Eddy Current Problems Research Articles

A discrete geometric formulation for eddy-current problems in fusion devices

All thermonuclear controlled fusion devices under construction or design have such high performances to require a special care in the dimensioning of various components, specifically from the electromagnetic point of view. To this purpose, it is fundamental to develop models which are both accurate (i.e. able to describe the physical phenomena) and predictive (i.e. useful not only to explain what happens in running experiments, but also to reliably extrapolate to other range of parameters).The dynamics of fusion plasmas is often conveniently described by Magneto-Hydro-Dynamics (MHD) equations, which predict that some unstable evolution modes may exist. On the other hand, the complexity of the intrinsically 3D model of the interactions between a realistic unstable plasma, the surrounding passive structures (important to guarantee a good MHD stability) and the active conductors (coils) require the numerical solution of challenging electromagnetic problems.In this work a discrete geometric formulation for eddy-current problems in the frequency domain is developed; the magnetic fields produced by a typical active coil system is calculated in the presence of 3D conductive structures.

A symmetric BEM–FEM method for an axisymmetric eddy current problem

This paper deals with the numerical solution of the time-harmonic eddy current model in an axisymmetric unbounded domain. To this end, a new symmetric BEM–FEM formulation is derived and also analyzed. Moreover, error estimates for the corresponding discretization are proven.

Frequency Domain Decomposition of 3-D Eddy Current Problems in Steel Laminations of Induction Machines

A novel method to reduce the computational burden of 3-D eddy current problems in steel laminations of induction machines is presented. Instead of a nonlinear time-stepping analysis, a set of linear time-harmonic simulations for known frequencies is used. The linear model considers the machines' saturation by a reluctivity obtained from a nonlinear 2-D field solution. Solving the time-harmonic problems in parallel results in a fast method. Comparisons with a time-stepping approach confirm the validity of the proposed method. Different eddy current formulations are investigated to accelerate the solution of the time-harmonic problems.

Comparison of Residual and Hierarchical Finite Element Error Estimators in Eddy Current Problems

The finite element computation of eddy current problems introduces numerical error. This error can only be estimated. Among all error estimators (EEs) already developed, two estimators, called residual and hierarchical EEs, proven to be reliable and efficient, are theoretically and numerically compared. Both estimators show similar behaviors and locations of the error.

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.

Contact Temperature Prediction in Three-Phase Gas-Insulated Bus Bars With the Finite-Element Method

Degradation of the electric contact in gas-insulated bus bars (GIBs) is closely related to the overheating phenomenon occurring at the contacts. This paper employs the finite-element method to solve the 3-D coupled eddy current, fluid, and thermal problems in a three-phase GIB. In the model, the contact resistance is measured and described as a resistor between the conductor and the contact. Computational fluid dynamics is used to predict the temperature rise of the components in the GIB. The temperature-dependent thermal properties of SF6 gas and air are considered in the thermal analysis. The developed methodology is implemented on a 126 kV three-phase GIB prototype and validated with the experimental temperature rise results.

Two-Scale Homogenization of the Nonlinear Eddy Current Problem With FEM

An efficient and accurate computation of the eddy current losses in laminated iron cores of electric devices is of great interest. Modeling each laminate individually by the finite element method requires many elements and leads to large systems of equations. Homogenization represents a promising method to overcome this problem. A two-scale finite element method is proposed to efficiently compute the eddy current losses in laminated media with nonlinear material properties. A rather coarse finite element grid suffices to approximate the losses accurately. The method based on the magnetic vector potential is described. The laminates are basically considered individually in the finite element assembly taking account of the nonlinearity. This is computationally very intensive. Some adapted integration rules are introduced and studied to accelerate the finite element assembly. The accuracy and the computational costs of the proposed method are shown by a numerical example.

Linear Representation of Saturation Characteristics Associated With Eddy Currents in Ferromagnetic Materials

Based on the linear field theory and the limiting nonlinear theory, respectively, the eddy current losses in thick iron plates and thin laminations are formulated. Two equivalent permeability calculation models are developed to take into account the saturation effects occurring in the corresponding ferromagnetic materials. By enabling the approximate treatment of the magnetic field saturation, the nonlinear eddy current problem is reduced to the linear one. The proposed approach to the linearization of the field analysis is investigated by comparing the results of the linear analysis using the equivalent permeability with those of the nonlinear finite element analysis and experimental data.

Voltage Excitation in Coil Rings Using Magnetically Coupled Inductor/Load For Axisymmetric Geometry

We present a method for solving eddy current problems for axisymmetric configurations of a simply connected induction coil where voltage excitation in coil rings is prescribed and the load is changing in time. We describe an algorithm for the electromagnetic induction problem which provides current continuity in the coil inductor and adapts the configuration to the generator mode-voltage limitation or power limitation, with only two runs for each time step. We take advantage of the electric-field approximation computed at the former time step to design the potentials which guarantee current continuity.

Asymptotic Boundary Element Methods for Thin Conducting Sheets in Two Dimensions

Shielding sheets are commonly used in the protection of electronic devices. With their large aspect ratios they become a serious issue for the direct application of BEMs due to the occurring almost singular integrals. Impedance transmission conditions (ITCs) allow for boundary element formulations where only sheet midline has to be triangulated. We propose and analyze BEMs of second kind for ITCs of two different types for the time-harmonic eddy-current problem in 2-D. In numerical experiments, we study the effect on the magnetic shielding efficiency of the discretization error due the boundary element discretization and the modeling error due to the impedance transmission condition.

Identification of Random Material Parameters in Eddy Current Problems

Stochastic identification of material parameters in eddy current problems is presented. A method has been developed for computing eddy currents in materials with random magnetic proper ...

A DIFFERENTIAL DPSM BASED MODELING APPLIED TO EDDY CURRENT IMAGING PROBLEMS

This paper deals with an innovative implementation of a semi-analytical modeling method, called the Distributed Points Source Method (DPSM), in the case of an eddy current problem. The DPSM has already shown great potentialities for the versatile and computationally efficient modeling of complex electrostatic, electromagnetic or ultrasonic problems. In this paper, we report a new implementation of the DPSM, called differential DPSM, which shows interesting prospects for the modeling of complex eddy current problems such as met in the non-destructive imaging of metallic parts. In this paper, the used eddy current imaging device is firstly presented. It is composed of an eddy current (EC) inducer and a magneto optical set-up used to translate the magnetic field distribution appearing at the surface of the imaged part, into a recordable optical image. In this study, the device is implemented for the time-harmonics (900 Hz) imaging of a two-layer aluminum based assembly, featuring surface-breaking and buried defects. Then, the basics of the time-harmonics DPSM modeling are recalled, and the differential approach is presented. It is implemented for the modeling of the interactions of the eddy current imaging device with the considered flawed assembly in the same operating conditions as the experimental implementation. The comparison between experimental and computed data obtained for millimetric surface and buried defects is presented in the form of complex magnetic cartographies and Lissajous plots. The obtained results show good agreement and open the way to the modeling of complex EC problems. Furthermore, the low computational complexity of the differential DPSM modelings makes it promising to consider for the solving of EC inverse problems.

Fast convolution quadrature based impedance boundary conditions

We consider an eddy current problem in time-domain relying on impedance boundary conditions on the surface of the conductor(s). We pursue its full discretization comprising (i) a finite element Galerkin discretization by means of lowest order edge elements in space, and (ii) temporal discretization based on Runge–Kutta Convolution Quadrature (CQ) for the resulting Volterra integral equation in time. The final algorithm also involves the fast and oblivious approximation of CQ.For this method we give a comprehensive convergence analysis and establish that the errors of spatial discretization, CQ and of its approximate realization add up to the final error bound.

A P-HIERARCHICAL ERROR ESTIMATOR FOR A FEM-BEM COUPLING OF AN EDDY CURRENT PROBLEM IN ℝ3-DEDICATED TO PROFESSOR WOLFGANG L. WENDLAND ON THE OCCASION OF HIS 75TH BIRTHDAY

We extend a p-hierarchical decomposition of the second degree finite element space of N<TEX>$\acute{e}$</TEX>d<TEX>$\acute{e}$</TEX>lec for tetrahedral meshes in three dimensions given in [1] to meshes with hexahedral elements, and derive p-hierarchical decompositions of the second degree finite element space of Raviart-Thomas in two dimensions for triangular and quadrilateral meshes. After having proved stability of these subspace decompositions and requiring certain saturation assumptions to hold, we construct a local a posteriori error estimator for fem and bem coupling of a time-harmonic electromagnetic eddy current problem in <TEX>$\mathbb{R}^3$</TEX>. We perform some numerical tests to underline reliability and efficiency of the estimator and test its usefulness in an adaptive refinement scheme.

NUMERICAL SOLUTION OF A TRANSIENT NONLINEAR AXISYMMETRIC EDDY CURRENT MODEL WITH NONLOCAL BOUNDARY CONDITIONS

This paper deals with an axisymmetric transient eddy current problem in conductive nonlinear magnetic media. This means that the relation between the magnetic field and the magnetic induction, the so-called H-B curve, is nonlinear. The source of the problem is the magnetic flux across a meridian section of the device, which leads to a parabolic nonlinear problem with nonlocal boundary conditions. First, by applying some abstract results, we prove the existence and uniqueness of the solution to a weak formulation written in terms of the magnetic field. Then, we compute the numerical solution of the problem by using a finite element method combined with a backward Euler time discretization. We derive error estimates in appropriate norms for both the semidiscrete (in space) and the fully discrete problems. Finally, we show numerical results which allow us to confirm the theoretical estimates and to assess the performance of the proposed scheme.

Finite Element Analysis of Nondestructive Testing Eddy Current Problems With Moving Parts

We present the logical expressions (LE) approach that allows fast computation of three-dimensional eddy current problems, including parts in motion. The approach applies time-dependent logical expressions to describe moving parts of the model on a fixed computational grid. The study is motivated by a novel nondestructive testing technique called Lorentz force eddy current testing (LET), which enables the detection of defects lying deep inside a conducting material. Depending on the definition of the frame of reference, we present two different implementations of the LE approach referred to as 1) moving magnet approach, and 2) moving defect approach. In order to demonstrate the advantages of the LE approach, we compare its results with the sliding mesh technique. The validation of the obtained results with experiments is also presented.

Unique shape detection in transient eddy current problems

Transient excitation currents generate electromagnetic fields which in turn induce electric currents in proximal conductors. For slowly varying fields, this can be described by the eddy current equations, which are obtained by neglecting the dielectric displacement currents in Maxwell’s equations. The eddy current equations are of parabolic–elliptic type. In insulating regions, the field instantaneously adapts to the excitation (elliptic behavior), while in conducting regions, this adaptation takes some time due to the induced eddy currents (parabolic behavior). The subject of this work is to locate the conductor(s) surrounded by a non-conducting medium from electromagnetic measurements, i.e., from knowledge of the excitation currents and measurements of the corresponding electromagnetic fields. We show that the conductors are uniquely determined by the measurements, and give an explicit criterion to decide whether a given point is inside or outside the conducting domain. This criterion serves as a base for rigorously justified non-iterative numerical reconstruction strategies.

A (T,ψ)‐ψe decoupled scheme for a time‐dependent multiply‐connected eddy current problem

The aim of this paper is to develop a fully discrete (T,ψ)‐ψe finite element decoupled scheme to solve time‐dependent eddy current problems with multiply‐connected conductors. By making ‘cuts’ and setting jumps of ψe across the cuts in nonconductive domain, the uniqueness of ψe is guaranteed. Distinguished from the traditional T‐ ψ method, our decoupled scheme solves the potentials T and ψ‐ ψe separately in two different simple equation systems, which avoids solving a saddle‐point equation system and leads to a remarkable reduction in computational efforts. The energy‐norm error estimate of the fully discrete decoupled scheme is provided. Finally, the scheme is applied to solve two benchmark problems—TEAM Workshop Problems 7 and IEEJ model. Copyright © 2013 John Wiley &amp; Sons, Ltd.

An Operator Splitting Finite Element Method for Eddy-Current Field Analysis in High-Speed Rotating Solid Conductors

Standard Galerkin finite element method (SGFEM) cannot solve eddy-current problems with dominated motional terms accurately and effectively. A numerical method for analyzing eddy-current magnetic field in high-speed rotating solid conductors using operator splitting finite element method (OSFEM) is presented. An artificial example and a benchmark transient eddy-current field problem, the problems 30A and 30B in testing electromagnetic analysis methods (TEAM) workshop, are used to test the accuracy and performance of the method. It is demonstrated that the numerical solution using the proposed method is nonoscillatory even for convection dominated cases. Besides, the calculated torque values are more accurate using the proposed OSFEM when compared with those obtained using SGFEM. It is concluded that the OSFEM can solve eddy-current problems with high-speed rotating conductors efficiently and accurately.

ERMES: A nodal-based finite element code for electromagnetic simulations in frequency domain

In this work we present a new finite element code in frequency domain called ERMES. The novelty of this computational tool rests on the formulation behind it. ERMES is the C++ implementation of a simplified version of the weighted regularized Maxwell equation method. This finite element formulation has the advantage of producing well-conditioned matrices and the capacity of solving problems in the low (quasi-static) and high frequency regimens. As a consequence of this versatility, ERMES has been applied successfully to microwave engineering, antenna design, electromagnetic compatibility and eddy currents problems. This paper describes the main features of ERMES and explains how to use this numerical tool for computing electromagnetic fields in frequency domain. Program summaryProgram title: ERMESCatalogue identifier: AEPV_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEPV_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 223528No. of bytes in distributed program, including test data, etc.: 35954099Distribution format: tar.gzProgramming language: C++.Computer: Any computer with Microsoft Windows (32-bits or 64-bits) installed.Operating system: Microsoft Windows 32-bits or 64-bits.RAM: Problem dependent. See [1] for examples of computational performance.Classification: 10.External routines:GiD [5] is used for geometrical modeling, data input, meshing and visualization of results.Nature of problem:Time-harmonic Maxwell equations.Solution method:Finite element formulation based on the weighted regularized Maxwell equation method [2, 1, 3, 4].Additional comments:!!!!! The distribution file for this program is approximately 35 Mbytes and therefore is not delivered directly when the download or Email is requested. Instead a html file giving details of how the program can be obtained is sent. !!!!!Running time:Problem dependent (see [1]).