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A new model for anisotropic magneto-electro-elastic Mindlin plates is developed by using an extended modified couple stress theory. The equations of motion and complete boundary conditions are simultaneously obtained by a variational formulation based on Hamilton’s principle. The new anisotropic magneto-electro-elastic plate model includes the models for orthotropic and transversely isotropic magneto-electro-elastic Mindlin plates and the model for isotropic Mindlin plates, all incorporating the microstructure effect, as special cases. To illustrate the new model, the static bending and free vibration problems of a simply supported transversely isotropic magneto-electro-elastic plate are analytically solved by directly applying the general formulas derived. For the static bending problem, the numerical results reveal that the deflection, rotation, electric potential, and magnetic potential of the simply supported plate predicted by the current non-classical model are always smaller than those predicted by the classical elasticity-based model, and the differences are significant when the plate thickness is very small but is diminishing as the thickness increases. For the free vibration problem, it is found that the natural frequency predicted by the new plate model is higher than that predicted by the classical model, and the difference is quite large for very thin plates.

The present paper deals with the study of the wave propagation in a homogeneous transversely isotropic magneto-thermo-piezoelectric cylindrical material, under the influence of rotation, in the context of the linear theory of thermoelasticity. The displacement, electric potential, magnetic potential, electrostatic displacement, and magnetic induction functions are introduced to uncouple the equations of motion, with boundary conditions. After deriving secular equations in closed form and isolated mathematical conditions, the effect of rotation, phase velocity, attenuation coefficient profiles and relative frequency shift on dispersion curves is studied. The study is very helpful in the development of rotation sensors and other piezoelectric devices. The numerical results of the physical quantities have been illustrated graphically.

The mean magnetic field equation describes the process of generating a magnetic field on a large scale as a result the EMF arising on a small scale. We consider the case where the large-scale magnetic field is also random and determine the density function of magnetic helicity. This function is invariant under gauge transformations of the magnetic vector potential. We study the equation for the magnetic helicity flux of a large-scale field and introduce a correction term related to the quadratic magnetic helicity invariant.

Simulation of eddy-current testing (ECT) with the scanning of a ferrite-core coil using the conventional finite-element method (FEM) requires re-meshing, as it uses one solution domain to cover all the conducting materials and magnetic materials, which is cumbersome and results in low computation efficiency. In this article, we present the hybrid formulation domain-decomposition FEM to resolve the problem. The solution domain is decomposed into two subdomains, and the test sample and the ferrite core are included in different subdomains. The subdomains are discretized separately, and the discretization is performed only once. The reduced magnetic vector potential formulation is used in the subdomain of the test sample, whereas the magnetic scalar potential formulation is used in the subdomain of the ferrite core. The proposed method is applied in simulating the ECT of the aluminum plates having through holes with or without defects, and the efficiency of the method is demonstrated. Finally, experimental validation is performed.

Neodymium magnets were independently discovered in 1984 by General Motors and Sumitomo. Today, they are the strongest type of permanent magnets commercially available. They are the most widely used industrial magnets with many applications, including in hard disk drives, cordless tools and magnetic fasteners. We use a vector potential approach, rather than the more usual magnetic potential approach, to derive the three-dimensional (3D) magnetic field for a neodymium magnet, assuming an idealized block geometry and uniform magnetization. For each field or observation point, the 3D solution involves 24 nondimensional quantities, arising from the eight vertex positions of the magnet and the three components of the magnetic field. The only unknown in the model is the value of magnetization, with all other model quantities defined in terms of field position and magnet location. The longitudinal magnetic field component in the direction of magnetization is bounded everywhere, but discontinuous across the magnet faces parallel to the magnetization direction. The transverse magnetic fields are logarithmically unbounded on approaching a vertex of the magnet. doi:10.1017/S1446181120000097

In this article, a fault tolerant deadbeat controller is proposed, which is able to compensate for both partial and uniform demagnetization faults in axial flux permanent-magnet synchronous machines with double rotor, even with asymmetric defects in both rotors. For this purpose, the system model used by the deadbeat controller is equipped with a look-up table of the back-emf, in function of the mechanical rotor position. This look-up table is generated by means of an analytical model based on the magnetic vector potential. With this information, the deadbeat controller can eliminate both the additional bias and ripple in the stator current components caused by demagnetization faults. The proposed control strategy is validated on a 4 kW test set-up of an axial flux permanent-magnet synchronous machine with yokeless and segmented armature topology fed by a three-phase two-level voltage source inverter, both by simulations and experiments. Key performance indicators concerning bias and ripple of the current components illustrate the superior performance of the proposed control strategy in comparison to a standard deadbeat controller without an up-to-date look-up table of the back-emf.

The most significant assumption of conventional analytical methods (e.g., the subdomain technique) is that the stator and rotor iron cores are infinitely permeable. The magnetic saturation is ignored, resulting in an overestimation of the flux density. This article presents an analytical model of permanent magnet (PM) machines with segmented Halbach arrays (PMMSHA). It considers the finite magnetic permeabilities of the stator teeth and tooth tips (STTT) and can study the tooth saturation effect. The permeability variation in the slots and teeth is solved via Cauchy’s product theorem. Based on magnetic vector potential and the Maxwell–Fourier method, Poisson’s and Laplace’s equations in the stator teeth/slots, tooth tips/slot openings, air gap, and PMs regions are solved. Consequently, the electromagnetic performance, including the magnetic field distribution in all regions, exciting force density, cogging torque, and electromagnetic torque, is calculated. The results of finite element analysis (FEA) show that the analytical model is highly accurate. Furthermore, the influences of the permeability of the STTT and the number of blocks per pole (NBPP) on the performance of the PMMSHA are studied.

As the main contribution, a nonlocal strain gradient theory (NSGT) is developed in conjugation with the generalized differential quadrature method (GDQM) to analyze the free vibration of a transver...

Recently, a novel code for the simulation of the plasma equilibrium and its evolution has been developed and fully implemented in ANSYS. We named it ANSYS Plasma Equilibrium and Evolution Code (APEC). It is based on the solution of the Grad–Shafranov equation and exploits the finite-element method using the magnetic potential vector formulation. In this article, we present a further development on the coupling of the axisymmetric plasma equilibrium equations, describing the plasma behavior, with the full 3-D eddy current equations, defining the surrounding 3-D structures. One test case, related to the International Thermonuclear Experimental Reactor (ITER) tokamak, has been fully analyzed and benchmarked with a validated code showing the validity of the proposed approach. In particular, the benchmark has concerned the comparison of the initial equilibrium and plasma evolution.

In this article, dyadic magnetic vector potential Green’s functions corresponding to electric dipole which illuminates homogenized metasurface have been presented. Metasurface is constructed by a periodic arrangement of planar metallic scatterers and located at the interface of two different dielectric half spaces. The metasurface is replaced by a thin sheet with the dyadic equivalent surface distribution of electric and magnetic susceptibilities. These vector potentials have been derived employing generalized sheet transition condition (GSTC) on the plane of the metasurface. The retrieval relations to obtain equivalent surface susceptibilities of such a surface are calculated by means of the calculated reflection and transmission coefficients due to TE polarized plane waves. Also, the transverse and longitudinal scalar electric potentials relative to the problem have been calculated with the aid of the Lorentz gauge. These vector and scalar potentials together are necessary to use in the mixed potential integral equation (MPIE) formulation of the method of moment (MoM) in an electromagnetic problem.

In this study, the effect of electric and magnetic potential on the free vibration of rotating circular cylindrical shell of functionally graded piezo electromagnetic composed with fiber reinforced polymers has been investigated. The laminated fiber reinforced polymer is in the core and is supported with the functionally graded material layer on the both sides, and two piezoelectric layers are mounted on the outside and inside of the cylindrical shells. Relationships between strain and displacement are expressed based on Love’s first approximation shell theory. Then according to Hamilton’s principle, the governing equations for the cylindrical shell are extracted. Also, the Navier’s solution is used to solve the equations of the cylindrical shell for the simply–simply supported boundary conditions. The numerical results of the analytical solution for the rotating functionally graded piezoelectric–fiber reinforced polymer cylindrical shells based on the effect of magnetic and electric potential, axial and circumferential wave number and speed rotation are obtained. These results are compared with other research and demonstrated a good agreement.

In a previous work, a hysteresis modeling, called G model, was proposed with the goal of describing this phenomenon by simple equations. In that work, we developed the direct model which uses the magnetic field as the independent variable. However, with the magnetic vector potential 2-D classical formulation, the magnetic induction is the independent quantity. The main contribution of this work is to present the inverse version of the G model adapted to vector potential formulation. In addition, a Mayergoyz vector hysteresis generalization is applied in order to consider rotational fields and couple the model with a 2-D vector potential finite-element method (FEM) formulation. In this approach, the anisotropy is inserted with the orientation distribution function (ODF).

The nonreciprocal properties of the modes of a backward spin wave in a tangentially magnetized ferrite plate have been investigated. It has been established that ratio R between the normalized amplitudes of the magnetic potential for two waves with the oppositely directed wave vectors oriented at angles φ and φ – π relative to the external magnetic field on the plate surface significantly depends on the φ value. The fR value divides the range of existence of the backward spin waves into two frequency bands: at frequencies below fR, the R(φ) dependence is monotonic (parameter R takes the minimum and maximum values at angles similar to the wave vector cutoff angles) and, at frequencies above fR, there are extreme points (maximum and minimum) at φ values equal the maximum cutoff angles of the surface spin wave. A formula has been derived for the wave vector orientation at which the m-th extreme point lying on one of the plate surfaces arises in the distribution of the amplitude of the magnetic potential of the mth wave mode in the ferrite plate cross section.

In the edge element discretization for magnetostatic problems, a gauge scheme, such as tree gauge, Lagrangian multiplier (LM) gauge, and auto gauge, is usually adopted to handle the singular curl-curl equation in terms of the magnetic vector potential. However, tree-gauge and LM-gauge schemes are not very efficient as a nonlinear problem with many increments has to be resolved. In order to overcome those difficulties, a mixed-gauge scheme based on current-splitting is proposed in this article. First, we make an improvement for LM-gauged formulation and adapt it to calculate the initial increment with a specially designed CG solver; then split the discrete source current density into a compatible part and incompatible part based on discrete Helmholtz decomposition; auto gauge cooperates with the compatible part to calculate the following increments. The mixed-gauge scheme combines the advantages of LM-gauge and auto-gauge schemes, which avoids the step to determine the source term, and at the same time, has high computational efficiency. The mixed-gauge scheme is tested in some examples against some other gauge schemes and found to be efficient and robust, where others possibly converge slowly or fail to produce a reliable solution.

Mục đích của nghiên cứu này được dựa trên phương pháp miền nhỏ hữu hạn với công thức véc tơ từ thế để phân tích lực điện từ tạo ra bởi sự phân bố của mật độ từ cảm tản trong khe hở không khí và mật độ dòng điện trong các cuộn dây, cái mà khó có thể thực hiện trực tiếp bằng phương pháp phần tử hữu hạn, khi mà một số vùng dẫn nghiên cứu có kích thước rất nhỏ so với toàn bộ miền nghiên cứu. Phương pháp bài toán nhỏ được áp dụng ở đây để liên kết các bài toán theo một vài bước: Một bài toán với mô hình đơn giản (các cuộn dây) được giải trước. Bài toán tiếp theo bao gồm một hoặc nhiều miền dẫn từ được đưa vào để hiệu chỉnh sai số do bài toán trước đó gây ra. Tất cả các bước đều được giải độc lập trong lưới và miền hình học khác nhau, điều này tạo thuận lợi cho việc chia lưới cũng như tăng tốc độ tính toán của mỗi một bài toán nhỏ.

A magnetostatic approximation is used to derive a formula for the wave vector orientation at which a nonreciprocal backward spin wave is effectively excited in a tangentially magnetized ferrite plate. The amplitude of the magnetic potential of the wave has an extremum on one of the surfaces of the plate for a given orientation of the wave vector, but this amplitude is low on the other surface of the plate.

The computation of three-dimensional (3-D) magnetic fields is a demanding task in the analysis of electrical machines and other electromagnetic devices. In this context, integral field calculation provides a smooth solution, high precision and resolution, ”on-demand”- calculation, and an origin-based formulation of the magnetic field and the magnetic vector potential. However, conventional elliptic methods lead to huge parallelizable computing efforts and significant errors. In this paper, a 3-D generic current-carrying arc segment with rectangular cross-section is studied. A new analytic formulation is proposed to speed up the computation of magnetic fields and reduce the error by more than three orders of magnitude. In addition, the proposed magnetic vector potential expression has similar accuracy as numerical integration. In fact, a significant reduction of the error level has been showcased clearly with respect to existing approaches. The present work is promising for improving the design methodology and optimization of large superconducting dipole magnets or arched end-winding geometries of large electrical machines. © 2020 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.

Quantitatively analyzing the role of turbulence in the magnetic fluctuation-induced self-generating-organization region and the plasma turbulence-induced self-feeding-sustaining region, which is closely related to the evolution of null points and the magnetic helical structure, represents the key issues in understanding the magnetic energy release-conversion, plasma heating, and charged particles energization and acceleration in three-dimensional large temporal-spatial scale turbulent magnetic reconnection (3D LTSTMR). The first part of this two-paper series developed and validated the continuous kinetic-dynamic-hydro fully coupled temporal-spatial scale relativistic hybrid particle-in-cell and lattice Boltzmann (RHPIC-LBM) model and code for investigating the fine structure evolution of the 2.5D solar atmosphere LTSTMR activities. Based on the model and code developed in Part I, in this paper, we investigate the turbulence of the magnetic helical structure, current density vector fields, self-generating-organization magnetic potential vector fields, self-feeding-sustaining plasma motion, and the ion and electron acceleration of 3D LTSTMR with 100,000 CPU cores on the Tianhe-2 from National Supercomputer Center in Guang Zhou (NSCC-GZ). According to the simulation evidence, we discovered and confirmed the following results: (i) Slipping magnetic reconnection (MR) exists in the adjacent magnetic field lines (MFLs) during the compress-stretch-slip process on the quasi-separatrix layers (QSLs and MFLs drastically change and form a linkage span) and the adjacent MFLs’ break-rejoin MR exists on the separatrix surfaces (SLs). Both are consistent to observations; (ii) The slipping MR (defined as 1st type MR) and the MFLs’ break-rejoin MR (defined as 2nd type MR) are closely linked with the oblique and resistive tearing instabilities, respectively. In the 3D model, the 1st type MR forms O -type null points, while the 2nd type MR forms X -type null points. The magnetic energy conversion is dominated by turbulence-induced oblique instabilities in the 3D model instead of the resistive tearing instabilities in the 2D/2.5D model, which is consistent with the 3D observations; (iii) Magnetic energy conversion occurs in the interaction of the plasmoid-to-flux rope, plasmoid-to-plasmoid, and flux rope-to-flux rope. The turbulent acceleration is an independent acceleration mechanism in LTSTMR, induced by the interaction of waves-to-waves and waves-to-particles, which is different from the original hybrid acceleration mechanism (composed of parallel electric fields, betatron, shock and Fermi) ; (iv) Particles can be energized and accelerated at a longer time scale (−30 s) and can be accelerated to relativistic energies after being pre-accelerated by a Fermi-Betatron-shock wave acceleration process. These results are in agreement with the observations.

A method using the coupling model combining the field and circuit equations is put forward to describe the wireless power transfer (WPT) system in this study. Hollow solenoidal resonant coils are usually excited by alternating currents in practice, doing studies related to the solutions to the solenoidal coils in alternating electromagnetic fields is important. In this study, the analytic expressions of the alternating magnetic vector potential are obtained from Maxwell's formulas for time-harmonic electromagnetic fields, and the mathematical expressions of the electric field intensity, the magnetic flux density, the induced voltage, and the mutual inductance are established for the coaxially symmetric solenoidal coils. The field-circuit coupling model is built for WPT, and the relationship between the input and output variables is obtained. The equations contain the dielectric information of the transmission space and the coils' radii and turn numbers, and can be used for analysing the dielectric effects of the space at lower or higher frequencies. The results are verified by experiments of 2-coil and 4-coil structures. The advantages of being able to analyse the distribution of the fields are illustrated by a comparison experiment with a laterally displaced coil, which is valuable to WPT design.

International audience; This paper presents an analytical subdomain model for the prediction of various electromagnetic parameters of magnetic-geared double-rotor machine (MGDRM). By dividing the MGDRM into six subdomains and solving the Maxwell equations in polar coordinates for each region, the vector magnetic potential distribution can be derived. Subsequently, the flux density distribution, back electromotive force (EMF), and output torque, can be calculated. Furthermore, based on the proposed model, the equivalent d-q axis is elaborated. Also, the corresponding d and q inductance can be obtained. It proves that the MGDRM can be regarded as a non-salient-pole synchronous machine. Besides, the power factor of the MGDRM under i d =0 control is deduced and optimized for the different thicknesses of modulators. In addition, the demagnetization capability is analysed by adopting the subdomain model. Finally, the accuracy of proposed subdomain model for MGDRMs is validated via the finite-element method (FEM).

Eddy currents (ECs) are simulated in a single laminate, representing the whole core of an electrical machine. Despite this drastic reduction in the complexity of the problem, a 3-D finite-element model turns out to be still too expensive for simulations. To overcome this difficulty, 2-D/1-D methods are used. This article presents a solution to consider both air gap and edge effect (EE) based on the multiscale finite-element method (MSFEM) using the magnetic vector potential (MVP) A. Linear material properties are assumed; therefore, this article is carried out in the frequency domain. The new 2-D/1-D MSFEM is discussed, and various simulation results are presented.

This article describes a fast characteristics calculation of large transformers with step-lap joints by dividing regions of finite-element analysis (FEA). The flux of the core in each divided region is given by fixing the magnetic vector potential at the divided surface due to Stokes' theorem. The exiting current of the coil is calculated from the magnetic field with Ampere's law. The validity and advantage of the proposed method are confirmed by comparing the calculated iron loss and exciting current with those of the FEA applied to the whole region. The usefulness of the proposed method is clarified by the application of characteristics improvement of the transformers.

With the increasing energy shortages and environmental degradation, electric logistics vehicle (ELVs) with energy conservation and environmental protection has become a research hotspot. The design of machine is the key to develop ELVs. Magnetic field analysis is the most critical issue since its accuracy affects the calculation of motor torque, loss, and other characteristics. To provide a calculation method for the field and performance analysis of the machine for ELVs, this paper presents an analytical model of air-gap field for hybrid excitation and interior permanent magnet machine. In the proposed model, it is taken into account for the shape of the stator and rotor teeth. The flux density on the rotor side is derived by equivalent magnetic circuit (EMC) with leakage magnetic flux. Taking the calculated flux density as one of the second boundary conditions, the air-gap field distribution is calculated by magnetic potential model with the eccentricity of the rotor. To verify the analytical method, we adopted the finite element method. The simulation results of the air-gap flux, back electromotive force, and cogging torque are in good agreement with the analytical results. Besides, applying the analytical model, the machine can be optimized for obtaining the optimal air-gap flux density distribution. The hybrid excitation machine with salient pole and interior magnets can provide a good flux density waveform. The study offers a helpful analytical method for design and optimization of the type of machine for ELVs.

Eccentric magnetic harmonic gears (EMHG) can achieve high speed ratio transmission and large torque output by modulating the air gap length between permanent magnets (PM). The analytical method is based on the boundary perturbation method. The vector magnetic potential perturbation equation in the air gap region is established, and the general solution is obtained using the boundary conditions. According to the superposition principle, the air gap flux density of stator and rotor PMs acting alone is synthesized, and the electromagnetic torque and unbalanced magnetic pull are calculated. Last, we compare magnetic field distributions, electromagnetic torque and unbalanced magnetic pull computed by the analytical method with those obtained from finite-element method (FEM).

A new fully discrete linearized $H^1$-conforming Lagrange finite element method is proposed for solving the two-dimensional magneto-hydrodynamics equations based on a magnetic potential formulation...

In this work, we study the stable determination of time-dependent coefficients appearing in the Schrodinger equation from partial Dirichlet-to-Neumann map measured on an arbitrary part of the boundary. Specifically, we establish stability estimates up to the natural gauge for the magnetic potential.

In this article, the proper orthogonal decomposition (POD) is applied for parametric analysis in the gauged potential formulation of the Darwin model, considering both capacitive and inductive effects. Due to the large contrast in material parameters, the resulting system matrix is ill-conditioned. In addition, the condition number of the corresponding snapshot complex matrices is very huge. To improve the stability of the POD method, a structured-preserving strategy is considered and implemented for different unknown potentials, namely, the magnetic vector potential (MVP) $\mathbf {A}$ , the electric scalar potential (ESP) $\varphi $ , and the Lagrange multiplier $p$ . In addition, a greedy algorithm is proposed to select the snapshots adaptively. Two numerical examples, including a parallel plate capacitor and a modified $RLC$ device structure, are provided to illustrate the capability of the proposed POD in model-order reduction in the frequency-domain solvers.

The human body is actuated by electric and magnetic energies where the neural systems are found to transfer their instructions by nerve impulses in the form of electric signals or as waves of electric potential. However; the absence of a clear definition of electric charge misled the neuroscientists in recognizing the nerve impulses as electric charges and they defined them as chemical ions which contradict their observation as waves of electric potential. According to a found definition of the electric charge as electromagnetic waves of electric potential that deletes confusions in electromagnetism, this innovative definition is found to be consistent with the neurologist’s observations of the nerve impulses and their found nature as electric signals. Similarly; the neurologists succeeded in stimulating the neural system by magnetic flux through the scull without clear explanation due to the unclear definitions of the magnetic flux as electromagnetic waves of magnetic potential. However; we succeed in this article in finding the plausible explanations of this stimulation by the innovative definitions of the electric charge and magnetic flux as electromagnetic waves of electric and magnetic potentials. We show the truth of these definitions by amble evidence.

We investigate the validity of Gaussian lower bounds for solutions to an electromagnetic Schrödinger equation with a bounded time-dependent complex electric potential and a time-independent vector magnetic potential. We prove that, if a suitable geometric condition is satisfied by the vector potential, then positive masses inside of a bounded region at a single time propagate outside the region, provided a suitable average in space–time cylinders is taken.

We develop a reduced-order model for thin plates made of hard magnetorheological elastomers (hard-MREs), which are materials composed of hard-magnetic particles embedded in a polymeric matrix. First, we propose a new magnetic potential, as an alternative to an existing torque-based 3D continuum theory of hard-MREs, obtained by reformulating the remnant magnetization of a deformed hard-MRE body. Specifically, the magnetizations in the initial and current configurations are related by the rotation tensor decomposed from the deformation gradient, independently of stretching deformation. This description is motivated by recently reported observations in microscopic homogenization simulations. Then, we derive a 2D plate model through the dimensional reduction of our proposed rotation-based 3D theory. For comparison, we also provide a second plate model derived from the existing 3D theory. Finally, we perform precision experiments to thoroughly evaluate the proposed 3D and 2D models on hard-magnetic plates under various magnetic and mechanical loading conditions. We demonstrate that our rotation-based modification of the magnetic potential is crucial in correctly capturing the behavior of plates subjected to an applied field aligned with the magnetization, and undergoing in-plane stretching. In all the tested cases, our rotation-based 3D and 2D models yield predictions in excellent quantitative agreement with the experiments and can thus serve as predictive tools for the rational design of hard-magnetic plate structures.

In this paper we investigate the existence of infinitely many solutions for nonlocal Schrödinger equation involving a magnetic potential $$ (-\triangle )_{A}^{s}u+V(x)u=f\bigl(x, \vert u \vert \bigr)u, \quad\text{in } {\mathbb {R}}^{N}, $$(−△)Asu+V(x)u=f(x,|u|)u,in RN, where $s\in (0,1)$s∈(0,1) is fixed, $N>2s$N>2s, $V:{\mathbb {R}}^{N}\rightarrow {\mathbb {R}}^{+}$V:RN→R+ is an electric potential, the magnetic potential $A:{\mathbb {R}}^{N}\rightarrow {\mathbb {R}}^{N}$A:RN→RN is a continuous function, and $(-\triangle )_{A}^{s}$(−△)As is the fractional magnetic operator. Under suitable assumptions for the potential function V and nonlinearity f, we obtain the existence of infinitely many nontrivial high energy solutions by using the variant fountain theorem.

In this article, we prove that the function Ψ(t, x) belongs to the domain of the definition of the operator H. We discuss that the operator H and its powers is essentially self-adjo...

Functionally graded magneto-electro-elastic (FGMEE) materials has been increasingly used in engineering applications, particularly in smart material or intelligent structure systems. This paper proposes a semi-analytical approach for sliding frictional contact problem between a rigid insulating sphere and a transversely isotropic FGMEE film and half-space based on frequency response functions (FRFs). Multilayered approximation is used to model the functionally graded material (FGM), and the FRFs for each MEE layer are derived explicitly. The unknown coefficients in FRFs are formulated by two matrix equations, and their efficient solution process is proposed. Based on the obtained FRFs, a highly efficient semi-analytical model (SAM) is developed which is able to solve the three-dimensional frictional contact of FGMEE materials with arbitrary layer designs. The model is validated with finite element method and the literature. Furthermore, the pressure/stress distribution and electric/magnetic potential are studied in different FGM designs to investigate the influence of material layout.

The bridge of the interior permanent magnet (IPM) machine can reduce the leakage of the machine, which makes the utilization rate of the permanent magnet (PM) improved. However, the asymmetric saturation in the bridges will cause the asymmetry distribution of the magnetic potential on the rotor surface, which will bring challenges to the accurate calculation of inductances. Excited by the armature current at different initial phase angles, the flux distributions in the bridges are analyzed. Meanwhile, the reason why the asymmetric saturation exists in the bridges is revealed theoretically. Based on the above analysis, the rotor magnetic potential (RMP) model for an IPM machine with the bridges is established and used to calculate the armature reaction magnetic field and inductances of the IPM machine. In this paper, the RMP distribution in the bridges is obtained by considering asymmetric saturation, so the RMP model can reflect the rotor surface magnetic potential distribution more accurately, which is evident from the adjacent degree between the calculation results and the finite-element analysis (FEA) ones.

We deal with the following nonlinear Schrodinger equation with magnetic field and critical growth: { ( e i ∇ − A ( x ) ) 2 u + V ( x ) u = f ( | u | 2 ) u + | u | 2 ⁎ − 2 u in R N , u ∈ H 1 ( R N , C ) , where e > 0 is a small parameter, N ≥ 3 , 2 ⁎ = 2 N N − 2 is the critical Sobolev exponent, A ∈ C 1 ( R N , R N ) is a magnetic vector potential, V : R N → R is a continuous positive potential having a local minimum and f : R → R is a superlinear continuous function with subcritical growth. Using penalization techniques and variational methods, we investigate the existence and concentration of nontrivial solutions for e > 0 small enough.

We present the results of computations of the trajectories (the asymptotic latitude and asymptotic longitude) and the magnetic cutoff rigidity of galactic cosmic ray (GCR) particles for the airport Deblin, Poland (geographical latitude 51°33′32′′ N, geographical longitude 21°50′53′′ E) based on the numerical integration of equations of motion of charged particles of cosmic rays in the Earth’s magnetic field. The set of allowed trajectories at a given site on the surface of the Earth is called the asymptotic cone of acceptance The initial distance from the center of the Earth was taken to be 20 km above the Earth’s surface. At about this altitude, most cosmic rays undergo nuclear collisions. Calculations were made for different of models of the International Geomagnetic Reference Field (IGRF) from 1965–2015. The IGRF is an internationally agreed and widely used mathematical model of the Earth’s magnetic field of internal origin. Each constituent model of the IGRF is a set of spherical harmonics of degree n and order m, representing a solution to Laplace’s equation for the magnetic potential arising from sources inside the Earth at a given epoch. In all asymptotic direction calculations, we used sixth-degree expansion of geomagnetic field model. Schmidt normalized spherical harmonic coefficients were taken for the epochs from 1965 to 2015. Knowledge of asymptotic directions and the magnetic rigidity cutoff is important from the point of view of study different classes of cosmic rays variations intensity and anisotropy.

Based on the solution of the hydrodynamic equations and Maxwell’s equations, we show that an external quasi-homogeneous magnetic field leads to the emergence of a secondary electric field that is resulted from a nonlinear effect over magnetic potential A. This field is proved to exist in the region with a depth of $$\delta {\text{/}}2$$ , where δ is the London penetration depth. The hydrodynamic flow velocity is estimated.

The resistance of heavy-gage current-carrying conductors to alternating current of industrial frequency increases due to the skin effect and proximity effect, which leads to increasing heat loss during transmission of electrical energy. When power cables for a voltage of 110 kV or more are fabricated, segmented Milliken-type current-carrying conductors are used, which make it possible to reduce the resistance to alternating current. The objective of this work is to carry out a numerical study of electromagnetic processes in heavy-gage Milliken-type current-carrying conductors using mathematical simulation. For this purpose, a geometric model is constructed that takes into account the complex spatial structure of current-carrying conductors. For comparison, a geometric model is considered in which the wires are arranged parallel to each other. The electromagnetic processes in the current-carrying conductors are described by differential equations for the complex amplitude of vector magnetic potential derived from Maxwell’s equations. The proposed set of differential equations is supplemented with corresponding boundary conditions. The given problem was solved numerically on the basis of the ANSYS Maxwell software package constructed based on the finite element method. As a result of calculations, the fields of vector magnetic potential in the whole volume under study and the distributions of the current density vector in conductors were obtained, from which the resistances of current-carrying conductors as a whole and of each wire separately were calculated. The calculation results revealed that using Milliken-type current-carrying conductors makes it possible to reduce the conductor resistance by equalizing the current density distribution over the cross sections of wires when a segment of them is twisted.

International audience; A non-classical model for transversely isotropic magneto-electro-elastic circular Kirchhoff plates is established based on the extended modified couple stress theory. The Gibbs-type variational principle is used to obtain the governing equations and boundary conditions. To illustrate the newly derived model, the static bending problem of a clamped circular plate subjected to a uniformly distributed constant load is solved numerically by Fourier-Bessel series. The numerical results show that the values of transverse displacement, electric and magnetic potentials predicted by the current model are always smaller than those of the classical model, and the differences are diminishing as the plate thickness increases. In addition, it is shown that the magneto-electro-elastic coupling effect plays an important role in the transverse displacement, electric potential and magnetic potential of the magneto-electro-elastic circular Kirchhoff plates. Furthermore, several reduced specific models are provided for simpler cases.

This work presents an efficient method based on asymptotic models in order to solve numerically eddy current problems in a bi-dimensional setting with a high contrast of linear relative magnetic permeability mu_r between a conductor and a dielectric subdomain. We describe a magnetic skin effect by deriving a multiscale expansion for the magnetic potential in power series of a small parameter delta which represents the skin depth. We make explicit the first asymptotics up to the order three. Some numerical results based on the finite element method will be presented to illustrate the magnetic skin effect and to validate the performance of the proposed asymptotic models in the dielectric medium. We confirm that the proposed asymptotics provide reduced computational costs for a wide range of the physical parameters introduced in our problem.

In this study, a magnetic field is analyzed using an analytical method and compared with the electromagnetic characteristics of 8-pole 9-slot and 8-pole 12-slot permanent magnet synchronous machines considering rotor eccentricity. The magnetic flux density and back electromotive force (EMF) are derived using perturbation theory and electromagnetic theory. First, the Fourier modeling of a permanent magnet is performed through magnetization modeling, and two analysis regions are set based on several assumptions for applying the analytical method. Accordingly, the governing equations of the analysis regions are derived in the form of Poisson and Laplace equations. In addition, the undefined coefficients of the general solutions are calculated through general solutions and appropriate boundary conditions, and the magnetic flux density and back EMF of the air gap region are derived based on the definition of the magnetic vector potential. The results obtained using the analytical method are compared with the finite element method and experimental results. In addition, we perform a torque analysis considering rotor eccentricity and analyze the torque ripple based on rotor eccentricity for two cases involving the pole/slot combination.

Magnetic induction communication and wireless power transmission based on magnetic coupling have significant application prospects in underwater environments. Mutual impedance is a key parameter particularly required for the design of the systems. However, mutual impedance is usually extracted from measurements when the coils are processed, which is obviously not conducive to the system optimization in the design phase. In this paper, a model of the mutual impedance of coils immersed in water is established. The magnetic vector potential is expressed in the form of series by artificially setting a boundary, and then the mutual impedance calculation formula of the coils immersed in water is derived. In the analysis, the effect of the conductivity of water, the excitation frequency, and the number of turns of the coils are mainly taken into account. In addition, the variation of the mutual impedance of coils in air and water with axial displacement is also compared. The models can be used to analyze the coil coupling characteristics in the presence of conductive medium, which is helpful for the design process.

The distant magnetic field of a magnetic dipole is usually derived via the magnetic vector potential and substantial vector calculus. This paper presents an alternate proof that is less mathematically intensive, and that ties together various problem-solving tricks (the principle of virtual work, observation that only instantaneous quantities matter and reciprocity of mutual inductance), functioning as an instructional exercise.

We investigate a fully discrete finite element scheme for the three-dimensional incompressible magnetohydrodynamic problem based on magnetic vector potential formulation that was introduced in Hiptmair et al. (MMMAS 28:659–695, 2018). The formulation enjoys the novel feature that it can always produce an exactly divergence-free magnetic induction discretized solution. Using a mixed finite element approach, we discretize the model by the fully discrete semi-implicit Euler scheme with the velocity and the pressure approximated by stable MINI finite elements and the magnetic vector potential by Nedelec edge elements. Under a reasonable regularity hypothesis for the exact solution, error estimates for the velocity and the magnetic vector potential are rigorously established. Finally, several numerical experiments are presented to illustrate the convergence properties of the numerical scheme.

Due to the longitudinal coupling of power transformer windings, the internal fault analysis become difficult and complicated. This paper presents a new approach base on magnetic potential balance principle calculation. When the internal fault occurs, it can extract useful information from the fault recorded diagram, analyze the fault type, and even lock the fault location. There is no need to build complex equations and models. In the end, the effectiveness of the method is proved by a case study.

Magnetic tweezers are crucial for single-molecule and atomic characterization, and biomedical isolation of microparticle carriers. The trapping component of magnetic tweezing can be relying on a magnetic potential well that can confine the relevant species to a localized region. Here, we report that magnetic microparticles with tailored anisotropy can transition from localized off-axis gyration to large-area locomotion in a rotating magnetic trap. The microparticles, consisting of assemblies of magnetic cores, are observed to either rotate about its structural geometric center or gyrate about one of the magnetic cores, the switching of which can be modulated by the external field. Raising the magnetic field strength above a threshold, the particles can go beyond the traditional synchronous-rotation and asynchronous-oscillation modes, and into a scenario of large-area circular motion. This results in peculiar retrograde locomotion related to the magnetization maxima of the microparticle. Our finding suggests the important role of the microparticle's magnetic morphology in the controlled transport of microparticles and developing smart micro-actuators and micro-robot devices., Comment: 4 figures