Asymptotic Method Research Articles

Theoretical computation of thermodynamic functions of sodium dimer with modified shifted Morse potential

In this article, the asymptotic iteration method and the Laguerre polynomials are employed to obtain the eigensolutions of the Schrödinger equation with modified shifted Morse potential model. Vibrational energy results for sodium dimer have been presented numerically and these results are compared with experimental data from available literatures. With the help of the energy expression, the partition function and other thermodynamic function expressions of this potential model for sodium dimer are obtained, using the Euler MacLaurin’s formula. For various vibrational quantum numbers considered, the thermodynamic functions show high level of dependence on the temperature parameter. These results are justified by the numerical results presented for the thermodynamic functions of sodium dimer. Our results also agree with the available results in literatures and can also serve as a guiding reference in further studies concerning chemical physics and spectroscopy.

Frequency experimental identification approach for single-phase induction motor common-mode parameters

Introduction. The presence of broad-spectrum and high-amplitude electromagnetic interference (EMI) within a single-phase induction motor (SPIM) drive poses a significant threat to both the system and other electronic equipment. High-frequency (HF) models of electrical motors play a critical role in overcoming these challenges, as they are essential for characterizing electromagnetic compatibility (EMC) in drives and designing effective EMI filters. The novelty of this study proposes an enhanced HF motor model based on transfer functions (TFs) to accurately represent the motor’s behavior at HFs for frequency-domain analyses in the range of 100 Hz to 30 MHz. Purpose. The equivalent HF model for a SPIM is discussed in this paper. The suggested equivalent circuit describes a motor’s common-mode (CM) properties. Methodology. HF model was developed by a frequency-domain analysis utilizing an experimental setup and MATLAB software. The motor impedance analysis is based on the measurement of variations in motor characteristics as a function of frequency in the CM setup. Originality. TF has been tuned using an asymptotic identification method of Bode to match the behavior of the real impedances of the motor parameters as a function of the frequency in the CM configuration. This tuned TFs are then synthesized into a comprehensive wideband EMC equivalent circuit model using the Foster network technique, which can be then simulated in any Spice-based simulator tools. Results. The proposed mathematical model was employed to conduct simulations, and the resulting predictions were validated against experimental data. CM response of the EMC equivalent circuit at low, medium, and HFs were compared between simulations and experimental measurements using Lt-Spice simulator software. Practical value. It is observed that results show satisfactory agreement with the measurements over a large frequency bandwidth [100 Hz–30 MHz], and the equivalent model of SPIM can be cascaded with other electronic and electrical modules to form a complete single-phase electric drive system model for fast analysis and prediction of system level EMI and electromagnetic sensitivity. References 37, table 5, figures 13.

A hybrid multiscale model for fluid flow in fractured rocks using homogenization method with discrete fracture networks

Fluid flow in subsurface tight reservoirs containing pores, microcracks and macrocracks is notably influenced by the characteristics of macro/micro-cracks. A novel hybrid multiscale model is proposed to address the response of macrocracks and pores/microcracks in different spatial scales. Specifically, an equivalent macroscopic model (EMM) deduced from locally periodic representative element volume (REV) is developed using the asymptotic homogenization method to represent the poroelastic behavior of porous medium with microcracks. Simultaneously, the macrocracks are modeled explicitly using the discrete fracture model (DFM), where the hydraulic properties of cracks influenced by fluid pressure gradient is represented by the nonlinear opening/closure behavior. The obtained hybrid model takes into account the heterogeneous nature of fractured rock masses containing pores, micro/macro-cracks, which is fundamental to describe fluid flow behavior in fracture-matrix system. Specialized finite elements, regular meshing technique and adaptive time stepping algorithm are adopted to improve the computational efficiency. The hybrid multiscale model is firstly validated step by step to demonstrate the accuracy and then used to simulate fluid flow in fractured rock reservoir, shedding light on the underlying mechanisms of the enhanced flow capacity resulting from microcrack distribution, connectivity, and macrocrack stimulation.

Steady secondary flow in a turbulent boundary layer past a slender axisymmetric body

The turbulent boundary layer in a viscous incompressible fluid developing longitudinally past the surface of a thin cone or cylinder at a finite distance from the laminar–turbulent transition zone is studied. The characteristic Reynolds number, determined from the external flow velocity and the length of the body, is assumed to be large, and the thickness of the boundary layer is small and comparable to the radius of the body. The asymptotic method of multiple scales is used to find solutions to the Navier–Stokes equations. Instead of the traditional decomposition of the solution into time-averaged values and their fluctuations, the velocities and pressure are expressed as an asymptotic series consisting of steady and perturbed terms. As a result, the viscous steady flow (‘secondary’) that arises in the boundary layer as a mandatory component of fast turbulent fluctuations was described. Analytical and numerical solutions for the radial steady velocity are presented, describing the self-induced suction of fluid from the external flow into the boundary layer. Further analytical solutions are obtained for the longitudinal and circumferential velocities, which differ markedly from the laminar regime. The solutions found are somewhat similar to the degenerate (one-dimensional) case of self-sustaining longitudinal thin structures in turbulent shear flows. A qualitative comparison with direct numerical simulations is presented.

Simultaneous confidence interval construction for many-to-one of proportion ratios of bilateral correlated data.

In ophthalmology and otolaryngology, data collected from paired body parts are typically reformatted into categorical bilateral data structures for subsequent research. This article applies Donner's equal correlation coefficient model and obtains nine simultaneous confidence intervals (SCI) of proportion ratios under three asymptotic statistical methods and three ways of multiplicity adjustment. The empirical coverage probability and mean interval width are evaluated through Monte Carlo simulations. A real example is used to demonstrate the proposed methods.

Symmetry Breaking and Modal Localization in a System of Parametrically Excited Microbeam Resonators

In this work, we study the nonlinear dynamics of parametrically excited bending vibrations of two weakly coupled beam microresonators under electrothermal excitation. A steady-state harmonic temperature distribution in the volume of the resonators in the frequency domain was obtained. A system of equations for mechanically coupled beam resonators is derived, considering the deposited particle on one of them. Using asymptotic methods of nonlinear dynamics, equations in slow variables were obtained, which were studied by methods of the theory of bifurcations. It is shown that in a perfectly symmetrical system in a certain frequency range, the effect of symmetry breaking is observed – the emergence of a mode with different amplitudes of oscillations of two beam resonators, which can be the basis for a new principle of high-precision measurements of weak disturbances of various physical natures, in particular – measurements of ultra-low masses of deposited particles.

Thermo-Magnetic properties of non-relativistic particles under the effect of energy-dependent Hellmann potential

This manuscript presents an analysis of the thermal and magnetic properties of particles moving in two dimensions under the influence of an external magnetic field and Aharanov-Bohm flux, considering an energy-dependent Hellmann potential. The energy eigenvalue is obtained using the asymptotic iteration method. Next, we model the system as a canonical ensemble and derive the partition function to acquire analytical expressions for the Helmholtz free energy, entropy, mean energy, and specific heat. We observe that an increase in the external magnetic field and AB flux generally increases the energy eigenvalues, with a stronger effect at lower flux values, while energy eigenvalues decrease with an increase in the screening parameter. Additionally, Helmholtz free energy and entropy functions are sensitive to temperature, particularly in the low-temperature regime, while specific heat exhibits a Schottky anomaly. We also compute the magnetisation and susceptibility functions, finding that both magnetisation and susceptibility increase with the external magnetic field strength and the AB flux.

Theoretical and experimental analysis of a nonlinear vibro-acoustic energy sink

This study is concerned with the experimental and theoretical analysis of a vibroacoustic nonlinear system devoted to absorption application. Based on the nonlinear energy sink principle, the dynamics of the system is reduced to two coupled degrees of freedom, one of which exhibits nonlinear behavior due to a large displacement mechanism. The set-up allows for precisely characterizing the nonlinear function of the mechanical system, first uncoupled and then coupled with an acoustic and linear resonator. Continuation procedures are performed experimentally and numerically, respectively using a phase-locked loop (PLL) and the asymptotic numerical method implemented in the MANLAB software package. The comparison between theoretical predictions and experimental results obtained on the uncoupled system allows for a parameter identification of the nonlinear resonator. Measurements on the coupled system highlight the energy pumping phenomenon, which can be predicted from numerical predictions in which identified parameters are introduced. The procedure aims at proposing analytical models for achieving efficient level attenuation, and providing design guidelines for sound reduction at high levels in confined spaces.

Acoustic properties of porous metamaterials featured by acoustic streaming, homogenization based modelling

We consider secondary effects induced by acoustic waves in viscous barotropic fluids saturating pores in elastic, or rigid scaffolds with elastically suspended particles. The asymptotic homogenization method is applied to derive models of the acoustic wave propagation and of the related acoustic streaming (AS) as the secondary effect. This phenomenon originating due to the nonlinearity (divergence of the Reynolds stress) in the Navier Stokes equation is retained using the perturbation analysis. This yields the first and the second order sub-problem enabling to linearize the model governing fluid dynamics. The local AS in the microstructure produces forces acting on the suspended particles, transforming the pore geometry, hence, modulating the acoustic properties. As the consequence, the permeability involved in both the subproblems is modified due to the slowly oscillating particles. The sensitivity analysis can be employed to introduce the homogenized coefficients of the model depending on the particle motion.

A novel engineered sandwich composite with ceramic-coated graphite fibre-reinforced face sheets and an auxetic core for enhanced broadband electromagnetic absorption: design using multiscale computational analysis

Abstract A novel sandwich composite is designed for enhanced broadband absorption using a multiscale computational analysis. The sandwich panel is configured with BaTiO3 coated graphite fiber reinforced polymer (C-GFRP) composite, while BaTiO3 based auxetic material is used as the core material. A computationally efficient in-house tool based on the variational asymptotic method is used to homogenise the electromagnetic properties and evaluate the reflection loss characteristics of the proposed sandwich structure using the scattering matrices. A detailed parametric study of 300 analyses is conducted to identify the optimal sandwich panel configurations for achieving broadband reflection loss under the normal incidence of transverse electric (TE) and transverse magnetic (TM) polarised waves. Two different configurations have been obtained, satisfying the requirements of broadband reflection loss in the X-band frequency range. Configuration (a) Vf = 15% without any ceramic coating, and configuration (b) Vf = 20% with ceramic coating volume fraction (Cf) of 70%. Configuration (a) and (b) maintained the desired reflection loss of greater than -10 dB up to an incidence angle of 40◦ and 60◦, respectively. With the demonstrated capability of covering the entire broadband frequency range, the proposed configurations can serve as baselines to explore novel ceramic-based engineered composite architectures for broadband absorption applications.

On the Dependence of the Surface Tension of a Spherical Droplet on its Charge

Qualitative analytical dependence of the value of the coefficient of the surface tension of a liquid on the density of the electric charge on its surface was derived from general theoretical positions by asymptotic methods. It is shown that the value of the surface tension coefficient of a liquid decreases with an increase in the density of the electric charge on the surface of the liquid.

Acoustics of wet porous media with evaporation/condensation

This paper investigates acoustic wave propagation in wet rigid-frame porous media accounting for evaporation and condensation. At equilibrium, the solid walls are covered by a thin water film, and water vapor in the air is at its temperature-dependent saturation pressure. Small acoustic perturbations cause water to vaporize or condense, which together with the reversibility of the phase change, lead to a linear problem where the usual local poro-acoustics physics is enriched with the (i) Clapeyron relation linking liquid-wall temperature, vapor pressure, and latent heat of vaporization, (ii) latent heat transfer in the solid frame, (iii) diffusion equation for water vapor in air, and (iv) water vapor's equation of state. The equilibrium temperature highly influences the vapor concentration and the physical parameters of saturated moist air. Using the two-scale asymptotic homogenization method, it is shown that the dynamic permeability is determined similarly to classical porous media, while the effective compressibility is modified by evaporation/condensation and the equilibrium temperature. This modification is influenced by vapor mass and heat flows associated with phase changes through a local fully coupled heat transfer and water vapor diffusion problem, with specific boundary conditions at the gas–water interface. The analysis identifies dimensionless parameters and characteristic frequencies defining the upscaled model's features. Depending on equilibrium temperature, the theory qualitatively and quantitatively determines the characteristics of acoustic waves propagating through the media. The results are illustrated and discussed with analytically developed models.

Three-dimensional resonant sloshing in an upright cylindrical container with a ring baffle

The effect of ring baffles on suppressing the three-dimensional (3D) resonant sloshing in an upright cylindrical container is experimentally investigated. The main objectives of this work are to examine the effectiveness of various baffle configurations, to establish the stability boundaries of the stable steady-state waves in the unbaffled and baffled containers, to provide accurate experimental data for the verification of the analytical and numerical models, and to prompt future investigations. For this purpose, hundreds of sloshing experiments are conducted in a cylindrical container with or without a ring baffle. An analytical potential-flow solution and an asymptotic multimodal method are used to elucidate the experimental results. It is found that the vertical location of the ring baffle has small influence on the fundamental natural frequency of the system; however, it has a significant influence on the viscous damping and the damping rate increases gradually with the ascension of the baffle. When the distance between the baffle and the free liquid surface is sufficiently large, the system exhibits three types of resonant wave patterns, namely stable planar, stable swirling, and irregular chaotic. These wave patterns are qualitatively and quantitatively similar to those in the unbaffled container. When the baffle is near the free liquid surface, neither the chaotic waves nor the swirling waves take place, but a new wave pattern with the characteristic of multiple wave crests is observed. Probably, this is the first time that the 3D resonant sloshing in the baffled cylindrical container has been systematically investigated.

Improved Solutions of OHAM Approximate Procedure for Classes of Nonlinear ODEs

The primary purpose of this study is to apply the Optimal Homotopy Asymptotic Method (OHAM) to various nonlinear initial value problems of different orders to evaluate its accuracy, convergence, and computational efficiency. OHAM is considered a highly effective technique for solving nonlinear differential equations and is commonly used in scientific and engineering disciplines. It combines the strengths of homotopy and asymptotic methods. OHAM offers a straightforward approach to controlling and adjusting the convergence of the series solution. This is achieved through the utilization of an auxiliary function that incorporates multiple convergent control parameters with one order of approximation, which are optimally determined. The OHAM approach heavily relies on the auxiliary function H(p) which allows for the flexible and efficient solving of nonlinear differential equations. By carefully constructing and optimizing the parameters of H( p), the convergence of the solution series can be effectively controlled. As a result, OHAM proves to be a versatile and effective approach for solving various mathematical and engineering problems. Several examples have been solved. Numerical comparisons, displayed in tables and shown graphically in figures, prove and confirm the capability, efficiency, and better accuracy with less computational work.

Mode II fracture behavior of glass fiber composite-steel bonded interface–experiments and CZM

The dominant failure mode was characterized as debonding in the novel non-welded wrapped composite joint made with GFRP composites wrapped around steel sections. Glass fiber composite-steel three-point end notched flexure (3ENF) and four-point end notched flexure (4ENF) specimens were utilized to experimentally investigate mode II fracture behavior of this composite-steel bonded interface. Two new methods were proposed with the help of digital image correlation (DIC) technique to quantify fracture data during the tests: 1) the “shear strain scaling method” to quantify the crack length a; 2) the asymptotic analysis method based on the longitudinal displacement distribution along the height of the specimen at the pre-crack tip to quantify the crack tip opening displacement (CTOD). To numerically simulate the mode II fracture behavior, a four-linear traction-separation law was proposed in the cohesive zone modeling (CZM) where the softening behavior with a plateau was defined by the authors between traditionally considered initiation and fiber bridging behavior. The experimental and numerical approaches were validated mutually through good matches between the test and FEA results. 3ENF test provided good insight into softening behavior while 4ENF contributed to quantification of fiber bridging. These findings contribute to a more comprehensive characterization and understanding of the ductile fracture behavior of bi-material bonded joints, especially in mode II failure scenarios.

SCATTERING OF THE E-POLARIZED ELECTROMAGNETIC WAVE BY SOLITUDE CONDUCTIVE STRIP WITH IMPEDANCE

The key problem of scattering of a plane E - polarized electromagnetic wave by a solitude electrically conductive impedance strip is studied in detail. A well-posed full-wave mathematical model of scattering in the form of the first order singular integral equation was obtained by applying a modifica- tion of a classical method if integral equations. To examine this singular integral equation two basic methods were used. The first one is remarkable Lord Rayleigh’s asymptotic method, which delivers an explicit analytical solution in a narrow frequency range. It deals with the long-wave case of the plane E - polarized electromagnetic wave scattering problem. The second method is a direct numerical method, dealing with approximate computa- tion of definite integrals, which gives an approximate solution as well, but in the full-frequency range. A series of numerical experiments on scattering a E-polarized electromagnetic wave by impedance strip were carried out and are presented in this paper. Comparing of computation results shows good agreement in long-wave case of the plane E - polarized electromagnetic wave scattering problem, which indicates the correctness of numeric compu- tation in full frequency range as well.

PROBLEMS OF MANAGING THE IMMUNE RESPONSE UNDER CONDITIONS OF COMPETITIVE ADSORPTION, DIFFUSION PERTURBATIONS AND TEMPERATURE RESPONSE OF THE ORGANISM

Analysis of the problems associated with the rapid spread of infectious diseases indicates the need for new approaches to the diagnosis and develop- ment of effective personalized treatment programs. One of the important aspects of solving this problem is the creation of a mathematical modeling toolkit for predicting the dynamics of infectious diseases, taking into account spatial effects, concentrated effects, various mechanisms of body protec- tion in the conditions of the application of various types of therapeutic methods of treatment. An important component of complex methods of treat- ment of a wide range of infectious diseases is the use of adsorption drugs, which, along with detoxification of the body, are able to provide an addi- tional mechanism of neutralization of viral elements. In the process of using such means, not only pathogenic elements and toxins will be adsorbed, but also various types of immune agents, which will affect the dynamics of the immune response. In the paper the infectious disease model is generalized to take into account the features of competitive adsorption of antigens and immune agents in the conditions of diffusion perturbations, concentrated influences and temperature reaction of the body. By synthesizing the ideas of step-by-step procedures, asymptotic and numerical methods, an effective computational technology of step-by-step approximation of the solution of the original model singularly perturbed problem was built. The results of the computer experiments presented in the paper illustrate the features of reducing the predicted concentration of viral elements during their competi- tive adsorption together with immune agents, in particular, the effect of reducing the efficiency of the use of non-specific adsorbents. It is emphasized that taking into account the features of the action of competitive adsorption is important for making decisions regarding the formation of rational treatment programs with the complex use of adsorbing substances and immunological drugs.

Laminar boundary layers in three‐dimensional MHD natural convective flow past a semi‐infinite vertical plate with variable sinusoidal suction

AbstractThe primary goal of this study is to investigate how the diffusion‐thermo and thermo‐diffusion affect a three‐dimensional hydromagnetic natural convective flow of a radiating fluid past a uniformly moving isothermal and isosolutal semi‐infinite vertical plate subjected to a sinusoidal suction velocity. The asymptotic series expansion method is adopted to solve the equations governing the flow model and to find the analytical solutions. The effect of variable suction on the flow and transport characteristics is mainly highlighted in the study. Both the diffusion‐thermo and thermo‐diffusion effects improve the fluid velocity.

Using Exact Tests from Algebraic Statistics in Sparse Multi-Way Analyses: An Application to Analyzing Differential Item Functioning

Asymptotic goodness-of-fit methods in contingency table analysis can struggle with sparse data, especially in multi-way tables where it can be infeasible to meet sample size requirements for a robust application of distributional assumptions. However, algebraic statistics provides exact alternatives to these classical asymptotic methods that remain viable even with sparse data. We apply these methods to a context in psychometrics and education research that leads naturally to multi-way contingency tables: the analysis of differential item functioning (DIF). We explain concretely how to apply the exact methods of algebraic statistics to DIF analysis using the R package algstat, and we compare their performance to that of classical asymptotic methods.

$$\delta '''$$-shock wave solution to a nonstrictly hyperbolic system of conservation laws using weak asymptotic method

$$\delta '''$$-shock wave solution to a nonstrictly hyperbolic system of conservation laws using weak asymptotic method