A minimax principle for nonoverdamped systems

A minimax characterization of a subset of the real eigenvalues ω of the quadratic eigenvalue problem (ω²A + ωB + C)ξ = 0 is presented, where A, B and C are linear Hermitian operators (matrices) on and into the n-dimensional complex Hilbert space E_n, and A > 0. This minimax characterization extends to all the eigenvalues of a semi-overdamped system, and Duffin's minimax theory for overdamped systems is recovered as a special case.     

Symplectic elasticity for bi-directional functionally graded materials

This paper suggests a symplectic framework for the analysis of plane problems of bi-directional functionally graded materials (FGMs), in which the elastic modulus varies exponentially both along the longitudinal and transverse coordinates while the Poisson’s ratio remains constant. First, by a procedure which has been used in symplectic elasticity for homogeneous materials, along with the introduction of some stress variables, the governing equations are rewritten in an appropriate state-space form. The method of separation of variables is then adopted to transform the original problem to an eigenproblem, in which the eigenvalues and eigensolutions are determined subsequently. From the physical essence, we can know that the Saint–Venant solutions correspond to two particular eigenvalues (e.g. zero and −α, where α is the gradient index along the longitudinal coordinate). The first-order eigenvector of Jordan normal form for the special eigenvalue −α must be solved in a different way than that for the homogenous materials or the uni-directional FGMs. The presented general eigensolutions, which are usually covered up by the Saint–Venant principle, play a significant role in the local behavior and may be crucial to the onset of failure of the materials/structures. Two numerical examples are considered to show the stress distributions in bi-directional FGM rectangular beams, and to indicate the importance of the developed solution in the local behavior.     

Comments on the eigenvalue formulation of problems with cracks,V-notched cracks,and corners

An eigenvector formulation is often used for configurations containing cracks, V-notched cracks, and corners. Coefficients associated with eigenvectors which give a stress singularity at the crack tip or corner are the stress intensities which can be used in a failure criterion. A complex variable formulation is often used to obtain eigenvalues and eigenvectors. In such a formulation, a solution is assumed in terms of a complex parameter, z, raised to the power , where is the eigenvalue. The quantity z is multi-valued. However, previous authors have only considered one of the possible values of the function. In doing so, they may have overlooked important eigenvalues and/or eigenvectors associated with the problem. This paper reformulates, for the general finite opening crack and real eigenvalues, the eigenvector problem so as to consider the multi-valued nature of z . For the zero opening crack, V-notched crack and corner problem, it is shown that with the new formulation, no new real eigenvalues exist and for the zero-opening crack problem, it is also shown that no new eigenvector mode shapes exist. The first eigenvalue for a 90° corner problem was considered. No new eigenvector mode shapes were found. These findings suggest that the linear elastic results found in the literature are complete and correct, even if their derivation is somewhat deficient in generality.     

Highly efficient general method for sensitivity analysis of eigenvectors with repeated eigenvalues without passing through adjacent eigenvectors

It is well known that the sensitivity analysis of the eigenvectors corresponding to multiple eigenvalues is a difficult problem. The main difficulty is that for given multiple eigenvalues, the eigenvector derivatives can be computed for a specific eigenvector basis, the so-called adjacent eigenvector basis. These adjacent eigenvectors depend on individual variables, which makes the eigenvector derivative calculation elaborate and expensive from a computational perspective. This research presents a method that avoids passing through adjacent eigenvectors in the calculation of the partial derivatives of any prescribed eigenvector basis. As our method fits into the adjoint sensitivity analysis , it is efficient for computing the complete Jacobian matrix because the adjoint variables are independent of each variable. Thus our method clarifies and unifies existing theories on eigenvector sensitivity analysis. Moreover, it provides a highly efficient computational method with a significant saving of the computational cost. Additional benefits of our approach are that one does not have to solve a deficient linear system and that the method is independent of the existence of repeated eigenvalue derivatives of the multiple eigenvalues. Our method covers the case of eigenvectors associated to a single eigenvalue. Some examples are provided to validate the present approach.     

Eigenvalues of damped structures: Vectoriteration in the original space of DOF

The equations of motion in structural dynamics as well as the corresponding eigenvalue problem are governed by 3 matrices for mass, damping and stiffness of order n which equals the number of degrees of freedom. High-performance eigenvalue-solvers are developed for only pairs A, B of matrices. Nevertheless, to benefit from these solvers, the original eigenvalue problem (λ² M+λD+K)x=0 is transformed into a linear eigenvalue representation with only two hypermatrices of double order 2n. Consequently the total numerical effort depends on this order 2n. This paper presents a vectoriteration process which actually works in the original space of order n and which needs no special actions like simultaneous iteration if complex conjugate eigenvalues λ, λˉ with identical norm have to be calculated. The theoretical foundation of this process still goes back to the pair of hypermatrices.     

An extended method of time-dependent fundamental solutions for inhomogeneous heat conduction

In this paper, the method of time-dependent fundamental solutions is extended to solving transient heat conduction problems in inhomogeneous media. The solution of the problem under investigation is split into two parts, namely the particular and homogenous solutions. The novelty of the proposed approach lies in that an approximation of the particular solution is derived by using the fundamental solutions of the associated eigenvalue equations. Numerical results for one- and two-dimensional geometries are presented to verify the efficacy of the proposed method. The effects of the numbers of source and collocation points, the eigenvalues and the parameter T on the accuracy of the numerical solution are also investigated.     

Calculation of eigenpair derivatives for asymmetric damped systems with distinct and repeated eigenvalues

An algorithm is derived for the computation of eigenpair derivatives of asymmetric quadratic eigenvalue problem with distinct and repeated eigenvalues. In the proposed method, the eigenvector derivatives of the damped systems are divided into a particular solution and a homogeneous solution. By introducing an additional normalization condition, we construct two extended systems of linear equations with nonsingular coefficient matrices to calculate the particular solution. The method is numerically stable, and the homogeneous solutions are computed by the second‐order derivatives of the eigenequations. Two numerical examples are used to illustrate the validity of the proposed method. Copyright © 2015 John Wiley & Sons, Ltd.     

Stability of a laminar premixed supersonic free shear layer with chemical reactions

We discuss the stability of a 2-dimensional compressible supersonic flow in the wake of a flat plate. The fluid is a multi-species mixture which is undergoing finite rate chemical reactions. We consider the spatial stability of an infinitesimal disturbance in the fluid. Numerical solutions of the eigenvalue stability equations for both reactive and non-reactive supersonic flows are presented and discussed. The chemical reactions have significant influence on the stability behavior. For instance, a neutral eigenvalue is observed near the freestream Mach Number M_∞ ～- 2.375 for non-reactive case, but disappears when the reaction is turned on. For reactive flows, the eigenvalues are not very dependant on the free stream Mach number. The disturbance amplification rate is higher and the wave speed, c_r is lower for most of the reactive cases when compared with the non-reactive cases.     

Plane analysis for functionally graded magneto-electro-elastic materials via the symplectic framework

By introducing the displacements, electric potential, magnetic potential and their dual counterparts as state variables, a symplectic analysis framework is established in the Hamiltonian system to solve the plane problem of functionally graded magneto-electro-elastic materials. The material properties are assumed to vary along the length direction in an identical exponential form. The method of separation of variables along with the eigenfunction expansion technique is employed to reduce the original problem to the eigenvalue/eigensolution analysis. The particular eigensolutions corresponding to eigenvalues of zero and –α are given, which while bearing definite physical interpretations exhibit some unique characteristics. A numerical example is presented to show the influence of material inhomogeneity on the β-group solutions of the problem.     

Solution of a complex quadratic eigenvalue problem

A complex quadratic eigenvalue problem of order n (n = 20, 30, 40,…) encountered in an investigation of pipe flow was reduced to that of computing the eigenvalues of either a real or a complex matrix of order 2n, depending on the linearization technique used. To get satisfactory results using the CDC 6000 series computer, double-precision versions of certain widely used and highly regarded eigenvalue subroutines were required. Although the modified subroutines used for the real matrix computations required less core than those used for the complex matrix, the calculation time was not significantly faster than that for the complex matrix. The use of ‘balancing’ subroutines merits further investigation.     

An efficient method for solving the eigenvalue problem for matrices having a skew-symmetric (or skew-Hermitian) component of low rank

In the numerical modelling of mechanical systems, eigenvalue problems occur in connection with the evaluation of resonance frequencies, buckling modes and other more esoteric calculations. The matrices whose eigenvalues are sought sometimes have a skew-symmetric component and the presence of this component adds significantly to the computational effort required. In many cases where there is a skew-symmetric component, this component has a much lower rank than the symmetric component which generally has rank equal to its dimension. Examples of such cases abound in the area of rotor dynamics where the stiffness and damping matrices associated with journal bearings have significant skew-symmetric components. The solution of the eigenvalue problem for an unsymmetric matrix takes more than twice the number of operations required for the solution of the eigenvalue problem for a symmetric matrix of the same dimension. This paper puts forward a new method for the solution of the eigenvalue problem for matrices having a skew-symmetric component of low rank and shows that it is faster than established methods of comparable accuracy for the general unsymmetric NxN matrix if the rank of the skew-symmetric component is less than N/7.3.     

Designing discriminative spatial filter vectors in motor imagery brain–computer interface

The problem of a volume conduction effect in electroencephalography is considered one of the challenging issues in brain–computer interface (BCI) community. In this article, we propose a novel method of designing a class‐discriminative spatial filter assuming that a combination of spatial pattern vectors, irrespective of the eigenvalues of the common spatial pattern (CSP), can produce better performance in terms of classification accuracy. We select discriminative spatial filter vectors that determine features in a pairwise manner, that is, eigenvectors of the K largest eigenvalue and the K smallest eigenvalue. Although the pair of the eigenvectors of the K largest and the K smallest eigenvalues helps extract discriminative features, we believe that a different set of eigenvector pairs is more appropriate to extract class‐discriminative features. In our experimental results using the publicly available dataset of BCI Competition IV, we show that the proposed method outperformed the conventional CSP methods and a filter‐bank CSP. © 2013 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 23, 147–151, 2013     

Reliable use of determinants to solve non-linear structural eigenvalue problems efficiently

A ‘multiple determinant parabolic interpolation method’ is described and evaluated, principally by using a plane frame test-bed program. It is intended primarily for solving the transcendental eigenvalue problems arising when the ‘exact’ member equations obtained by solving the governing differential equations of members are used to find the eigenvalues (i.e. critical buckling loads or undamped natural frequencies) of structures. The method has five stages which together ensure successful convergence on the required eigenvalues in all circumstances. Thus, whenever checks indicate its suitability, parabolic interpolation is used to obtain eigenvalues more rapidly than would the popular bisection alternative. The checks also ensure a wise choice of the determinant used by the interpolation. The determinants available are all usually zero at eigenvalues, and comprise the determinant of the overall stiffness matrix K _n and the determinants which result, with negligible extra computation, from effectively considering all except the last m (m=1, 2,…, n?1) freedoms to which K _n corresponds as internal substructure freedoms. Tests showed time savings compared to bisection of 31 per cent when finding non-coincident eigenvalues to relative accuracy ? = 10^?4, increasing to 62 per cent when ? = 10^?8. The tests also showed time savings of about 10 per cent compared with an earlier Newtonian approach. The method requires no derivatives and its use in the widely available space frame program BUNVIS-RG has demonstrated how easily it can replace bisection, which was used in the earlier program BUNVIS.     

A Fourier series solution for the longitudinal vibrations of a bar with viscous boundary conditions at each end

This paper presents the generalized Fourier series solution for the longitudinal vibrations of a bar subjected to viscous boundary conditions at each end. The model of the system produces a non-self-adjoint eigenvalue problem which does not yield a self-orthogonal set of eigenfunctions with respect to the usual inner product. Therefore, these functions cannot be used to calculate the coefficients of expansion in the Fourier series. Furthermore, the eigenfunctions and eigenvalues are complex-valued. The eigenfunctions can be utilized if the space of the wave operator is extended and a suitable inner product is defined. It is further demonstrated that the series solution contains the solutions for free–free, fixed–damper, fixed–fixed, and fixed–free bar cases. The presented procedure is applicable in general to other problems of this type. As an illustration of the theoretical discussion, the results from numerical simulations are presented.     

A method for solving high-order real symmetric eigenvalue problems

A new method for solving structural dynamics problems has been proposed by the author in a separate paper.¹ This method of solution produces a high order real symmetric eigenvalue problem of the form ( A ? λ B ? λ² C ? λ³ D …) U = 0. An algorithm for solving such an eigenvalue problem using simultaneous iteration is presented in this paper. Methods of accelerating the convergence and reducing the amount of computation are also described. A numerical example is given in which the algorithm is used to calculate the eigenvalues and eigenvectors of a framed structure.     

A Padé approximate linearization algorithm for solving the quadratic eigenvalue problem with low‐rank damping

The low‐rank damping term appears commonly in quadratic eigenvalue problems arising from physical simulations. To exploit the low‐rank damping property, we propose a Padé approximate linearization (PAL) algorithm. The advantage of the PAL algorithm is that the dimension of the resulting linear eigenvalue problem is only n + ?m, which is generally substantially smaller than the dimension 2n of the linear eigenvalue problem produced by a direct linearization approach, where n is the dimension of the quadratic eigenvalue problem, and ? and m are the rank of the damping matrix and the order of a Padé approximant, respectively. Numerical examples show that by exploiting the low‐rank damping property, the PAL algorithm runs 33–47% faster than the direct linearization approach for solving modest size quadratic eigenvalue problems. Copyright © 2015 John Wiley & Sons, Ltd.     

Convergence studies of eigenvalue solutions using two finite plate bending elements

The convergence rates of eigenvalue solutions using two finite plate bending elements are studied. The elements considered are the well-known 12 degree of freedom, non-conforming rectangular element and the 16 degree of freedom, conforming rectangular element. Three problems are analysed, a square plate simply supported on two opposite sides with the other two sides clamped, simply supported, or free. Closed form, finite element solutions for these problems are obtained by using shifting E-operators. With few exceptions, eigenvalue solutions found with the non-conforming element converge from below the exact answers at an asymptotic rate of n^?2, where n is the number of elements on a side. However, since the array size needed for such convergence is very large, little can be said about the convergence rates for practical arrays. The conforming element solutions converge from above at an asymptotic rate of n^?4. A comparison of the errors involved in using these two elements shows that the conforming element is far superior to the non-conforming element.     

A Simple Necessary and Sufficient Condition on Physically Realizable Mueller Matrices

Abstract

Development of simple tools to test physical realizability of measured or computed Mueller matrices is the subject of this paper. In particular, the overpolarization problem, i.e., the problem of ensuring that the output degree of polarization does not exceed unity is solved by finding an easily implementable necessary and sufficient condition. With G being the Lorentz metric, it states that a given matrix M is not overpolarizing if and only if the spectrum of GM ^T GM is real and an eigenvector associated with the largest eigenvalue is a physical Stokes vector. This result is used to characterize some M classes of special interest, and is used to test several examples from recent literature.     

The Trefftz method for solving eigenvalue problems

For Laplace's eigenvalue problems, this paper presents new algorithms of the Trefftz method (i.e. the boundary approximation method), which solve the Helmholtz equation and then use an iteration process to yield approximate eigenvalues and eigenfunctions. The new iterative method has superlinear convergence rates and gives a better performance in numerical testing, compared with the other popular methods of rootfinding. Moreover, piecewise particular solutions are used for a basic model of eigenvalue problems on the unit square with the Dirichlet condition. Numerical experiments are also conducted for the eigenvalue problems with singularities. Our new algorithms using piecewise particular solutions are well suited to seek very accurate solutions of eigenvalue problems, in particular those with multiple singularities, interfaces and those on unbounded domains. Using piecewise particular solutions has also the advantage to solve complicated problems because uniform particular solutions may not always exist for the entire solution domain.     

Multiplicity of periodic solutions for a boundary eigenvalue problem

In this paper we have used Ricceri's Three Critical Points Theorem to establish multiple existence of eigenvector solutions to a class of eigenvalue problems for each eigenvalue within a specified open interval. We assume neither uniform nor local coercivity of any kind on the potential.