Variation Of Eigenvalues Research Articles

An eigenvalue variation problem of magnetic Schrödinger operator in three dimensions

This paper concerns the lowest eigenvalue $\mu(b\N^Q)$ of the Schrodinger operator in three-dimensions with a magnetic potential $b\N^Q$, where the vector field $\N^Q$ depends on a matrix $Q$ varying in $SO(3)$ and $b$ is a real parameter. The eigenvalue variation problem is to minimize the lowest eigenvalue among all $Q$ in $SO(3)$. This problem arises in the phase transitions of smectic liquid crystals. We give an estimate of the minimum value inf${\mu(b\N^Q):~Q\in SO(3)\}$ for large $b$, and examine its dependence on geometry of the domain surface.

Full-order and multimode flutter analysis using ANSYS

This paper presents the full-order and multimode methods for analyzing coupled flutter of long-span bridges using commercial finite element (FE) package ANSYS. In the full-order method of flutter analysis, a novel FE model is developed to model the coupled wind-bridge system, in which a specific user-defined Matrix27 element in ANSYS is adapted to model the aeroelastic forces and its stiffness or damping matrices are parameterized by wind velocity and vibration frequency. Variation of complex eigenvalues of the coupled system with wind velocity is then depicted by using this model together with complex eigenvalue analysis, and flutter instability can be determined from the variation diagram. In the multimode method, equations of motion for the coupled wind-bridge system are first represented using a modal superposition technique. This formulation leads to a single-parameter searching technique without iteration to determine the conditions of flutter instability when structural damping is not considered in solution. The contribution of participating modes to flutter instability is given in terms of modal amplitude and modal energy in the multimode method. Numerical studies are provided to validate the developed methods as well as to demonstrate both the procedures for flutter analysis using ANSYS. The proposed methods enable the bridge designers and engineering practitioners to analyze bridge flutter in commercial FE package ANSYS.

이산 전력시스템에서 TCSC의 주기적 스위칭 동작에 의한 진동모드의 감도해석

본 논문에서는 RCF 해석법을 싸이리스터 제어 FACTS 설비인 TCSC를 포함하는 전력계통의 미소신호안정도 해석에 적용하였다. 이산시스템에서 RCF 해석법에 기초한 고유치 감도해석 알고리즘을 제시하고 TCSC를 포함하는 전력계통에 적용하였다. 사례연구를 통해서 RCF 해석법이 TCSC의 주기적 스위칭 동작에 의해 발생하는 진동모드의 변화와 새로이 발생되는 불안정 진동모드의 정확한 해석에 매우 유용한 해석방법임을 보였다. 또한 RCF 해석법에 기초한 고유치 감도해석 방법을 사용하여 이산시스템에서 주기적 스위칭 동작에 의해 발생되는 중요 진동모드에 대한 제어기 감도계수를 정확히 구할 수 있음을 보였다. 이러한 사례연구 결과는 기존의 연속시스템에서의 상태방정식에 의한 해석결과와 크게 다른 것이며, RCF 해석법이 TCSC와 같이 주기적 스위칭 동작을 하는 설비를 포함하는 이산전력계통의 해석에 매우 유용한 방법임을 보여준다.

Network structure and dynamics of hydrogenated amorphous silicon

In this paper we discuss the application of current ab initio computer simulation techniques to hydrogenated amorphous silicon (a-Si:H). We begin by discussing thermal fluctuation in the number of coordination defects in the material, and its temperature dependence. We connect this to the ‘fluctuating bond-center detachment’ mechanism for liberating H bonded to Si atoms. Next, from extended thermal MD simulation, we illustrate various mechanisms of H motion. The dynamics of the lattice is then linked to the electrons, and we point out that the squared electron-lattice coupling (and the thermally-induced mean square variation in electron energy eigenvalues) is robustly proportional to the localization of the conjugate state, if localization is measured with inverse participation ratio. Finally we discuss the Staebler–Wronski effect using these methods, and argue that a sophisticated local heating picture (based upon reasonable calculations of the electron-lattice coupling and molecular dynamic simulation) explains significant aspects of the phenomenon.

Genetic Algorithm Applied to the Eigenvalue Equalization Filtered‐x LMS Algorithm (EE‐FXLMS)

The FXLMS algorithm, used extensively in active noise control (ANC), exhibits frequency‐dependent convergence behavior. This leads to degraded performance for time‐varying tonal noise and noise with multiple stationary tones. Previous work by the authors proposed the eigenvalue equalization filtered‐x least mean squares (EE‐FXLMS) algorithm. For that algorithm, magnitude coefficients of the secondary path transfer function are modified to decrease variation in the eigenvalues of the filtered‐x autocorrelation matrix, while preserving the phase, giving faster convergence and increasing overall attenuation. This paper revisits the EE‐FXLMS algorithm, using a genetic algorithm to find magnitude coefficients that give the least variation in eigenvalues. This method overcomes some of the problems with implementing the EE‐FXLMS algorithm arising from finite resolution of sampled systems. Experimental control results using the original secondary path model, and a modified secondary path model for both the previous implementation of EE‐FXLMS and the genetic algorithm implementation are compared.

Theoretical investigation of the effect of asymmetry on optical anisotropy and electronic structure of Stranski-Krastanov quantum dots

The effect of size and shape anisotropy on the optical properties of Stranski-Krastanov quantum dots QDs is theoretically investigated. The QD is modeled using anisotropic parabolic confinement potential. The complex structure of the valence band is described by Luttinger Hamiltonian. The energy spectra and eigenfunctions of hole states are calculated by numerical diagonalization of the Hamiltonian. The dipole matrix elements are obtained for the interband transitions and hence the degree of linear polarization is calculated. The formulation is applied to self-assembled CdSe quantum dots for numerical analysis. The variation of energy eigenvalues with the QD shape anisotropy parameter is studied and the effect of valence subband mixing is clearly identified. The crossings and anticrossings of the valence subbands have been explained in terms of the symmetries of the corresponding eigenstates. It is worthy to note that these symmetry properties of the energy states are responsible for the specific types of dipole selection rules for the anisotropic QDs. The degree of linear polarization is found to increase almost linearly with anisotropy parameter for the transitions from heavy-hole ground states. On the contrary, for the excited hole states, the change is nonmonotonic due to strong anisotropy-dependent mixing effects.

The First Eigenvalue of the Laplacian and the Conductance of a Compact Surface

We present some results whose central theme is the phenomenon of the first eigenvalue of the Laplacian and conductance of the dynamical system. Our main tool is a method for studying how the hyperbolic metric on a Riemann surface behaves under deformation of the surface. With this model, we show that there are variation of the first eigenvalue of the laplacian and the conductance of the dynamical system, with the Fenchel–Nielsen coordinates, that characterize the surface.

Optimization of the first Steklov eigenvalue in domains with holes: a shape derivative approach

The best Sobolev trace constant is given by the first eigenvalue of a Steklov-like problem. We deal with minimizers of the Rayleigh quotient ‖u‖2 H 1 (Ω) 2/‖u‖2 L 2 (∂Ω) for functions that vanish in a subset A⊂ Ω, which we call the hole. We look for holes that minimize the best Sobolev trace constant among subsets of Ω with prescribed volume. First, we find a formula for the first variation of the first eigenvalue with respect to the hole. As a consequence of this formula, we prove that when Ω is a ball the symmetric hole (a centered ball) is critical when we consider deformations that preserves volume but is not optimal. Finally, we prove that by the Finite Element Method we can approximate the optimal configuration and, by means of the shape derivative, we design an algorithm to compute the discrete optimal holes.

Optimal Proliferation Rate in a Cell Division Model

We consider a size structured cell population model where a mother cell gives birth to two daughter cells. We know that the asymptotic behavior of the density of cells is given by the solution to an eigenproblem. The eigenvector gives the asymptotic shape and the eigenvalue gives the exponential growth rate and so the Maltusian parameter. The Maltusian parameter depends on the division rule for the mother cell, i.e., symmetric (the two daughter cells have the same size) or asymmetric. We use a min-max principle and a differentiation principle to find the variation of the first eigenvalue with respect to a parameter of asymmetry of the cell division. We prove that the symmetrical division is not always the best fitted division, i.e., the Maltusian parameter may be not optimal.

Three-dimensional asymptotic stress field at the front of an unsymmetric bimaterial wedge associated with matrix cracking or fiber break

A combined approach for prediction of the singular stress fields at the interfacial fronts of fiber breaks and matrix cracks is presented. A recently developed eigenfunction expansion method is employed for obtaining three-dimensional asymptotic displacement and stress fields in the vicinity of a point located at the front of a bimaterial wedge of general (unsymmetric) geometrical configuration (with respect to bimaterial interface), and subjected to extension/bending (mode I) and in-plane shear/twisting (mode II) far field loading and free–free wedge-side boundary condition. Each material is isotropic and elastic, but with different material properties. The material 2 (fiber in the case of matrix cracking or matrix in the case of fiber break) is always taken to be a half-space, while the wedge aperture angle of the material 1 is varied to represent varying matrix cracking or fiber break incidence angles at the interfaces. Numerical results pertaining to the variation of the lowest eigenvalues (or stress singularities) for various wedge aperture angles of the material 1, subjected to the afore-mentioned wedge-side boundary condition, are also presented. Variation of the same with the shear moduli ratio of the component material phases is also an important part of the present investigation. The conclusion drawn from the present asymptotic analysis of the matrix cracking problem is in agreement with that of an earlier investigation based on the energy based criterion. Similar conclusion drawn from the present asymptotic analysis of fiber break problem is in agreement with that of an energy based criterion derived here in analogy to the corresponding matrix crack problem by another investigator.

Spectral proper transformation of wind pressure fluctuations: application to a square cylinder and a bridge deck

The results of spectral proper transformation (SPT) analyses applied to the pressure fluctuations on a square cylinder and on a bridge box deck are presented in this paper. The main objective of the paper is that of comparing the results obtained using SPT with those obtained through a covariance proper transformation (CPT) analysis, to understand whether additional information is provided by SPT with respect to CPT. The meaning of the frequency dependency of the spectral eigenvalues and eigenvectors is also investigated. It is observed that a correspondence between SPT and CPT modes can be found in some cases, if the phase shift between the components of the SPT complex modes is properly accounted for. It is also found that the frequency variation of the SPT eigenvalues is related to the characteristic frequencies of the different physical phenomena causing the pressure fluctuations.

The relative effects of sound-speed variability in the water column versus the seabed on normal-mode propagation in shallow water

Normal-mode propagation in shallow water is influenced by variations in the acoustic properties of both the water column and the seabed. Specifically, the sound speed in the water may fluctuate due to the passage of an internal wave, while the sound speed in the bottom may change due to variable geological features. Assessing the relative effects of the water column versus the seabed on the characteristics of modal propagation is critical to the understanding of both the forward and inverse problems in shallow-water acoustics. In this paper, perturbation theory is combined with a Pekeris waveguide model to provide an analytic formalism for evaluating the delicate interplay between the water column and the seabed in shallow water propagation. In particular, the relative contributions of sound-speed fluctuations in the water and the bottom to variations in the modal eigenvalues are determined. The results are affected by both the strengths of the fluctuations and the magnitudes of the background modal eigenfunctions in the water and the seabed. The complexity of the problem is illustrated with synthetic and experimental data. [Work supported by ONR.]

Linear active structures and modes (II) —Discrete systems and beams

The basic concepts about the active structures and some attributes of the modes were presented in paper “Liner Active Structures and Modes (I)”. The characteristics of the active discrete systems and active beams were discussed, especially, the stability of the active structures and the orthogonality of the eigenvectors. The notes about modes were portrayed by a model of a seven-storeyed building with sensors and actuators. The concept of the adjoint active structure was extended from the discrete systems to the beams that were the representations of the continuous structures. Two types of beams with different placements of the measuring and actuating systems were discussed in detail. One is the beam with the discrete sensors and actuators, and the other is the beam with distributed sensor and actuator function. The orthogonality conditions were derived with the modal shapes of the active beam and its adjoint active beam. An example shows that the variation of eigenvalues with feedback amplitude for the homo-configuration and non-homo-configuration active structures.

ULF geomagnetic anomaly associated with 2000 Izu Islands earthquake swarm, Japan

Electromagnetic phenomena associated with large earthquakes have been investigated to establish any method to monitor the crustal activity such as earthquakes and volcanic eruptions. The ULF geomagnetic approach is considered to be one of the most promising methods for the imminent prediction. In this paper, a new method of principal component analysis (PCA) has been applied to the ULF geomagnetic data associated with 2000 Izu Islands earthquake swarm. This swarm activity started on June 26, 2000 and terminated in September, 2000, and during this activity there were observed 5 large earthquakes whose magnitude was grater than 6 (July 1, 8, 15, 30, and August 18). During this period our ULF stations were fortunately in operation, and there are three stations closely distributed (about 5 km distances). The epicentral distances are about 80–100 km. The PCA has been applied to the ULF horizontal NS component, because we can distinguish a few noises sources based on the orthogonal decomposition in the PCA. We investigate the temporary variations of eigenvalues and eigenvectors of PCA results, and the results are summarized as follows. The first principal component is found to be the signal originated in the solar–terrestrial effects such as geomagnetic pulsation. The variation of eigenvector for the first principal component suggests that this signal is very stable over the whole analyzed period. As for the second principal component, the local artificial noise is included. As for the smallest third component in the local midnight at the Izu Peninsula, it indicated an apparent increase in the third eigenvalue a few days before the large earthquakes. Also about three months before the beginning of swarm activity, the level of the third eigenvalue was slightly enhanced. Correspondingly, the pattern of eigenvector direction in the signal subspace is changed simultaneously and it recovered to the original position after the swarm activity. These features are likely to be correlated with large earthquakes. Finally we want to emphasize that PCA approach is promising for monitoring the crustal activity.

Self-oscillating and chaotic behaviour of a PI-controlled CSTR with control valve saturation

Abstract The time response of a PI-controlled continuous stirred tank reactor (CSTR), where a first order irreversible reaction occurs, is analysed in this paper. The parameters of the PI controller and the saturation of the cooling water temperature flow are used as parameters to show that self-oscillations and chaotic dynamics can appear. The conditions under which self-oscillation occurs are investigated taking into account heat generation, heat removal and the variation of eigenvalues of the linearized system at the equilibrium points, respectively. Also, the appearance of a Silnikov type homoclinic orbit, giving chaotic dynamics, is realised. The existence of a new set of strange attractors with PI control has been verified and the sensitivity to the initial conditions is also analysed.

Curve Veering of Eigenvalue Loci of Bridges with Aeroelastic Effects

The eigenvalues of bridges with aeroelastic effects are commonly portrayed in terms of a family of frequency and damping loci as a function of mean wind velocity. Depending on the structural dynamic and aerodynamic characteristics of the bridge, when two frequencies approach one another over a range of wind velocities, their loci tend to repel, thus avoiding an intersection, whereas the mode shapes associated with these two frequencies are exchanged in a rapid but continuous way as if the curves had intersected. This behavior is referred to as the curve veering phenomenon. In this paper, the curve veering of cable-stayed and suspension bridge frequency loci is studied. A perturbation series solution is utilized to estimate the variations of the complex eigenvalues due to small changes in the system parameters and establish the condition under which frequency loci veer, quantified in terms of the difference between adjacent eigenvalues and the level of mode interaction. Prior to the discussion of bridge frequency loci, the curve veering of a two-degree-of-freedom system comprised of a primary structure and tuned mass damper is discussed, which not only provides new insight into the dynamics of this system, but also helps in understanding the veering of bridge frequency loci. To study this more complicated dynamic system, a closed-form solution of a two-degree-of-freedom coupled flutter is obtained, and the underlying physics associated with the heaving branch flutter is discussed in light of the veering of frequency loci. It is demonstrated that the concept of curve veering in bridge frequency loci provides a correct explanation of multimode coupled flutter analysis results for long span bridges and helps to improve understanding of the underlying physics of their aeroelastic behavior.

Feature Variation and its Impact on Structural Acoustic Response Predictions

This paper presents a screening technique to assess the impact on model fidelity introduced by variations in the properties or positions of features in harmonically forced fluid-loaded structural acoustic models. The perspective taken is one of knowledge of a reference state, with a desire to determine the impact on the total radiated acoustic power due to perturbations in the reference state. Such perturbations change the predicted resonance frequencies of a structure under consideration, and hence, change the predicted response amplitudes. The method uses a single degree of freedom response model in the local region of each fluid-loaded resonance, coupled with eigenvalue sensitivities or variations, to estimate the perturbation impact. The perturbation is scaled by the degree to which each given mode participates in the response quantity of interest. The SDOF model yields results that indicate that proportional bandwidth analysis will be less sensitive to perturbation than constant bandwidth analysis. This is demonstrated through comparison of a constant bandwidth analysis and a 1/3 octave analysis applied to the same system. Elements of the analysis method are not necessarily restricted to model perturbations nor acoustic power, rather they may be used to assess the perturbation of any quadratic response quantity of interest due to changes in resonance frequency.

Three-dimensional asymptotic stress field at the front of an unsymmetric bimaterial pie-shaped wedge under antiplane shear loading

Three-dimensional asymptotic stress field in the vicinity of the front of a bimaterial pie-shaped wedge of general (unsymmetric) geometry is obtained by a recently developed eigenfunction expansion method. The bimaterial pie-shaped wedge is subjected to four combinations of wedge-side boundary conditions – free–free, clamped–clamped, free–clamped and clamped–free. Each material is isotropic and elastic, but with different material properties. Additionally, numerical results pertaining to the variation of the lowest eigenvalues (or stress singularities) with respect to the wedge opening angle of the bimaterial wedge, subjected to the aforementioned wedge-side boundary conditions, are also presented. Dependence of the same on the material properties of the component material phases is also an important part of the present investigation.

Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains

We establish an asymptotic estimate of the lowest eigenvalue μ ( b F ) \mu (b\mathbf {F}) of the Schrödinger operator − ∇ b F 2 -\nabla _{b\mathbf {F}}^{2} with a magnetic field in a bounded 2 2 -dimensional domain, where curl F \mathbf {F} vanishes non-degenerately, and b b is a large parameter. Our study is based on an analysis on an eigenvalue variation problem for the Sturm-Liouville problem. Using the estimate, we determine the value of the upper critical field for superconductors subject to non-homogeneous applied magnetic fields, and localize the nucleation of superconductivity.

Regular self-oscillating and chaotic dynamics of a continuous stirred tank reactor

A continuous stirred tank reactor for propylene glycol production is shown in this paper, analyzing the regular, self-oscillating and chaotic dynamics, which can appear under specific operating conditions. The paper shows that there is a small interval of concentration and temperature of the inlet stream to the reactor, that produces self-oscillating behavior, taking into account the variation of eigenvalues of the linearized system at the equilibrium point, both the multi-dimensional and two-dimensional simplified models of the reactor. The conditions that produce chaotic dynamics are researched when the reactor is in self-oscillating regime, showing that a coolant flow rate periodic variation can produce chaotic behavior. Through the determination of chaotic maps, bifurcation diagrams, the existence of a new set of strange attractors has been tested, showing that the mechanism of chaos is due to the transition from quasi-periodic initial state. A short explanation of the simulation programs is also presented.