Mode Solutions Research Articles

Refraction of the Two-Photon Multimode Field via a Three-Level Atom

Classically, the refractive index of a medium is due to a response on said medium from an electromagnetic field. It has been shown that a single two-level atom interacting with a single photon undergoes dispersion. The following extends that analyses to a three-level system interacting with two photons. Analysis of the system is completed both numerically for all photonic field modes, and analytically for an adiabatic solution of a single field mode. The findings are not only interesting for understanding additional physical phenomena due to the increased complexity of a three-level, two-photon system, but are also necessary for advancing applications such as quantum communications, quantum computation, and quantum information.

The Factorial Fundamental Frequency Explicit Formulas for Homogenous Rectangular Thin Plates with Four Free Edges

PurposeThis study aims to investigate fundamental modal solutions for rectangular thin plates with four free edges: (1) to provide a novel criterion for fundamental mode identification, and (2) to derive high accurate explicit formulas for calculating the fundamental frequency.MethodTo achieve the study objectives, an investigation into the fundamental mode space was conducted, varying structural and material parameters including the plate’s dimensions (a, b), thickness (h), and Poisson’s ratio (ν). A novel, explicit criterion for fundamental mode identification was formulated, incorporating a six sigma tolerance tailored for specific aspect ratios. Ordinary least squares (OLS) multiple linear regression methodology was integrated within a dedicated process to identify optimal factor combinations for factorial frequencies based on 56 features. This methodology allowed for the derivation of factorial frequency formulas as a function of a, b, h, and ν, providing explicit expressions for calculating the fundamental frequency.ConclusionThe study concludes that the global fundamental mode shapes of rectangular thin plates with four free edges are primarily governed by the aspect ratio a/b, Poisson’s ratio ν, and the thickness-induced shearing effect h on both dimensions. The derived explicit formulas for calculating the fundamental frequency exhibit exceptional accuracy, with a maximum error of −0.217% compared to extensive finite element analysis results, surpassing current state-of-the-art accuracy. This work presents physically meaningful, explicit formulations and introduces a novel perspective in plate and shell research, particularly relevant in scenarios with adjacent free edges and similar global modes.

Alfvén continuum modes around an X-point with a guide field

Context. Oscillations around X-points are important for the local heating of the coronal plasma by magnetic reconnection and for the generation of quasi-periodic pulsations in solar flares. The processes of phase mixing and resonant absorption are of particular interest in this context. Aims. The aim of this paper is to find the Alfvén continuum eigenmodes around an X-point with a guide field, as these modes are particularly important for phase mixing and resonant absorption. Methods. We studied the linear oscillations in the neighborhood of an X-point with the use of flux coordinates. Reduced equations that describe continuum oscillations were used to determine the Alfvén continuum modes both with and without a guide field in the direction normal to the X-point plane. Results. We determined Alfvén continuum mode solutions in the form of analytical solutions for the special case without a guide field. We also derived numerical solutions in the more general case with a guide field under the assumption of pressureless plasma.

Blast response of rectangular graded foam sandwich plate with fiber-metal laminate face-sheets

In this paper, the dynamic response of a fully clamped rectangular graded foam sandwich plate (GFSP) with fiber-metal laminate (FML) face-sheets under impulse loading is studied analytically and numerically. Due to the fibers having intricate layups, there are various difficulties in developing an analytical model that can accurately analyze the structural properties of fiber layups. In this work, the bound and membrane mode solutions for the blast response of a rectangular GFSP with FML face-sheets are obtained on the basis of the modified rigid-plastic model by introducing the number of fiber layups, metal volume fraction, composite density coefficient, and composite strength coefficient. In addition, finite element analysis is performed using ABAQUS/Explicit software. Analytical predictions agree very well with numerical results. The effects of metal volume fraction, composite coefficients of density and strength, thickness and average yield strength of the graded foam, and gradient property of the foam on the blast response of GFSP with FML face-sheets are discussed. It is demonstrated that the proposed analytical model can be efficient in predicting the blast response of rectangular GFSP with FML face-sheets.

Discussion on the analytical solutions of graphene surface modes and relationship with quantum systems

Discussion on the analytical solutions of graphene surface modes and relationship with quantum systems

Effects of modal coupling on the nonlinear dynamics of hyperelastic circular cylindrical shells

The nonlinear dynamics of a simply supported circular cylindrical shell is analyzed using an analytical model that considers both physical and geometric nonlinearities. The material that composes the shell is assumed as homogeneous, isotropic, hyperelastic and incompressible, being described by the Mooney-Rivlin hyperelastic constitutive law. The geometric nonlinearity is introduced into the analytical model by applying the Sanders-Koiter’s nonlinear shell theory. The Rayleigh-Ritz method and Hamilton's principle are employed to obtain the nonlinear equilibrium equations. For that, the energy density function is expanded in Taylor series up to the fourth order and the transversal displacement field is described in Fourier series that considers both asymmetric and axisymmetric modes. This works focusses on evaluating the influence of the asymmetric and axisymmetric modes, selected in the modal solution of the transversal displacement field, on the frequency-amplitude relationships of the shell and resonance curves. It can be observed that the asymmetric modes provoke nonlinear hardening behavior of the hyperelastic shell. On the other hand, when the axisymmetric modes are added, there is a significant change in the nonlinear behavior of the hyperelastic shell - which indicates the importance of this modal coupling in the nonlinear dynamic analysis.

Influence of the unilateral elastic base on the backbone curves of an imperfect cylindrical panel

This work evaluates the influence of a discontinuous unilateral elastic base and an initial geometrical imperfection on the nonlinear free vibration of cylindrical panels. The Donnell’s nonlinear shallow shell theory is considered to describe the cylindrical panel, then the equations are discretized by the Galerkin method. The unilateral elastic base is represented by the Signum function, and the Heaviside function describes the discontinuity of the elastic base. The results show the analysis of the nonlinear free vibrations of the cylindrical panel through the backbone curves, investigating the influence of the hypothesis of contact of the unilateral elastic base and the initial geometrical imperfection of the cylindrical panel. The modal solution employed has with five degree-of-freedom, being sufficient to describe the nonlinear softening behavior of the imperfection cylindrical panel in contact with the discontinuous unilateral elastic base. The numerical results reveal that the imperfect cylindrical panel in contact with unilateral elastic base presents less structural stiffness than in contact with bilateral elastic base, with decreasing the natural frequencies. In conclusion, the backbone curves are strongly influenced by discontinuous unilateral elastic base and the imperfections of the cylindrical panel.

Nonlinear normal modes to analyze the nonlinear forced vibrations of cylindrical shells with internal resonances

In this work, the nonlinear normal modes are applied to analyze the nonlinear forced vibrations of a simply supported cylindrical shell with internal resonances. The nonlinear equilibrium equations are obtained considering the Donnell's nonlinear shallow shell theory. The modal solution to the transversal displacement field, used to discretize the equilibrium equations, is obtained by perturbation techniques that consider an internal resonance between the linear vibration modes. The discretized equations of the reduced order model are obtained using an invariant manifold approach. The nonlinear dynamic behavior is analyzed from the resonance curves that are obtained by the continuation method. The resonance curves are obtained for both full and reduced order models, and these results are compared to determine the level of a harmonic load that can be reliably represented by a reduced order model. Several multi-modes are considered to assemble the best nonlinear normal modes basis that contains the most important information of the interactions that occur between the modes of the transversal displacement field. Time responses and phase portraits, with mapping Poincaré sections, are also used to analyze the nonlinear dynamic behavior of the cylindrical shell and to check the accuracy of the reduced order models.

Shear mode solutions to penny-shaped crack problems in two-dimensional hexagonal piezoelectric quasicrystal media

This study investigates shear mode penny-shaped crack problems in an infinite three-dimensional body composed of a two-dimensional hexagonal quasicrystal medium with piezoelectric effect. The crack is subjected to a set of shear phonon and phason loadings within the crack plane. This shear mode crack problem is transformed into a mixed boundary value problem in the upper half space. Subsequently, it is elegantly solved utilizing Fabrikant’s potential theory method. The boundary integral–differential equations governing three-dimensional shear mode crack problems in two-dimensional hexagonal piezoelectric quasicrystals are derived with the phonon and phason displacement discontinuities serving as unknown variables. Closed-form solutions for all physical field quantities are presented, not merely limited to the crack surface, but rather extended comprehensively to the entire space. Key fracture mechanics parameters, such as phonon and phason displacement discontinuities, stress intensity factors at the crack tip, and energy release rate, are explicitly derived. Numerical results are provided to validate the obtained analytical solutions and illustrate the distribution of the electric-phason-phonon coupling field around the crack in graphical form. Additionally, these numerical results also compare the fracture mechanics parameters of the chosen piezoelectric quasicrystal with its corresponding non-piezoelectric quasicrystal, thereby investigating the influence of piezoelectric effect on quasicrystals. The obtained solution can be used as a benchmark for the experimental and numerical study of shear mode cracks in piezoelectric quasicrystals.

Dynamic response of double-layer rectangular sandwich plates with graded foam cores under blast loading

In this paper, dynamic response of a fully clamped double-layer (DL) rectangular sandwich plate with graded foam cores (GFC) under blast loading is studied theoretically and numerically. Based on the yield criterion, an analytical model for dynamic response of a DL rectangular sandwich plate with GFC is established under blast loading. With the use of the inscribing and circumscribing squares of the exact yield locus, the bound of the analytical solution of the dynamic response of a double-layer rectangular sandwich plate with GFC is obtained. By neglecting the effect of bending moments, the membrane mode solution of a DL rectangular sandwich plate with GFC in large deflection is obtained. Finite element analysis (FEA) is performed using ABAQUS/Explicit software. The effects of interlayer factor, average yield strength of graded foams, and gradient properties of graded foams on the dynamic response of DL rectangular sandwich plate with GFC are considered. The analytical predictions are in excellent accord with the numerical ones. It is demonstrated that the proposed analytical model is effective to predict the blast response of a DL rectangular sandwich plate with GFC.

Mode analysis for the linearized Einstein equations on the Kerr metric: the large $\mathfrak{a}$ case

We give a complete analysis of mode solutions for the linearized Einstein equations and the 1 -form wave operator on the Kerr metric in the large \mathfrak{a} case. By mode solutions we mean solutions of the form e^{-it_*\sigma}\tilde{h}(r,\theta,\varphi) where t_{*} is a suitable time variable. The corresponding Fourier-transformed 1 -form wave operator and linearized Einstein operator are shown to be Fredholm between suitable function spaces and \tilde{h} has to lie in the domain of these operators. These spaces are constructed following the general framework of Vasy (2013, 2021) along the lines of Häfner et al. (2021). No mode solutions exist for \Im \sigma\ge 0,\, \sigma\neq 0 . For \sigma=0 mode solutions are Coulomb solutions for the 1 -form wave operator and linearized Kerr solutions plus pure gauge terms in the case of the linearized Einstein equations. If we fix a De Turck/wave map gauge, then the zero mode solutions for the linearized Einstein equations lie in a fixed seven-dimensional space. The proof relies on the absence of modes for the Teukolsky equation shown by Whiting (1989) and Anderson et al. (2019) and a complete classification of the gauge invariants of linearized gravity on the Kerr spacetime by Aksteiner and Bäckdahl (2018) and Aksteiner et al. (2021).

Nuclear acoustic envelope soliton in a relativistically degenerate magneto-rotating stellar plasma

The study explores a model called Relativistic Degenerate Magneto-Rotating Quantum Plasma (RDMRQP), which comprises a static heavy nucleus, an inertial non-degenerate light nucleus, and warm non-relativistic or ultra-relativistic electrons. The focus is on observing the emergence of Nucleus-Acoustic Envelope Solitons (NAESs). Using the reductive perturbation method, a Nonlinear Schrödinger equation (NLSE) is derived to characterize the properties of NAESs. We have analysed the Peregrine breather soliton solution. The investigation reveals that the temperature of warm degenerate species, plasma system’s rotational speed, and the presence of heavy nucleus species can alter the fundamental features (height and width) of NAESs in the WDMRQP system. The study emphasizes the existence of only positive NA wave potential. Additionally, a phase plane analysis is conducted to gain a deeper understanding of the parametric dependencies. Through detailed mathematical and numerical analysis, the study avoids overloading with complex mathematics while demonstrating parametric dependence via phase portrait analysis. The research augments the envelop soliton model with breather mode solutions and discusses modulation instability using the Benjamin-Feir Index, highlighting the significance of solitons in star formation. Envelop solitons, stable waves within stars and proto-stars, influence energy transport and stellar evolution, playing a crucial role in the accretion process and formation of stable structures. Key findings include the effects of various parameters on NAESs’ generation and propagation in which non-relativistic and ultra-relativistic electrons support NAESs, with amplitude and width influenced by temperature, rotational frequency, inclination angle, and the presence of a static heavy nucleus. The research is applicable to hot white dwarfs and neutron stars, suggesting further exploration of quantum effects and non-planar or arbitrary amplitude NAESs.

Multi-Variable Optimization to Enhance the Performance of a Cantilevered Piezoelectric Harvester

Over the past decade, various researchers have shown the practicality of extracting electrical energy from vibrating structures. The primary objective of this research was to optimize the parameters of a basic cantilever harvester to maximize the energy production of electricity from surrounding mechanical energy. A distributed parameter model and its modal solution were utilized to determine the design variables via a parametric investigation. It is important to ensure sufficient power generation when there is a wide range of frequencies being excited, without any specific frequency being emphasized. The cost function for the optimization problem was defined as the average power within a specific frequency range, considering geometric and physical restrictions. It was discovered that different parameters’ values can be utilized to generate varying maximum power levels for different frequency ranges. Furthermore, a comparative analysis was performed between the finite element method (FEM) using ANSYS and the current harvesting model. The comparison demonstrated a high level of concordance, enabling the utilization of the finite element method (FEM) for subsequent investigation.

Fokker-Planck modeling of the stochastic dynamics of a Rijke tube.

We derive and numerically validate a low-order oscillator model to capture the stochastic dynamics of a prototypical thermoacoustic system (a Rijke tube) undergoing a subcritical Hopf bifurcation in the presence of additive noise. We find that on the fixed-point branch before the bifurcation, the system is dominated by the first duct mode, and the Fokker-Planck solution for the first Galerkin mode can adequately predict the stochastic dynamics of the overall system. We also find that this analytical framework predicts well the dominant mode on the limit-cycle branch, but underperforms in the hysteretic bistable zone where the role of nonlinearities is more pronounced. Besides offering new insights into stochastic thermoacoustic behavior, this study shows that an analytical framework based on the Fokker-Planck equation can facilitate the early detection of thermoacoustic instabilities in a Rijke-tube model subjected to noise.

A person-centered, transdiagnostic schema and mode profile approach to predict outcome in time-limited schema group therapy

Objective This study employs a person-centered transdiagnostic approach to examine how schema and mode profiles predict symptom severity reduction in schema group therapy for patients with personality disorders and enduring clinical syndromes. Method We analyzed symptom reduction in 248 patients across three formats of manualized, time-limited schema group therapy. Latent profile analysis and mixed multilevel modeling were used to determine the extent to which schema/mode classes predict symptom reduction, and whether the inclusion of individual schemas and modes enhances these predictions. Results No significant differences in treatment outcomes were found across the group modalities. A three latent profile solution for schemas and modes showed external validity with clinical variables and demonstrated that declines in symptom severity varied by schema and mode class, even after adjusting for baseline symptom severity. Adding the Vulnerability to Harm schema and Vulnerable Child mode to the model increased the explained variance. Conclusion Patients with more severe personality problems show more substantial symptom reduction. Both schema and mode profiles significantly contribute to predicting post-treatment symptom levels. Understanding these profiles may help therapists tailor interventions more effectively, consistent with Young’s theoretical model. Trial registration: ISRCTN.org identifier: ISRCTN17262253.

Design of a 1 × 3 Power Splitter Based on Multimode Interference in a Parabolic-Type Slot-Waveguide Structure

Multimode interference (MMI) couplers based on silicon slot-waveguide structures have received widespread attention in recent years. The key issues that need to be addressed are the size and loss of such devices. This study introduces a 1 × 3 silicon-based slot-waveguide multimode interference power splitter. The device uses a gallium-nitride slot-waveguide structure to reduce the length of the coupling region and decrease additional losses. To reduce the width of the coupling region, the multimode interference coupling area is designed with a parabolic-shaped structure. The introduction of a tapered structure between the input/output waveguides and the coupling region improves additional losses and non-uniformity. Furthermore, we conducted an analysis of the fabrication tolerances of the coupling region. In this paper, we use mode solution to simulate the design of the device in the 1550 nm optical wavelength range. The eigenmode expansion method is used to simulate and optimize the parameters of the device. The device is simulated using the eigenmode expansion solver. The simulation results show that the total length of the coupling region for the device is only 4 μm. The normalized transmission of the device is 0.992, and its excess loss and imbalance are 0.036 dB and 0.003 dB, respectively. The proposed power splitter can be applied to integrated optical circuit design, optical sensing, and optical power measurement.

Operational strategy optimization of an existing ground source heat pump (GSHP) system using an XGBoost surrogate model

Ground source heat pumps (GSHPs) are experiencing a surge in popularity due to their efficient use of geothermal energy for building heating and cooling. Optimizing GSHP systems is crucial for maximizing heat extraction efficiency. However, existing research primarily focuses on design-stage optimization, neglecting the potential of improving operational strategies in already deployed systems. This study proposes a novel method for optimizing the operational strategy of existing GSHPs with machine-learning-based surrogate models. Our approach considers both demanded heat extraction and the variability of GSHP operation, including running and idle times. First, a numerical model is established with a closed-loop procedure replicating real-world GSHP operation, capturing dynamic inlet temperature changes. The Latin hypercube sampling is subsequently employed to generate sufficient data for XGBoost model development, where a designed disassembly method is proposed to shed light on the problem of a fixed number of input parameters. The operational strategy of the GSHP system is then implemented to offer solutions for both regular and irregular modes. The effectiveness of the method is validated using 40 days of real-case monitoring data. The results demonstrate that the surrogate model achieves high accuracy, with an absolute temperature difference error below 0.1 K and a heat extraction absolute percentage error (APE) less than 2.0 %. Additionally, the optimized operational strategy reduces total operation time by 2–9 % solely through scheduling adjustments, showcasing its significant practical potential. This study presents an efficient method for optimizing the operational strategies of existing GSHPs while highlighting the value of process-accelerated optimization using numerical simulation-based surrogate models.

Computation and analysis of surface wave dispersion and attenuation in layered viscoelastic-vertical transversely isotropic media by the generalized R/T coefficient method

SUMMARY In this study, we propose a systematic and effective method, that is, an extended version of the generalized reflection/transmission (R/T) coefficient method, for computing the phase-velocity (${c}_r$) dispersion curves, attenuation coefficient ($\alpha $) curves, and eigenfunctions of both Rayleigh and Love waves as well as the ellipticity of Rayleigh waves in layered viscoelastic-vertical transversely isotropic (VTI) media. The numerical scheme of combining the root-searching method with the local optimization method is designed for determining the complex-valued modal solutions (i.e. complex wavenumber $k = {\omega {/ {\vphantom {\omega {{c}_r - i\alpha }}}} {{c}_r - i\alpha }}$) of surface waves. The near-surface sedimentary geological environment is taken as the model example because it is typical viscoelastic-VTI media. Besides the anisotropic-viscoelastic (AV) media, our algorithm can also compute surface waves in isotropic-elastic (IE), isotropic-viscoelastic (IV) and anisotropic-elastic (AE) media by resetting the corresponding parameters. Using the six-layer half-space models and in these four media, we verify the correctness of our algorithm by benchmarking the modal solutions against those from other methods. In the four-layer half-space model, by comparing the results of IE, IV, AE and AV media, we analyse the effects of velocity anisotropy, viscoelasticity and attenuation anisotropy on the dispersion and attenuation characteristics of both Rayleigh and Love waves in detail. Our study can provide a theoretical basis and useful tool for surface wave imaging considering the anisotropy and/or viscoelasticity of the medium, which has the potential to better investigate the solid Earth's internal structure.

An optimal variable importance for machine learning classification models using modified simulated annealing algorithm

Abstract Each machine learning model will generate a different importance variable even though the method used is the same. Interpreting the variable significance is confusing. This study proposes combining several variable importance measures using a simulated annealing algorithm with an initial solution of mean and mode. The study uses simulation and empirical data. The simulation data are divided into three scenarios: no correlation, moderate correlation, and high correlation among predictor variables. The empirical data consist of 24 predictor variables. The machine learning models are classification models of random forest, extreme gradient boosting, neural network, and support vector machine. Based on the simulation data study, the combined variable importance will be optimal when predictor variables have low correlation. The simulated annealing algorithms show convergent objective values around the 25th iteration in empirical data. The more predictor variables, the higher the accuracy of this variable importance. Accuracy is optimal when the number of predictors exceeds ten. The five most important variables in explaining family food insecurity are the education of the family head, the floor type of the house, the number of family members who have a savings account, ownership of land, and decent drinking water.

FE/PDE: a novel approach applied to PC plate structure with multi-scale and multi-physics field coupling

For the more straightforward and more efficient solution of phononic crystal (PC) plate frequency band structure, transmission curve, and vibration mode, in this paper, related theories based on spatial Fourier series expansions, combined with Bloch’s theorem, a novel approach to solve the structural governing equations of PC plate is proposed by using the partial differential equations (PDE) module in the finite element software COMSOL. It is named the FE/PDE (Finite element and partial differential equations) method. The method’s accuracy is verified by comparing the results with those obtained from the traditional method. Systematic elucidation of the application of the method to probe the properties of multi-scale, multi-physics field coupled PC plate. In order to demonstrate the flexibility and scientific validity of the method, a novel nano-piezoelectric PC plate structure is proposed and solved. The method is simple, computationally efficient, and applicable, and provides a new method for investigating the properties of PC plates.