Asymptotic Method Research Articles

Flexural-gravity wave interaction with undulating bottom topography in the presence of uniform current: An asymptotic approach

Using an asymptotic method, this article deals with flexural-gravity wave scattering with undulating bottom topography, including the effect of uniform currents. The interest in this problem lies in developing second-order solutions using the Fourier transform, which minimises the error gap between first and second-order solutions. The present method allows the physical processes involved in the sea-bed topography, uniform current, plate-covered surface, and wave interaction to be studied. Specifically, we observe Bragg resonance between the flexural-gravity waves and the bottom ripples, which are associated with the reflection of incident wave energy. We examine the effects of wave current and emphasise how crucial the asymptotic expansion method is to the emergence of the current response. We demonstrate that bottom topography dominates the effects of Bragg resonance for depth Froude numbers valued at 0.8 or less. Further, most reflected wave components have their frequencies shifted by the current, and wave action conservation causes reflected wave energy to be enhanced for following currents. Using the Joint North Sea Wave Observation Project spectrum and the discrete Fourier transform, the theory derived in the frequency domain is shown in the time domain to analyse wave propagation through the whole system.

Utilizing the Optimal Auxiliary Function Method for the Approximation of a Nonlinear Long Wave System considering Caputo Fractional Order

In this article, we explore the utilization of the Caputo derivative and the Riemann–Liouville (R–L) fractional integral to analyze the optimal auxiliary function method for approximating fractional nonlinear long waves. Approximate long wave equation with a distinct dispersion relation offers the most accurate description of shallow water wave properties. Various methods, including the Adomian decomposition technique, the variational iteration method, the optimum homotopy asymptotic method, and the new iterative technique, have been employed and compared to those obtained using the fractional-order approximate long wave equation. The results of our study indicate that the optimal auxiliary function method is highly successful and practically simple, achieving better and more rapid convergence after just one repetition. This method is recognized as an efficient approach, demonstrating high precision in solving intriguing and intricate problems. Furthermore, it proves to be more time and resource efficient than other relevant analytical techniques, leading to significant savings in both volume and time. Compared to the Adomian decomposition technique, the new iterative technique, the variational iteration method, and the optimum homotopy asymptotic method, the suggested technique is extremely accurate computationally. It is also easy to analyze and solve fractionally linked nonlinear complex phenomena that arise in science and technology. We present the numerical and graphical findings that support these conclusions.

Modified Optimal Homotopy Asymptotic Method for KdV Family of Equations

Modified Optimal Homotopy Asymptotic Method for KdV Family of Equations

Tangent hyperbolic nanofluid flow through a vertical cone: Unraveling thermal conductivity and Darcy–Forchheimer effects

Purpose: This paper demonstrates the way tangent hyperbolic nanofluid flow through a vertical cone is influenced by varying viscosity and varying thermal conductivity. This study also seeks to illustrate the impact of convective boundary conditions on a fluid. The mathematical modeling also takes the Darcy–Forchheimer effect into account. Methodology: Using the appropriate similarity transformation, the fluid problem is reduced into a set of nonlinear ordinary differential equations. These systems of differential equations are evaluated numerically by applying the Optimal Homotopy Asymptotic Method. Findings: The nature of emergent parameters is examined in relation to the temperature distribution, nanoparticle concentration profile, and velocity profile. An increase in variable viscosity corresponds to a decrease in fluid velocity, while enhanced thermal conductivity results in elevated fluid temperature. The skin friction coefficient, Sherwood, and Nusselt numbers are numerically examined for active concerned parameters. These findings can be put into practice in a variety of fields such as polymer cooling systems and medication. Originality: Existing literature has yet to explore the combination of tangent hyperbolic nanofluids with varying viscosity and thermal conductivity under convective boundary conditions.

Corrigendum to “Study on equivalent mechanical properties of U-shaped bellows based on novel implementation of asymptotic homogenization method” [Reg Stud Mar. Struct. 103622, March 2024]

Corrigendum to “Study on equivalent mechanical properties of U-shaped bellows based on novel implementation of asymptotic homogenization method” [Reg Stud Mar. Struct. 103622, March 2024]

A Constitutive Law Modeling the Mixed Hardening Behavior of Particle-Reinforced Composites

Abstract A homogenized elasto-plastic constitutive law (including both constitutive equations and inversed constitutive equations) is proposed in an incremental form for the particle-reinforced composites based on the flow theory of plasticity and the asymptotic homogenization method. The constitutive law can be used to predict the mixed hardening behavior of particle-reinforced composites under arbitrary loading conditions if the uniaxial tension test curve of matrix materials is known. It is found that the constitutive law of particle-reinforced composites is similar in form to the law of matrix materials. There is a simple proportional relationship between the yield stress, the plastic modulus, and the deviatoric back stress of particle-reinforced composites and the corresponding parameters of matrix materials, which is equal to the ratio of the shear modulus of composites to the shear modulus of matrix materials. The tangent modulus of particle-reinforced composites can be calculated using a simple arithmetic formula according to the tangent modulus of matrix materials. A numerical algorithm is suggested.

An Asymptotic Preserving Discontinuous Galerkin Method for a Linear Boltzmann Semiconductor Model

An Asymptotic Preserving Discontinuous Galerkin Method for a Linear Boltzmann Semiconductor Model

Abstract PO1-10-09: The Performance Of Mobile Units Versus Hospital Units In National Breast Cancer Screening Program In Taiwan, 2010-2020

Abstract Purpose After the implementation of mobile units, the screened population increased from 3.9% to 40% of all women from 2004 to 2019 in Taiwan’s Breast Cancer Screening Program. There are limited studies comparing mobile units to hospital units. This study evaluated and compared the performance of the biennial mammography breast cancer screening in mobile units to hospital units in Taiwan over the past decade. Methods A cohort of 3 062 190 women aged 40 to 69 years who underwent biennial breast cancer mammography screenings from 2010 to 2020 was conducted, and data were obtained from the Health Promotion Administration, Ministry of Health and Welfare of Taiwan. Comparing participation of screened women mobile units and hospital units across national screening. Performance measures were determined as percentages and 95% confidence intervals (CIs) using the Wald asymptotic method. Results Among 3 062 190 women who underwent 7 444 450 examinations in the study population, the screened population by mobile units accounted for approximately 45.45 % of total examinations. The volume of screening mammograms performed in the mobile unit increased the number of people reached by this program. Since 2015, screenings performed in the mobile units have outnumbered those performed at brick-and-mortar hospitals: they comprised 480 456 of the 859 221 examinations (55.9%) performed in 2019. During the 10-year periods, the screen-detected cancers were 12 000 and 18 370 for mobile units and hospital units, respectively. Young women preferred screening in hospitals, resulting in a greater proportion of screen-detected cancers among those attending hospitals (49.4%) compared to the mobile unit (34.3%). The mean recall rate was higher in the hospitals at 8.95% (95% CI, 8.93%-8.99%) than in the mobile units at 7.27% (95% CI, 7.25%-7.32%). Conclusion Implementing mobile units increased the coverage of the screening program across the population. Hospital units had a younger patient population and higher recall rate than those mobile units. We can enhance mammography screening rates and follow-up among these groups by identifying these features, which can use to develop programs and materials fitting populations' specific requirements. Summary statement In Taiwan, the development of breast cancer screening has shown beneficial outcomes, especially with the deployment of mobile unit services to increase screened women. Our results could serve as an informative reference for other nations. Keywords: Breast cancer; screening; mobile units; hospital units. Table 1. Clinical Demographics for Mammographic Screening Examinations Table 2. Performance Measures of Digital Mammographic Screenings for Breast Cancer Citation Format: Vy Vu, Amy Ming-Fang Yen, Melissa Min-Szu Yao, Wing P. Chan. The Performance Of Mobile Units Versus Hospital Units In National Breast Cancer Screening Program In Taiwan, 2010-2020 [abstract]. In: Proceedings of the 2023 San Antonio Breast Cancer Symposium; 2023 Dec 5-9; San Antonio, TX. Philadelphia (PA): AACR; Cancer Res 2024;84(9 Suppl):Abstract nr PO1-10-09.

Understanding Stokes Drift Mechanism via Crest and Trough Phase Estimates

Abstract By providing mathematical estimates, this paper answers a fundamental question—“what leads to Stokes drift”? Although overwhelmingly understood for water waves, Stokes drift is a generic mechanism that stems from kinematics and occurs in any nontransverse wave in fluids. To showcase its generality, we undertake a comparative study of the pathline equation of sound (1D) and intermediate-depth water (2D) waves. Although we obtain a closed-form solution x(t) for the specific case of linear sound waves, a more generic and meaningful approach involves the application of asymptotic methods and expressing variables in terms of the Lagrangian phase θ. We show that the latter reduces the 2D pathline equation of water waves to 1D. Using asymptotic methods, we solve the respective pathline equation for sound and water waves, and for each case, we obtain a parametric representation of particle position x(θ) and elapsed time t(θ). Such a parametric description has allowed us to obtain second-order accurate expressions for the time duration, horizontal displacement, and average horizontal velocity of a particle in the crest and trough phases. All these quantities are of higher magnitude in the crest phase in comparison to the trough phase, leading to a forward drift, i.e., Stokes drift. We also explore particle trajectory due to second-order Stokes waves and compare it with linear waves. While finite amplitude waves modify the estimates obtained from linear waves, the understanding acquired from linear waves is generally found to be valid.

Determining the parameters of the functioning for a nonlinear ballistic system in a real external environment

Determining the parameters of the functioning for a nonlinear ballistic system in a real external environment

Conformal and Non-Minimal Couplings in Fractional Cosmology

Fractional differential calculus is a mathematical tool that has found applications in the study of social and physical behaviors considered “anomalous”. It is often used when traditional integer derivatives models fail to represent cases where the power law is observed accurately. Fractional calculus must reflect non-local, frequency- and history-dependent properties of power-law phenomena. This tool has various important applications, such as fractional mass conservation, electrochemical analysis, groundwater flow problems, and fractional spatiotemporal diffusion equations. It can also be used in cosmology to explain late-time cosmic acceleration without the need for dark energy. We review some models using fractional differential equations. We look at the Einstein–Hilbert action, which is based on a fractional derivative action, and add a scalar field, ϕ, to create a non-minimal interaction theory with the coupling, ξRϕ2, between gravity and the scalar field, where ξ is the interaction constant. By employing various mathematical approaches, we can offer precise schemes to find analytical and numerical approximations of the solutions. Moreover, we comprehensively study the modified cosmological equations and analyze the solution space using the theory of dynamical systems and asymptotic expansion methods. This enables us to provide a qualitative description of cosmologies with a scalar field based on fractional calculus formalism.

Equivalent single-layer model for hierarchical diamond honeycomb sandwich panels using variational asymptotic method

Sandwich panels with a hierarchical core facilitate the optimal utilization of limited materials to enhance bearing capacity in strategic locations. Nevertheless, the complex sub-structural cells have hindered extensive investigation into their static and modal characteristics. The objective of this study is to obtain the effective plate properties by homogenizing the representative unit cell and subsequently applying them to a two-dimensional equivalent single layer model (2D-ESL) based on the variational asymptotic method (VAM). Furthermore, three-point bending test of 3D-printed specimen and 3D detailed finite element results are employed to confirm the predictions of 2D-ESL under various load and boundary conditions. The effects of material factors (such as fiber volume fraction and layup configuration) and structural parameters (including aspect ratio, hierarchy ratio, and slenderness ratios of ribs and ligaments) on the equivalent stiffness and modal characteristics are also examined. Moreover, sandwich panels with different hierarchical configurations (including 2, 3 and infinite segments within the inclined ribs) are investigated to provide valuable insights for selecting appropriate hierarchies based on specific conditions. Notably, the transition from polyline to arc-shaped segments has been proven to effectively mitigate local stress concentration, leading to a more uniform distribution of local displacement.

On one solution of the problem of vibrations of mechanical systems with moving boundaries

An analytical method of solving the wave equation describing the oscillations of systems with moving boundaries is considered. By changing the variables that stop the boundaries and leave the equation invariant, the original boundary value problem is reduced to a system of functional-difference equations, which can be solved using direct and inverse methods. An inverse method is described that makes it possible to approximate quite diverse laws of boundary motion by laws obtained from solving the inverse problem. New particular solutions are obtained for a fairly wide range of laws of boundary motion. A direct asymptotic method for the approximate solution of a functional equation is considered. An estimate of the errors of the approximatemethod was made depending on the speed of the boundary movement.

Multiscale modeling of gas-induced fracturing in anisotropic clayey rocks

In the context of repositories for nuclear waste, understanding the behavior of gas migration through clayey rocks with inherent anisotropy is crucial for assessing the safety of geological disposal facilities. The primary mechanism for gas breakthrough is the opening of micro-fractures due to high gas pressure. This occurs at gas pressures lower than the combined strength of the rock and its minimum principal stress under external loading conditions. To investigate the mechanism of microscale mode-I ruptures, it is essential to incorporate a multiscale approach that includes subcritical microcracks in the modeling framework. In this contribution, we derive the model from microstructures that contain periodically distributed microcracks within a porous material. The damage evolution law is coupled with the macroscopic poroelastic system by employing the asymptotic homogenization method and considering the inherent hydro-mechanical (HM) anisotropy at the microscale. The resulting permeability change induced by fracture opening is implicitly integrated into the gas flow equation. Verification examples are presented to validate the developed model step by step. An analysis of local macroscopic response is undertaken to underscore the influence of factors such as strain rate, initial damage, and applied stress, on the gas migration process. Numerical examples of direct tension tests are used to demonstrate the model's efficacy in describing localized failure characteristics. Finally, the simulation results for preferential gas flow reveal the robustness of the two-scale model in explicitly depicting gas-induced fracturing in anisotropic clayey rocks. The model successfully captures the common behaviors observed in laboratory experiments, such as a sudden drop in gas injection pressure, rapid build-up of downstream gas pressure, and steady-state gas flow following gas breakthrough.

Cross-sectional analysis of initially curved composite beams with rectangular and circular sections using an asymptotic meshless method

An analytical meshless cross-sectional dimensional reduction method is developed to perform structural analysis of initially curved composite beams. By introducing the Pascal polynomials in the Cartesian coordinates system for beams with rectangular cross-sections and in the polar coordinates system for the beams with circular cross-sections to the warping field, a three-dimensional beam problem can be split into a two-dimensional cross-sectional analysis and a one-dimensional along the reference line of the beam. The obtained cross-sectional stiffness matrix based on higher-order of the strain energy can be incorporated into a one-dimensional finite element solution for initially curved composite beams, for the purpose of strain and stress analysis. Finally, the three-dimensional strain through the thickness of the beam can be achieved by the recovery analysis. The present method takes advantage of Pascal polynomials so that the solution procedure will be more simplified and computationally more efficient compared to three-dimensional finite element (3D FE) method. The proposed method for different isotropic and anisotropic beams is examined and compared with the literature, 3D FE, and Variational Asymptotic Beam Sectional (VABS) package which is a finite element based cross-sectional analysis tool for composite beams.

Numerical solutions of thin film flows of non-Newtonian fluids via optimal homotopy asymptotic approach

The objective of this research is to recover new solutions in the lifting and drainage cases of thin film flows involving non-Newtonian fluid models namely Pseudo-Plastic (PP) and Oldroyd 6-Constant (O6C). Both of the considered fluids exhibit numerous uses in industry when coupled with thin film phenomena. Some of the industrial applications include decorative and optical coatings, prevention of metallic corrosion and lithography of various diodes, sensors and detectors. For solution purpose, a modified version of Optimal Homotopy Asymptotic Method (OHAM) is proposed in which Daftardar–Jafari polynomials will replace the classical OHAM polynomials in nonlinear problems and provide better results in terms of accuracy. The paper includes a comprehensive application of modified algorithm in the case of thin film phenomena. To validate the obtained series solutions, the paper employs a rigorous assessment of convergence and validity by computing the residual errors in each scenario. For showing the effectiveness of modified algorithm, numerical comparison of classical and modified OHAMs is also presented in this study. Furthermore, the study conducts an in-depth graphical analysis to assess the impact of fluid parameters on velocity profiles both in lifting and drainage scenarios. The results of this investigation demonstrate that the proposed modification of OHAM ensures better accuracy of solutions than the classical OHAM. Consequently, this method can be effectively utilized for tackling more advanced situations.

A mathematical model of the gravitational potential of the planet taking into account tidal deformations

Objectives. This paper investigates the gravitational potential of a viscoelastic planet moving in the gravitational field of a massive attracting center (star), a satellite and one or more other planets moving in Keplerian elliptical orbits relative to the attracting center. Celestial bodies other than a viscoelastic planet are modeled by material points. Within the framework of the linear model of the theory of viscoelasticity, the problem of finding the vector of elastic displacement has been resolved. Traditionally, a solid body model is used to determine the Earth’s gravitational field, while tidal deformations are taken into account in the form of small corrections to the coefficients of the geopotential model. In this work, the viscoelastic ball model is used to take into account tidal effects. The relevance of the research topic is associated with high-precision forecasting of the movement of artificial satellites of the Earth, high-precision measurement of the Earth’s gravitational field.Methods. In this study the asymptotic and analytical methods developed by V.G. Vilke are used for mechanical systems containing viscoelastic elements of high rigidity, as well as methods of classical mechanics, mathematical analysis. The graphs were plotted using the Octave mathematical package.Results. After resolving the quasi-static problem of elasticity theory by calculating triple integrals over a spherical area, a formula for the gravitational potential of a deformable planet was obtained. In addition, the gravitational potential of the Earth was also calculated taking into account solid-state tidal effects from the Moon, Sun, and Venus at an external point. Graphs were constructed to show the dependence of the Earth’s gravitational potential on time.Conclusions. The theoretical and numerical results established herein show that the main contribution to the gravitational potential of the Earth is made by the Moon and the Sun. The influence of other planets in the solar system is small. The value of the gravitational potential at the outer point of the Earth, taking into account tidal effects, depends both on the position of the point in the moving coordinate system and on the relative position of celestial bodies.

Asymptotic Ray Method for the Double Square Root Equation

The parabolic wave equation describes wave propagation in a preferable direction, which is usually horizontal in underwater acoustics and vertical in seismic applications. For dense receiver arrays (receiver spacing is less than signal wavelength), this equation can be used for propagating the recorded wavefield back into the medium for imaging sources and scattering objects. Similarly, for multiple source and receiver array acquisition, typical for seismic exploration and potentially beneficial for ocean acoustics, one can model data in one run using an extension of the parabolic equation—the pseudo-differential Double Square Root (DSR) equation. This extended equation allows for the modeling and imaging of multi-source data but operates in higher-dimensional space (source, receiver coordinates, and time), which makes its numerical computation time-consuming. In this paper, we apply a faster ray method for solving the DSR equation. We develop algorithms for both kinematic and dynamic ray tracing applicable to either data modeling or true-amplitude recovery. Our results can be used per se or as a basis for the future development of more elaborated asymptotic techniques that provide accurate and computationally feasible results.

3D equivalent Cauchy model for serrated re-entrant auxetic honeycombs based on variational asymptotic method

The presence of serrated ligaments in the re-entrant auxetic honeycomb (abbreviated as serrated RAH) effectively enhances the stiffness while maintaining auxeticity in the re-entrant direction. To investigate its auxetic characteristics, a 3D equivalent Cauchy model (3D-ECM) was established using the variational asymptotic method. The unique aspect is that the engineering constants are obtained by homogenizing the representative unit-cell and then utilized in the 3D-ECM for global analysis. Subsequently, the resulting global responses are fed into the recovery relationships for localized field analysis. The accuracy of the 3D-ECM and the recovered local field distributions were confirmed by comparing them with the results obtained from the three-dimensional FE model (3D-FEM) and the experimental data collected from the 3D-printed specimen. The effects of the geometric parameters on the negative Poisson’s ratio and engineering constants were in-depth discussed. Specifically, an optimal value of 0.5 for the re-entrant ratio and d-value (corresponding to the re-entrant angle of 90 degree) yield maximum auxeticity and comparatively higher elastic moduli. In addition, the 3D-ECM greatly enhanced the computational efficiency without compromising the accuracy in predicting global behaviors and local field distributions. The unit-cell tailorability in the proposed model offers valuable guidance for optimizing the design of serrated RAHs.

Stability analysis of premixed flames with downstream heat gain

Pulsating instability is a kind of combustion intrinsic instability, that can affect the flammable limit and may couple with other processes to generate more complex combustion instability phenomena. Pulsating instability was found in the premixed flame of the double-flame partially premixed flame (PPF), in which the flame patterns change from stable to pulsating unstable, and to stable again as the equivalence ratio or stretch rate increases. To further investigate this phenomenon, a simpler model, stretched premixed flames with downstream heat gain, is analyzed by the asymptotic expansion method in this paper. The downstream heat gain inhibits the steady extinction of stretched premixed flames with a large Zeldovich number. The stability of premixed flames changes in the zero-limit mode, two-limits mode and one-limit mode sequentially as the Zeldovich number increases. The zero-limit mode represents the premixed flame is stable for all stretch rates. The two-limit mode and one-limit represent the premixed flame changes in the pattern of stable-pulsating-stable and pulsating-stable as the stretch rate increases, respectively. The downstream heat gain suppresses the pulsating instability to yield the stable region at the large stretch rate and large Zeldovich number. The influence of other factors, including the Lewis number, the downstream temperature and the downstream boundary position are also discussed. The current theoretical method is also applied to the extracted parameters from numerical calculations of hydrogen/air double-flame PPF. A qualitative similarity is found between the theoretical results and numerical results.