Scalar Potential Formulation Research Articles

Inclusion of a Direct and Inverse Energy-Consistent Hysteresis Model in Dual Magnetostatic Finite-Element Formulations

This paper deals with the implementation of an energy-consistent ferromagnetic hysteresis model in 2-D finite-element computations. This vector hysteresis model relies on a strong thermodynamic foundation and ensures the closure of minor hysteresis loops. The model accuracy can be increased by controlling the number of intrinsic cell components, while parameters can be easily fitted on common material measurements. Here, the native $h $ -based material model is inverted using the Newton–Raphson method for its inclusion in the magnetic vector potential formulation. Simulations are performed on a 2-D T-shaped magnetic circuit exhibiting rotational flux. By way of validation, the results are compared with those obtained with the dual magnetic scalar potential formulation. A very good agreement for global quantities is observed.

Experimental study of molten metal flow and numerical simulation of magnetic field during permanent magnet stirring and its application in continuous casting

Molten metal flow and magnetic field of the permanent magnet stirring (PMS) were investigated by experimental study and numerical simulation, respectively. A laboratory experiment was first performed to study the effects of PMS on the flow velocity of the Sn-20%Pb hypereutectic alloy during solidification. The results showed that the center magnetic flux density and stirring frequency were the two key factors affecting the electromagnetically-driven convection of the molten alloy. The electromagnetic field analysis software Opera-3D was used to develop a materials-saving and highly effective final permanent magnet stirring (FPMS) apparatus comprising a reasonable magnetic field design. Based on the scalar potential formulation, the stationary magnetic fields of the PMS were calculated and the number, arrangement and size of the permanent magnets were optimized. Thereafter, a pair of arc NdFeB permanent magnets with optimally designed geometrical dimensions was developed for industrial use in the FPMS apparatus. Consequently, with the industrial application of the new equipment in continuous casting, an improvement of the macrosegregation and other macro-defects of the rectangular billet were obtained.

Nondestructive Determination of the Permittivity Tensor of a Uniaxial Material Using a Two-Port Clamped Coaxial Probe

A two-port coaxial probe is introduced to nondestructively determine the permittivity tensor of a uniaxial material. The proposed approach possesses several advantages over existing techniques, e.g., only a single sample is required, the sample does not need to be rotated, and only a single measurement system is needed. The derivation of the theoretical scattering parameters is shown. This is accomplished by applying Love’s equivalence theorem and the continuity of transverse magnetic fields to formulate a system of coupled integral equations. A necessary step in this approach is the derivation of the magnetic-current-excited uniaxial parallel-plate Green’s function. The development of this Green’s function is presented here using a new scalar potential formulation, which significantly reduces the difficulty of the probe’s theoretical development. The system of coupled integral equations is solved using the method of moments to yield the theoretical scattering parameters. The permittivity tensor is found by minimizing the two-norm of the vector difference between the theoretical and measured scattering parameters via nonlinear least squares. To validate the probe, measurement results of a uniaxial absorber are presented and compared to those obtained using a focused-beam (free-space) measurement system. The probe’s sensitivity to uncertainties in measured scattering parameters, sample thickness, and coaxial line properties is also investigated.

3-D Analytical Model for Axial-Flux Eddy-Current Couplings and Brakes Under Steady-State Conditions

This paper presents a 3-D analytical model for axial-flux eddy-current couplings and brakes, leading to closed-form expressions for the torque and the axial force. The proposed model is valid under a steady-state condition (constant speed operation). It takes into account the reaction field due to induced currents in the moving conducting part. In order to simplify the analysis, we adopt the assumption of linearization at the mean radius, and the problem is then solved in 3-D Cartesian coordinates (curvature effects are neglected). The solution is obtained by solving the Maxwell equations with a magnetic scalar potential formulation in the nonconductive regions (magnets and air gap) and a magnetic field strength formulation in the conductive region (copper). Magnetic field distribution, axial force, and torque computed with the 3-D analytical model are compared with those obtained from the 3-D finite-element simulations and experimental results.

Eddy Currents and Non-Conforming Sliding Interfaces for Motion in 3-D Finite Element Analysis of Electrical Machines

This paper presents non-conforming sliding interfaces for motion in 3-D finite element simulations. Sliding interfaces are favorable, especially for field circuit coupling in comparison to other approaches such as the Lockstep method because an arbitrary position of the rotor is possible. A previously presented approach by the authors is extended to take eddy-currents into account. The sliding interfaces approach utilizes specific Lagrange multiplier to handle the relative motion between stator and rotor which require a magnetic scalar potential formulation. The formulation is presented as well as methods to compute the mandatory cohomology basis functions.

Fast Computation of Cuts With Reduced Support by Solving Maximum Circulation Problems

We present a technique to efficiently compute optimal cuts required to solve 3-D eddy current problems by magnetic scalar potential formulations. By optimal cuts, we mean the representatives of (co)homology generators with minimum support among the ones with a prescribed boundary. In this paper, we obtain them by starting from the minimal (co)homology generators of the combinatorial two-manifold representing the interface between conducting and insulating regions. Optimal generators are useful because they reduce the fill-in of the sparse matrix and ease human-guided basis selection. In addition, provided that the mesh is refined enough to allow it, they are not self-intersecting. The proposed technique is based on a novel graph-theoretic algorithm to solve a maximum circulation network flow problem in unweighted graphs that typically runs in linear time.

One Stroke Complementarity for Poisson-Like Problems

Taking electrokinetics as a paradigm problem for the sake of simplicity, complementarity originates when an irrotational electric field and a solenoidal current density satisfying boundary conditions are in hand. We first compare three formulations to obtain a solenoidal current density, both in terms of pure computational advantage and in the ability to pursue symmetric energy bounds with respect to the standard electric scalar potential formulation. For these formulations, we devise post-processing techniques that promise to provide bilateral bounds in one stroke, hence requiring the solution of just one linear system.

Magnetic force and torque on particles subject to a magnetic field

Abstract Materials that are sensitive to an applied magnetic field are of increased interest and use to industry and researchers. The realignment of magnetizable particles embedded within a substrate results in a deformation of the material and alteration of its intrinsic properties. An increased understanding of the influence of the particles under magnetic load is required to better predict the behaviour of the material. In this work, we examine two distinct approaches to determine the resulting magnetic force and torque generated within a general domain. The two methodologies are qualitatively and quantitatively compared, and we propose scenarios under which one is more suitable for use than the other. We also describe a method to compute the generated magnetic torque. These post-processing procedures utilize results derived from a magnetic scalar-potential formulation for the large deformation magneto-elastic problem. We demonstrate their application in several examples involving a single and two particle system embedded within a carrier matrix. It is shown that, given a chosen set of boundary conditions, the magnetic forces and torques acting on a particle are influenced by its shape, size and location within the carrier.

Improvised Asymptotic Boundary Conditions for Electrostatic Finite Elements

Improvised asymptotic boundary conditions are a sparse but approximate method of solving open boundary problems. Since this method approximates unbounded space as series of isotropic shells, it can be readily implemented in virtually any finite element solver without the need for additional code. A previous paper addressed application of this technique to magnetic problems described with a vector potential formulation. This paper extends the method to electrostatics and related scalar potential formulations.

Linear Subspace Reduction for Quasistatic Field Simulations to Accelerate Repeated Computations

The design of electrical machines and high-voltage devices requires the simulation of non-linear low frequency electromagnetic problems in time domain. Typically magneto- and electro-quasistatic problems in magnetic vector/electric scalar potential formulation lead after space discretization to differential-algebraic or ordinary differential equations, respectively. Therefore, huge systems of equations have to be solved in both cases. Typically a large number of degrees of freedom (DoF) is in domains with constant material parameters, i.e., linear subdomains, e.g., air or vacuum in exterior domains. In this paper, we present a method for low frequency simulations based on the proper orthogonal decomposition to reduce the DoF in these linear subspaces. The application of the method will be shown for a simple transformer model and within a global sensitivity analysis (uncertainty quantification) of the switching point in a non-linear resistive material used in a 11 kV standard insulator model.

Lazy Cohomology Generators: A Breakthrough in (Co)homology Computations for CEM

Computing the first cohomology group generators received great attention in computational electromagnetics as a theoretically sound and safe method to produce cuts required when eddy-current problems are solved with the magnetic scalar potential formulations. This paper exploits the novel concept of lazy cohomology generators and a fast and general algorithm to compute them. This graph-theoretic algorithm is much faster than all competing ones being the typical computational time in the order of seconds even with meshes formed by millions of elements. Moreover, this paper introduces the use of minimal boundary generators to ease human-based basis selection and to obtain representatives of generators with compact support. We are persuaded that this is the definitive solution to this long-standing problem.

VECTOR POTENTIAL ELECTROMAGNETICS WITH GENERALIZED GAUGE FOR INHOMOGENEOUS MEDIA: FORMULATION (Invited Paper)

The mixed vector and scalar potential formulation is valid from quantum theory to classical electromagnetics. The present rapid development in quantum optics applications calls for electromagnetic solutions that straddle both the quantum and classical physics regimes. The vector potential formulation using A and ' (or A-' formulation) is a good candidate to bridge these two regimes. Hence, there is a need to generalize this formulation to inhomogeneous media. A generalized gauge is suggested for solving electromagnetics problems in inhomogenous media that can be extended to the anistropic case. An advantage of the resulting equations is their absence of catastrophic breakdown at low-frequencies. Hence, the usual difierential equation solvers can be used to solve them over a wide range of scales and bandwidth. It is shown that the interface boundary conditions from the resulting equations reduce to those of classical Maxwell's equations. Also, the classical Green's theorem can be extended to such a formulation, resulting in an extinction theorem and a surface equivalence principle similar to the classical case. Moreover, surface integral equation formulations can be derived for piecewise homogeneous scatterers. Furthermore, the integral equations neither exhibit the low-frequency catastrophe nor the frequency imbalance observed in the classical formulation using E-H flelds. The matrix representation of the integral equation for a PEC (perfect electric conductor) scatterer is given.

Scalar Potential Formulations for Magnetic Fields Produced by Arbitrary Electric Current Distributions in the Presence of Ferromagnetic Bodies

In this paper it is shown how to analyze stationary or quasistationary magnetic fields due to arbitrary distributions of electric current in terms of single-valued scalar potentials. The total scalar potential in a specified region outside the magnetic material bodies, not containing electric currents, is obtained by superposing two single-valued Laplacian scalar potentials. One is produced by the given current distribution in an unbounded space and is determined from the free-space values of the normal component of the magnetic field intensity over the region boundary. The other one is defined in the entire region outside the magnetic bodies and is employed in order to satisfy the boundary conditions. When the magnetic bodies can be considered to be linear and homogeneous or piecewise homogeneous, with no electric current within them, the field inside the bodies can also be expressed in terms of Laplacian scalar potentials. If the material of the bodies can be approximated to be ideal ferromagnetic, then the exterior field problem is a Dirichlet problem and is solved independently of the field inside the bodies. The Neumann boundary condition for the interior field problem is provided from the solution of the exterior problem. Differential equation and integral equation formulations, as well as an illustrative application example, are presented. Since the resultant field is computed in terms of only single-valued scalar potentials, the solution techniques based on the formulations presented in this paper are considerably more efficient than those in usual solution methods.

Direct computation of current density to solve 3D electric conduction problems using facet elements with FEM

In this paper, an original alternative method to compute the current densityJ in two steps with the FEM is proposed, called the J0-T formulation. The goal of this formulation is to respect the conservation of the normal component of the current density. The first step computes an initial source current densityJ0, and the second step computes an electric vector potential T. Finally, the current densityJ is computed usingJ=J0+curlT. The first step ensures the conservation of the normal component ofJ0 between the facets of the mesh. To reach this condition, J0 is computed with facet finite elements. However, J0 is not unique, which is why a second step is added. In addition to the conditions of the first step, the second step corrects the tangential component ofJ. To reach these conditions, T is computed with edge finite elements. The J0-T formulation is then validated on a right angle conductor with section and direction changes and on a 21-turn copper coil. A conservation of the current is obtained with this formulation that is better than the classical electric scalar potential formulation. Copyright © 2013 John Wiley & Sons, Ltd.

A hybrid method BEM-NRM for magnetostatics problems

This article proposes a new method to solve non-linear magnetostatic problems applied to the modeling of electromagnetic devices. A reduced scalar potential formulation is presented and solved by a hybrid BEM-NRM. It has been implemented and tested in a software called MaGot. We will show that the computation time of this new method is very low. Thus, it could be easily and efficiently used for the pre-sizing of actuators.

A Hybrid Boundary Element Method-Reluctance Network Method for Open Boundary 3-D Nonlinear Problems

A scalar potential formulation coupling a hybrid boundary element method and a reluctance network method is presented to solve 3-D nonlinear magnetostatic problems. The computation times and the accuracy of this approach are compared with those of the finite element method in the context of force and torque evaluations. It is shown that this method could efficiently be used for the presizing of actuators. Discussion concerning the advantages of this approach compared to a pure reluctance network method is also presented.

Modeling of Magneto-Mechanical Coupling Using Magnetic Volume Integral and Mechanical Finite-Element Methods

The magneto-mechanical coupling with deformable bodies is studied using different numerical methods for both physics. The magnetic behavior is considered linear and without magnetostrictive effects; then equations of the magneto-mechanical coupling can be solved separately. A volume integral method using the magnetic scalar potential formulation is used. This method is particularly well suited to compute fields without meshing the air domain. A local application of the virtual work principle, adapted to the integral formulation, is used to compute the nodal magnetic force distribution at the origin of the magneto-mechanical coupling with deformable bodies. An advantage of this approach is that the same mesh can be used by both methods during the resolution steps.

Multilevel fast multipole method based on a potential formulation for 3D electromagnetic scattering problems

A combination of the multilevel fast multipole method (MLFMM) and boundary element method (BEM) can solve large scale photonics problems of arbitrary geometry. Here, MLFMM-BEM algorithm based on a scalar and vector potential formulation, instead of the more conventional electric and magnetic field formulations, is described. The method can deal with multiple lossy or lossless dielectric objects of arbitrary geometry, be they nested, in contact, or dispersed. Several examples are used to demonstrate that this method is able to efficiently handle 3D photonic scatterers involving large numbers of unknowns. Absorption, scattering, and extinction efficiencies of gold nanoparticle spheres, calculated by the MLFMM, are compared with Mie's theory. MLFMM calculations of the bistatic radar cross section (RCS) of a gold sphere near the plasmon resonance and of a silica coated gold sphere are also compared with Mie theory predictions. Finally, the bistatic RCS of a nanoparticle gold-silver heterodimer calculated with MLFMM is compared with unmodified BEM calculations.

A Priori Error Indicator in the Transformation Method for Problems With Geometric Uncertainties

To solve stochastic problems with geometric uncertainties, one can transform the original problem in a domain with stochastic boundaries and interfaces to a problem defined in a deterministic domain with uncertainties in the material behavior. The latter problem is then discretized. There exist infinitely many random mappings that lead to identical results in the continuous domain but not in the discretized domain. In this paper, an a priori error indicator is proposed for electromagnetic problems with scalar and vector potential formulations. This leads to criteria for selecting random mappings that reduce the numerical error. In an illustrative numerical example, the proposed a priori error indicator is compared with an a posteriori estimator for both potential formulations.

Non-Conforming Sliding Interfaces for Relative Motion in 3D Finite Element Analysis of Electrical Machines by Magnetic Scalar Potential Formulation Without Cuts

This paper discusses non-conforming sliding interfaces for motion in combination with a magnetic scalar potential formulation. Lagrange multiplier are used to implement the relative motion of stator and rotor. The utilization of the specific Lagrange multiplier approach implies the application of a magnetic scalar potential formulation in 3D Finite Element (FE) modeling of electrical machines because up to the present a canonical definition of biorthogonal basis functions for the magnetic vector potential is not available.