Eigenvalues Of Modes Research Articles

Pulsed-Neutron Die-Away Experiments for Plastics and Neutron Thermal Scattering Laws

Pulsed-neutron die-away (PNDA) experiments can be useful benchmarks to validate neutron thermal scattering laws (TSLs). The experiment uses a neutron generator to impinge a short (∼10−4 s), mono-energetic neutron pulse on a target sample. After the pulse, the neutron population within the sample moderates and reaches thermal equilibrium with a fundamental spatial mode and characteristic decay-time eigenvalue. The eigenvalue can be extracted from the experimental measurements of the neutron flux and used as an integral parameter in validation. For certain materials and geometric configurations, the eigenvalue is heavily influenced by thermal neutron scattering of only the target material. For that reason, a PNDA experiment can have a higher sensitivity to TSLs than is commonly available in critical experiments. Herein, we present results for a series of new PNDA experiments conducted at Lawrence Livermore National Laboratory with plastic materials, e.g. high-density polyethylene and Lucite. We compare the experimental integral parameters to simulated results and report trends in the biases. We evaluate the bias with different Monte Carlo transport codes (MCNP6.2 and Mercury) and show no significant differences between the results of the two codes.

Mode coupling in a coastal wedge due to quasi-crossing of range-dependent modal eigenvalues.

A specific mechanism of mode coupling in a waveguide propagation is studied when two range-dependent eigenvalues approach each other. This phenomenon is analogous to the so-called quasi-crossing of states in atomic physics (Landau-Zener theory). It is considered for the sound wave propagation in a coastal wedge in the presence of a sound-speed profile. The change in mode composition and the corresponding spatial variability of the sound field are analyzed by using modes coupling equations and the parabolic equation with a field decomposition over adiabatic modes, respectively.

Multiple Stacks of Rotated Metal Strip Gratings for High-Gain Bandwidth Enhancement and Gain Control of Overlying Dipoles—Analysis by Asymptotic Strips Boundary Conditions

As an amenable and efficient alternative to formulaically cumbersome and computationally demanding full-wave modal approaches, the herein analytical method uses the asymptotic strips boundary conditions along with classical vector potentials and coordinate transformations to treat multiple stacks of rotated metal strip gratings with general dielectric layers between them. For all cases, either with conducting terminal planes on just one side, on both, or none at all, solutions to plane-wave illuminated and/or modal dispersion eigenvalue problems are all obtainable. With the additional degrees of parametric freedom afforded by the angles of rotation between gratings, not only may wider ranges of special properties such as artificial surface impedances be achieved at fixed frequencies, they can also be reconfigured by the rotation of the gratings, being a simple, single-sweep-type mechanical motion, as opposed to electronic means entailing complex networks and intricate circuitries. In addition to the presentation of the formulation details and validation results, this article demonstrates the importance of multiple stacks of rotated strip gratings in widening the otherwise compromised bandwidths of artificial magnetic conductor (AMC) surface characteristics subjected to strict reflection-phase criterion for preserving high maximal broadside gains of overlying prostrate dipole antennas as well as attaining full-range gain control. Prototypes are manufactured and measured with a good agreement with theory.

Physics-Informed Neural Network Method and Application to Nuclear Reactor Calculations: A Pilot Study

In this paper, the physics-informed neural network (PINN) method is investigated and applied to nuclear reactor physics calculations with neutron diffusion models. The reactor problems were introduced with both fixed-source and eigenvalue modes. For the fixed-source problem, the loosely coupled reactor model was solved with the forward PINN approach, and then, the model was used to optimize the neural network hyperparameters. For the k-eigenvalue problem, which is unique for reactor calculations, the forward PINN approach was modified to expand the capability of solving for both the fundamental eigenvalue and the associated eigenfunction. This was achieved by using a free learnable parameter to approximate the eigenvalue and a novel regularization technique to exclude null solutions from the PINN framework. Both single-energy-group and multiple-energy-group diffusion models were examined in the work to demonstrate the PINN capabilities of solving systems of coupled partial differential equations in reactor problems. A series of numerical examples was tested to demonstrate the viability of the PINN approach. The PINN solution was compared against the finite element method solution for the neutron flux and the power iteration solution for the k-eigenvalue. The error in the predicted flux ranged from 0.63% for simple fixed-source problems up to about 15% for the two-group k-eigenvalue problem. The deviations in the predicted k-eigenvalues from the power iteration solver ranged from 0.13% to 0.92%. These generally acceptable results preliminarily justified the feasibility of PINN applications in reactor problems. The advantageous application potentials as well as the observable computational deficits of the PINN approaches are discussed along with the pilot study.

Electromagnetic Parametric Modeling Using Combined Neural Networks and RLGC-Based Eigenfunctions for Two-Port Microstrip Structures

This article proposes a novel electromagnetic (EM) parametric modeling technique using combined neural networks and resistance, inductance, conductance, and capacitance (RLGC)-based eigenfunctions (neuro-EF) for microwave components with two-port microstrip structures. In the proposed technique, neural networks are used to learn the unknown relationship between the parameters of RLGC of the eigenfunction and geometrical parameters. The generation of training data depends on obtaining the correct eigenvalues of different modes. However, for different geometrical parameter samples, there is no uniform correspondence between the calculated eigenvalues and the modes. The incorrect correspondence may cause the two modes to be swapped, defined as the mode-swap issue. We propose a mode-matching method based on eigenvectors to solve this mode-swap issue. After the training data of eigenfunction parameters is obtained, a preliminary training of the neural networks and a two-step refinement training of the neuro-EF model are proposed to develop the overall EM parametric model. By the proposed modeling technique, the trained model can provide a fast and accurate prediction of EM responses for two-port microstrip structures as geometrical parameters change. For the parametric modeling of microwave components with microstrip structures, the proposed technique can obtain better accuracy in larger geometrical variations compared with the existing methods. Two examples of microstrip structures are used to illustrate the proposed technique.

Quantitative Assessment of Dynamic Stability Characteristics for Jet Transport in Sudden Plunging Motion

In this paper, we present a monitoring program of loss control prevention for airlines to enhance aviation safety and operational efficiency. The assessments of dynamic stability characteristics based on the approaches of oscillatory motion and eigenvalue motion modes for jet transport aircraft response to sudden plunging motions are demonstrated. A twin-jet transport aircraft encountering severe clear-air turbulence in transonic flight during the descending phase was examined as the study case. The flight results in sudden plunging motions with abrupt changes in attitude and gravitational acceleration (i.e., the normal load factor) are provided. Development of the required thrust and aerodynamic models with the flight data mining and the fuzzy logic modeling techniques was carried out. The oscillatory derivatives extracted from these aerodynamic models were then used in the study of variations in stability characteristics during the sudden plunging motion. The fuzzy logic aerodynamic models were utilized to estimate the nonlinear unsteady aerodynamics while performing numerical integration of flight dynamic equations. The eigenvalues of all motion modes were obtained during time integration. The positive real part of the eigenvalues is to indicate unstable motion. The dynamic stability characteristics during sudden plunging motion are easily judged by the values in positive or negative. The present quantitative assessment method is an innovation to examine possible mitigation concepts of accident prevention and promote the understanding of aerodynamic responses of the jet transport aircraft.

Dynamic mode decomposition for subcriticality measurement using measured data at single location

Dynamic mode decomposition (DMD) is an effective approach for extracting a fundamental mode eigenvalue in subcriticality measurements. According to several previous studies, data for DMD must be obtained at a sufficient number of locations. However, the number of measurement locations is typically limited owing to experimental constraints. In this study, a time-shifting augmentation technique for DMD is employed, where an augmented data matrix is constructed by stacking multiple time-shifted copies of time-series data at a single location. The proposed DMD technique is applied to numerical simulations of subcriticality measurements, such as those involving the pulsed neutron method and Rossi-α method. The fundamental mode eigenvalues of these measurements are accurately extracted via DMD using the augmented data matrix for a single location without introducing the masking time technique.

Investigation of the AGN-201M Research Reactor’s Unique Dominance Ratio

The dominance ratio is the ratio of the first higher-order mode eigenvalue of a system to the fundamental eigenvalue, k1/k0. It can be used to determine how well coupled the neutrons in a multiplying system are, as well as the computational difficulty of the power iteration method in a Monte Carlo simulation. The purpose of this study is to investigate the University of New Mexico’s (UNM’s) AGN-201M reactor’s unusually low dominance ratio of 0.632. The AGN-201M reactor is a small, thermal spectrum reactor located at the UNM. It is moderated by polyethylene, reflected by graphite, and uses fuel comprised of uranium microspheres embedded in polyethylene plates that are separated by an aluminum baffle. The investigation included a parametric study of the reactor’s fuel geometry, fuel density, and reflector thickness to examine their impact on the reactor’s dominance ratio. In addition, neutronically similar systems were examined to identify common causes for systems with low dominance ratios. The reason for the small dominance ratio of the AGN-201M reactor when compared to large thermal reactors was determined to be because of its size and fuel plate composition. The reflector’s effect on the dominance ratio is small in comparison to the other factors but was found to have a nonzero effect. Furthermore, the AGN-201M was found to have a significantly lower dominance ratio than systems with which it shares a very high ( > 95%) degree of neutronic similarity. However, the two most similar systems were close in size to the core of the AGN-201M reactor and were moderated with polyethylene as well.

Computing the quasinormal modes and eigenfunctions for the Teukolsky equation using horizon penetrating, hyperboloidally compactified coordinates

We study the quasinormal mode eigenvalues and eigenfunctions for the Teukolsky equation in a horizon penetrating, hyperboloidally compactified coordinate system. Following earlier work by Zenginoğlu (2011 Phys. Rev. D 83 127502), we show that the quasinormal eigenfunctions (QNEs) for the Teukolsky equation are regular from the black hole horizon to future null infinity in these coordinates. We then present several example QNE solutions, and study some of their properties in the near-extremal Kerr limit.

Assessment of structural health of an existing prestressed concrete bridge by finite element analysis

ABSTRACT This study investigates the feasibility of an alternative structural health monitoring (SHM) technique for an existing, post-tensioned, prestressed concrete bridge in Niigata, Japan. Currently, a static SHM system is in place to detect the progress of damages within the bridge. However, the existing system cannot properly monitor the structural health of the bridge including the periodic vibration, which is one of the damage-sensitive features of interest. Therefore, to effectively detect the real-time performances of the bridge, a three-dimensional (3D) Finite Element (FE) model was developed as a reference and verified using on-site load-deflection test results. After validating the reference FE model, different damage scenarios, such as degradation of concrete, corrosion & rupture of steel tendons and missing tendons, were incorporated in the FE models. Based on non-linear structural and Eigenvalue analyses, natural frequencies and mode shapes of the bridge remain constant even after careful consideration of all types of damages in the FE model. However, vertical displacements are observed to increase for the damage scenarios. Although the effect of tendon rupture and corrosion showed negligible influences on the vertical displacement, the deterioration of the concrete largely influenced the vertical displacement. Additionally, crack widths were found to vary with damage types. Brifely, this study recommends some effective indicators to monitor the structural conditions of the bridge using FE analysis (FEA).

The effect of temperature on frequency and instability variations in a smooth-bore relativistic magnetron

This paper analyzes the extraordinary mode eigenvalue equation to investigate the effects of temperature on frequency and growth rate of instability in a cylindrical smooth-bore relativistic magnetron. This analysis is based on the framework of the macroscopic fluid model as well as Maxwell's equations, which include electromagnetic and relativistic effects comprehensively. We applied linear perturbation theory around the steady state profiles with the local approximation for perturbed density along the radial direction to derive the eigenvalue equation. The derived eigenvalue equation was solved numerically using shooting to a fitting point method. Due to explosive emission, temperature of about 8 eV is reported [Andreev and Hendricks, IEEE Trans. Plasma Sci. 40, 1551 (2012)]. According to the findings of the current study for the first six azimuthal modes, temperature rise can lead to increasing frequency and decreasing instability in a relativistic magnetron. In addition, after a large number of pulses and rising temperature in the system, the effect of temperature should be considered as an effective element in the oscillations of frequency.

Wave mode computing method using the step-split Padé parabolic equation

Models based on a parabolic equation (PE) can accurately predict sound propagation problems in range-dependent ocean waveguides. Consequently, this method has developed rapidly in recent years. Compared with normal mode theory, PE focuses on numerical calculation, which is difficult to use in the mode domain analysis of sound propagation, such as the calculation of mode phase velocity and group velocity. To broaden the capability of PE models in analyzing the underwater sound field, a wave mode calculation method based on PE is proposed in this study. Step-split Padé PE recursive matrix equations are combined to obtain a propagation matrix. Then, the eigenvalue decomposition technique is applied to the matrix to extract sound mode eigenvalues and eigenfunctions. Numerical experiments on some typical waveguides are performed to test the accuracy and flexibility of the new method. Discussions on different orders of Padé approximant demonstrate angle limitations in PE and the missing root problem is also discussed to prove the advantage of the new method. The PE mode method can be expanded in the future to solve smooth wave modes in ocean waveguides, including fluctuating boundaries and sound speed profiles.

Extraction Method andApplication of Unit Mode to Simplify NVH Problem

Conducting transmission path analysis using unit modes with simple mode shapes, the vibration mechanism, which is generally difficult to grasp, can be clarified. This paper explains how to extract multiple unit modes in order of importance. In addition, an application example to a vehicle FE model is introduced, in which road noise is reduced by shifting the eigenvalues of unit modes and thereby changing the vibration path.

Effect of vortex dynamics and instability characteristics on the induced drag of trailing vortices

In this paper, trailing vortices generated by three wingtip configurations, namely the M6 wing and the M6 wing with a blended or split winglet, are experimentally investigated using the Stereo Particle Image Velocimetry (SPIV) technology. Then, linear stability analysis is performed to investigate instability characteristics. Three corresponding trailing vortex patterns, including the isolated trailing vortex without wake (pattern v) and with wake (pattern v-w), co-rotating vortex pair (pattern v-v), are observed in experiments. The strength of trailing vortices, characterized by circulation, is reduced after installing winglets as expected, and the strength of pattern v-v can be further suppressed compared with pattern v-w. Moreover, instability characteristics, such as the eigenvalue spectrum and perturbation mode, are distinctive among these three vortex patterns. The distribution of eigenvalue spectrums indicates that pattern v and pattern v-w are temporally “marginally stable”, while pattern v-v is temporally “unstable.” The primary perturbation mode of pattern v and pattern v-w is the m=-1 helical mode, while |m|>1 for the case of pattern v-v. The effect of vortex dynamics and instability characteristics can be concluded in two aspects. Firstly, the value of induced drag is polluted by about 3% from vortex wandering since vortex wandering affects the tangential velocity and streamwise vorticity of trailing vortices. Secondly, the growth rate and penetration depth perturbation mode affect trailing vortex evolution and further affect induced drag. Specifically, the larger the growth rate and penetration depth are, the more turbulence injects inside the vortex core, thus leading to a quicker and more intense attenuation of trailing vortex, as well as a smaller induced drag. This finding can guide us to manipulate the induced drag in flow control.

Analysis of Synchronous Generators’ Local Mode Eigenvalues in Modern Power Systems

New energy sources, storage facilities, power electronics devices, advanced and complex control concepts, economic operating doctrines, and cost-optimized construction and production of machines and equipment in power systems adversely affect small-signal stability associated with local oscillations. The objective of the article is to analyze local oscillations and the causes that affect them in order to reduce their negative impact. There are no recognized analyses of the oscillations of modern operating synchronous generators exposed to new conditions in power systems. The basic idea is to perform a numerical analysis of local oscillations of a large number of synchronous generators in the power system. The paper represents the local mode data obtained from a systematic analysis of synchronous generators in the Slovenian power system. Analyzed were 74 synchronous generators of the Slovenian power system, plus many additional synchronous generators for which data were accessible in references. The mathematical models convenient for the study of local oscillations are described first in the paper. Next, the influences of transmission lines, size of the synchronous generators, operating conditions, and control systems were investigated. The paper’s merit is the applicable rules that have been defined to help power plant operators avoid stability-problematic situations. Consequently, boundaries were estimated of the eigenvalues of local modes. Finally, experiments were performed with a laboratory-size synchronous generator to assess the regularity of the numerically obtained conclusions. The obtained results enable the prediction of local oscillations’ frequencies and dampings and will be useful in PSS planning.

Nonlinear Aeroelastic Simulations and Stability Analysis of the Pazy Wing Aeroelastic Benchmark

The Pazy wing aeroelastic benchmark is a highly flexible wind tunnel model investigated in the Large Deflection Working Group as part of the Third Aeroelastic Prediction Workshop. Due to the design of the model, very large elastic deformations in the order of 50% span are generated at highest dynamic pressures and angles of attack in the wind tunnel. This paper presents static coupling simulations and stability analyses for selected onflow velocities and angles of attack. Therefore, an aeroelastic solver developed at the German Aerospace Center (DLR) is used for static coupling simulations, which couples a vortex lattice method with the commercial finite element solver MSC Nastran. For the stability analysis, a linearised aerodynamic model is derived analytically from the unsteady vortex lattice method and integrated with a modal structural model into a monolithic aeroelastic discrete-time state-space model. The aeroelastic stability is then determined by calculating the eigenvalues of the system’s dynamics matrix. It is shown that the stability of the wing in terms of flutter changes significantly with increasing deflection and is heavily influenced by the change in modal properties, i.e., structural eigenvalues and eigenvectors.

Mixed mode equilibrium analysis of asymptotic fields around stationary sharp V-notches in polymer gels: A linear poroelastic study

In this paper, the equilibrium asymptotic stress and solvent concentration fields around stationary sharp V-notches are extracted for in-plane mixed mode loadings employing the linear poroelasticity theory. It is shown that at thermodynamic equilibrium, mechanical and chemical equilibrium equations are uncoupled and thus can be solved independently. The mechanical equilibrium equations are then solved using the Airy stress function technique. This technique replaces the coupled mechanical equilibrium relations with the compatibility biharmonic equation which can be solved using the separation of variables method. Applying traction free boundary conditions on the notch edges leads to an eigenvalue problem which gives both mode I and mode II eigenvalues. These eigenvalues can be real or complex numbers depending on the notch opening angle. Since there are infinite eigenvalues for both modes of deformation, series solutions will be obtained for the equilibrium fields. From these asymptotic solutions, it is found that the obtained stress fields are similar to their corresponding linear elasticity solution. Furthermore, from the solutions, it can be observed that the opening and shear deformations near the notch tip lead to cosine and sine variations of the solvent concentration field with respect to the angular coordinates, respectively. The accuracy of these asymptotic results is finally verified using finite element analyses of a single-edge notched (SEN) specimen subjected to far field applied small displacements. The numerical analyses are performed for two notch opening angles of 30° and 60° for both plane stress and plane strain conditions. The comparative study between the finite element results and the asymptotic solution proves that the present solution can accurately capture the near notch tip stress and solvent concentration fields by considering only the first few terms of the series solutions.

Exact closed-form solution for free vibration of Euler-Bernoulli and Timoshenko beams with intermediate elastic supports

This paper presents exact closed-form solutions for free vibration of discretely supported Euler-Bernoulli (DEB) and Timoshenko beams (DTB) in the presence of an arbitrary number of intermediate elastic constraints. The exact eigenvalue equations and mode shapes of the DEB and DTB are derived using the generalized function method for the first time, which expands the general solution of mode shapes as a combination of the standard trigonometric/hyperbolic functions with integration constants extended to generalized functions. The second-order eigenvalue equation is formulated from the perspective of the entire domain of the beam without enforcement of any continuity conditions. Several typical constraints of interest in practical engineering are investigated and other cases can be achieved by following similar methodologies elucidated in this paper. The accuracy of present solutions is validated by numerical results obtained from the finite element method (FEM) and previous literature adopting the transfer matrix method (TMM) and Green's function method (GFM). The exiting frequency equation and mode shape of a beam with elastic end constraints are proved to be the degenerate solutions of current study. Courant's maximum-minimum principle is reproduced with detailed discussions on the support stiffnesses and positions, bimodal behavior, and sensitivity to geometrical variations. As an extended application, a new method based on rigorous mathematical deduction is proposed to efficiently determine the critical stiffness leading to the maximum fundamental frequency of a beam with multiple intermediate supports. This study is of significance to the support optimization and dynamic analysis of discretely supported beam-like structures.

A Simulation Study of Ultra-relativistic Jets–I. A New Code for Relativistic Hydrodynamics

Abstract In an attempt to investigate the structures of ultra-relativistic jets injected into the intracluster medium (ICM) and the associated flow dynamics, such as shocks, velocity shear, and turbulence, we have developed a new special relativistic hydrodynamic (RHD) code in the Cartesian coordinates, based on the weighted essentially non-oscillatory (WENO) scheme. It is a finite difference scheme of high spatial accuracy, which has been widely employed for solving hyperbolic systems of conservation equations. The code is equipped with different WENO versions, such as the fifth-order accurate WENO-JS, WENO-Z, and WENO-ZA, and different time-integration methods, such as the fourth-order accurate Runge–Kutta (RK4) and strong stability preserving RK (SSPRK), as well as the implementation of the equations of state (EOSs) that closely approximate the EOS of the single-component perfect gas in relativistic regimes. In addition, it incorporates a high-order accurate averaging of fluxes along the transverse directions to enhance the accuracy of multidimensional problems, and a modification of eigenvalues for the acoustic modes to effectively control the carbuncle instability. Through extensive numerical tests, we assess the accuracy and robustness of the code, and choose WENO-Z, SSPRK, and the EOS suggested in Ryu et al. as the fiducial setup for simulations of ultra-relativistic jets. The results of our study of ultra-relativistic jets using the code is reported in an accompanying paper.

Mode coupling in a coastal wedge due to quasi crossing of range-dependent modal eigenvalues and spatial sound field variability

In the paper sound waves propagation in a coastal wedge with sound speed profile (the presence of thermocline) is studied. In down- and up-slope propagation each adiabatic waveguide mode is transformed from the “bottom—surface reflected” shape to the “bottom-bottom refracted” one at some distance from the edge, depending on mode number. Analysis of mode composition and the corresponding spatial variability of the sound field is carried out using Parabolic Equation with the following decomposition of the field over adiabatic modes. It is shown, that there is significant change of amplitudes of adiabatic modes which is interpreted as manifestation of mode coupling. It is shown that mode coupling has local character, at the definite distance from the source, where two range dependent eigenvalues convergent to each other. This phenomenon is analog to the so called quasi crossing of states in atomic physics (Landau–Zener theory of nonadiabatic transitions in two-level system). This leads to sequential excitation of more and more higher modes in down-slope propagation and even creation of higher modes which did not exist at the position of the source. Results of modeling are presented, possible experimental observations are discussed. [Work was supported by RFBR, Grant 20-55-S52005.]