Relaxed Boundary Conditions Research Articles

On multiplicative generators of the unified form of 0-overlap and 1-grouping functions

Recently, Qiao introduced a unified form for 0-overlap and 1-grouping functions, called Θ−Ξ functions. With more relaxed boundary conditions, Θ−Ξ functions can be served as a generalization of overlap and grouping functions. As a result, the flexibility and generality of Θ−Ξ functions allow for a wider range of applications, with a potential for application that exceeds that of overlap and grouping functions. This paper focuses on the study of the multiplicative generator pairs (MGPs) of Θ−Ξ functions. Firstly, we propose two concepts of an MGP for a Θ−Ξ function, which are based on the correlation between the value of boundary elements ℓ and ℘, specifically, ℓ<℘ and ℓ>℘. And then, for each of the two cases in turn, we provide the conditions for deriving a Θ−Ξ function from the MGP and determine its block structures. Moreover, we discuss such multiplicatively generated Θ−Ξ functions with neutral elements and examine the actions of (ℓ,℘)-pseudo-automorphisms on them. Finally, we establish two equivalent relationships between MGPs of Θ−Ξ functions with ℓ<℘ and those with ℓ>℘.

Studies on Ionic Flows via Poisson–Nernst–Planck Systems with Bikerman’s Local Hard-Sphere Potentials under Relaxed Neutral Boundary Conditions

We examine the qualitative properties of ionic flows through ion channels via a quasi-one-dimensional Poisson–Nernst–Planck model under relaxed neutral boundary conditions. Bikerman’s local hard-sphere potential is included in the model to account for finite ion size effects. Our main interest is to examine the boundary layer effects (due to the relaxation of electroneutrality boundary conditions) on both individual fluxes and current–voltage relations systematically. Critical values of potentials are identified that play significant roles in studying internal dynamics of ionic flows. It turns out that the finite ion size can either enhance or reduce the ionic flow under different nonlinear interplays between the physical parameters in the system, particularly, boundary concentrations, boundary potentials, boundary layers, and finite ion sizes. Much more rich dynamics of ionic flows through membrane channels is observed.

Residual gauge symmetry in light-cone electromagnetism

We analyze the residual gauge freedom in light-cone electromagnetism in four dimensions. The standard boundary conditions involved in the so-called lc2formalism, which contains only the two physical degrees of freedom, allow for a subset of residual gauge transformations. We relax the boundary conditions imposed on the fields in order to obtain all the residual gauge transformations. We compute the canonical generators for Poincaré and gauge transformations with these relaxed boundary conditions. This enables us to distinguish between the trivial (proper) and large (improper) gauge transformations in light-cone electromagnetism. We then employ the Newman-Penrose formalism to identify the incoming and outgoing radiation fields. We comment on the quadratic form structure of light-cone Hamiltonians, often encountered in lc2 gauge theories.

Mathematical Analysis on Current–Voltage Relations via Classical Poisson–Nernst–Planck Systems with Nonzero Permanent Charges under Relaxed Electroneutrality Boundary Conditions

We focus on a quasi-one-dimensional Poisson-Nernst-Planck model with small permanent charges for ionic flows of two oppositely charged ion species through an ion channel. Of particular interest is to examine the dynamics of ionic flows in terms of I-V (current-voltage) relations with boundary layers due to the relaxation of neutral conditions on boundary concentrations. This is achieved by employing the regular perturbation analysis on the solutions established through geometric singular perturbation analysis. Rich dynamics are observed, particularly, the nonlinear interplays among different physical parameters are characterized. Critical potentials are identified, which play critical roles in the study of ionic flows and can be estimated experimentally. Numerical simulations are performed to further illustrate and provide more intuitive understandings of our analytical results.

Non-Fourier Heat Conduction and Thermal-Stress Analysis of a Spherical Ice Particle Subjected to Thermal Shock in PEM Fuel Cell at Quick Cold Start-Up

Quick start at low temperature is one of the key bottleneck technologies that restrict the large-scale commercialization of proton exchange membrane (PEM) fuel cell vehicles. Ice and devices in battery systems based on thermal deicing are inevitably subjected to thermal shock. In order to study this problem, a non-Fourier heat conduction model is established to study the temperature response of a spherical ice particle subjected to thermal shock with different boundary conditions on the surface. Furthermore, distribution of thermal stress in the particle is obtained using the calculated temperature field, and the effect of thermal relaxation time and boundary conditions on the temperature response as well as the thermal stress field are also analyzed. The results, which are significantly different from that obtained using Fourier law of heat conduction, show that the mechanical stresses and the serious expansion of the ice particle may lead to the severe deformation of the devices connected to the ice during the cold start-up, imposing a great challenge in controlling structural reliability of PEM fuel cells. The numerical results are expected to provide a scientific theoretical basis for PEM fuel cell design and low-temperature start-up control.

Influence of the boundary relaxation on the free vibration of rotating composite laminated Timoshenko beams

In this paper, the influence of boundary relaxation on the free vibration characteristics of a rotating composite laminated Timoshenko beam is studied. Based on the first order shear theory, the theoretical model of the rotating composite laminated Timoshenko beam is established, in which the Possion’s effect is also considered. The relaxed boundary conditions of the beam are simulated using a set of artificial springs. By adjusting the stiffness of the springs different extent relaxation boundary conditions of the beam can be obtained. Relaxation parameters are introduced to evaluate the extent of boundary relaxation. A uniformed formula for the centrifugal force of the rotating beam with relaxed boundary conditions is deduced. By adopting the Rayleigh-Ritz method together with the artificial spring technique the solution for the free vibration characteristics of the rotating beam with relaxed boundary is carried out. The results achieved by the present approach are compared with the results obtained by the finite element method and those in the existing literature. Some new results about the influences of boundary relaxation on the first order natural frequency and mode shape of the rotating beam are presented. The influences of parameters, such as the rotating speed, the fiber orientation angles, the slenderness ratios, and the hub ratios on the free vibration behaviors of the beam with relaxed boundary conditions are discussed.

Quasinormal modes and black hole hairs in AdS

Holography relates the quasinormal modes frequencies of AdS black holes to the pole structure of the dual field theory propagator. These modes thus provide the timescale for the approach to thermal equilibrium in the CFT. Here, we study how such pole structure and, in particular, the time to equilibrium can get modified in the presence of a black hole hair. More precisely, we consider in AdS a set of relaxed boundary conditions that allow for a low decaying graviton mode near the boundary, which triggers an additional degree of freedom. We solve the scalar field response on such background analytically and non-perturbatively in the hair parameter, and we obtain how the pole structure gets affected by the presence of a black hole hair, relative to that of the usual AdS black hole geometry. The setup we consider is a massive 3D gravity theory, which admits a one-parameter family deformation of BTZ solution and enables us to solve the problem analytically. The theory also admits an AdS$_3$ soliton which gives a family of vacua that can be constructed from the hairy black hole by means of a double Wick rotation. The spectrum of normal modes on the latter geometry can also be solved analytically; we study its properties in relation to those of the AdS$_3$ vacuum.

Size Scaling of Plastic Deformation in Simple Shear: Fractional Strain-Gradient Plasticity and Boundary Effects in Conventional Strain-Gradient Plasticity

Abstract A recently developed model based on fractional derivatives of plastic strain is compared with conventional strain-gradient plasticity (SGP) models. Specifically, the experimental data and observed model discrepancies in the study by Mu et al. (2016, “Dependence of Confined Plastic Flow of Polycrystalline Cu Thin Films on Microstructure,” MRS Com. Res. Let. 20, pp. 1–6) are considered by solving the constrained simple shear problem. Solutions are presented both for a conventional SGP model and a model extension introducing an energetic interface. The interface allows us to relax the Dirichlet boundary condition usually assumed to prevail when solving this problem with the SGP model. We show that the particular form of a relaxed boundary condition does not change the underlying size scaling of the yield stress and consequently does not resolve the scaling issue. Furthermore, we show that the fractional strain-gradient plasticity model predicts a yield stress with a scaling exponent that is equal to the fractional order of differentiation.

NOMA-Based 802.11n for Industrial Automation

Industry 4.0 and Industrial Internet refer to the expected revolution in production, utility management and, in general, fully automated, interconnected and digitally managed industrial ecosystems. One of the key enablers for Industry 4.0 lies on reliable and timely exchange of information and large scale deployment of wireless communications in industry facilities. Wireless will bring solutions to overcome the main drawbacks of the current wired systems: lack of mobility, deployment costs, cable damage dependency and scalability. However, the strict requirements in reliability and latency of use cases such as Factory Automation (FA) and Process Automation (PA) are still a major challenge and a barrier for massive deployment of currently available wireless standards. This paper proposes a PHY/MAC wireless communication solution for FA and PA based on Non-Orthogonal Multiple Access (NOMA) in combination with the 802.11n standard. The communication system proposed aims at delivering two different sets of services. The first service class is composed of Critical Services (CS) with strict restrictions in reliability and latency. The same communication system should convey also a second group of services, referred as Best Effort (BE) with more relaxed boundary conditions. The proposal theoretical background, a detailed transmission-reception architecture, the physical layer performance and the MAC level system reliability are presented in this paper. The solution provides significantly better reliability and higher flexibility than TDMA systems, jointly with a predictable control-cycle latency.

Extension of classical stability theory to viscous planar wall-bounded shear flows

A viscous extension of Arnold’s inviscid theory for planar parallel non-inflectional shear flows is developed and a viscous Arnold’s identity is obtained. Special forms of the viscous Arnold’s identity have been revealed that are closely related to the perturbation’s enstrophy identity derived by Synge (Proceedings of the Fifth International Congress for Applied Mechanics, 1938, pp. 326–332, John Wiley) (see also Fraternaleet al.,Phys. Rev. E, vol. 97, 2018, 063102). Firstly, an alternative derivation of the perturbation’s enstrophy identity for strictly parallel shear flows is acquired based on the viscous Arnold’s identity. The alternative derivation induces a weight function. Thereby, a novel weighted perturbation’s enstrophy identity is established, which extends the previously known enstrophy identity to include general streamwise translation-invariant shear flows. Finally, the validity of the enstrophy identity for parallel shear flows is rigorously examined and established under global nonlinear dynamics imposed with two classes of wall boundary conditions. As an application of the enstrophy identity, we quantitatively investigate the mechanism of linear instability/stability within the normal modal framework. The investigation reveals a subtle interaction between a critical layer and its adjacent boundary layer, which determines the stability nature of the disturbance. As an implementation of the relaxed wall boundary conditions imposed for the enstrophy identity, a control scheme is proposed that transitions the wall settings from the no-slip condition to the free-slip condition, through which a flow is stabilized quickly in an early stage of the transition.

Analysis for Contact Angle Hysteresis on Rough Surfaces by a Phase-Field Model with a Relaxed Boundary Condition

In this paper, we study the contact angle hysteresis on a slowly moving rough boundary using a phase-field model with a relaxed boundary condition. In particular, we want to model the recent experi...

Solid-state dewetting on curved substrates

Based on the thermodynamic variation to the free energy functional, we propose a sharp-interface model for simulating solid-state dewetting of thin films on rigid curved substrates in two dimensions. This model describes the interface evolution which occurs through surface diffusion-controlled mass transport and contact point migration along the curved substrate. Furthermore, the surface energy anisotropy is easily included into the model, and the contact point migration is explicitly described by the relaxed contact angle boundary condition. We implement the mathematical model by a semi-implicit parametric finite element method to study several interesting phenomena, such as "small" particle migration on curved substrates and templated solid-state dewetting on a pre-patterned substrate. Based on ample numerical simulations, we demonstrate that, the migration velocity of a "small" solid particle is proportional to the substrate curvature gradient $\hat{\kappa}'$ and inversely proportional to the square root of the area of the particle $\sqrt{A}$, and it decreases when the isotropic Young angle $\theta_i$ increases. In addition, we also observe four periodic categories of dewetting on a pre-patterned sinusoidal substrate. Our approach can provide a convenient and powerful tool to exploring how to produce well-organized nanoparticles by making use of template-assisted solid-state dewetting.

A three-dimensional phonon energy transport model based on the non-dimensional lattice Boltzmann method

A new three-dimensional phonon energy transport model based on the non-dimensional lattice Boltzmann method (NDLBM) is developed to be applicable in the sub-continuum regime for diffusive and ballistic phonon transport. Two Knudsen numbers based on the base frequency and variable frequency, Knω0 and Knω, are defined in order to isolate the effects of material size and phonon transfer frequency. Both the relaxation time and boundary conditions of the phonon Boltzmann transport equation are modeled in dimensionless form. The physical meaning of each component in the relaxation time and discontinuous boundary models is interpreted. By introducing the Boltzmann constant and Planck’s constant, all other coefficients can be simplified into order of one. The transient dimensionless phonon probability distribution functions are solved by the Parallel NDLBM with D3Q27 grids. The dimensionless temperature distributions, i.e. the dimensionless phonon energy density distributions, compare favorably with previous experimental and numerical results for various Knω0. The effects of material size, temperature, and phonon transfer frequency are investigated.

Regimes of Axisymmetric Flow and Scaling Laws in a Rotating Annulus with Local Convective Forcing

We present a numerical study of axisymmetric flow in a rotating annulus in which local thermal forcing, via a heated annular ring on the outside of the base and a cooled circular disk in the centre of the top surface, drives convection. This new configuration is a variant of the classical thermally-driven annulus, where uniform heating and cooling are applied through the outer and inner sidewalls respectively. The annulus provides an analogue to a planetary circulation and the new configuration, with its more relaxed vertical thermal boundary conditions, is expected to better emulate vigorous convection in the tropics and polar regions as well as baroclinic instability in the mid-latitude baroclinic zone. Using the Met Office/Oxford Rotating Annulus Laboratory (MORALS) code, we have investigated a series of equilibrated, two dimensional axisymmetric flows across a large region of parameter space. These are characterized in terms of their velocity and temperature fields. When rotation is applied several distinct flow regimes may be identified for different rotation rates and strengths of differential heating. These regimes are defined as a function of the ratio of the horizontal Ekman layer thickness to the non-rotating thermal boundary layer thickness and are found to be similar to those identified in previous annulus experiments. Convection without rotation is also considered and the scaling of the heat transport with Rayleigh number is calculated. This is then compared with existing work on the classical annulus as well as horizontal and Rayleigh-Bénard convection. As with previous studies on both rotating and non-rotating convection the system’s behaviour is found to be aspect ratio dependent. This dependence is seen in the scaling of the non-rotating Nusselt number and in transitions between regimes in the rotating case although further investigation is required to fully explain these observations.

The Dynamics of Three-Phase Triple Junction and Contact Points

We use the method of matched asymptotic expansions to study the sharp interface limit of the three-phase system modeled by the Cahn--Hilliard equations with the relaxation boundary condition. The dynamic laws for the interfaces, the triple junction, and the contact points are derived at different time scales. In particular, we show, at $O(t)$ time scale, the dynamic of the triple junction is determined by the balance of the chemical potential gradient along the three interfaces meeting at the triple junction. At faster time scale $O(\epsilon t)$, the motion of the triple junction is controlled by the contact point motions and geometric constraints.

A dynamic theory for contact angle hysteresis on chemically rough boundary

We study the interface dynamics and contact angle hysteresis in a two dimensional, chemically patterned channel described by the Cahn-Hilliard equation with a relaxation boundary condition. A system for the dynamics of the contact angle and contact point is derived in the sharp interface limit. We then analyze the behavior of the solution using the phase plane analysis. We observe the stick-slip of the contact point and the contact angle hysteresis. As the size of the pattern decreases to zero, the stick-slip becomes weaker but the hysteresis becomes stronger in the sense that one observes either the advancing contact angle or the receding contact angle without any switching in between. Numerical examples are presented to verify our analysis.

Phase Field Model for Solidification with Boundary Interface Interaction

By incorporation the surface free energy in the free energy functional, a phase field model for solidification with boundary interface intersection is developed. In this model, the bulk equation is appropriately modified to account for the presence of heat diffusion inside the diffuse interface, and a relaxation boundary condition for the phase field variable is introduced to balance the interface energy and boundary surface energy in the multiphase contact region. The asymptotic analysis is applied on the phase field model to yield the free interface problem with dynamic contact point condition.

Forced Horizontal Vibration of Rigid Disc in Transversely Isotropic Trimaterial Full-Space

AbstractA theoretical investigation is presented for the dynamic interaction of a rigid circular disc and its surrounding space, which is a trimaterial transversely isotropic full-space, and the disc is undergoing a prescribed horizontal harmonic vibration. Because the equations of motion form a system of coupled partial differential equations in a cylindrical coordinate system, a system of two scalar potential functions are used to decouple the equations of motion. This approach results in a set of two separated partial differential equations, which may be transformed to some ordinary differential equations by using Fourier series along the angular coordinate and the Hankel integral transformations with respect to the radial coordinate in a cylindrical coordinate system. After solving the ordinary differential equations, the unknown functions are determined by satisfying relaxed boundary conditions, which themselves are transformed to a set of four coupled integral equations. These coupled integral equat...

A Phase Field Model for Binary Alloy Solidification with Boundary Interface Intersection

A phase field model for binary alloy solidification with boundary interface intersection is developed. In the phase field model, the heat and solute conservation equations are appropriately modified to account for the presence of heat and solute rejection inside the diffuse interface, and a relaxation boundary condition for the phase field variable is introduced to balance the interface energy and boundary surface energy in the multiphase contact region. The thin interface asymptotic analysis is applied on the phase field model to yield the free interface problem with dynamic contact point condition.

Mathematical modeling of a rigid circular membrane in a trimaterial transversely isotropic full-space undergoing a prescribed displacement

A horizontal layer between two transversely isotropic half-spaces forms a trimaterial full-space, involved in this paper. A mathematical formulation is presented to determine the response of a rigid circular membrane, which is laid down at an interface of the tri-material transversely isotropic full-space and is considered to be under a prescribed horizontal displacement. The governing equations are expressed in the cylindrical coordinate system. With the aid of a system of two scalar potential functions, the governing equations of motion can be uncoupled into two separated partial differential equations, which may be transformed to some ordinary differential equations by applying the Hankel integral transforms in the radial direction and Fourier series along the angular coordinate. After determining the unknown functions by imposing the relaxed boundary conditions, they are transformed to a set of four coupled integral equations, which are reduced to two coupled Fredholm-Volterra integral equations of the second kind, from which both displacement and the stress fields are computed. To confirm the accuracy of the numerical evaluation of the integrals involved, the numerical results are compared with the solutions exists for a transversely isotropic half-space. In order to investigate the degree of material anisotropy, some numerical evaluations are given for different combinations of transversely isotropic region.