Analytical Results Research Articles

Gaussian smoothed particle hydrodynamics: A high-order meshfree particle method

Smoothed particle hydrodynamics (SPH) has attracted significant attention in recent decades, and exhibits special advantages in modeling complex flows with multiphysics processes and complex phenomena. Its accuracy depends heavily on the distribution of particles, and will generally be lower if the particles are distributed non-uniformly. A high-order SPH scheme is proposed in the present work for simulating both compressible and incompressible flows. It uses a Gaussian quadrature rule to perform the particle approximation of SPH by introducing Gaussian nodes. Unfortunately, the Gaussian nodes hardly overlap with SPH particles due to the Lagrangian feature, and thus we use a high-order interpolation method to obtain the corresponding physical quantities at the Gaussian nodes. The accuracy and robustness of the proposed Gaussian SPH are demonstrated by several numerical tests, including the Sod problem, Poiseuille flow, Couette flow, cavity flow, Taylor–Green vortex and dam break flow, and a convergence analysis is also conducted to evaluate the effects of particle resolution and distribution for reconstructing a given function. The simulation results for each test case are in good agreements with the available analytical, experimental or numerical results, showing that the proposed Gaussian SPH method is accurate and reliable but expensive for simulating compressible and incompressible flow problems.

Input-to-state hybrid impulsive formation stabilization for multi-agent systems with impulse delays

This paper addresses the input-to-state formation stabilization problem of nonlinear multi-agent systems within a hybrid impulsive framework, considering delay-dependent impulses, strong nonlinearity, and deception attack signals. By leveraging Lyapunov functionals, impulsive comparison theory, average impulsive interval methods, and graph theory, we develop novel criteria for possessing locally input-to-state and integral input-to-state formation stabilization across different impulse sequence classes. These criteria are expressed in terms of continuous/impulsive feedback gains, time delay size, nonlinearity strength, uniform upper bound of impulsive interval, and length of average impulsive interval. Notably, the design of control impulses benefit the destabilizing continuous dynamics in the formation stabilization process. To demonstrate the effectiveness and validity of the analytical results, we provide numerical simulation examples involving various types of external attack signals.

Heisenberg spin chain with random-sign couplings

We study the 1D quantum Heisenberg chain with randomly ferromagnetic or antiferromagnetic couplings [a model previously studied by approximate strong-disorder renormalization group (RG)]. We find that, at least for sufficiently large spin S, the ground state has "spin glass" order. The spin waves on top of this state have the dynamical exponent [Formula: see text], intermediate between the values z = 1 of the antiferromagnet and z = 2 of the ferromagnet. Density matrix renormalization group (DMRG) simulations are in good agreement with the analytical results for spins S = 1 and [Formula: see text]. The case [Formula: see text] shows large finite size effects: We suggest that this case is also ordered, but with a small ordered moment.

Approximation for probability of coverage in softly inhibitive cellular networks

AbstractThe Strauss point process is a very popular model for describing the random cellular networks, yet several key statistical properties such as intensity, empty space function, and probability generating functional have remained elusive. This article addresses these issues by first leveraging the Poisson saddle point method to approximate the distance‐conditioned intensity for Strauss point processes. Subsequently, the author derives an analytically tractable expression for the distribution of empty space distance based on a conditional thinning mechanism. Additionally, the author establishes an upper bound for the probability generating functional in Strauss point processes, which is crucial for evaluating the Laplace transform of cumulative interference in relevant cellular networks. These findings facilitate the systematic derivation of spatially averaged probability of coverage, and the accuracy of analytic results is validated through simulations.

Universal distribution of the number of minima for random walks and Lévy flights.

We compute exactly the full distribution of the number m of local minima in a one-dimensional landscape generated by a random walk or a Lévy flight. We consider two different ensembles of landscapes, one with a fixed number of steps N and the other till the first-passage time of the random walk to the origin. We show that the distribution of m is drastically different in the two ensembles (Gaussian in the former case, while having a power-law tail m^{-3/2} in the latter case). However, the most striking aspect of our results is that, in each case, the distribution is completely universal for all m (and not just for large m), i.e., independent of the jump distribution in the random walk. This means that the distributions are exactly identical for Lévy flights and random walks with finite jump variance. Our analytical results are in excellent agreement with our numerical simulations.

Elastic Local Buckling Analysis of a Sandwich Corrugated Steel Plate Pipe-Arch in Underground Space

In underground spaces, corrugated steel plate (CSP) pipe-arches may experience local buckling instability, which can subsequently lead to the failure of the entire structure. Recently, sandwich CSP pipe-arches have been used to enhance the stability of embedded engineering outcomes, and their buckling behaviors require in-depth research. In this paper, we establish a theoretical model by simplifying soil support and using Hoff sandwich plate theory to focus on the local buckling stability of the straight segment in embedded sandwich CSP pipe-arches using the Rayleigh–Ritz method. Through stability analysis, the instability criteria for embedded sandwich CSP pipe-arches are analytically determined. Numerical calculations reveal that the critical buckling load of a sandwich CSP pipe-arch is affected by several factors, including the elastic modulus, thickness, Poisson’s ratio, rotational constraint stiffness, and the length of the straight segment. Specifically, increasing the thickness of the sandwich CSP pipe-arch can substantially enhance the critical buckling load. Meanwhile, the wavenumber is affected by the elastic modulus and the length of the straight segment. The analytical results are in agreement with those obtained from finite element analysis. These findings provide a theoretical basis and guidance for the application of sandwich CSP pipe-arches in fields such as subway stations, tunnel construction, underground passages, and underground parking facilities.

Oscillatory systems with two degrees of freedom and van der Pol coupling: Analytical approach

In this paper steady‐state vibrations of the two‐degrees‐of‐freedom oscillatory systems with van der Pol coupling are investigated. The model is a system of two differential equations with weak nonlinearity. A new solving procedure based on D′Alembert's method and the method of time‐variable amplitude and phase is developed. The main advantage of the method in comparison to others is that it gives the solution of the system of two coupled weak nonlinear equations in the form that is simple to analyze, as it has the same form as the solution of the corresponding system of linear equations. In the paper two types of systems are considered: one, a two‐mass system with two degrees of freedom, and second, the one‐mass system with two degrees of freedom. The torsional vibrations of a two‐mass system and vibrations of a Jeffcott rotor with two‐degrees‐of‐freedom are analyzed. Analytically obtained results are numerically tested. It is obtained that the difference between analytic and numeric results is small and almost negligible. As the accuracy of the analytic solution is high, it is suitable for application in technics and engineering. Conclusions about steady‐state self‐sustainable oscillators, orbital, and limit cycle motions are given.

IDA-Based Seismic Fragility Analysis of a Concrete-Filled Square Tubular Frame

Based on the incremental dynamic analysis (IDA) method, this paper conducts seismic fragility analysis of a CFST plane frame, a CFST spatial frame under 1D (one-dimensional) ground motions, and a CFST spatial frame under 2D (two-dimensional) ground motions, with different attacking angles. Firstly, nine-story, three-span CFST frame structures (including the plane frame and spatial frame) were modeled in OpenSees, based on the accurate simulation of the hysteresis performance of the test CFST frames. Then, twenty-five groups of ground motions were employed to analyze the seismic response. Lastly, the IDA curve clusters, probabilistic demand models, and seismic fragility curves of frame structures were researched, respectively. The analytical results showed that the exceeding probability of the spatial frame under 2D ground motions was successively greater than that under 1D ground motions, and greater than the plane frame, and the maximum difference at each performance level was up to 6% and 16%, respectively. The fragility analysis result of the spatial frame was sensitive to the attacking angle of ground motion, and the exceeding probability of the 135°, 150°, and 165° fragility curves was larger than that of the 0° (original attacking angle) fragility curve at each performance level. The research results provide a reference for seismic fragility analysis of CFST frame structures employing the IDA method.

Drying of a Clayey Ceramic Flat Plate: Simulation Studies Using the Galerkin-Based Integral Method

The ceramics sector is crucial to the global economy. This research is devoted to studying the drying process of ceramic parts with arbitrary shapes based on Fick’s second law of diffusion and energy conservation. Herein, the mathematical procedure to obtain the exact solutions of the model equations using the Galerkin-based integral method is provided. In the mathematical modeling are considered constant properties and equilibrium conditions at the surface of the material. Emphasis is given to clay ceramic flat plate. Analytical results of the average moisture content, local temperature, and moisture content and temperature fields within the ceramic parts are presented, followed by an in-depth discussion.

Moderating Effects of Visual Order in Graphical Symbol Complexity: The Practical Implications for Design

In the field of visual graphic design, complexity plays a crucial role in visual information processing, and it is assumed to be an absolute quantity based on the number of the presenting features and components. However, it remains unclear whether the visual order of the constituent elements in graphical symbol complexity affects cognitive processing, especially memory processing. Our research innovatively generated four groups of novel, meaningless graphical symbols (complex and ordered, complex and disordered, simple and ordered, and simple and disordered) and experimentally manipulated the level of complexity and order in these stimuli. Before the formal experiment, a five-point scale was used to further rule out differences between objective and subjective definitions of these graphical symbols on ratings of complexity, order, concreteness, and familiarity. Then, we used a cue-recall task to compare subjects’ memory performance of those four graphical symbol groups. The analytical results showed a significant interaction between visual order and graphical symbol complexity, with the complexity effect appearing only when the stimuli were in disordered condition and disappearing once the stimuli were ordered. In addition, this study conducted a practical application validation to confirm that increasing the level of visual order is an effective way to improve user experience while maintaining the same level of complexity. The findings can serve as a reference for graphical symbol design, graphic design, and visual communication design.

Calculation of the Second EXAFS Cumulant of Si Using the Anharmonic Correlated Einstein Model

The second expanded X-ray absorption fine structure (EXAFS) cumulant of the crystalline silicon (Si) has been studied in the temperature-dependent. This is calculated in explicit forms using the anharmonic correlated Einstein (ACE) model developed from the correlated Einstein model based on the anharmonic effective potential and the quantum statistical theory. The numerical results of Si in the temperature range from 0 to 1200 K are in good agreement with those obtained by the other theoretical models and experiments at several temperatures. The analytical results show that the ACE model is suitable for analyzing the experimental EXAFS data of diamond cubics.

Field Theory of Active Brownian Particles with Dry Friction

Abstract We present a field theoretic approach to capture the motion of a particle with dry friction for one- and two-dimensional diffusive particles, and further expand the framework for two-dimensional active Brownian particles. Starting with the Fokker-Planck equation and introducing the Hermite polynomials as the corresponding eigen-functions, we obtain the actions and propagators. Using a perturbation expansion, we calculate the effective diffusion coefficient in the presence of both wet and dry frictions in a perturbative way via the Green-Kubo relation. We further compare the analytical result with the numerical simulation. Our result can be used to estimate the values of dry friction coefficient in experiments.

An analytical model for predicting residual stress in shot peening with strain energy method

In this study a model by novel analytical approach is developed and experimentally verified for shot peening residual stress distribution. The residual stress field induced by single shot impact is calculated by using Glinka–Molski energy-based method and kinematic hardening model. The formulation of the compressive residual stress (CRS) distribution is often based on plane strain or plane stress. It can be determined from the derived relation presented in this paper, the final residual stress in the full coverage conditions is the average of the two strain and stress plane expressions proposed by previous researchers. The distribution of residual stress is one of the key differences between the profiles produced by the results of the current model. There is a significant distinction between surface residual stress and maximum CRS, because the CRS profile near the surface is more curved compared to profiles obtained in earlier analytical models. The experimental data obtained by XRD analysis indicate the correctness and precision of the current model. Another goal of this study is to increase the fatigue life of GTD-450 stainless steel by shot peening at two different peening intensities. The fatigue life of samples were obtained by rotary bending test. Analytical results that confirmed by experimental findings shown bigger maximum compressive residual stresses occurred in higher shot peening intensities. This incident can improved fatigue life by deeper plastically deformed layer.

Analytic evolution for complex coupled tight-binding models: Applications to quantum light manipulation

We present analytic solutions to the evolution in generalized tight-binding models, which consider complex first-neighbor couplings with equal amplitude and arbitrary phases. Our findings provide a powerful tool to efficiently calculate expectation values and correlations within the system, which are otherwise difficult to compute numerically. We apply our results to relevant examples in quantum light manipulation using -port linear couplers, describing the evolution of single (multi)-mode squeezing, single-photon added (subtracted) Gaussian states, and second-order site-to-site photon correlations. Significantly, our analytic results outperform standard numerical calculations. Our study paves the way for a comprehensive mathematical framework describing the spatial evolution of quantum states across a wide range of physical systems governed by the tight-binding model. Published by the American Physical Society 2024

Inquisition of optical soliton structure and qualitative analysis for the complex-coupled Kuralay system

This study seeks to explore the integrable dynamics of induced curves through the utilization of the complex-coupled Kuralay system. The importance of the coupled complex Kuralay equation lies in its role as a fundamental model that contributes to the understanding of intricate physical and mathematical concepts, making it a valuable tool in scientific research and applications. The soliton solutions originating from the Kuralay equations are believed to encapsulate cutting-edge research in various important domains such as optical fibers, nonlinear optics, and ferromagnetic materials. Analytical techniques are employed to derive traveling wave solutions for this model, given that the Cauchy problem cannot be resolved using the inverse scattering transform. In the quest for solitary wave solutions, the extended modified auxiliary equation mapping (EMAEM) method is employed. We derive several novel families of precise traveling wave solutions, encompassing trigonometric, hyperbolic, and exponential forms. Moreover, the planar dynamical system of the concerned equation is created, all probable phase portraits are given, and sensitive inspection is applied to check the sensitivity of the considered equation. Furthermore, after adding a perturbed term, chaotic and quasi-periodic behaviors have been observed for different values of parameters, and multistability is reported at the end. Numerical simulations of the solutions are incorporated alongside the analytical results to enhance comprehension of the dynamic characteristics of the solutions obtained. This study’s outcomes can offer valuable insights for addressing other nonlinear partial differential equations (NLPDEs). The soliton solutions obtained in this study offer important insights into the intricate nonlinear equation being examined.

Kink phenomena of the time-space fractional Sharma-Tasso-Olver (STO) equation

Abstract This paper aims to obtain exact solutions of solitary waves for the conformable fractional Sharma-Tasso-Olver (STO) equation which plays an important role in nuclear physics to describe the physical occurrences such as the fission and fusion processes. Solitary waves operate central parts in different areas of study such as electromagnetism, atomic quantum theory, as well as special relativity. By means of sub-ode approach with the aid of the modified fractional Riccati-Bernoulli equation, the exact forms of generalized solitary solution of the fractional (STO) equation are found and specified in hyperbolic, trigonometric, and rational functions. This makes the visualization of the fractional effects and the dynamic behaviors of these solutions in 3D and 2D help in establishing practicality for application of the results. The novel analytical results benefit general engineering and mathematical physics in demonstrating that the proposed employment of the given technique allows solving nonlinear problem analytically. These findings are significant for the progress of wave proceedings in the number of applications.

Optimization of the Transmission X‐Ray Target Toward Obtaining Monochromatic Radiation

AbstractThe article presents a process of designing and optimizing a transmission X‐ray target allowing to obtaining of near monochromatic radiation. It describes how the choice of materials, their thickness, and the energy of electrons influence the generated X‐ray spectra and which of the physical phenomena plays the most important role in this process. Due to the analytical, experimental, and simulation results, the authors prepared a Ni‐Si bilayer structure, which produces radiation confined to more than 50% within a single 7.2 keV emission peak.

An analytical solution of direction evolution of crack growth during progressive failure in brittle rocks

AbstractMicrocrack growth during progressive compressive failure in brittle rocks strongly influences the safety of deep underground engineering. The external shear stress τxy on brittle rocks greatly affects microcrack growth and progressive failure. However, the theoretical mechanism of the growth direction evolution of the newly generated wing crack during progressive failure has rarely been studied. A novel analytical method is proposed to evaluate the shear stress effect on the progressive compressive failure and microcrack growth direction in brittle rocks. This model consists of the wing crack growth model under the principal compressive stresses, the direction correlation of the general stress, the principal stress and the initial microcrack inclination, and the relationship between the wing crack length and strain. The shear stress effect on the relationship between y‐direction stress and wing crack growth and the relationship between y‐direction stress and y‐direction strain are analyzed. The shear stress effect on the wing crack growth direction during the progressive compressive failure is determined. The initial crack angle effect on the y‐direction peak stress and the wing crack growth direction during the progressive compressive failure considering shear stress is also discussed. A crucial conclusion is that the direction of wing crack growth has a U‐shaped variation with the growth of the wing crack. The rationality of the analytical results is verified by an experiment and from numerical results. The study results provide theoretical support for the evaluation of the safety and stability of surrounding rocks in deep underground engineering.

Improved forming height and reduced energy consumption through an optimized hybrid quasi-static and high-speed forming strategy

It is widely accepted that aluminum alloy-based materials exhibit improved formability and enhanced ductility under high-speed impact. However, in this study, the experimental results were not entirely similar to the traditional theoretical results. Thus, this study once again confirms the conclusion that high-rate forming can increase the forming limit of materials, and simultaneously supplements the conclusion. Herein, a hybrid process combining quasi-static hydraulic forming and electromagnetic hydraulic forming was proposed. The effects of the forming sequence and pre-deformation amount on sheet bulging and fracture morphology were analyzed via experiments and simulation. It was found that when the electromagnetic hydraulic forming is pre-deformation and the quasi-static hydraulic forming is post-deformation, the forming height of aluminum alloy does not improve significantly. Conversely, when the quasi-static hydraulic forming is pre-deformation and the electromagnetic hydraulic forming is post-deformation, the forming height of aluminum alloy improves significantly. In case of pre-deformation quasi-static liquid pressure P0 = 2 MPa, and post-deformation electromagnetic hydraulic forming, the limit forming height is 22.4 % higher than that under quasi-static hydraulic forming. Moreover, the limiting voltage decreases with increasing pre-deformation quasi-static liquid pressure P0, and the energy consumption reduces by 42.9 %. The deformation behavior and damage characteristics of quasi-static hydraulic forming, electromagnetic hydraulic forming, and hybrid forming were accurately predicted by multi-physics coupling analysis. Compared with quasi-static hydraulic forming, void nucleation and growth are inhibited due to high-speed impact. In particular, when the sheet is about to crack during high-speed forming, the voids that should have grown sharply are significantly inhibited. Therefore, the improved formability mainly acts at the post-deformation stage during high-speed forming, with the analytical results corroborating the experimental and simulation ones.

Viscoelastic-adhesive modeling of hybrid adhesively bonded double-lap joints: analytical and numerical studies

ABSTRACT There are a limited number of studies examining the viscoelastic and viscoplastic behavior of adhesive joints. If there is more than one adhesive in the overlap region, the problem becomes more complex and viscoelastic modeling of the overlap region becomes difficult. There is no study in the literature that covers such a bonding (overlap) scheme and examines the viscoelastic behavior of the double lap joint (DLJ). In this study, the effect of viscoelasticity of polymeric adhesives on shear stress and strain distributions in the hybrid-adhesive layer of a DLJ is studied. Maxwell-Wiechert (MW) viscoelastic material model is used to define the viscoelastic behavior of the adhesives. Analytical and finite element (FE) results are compared for both mono- and hybrid-adhesive joints under the step load. Good matches are observed between the analytical and numerical results. Four different hybrid-adhesive joints with different combinations of the adherend thicknesses are investigated by considering only the analytical method. In order to investigate the effect of the loading type on the viscoelastic behavior, step and ramp loads were applied to the joint separately. It is observed that applying the ramp load to the hybrid-adhesive joints affects the shear stresses occurred at the ends of the overlap more than the mono-adhesive joints.