Small Parameter Research Articles

Experiment and simulation study on the functional fatigue of shape memory alloy beam actuators

The functional fatigue behavior of shape memory alloy (SMA) beam actuators is gaining importance as their utilization in engineering applications becomes more widespread. However, research on the functional fatigue behavior of SMA beam actuators under bending conditions is not as extensive as that on SMA wires. In this paper, an experimental study and theoretical analysis of the functional fatigue behavior of SMA beam actuators were conducted. A measuring method for bending deflection and an automatic thermal cyclic test bench was designed. A series of functional fatigue tests were conducted on SMA beam actuators under different bias load conditions and the functional fatigue patterns were obtained. The material damage factor is defined and calculated through specimen tests. By incorporating the damage factor to modify the existing constitutive model, a model considering functional fatigue and tension-compression asymmetry is obtained. Finite element analysis (FEA) is performed based on this modified model to simulate the actuating performance of SMA beam actuators and to compute the deflection at different numbers of cycles. Additionally, a small parameter study on the actuator shape is conducted using FEA. By comparing the FEA results with functional fatigue experimental data, the effectiveness of the modified model is validated, demonstrating its capability to describe functional fatigue and predict actuator lifespan.

The role of nanosize dependence of dislocation and surface roughness concentrations to calculate lattice thermal conductivity in In0.53Ga0.47As nanofilms

ABSTRACT Systematic empirical relations were established for the nanosize-dependent surface roughness, ε ( D ) , dislocations, N D ( D ) , and effective length to examine their effects on the temperature-dependent lattice thermal conductivity (LTC) within the framework of the Morelli-Callaway model for In0.53Ga0.47As alloy films with thicknesses ranging from 10 to 70 and 1400 nm, the latter representing the material’s bulk state. The results were analyzed for film thicknesses of 2, 5, 200, and 500 nm, under the assumption that electrons and impurities are unaffected by nanosize, while a small alloy parameter significantly and sensitively alters the LTC curve in medium and high-temperature regions. According to these new findings, and considering phonon scattering processes, the LTC’s temperature dependence is sufficiently reduced at the nanoscale and fits well with the available reported experimental findings. It was also discovered that the LTC’s peak maximum value and its corresponding temperature are influenced by the film’s dislocation concentration, surface roughness, and thickness. Unravelling the complex interplay between nanosize-dependent dislocation and surface roughness concentrations becomes critical for a thorough knowledge when examining the delicate nuances of LTC in In0.53Ga0.47As nanofilms. Finally, the Morelli-Callaway-based model has been modified to calculate LTC for nanofilms without fitting parameters, provided their related bulk state values are known.

One-dimensional, multi-fluid model of the plasma-wall transition. II. Negative ions

The plasma-wall transition is investigated by a one-dimensional steady-state multifluid model, which was presented in detail in Part I [T. Gyergyek et al., AIP Adv. 14, 045201 (2024)]. In this work, the plasma-wall transition is analyzed for the case where the plasma consists of singly charged positive ions, electrons, and singly charged negative ions. When the temperature and initial density of the negative ions are varied, a transition between two types of solutions of the model is observed. We call them the low and high solution, with respect to the absolute value of the potential drop. When the density and temperature of the negative ions are above a critical value, the low solution is observed. As the mass of the positive ions increases, these critical values also increase, but only until the ion mass is below about 1000 electron masses. With larger ion masses, the critical density of the negative ions and the temperature no longer change. In the low solution, the potential drop in front of the sheath is determined by the negative ions and is smaller in absolute terms than in the case of the high solution, where the potential drop in front of the sheath is determined by the electrons. If the problem is analyzed on the pre-sheath scale, the transition between the low and high solution is very sharp. However, when the neutrality condition is replaced by the Poisson equation, this transition becomes blurred and the solutions of the model equations exhibit oscillations. The role of the smallness parameter is highlighted. It is shown how the initial electric field is determined. Deviation of the negative ion density profile from the Boltzmann relation is discussed.

Exploring the geometry of the bifurcation sets in parameter space

By studying a nonlinear model by inspecting a p-dimensional parameter space through (p-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(p-1)$$\\end{document}-dimensional cuts, one can detect changes that are only determined by the geometry of the manifolds that make up the bifurcation set. We refer to these changes as geometric bifurcations. They can be understood within the framework of the theory of singularities for differentiable mappings and, in particular, of the Morse Theory. Working with a three-dimensional parameter space, geometric bifurcations are illustrated in two models of neuron activity: the Hindmarsh–Rose and the FitzHugh–Nagumo systems. Both are fast-slow systems with a small parameter that controls the time scale of a slow variable. Geometric bifurcations are observed on slices corresponding to fixed values of this distinguished small parameter, but they should be of interest to anyone studying bifurcation diagrams in the context of nonlinear phenomena.

Secondary massive quarks with the Mellin-Barnes expansion

Processes involving only massless or massive quarks at tree-level get corrections from massive (lighter, heavier, or equal-mass) secondary quarks starting at two-loop order, generated by a virtual gluon splitting into a massive quark anti-quark pair. One convenient approach to compute such two-loop corrections is starting with the one-loop diagram considering the virtual gluon massive. Carrying out a dispersive integral with a suitable kernel over the gluon mass yields the desired two-loop result. On the other hand, the Mellin-Barnes representation can be used to compute the expansion of Feynman integrals in powers of a small parameter. In this article we show how to combine these two ideas to obtain the corresponding expansions for large and small secondary quark masses to arbitrarily high orders in a straightforward manner. Furthermore, the convergence radius of both expansions can be shown to overlap, being each series rapidly convergent. The advantage of our method is that the Mellin representation is obtained directly for the full matrix element from the same one-loop computation one needs in large-β0 computations, therefore many existing results can be recycled. With minimal modifications, the strategy can be applied to compute the expansion of the one-loop correction coming from a massive gauge boson. We apply this method to a plethora of examples, in particular those relevant for factorized cross sections involving massless and massive jets, recovering known results and obtaining new ones. Another bonus of our approach is that, postponing the Mellin inversion, one can obtain the small- and large-mas expansions for the RG-evolved jet functions. In many cases, the series can be summed up yielding closed expressions.

Численно-аналитическая модель для интерпретации результатов индикаторных исследований нефтяных пластов: решение прямой задачи при наличии каналов низкого фильтрационного сопротивления

The article considers the problem of interpretation of tracer tests in oil reservoirs, into which a specific chemical substance - tracer - is injected via an injection well for flow monitoring and tracer concentration measurements in surrounding production wells. The testing data provide evidence of the existence of low resistance channels in the reservoirs, along which the tracer moves at velocities several orders of magnitude higher than in the reservoir. The nature and properties of these channels are unknown, and a semi-analytical model involving two small parameters is proposed to solve this problem. This makes it possible to construct two non-conjugate solutions: universal numerical and analytical. Such an approach allows us to solve both direct and inverse problems. The present article is devoted to solving the direct problem. The problem of pressure distribution in a rectangular reservoir pattern is solved in dimensionless coordinates and is universal, i.e., it does not depend on the size of the area and the pressure drop. This solution is approximated by power functions and is further used to determine the mass transfer between channel and reservoir. The problem of tracer slug propagation through the channel obeys the law of conservation of tracer mass and Darcy's law. Its solution is found by the methods of characteristics under the condition of smallness of the parameter reflecting the ratio of reservoir permeability to channel permeability. The results of the semi-analytical solution are compared with the numerical solutions previously obtained by other authors. The fundamental difference between these solutions is the presence of numerical diffusion when using the finite-difference method of solution, which leads to the «dissipation» of the trace slug edges.

Pitchfork bifurcation along a slow parameter ramp: Coherent structures in the critical scaling

AbstractWe investigate the slow passage through a pitchfork bifurcation in a spatially extended system, when the onset of instability is slowly varying in space. We focus here on the critical parameter scaling, when the instability locus propagates with speed , where is a small parameter that measures the gradient of the parameter ramp. Our results establish how the instability is mediated by a front traveling with the speed of the parameter ramp, and demonstrate scalings for a delay or advance of the instability relative to the bifurcation locus depending on the sign of , that is on the direction of propagation of the parameter ramp through the pitchfork bifurcation. The results also include a generalization of the classical Hastings–McLeod solution of the Painlevé‐II equation to Painlevé‐II equations with a drift term.

A BDF2 method for a singularly perturbed transport equation

A singularly perturbed transport equation is considered. A variable two-step backward differentiation formulas (BDF2) on a Shishkin-type mesh is used to discrete the first-order derivatives of the singularly perturbed transport equation. The stability and error analysis are derived by using the discrete orthogonal convolution kernels. It is proved that the scheme is second-order uniformly convergent with respect to the small parameter, which improves previous results. Numerical experiments are presented to support the theoretical result.

Asymptotic Analysis of an Elastic Layer under Light Fluid Loading

Asymptotic analysis for an elastic layer under light fluid loading was developed. The ratio of fluid and solid densities was chosen as the main small parameter determining a novel scaling. The leading- and next-order approximations were derived from the full dispersion relation corresponding to long-wave, low-frequency, antisymmetric motions. The asymptotic plate models, including the equations of motion and the impenetrability condition, motivated by the aforementioned shortened dispersion equations, were derived for a plane-strain setup. The key findings included, in particular, the necessity of taking into account transverse plate inertia at the leading order, which is not the case for heavy fluid loading. In addition, the transverse shear deformation, rotation inertia, and a number of other corrections appeared at the next order, contrary to the previous asymptotic developments for fluid-loaded plates not assuming a light fluid loading scenario.

Asymptotic analysis of the nonsteady micropolar fluid flow through a system of thin pipes

In this paper, we analyze the time‐dependent flow of an incompressible micropolar fluid in a multiple‐pipe system. Motivated by the applications, we assume that the pipes have circular cross‐section and that the ratio between pipes' thickness and its length is small, denoted by the parameter . Far from the junction, the fluid exhibits different behavior depending on the magnitude of the viscosity coefficients with respect to the small parameter . Focusing on the critical case described by the strong coupling between velocity and microrotation, the complete asymptotic expansion of the solution (up to an arbitrary order) is built. To improve the accuracy of the asymptotic approximation, we introduce the boundary layer correctors near the pipes' ends and take into account the interior layer correction in the vicinity of the junction as well. The convergence is also proved via error estimates, providing the rigorous justification of the proposed effective model.

Serum neurofilament light chain levels correlate with small fiber related parameters in patients with hereditary transthyretin amyloidosis with polyneuropathy (ATTRv-PN).

Recent evidence suggests that both serum neurofilament light chain (sNfL) levels and small fiber related diagnostic variables may be valuable disease biomarkers of hereditary transthyretin amyloidosis with polyneuropathy (ATTRv-PN). Our study aimed to explore the relations between sNfL and small fiber related skin biopsy and quantitative sensory testing (QST) parameters in a cohort of ATTRv-PN patients and pre-symptomatic carriers. We retrospectively analyzed data from 13 ATTRv patients and 21 pre-symptomatic carriers who underwent sNfL dosage, skin biopsy, and QST, and analyzed correlations between sNFL, intraepidermal nerve fiber density (IENFD), and cold (CDT) and warm detection thresholds (WDT). Both sNfL and small fiber related parameters significantly differed between carriers and patients (sNfL: p < 0.0001; IENFD: p = 0.0008; CDT, WDT: < 0.0001). sNFL levels were normal in all carriers, altered in 85% of patients, negatively correlated with distal IENFD (r = -0.47, p = 0.005), and significantly correlated with CDT (r = -0.68; p < 0.0001) and WDT (r = 0.57; p < 0.0001). Our study showed that sNfL reliably discriminates symptomatic ATTRv-PN patients from pre-symptomatic carriers, and found significant relations between sNfL, skin biopsy, and QST small fiber related parameters, suggesting that sNfL might be a valuable biomarker of peripheral nerve involvement in ATTRv-PN and a supportive criterion for symptomatic disease transition.

Nonintegrability of time-periodic perturbations of single-degree-of-freedom Hamiltonian systems near homo- and heteroclinic orbits

We consider time-periodic perturbations of single-degree-of-freedom Hamiltonian systems and study their real-meromorphic nonintegrability in the Bogoyavlenskij sense using a generalized version due to Ayoul and Zung of the Morales–Ramis theory. The perturbation terms are assumed to have finite Fourier series in time, and the perturbed systems are rewritten as higher-dimensional autonomous systems having the small parameter as a state variable. We show that if the Melnikov functions are not constant, then the autonomous systems are not real-meromorphically integrable near homo- and heteroclinic orbits. Our result is not just an extension of previous results for homoclinic orbits to heteroclinic orbits and provides a stronger conclusion than them for the case of homoclinic orbits. We illustrate the theory for two periodically forced Duffing oscillators and a periodically forced two-dimensional system.

Periodic perturbations of central force problems and an application to a restricted 3-body problem

We consider a perturbation of a central force problem of the formx¨=V′(|x|)x|x|+ε∇xU(t,x),x∈R2∖{0}, where ε∈R is a small parameter, V:(0,+∞)→R and U:R×(R2∖{0})→R are smooth functions, and U is τ-periodic in the first variable. Based on the introduction of suitable time-maps (the radial period and the apsidal angle) for the unperturbed problem (ε=0) and of an associated non-degeneracy condition, we apply an higher-dimensional version of the Poincaré–Birkhoff fixed point theorem to prove the existence of non-circular τ-periodic solutions bifurcating from invariant tori at ε=0. We then prove that this non-degeneracy condition is satisfied for some concrete examples of physical interest (including the homogeneous potential V(r)=κ/rα for α∈(−∞,2)∖{−2,0,1}). Finally, an application is given to a restricted 3-body problem with a non-Newtonian interaction.

Two-dimensional transient contrasting structure evolution in an inhomogeneous media with the advection

The inner transition layer evolution for two-dimensional quasi-linear initial-boundary value problem for the reaction-advection-diffusion equation in an inhomogeneous media with a small parameter within the high order derivatives is considered. Within the framework of the main (zero order) sum of the asymptotic series, the position of the inner transition layer is described by the Hamilton–Jacobi equation. The potential is calculated as an integrated density function of the reaction sources within the limits of the equilibrium levels. The front line of the transition layer evolves in the same way as the constant-eikonal line (the wavefront line in the other words) for the problem of wave propagation in an inhomogeneous medium in short-wave (geometro-optical) asymptotics. The sum of the asymptotic series of zero order and first order is found, the existence gap of a smooth front line, the time of destruction of the contrasting structure are calculated.

Optimal quantum metrology of two-photon absorption

Abstract Two-photon absorption (TPA) is a nonlinear optical process with wide-ranging applications from spectroscopy to super-resolution imaging. Despite this, the precise measurement and characterisation of TPA parameters are challenging due to their inherently weak nature. We study the potential of single-mode quantum light to enhance TPA parameter estimation through the quantum Fisher information (QFI). Discrete variable (DV) quantum states (defined to be a finite superposition of Fock states) are optimised to maximise the QFI for given absorption, revealing a quantum advantage compared to both the coherent state (classical) benchmark and the single-mode squeezed vacuum state. For fixed average energy ̄n ∈ 2N, the Fock state is shown to be optimal for large TPA parameters, while a superposition of vacuum and a particular Fock state is optimal for small absorption for all ̄n. This differs from single-photon absorption where the Fock state is always optimal. Notably, photon counting is demonstrated to offer optimal or nearly optimal performance compared to the QFI bound for all levels of TPA parameters for the optimised quantum probes, and their quantum advantage is shown to be robust to single-photon loss. Our findings provide insight into known limiting behaviours of Gaussian probes and their different Fisher information (FI) scalings under photon counting (∝ ̄n2 for squeezed vacuum states versus ̄n3 for coherent states). The squeezed state outperforms coherent states for small TPA parameters but underperforms in the intermediate regime, becoming comparable in the large absorption limit. This can be explained through fundamental differences between behaviours of even and odd number Fock states: the former’s QFI diverges in both large and small absorption limits, while the latter diverges only in the small absorption limit, dominating at intermediate scales.

Nonlocal quasi-3d vibration/ analysis of three-layer nanoplate surrounded by Orthotropic Visco-Pasternak foundation by considering surface effects and neutral surface concept

The present article deals with the application of refined plate theory and surface effects to investigate the nonlocal free vibration behavior of a functionally graded nano-plate composed of two piezoelectric face sheets resting on Orthotropic Visco-Pasternak. Surface and nonlocal effects are applied using the surface piezoelectricity and stress-strain gradient theories. Surface effects, refined quasi-3D higher-order theory, neutral surface position and in-plane mechanical preloads are considered to study piezoelectric sandwich nano-plate with a functionally graded core in this study for the first time, simultaneously. The size-dependent governing equations are derived based on 2D and 3D refined theory using Hamilton’s principle and different boundary conditions are studied and investigated according to the semi-analytical solution method. Comparing the results of the vibration behavior of 2D and quasi-3D models, local and non-local models, and also, examining different foundations simultaneously can be said as one of the innovations of the current research. Finally, a parametric study is presented to examine the effects of different parameters such as surface effects, small scale parameters, mechanical preloads, thickness stretching effects, neutral surface position, non-homogeneous coefficient, applied voltage, different foundation constants, boundary conditions, geometrical ratios and different theories in details. It is indicated that the inclusion of surface effects and thickness stretching effects leads to a significant influence on the vibration behavior of three-layered nano-scale plates. The presented results in this article may provide useful guidance for the design and development of the next generation of nano-devices.

Nonlinear statics of magneto-electro-elastic nanoplates considering flexomagnetoelectric effect based on nonlocal strain gradient theory

The nonlinear static behavior of hygrothermal magnetoelectroelastic(MEE) nanoplates considering the flexomagnetoelectric (FME) effect is researched by engaging the first-order shear deformation theory (FSDT). The constitutive equations of MEE nanoplates take into account the FME effect and hygrothermal effect. Leveraging the FSDT, von Karman's nonlinear equation and Hamilton's principle, the nonlinear control equation for hygrothermal MEE nanoplates can be derived by using the variational approach. The nonlocal strain gradient theory (NSGT) has application in the size effect of MEE nanoplates. The nonlocal nonlinear term in the control equation is handled by employing the Airy stress function, and then the non-linear mechanical model is solved by the Galerkin method. Thus, the nonlinear load-deflection curves (NLDC) of the MEE nanoplate are obtained via the introduction of material parameters, enabling an examination of the impact of various factors such as the FME effect, two small size parameters of NSGT and other parameters on the nonlinear bending behavior of MEE nanoplates. In conclusion, this study offers a theoretical foundation for taking into account the FME effect in the development of nanodevices, thereby contributing to advancements in this field.

Numerical method for second order singularly perturbed delay differential equations with fractional order in time via fitted computational method

The numerical solution of time fractional parabolic differential equations with singular perturbations and delay is the subject of this article. An arbitrarily small perturbation parameter ɛ(0<ɛ<<1) is multiplied the highest order derivative term. The problem’s solution shows an exponential boundary layer on the right side of the spatial domain for small values of ɛ. The solution and its derivatives are discussed together with their properties and bounds. The Crank–Nicolson approach for time discretization and the fitted operator non-standard finite difference method for space discretization are used to solve the time fractional singularly perturbed time delay problem under consideration. The comparison principle and solution bound are used to examine and analyze the scheme’s stability. The scheme’s uniform convergence is examined and demonstrated. The proposed scheme has a uniform convergence order of Mx−1+(Δt)2−γ. Two numerical illustrations for various values of ɛ are taken into consideration to validate the theoretical analysis of the scheme.The constructed scheme provides a better order of convergence and more accurate results than methods found in the literature.

A two-grid characteristic block-centered finite difference method for Darcy-Forchheimer slightly compressible miscible displacement problem

In this paper, a two-grid characteristic block-centered finite difference method is developed for solving a nonlinear Darcy-Forchheimer slightly compressible miscible displacement problem in porous media. The method is proposed for solving the nonlinear system in two steps. In the first step, the nonlinear equations are approximated on a coarse grid using the characteristic block-centered finite difference method. In the second step, the nonlinear system is linearized using Newton's method with a small positive parameter to ensure the differentiability of the nonlinear term |u|u in the Darcy-Forchheimer equation. The proposed two-grid method is rigorously analyzed, in which a priori error estimates of the velocity, pressure, concentration and its flux are provided, and the rates of convergence are O(ε+Δt+H3+h2). Finally, numerical experiments are conducted to demonstrate the effectiveness of the proposed methods by comparing the efficiency, especially in terms of CPU time.

Imaging Rovibrational Excitation of Scattered YO Molecules in Inelastic Collisions with Kr and Ne.

Energy transfer between atoms and molecules is fundamental to many physical and chemical processes, and understanding the mechanisms and outcomes of energy transfer is crucial for various applications in physics and chemistry. Here, the rovibrational excitation of YO(X 2Σ+) molecules with the collision of Kr and Ne has been studied in the laser-ablation crossed beam and time-sliced ion velocity map imaging setup in combination with the resonance enhanced multiphoton ionization scheme. Significant changes in the angular distribution for different rovibrational excitations of YO molecules are observed with the collision of Kr. The sharp forward distribution for low rovibrational excitation of YO(v' = 0, 1) molecules suggest that the weak attractive potential between Kr and YO is dominant at large impact parameters. Comparatively, the strong sideway distribution for highly rovibrationally excited YO(v' = 1, 2, 3, and 5) is due to rainbow scattering from the stronger attractive potential of Kr···YO at relatively small impact parameters. The more isotropic angular distribution in the highly rovibrationally excited YO(v' = 11) indicates the formation of a short-lived complex. A change in the angular distribution of scattered YO with different rovibrational excitations was also observed in the collisions of Ne. For YO as a heteronuclear diatomic molecule, collisions of the Y- and the O-end of YO with rare gases would affect the contribution of inelastic processes at different impact parameters.