell theory, and the aerodynamic pressure is evaluated using the two-dimensional static aerodynamicapproximation.Thethermaleffectduetoaerodynamicheatingisnotconsidered.Theanalysisconsists of two stages. In the e rst stage, the free-vibration problem of the panel is performed by using a superstrip concept and a multilevel substructuring technique. The natural frequencies and vibration modes are, thus, obtained by employing the highly reliable bisection Sturm sequenceapproach with great accuracy and efe ciency. In thesecond stage, the obtained natural frequencies and the corresponding mode shapes are utilized to reduce considerably the size of the original aeroelastic eigenvalue problem, which is then solved to determine the critical dynamic pressure and the corresponding e utter frequency. A variety of numerical experiments, which are conducted to validate and test the performance of the formulation, demonstrates the reliability, efe ciency, accuracy, and versatility of the method.