Field Distribution Function Research Articles

Axisymmetric hybrid Vlasov equilibria with applications to tokamak plasmas

We derive axisymmetric equilibrium equations in the context of the hybrid Vlasov model with kinetic ions and massless fluid electrons, assuming isothermal electrons and deformed Maxwellian distribution functions for the kinetic ions. The equilibrium system comprises a Grad–Shafranov partial differential equation and an integral equation. These equations can be utilized to calculate the equilibrium magnetic field and ion distribution function, respectively, for given particle density or given ion and electron toroidal current density profiles. The resulting solutions describe states characterized by toroidal plasma rotation and toroidal electric current density. Additionally, due to the presence of fluid electrons, these equilibria also exhibit a poloidal current density component. This is in contrast to the fully kinetic Vlasov model, where axisymmetric Jeans equilibria can only accommodate toroidal currents and flows, given the absence of a third integral of the microscopic motion.

Bond strength theory between rebar and recycled aggregate concrete after freeze-thaw cycles under stress state Ⅰ: Uniaxial lateral compression

In recent years, the utilization of recycled aggregate concrete (RAC) as an environmental-friendly material in the construction industry has been actively promoted. A sufficient bond between rebar and RAC is crucial for the load carrying capacity and long-term durability of RAC structure. However, many connection sections between rebar and RAC may be destructed, especially in cold regions where freeze-thaw cycles can worsen the deterioration of structures, under uniaxial compression. To investigate the coupling effect of freeze-thaw cycles and uniaxial lateral pressure on bonding properties of rebar and RAC, variables, such as number of freeze-thaw cycles (F), uniaxial lateral pressure stress ratio (ξ), and recycled aggregate replacement ratio (R), are considered in this study. 117 specimens were fabricated and divided into 39 groups for pull-out test. Different bond-slip curves were obtained, and corresponding characteristics were analyzed. Moreover, this paper proposed a novel bond strength theory by combining a superposed stress field distribution function with a splitting surface stress analysis. The distribution function consisted of the stress distribution of uniaxial lateral pressure loading state without pullout and pullout process without lateral pressure, while the stress analysis amended the softened sleeve theory with a spatial intensity model. The results show that both bond strength and peak slip are positively correlated with uniaxial lateral compression but negatively correlated with the accumulation of freeze-thaw damage. The recycled aggregate (RA) replacement ratio has a minor effect on bond strength and peak slip. The factor of uniaxial lateral pressure is dominant and mitigates the adverse effects of freeze-thaw damage and RA substitution on bonding properties. Notably, the methods of decomposition and stress synthesis break through the limitations of traditional bond strength theory, resulting in a wider scope of application. The bond strength theory applies not only to elastic sleeve bonding part but also to softened sleeve bonding part in reinforced concrete building structure. This theory can more accurately predict the failure morphology and ultimate bond strength in freeze-thaw conditions under uniaxial lateral pressure load.

A generally variational phase field model of fracture

Although the phase field method has been widely used in the field of fracture, various models exist, such as the second-order and fourth-order phase field model in the field of fracture mechanics, and the phase field evolution model in the field of physics, and it is not yet possible to establish these models under a unified framework. In this paper, a thermodynamically generalized framework of variational phase field model of fracture is presented, which can effectively unify different models under this framework. Based on this variational framework, not only the current second-order and fourth-order phase field model can be unified, but also new second-order, fourth-order, sixth-order, and eighth-order phase field model can be obtained from this framework. In addition, a method is proposed to convert the lower-order model into a higher-order model, which can convert a phase field distribution function of the lower-order model into a high-order phase field distribution function, and then obtain a higher-order phase field model. Finally, the new model is verified by some representative examples, and the second-order and fourth-order phase field models are compared, which shows the effectiveness of building a new phase field model from the variational framework.

Three-dimensional Analytical Modeling of Axial Flux Permanent Magnets Maglev Motor

The three-dimensional(3D) analytical model of the magnetic field in an Axial Flux Permanent Magnets Maglev Motor(AFPMMM) is proposed and investigated the influence of the structural parameters on electromagnetic characteristics. Firstly, the topology and working principle of the AFPMMM is introduced, and the model is transferred into a mathematical model in 3D cartesian coordinate. Then, the volume integral method and equivalent current sheets model is applied to find the 3D magnetic field distribution function of Halbach rotor. A unified form expression can be obtained by two dimensional discrete fourier transform (2-D DFT) is applied on the 3D magnetic field distribution function. Thirdly, the conductive and nonconductive regions of AFPMMM will be formulated by the second order vector potential (SOVP) to built the 3D analytic model. The expression of the lift force, torque and power losses was derived. Besides, the relationship between electromagnetic characteristics and structural parameters of the AFPMMM were analyzed based on 3D analytic model and validated using the 3D finite element analysis(FEA). Finally, the experiments based on a small scale prototype are carried out to verify the analytical results.

Magnetophoretic Control of Diamagnetic Particles Inside an Evaporating Droplet.

The present study reports the magnetophoresis of diamagnetic particles in an evaporating ferrofluid droplet. Both solid and ring magnet arrangements are used to investigate the effect of magnetic field distribution. The distance of the magnet from the droplet is varied to study the effect of magnetic field strength. The magnetic field distribution is computed using COMSOL multiphysics software. Magnetometer measurements have been carried out to validate the simulation results. The motion of particles and the drying pattern of evaporating ferrofluid droplets are visualized using the confocal microscopy technique. Both bright-field and fluorescence imagings have been carried out to observe the differential deposition of the fluorescent particle (microparticle) and magnetic nanoparticles in the absence and presence of a magnetic field. The velocity of diamagnetic particles as a function of magnetic field distribution and strength has been studied using the micro-PIV technique. In the absence of the magnetic field, a ring-shaped deposition pattern is observed. The mixture of microparticles (diamagnetic) and nanoparticles (magnetic) is deposited between the outer and inner edges of the ring. The diamagnetic particles occupy the inner and outer edges of the ring. Magnetic particles travel toward the higher magnetic field zone and diamagnetic particles move toward the smaller magnetic field zone when a magnetic field is applied by a solid magnet placed over the droplet. This can be attributed to the negative magnetic force originating from the difference between the susceptibility of magnetic and nonmagnetic particles. The negative magnetic force on the microparticle increases as the magnetic field intensity increases, causing the microparticle to convect faster toward the contact line. The deposition behavior can be reversed or suppressed using a ring magnet in place of a solid magnet. In this case, the negative magnetic force is stronger at the contact line region of the droplet and decreases as it approaches the center region of the droplet. The deposition behavior of diamagnetic particle depends on the balance between the Marangoni force and the magnetophoretic force. Overall, the present study demonstrates the capability of the controlled deposition of diamagnetic polystyrene particles by suitable arrangement of the solid and ring magnet.

Out-of-plane effect on the sharp V-notch tip fields in power-law creeping solids: A three-dimensional asymptotic analysis

Understanding the three-dimensional (3D) effect on the notch tip field is important for the fracture mechanics analysis of engineering materials and structures. This paper presents an asymptotic solution for 3D sharp V-notched structures subjected to mode Ⅰ creep loading condition. The asymptotic solution, including the singularities and stress field distribution functions, is obtained by introducing the out-of-plane constraint factor. It is found that the singular behavior is related to the notch angle, material creeping exponents, and the out-of-plane constraint factor. Specifically, singularity changes sharply near the free surface. KNt-Tz solution is presented to describe the mode Ⅰ stress field around the tip of 3D V-notch under creep condition. Its accuracy and advantages compared with plane strain or plane stress solutions are validated by finite element analysis. The difference between effective ranges of KNt-Tz solution and that of plane strain solution is clarified. Moreover, the effect of the creep time on the stress field is also shown. The solutions given in this paper could be beneficial for understanding the 3D effect on notch considering creep loading conditions.

Theoretical and methodological approach to the study of technological processes

The article investigates some issues of complex systems analysis and synthesis that contain local, discrete sources of temperature fields. The emphasis of the author's research lies in the calculation and optimization of the parameters of the laser action on the embryo. The biotechnological process is described by the boundary value problem of a non-stationary, multidimensional differential equation of thermal conductivity which satisfies the boundary conditions of heat flux, beginning and end of the laser action process. The author proposes an algorithm for calculation and optimization of the control parameters of laser action on multilayer microbiological material. The object of the study is the embryo under the action of a laser beam for fission. As a first approximation, at the stage of calculation and optimization of the parameters of the biotechnological process, the embryo is considered as a homogeneous body. The values of thermophysical characteristics are calculated by the method of expert evaluation of the parameters of the emitters. The correctness of the boundary value problem of the process of laser action on the embryo is proved by the author using the method of pseudodifferential operators. Seeking the solution of the boundary value problem in the form of the series, the analytical function of the temperature field distribution is obtained using the Fourier method of the separated variables in the article. Using the method of indeterminate coefficients, the author found the temperature of the laser action on the embryo. Using the approximate gradient method of finding local extrema and the method of directed search of local extrema, it is possible to obtain rational values of optimized parameters of the biotechnological process. The author outlines possible approaches for optimization of technical parameters of laser emitters. In his opinion, the results of research can be considered fundamental for the calculation and optimization of the parameters of the laser action on the embryo, taking into account the multilayer structure of the latter.

The Influence of O2 on Positive Streamer Initiation in Supercritical CO2 With Field Ionization Using a 3-D Particle Model

In this work, the influence of varying mixing ratios of O2 on positive streamer initiation in supercritical (SC) CO2 at different applied voltages was investigated using a 3-D particle model. Zener’s model was included in the 3-D particle model to calculate the field ionization rate during streamer formation and propagation. The evolution of positive streamer was classified into three main stages namely inception cloud, primary streamer channels, and successive streamer branching. The effect of O2 with different mixing ratios (2%, 10%, and 20%) on the morphology, average electron density, electric field distribution, streamer length, and electron energy distribution function (EEDF) for positive streamer in SC-CO2 at four different voltages (12, 12.5, 13, and 13.5 kV) is then investigated. The 2-D cross sections of the initial stage of the streamer and the electric field distribution are presented in detail. At the same applied voltage, the size of inception cloud, average electron density, electric field, and streamer length with the maximum electric field decreased with increasing oxygen concentration in SC-CO2. With increasing applied voltage, the apparent size of inception cloud, primary streamer channels, successive streamer branching, average electron density, electric field, and streamer length increased. The simulation results showed that adding O2 to SC-CO2 can significantly affect the streamer initiation and propagation.

Dynamic exponential model of ferromagnetic hysteresis

We introduce a dynamic exponential model of ferromagnetic hysteresis that unifies the cooperative exponential model with the Néel fluctuating field, Jiles-Atherton approach to the anhysteretic and a Preisach convolution integral over a field distribution function. Dynamic effects such as viscous field advance and eddy current field delay are modeled directly by the model parameters. The dynamic effect of creeping in the direction of the previous maximum susceptibility due to viscosity as predicted by Street is observed and is modeled using the proposed method. Negative tilting of asymmetric minor hysteresis loops (lower susceptibility) with cycling is also observed and is also modeled using the proposed model as increased Néel's “opposition” in our cooperative field, Hcoop. Shearing of hysteresis loops is modeled as an averaging of the exponential model <Xexp> over viscous advance and eddy current delay fields, causing truncation and broadening. Detailed measurement procedures are described for a hollow cylinder and thin and thick rings of Fe94Ni3Cr2Mo1 steel with variable frequency sine, triangle and step field waveforms.

A Simplified Predictive Control Method of Power Frequency Electric Field on Roof near 500 kV Transmission Lines

In order toget a simplified predictive control method of power frequency electric field on roof near 500 kV transmission lines, a three-dimensional electric field simulation model including the lines, tower and house isestablished based on the finite element method. Based on the knowledge that obtained from existing studies, the spatial distance between the house and the transmission line is the main factor affecting the power frequency electric field intensity on the roof. Therefore, the spatial distance is determined as a variable, and the least squares method is used to fit the optimal electric field distribution function. A simplified prediction model of the power frequency electric field at fixed monitoring point on the roof near the transmission line is derived. The results show that the fitting function of the third-order polynomial can achieve better predicting effect. The specific mathematical expression of the prediction model is provided.A deductive process of the application of the proposed method is given. The proposed power frequency electric field predictive control method can provide reference for the planning, design and environmental impact assessment of transmission line projects.

On relaxation processes in a completely ionized plasma

Relaxation of the electron energy and momentum densities is investigated in spatially uniform states of completely ionized plasma in the presence of small constant and spatially homogeneous external electric field. The plasma is considered in a generalized Lorentz model which contrary to standard one assumes that ions form an equilibrium system. Following to Lorentz it is neglected by electron-electron and ion-ion interactions. The investigation is based on linear kinetic equation obtained by us early from the Landau kinetic equation. Therefore long-range electron-ion Coulomb interaction is consequentially described. The research of the model is based on spectral theory of the collision integral operator. This operator is symmetric and positively defined one. Its eigenvectors are chosen in the form of symmetric irreducible tensors which describe kinetic modes of the system. The corresponding eigenvalues are relaxation coefficients and define the relaxation times of the system. It is established that scalar and vector eigenfunctions describe evolution of electron energy and momentum densities (vector and scalar system modes). By this way in the present paper exact close set of equations for the densities valid for all times is obtained. Further, it is assumed that their relaxation times are much more than relaxation times of all other modes. In this case there exists a characteristic time such that at corresponding larger times the evolution of the system is reduced described by asymptotic values of the densities. At the reduced description electron distribution function depends on time only through asymptotic densities and they satisfy a closed set of equations. In our previous paper this result was proved in the absence of an external electric field and exact nonequilibrium distribution function was found. Here it is proved that this reduced description takes also place for small homogeneous external electric field. This can be considered as a justification of the Bogolyubov idea of the functional hypothesis for the relaxation processes in the plasma. The proof is done in the first approximation of the perturbation theory in the field. However, its idea is true in all orders in the field. Electron mobility in the plasma, its conductivity and phenomenon of equilibrium temperature difference of electrons and ions are discussed in exact theory and approximately analyzed. With this end in view, following our previous paper, approximate solution of the spectral problem is discussed by the method of truncated expansion of the eigenfunctions in series of the Sonine polynomials. In one-polynomial approximation it is shown that nonequilibrium electron distribution function at the end of relaxation processes can be approximated by the Maxwell distribution function. This result is a justification of Lorentz–Landau assumption in their theory of nonequilibrium processes in plasma. The temperature and velocity relaxation coefficients were calculated by us early in one- and two-polynomial approximation.

Spatial characteristics analysis of Eaves Gallery along the street in China: take Hubei, Bashu, Jiangzhe areas as examples

ABSTRACTThe Eaves Gallery (EG) along the street as a form of Chinese traditional dwelling was special architecture form for preventing pedestrians from rains and snows while walking through or shopping which has played important roles in the ancient commercial activities. In the past, the studies about EG focused on architectural history, style, evolution in the separated specific local area. It helped the promotion of social attention to EG for protection and renewal. However, the protection and renewal conditions of EG are different in various regions of China. Few studies have done to compare and overview the characteristics of EG in these various regions. In this study, typical EG along the streets in Hubei, Bashu, Jiangzhe along Yangtze River regions from east to west of China were taken as case studies and are explicated based on the comparison and analyzing field distribution mode, space combination, spatial scale, and function. Thus, the spatial functional characteristics of EG in these three regions are divided into unitary type, conversion type and mixed type. The outcome will serve as a comparative basis for the existential form and spatial characteristics of extant traditional EG accompanied by optimum protection and renewal policy discussion in the future.

Lattice Boltzmann simulation of tree-shaped fins enhanced melting heat transfer

A thermal lattice Boltzmann model is developed to simulate the melting process with natural convection in a cavity filled with tree-shaped solid fins, in which the velocity field and temperature field distribution functions are considered. The present model incorporates the total enthalpy and a free parameter in the equilibrium distribution function to handle conjugate heat transfer. The results indicate that natural convection of liquid phase change material (PCM) plays a significant role in the melting heat transfer of PCM. Increasing the number of branching levels leads to a more rapid melting process, and selecting appropriate bifurcation angle has more efficient heat transfer performance.

Mathematical models and nonlinear dynamics of a linear electromagnetic motor

The aim of this paper was to construct a mathematical model of a drive system with ironless permanent magnet synchronous linear motor and to animalize the influence of the magnetic field distribution function on the model’s accuracy and ease of simulation computation. The studied motor employs an U-shaped stationary guideway with permanent magnets placed perpendicularly to the motor’s direction of motion and a forcer with three sets of rectangular coils subjected to alternating external electrical voltage. The system’s parameters are both mechanical (number of magnets and coils, size of magnets, distances between magnets, size of coils) and electromagnetic (auxiliary magnetic field, permeability, coil’s resistance). Lorentz force allows for the transition from electromagnetic parameters to mechanical force, and Faraday’s law of induction creates a feedback between the forcer’s speed and coils voltage. An Ampere’s model of permanent magnet is used to determine the function of auxiliary magnetic field distribution throughout the stator. Two simplified distribution functions are introduced and studied. During validation, external current function is applied to each coil (serving as excitation), while the displacement of forcer in time is the output function. Model parameters are found via genetic algorithms such that the numerical solution of the model best fits experimental data. Several cases of motor operation are compared against simulation results showing good coincidence between computation and experiment.

Analytic solution of field distribution and demagnetization function of ideal hollow cylindrical field source

The Halbach type hollow cylindrical permanent magnet array (HCPMA) is a volume compact and energy conserved field source, which have attracted intense interests in many practical applications. Here, using the complex variable integration method based on the Biot-Savart Law (including current distributions inside the body and on the surfaces of magnet), we derive analytical field solutions to an ideal multipole HCPMA in entire space including the interior of magnet. The analytic field expression inside the array material is used to construct an analytic demagnetization function, with which we can explain the origin of demagnetization phenomena in HCPMA by taking into account an ideal magnetic hysteresis loop with finite coercivity. These analytical field expressions and demagnetization functions provide deeper insight into the nature of such permanent magnet array systems and offer guidance in designing optimized array system.

Structure of the current sheets in the near-Mars magnetotail. Maven observations

During the last 15 years, the Current Sheets (CSs) have been intensively studied in the tail of the terrestrial magnetosphere, where protons are the dominated ion component. On the contrary, in the Martian magnetotail heavy ions (O+ and+ 0) usually dominate while the abundance of protons can be negligible. Hence it is interesting to study the spatial structure and plasma characteristics of such “oxygen” CSs. MAVEN spacecraft (s/c) currently operating on the Martian orbit with a unique set of scientific instruments allows observation of the magnetic field and three-dimensional distribution functions of various ion components and electrons with a high time resolution. In this paper, we analyse nine intervals of the CSs observed by MAVEN in the near-Mars tail at the distances from the planet ~1.5–1R M , where R M is the radius of Mars. We analyse the spatial structure of the CSs and estimate their thickness for different magnetic configurations and relative abundance of the heavy and light ions in the sheets. It is shown that, similarly to the CSs in the Earth’s magnetotail, the thickness and complexity of the spatial structure of the Maritan CSs (i.e. the presence of embedded and / or peripheral current structures) depend on the magnetic configuration of the sheets, which, in turn, affects the fraction of the quasi-adiabatic particles in the CSs.

Effective temperature of the non-equilibrium electrons in a degenerate semiconductor at low lattice temperature

The energy balance equation for the electron–phonon system is recast taking the degeneracy of the carrier ensemble into account. The effect of degeneracy on the field dependence of the temperature of the non-equilibrium carriers has been studied by solving the same equation. The high field distribution function of the carriers is assumed to be given by the Fermi Dirac function at the field dependent carrier temperature. The distribution function has been approximated in a way that facilitates analytical solution of the problem without any serious loss of accuracy. The field dependence of the electron temperature thus obtained seems to be significantly different from what follows had the degeneracy not been taken into account. The agreement of the results obtained from the present analysis with the available experimental data for Ge and InSb are quite satisfactory. The scope of further refinement of the present theory is highlighted.

Helmholtz HSH-Based Basis: A Compact Phenomenological Representation of Arbitrary Reflectance

Visual appearance can be phenomenologically modeled through an integral equation, known as reflectance equation. It describes the surface radiance which depends on the interaction between incident light field and surface Bidirectional Reflectance Distribution Function (BRDF). Being defined on the Cartesian product of the incident and outgoing hemispheres, hemispherical basis is the natural way to represent surface BRDFs. Nonetheless, due to their compactness in the frequency space, spherical harmonics have been extensively used for this purpose. Addressing the geometrical compliance of hemispherical basis, this paper proposes a Cartesian product of the hemispherical harmonics to provide a compact representation of plausible BRDFs, while satisfying the Helmholtz reciprocity property. We provide an analytical analysis and experimental justification that our basis provides better approximation accuracy when compared to similar bases in literature.

Analysis of magnetic electron lens with secant hyperbolic field distribution

Electron-optical imaging instruments like Scanning Electron Microscope (SEM) and Transmission Electron Microscope (TEM) use specially designed solenoid electromagnets for focusing of the electron beam. Indicators of imaging performance of these instruments, like spatial resolution, have a strong correlation with the focal characteristics of the magnetic lenses, which in turn have been shown to be sensitive to the details of the spatial distribution of the axial magnetic field. Owing to the complexity of designing practical lenses, empirical mathematical expressions are important to obtain the desired focal properties. Thus the degree of accuracy of such models in representing the actual field distribution determines accuracy of the calculations and ultimately the performance of the lens. Historically, the mathematical models proposed by Glaser [1] and Ramberg [2] have been extensively used. In this paper the authors discuss another model with a secant-hyperbolic type magnetic field distribution function, and present a comparison between models, utilizing results from finite element-based field simulations as the reference for evaluating performance.

Maximum magnetic recording density and switching field distribution

An analytical calculation of the parameters of magnetic recording on a square lattice of magnetic granules of the patterned medium has been carried out. The maximum density of the longitudinal and perpendicular recording, which is limited by the thermal instability of the magnetic state and by the magnetic dipole interaction of magnetic granules (bits), has been calculated. The switching field distribution function with the parameters determined by the magnetic dipole interaction of granules and their geometry has been determined for perpendicular recording on rod-shaped granules.