3D Eddy Currents Research Articles

Data-driven simulation of transient fields in air–coil magnets for accelerators

Time-varying fields in fast-ramping magnets for accelerators are difficult to compute in the range of accuracy required for magnet operation. This is due to the complexity of the dynamic phenomena such as hysteresis and 3D eddy currents. On the other hand, magnetic measurements that intercept all these physical phenomena are often limited to a subset of excitation cycles and restricted spatial domains. The measurement results are therefore difficult to extrapolate without a validated physical model of the device.This paper proposes measurement-updated field simulations to characterize dynamic effects in accelerator magnets. The main idea is to construct a reduced-order model, whose variables are retrievable from measurements by means of a state estimator, and to update the model by minimizing the error between simulations and measurements. The proposed method is applied to a linear, time-transient electromagnetic-field problem of an air–coil corrector magnet with aluminium collars. The proposed method is a first step towards a hybrid twin of an accelerator magnet.

Calculation of 3D transient Eddy current by the face‐smoothed finite element–boundary element coupling method

To calculate the distribution of the magnetic field and eddy current density on the surface of an aluminium plate, a method that couples the face-smoothed finite element method (FS-FEM) to the boundary element method (BEM) is proposed in this study. This method combines the advantages of the FS-FEM and BEM, which can rapidly and accurately calculate the distributions of vertical magnetic field and eddy current field on the surface of an aluminium plate. The structural parameters and material properties of the coil and aluminium plate are considered. An accurate three-dimensional calculation model is established. Then, the vertical magnetic field and eddy current field distributions are calculated in this study. In the case of the same grid density, compared with the finite element–boundary element coupling algorithm, the simulation results show that the FS-FEM and the boundary element coupling method have obvious advantage in improving the calculation accuracy. The maximum relative error between the calculated results and measured results is only 4%. The proposed method in this study is available for reference to the transient open-domain eddy current field analysis.

Distributed current source modeling method for 3D eddy current problem in magnetic conductor with discrete state-space J-φ formulation

This paper presents a distributed-current-source (DCS) modeling method to analyze the eddy-current (EC) induced in an electrically conductive material with a weak magnetic permeability. Formulated in state space to solve for an unknown EC density J and electric potential φ, the method decomposes the conductor into elemental volume/surface current sources to model the eddy-current and magnetization effects. The closed-form solutions to a three-dimension (3D) EC problem with geometrical boundary effects are derived and verified by comparing with finite-element methods and published data, both numerically and experimentally. An element-refinement method, which accounts for the skin-depth effects and non-uniform boundary EC, is proposed and validated. The DCS modeling method has been applied to a field-reconstruction application for parametric analysis demonstrating its effectiveness and efficiency.

3D eddy currents computation by BEM using the modified magnetic vector potential and the reduced magnetic scalar potential

AbstractA new boundary element method (BEM) formulation has been developed for the computation of 3D eddy currents in both magnetic and conductive regions. The modified magnetic vector potential A∗ in the active regions and the reduced magnetic scalar potential φ in the air are used as unknown variables. The formulation is limited to low‐frequency magneto‐harmonic problems associated to linear, homogeneous, isotropic, and simply‐connected regions. In such a configuration, it leads to accurate results with a reduced number of degrees of freedom located on a surface mesh delimiting the air/material interface.

A novel code for the simulation of plasma equilibrium and evolution implemented in ANSYS

The simulation of the plasma equilibrium and its evolution is important for the study of plasma physics and for the correct design of fusion devices. For this purpose, a novel code, based on the solution of the Grad-Shafranov equation, has been fully implemented in ANSYS. It exploits the finite element method using the magnetic potential vector formulation. In this approach plasma pressure and current density profiles are described by means of two main parameters: the internal plasma inductance li and the poloidal beta βp. Several ITER equilibria of limiter and diverted plasma configurations have been reproduced and benchmarked with validated codes (e.g. DINA). Being fully implemented in ANSYS this code will easily allow the coupling of the 2D axisymmetric plasma equilibrium equations, describing the plasma behavior, with the full 3D eddy currents equations, describing the surrounding three-dimensional structures. Moreover it will also allow the coupling with the ANSYS mechanical module to perform self-consistent coupled electro-mechanical analyses.

3D Modeling of the Magnetization of Superconducting Rectangular-Based Bulks and Tape Stacks

In recent years, numerical models have become popular and powerful tools to investigate the electromagnetic behavior of superconductors. One domain where this advances are most necessary is the 3D modeling of the electromagnetic behavior of superconductors. For this purpose, a benchmark problem consisting of superconducting cube subjected to an AC magnetic field perpendicular to one of its faces has been recently defined and successfully solved. In this work, a situation more relevant for applications is investigated: a superconducting parallelepiped bulk with the magnetic field parallel to two of its faces and making an angle with the other one without and with a further constraint on the possible directions of the current. The latter constraint can be used to model the magnetization of a stack of high-temperature superconductor tapes, which are electrically insulated in one direction. For the present study three different numerical approaches are used: the Minimum Electro-Magnetic Entropy Production (MEMEP) method, the $H$-formulation of Maxwell's equations and the Volume Integral Method (VIM) for 3D eddy currents computation. The results in terms of current density profiles and energy dissipation are compared, and the differences in the two situations of unconstrained and constrained current flow are pointed out. In addition, various technical issues related to the 3D modeling of superconductors are discussed and information about the computational effort required by each model is provided. This works constitutes a concrete result of the collaborative effort taking place within the HTS numerical modeling community and will hopefully serve as a stepping stone for future joint investigations.

3D-FEM computation and experimental study of eddy currents for characterization of surface cracks

PurposeThe purpose of this paper is to present a new approach for computing by measuring and testing related 3D Eddy currents. In the process, a magnetic vector is formulated from the theoretical setup and obtained results from relevant applications are checked for the consistency of the theory. Besides, cracks detection as well as its propagation is studied through the two parameters: SIF and J-integral. A simulation by a numerical approach using finite-element discretization of 3D governing equations is employed to detect damaged zones and cracks. This approach has been used in the aircraft industry to control cracks. Besides, it makes it possible to highlight the defects of parts while preserving the integrity of the controlled products. Obtained results are compared and agreed with those of other researchers.Design/methodology/approachFinite-element discretization of 3D for solving problem in eddy current testing is presented in this paper. The main idea is the introduction of categorization for the shape reconstruction using the non-destructive testing by 3D-EC. The results are presented for a simple eddy current problem using the finite-element method as an experimental support.FindingsIn this research work, results of the various cases of simulation have been obtained. From these results of various boxes of simulation, one can conclude that the calculation of the impedance in only one point is not enough to confirm the presence or the absence of a defect for materials. Then, this confirmation leads us to the calculation of the impedance along the plate. The detection of an external defect requires the energy of the sensor by high frequencies .The position of defect (internal, in the middle, external) has a large effect on the impedance. The use of this sensor type in industrial application is frequent because of its precision (minimal error) and its low costs. The major disadvantage of this type of sensor lies in the fact that it is unable to detect a defect.Originality/valueThis paper fulfills an identified need to detect cracks in materials and eventually to study their propagation.

Equilibrium reconstruction with 3D eddy currents in the Lithium Tokamak eXperiment

Axisymmetric free-boundary equilibrium reconstructions of tokamak plasmas in the Lithium Tokamak eXperiment (LTX) are performed using the PSI-Tri equilibrium code. Reconstructions in LTX are complicated by the presence of long-lived non-axisymmetric eddy currents generated by a vacuum vessel and first wall structures. To account for this effect, reconstructions are performed with additional toroidal current sources in these conducting regions. The eddy current sources are fixed in their poloidal distributions, but their magnitude is adjusted as part of the full reconstruction. Eddy distributions are computed by toroidally averaging currents, generated by coupling to vacuum field coils, from a simplified 3D filament model of important conducting structures. The full 3D eddy current fields are also used to enable the inclusion of local magnetic field measurements, which have strong 3D eddy current pick-up, as reconstruction constraints. Using this method, equilibrium reconstruction yields good agreement with all available diagnostic signals. An accompanying field perturbation produced by 3D eddy currents on the plasma surface with a primarily n = 2, m = 1 character is also predicted for these equilibria.

3D Eddy Current Testing by FEM for Detection of Crack’s in Materials

In this paper, we present a nondestructive Testing by sensor Eddy current is used as a tool to control cracks and micro-cracks in materials. A new method for computing by measuring and testing related 3D Eddy currents is considered. In the process, a Potential Magnetic Vector is provided on the basis of formulations taken from the theoretical set up. Thus, results of relevant applications are obtained to check the theory consistency. A simulation by a numerical approach using Finite element discretization of 3-D Eddy Current governing equations is employed to detect cracks and damaged zones in materials and eventually to study their propagation.

Multi-scale FEM and magnetic vector potential A for 3D eddy currents in laminated media

Purpose – The purpose of this paper is an accurate computation of eddy currents in laminated media with minimal computer resources. Design/methodology/approach – Modeling each laminate of the laminated core of electrical devices requires prohibitively many finite elements (FEs). To overcome this restriction a higher order multi-scale FE method with the magnetic vector potential A has been developed to cope with 3D problems considering edge effects. Findings – The multi-scale FE approach facilitates an accurate simulation of the eddy current losses with minimal computer resources. Numerical simulations demonstrate a remarkable accuracy and low computational costs. The effect of regularization on the results is shown. Practical implications – The eddy current losses are of great interest in the design of electrical devices with laminated cores. Originality/value – The multi-scale FE approach takes also into account of the edge effects in 3D.

An Efficient Eddy Current Model for Nonlinear Maxwell Equations with Laminated Conductors

In this paper, we propose a new eddy current model for the nonlinear Maxwell equations with laminated conductors. Direct simulation of three-dimensional (3D) eddy currents in grain-oriented (GO) silicon steel laminations is very challenging since the coating film over each lamination is only several microns thick and the magnetic reluctivity is nonlinear and anisotropic. The system of GO silicon steel laminations has multiple sizes, and the ratio of the largest scale to the smallest scale can amount to $10^6$. The new model omits coating films and thus reduces the scale ratio by 2--3 orders of magnitude. It avoids very fine or very anisotropic mesh in coating films and can reduce computations greatly in computing 3D eddy currents. We establish the well-posedness of the new model and prove the convergence of the solution of the original problem to the solution of the new model as the thickness of coating films tends to zero. The new model is validated by finite element computations of an engineering benchmark problem---Team Workshop Problem $21^c$--M1.

Transient heating effects in electromagnetic launchers with complex geometries: A 3D hybrid FE/BE analysis

A time-stepping finite-element (FE) and boundary-element (BE) coupling method was implemented to include 3D transient eddy current under electromagnetic heating effects in electromagnetic launchers. Lagrangian formulations were used to model electromagnetic and thermal diffusive processes with moving conductors. The finite-element formulation was employed for the general diffusion equation in a bounded conductor region including rail and armature; while, in the exterior region, the boundary element method was developed for Laplace's equation. The coupled boundary was employed directly on the surface of conductors and no special treatment on boundary was required. The electromagnetic and temperature fields were coupled through iterative calculations. This method results in considerable time saving in modeling as it eliminates the surrounding air region.

Convergence of Adaptive Edge Element Methods for the 3D Eddy Currents Equations

We consider an Adaptive Edge Finite Element Method (AEFEM) for the 3D eddy currents equations with variable coe‐cients using a residual-type a posteriori error estimator. Both the components of the estimator and certain oscillation terms, due to the occurrence of the variable coe‐cients, have to be controlled properly within the adaptive loop which is taken care of by appropriate bulk criteria. Convergence of the AEFEM in terms of reductions of the energy norm of the discretization error and of the oscillations is shown. Numerical results are given to illustrate the performance of the AEFEM.

Multigrid for transient 3D eddy current analysis

A multigrid (MG) algorithm for the calculation of transient 3D eddy currents is presented. The finite element method with edge elements is used for the discretization of the problems. For the MG smoother, a multiplicative Schwarz algorithm and a special Gauss‐Seidel iteration are compared for different formulations of the eddy‐current problem. It appears that, using the right combination of smoother and eddy‐current formulation, the computation times can be substantially reduced.

Effect of 3d eddy currents on stray field in plasma breakdown region

Abstract 2D axisymmetrical code TRANSMAC was developed to provide a detailed analysis of transients during plasma initiation stage in ITER. Due to nonaxisymmetrical effects (holes and ports in blanket and vacuum vessel, etc.) a 2D model needs to be corrected. An approach to a 2D model correction on the basis of the 3D code TYPHOON (A.V. Belov et al., Fusion Engineering and Design, 31, 167–180, 1996) is verified and discussed.

Gauged current vector potential and reentrant corners in the FEM analysis of 3D eddy currents

The nodal finite element realization of the T-/spl Omega/ method involving a gauged current vector potential, T, is shown to yield erroneous results if applied to 3D eddy current problems with reentrant corners in the conducting region. The reason for the problem is pinpointed and a remedy is suggested.

A FEM formulation to treat 3D eddy currents in laminations

A method is presented to compute the eddy current losses in laminated nonsaturated iron cores based on the finite element method. In the first step, a three-dimensional model of the core treated as a bulk of anisotropic material with low conductivity in the direction normal to the lamination is analyzed. Each laminate is taken into account in the second step by writing the field components in it as fundamental solutions of the diffusion equation so that the averages of the tangential electric and magnetic field components over the thickness of the sheet equal the averages obtained in the first step.

Surface impedances, BIEM and FEM coupled with 1D non-linear solutions to solve 3D high frequency eddy current problems

The BIEM coupled with surface impedances gives good results at a low cost for 3D high frequency eddy currents calculation with linear materials. We review the possible extensions to non-linear materials. We propose (and test in 3D) an original method. Based on 1D computations of the complex surface impedance. Results are applied to induction hardening processes. This paper presents essentially validations of the method of non-linear surface impedances.

3D eddy currents modelling by means of a particular reduced scalar potential technique

The paper uses a particular reduced scalar potential technique enabling 3D eddy currents modelling in cases where the currents are developed in well defined paths (such as shading rings, cages etc.). This method involves a scalar potential in the whole solution domain and a single component vector quantity in the conducting parts. Experimental verification is performed by comparing computed and measured forces, developed in a contactor consisting of an E-shape electromagnet with shading rings.

Mechanical forces simulation and stress analysis of the TEXTOR vacuum vessel during plasma disruption under 3d eddy currents load

The TEXTOR vacuum vessel represents a steel torus shell with numerous radial and vertical ports. The induced eddy currents as well as electromagnetic forces in the vessel during plasma disruption have been calculated using the TYPHOON code. For the purposes of the stress analysis the vessel shells are modeled with shell elements. The bellows and flanges are built with 3D anisotropic solid elements. To apply the calculated electromagnetic forces to this model a special interface code has been developed. Stress analysis has been performed in two steps of loading in reference to symmetry and antisymmetry boundary conditions and the results have been superimposed.