Generalized Eigenvalue Equation Research Articles

Euclidean time method in generalized eigenvalue equation

Euclidean time method in generalized eigenvalue equation

Integral Operator-Based Characteristic Mode Theory for Conducting, Material, and Lossy Structures

Integral operator-based theory of characteristic modes (CMs) for conducting, material, and lossy structures is reviewed. CMs are defined as solutions of a generalized eigenvalue equation (GEE). These GEEs are presented for various surface and volume integral operators and material structures. Interpretation of the characteristic eigenvalues in terms of electromagnetic power is studied based on the Mie expansion and integral operator formalism. Orthogonality and diagonalizing properties of the CMs are summarized. Challenges related to dielectric-magnetic bodies, lossy structures and spurious modes are discussed, as well as differences between the surface and volume operator approaches.

Research on Reradiation Interference Resonance Mechanism of Power Transmission Lines in Medium-Wave Band Based on Characteristic Mode Theory

The key to decreasing reradiation interference (RRI) of power transmission lines on adjacent radio stations is to clarify the RRI resonance mechanism. Aiming at the defects of existing RRI resonance analysis methods, including frequency limitation and lack of physical explanation, a RRI resonance analysis method based on characteristic mode (CM) theory is proposed in this paper. Firstly, based on the generalized eigenvalue equation, a set of characteristic currents with orthogonal relationship and their eigenvalues for power transmission lines are solved, and combined with Poynting’s theorem, the CM radiation characteristics are analyzed. Secondly, under specific external excitation, CM-related parameters are obtained through modal decomposition. Finally, the total energy radiated by power transmission lines is decomposed into the superposition of energy radiated by each CM, and the mechanism of RRI resonance is clarified from the perspective of CM. The simulation results show that compared with the IEEE guide and the generalized resonance theory, the method in this paper is effective and independent of observation points, which can provide a theoretical support for further research on the RRI suppression measures.

Characteristic Modes Analysis of Aperiodic Arrays Using Entire Domain Basis Function Method With Adaptive Cross Approximation Algorithm

The characteristic modes of the aperiodic arrays are analyzed by combining the entire domain basis function method with the adaptive cross approximation (ACA) algorithm. The array cells are identical. The entire domain basis functions defined on the array cells are obtained by analyzing the characteristic modes of the reference cell firstly. Then, the generalized eigenvalue equation whose size is smaller than the conventional one can be obtained. After solving the equation, the characteristic modes of the whole array can be analyzed. Calculation of the matrices is very time-consuming. To speed up the computation, the ACA algorithm is applied. Several numerical examples are performed. The accuracy and computational efficiency of the proposed technique are validated by simulation results.

MoM Solution of Broadband Radiation and Scattering Problems Based on Entire-Domain Hyperbasis Functions Approach

A new method of moments (MoM)-based approach is proposed to efficiently solve broadband radiation and scattering problems on triangulated, quadrilateral, and mixed mesh geometries. This approach is based on expanding the total current on a conducting body at any desired frequency in terms of entire-domain hyperbasis functions (HBFs), calculated at only one frequency. These HBFs must be found as eigenfunctions of the generalized eigenvalue equation (GEE), defined at that frequency. The developed approach is validated by comparing results with those obtained by direct MoM for different types of mesh. Finally, it is demonstrated for solving practical antenna problems.

Orthogonality Properties of Characteristic Modes for Lossy Structures

Orthogonality of the characteristic modes with respect to the weight operator of the generalized eigenvalue equation (GEE), and in the far-field, is investigated in the case of lossy conducting and dielectric objects. Linking the weight operator to radiated power is shown to provide orthogonal far fields in the lossless case. In the lossy case, both the orthogonality of the characteristic far fields and the weight operator orthogonality of the modal currents are satisfied with respect to the Hermitian inner products only for sufficiently symmetric geometries, such as a sphere. For irregular lossy shapes, independent of the symmetry of the formulation, the far-field orthogonality can be obtained only with respect to the symmetric (non-Hermitian) product. The weight operator orthogonality can be satisfied with (complex) symmetric formulations, but again only with respect to the symmetric product. Since the symmetric products are not related to any physical power quantity, the modes do not form a (radiated) power orthogonal set in the lossy case. Hence, for lossy structures, characteristic modes (CMs) do not satisfy their classical definition and they need to be redefined.

Stochastic natural frequency analysis of varying diameter functionally graded material pipe conveying fluid

This paper investigates the uncertainty of the structure and material parameters on the natural frequency of the varying diameter functionally graded material (FGM) pipe conveying fluid. Firstly, concentrating on the spatially dependent uncertainty of geometry varying in axial direction as well as the material varying in radial direction, the random field is employed to describe the uncertainty parameters. Then, a generalized eigenvalue equation of the varying diameter FGM pipe is derived by the dynamic stiffness method while introducing the random field and variables into the deterministic governing equation. Due to the input uncertainty, the natural frequencies of the varying diameter FGM pipe change to random variables. In this respect, the stochastic characteristics of natural frequencies can be approximated by the Monte Carlo simulation. However, the enormous computational cost is unbearable. The efficient artificial neural network surrogate model is introduced and incorporated for the natural frequency uncertainty quantification. Consequently, a stochastic natural frequency analysis approach based on the neural network is proposed for the varying diameter FGM pipe conveying fluid. Some examples are investigated to illustrate the effectiveness of the proposed method and the influence of input uncertainty on the probabilistic characterization of the natural frequencies.

Subordinacy theory for extended CMV matrices

We develop subordinacy theory for extended CMV matrices. That is, we provide explicit supports for the singular and absolutely continuous parts of the canonical spectral measure associated with a given extended CMV matrix in terms of the presence or absence of subordinate solutions of the generalized eigenvalue equation. Some corollaries and applications of this result are described as well.

Probing anharmonic phonons by quantum correlators: A path integral approach.

We devise an efficient scheme to determine vibrational properties from Path Integral Molecular Dynamics (PIMD) simulations. The method is based on zero-time Kubo-transformed correlation functions and captures the anharmonicity of the potential due to both temperature and quantum effects. Using analytical derivations and numerical calculations on toy-model potentials, we show that two different estimators built upon PIMD correlation functions fully characterize the phonon spectra and the anharmonicity strength. The first estimator is associated with the force-force quantum correlators and, in the weak anharmonic regime, yields reliable zero-point motion frequencies and thermodynamic properties of the quantum system. The second one is instead connected to displacement-displacement correlators and accurately probes the lowest-energy phonon excitations, regardless of the anharmonicity strength of the system. We also prove that the use of generalized eigenvalue equations, in place of the standard normal mode equations, leads to a significant speed-up in the PIMD phonon calculations, both in terms of faster convergence rate and smaller time step bias. Within this framework, using ab initio PIMD simulations, we compute phonon dispersions of diamond and of the high-pressure I41/amd phase of atomic hydrogen. We find that in the latter case, the anharmonicity is stronger than previously estimated and yields a sizeable red-shift in the vibrational spectrum of atomic hydrogen.

Derivation and Computation of Surface Differential Equations for the Induced Current on Perfectly Electrically Conducting Objects in Electromagnetics

Surface differential equations (SDEs) are derived for the induced current on the outer surface of a perfectly electrically conducting (PEC) object for the first time. The derived equations are wave equations that describe the propagation of surface current waves along a PEC surface. A numerical method based on SDEs is proposed, which is named the eigenmode theory in this article. In this method, SDEs are discretized into a generalized eigenvalue equation, which can be used to obtain a series of modal currents and expand the surface current induced by any excitation. The modal currents are only dependent on the shape and size of the PEC object and are independent of any specific excitation, including frequency, feeding location, and so on. This property results in an eigenmode theory with great potential in many challenging antenna designs. Numerical examples for both scattering and radiation analyses demonstrate the effectiveness and accuracy of eigenmode theory and, thus, give evidence for the correctness of SDEs.

Fast analysis of scattering from metallic‐dielectric composite large antenna arrays using characteristic modes

AbstractIn this article, characteristic modes for metallic‐dielectric composite structures (MDCS) are explored for fast and accurate analysis of scattering from large antenna arrays. Specifically, a generalized eigenvalue equation is formulated from a method of moments (MoM) impedance matrix of the electric field integral equation and Poggio, Miller, Chang, Harrington, Wu, and Tsai formulation. Modal currents for an antenna element with MDCS are then used as entire domain basis functions in the MoM for expanding equivalent currents over a large antenna array with MDCS. Only a small number of the modal currents are required to approximate the induced current over the antenna array with sufficient accuracy. Thus, the number of the unknowns and the corresponding CPU time in the MoM are significantly reduced in the proposed method. Furthermore, as the periodicity and intrinsic relationship of the matrix elements are considered when using the modal currents as entire domain basis functions, the computational efficiency of the proposed method is further improved. Numerical results of typical antenna arrays with MDCS are presented to illustrate the efficiency and accuracy of the proposed method.

Variant Characteristic Mode Equations Using Different Power Operators for Material Bodies

This paper is concerned with the electromagnetic theory of characteristic modes for general homogeneous material bodies using surface integral equations (SIEs), which are computationally more efficient than using volume integral equations (VIEs). However, the generalized eigenvalue equations (GEEs) based on SIEs are prone to generate spurious or extra modes, unlike their counterparts based on VIEs. In this work, several variant SIE-based GEEs are proposed to eliminate the spurious/extra modes and improve the numerical performances. These variant GEEs share a common point that the reactive power operator related to the material domain should be removed from the left-hand sides of GEEs by explicitly enforcing the boundary condition, because it seems sensitive to incur unwanted modes. These variant GEEs are diverse in choosing the active power operator related to the material domain on the right-hand sides to suit different medium parameters, which even can be dropped directly for lossless cases. Numerical results show that these variant GEEs do not generate spurious/extra resonant modes in observed frequency range, and at least one of the variants would have acceptable numerical accuracy.

Analysis of Quadruple Corner-Cut Ridged Elliptical Waveguide by NURBS Enhanced Scaled Boundary Finite Element Method

The scaled boundary finite element method(SBFEM) is a semi-analysis method, combing the advantages of boundary element method and finite element method. However, in solving the quadruple corner-cut ridged elliptical(QCRE) waveguide, the traditional SBFEM employ the Lagrange polynomials as the basis functions which leads to the curved boundaries cannot be exactly represented and the continuity order across element is low. In this paper, a non-uniform rational B-spline (NURBS) enhanced SBFEM is firstly extended to solve the QCRE waveguide, which can exactly describe the curved boundaries, reduce the spatial dimensions by one and obtain analytically results in the radial direction. According to its symmetry, only a quarter of the ridged elliptical waveguide needs to be simulated and subdivided into several subdomains. The curved boundaries and straight boundaries of the subdomains are described and discretized by the NURBS and Lagrange basis functions, respectively. The side-face boundaries do not need to be discretized. Then, the NURBS enhanced SBFEM governing equation of the waveguide eigenvalue problem is derived based on the vibrational principle and scaled boundary coordinate transforming. Finally, a generalized eigenvalue equation respecting to the cut-off wave numbers is established by introducing the boundary dynamic stiffness and employing a continued fraction solution. Numerical results verify the high computational efficiency and accuracy of the NURBS enhanced SBFEM with exactly describing the curved boundaries. The influence of the corner-cut on the cut-off wave numbers of several modes and single-mode bandwidth are investigated in details.

Vibration Analysis of Vessels Conveying Blood Flow Embedded in Viscous Fluid

Vibration analysis of vessels conveying blood flow embedded in viscous fluid is studied based on the modified strain gradient theory. The viscoelastic vessels are simulated as a non-classical Euler-Bernoulli beam theory. Employing Hamilton’s principle, the governing equations for size-dependent vessels are derived. The Galerkin method is used in order to transform the resulting equations into general eigenvalue equations. The effects of the blood flow profile and its modification factors, red blood cells (RBCs) and hematocrit are considered in the blood flow. Besides, the influences of the constitutional material gradient scale, blood flow, internal pressure, structural damping coefficient, viscous fluid substrate and various boundary conditions on the natural frequencies and critical buckling velocities are studied. It is revealed that as the hematocrit, fluid viscosity of substrate, internal pressure and mass ratio increase, the natural frequencies and critical buckling velocities decrease. Furthermore, the results indicated that the strain gradient theory predicts the highest natural frequencies and critical buckling velocities among others. The results are compared with those available in the literature and good agreement has been observed.

An efficient stochastic natural frequency analysis method for axially varying functionally graded material pipe conveying fluid

This study proposes an effective approach for stochastic natural frequency analysis of axially varying functionally graded material (FGM) pipes conveying fluid. The Karhunen-Loève expansion is first employed to generate a random filed with axially varying FGM properties. Then the input uncertainty is substituted into the governing equation of axially varying FGM pipes conveying fluid. The stochastic differential quadrature method is introduced to derive the generalized eigenvalue equation and obtain the natural frequencies. Based on the established stochastic mechanical model, the statistical characteristics of the stochastic natural frequency response can be estimated. In order to mitigate the calls of the mechanical model, a Kriging model requiring a rather small design of experiment is utilized to mimic the true mechanical model. An approach combining the material random field, the derived generalized eigenvalue equation and the Kriging model is finally proposed. The example investigations have validated the effectiveness as well as the efficiency of the proposed strategy for stochastic natural frequency analysis of axially varying FGM pipes conveying fluid with random material properties.

Characteristic Mode Formulations for Penetrable Objects Based on Separation of Dissipation Power and Use of Single Surface Integral Equation

Novel characteristic mode (CM) formulations for penetrable objects based on explicit separation of dissipation power from the total active power and use of single surface integral equation (SSIE) are proposed. Due to the separation of the dissipation power, the right-hand side of the generalized eigenvalue equation can be set to be exclusively related to the radiated power, which permits the far-field pattern by different CMs to be orthogonal whether the material is lossless or not. The use of SSIE allows both the equivalent electric and magnetic currents to be expressed by a single effective electric or magnetic current, so that the conventional inverse to express the equivalent electric current by the equivalent magnetic current or vice versa is avoided, which is favorable for frequency sweeping simulations. Numerical examples for verifications are provided by comparing with acknowledged results or measured data.

Theory of Characteristic Modes for Nonsymmetric Surface Integral Operators

The theory of characteristic modes is formulated with nonsymmetric surface integral operators for perfect electric conductors, impedance surfaces, and homogeneous dielectric bodies. For nonsymmetric (nonself-adjoint) operators, the eigenvectors are not orthogonal with respect to the weighted inner product defined with the weighting operator of the generalized eigenvalue equation. Rather, this orthogonality holds between the eigenvectors of the original equation and the adjoint equation, including adjoint operators. This implies that the modal expansion, used to express any scattering or radiation solution as a linear combination of the modes, requires these two sets of eigenvectors. For matrix equations, the eigenvectors of the adjoint equation correspond to the left eigenvectors of the original equation.

PMCHWT-Based Characteristic Mode Formulations for Material Bodies

By studying solutions of various forms of the Poggio–Miller–Chang–Harrington–Wu–Tsai (PMCHWT)-based characteristic mode (CM) formulations, we have found that the proper choice of the weighting operator of the generalized eigenvalue equation is crucial in developing formulations that are immune from spurious solutions. Spurious modes of the Chang–Harrington surface formulation for CMs of lossless material bodies are shown to be solutions of a reverse problem where the material parameters of the background and the object are swapped. These unwanted modes can be avoided by removing the integral operators of the interior of the object from the weighting operator of the generalized eigenvalue equation. We also illustrate that to obtain physically meaningful eigenvalues with orthogonal far-field patterns, it is necessary to define that weighting operator so that it connects the eigensolutions to radiated power.

First-principle calculation of Chern number in gyrotropic photonic crystals.

As an important figure of merit for characterizing the quantized collective behaviors of the wavefunction, Chern number is the topological invariant of quantum Hall insulators. Chern number also identifies the topological properties of the photonic topological insulators (PTIs), thus it is of crucial importance in PTI design. In this paper, we develop a first principle computatioal method for the Chern number of 2D gyrotropic photonic crystals (PCs), starting from the Maxwell's equations. Firstly, we solve the Hermitian generalized eigenvalue equation reformulated from the Maxwell's equations by using the full-wave finite-difference frequency-domain (FDFD) method. Then the Chern number is obtained by calculating the integral of Berry curvature over the first Brillouin zone. Numerical examples of both transverse-electric (TE) and transverse-magnetic (TM) modes are demonstrated, where convergent Chern numbers can be obtained using rather coarse grids, thus validating the efficiency and accuracy of the proposed method.

Polarization resolved analysis of phonon transport in a multi-terminal system**Project supported by the National Natural Science Foundation of China (Grant No. 51376094) and Jiangsu Overseas Visiting Scholar Program for University Prominent Young & Middle-aged Teachers and Presidents, China.

The atomistic Green’s function method is improved to compute the polarization resolved phonon transport in a multi-terminal system. Based on the recent developments in literature, the algorithm is simplified. The complex phonon band structure of a semi-infinite periodic terminal is obtained by the generalized eigenvalue equation. Then both the surface Green’s function and phonon group velocity in the terminal are determined from the wave modes propagating away from the scattering region along the terminal. With these key ingredients, the individual phonon mode transmittance between the terminals can be calculated. The feasibility and validity of the method are demonstrated by the chain example compared with the wave packet method, and an example of graphene nanojunction with three terminals.