Mode Approximation Research Articles

Subcriticality measurement using time-domain decomposition-based integral method for simultaneous reactivity and source changes

ABSTRACT To estimate the subcriticality in dollar units for an arbitrary state-change, the time-domain decomposition-based integral method (TDDI) is proposed using the point kinetics theory based on the fundamental mode approximation. In a general transient subcritical system, reactivity, neutron source intensity, and point kinetics parameters can vary simultaneously. Furthermore, the state-change may not necessarily be a stepwise change. For such a transient, the TDDI method can estimate the subcriticality after the transient using only the time variation of the neutron count rate. Therefore, the proposed method is useful to approximately estimate the subcriticality in a system where a detailed core configuration is unknown. To investigate the applicability of the TDDI method, transient experiments with simultaneous reactivity and source changes or to two successive safety rods dropping were performed at the Kindai University Training and Research Reactor (UTR-KINKI). By comparing with reference values using excess reactivity and control rod worth, it was validated that the subcriticality values obtained by the TDDI method better agree with the reference values than the previous integral method.

Thermodynamic properties of graphene bilayers

Thermodynamic properties of graphene bilayers are studied by path-integral molecular dynamics (PIMD) simulations, considering quantization of vibrational modes and anharmonic effects. Bilayer graphene has been studied at temperatures between 12 and 1500 K for zero external stress, using the LCBOPII effective potential. We concentrate on the thermal expansion, in-plane and out-of-plane compressibility, and specific heat. Additional insight into the meaning of our results for bilayer graphene is obtained from a comparison with data obtained from PIMD simulations for monolayer graphene and graphite. They are also analyzed in view of experimental data for graphite. Zero-point and thermal effects on the in-plane and ``real'' area of bilayer graphene are studied. The thermal expansion coefficient ${\ensuremath{\alpha}}_{xy}$ of the in-plane area is negative at low temperatures and positive for $T\ensuremath{\gtrsim}800\phantom{\rule{0.16em}{0ex}}\mathrm{K}$. The minimum ${\ensuremath{\alpha}}_{xy}$ is $\ensuremath{-}6.6\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}\phantom{\rule{4pt}{0ex}}{\mathrm{K}}^{\ensuremath{-}1}$ at $T\ensuremath{\approx}220\phantom{\rule{4pt}{0ex}}\mathrm{K}$. Both in-plane (${\ensuremath{\chi}}_{xy}$) and out-of-plane (${\ensuremath{\chi}}_{z}$) compressibilities of graphene bilayers are found to increase for rising temperature, and turn out to be lower than that corresponding to monolayer graphene and higher than those found for graphite. At 300 K, we find for the bilayer ${\ensuremath{\chi}}_{xy}=9.5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}\phantom{\rule{4pt}{0ex}}{\AA{}}^{2}/\mathrm{eV}$ and ${\ensuremath{\chi}}_{z}=2.97\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}\phantom{\rule{4pt}{0ex}}{\mathrm{GPa}}^{\ensuremath{-}1}$. Results for the specific heat obtained from the simulations are compared with those given by a harmonic approximation for the vibrational modes. This approach is noticeably accurate at temperatures lower than 200 K.

Variational Bethe ansatz approach for dipolar one-dimensional bosons

We propose a variational approximation to the ground state energy of a one-dimensional gas of interacting bosons on the continuum based on the Bethe Ansatz ground state wavefunction of the Lieb-Liniger model. We apply our variational approximation to a gas of dipolar bosons in the single mode approximation and obtain its ground state energy per unit length. This allows for the calculation of the Tomonaga-Luttinger exponent as a function of density and the determination of the structure factor at small momenta. Moreover, in the case of attractive dipolar interaction, an instability is predicted at a critical density, which could be accessed in lanthanide atoms.

On natural frequencies of three-dimensional moonpool of vessels in the fixed and free-floating conditions

The hydrodynamic interactions between the vessel motion and resonant wave response inside the three-dimensional moonpool are studied in the present paper. In particular, the shift of natural frequencies of a three-dimensional moonpool in the fixed and free-floating conditions is computed and examined systematically. Barges with different moonpool dimensions are analyzed to examine how the moonpool configurations affect the shift of resonant frequencies. Detailed analyses are performed for cases with and without recess. Numerical computations are performed for regular waves with wave headings θ=0∘, 45∘ and 90∘. Free-surface elevations inside the moonpool with respect to the incident wave frequencies are computed for both fixed and free-floating conditions. It is observed that natural frequencies of the moonpool in the free-floating condition are higher than those in the fixed condition. For cases without recess, the shift of resonant frequency is more prominent in piston mode resonance, in particular for shallow or long moonpools. In contrast, for cases with a recess, the shift of resonant frequency in the first sloshing mode is relatively larger than that in piston mode. In addition, the effects of motion RAOs on the free-surface elevation inside the moonpool in the free-floating condition are also examined. It is found that the moonpool resonance can induce a local hump in heave and pitch responses. Moreover, through deriving a modified frozen mode approximation model, we reveal that, comparing to the fixed condition, the vessel motion in the free-floating condition yields a reduction in the added mass due to the fluid underneath the moonpool so that the piston mode resonant frequency increases.

A comparison of nonlinear modal synthesis using a time varying linear approximation and direct computation

Modal synthesis is a common technique to model linear musical systems efficiently using a parallel bank of second-order digital filters to represent the modes of a resonating instrument. However, many instruments exhibit nonlinear behavior when driven at high amplitudes, often resulting in amplitude dependent mode center frequencies. This behavior can be modeled in a simplified manner using a time varying linear modal structure or could be directly modeled using a nonlinear mode approximation such as that of a Duffing oscillator. The time-varying linear approach is compared to the direct approach to gauge if the approximation is valid within perceptual limits. Particular emphasis is put on weakly nonlinear modes which may be found in a string instrument with thin vibrating plates.

Theoretical study of electron-induced vibrational excitation of NO2

In this study, we compute cross sections for vibrational excitation of NO2 by electron impact. Calculations are performed using a theoretical approach based on a combination of the normal mode approximation for vibrational states of the target molecule, fixed-nuclei electron-NO2 scattering matrices and the vibrational frame transformation employed to evaluate the scattering matrix for vibrational transitions. Thermally-averaged rate coefficients are derived from the obtained cross sections for temperatures in the 10–10000 K interval for excitation of each normal mode of the target molecule. Analytical fits for the rate coefficients for singlets and triplets are provided. In addition, a comprehensive set of calculations are performed for assessing the uncertainty of the present calculations. The uncertainty assessments indicate that the computed observables for vibrational (de-)excitation is reasonable for later use in NO2-containing plasma kinetics modeling.

Simplified integrated nested Laplace approximation

SummaryIntegrated nested Laplace approximation provides accurate and efficient approximations for marginal distributions in latent Gaussian random field models. Computational feasibility of the original Rue et al. (2009) methods relies on efficient approximation of Laplace approximations for the marginal distributions of the coefficients of the latent field, conditional on the data and hyperparameters. The computational efficiency of these approximations depends on the Gaussian field having a Markov structure. This note provides equivalent efficiency without requiring the Markov property, which allows for straightforward use of latent Gaussian fields without a sparse structure, such as reduced rank multi-dimensional smoothing splines. The method avoids the approximation for conditional modes used in Rue et al. (2009), and uses a log determinant approximation based on a simple quasi-Newton update. The latter has a desirable property not shared by the most commonly used variant of the original method.

Reduced-order models for the analysis of a vertical rod under parametric excitation

This paper focuses on the analysis of the parametric excitation of a vertical and immersed flexible rod, showing the influence of the choice of the shape function used in the Galerkin’s method. For this, three different reduced-order models (ROMs) are obtained from the continuous equation of transverse motion employing different shape functions. The first model (ROM(i)) uses an approximation of the actual vibration mode of the rod, written as a “Bessel-like” function. The second model (ROM(ii)) is based on a single trigonometric function as the shape function. Finally, a multi-modal ROM (ROM(iii)) is obtained using three trigonometric functions as a set of shape functions. Simulations are carried out aiming at verifying the capability of each model to properly represent the dynamics of the rod under parametric excitation. The quality of the numerical results obtained from the integration of the aforementioned ROMs is assessed by means of a comparison with a solution based on the finite element method (FEM). In addition to the numerical analysis, an analytical solution for the steady-state amplitude of a generic Duffing-Mathieu-Morrison oscillator is obtained using the method of multiple scales for the one degree of freedom ROMs. A case study is developed using the data of a vertical riser as an example of an engineering application. Maps of the steady-state amplitude as a function of the excitation amplitude and frequency are plotted using both the numerical simulations and the multiple scales solution. The results show that ROM(i) and ROM(iii) are in good agreement with the finite element solution. ROM(i) has the advantage of having only one degree of freedom and, consequently, can be studied using the analytical solution aforementioned. The use of a ROM with one degree of freedom using “Bessel-like” functions in the Galerkin’s scheme is concluded to have clear advantages from the practical point of view. The analytical solution allows this kind of ROM to give a post-critical amplitude map with low computational effort and that is in good agreement with the maps obtained with the simulation of the ROMs.

Variation of the spin textures of 2-species spin-1 condensates studied beyond the single spatial mode approximation and the experimental identification of these textures

Based on the numerical solutions of the coupled Gross–Pitaevskii equations, the spin-textures of the ground state of a Bose–Einstein condensate with two kinds of spin-1 atoms have been studied. Besides, making use of the fractional parentage coefficients, the probabilities of an atom in spin component μ and of two correlated atoms one in μ and one in ν have been calculated. The previous results under the single spatial mode approximation (SMA) are found to be qualitatively consistent with the present results beyond the SMA. The variation of the probabilities against the input parameters is found to match the corresponding variation of the spin-textures excellently. In particular, once a spin-texture transition occurs, the associated probabilities will change suddenly. Since the probabilities are observable, the observation provides a way for the experimental identification of the spin-textures. Besides, sine the critical points (where a transition occurs) can be accurately identified, information on the input parameters (say, the very weak spin-dependent forces) can be extracted.

Stability analysis of split-explicit free surface ocean models: Implication of the depth-independent barotropic mode approximation

The evolution of the oceanic free-surface is responsible for the propagation of fast surface gravity waves, which approximatively propagate at speed gH (with g the gravity and H the local water depth). In the deep ocean, this phase speed is roughly two orders of magnitude faster than the fastest internal gravity waves. The very strong stability constraint imposed by those fast surface waves on the time-step of numerical models is handled using a mode splitting between slow (internal/baroclinic) and fast (external/barotropic) motions to allow the possibility to adopt specific numerical treatments in each component. The barotropic mode is traditionally approximated by the vertically integrated flow because it has only slight vertical variations. However the implications of this assumption on the stability of the splitting are not well documented. In this paper, we describe a stability analysis of the mode splitting technique based on an eigenvector decomposition using the true (depth-dependent) barotropic mode. This allows us to quantify the amount of dissipation required to stabilize the approximative splitting. We show that, to achieve stable integrations, the dissipation usually applied through averaging filters can be drastically reduced when incorporated at the level of the barotropic time stepping. The benefits are illustrated by numerical experiments. In addition, the formulation of a new mode splitting algorithm using the depth-dependent barotropic mode is introduced.

Black hole quantum atmosphere for freely falling observers

We analyze Hawking radiation as perceived by a freely-falling observer and try to draw an inference about the region of origin of the Hawking quanta. To do so, first we calculate the energy density from the stress energy tensor, as perceived by a freely-falling observer. Then we compare this with the energy density computed from an effective temperature functional which depends on the state of the observer. The two ways of computing these quantities show a mismatch at the light ring outside the black hole horizon. To better understand this ambiguity, we show that even taking into account the (minor) breakdown of the adiabatic evolution of the temperature functional which has a peak in the same region of the mismatch, is not enough to remove it. We argue that the appearance of this discrepancy can be traced back to the process of particle creation by showing how the Wentzel–Kramers–Brillouin approximation for the field modes breaks down between the light ring at 3M and 4M, with a peak at r=3.3M exactly where the energy density mismatch is maximized. We hence conclude that these facts strongly support a scenario where the Hawking flux does originate from a “quantum atmosphere” located well outside the black hole horizon.

Adaptive short-time Fourier transform and synchrosqueezing transform for non-stationary signal separation

The synchrosqueezing transform, a kind of reassignment method, aims to sharpen the time-frequency representation and to separate the components of a multicomponent non-stationary signal. In this paper, we consider the short-time Fourier transform (STFT) with a time-varying parameter, called the adaptive STFT. Based on the local approximation of linear frequency modulation mode, we analyze the well-separated condition of non-stationary multicomponent signals using the adaptive STFT with the Gaussian window function. We propose the STFT-based synchrosqueezing transform (FSST) with a time-varying parameter, named the adaptive FSST, to enhance the time-frequency concentration and resolution of a multicomponent signal, and to separate its components more accurately. In addition, we also propose the 2nd-order adaptive FSST to further improve the adaptive FSST for the non-stationary signals with fast-varying frequencies. Furthermore, we present a localized optimization algorithm based on our well-separated condition to estimate the time-varying parameter adaptively and automatically. Simulation results on synthetic signals and the bat echolocation signal are provided to demonstrate the effectiveness and robustness of the proposed method.

Adaptive synchrosqueezing transform with a time-varying parameter for non-stationary signal separation

The continuous wavelet transform (CWT)-based synchrosqueezing transform (SST) is a special type of the reassignment method which not only enhances the energy concentration of CWT in the time-frequency plane, but also separates the components of multicomponent signals. The “bump wavelet” and Morlet's wavelet are commonly used continuous wavelets for SST. There is a parameter in these wavelets which controls the widths of the time-frequency localization window. In most literature on SST, this parameter is a fixed positive constant. In this paper, we consider the CWT with a time-varying parameter (called the adaptive CWT) and the corresponding SST (called the adaptive SST). We also introduce the 2nd-order adaptive SST. We analyze the separation conditions for non-stationary multicomponent signals with the local approximation of linear frequency modulation mode. We derive well-separated conditions of a multicomponent signal based on the adaptive CWT. We propose methods to select the time-varying parameter so that the corresponding adaptive SSTs of the components of a multicomponent signal have sharp representations and are well-separated. We provide comparison experimental results to demonstrate the efficiency and robustness of the proposed adaptive SST in separating components of multicomponent signals with fast varying frequencies.

Nuclear quantum effects in graphene bilayers.

Graphene bilayers display peculiar electronic and mechanical characteristics associated with their two-dimensional character and relative disposition of the sheets. Here, we study nuclear quantum effects in graphene bilayers by using path-integral molecular dynamics simulations, which allow us to consider quantization of vibrational modes and study the effect of anharmonicity on physical variables. Finite-temperature properties are analyzed in the range from 12 to 2000 K. Our results for graphene bilayers are compared with those found for graphene monolayers and graphite. Nuclear quantum effects turn out to be appreciable in the layer area and interlayer distance at finite temperatures. Differences in the behavior of in-plane and real areas of the graphene sheets are discussed. The interlayer spacing has a zero-point expansion of 1.5 × 10-2 Å with respect to the classical minimum. The compressibility of graphene bilayers in the out-of-plane direction is found to be similar to that of graphite at low temperatures and increases faster as the temperature is raised. The low-temperature compressibility increases by 6% due to zero-point motion. Special emphasis is placed on atomic vibrations in the out-of-plane direction. Quantum effects are present in these vibrational modes, but classical thermal motion becomes dominant over quantum delocalization for large system size. The significance of anharmonicities in this atomic motion is estimated by comparing with a harmonic approximation for the vibrational modes in graphene bilayers.

Spin-Textures of the Condensates with Two Kinds of Spin-1 Atoms Studied Beyond the Single Spatial Mode Approximation

This paper is based on the coupled Gross–Pitaevskii equations (CGP) for two-species BEC, where both the central and spin-dependent forces are included. Under the Thomas–Fermi approximation (TFA), an approach is proposed to obtain analytical solutions of the CGP. Recall that the spin-textures depend seriously on three factors $$I_A=\int \varphi _A^4\mathrm{d}\mathbf {r}$$, $$I_B=\int \varphi _B^4\mathrm{d}\mathbf {r}$$, and $$I_{AB}=\int \varphi _A^2\varphi _B^2\mathrm{d}\mathbf {r}$$, where $$\varphi _A$$ and $$\varphi _B$$ are the spatial wave functions of the two species. In the single spatial mode approximation (SMA), these three factors remain unknown. However, in this paper, these factors have been obtained and analytically expressed by the input parameters (in which those for the central forces are essential). Besides, a number of formulae have been derived to describe the critical points of spin-transition. Thus, the present theory is a complement of the SMA so that the theoretical result can be compared with the experimental result quantitatively, and the effect of each parameter involved in experiments can be clarified. Finally, the results under the TFA are compared with those from the exact numerical solution of the CGP.

Normal Modes of Transverse Coronal Loop Oscillations from Numerical Simulations. I. Method and Test Case

Abstract The purpose of this work is to develop a procedure to obtain the normal modes of a coronal loop from time-dependent numerical simulations with the aim of better understanding observed transverse loop oscillations. To achieve this goal, in this paper we present a new method and test its performance with a problem for which the normal modes can be computed analytically. In a follow-up paper, the application to the simulations of Rial et al. is tackled. The method proceeds iteratively and at each step consists of (i) a time-dependent numerical simulation followed by (ii) the Complex Empirical Orthogonal Function (CEOF) analysis of the simulation results. The CEOF analysis provides an approximation to the normal mode eigenfunctions that can be used to set up the initial conditions for the numerical simulation of the following iteration, in which an improved normal mode approximation is obtained. The iterative process is stopped once the global difference between successive approximate eigenfunctions is below a prescribed threshold. The equilibrium used in this paper contains material discontinuities that result in one eigenfunction with a jump across these discontinuities and two eigenfunctions whose normal derivatives are discontinuous there. After six iterations, the approximations to the frequency and eigenfunctions are accurate to ≲0.7% except for the eigenfunction with discontinuities, which displays a much larger error at these positions.

Determining efficient values for the thermophysical properties of bulk materials

A procedure has been devised for determining the effective thermophysical properties of bulk materials with different granulometric and material composition, based on the integration of discrete and continuous models of media. The problem on the mechanical-thermal state of a cylindrical layer of bulk material has been stated in order to determine its effective thermophysical properties. Based on the discrete-continuous perceptions of bulk media, an approach has been suggested and a procedure has been devised for solving the problem set. The algorithm for determining effective values of thermophysical properties of bulk materials has been constructed. Numerical implementation of the developed procedure was performed using free open source software (LIGGGHTS, ParaView). The proposed procedure makes it possible to determine effective values for the thermophysical properties of a bulk material (bulk density, effective thermal conductivity coefficient and the effective value for isobaric mass heat capacity) with arbitrary material and granulometric composition. In this case, there is a need for a minimum volume of complex and costly experimental studies with subsequent numerical simulation of the process of the mechanical-thermal state of the examined bulk material. In this case, the true physical properties can be acquired from reference books. Using an example of model material, its effective thermophysical properties have been defined for different granulometric composition and the verification of the developed procedure has been performed. It was established that data on the effective thermal conductivity calculation based on the devised procedure differ from data obtained based on the theoretical averaged dependences, within 0.8‒9.0 %. The reported results are useful for numerical analysis in the continual approximation of thermal modes of the processes and equipment where bulk materials are used

Dynamic Modeling of Two-Phase Gas/Liquid Flow in Pipelines

Summary Presented is a reduced–order thermal fluid dynamic model for gas/liquid two–phase flow in pipelines. Specifically, a two–phase–flow thermal model is coupled with a two–phase–flow hydraulics model to estimate the gas and liquid properties at each pressure and temperature condition. The proposed thermal model estimates the heat–transfer coefficient for different flow patterns observed in two–phase flow. For distributed flows, where the two phases are well–mixed, a weight–based averaging is used to estimate the equivalent fluid thermal properties and the overall heat–transfer coefficient. Conversely, for segregated flows, where the two phases are separated by a distinct interface, the overall heat–transfer coefficient is dependent on the liquid holdup and pressure drop estimated by the fluid model. Intermittent flows are considered as a combination of distributed and segregated flow. The integrated model is developed by dividing the pipeline into segments. Equivalent fluid properties are identified for each segment to schedule the coefficients of a modal approximation of the transient single–phase–flow pipeline–distributed–parameter model to obtain dynamic pressure and flow rate, which are used to estimate the transient temperature response. The resulting model enables a computationally efficient estimation of the pipeline–mixture pressure, temperature, two–phase–flow pattern, and liquid holdup. Such a model has utility for flow–assurance studies and real–time flow–condition monitoring. A sensitivity analysis is presented to estimate the effect of model parameters on the pipeline–mixture dynamic response. The model predictions of mixture pressure and temperature are compared with an experimental data set and OLGA (2014) simulations to assess model accuracy.

Forty-year review of developments in underwater acoustic modeling

This is the sixth paper in a series of reviews presented at eight-year intervals starting in 1979 [J. Acoust. Soc. Am. 65, S42 (1979); 82, S102 (1987); 97, 3312 (1995); 114, 2430 (2003); 129, 2631 (2011)]. All surveys covered basic acoustic models and sonar performance models. Basic acoustic models included 143 propagation, 29 noise, and 32 reverberation models. Propagation models were categorized according to ray theory, multipath expansion, normal mode, fast field, and parabolic approximation formulations; further distinctions were made between range-independent and range-dependent solutions. Noise models included ambient noise and beam-noise statistics models. Reverberation models included cell and point-scattering formulations. The 44 sonar performance models included active sonar models, model operating systems, and tactical decision aids. Active sonar models integrated basic acoustic models, signal-processing models, and supporting databases into cohesive operating systems organized to solve the sonar equations. Model operating systems provided a framework for the direct linkage of data-management software with computer-implemented codes of acoustic models. Tactical decision aids represented a form of engagement-level simulation that blended environmental and sonar performance information with tactical rules. The overall inventory increased by 5 models per year; historical references are maintained in the fifth edition of Underwater Acoustic Modeling and Simulation.

Pressure response of the THz spectrum of bulk liquid water revealed by intermolecular instantaneous normal mode analysis.

The radial distribution functions of liquid water are known to change significantly their shape upon hydrostatic compression from ambient conditions deep into the kbar pressure regime. It has been shown that despite their eye-catching changes, the fundamental locally tetrahedral fourfold H-bonding pattern that characterizes ambient water is preserved up to about 10 kbar (1 GPa), which is the stability limit of liquid water at 300 K. The observed increase in coordination number comes from pushing water molecules into the first coordination sphere without establishing an H-bond, resulting in roughly two such additional interstitial molecules at 10 kbar. THz spectroscopy has been firmly established as a powerful experimental technique to analyze H-bonding in aqueous solutions given that it directly probes the far-infrared lineshape and thus the prominent H-bond network mode around 180 cm-1. We, therefore, set out to assess pressure effects on the THz response of liquid water at 10 kbar in comparison to the 1 bar (0.1 MPa) reference, both at 300 K, with the aim to trace back the related lineshape changes to the structural level. To this end, we employ the instantaneous normal mode approximation to rigorously separate the H-bonding peak from the large background arising from the pronounced librational tail. By exactly decomposing the total molecular dynamics into hindered translations, hindered rotations, and intramolecular vibrations, we find that the H-bonding peak arises from translation-translation and translation-rotation correlations, which are successively decomposed down to the level of distinct local H-bond environments. Our utmost detailed analysis based on molecular pair classifications unveils that H-bonded double-donor water pairs contribute most to the THz response around 180 cm-1, whereas interstitial waters are negligible. Moreover, short double-donor H-bonds have their peak maximum significantly shifted toward higher frequencies with respect to such long H-bonds. In conjunction with an increasing relative population of these short H-bonds versus the long ones (while the population of other water pair classes is essentially pressure insensitive), this explains not only the blue-shift of the H-bonding peak by about 20-30 cm-1 in total from 1 bar to 10 kbar but also the filling of the shallow local minimum of the THz lineshape located in between the network peak and the red-wing of the librational band at 1 bar. Based on the changing populations as a function of pressure, we are also able to roughly estimate the pressure-dependence of the H-bond network mode and find that its pressure response and thus the blue-shifting are most pronounced at low kbar pressures.