Eigenvalue Perturbation Research Articles

Dynamics of non-axisymmetric Beltrami flows

In previous works [R. González, G. Sarasua, and A. Costa, “Kelvin waves with helical Beltrami flow structure,” Phys. Fluids 20, 024106 (2008) and R. González, A. Costa, and E. S. Santini, “On a variational principle for Beltrami flows,” Phys. Fluids 22, 074102 (2010)], we analyzed the Beltrami flow structure of Kelvin waves in an ideal fluid. As a result, we were able to show an important feature of Beltrami flows: their stability for Beltrami perturbations with the same eigenvalue as the basic flow. Here, instead, we study the dynamics of Beltrami perturbations by performing a modal analysis. In the first place, we study the modes that are generated by perturbing a uniformly translating and solidly rotating basic flow. In order to simplify the analysis, we consider the non-inertial frame in which this basic flow is at rest. In the second place, we analyze a basic Beltrami flow that is stationary in the non-inertial frame considered and is perturbed with Beltrami modes. We find that the last case is only possible when the perturbation eigenvalue is the same as that of the basic Beltrami flow. This is what we have called dynamical property. In both cases, the dynamics are represented by progressive waves in the moving frame. We apply this analysis to a rotating flow in an infinite cylinder and to an axisymmetric rotating Beltrami flow in a semi-infinite cylinder. In both cases, the development of secondary Beltrami modes is possible due to the dynamical property.

Efficient Localization Methods for Passivity Enforcement of Linear Dynamical Models

This paper describes a novel approach for passivity enforcement of compact dynamical models of electrical interconnects. The proposed approach is based on a parameterization of general state-space scattering models with fixed poles. We formulate the passivity constraints as a unitary boundedness condition on the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> norm of the system transfer function. When this condition is not verified, we use it as an explicit constraint within an iterative perturbation loop of the system state-space matrices. Since the resulting optimization framework is convex but nonsmooth, we solve it via localization based algorithms, such as the ellipsoid and the cutting plane methods. The proposed technique solves two critical bottleneck issues of the existing approaches for passivity enforcement of linear macromodels. Compared to quasi-optimal schemes based on singular value or Hamiltonian eigenvalue perturbation, we are able to guarantee convergence to the optimal solution. Compared to convex formulations based on direct Bounded Real Lemma constraints, we are able to reduce both memory and time requirements by orders of magnitude. We demonstrate the effectiveness of our approach on a number of cases for which existing algorithms either fail or exhibit very slow convergence.

Modified perturbation method for eigenvalues of structure with interval parameters

In overcoming the drawbacks of traditional interval perturbation method due to the unpredictable effect of ignoring higher order terms, a modified parameter perturbation method is presented to predict the eigenvalue intervals of the uncertain structures with interval parameters. In the proposed method, interval variables are used to quantitatively describe all the uncertain parameters. Different order perturbations in both eigenvalues and eigenvectors are fully considered. By retaining higher order terms, the original dynamic eigenvalue equations are transformed into interval linear equations based on the orthogonality and regularization conditions of eigenvectors. The eigenvalue ranges and corresponding eigenvectors can be approximately predicted by the parameter combinatorial approach. Compared with the Monte Carlo method, two numerical examples are given to demonstrate the accuracy and efficiency of the proposed algorithm to solve both the real eigenvalue problem and complex eigenvalue problem.

Robust Estimator-Correlator for Spectrum Sensing in MIMO CR Networks with CSI Uncertainty

This work introduces novel detection schemes which are robust with respect to the uncertainty in the estimate of the signal covariance matrix for non-coherent spectrum sensing in multiple-input multiple-output (MIMO) cognitive radio networks. We employ an eigenvalue perturbation theory based approach to model the uncertainty in the estimated signal covariance matrix. Subsequently, we derive an optimization framework for the generalized likelihood ratio test (GLRT) based robust test statistic detector (RTSD) and robust estimator-correlator detector (RECD) towards primary user detection, which incorporate the channel state information (CSI) uncertainty inherent in such scenarios. Further, employing the Karush-Kuhn-Tucker (KKT) conditions, we derive closed form expressions for the proposed robust spectrum sensing schemes. Simulation results demonstrate the superior performance of the proposed robust detectors in comparison to the uncertainty agnostic estimator-correlator (EC) detector for spectrum sensing in MIMO cognitive radio networks with CSI uncertainty.

A direct-variance-analysis method for generalized stochastic eigenvalue problem based on matrix perturbation theory

It has been extensively recognized that the engineering structures are becoming increasingly precise and complex, which makes the requirements of design and analysis more and more rigorous. Therefore the uncertainty effects are indispensable during the process of product development. Besides, iterative calculations, which are usually unaffordable in calculative efforts, are unavoidable if we want to achieve the best design. Taking uncertainty effects into consideration, matrix perturbation methodpermits quick sensitivity analysis and structural dynamic re-analysis, it can also overcome the difficulties in computational costs. Owing to the situations above, matrix perturbation method has been investigated by researchers worldwide recently. However, in the existing matrix perturbation methods, correlation coefficient matrix of random structural parameters, which is barely achievable in engineering practice, has to be given or to be assumed during the computational process. This has become the bottleneck of application for matrix perturbation method. In this paper, we aim to develop an executable approach, which contributes to the application of matrix perturbation method. In the present research, the first-order perturbation of structural vibration eigenvalues and eigenvectors is derived on the basis of the matrix perturbation theory when structural parameters such as stiffness and mass have changed. Combining the first-order perturbation of structural vibration eigenvalues and eigenvectors with the probability theory, the variance of structural random eigenvalue is derived from the perturbation of stiffness matrix, the perturbation of mass matrix and the eigenvector of baseline-structure directly. Hence the Direct-Variance-Analysis (DVA) method is developed to assess the variation range of the structural random eigenvalues without correlation coefficient matrix being involved. The feasibility of the DVA method is verified with two numerical examples (one is truss-system and the other is wing structure of MA700 commercial aircraft), in which the DVA method also shows superiority in computational efficiency when compared to the Monte-Carlo method.

Eigenvector perturbations of complex networks

Recently spectral perturbations, involving eigenvalue and eigenvector perturbations, which have attracted more attentions than conventional topological perturbations, are used to analyze and promote the robustness of complex networks. However, to the best of our knowledge, the study of eigenvector perturbation and the equivalence between it and topological perturbation has not been found yet. In this paper, we first deduce the mathematical relationship between eigenvalue perturbation and its corresponding eigenvector perturbation for network reconstructions. Afterwards, two perturbed forms of eigenvector spectrum, global perturbation and local perturbation, are defined, such that we can examine the impacts of eigenvector perturbations on network robustness, and compare those to the impacts of topological perturbations on robustness. Meanwhile, the theoretical ranges of the allowed values of two eigenvector perturbation factors are derived in terms of the accurate reconstruction condition of networks, and validated by experimental simulations. By comparison our finding is that the eigenvector perturbations we define seem equivalent to topological perturbations.

Vector Finite Elements and the Step by Step Eigenvalue Perturbation Methods

The electromagnetic resonant structures and their perturbations have been found their way in many applications in microwave engineering. Analysis of these structures using finite element method results in a generalized eigenvalue problem, where the eigenvalues correspond to the resonant frequencies, and the eigenvectors correspond to the resonant modes. The perturbations of resonant structures yield perturbed eigenvalue problem and can be solved by eigenvalue perturbation methods, effectively. Combining finite element method with step by step eigenvalue perturbation method yields parametric history with respect to perturbation parameter. In this study perturbation of a microwave ring resonator placed in a metalic enclosure has been examined, combining the vector finite element and the step-by-step generalized eigenvalue perturbation methods.

Optimal topology design of replaceable bar dampers of a reticulated shell based on sensitivity analysis

In this study, a method for control of reticulated shells is proposed and its practicality is demonstrated. The control is implemented by replacing selected bars of the shell with passive viscoelastic dampers. By applying the eigenvalue perturbation technique and the earthquake spectrum concept, the sensitivities of various topologies of the shell are analyzed, and the optimal topology is determined by taking their symmetries into consideration. The results of this research show that common damper topologies are not effective for all types of responses and recorded earthquakes. The optimal topology identifi ed requires a minimal number of dampers for each type of earthquake record. The displacement control effect of the dynamic responses of the optimal topology is 10%–20%; the acceleration control effect is also about 10%–20%; and the axial force control effect is as much as 30%–45%. Furthermore, the incremental dynamic analysis (IDA) method is used to investigate the stability of the controlled shell. The results show that the dynamic stability of the controlled shell is well preserved when it is vibrated under vibration and is better than the uncontrolled shell. The ultimate load increased by 10% and the elements entered into the plastic stage when the peak acceleration reached 580 Gal, which is 200 Gal larger than the uncontrolled shell.

Perturbation Bound for the Eigenvalues of a Singular Diagonalizable Matrix

AbstractIn this short note, we present a sharp upper bound for the perturbation of eigenvalues of a singular diagonalizable matrix given by Stanley C. Eisenstat [3].

Application and Generalization of Eigenvalues Perturbation Bounds for Hermitian Block Tridiagonal Matrices

The paper contains two parts. First, by applying the results about the eigenvalue perturbation bounds for Hermitian block tridiagonal matrices in paper [1], we obtain a new efficient method to estimate the perturbation bounds for singular values of block tridiagonal matrix. Second, we consider the perturbation bounds for eigenvalues of Hermitian matrix with block tridiagonal structure when its two adjacent blocks are perturbed simultaneously. In this case, when the eigenvalues of the perturbed matrix are well-separated from the spectrum of the diagonal blocks, our eigenvalues perturbation bounds are very sharp. The numerical examples illustrate the efficiency of our methods.

Design optimization of a carbon fiber reinforced composite automotive lower arm

Substituting composites for the metallic structures has many advantages because of the higher specific stiffness and higher specific strength of the composite materials. In this paper, we designed an automotive composite lower arm using carbon-epoxy composite materials. To optimize the stacking sequence of the composite layer, we used a micro-genetic algorithm and investigated its effects on the performances of a lower arm, such as static/buckling load capability and stiffness. To maximize the buckling load capability, we performed the design optimization with the linear perturbation eigenvalue analysis, targeting a 50% weight reduction of conventional steel lower arm. We verified again the performance of the optimized composite lower arm using the static Riks analysis technique. Finally, we found that our composite lower arm had two times higher stiffness and buckling strength compared to the conventional steel lower arm while having 50% less weight.

A Perturbation Solution of the Thickness Natural Vibrations of a Piezoelectric Plate

This paper discusses a perturbation approach to the calculation of the natural frequencies and mode shapes for both the displacement and the electrostatic potential through-thickness vibration of an infinite piezoelectric plate. The problem is formulated within the coupled theory of linear piezoelectricity. It is described by a set of two coupled differential equations with unknown thickness displacement, the electrostatic potential and a general form of boundary conditions. A consistent perturbation solution to the natural vibration problem is described. An important element not present in the classical eigenvalue perturbation solution is that the small parameter appears in the boundary conditions; a way to handle this complication has been discussed. The natural frequencies and mode shapes obtained using the perturbation approach are compared to exact solutions, demonstrating the effectiveness of the proposed method. The advantage of the perturbation method derives from the fact that coupled piezoelectric results can be obtained from the elastic solution during the postprocessing stage.

Eigenvalue perturbation theory of symplectic, orthogonal, and unitary matrices under generic structured rank one perturbations

We study the perturbation theory of structured matrices under structured rank one perturbations, with emphasis on matrices that are unitary, orthogonal, or symplectic with respect to an indefinite inner product. The rank one perturbations are not necessarily of arbitrary small size (in the sense of norm). In the case of sesquilinear forms, results on selfadjoint matrices can be applied to unitary matrices by using the Cayley transformation, but in the case of real or complex symmetric or skew-symmetric bilinear forms additional considerations are necessary. For complex symplectic matrices, it turns out that generically (with respect to the perturbations) the behavior of the Jordan form of the perturbed matrix follows the pattern established earlier for unstructured matrices and their unstructured perturbations, provided the specific properties of the Jordan form of complex symplectic matrices are accounted for. For instance, the number of Jordan blocks of fixed odd size corresponding to the eigenvalue 1 or −1 have to be even. For complex orthogonal matrices, it is shown that the behavior of the Jordan structures corresponding to the original eigenvalues that are not moved by perturbations follows again the pattern established earlier for unstructured matrices, taking into account the specifics of Jordan forms of complex orthogonal matrices. The proofs are based on general results developed in the paper concerning Jordan forms of structured matrices (which include in particular the classes of orthogonal and symplectic matrices) under structured rank one perturbations. These results are presented and proved in the framework of real as well as of complex matrices.

Asymptotic expansions for transmission eigenvalues for media with small inhomogeneities

We consider the transmission eigenvalue problem for an inhomogeneous medium containing a finite number of diametrically small inhomogeneities of different refractive index. We prove a convergence result for the transmission eigenvalues and eigenvectors corresponding to media with small homogeneities as the diameter of small inhomogeneities goes to zero. In addition we derive rigorously a formula for the perturbations in the real transmission eigenvalues caused by the presence of these small inhomogeneities.

Output feedback stabilisation of single-input single-output linear systems with I/O network-induced delays. An eigenvalue-based approach

This work addresses the output feedback stabilisation problem for a class of linear single-input single-output systems subject to I/O network delays. More precisely, we are interested in the characterisation of the set of delay and gain parameters guaranteeing the stability of the closed-loop system. To perform such an analysis, we adopt an eigenvalue perturbation based approach. Various illustrative numerical examples complete the presentation.

Modulation of Shallow-Water Equatorial Waves due to a Varying Equivalent Height Background

Abstract The dynamics of convectively coupled equatorial waves (CCEWs) is analyzed in an idealized model of the large-scale atmospheric circulation. The model is composed of a linear rotating shallow-water system with a variable equivalent height, or equivalent gravity wave speed, which varies in space. This model is based on the hypothesis that moist convection acts to remove convective instability, therefore modulating the equivalent height of a shallow-water system. Asymptotic solutions are derived in the case of a small perturbation around a constant coefficient, which is assumed to be a mean moist equivalent height derived from satellite observations. The first-order solutions correspond to the free normal modes of the linear shallow-water system and the second-order flow is derived solving a perturbation eigenvalue problem. The asymptotic solutions are documented in the case of a zonally varying equivalent height and for wavenumbers and frequencies that are consistent with observations of CCEWs. This analysis shows that the dynamics of the secondary divergence and its impact on the full divergence varies mode by mode. For instance, for a negative equivalent height anomaly, which is interpreted as a moister background, the secondary divergence is nearly in phase with the primary divergence in the case of Kelvin waves—in contrast to mixed Rossby–gravity waves where the secondary divergence acts to attenuate the primary divergence. While highly idealized, the modeled waves share some features with observations, providing a mechanism for the relationship between CCEWs phase speed, amplitude, and horizontal structure.

Structural Damage Diagnosis Using an Improved Eigenvalue Perturbation Method

This paper presents an improved method to identify structural damages only by changes of natural frequencies. The underlying principle of the proposed technique is to measure the natural frequencies of the damaged system, and then use this set of data as well as the original test data of undamaged system to identify structural damages with the help of eigenvalue perturbation method. In this contribution, a simple accelerated formula is developed to improve the accuracy of the eigenvalue perturbation method. With the introduction of the accelerated formula, the proposed method is able to achieve more accurate results than that obtained by the original eigenvalue perturbation method without any high-order analysis or multi-iterations. The effectiveness of the proposed method is illustrated using simulated data on a published numerical example. From the numerical results, it can be conclued that the proposed approach is simple to implement and can identify structural damages very accurately only by the first few natural frequencies.

Adjoint-Based Sensitivity Analysis of Coupled Criticality Problems

Sensitivity analysis is a technique that is widely used in reactor physics calculations to efficiently obtain first-order changes in responses of interest due to variations of input parameters. This paper presents an extension of the well-known perturbation procedures for the critical eigenvalue and flux functionals. The extended method makes it possible to determine sensitivities in coupled criticality problems with mutual feedback between neutronics and one or more augmenting systems (e.g., thermal hydraulics or fission product poisoning). The technique uses appropriate neutronic and augmenting adjoint functions, which can be obtained by solving a system of coupled adjoint equations.Three different approaches are presented for considering the effects of perturbations in coupled criticality problems with feedback: The steady-state power level is allowed to adjust to maintain criticality with the perturbed parameters (power perturbation), a change is allowed in the critical eigenvalue while the flux is constrained (eigenvalue perturbation), or simultaneous perturbations are made to ensure criticality at the unperturbed power level (control parameter perturbation). In the case of power and eigenvalue perturbations, sensitivities can be obtained with or without power- and k-reset procedures, respectively, yielding identical results to control parameter perturbation.The paper presents the theoretical background, an application to a one-dimensional slab problem with thermal and fission product feedback, and a numerical procedure to obtain the necessary adjoint functions. The proposed technique relies on using the neutronics and augmenting codes separately as a preconditioner for Krylov methods employed to the coupled adjoint problem. This makes the development of new codes unnecessary and provides a means of large-scale implementation.

On sharp frame diagonalization

It was recently shown that if M=Γ1⊕⋯⊕Γk∈Cn×n is a Jordan matrix with k nontrivial Jordan blocks Γi, then M can be frame diagonalized by embedding M into a diagonalizable matrix in C(n+ℓ)×(n+ℓ) with ℓ=k. This naturally motivates a pursuit of the best possible value of ℓ for which this is possible. Here, we use Lidskii’s Theorem on eigenvalue perturbations to construct diagonalizing frames for ℓ=k☆:max{gmM(λ)|λ∈σ(M)}. Moreover, we verify that k☆ is sharp.

The Effects of Perturbing Eigenvalues In The Presence of Multicollinearity

Multicollinearity is a linear dependency between two or more explanatory variables in the regression models which can seriously distort the least squares estimates. The Ordinary Least Squares Estimator is an unbiased estimator that is used to estimate the unknown parameters in the model. The variance of the Ordinary Least Squares Estimates would be inflated and the regression coefficients often indetermined in the presence of multicollinearity. Therefore, biased estimators are suggested as alternatives to the Ordinary Least Squares Estimator. In this study, a new method of solving multicollinearity problem through perturbation of eigenvalues is proposed. The performance of this estimator is evaluated by comparing it with some existing estimators in terms of mean squared error. The first new estimator is compared with the Ordinary Least Squares Estimator, Principal Component regression and the Ordinary Ridge Regression Estimator.