Scalar Potential Formulation Research Articles

Incorporation of a 3-D Energy-Based Vector Hysteresis Model Into the Finite Element Method Using a Reduced Scalar Potential Formulation

Incorporation of a 3-D Energy-Based Vector Hysteresis Model Into the Finite Element Method Using a Reduced Scalar Potential Formulation

3D electromagnetic analytical model for radial-flux eddy-current couplers

PurposeThis paper aims to develop a new 3D analytical model in cylindrical coordinates to study radial flux eddy current couplers (RFECC) while considering the magnetic edge and 3D curvature effects, and the field reaction due to the induced currents.Design/methodology/approachThe analytical model is developed by combining two formulations. A magnetic scalar potential formulation in the air and the magnets regions and a current density formulation in the conductive region. The magnetic field and eddy currents expressions are obtained by solving the 3D Maxwell equations in 3D cylindrical coordinates with the variable separation method. The torque expression is derived from the field solution using the Maxwell stress tensor. In addition to 3D magnetic edge effects, the proposed model takes into account the reaction field effect due to the induced currents in the conducting part. To show the accuracy of the developed 3D analytical model, its results are compared to those from the 3D finite element simulation.FindingsThe obtained results prove the accuracy of the new developed 3D analytical model. The comparison of the 3D analytical model with the 2D simulation proves the strong magnetic edge effects impact (in the axial direction) in these devices which must be considered in the modelling. The new analytical model allows the magnetic edge effects consideration without any correction factor and also presents a good compromise between precision and computation time.Practical implicationsThe proposed 3D analytical model presents a considerably reduced computation time compared to 3D finite element simulation which makes it efficient as an accurate design and optimization tool for radial flux eddy current devices.Originality/valueA new analytical model in 3D cylindrical coordinates has been developed to find the electromagnetic torque in radial flux eddy current couplers. This model considers the magnetic edge effects, the 3D curvature effects and the field reaction (without correction factors) while improving the computation time.

A Generalized Scalar Potential Integral Equation Formulation for the DC Analysis of Conductors

The electrostatic modeling of conductors is a fundamental challenge in various applications, including the prediction of parasitic effects in electrical interconnects, the design of biasing networks, and the modeling of biological, microelectromechanical, and sensing systems. The boundary element method (BEM) can be an effective simulation tool for these problems because it allows modeling three-dimensional objects with only a surface mesh. However, existing BEM formulations can be restrictive because they make assumptions specific to particular applications. For example, capacitance extraction formulations usually assume a constant electric scalar potential on the surface of each conductor and cannot be used to model a flowing current, nor to extract the resistance. When modeling steady currents, many existing techniques do not address mathematical challenges such as the null space associated with the operators representing the internal region of a conductor. We propose a more general BEM framework based on the electric scalar potential for modeling conductive objects in various scenarios in a unified manner. Restrictive application-specific assumptions are not made, and the aforementioned operator null space is handled in an intuitive and rigorous manner. Numerical examples drawn from diverse applications confirm the accuracy and generality of the proposed method.

Modeling of Magnetoelectric Effects in Composite Structures by FEM–BEM Coupling

In this paper, a coupling of the Finite Element Method (FEM) and the Boundary Element Method (BEM) is used to model the behavior of magnetoelectric effects in composite structures. This coupling of numerical methods makes it possible not to have to consider a free space domain, and thus to use a single mesh for the magnetic, mechanical and electrical problems. This results in a consequent reduction of the number of unknowns which is accompanied by shorter computation times compared to a classical FEM approach. A mixed magnetic vector potential, reduced magnetic scalar potential formulation is used for the magnetic problem, and classical FEM formulations are used for electrical and mechanical problems. The resulting global algebraic system is solved by a block Gauss-Seidel solver.

Asymptotic homogenization in the determination of effective intrinsic magnetic properties of composites

We present a computational framework for two-scale asymptotic homogenization to determine the intrinsic magnetic permeability of composites. To this end, considering linear magnetostatics, both vector and scalar potential formulations are used. Our homogenization algorithm for solving the cell problem is based on the displacement method presented in Lukkassen et al. 1995, Composites Engineering, 5(5), 519-531. We propose the use of the meridional eccentricity of the permeability tensor ellipsoid as an anisotropy index quantifying the degree of directionality in the linear magnetic response. As application problems, 2D regular and random microstructures with overlapping and nonoverlapping monodisperse disks, all of which are periodic, are considered. We show that, for the vanishing corrector function, the derived effective magnetic permeability tensor gives the (lower) Reuss and (upper) Voigt bounds with the vector and scalar potential formulations, respectively. Our results with periodic boundary conditions show an excellent agreement with analytical solutions for regular composites, whereas, for random heterogeneous materials, their convergence with volume element size is fast. Predictions for material systems with monodisperse overlapping disks for a given inclusion volume fraction provide the highest magnetic permeability with the most increased inclusion interaction. In contrast, the disk arrangements in regular square lattices result in the lowest magnetic permeability and inadequate inclusion interaction. Such differences are beyond the reach of the isotropic effective medium theories, which use only the phase volume fraction and shape as mere statistical microstructural descriptors.

Electromagnetic Tracking of Elongated Sensors for Endoscopic Navigation

As the prevalence of image-guided interventions increases, electromagnetic tracking (EMT) systems play an important role in modern patient care, as they enable real-time instrument positioning and navigation inside the human body without line-of-sight restrictions. Miniature-size inductive coils are the gold standard in clinical settings, as they provide accurate, passive sensing of the magnetic field. To compensate for their small dimensions, such sensors are designed with an elongated shape, where the coil length is usually 10 to 20 times larger than the diameter. In this article, the benefits of a field model based on the magnetic scalar potential formulation are demonstrated for EMT applications where elongated tracking sensors are used. The novel method resolves the single-point approximation error when the coil length is not negligible, and demonstrates improvements in terms of speed and storage requirements. A detailed analysis is proposed where alternative formulations of the magnetic model used in the tracking algorithm are compared. Although this work does not resolve any substantial limits of EMT used in a clinical environment, which are mainly caused by the presence of magnetic distortions, the proposed method is an improvement over existing EMT systems because it enables more accurate and faster tracking. The method might facilitate the use of longer tracking sensor coils which can achieve high sensitivities without the requirement of a magnetic core. In the envisioned application, such coils may be wound around flexible instruments, such as endoscopes or catheters.

3D fast analytical computation of a permanent magnets axial coupler

PurposeThe purpose of this paper is to develop a new and fast three-dimensional (3D) analytical model to study a synchronous axial magnetic coupling with rectangular shaped magnets. This model takes into account edge and curvature 3D effects.Design/methodology/approachThis paper firstly introduces a 3D analytical model for an axial coupler with sector shaped permanent magnet (PM) based on magnetic scalar potential formulation in cylindrical coordinates. The magnetic field in PM, air gap and iron disks is computed by solving Laplace’s and Poisson’s partial differential equation. This solution is then used to compute the field in rectangular shaped magnets. To do so, the adopted approach consists to divide the rectangular magnet into sector radial slices each of which the 3D model allows the determination of the magnetic field distribution. The results obtained by the proposed 3D analytical model are validated through 3D finite element computations. Furthermore, a prototype axial magnetic coupler has been constructed so air gap flux density and static torque measurements are compared to the analytical predictions.FindingsThe results obtained by the analytical model show the effectiveness of the proposed geometry transformation approach. The developed model takes into account all the 3D effects without needing any correction factor.Research limitations/implicationsThe developed method provides an efficient and rapid tool for evaluating the influence of geometric and physical parameters of a synchronous magnetic coupling as part of a design optimization process.Practical implicationsThe developed method provides an efficient and rapid tool for evaluating the influence of geometric and physical parameters of a synchronous magnetic coupling as part of a design optimization process.Originality/valueA new and fast 3D analytical model, to improve the computation of the electromagnetic torque developed by a synchronous magnetic coupler with rectangular shaped magnets, has been developed. The proposed approach is really effective as it leads to consistent results when compared to 3D finite element method ones without any need for correction factor.

Passive seismic reflection imaging based on acoustic and elastic reverse time migration without source information: theory and numerical simulations

To explore crustal structures using seismic records of earthquakes, we proposed a new seismic reflection imaging method based on reverse time migration (RTM) without source information. In the proposed RTM method using multiple reflections between surface and subsurface boundaries, we correlate forward- and backward-extrapolated wavefields of the same observation records from all receivers at once. We implemented acoustic RTM based on a scalar wave equation and elastic RTM based on a velocity-stress formulation for wavefield modelling. Furthermore, we adopted scalar and vector potential formulations in elastic RTM to obtain separate P- and S-wave reflection images. In our study, numerical simulations with a regional crustal structure model were conducted to compare combinations among the synthetic data by acoustic or elastic wave modelling, observation conditions, and RTM formulations. We demonstrated the validity of the proposed method with idealised dense observations and confirmed a limitation due to quasi-realistic sparse observations. Artefacts due to the mixture of P- and S-waves in the acoustic RTM images from elastic synthetic data were attenuated by elastic RTM, which addresses the proper illumination of P- and S-wave reflections.

A hybrid FPM/BEM scalar potential formulation for field calculation in nonlinear magnetostatic analysis of superconducting accelerator magnets

A new hybrid numerical method for the solution of nonlinear magnetostatic problems in accelerator magnets is proposed. The methodology combines the Fragile Points Method (FPM) and the Boundary Element Method (BEM). The FPM is a Galerkin-type meshless method employing for field approximation simple, local, and discontinuous point – based interpolation functions. Because of the discontinuity of these functions, the FPM can be considered as a meshless discontinuous Galerkin formulation where numerical flux corrections are employed for the treatment of local discontinuity inconsistencies. The FPM possesses the advantages of a mesh-free method, evaluates with high accuracy field gradients, and treats nonlinear magnetostatic problems by utilizing, for the same problem, fewer nodal points than the Finite Element Method (FEM). In the present work, the nonlinear ferromagnetic material of an accelerator magnet is treated by the FPM, while the BEM is employed for the infinitely extended, surrounding air space. The proposed hybrid scheme is based on the scalar potential formulation of Mayergouz et.al (1987), also used in the BEM/BEM and FEM/BEM schemes of Rodopoulos et al. (2019, 2020). The applicability of the method is demonstrated with the solution of representative 2-D magnetostatic problems and the obtained numerical results are compared to those provided by the FEM/BEM scheme of Rodopoulos et al. (2020), as well as by the commercial FEM package ANSYS. Finally, the magnetic field utilized for stable bending of the particles’ trajectory in a 16 Tesla dipole magnet design for the Future Circular Collider (FCC) project of CERN is accurately evaluated with the aid of the proposed here FPM/BEM scheme.

A General Method to Compute the Electric Flux Lines between Two Magnet Wires in Close Contact and Its Application for the Evaluation of Partial Discharge Risks in the Slots of Electric Machines Embedded in Future Transportation Systems

The sizing of the Electrical Insulation System (EIS) is an important challenge in electric motors of higher specific power driven by faster inverters. That keeps increasing the electric stress which the winding is submitted in the stator slot. Consequently, Partial Discharges (PD) are more likely to occur. Nowadays, the Paschen’s criterion is widely used to evaluate the risk of partial discharge. It requires the knowledge of electric field lines. This paper presents a method to precisely compute the electric field lines in a two-dimensional (2D) electrostatic problem. The field of study is composed of two magnet wires in close contact. Such configuration is representative of the turn-to-turn interaction in an electric motor slot. The problem is solved using the scalar potential formulation only. The notion of flux tubes is used for the post process of the electric field lines in a developed numerical code on Matlab. The developed method is compared to a ballistic method already included on Matlab. The work presented here is included in an automatic tool to suppress or reduce the partial discharge risk in a stator slot of high power density motor destined for future transportation systems.

Improved 3D electromagnetic analytical model for planar induction heater with consideration of transverse edge effects

PurposeThis paper aims to propose a new 3D electromagnetic model to compute translational motion eddy current in the conducting plate of a novel linear permanent magnet (PM) induction heater. The movement of the plate in a DC magnetic field created by a PM inductor generates induced currents that are at the origin of a heating power by Joule effect. These topologies have strong magnetic end effects. The analytical model developed in this work takes into account the finite length extremity effects of the conducting plate and the reaction field because of induced currents.Design/methodology/approachThe developed model is based on the combination of the sub-domain’s method and the image’s theory. First, the magnetic field expressions because of the PMs are obtained by solving the three-dimensional Maxwell equations by the method of separation of variables, using a magnetic scalar potential formulation and a magnetic field strength formulation. Then, the motional eddy currents are computed using the Ampere law, and the finite length extremity effects of the conducting plate are taken into account using the image’s method. To analyze the accuracy of the proposed model, the obtained results are compared to those obtained from 3D finite element model (FEM) and from experimental tests performed on a prototype.FindingsThe results show that the developed analytical model is very accurate, even for geometries where the edge effects are very strong. It allows directly taking into account the finite length extremity effects (the transverse edge effects) of the conducting plate and the reaction field because of induced currents without the need of any correction factor. The proposed model also presents an important reduction in computation time compared to 3D finite element simulation, allowing fast analysis of linear PM induction heater.Practical implicationsThe proposed electromagnetic analytical model can be used as a quick and accurate design tool for translational motion PM induction heater devices.Originality/valueA new 3D analytical electromagnetic model, to find the induced power in the conducting plate of a novel translational motion induction heater has been developed. The studied heating device has a finite length and a finite width, which create edge effects that are not easily considered in calculation. The novelty of the presented method is the accurate 3D analytical model, which allows finding the real power heating and real distribution of the induced currents in the conducting plate without the need to use correction factor. The proposed model also takes into account the reaction field because of induced currents. In addition, the developed model improves an important reduction in the computation time compared with 3D FEM simulation.

Hybrid Formulation Domain-Decomposition Finite-Element Method for Simulating Eddy-Current Testing

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.

Nonlinear BEM/FEM scalar potential formulation for magnetostatic analysis in superconducting accelerator magnets

Hybrid schemes that combine the Finite Element Method (FEM) and the Boundary Element Method (BEM) have been extensively used for the solution of nonlinear magnetostatic problems. The majority of those schemes rely either on reduced and total scalar or vector potential formulations. One main issue regarding the scalar potential formulations is the cancelation errors and difficulties on the definition of potential jump condition on the surface of a magnet, while in the case of vector potential formulations, the size of the simulated problem increases drastically especially in three dimensions. An alternative formulation has been proposed by Mayergoyz et al. (1987), which is based on scalar potentials and provides assistance in ameliorating the aforementioned problems. In the present work that formulation is implemented by employing a BEM/FEM scheme, for the solution of nonlinear magnetostatic problems. The proposed BEM/FEM formulation is employed for the solution of representative magnetostatic problems dealing with different types of superconducting accelerator magnets, while the provided accuracy is assessed through comparisons made with corresponding results obtained by a well-known commercial FEM package.

Accuracy Assessment of Numerical Dosimetry for the Evaluation of Human Exposure to Electric Vehicle Inductive Charging Systems

In this article, we discuss numerical aspects related to the accuracy and the computational efficiency of numerical dosimetric simulations, performed in the context of human exposure to static inductive charging systems of electric vehicles. Two alternative numerical methods based on electric vector potential and electric scalar potential formulations, respectively, are here considered for the electric field computation in highly detailed anatomical human models. The results obtained by the numerical implementation of both approaches are discussed in terms of compliance assessment with ICNIRP guidelines limits for human exposure to electromagnetic fields. In particular, different strategies for smoothing localized unphysical outliers are compared, including novel techniques based on statistical considerations. The outlier removal is particularly relevant when comparison with basic restrictions is required to define the safety of electromagnetic fields exposure. The analysis demonstrates that it is not possible to derive general conclusions about the most robust method for dosimetric solutions. Nevertheless, the combined use of both formulations, together with the use of an algorithm for outliers removal based on a statistical approach, allows to determine final results to be compared with reference limits with a significant level of reliability.

New BEM/BEM and BEM/FEM scalar potential formulations for magnetostatic problems

The majority of the numerical methods applied so far for the simulation of a magnetostatic problem is based either on reduced and total scalar or vector potential formulations. The first one suffers from cancellation errors, the second one from the difficulty of defining the jump condition for potential on the surface of a magnet. Furthermore, the vector potential introduces three Degrees of Freedom per node increasing the computational effort for the solution of a three dimensional problem. Mayergoyz et al. (1987) proposed an alternative formulation, which is based on scalar potentials and is free of the aforementioned problems. The present work implements this formulation for the solution of magnetostatic problems by utilizing either Boundary Element Method (BEM) or its combination with Finite Element Method (FEM). Both numerical schemes are explained for two dimensional and linear problems. Two representative magnetostatic problems are solved and their solutions are compared to the corresponding ones taken by well-known commercial package.

3D magnetotelluric modeling by using finite-difference method: Comparison study of different forward modeling approaches

We have investigated different approaches to solve magnetotelluric forward modeling problems. Besides classical direct electromagnetic (EM) formulation, ungauged and gauged (Lorenz, Coulomb, and axial) vector and scalar potential formulations are discretized using the finite-difference (FD) numerical solution technique. Linear matrix equations obtained from FD discretization for each EM field formulation are solved by using the conjugate-gradient iterative solution method. We compared FD solutions of each approach with respect to accuracy and speed. We found that all approaches generate accurate results. However, an ungauged approach gave solutions with a lower conjugate-gradient iteration and accordingly less computer time. All stiffness matrices arising from each formulation are examined in terms of their size and type. FD solution of the Coulomb-gauge and ungauged approaches generates larger stiffness matrices than the other approaches. Direct EM, ungauged, and axial-gauge approaches generate a symmetric stiffness matrix. We recommend the use of the ungauged approach in inversion algorithms, which gives faster forward solution, amongst other things.

Dissipative Quantum Electromagnetics

The dissipative quantum electromagnetics is introduced in a comprehensive manner as a field-matter-bath coupling problem. First, the matter is described by a cluster of Lorentz oscillators. Then the Maxwellian free field is coupled to the Lorentz oscillators to describe a frequency dispersive medium. The classical Hamiltonian is derived for such a coupled system, using Lorenz gauge and decoupled scalar and vector potential formulations. The classical equations of motion are derivable from the Hamiltonian using Hamilton equations. Then the Hamiltonian is quantized with all the pertinent variables with the introduction of commutators between the variables and their conjugate pairs. The quantum equations of motion can be derived using the quantum Hamilton equations. It can be shown that such a quantization scheme preserves the quantum commutators introduced. Then a noise bath consisting of simple harmonic oscillators is introduced and coupled to the matter consisting of Lorentz oscillators to induce quantum loss. Langevin source emerges naturally in such a procedure, and it can be shown that the results are consistent with the fluctuation dissipation theorem, and the quantization procedure of Welsch's group. The advantage of the present procedure is that no diagonalization of the Hamiltonian is necessary to arrive at the quantum equations of motion. Finally, we apply the quantization scheme to model spontaneous emission of a two-level polarized atom placed above a dielectric cylinder that supports a bound state in the continuum.

Hybrid Model: Permeance Network and 3-D Finite Element for Modeling Claw-Pole Synchronous Machines

Designing a claw-pole synchronous machine implies solving many 3D nonlinear magnetostatic problems which makes the computation (CPU) time very long. In our model, the mesh is refined to reach the desired level of precision on global quantities such as torque. Since the airgap is very thin (around 0.3 mm for a 100 mm diameter) and a Newton Raphson algorithm requires several iterations to converge, CPU time may be too high. Nowadays, many researches are ongoing to reduce the CPU time, while preserving an acceptable accuracy. One of the most efficient methods is permeance networks but this method is not suitable for complex geometries. Our main contribution is to use a permeance network in the areas where flux lines are easy to guess and to solve a 3D FEM problem in complex geometry areas of the magnetic device: the claw poles and the air gap for example. Moreover, current sources belong to the permeance network model, so that there are no current sources in the 3D FEM problem. Then, a 3D scalar magnetic potential formulation can be used easily. The two classical magnetostatic formulations (magnetic scalar potential formulation (Um – hs) and vector potential formulation (a – j)) are presented in this paper. Then, the hybridization of 3D FEM formulation and the permeance network, is presented. Numerical results are compared with experimental measurements and a good agreement is obtained while reducing the CPU time. Index Terms—Claw-pole synchronous machine, 3D FEM, permeance networks, hybrid models.

Optimisation and measurement of eddy current damping in a passive tuned mass damper

This paper describes the optimisation of the eddy current damping, applied in a passive tuned mass damper (TMD). A semi-analytical model based on scalar potential formulation is extended for different permanent magnet (PM) topologies. Optimal design parameters are acquired by particle swarm optimisation. A 3-DoF mechanical model is presented to predict the quality reduction factor of the TMD. Measurements are performed for validating the semi-analytical model. Furthermore, an experiment is carried out to test a TMD with two optimal PM topologies for sinusoidal excitations.

Analogies Between Finite-Difference and Finite-Element Methods for Scalar and Vector Potential Formulations in Magnetic Field Calculations

Numerical 3-D formulations using scalar $\Omega $ and vector A potentials are examined for magnetic fields with an emphasis on the finite-difference method (FDM) and finite-element method (FEM) using nodal and facet elements. It is shown that for hexahedral elements, the FDM equations may be presented in a form similar to the FEM equations; to accomplish this, the coefficients defining volume integrals in the FEM need to be expressed in an approximate manner, while the nodes in the FDM require supplementary association with middle points of edges, facets, and volumes. The analogy between a description of magnetic field sources arising from the classical MMF distribution approach, and when expressed in terms of edge values of vector potential, $T_{0}$ is emphasized. Comparisons are made between the results obtained using the FDM and the FEM for both the scalar and vector potential formulations. Forces in systems containing permanent magnets and torques in permanent magnet machines are calculated and compared using both the approaches for scalar and vector formulations. A unified form of the stress tensor has been applied to the FDM and FEM.