3D Eddy Current Problems Research Articles

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.

Boundary element method for 3D conductive thin layer in eddy current problems

PurposeThin conducting sheets are used in many electric and electronic devices. Solving numerically the eddy current problems in presence of these thin conductive sheets requires a very fine mesh which leads to a large system of equations, and it becomes more problematic in case of higher frequencies. The purpose of this paper is to show the numerical pertinence of equivalent models for 3D eddy current problems with a conductive thin layer of small thickness e based on the replacement of the thin layer by its mid-surface with equivalent transmission conditions that satisfy the shielding purpose, and by using an efficient discretization using the boundary element method (BEM) to reduce the computational work.Design/methodology/approachThese models are solved numerically using the BEM and some numerical experiments are performed to assess the accuracy of the proposed models. The results are validated by comparison with an analytical solution and a numerical solution by the commercial software Comsol.FindingsThe error between the equivalent models and analytical and numerical solutions confirms the theoretical approach. In addition to this accuracy, the computational work is reduced by considering a discretization method that requires only a surface mesh.Originality/valueBased on a hybrid formulation, the authors present briefly a formal derivation of impedance transmission conditions for 3D thin layers in eddy current problems where non-conductive materials are considered in the interior and the exterior domain of the sheet. BEM is adopted to discretize the problem as there is no need for volume discretization.

Multilevel adaptive cross approximation for efficient modeling of 3D arbitrary shaped eddy current NDE problems

In this article, the multilevel adaptive cross approximation (MLACA) algorithm is presented to accelerate the boundary element method (BEM) for eddy current nondestructive evaluation (NDE) 3D problems involving arbitrary shapes. The Stratton-Chu formula, which does not have the low frequency breakdown issue, has been selected for modeling. The equivalent electric and magnetic surface currents are expanded with Rao-Wilton-Glisson (RWG) vector basis functions while the normal component of the magnetic field is expanded with pulse basis functions. The MLACA compresses the rank deficient matrices with the ACA and the butterfly algorithm. We improve the efficiency of MLACA by truncating the integral kernels after a certain distance and applying the multi-stage (level) algorithm adaptively based on the criteria for different operators to further decrease the memory and CPU time requirements while keeping almost the same accuracy comparing with the traditional MLACA. The proposed method is especially helpful to deal with the large solution domain issue of the BEM for eddy current problems. Numerical predictions are compared with the analytical, the semi-analytical predictions and the experimental results for 3D eddy current NDE problems of practical interest to demonstrate the robustness and efficiency of the proposed method.

A nonlinear magnetic circuit model for periodic eddy current problems using T,ϕ-ϕ formulation

PurposeA transformer model is used as a benchmark for testing various methods to solve 3D nonlinear periodic eddy current problems. This paper aims to set up a nonlinear magnetic circuit problem to assess the solving procedure of the nonlinear equation system for determining the influence of various special techniques on the convergence of nonlinear iterations and hence the computational time.Design/methodology/approachUsing the T,ϕ-ϕ formulation and the harmonic balance fixed-point approach, two techniques are investigated: the so-called “separate method” and the “combined method” for solving the equation system. When using the finite element method (FEM), the elapsed time for solving a problem is dominated by the conjugate gradient (CG) iteration process. The motivation for treating the equations of the voltage excitations separately from the rest of the equation system is to achieve a better-conditioned matrix system to determine the field quantities and hence a faster convergence of the CG process.FindingsIn fact, both methods are suitable for nonlinear computation, and for comparing the final results, the methods are equally good. Applying the combined method, the number of iterations to be executed to achieve a meaningful result is considerably less than using the separated method.Originality/valueTo facilitate a quick analysis, a simplified magnetic circuit model of the 3D problem was generated to assess how the different ways of solutions will affect the full 3D solving process. This investigation of a simple magnetic circuit problem to evaluate the benefits of computational methods provides the basis for considering this formulation in a 3D-FEM code for further investigation.

A novel approach to deal with periodic boundary conditions for 3D eddy current field problems

A novel approach to deal with periodic boundary conditions for 3D eddy current field problems

A T-ψ formulation with the penalty function term for the 3D eddy current problem in laminated structures

In this paper we present a new eddy current model by means of the T-ψ formulation with the penalty function term for the three-dimensional(3D) nonlinear eddy current equations with laminated conductors. By omitting the insulating films between the neighboring laminations, the approximate model reduces the scale ratio by 2–3 orders of magnitude. Then it avoids very fine or very anisotropic mesh in coating films and can save computational work greatly in numerical approximations. We study the weak formulations of the original and the approximate problem to obtain their well-posedness. Further, we prove the convergence for the solution of the new model to the solution of the original problem as the thickness of coating films approaches zero.

A novel approach for solving three dimensional eddy current problems in fusion devices

We present a technique to efficiently solve 3D eddy current problems in fusion devices, whose structures exhibit a geometrical symmetry. The proposed approach is based on the exploitation of symmetry via harmonic analysis and it is suitable to treat also problems where the field sources (plasma or external coils) do not share the domain symmetry. A simple test case is presented to describe the methodology. Then, a more complicated geometry is considered, which represents a realistic vacuum vessel of an ITER like fusion device (double layer structure with several portholes). The results are discussed for both axisymmetric and non-axisymmetric excitations in frequency domain.

Spectral stochastic finite element method for solving 3D stochastic eddy current problems

In this paper, a spectral stochastic finite element method (SSFEM) is described and applied to quantify uncertainties in an industrial eddy current non destructive testing (NDT) problem. A new iterative solver based on the fixed-point method is proposed to solve the large scale linear system. The efficiency of the suggested iterative solver to tackle the SSFEM computational complexity is demonstrated. The SSFEM results and the computation times obtained with 2 and 4 random variables are compared to those given by the anisotropic adaptive sparse grid non intrusive spectral projection (NISP). On this application, the SSFEM dedicated solver is competitive with the adaptive sparse grids strategy.

Optimal fixed‐point method for solving 3D nonlinear periodic eddy current problems

Purpose – The purpose of the paper is to present a method for efficiently obtaining the steady‐state solution of the quasi‐static Maxwell's equations in case of nonlinear material properties and periodic excitations.

Solution of the inverse problem of magnetic induction tomography (MIT)

Magnetic induction tomography (MIT) of biological tissue is used to reconstruct the changes in the complex conductivity distribution inside an object under investigation. The measurement principle is based on determining the perturbation ΔB of a primary alternating magnetic field B0, which is coupled from an array of excitation coils to the object under investigation. The corresponding voltages ΔV and V0 induced in a receiver coil carry the information about the passive electrical properties (i.e. conductivity, permittivity and permeability). The reconstruction of the conductivity distribution requires the solution of a 3D inverse eddy current problem. As in EIT the inverse problem is ill-posed and on this account some regularization scheme has to be applied. We developed an inverse solver based on the Gauss–Newton-one-step method for differential imaging, and we implemented and tested four different regularization schemes: the first and second approaches employ a classical smoothness criterion using the unit matrix and a differential matrix of first order as the regularization matrix. The third method is based on variance uniformization, and the fourth method is based on the truncated singular value decomposition. Reconstructions were carried out with synthetic measurement data generated with a spherical perturbation at different locations within a conducting cylinder. Data were generated on a different mesh and 1% random noise was added. The model contained 16 excitation coils and 32 receiver coils which could be combined pairwise to give 16 planar gradiometers. With 32 receiver coils all regularization methods yield fairly good 3D-images of the modelled changes of the conductivity distribution, and prove the feasibility of difference imaging with MIT. The reconstructed perturbations appear at the right location, and their size is in the expected range. With 16 planar gradiometers an additional spurious feature appears mirrored with respect to the median plane with negative sign. This demonstrates that a symmetrical arrangement with one ring of planar gradiometers cannot distinguish between a positive conductivity change at the true location and a negative conductivity change at the mirrored location.

Numerical solution of the general 3D eddy current problem for magnetic induction tomography (spectroscopy)

Magnetic induction tomography (MIT) is used for reconstructing the changes of the conductivity in a target object using alternating magnetic fields. Applications include, for example, the non-invasive monitoring of oedema in the human brain. A powerful software package has been developed which makes it possible to generate a finite element (FE) model of complex structures and to calculate the eddy currents in the object under investigation. To validate our software a model of a previously published experimental arrangement was generated. The model consists of a coaxial coil system and a conducting sphere which is moved perpendicular to the coil axis (a) in an empty space and (b) in a saline-filled cylindrical tank. The agreement of the measured and simulated data is very good when taking into consideration the systematic measurement errors in case (b). Thus the applicability of the simulation algorithm for two-compartment systems has been demonstrated even in the case of low conductivities and weak contrast. This can be considered an important step towards the solution of the inverse problem of MIT.

Preconditioning of the second‐kind boundary integral equations for 3D eddy current problems

AbstractStarting from the time‐harmonic Maxwell equations at low‐frequency eddy current approximation the H–ϕ formulation is presented. An equivalent system of boundary integral equations of the second kind on the conductor surface (resp. the conductor/dielectric) is derived. Discretizing these equations with a boundary element method (BEM) yields a block linear equation system \[ \left[\begin{array}{cc} A_{1} & B_{1} \\ \tfrac{\mu}{\mu_{0}}\tilde{B}_{2} & A_{2}\end{array} \right] \left( \begin{array}{c} j \\ \sigma \end{array} \right)=\left(\begin{array}{c} b_{1} \\ b_{2}\end{array} \right) \] with a fully populated stiffness matrix. This system is non‐symmetric and, for large μ/μ0 (of interest in practice) ill‐conditioned. Iterative solvers like GMRES converge very slowly, in general. We propose a new preconditioner which depends on μ. We present numerical results with a serial and parallel version of this preconditioner also on large industrial eddy current problems with compex geometry. The performance of preconditioned GMRES is found to be practically independent of μ/μ0. Copyright © 2001 John Wiley & Sons, Ltd.

Movement simulation in 3D eddy current transient problem

An algorithm for the simultaneous solving of the equations of the 3D transient electromagnetic field and equations of electric circuits of an electromagnetic device containing moving conducting parts has been elaborated. The A‐T formulation has been applied to the description of 3D eddy current problem. In order to solve the 3D problem, the iterative procedure, in which the 3D task is substituted with a sequence of 2D problems, has been applied. Different procedures for movement modelling have been considered. As an example, the dynamics of a linear induction motor have been investigated.

An adaptive-step time integration method applied to transient magnetic field problems

An adaptive-step time integration method is presented for solving 3D magnetic field and eddy current problems. It is based on an implicit Runge-Kutta method that is a well-known approach for the problems of differential-algebraic equations. The method has been applied to both linear and nonlinear cases and its advantages over the conventional B-method with fixed step are demonstrated.

On the use of the new edge based A/spl I.oarr/-A/spl I.oarr/, T/spl I.oarr/ formulation for the calculation of time-harmonic, stationary and transient eddy current field problems

Most papers concerning the calculation of 3D eddy current problems are using a combination of a vector potential and a scalar potential to solve the electromagnetic field in the eddy current regions. This paper uses the A/spl I.oarr/-A/spl I.oarr/, T/spl I.oarr/ formulation with both the magnetic vector potential A/spl I.oarr/ and the electric vector potential T/spl I.oarr/ in the eddy current regions. Since nodal vector potentials with continuous normal components have accuracy problems at interfaces of regions with different permeabilities, edge elements can be used for both potentials. The formulation is applied to the calculation of stationary eddy current field problems induced by motion, time-harmonic and transient eddy current field problems.

Solving 3D eddy current problems using second order nodal and edge elements

Several second order nodal and edge elements have been applied in a potential formulation to solve 3D eddy current problems. The asymmetry of the facet related functions in the edge element basis is discussed. A new basis is proposed. Application of a gauge condition for the uniqueness of vector potential is cumbersome in the case of high order elements. This work shows that the system converges without explicit gauge condition when using the bi-conjugate gradient method. The performance of different elements is compared through an example.

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.

3D eddy current computation with edge elements in terms of the electric intensity

A finite element formulation for the solution of 3D eddy current problems in terms of the electric intensity E is presented. A weak formulation, based on a Galerkin weighted residual procedure, is presented and edge elements, that impose only tangential continuity across element interfaces of the approximated field, are employed for the discretization of the problem with the finite element method. The reliability and validity of the suggested method is verified by its application to the calculation of the 3D eddy current distribution in two conducting systems.

A novel formulation for 3D eddy current problems with moving bodies using a Lagrangian description and BEM-FEM coupling

This paper summarizes the theoretical background of 3D eddy current problems with moving bodies. A novel 3D formulation combining a Lagrangian description and BEM-FEM coupling is subsequently developed. The conducting and permeable bodies are described by the FEM in their respective rest frame, whereas the surrounding space is treated by the BEM in the laboratory frame. The FEM and BEM descriptions are coupled together taking into account the transformation between the different frames. The proposed formulation contains no explicit velocity terms.

Complex representation in nonlinear time harmonic eddy current problems

Several possibilities are presented to deal with nonlinearity in ferromagnetic media in the case of time harmonic excitation in steady state, without losing simplicity in describing the potentials by means of complex peak values. The main idea is to introduce a fictitious time independent and inhomogeneous material to take into account the nonlinear relationship between the field quantities. Four methods are shown and investigated on a 3D time harmonic eddy current problem, using the T,/spl Phi/-/spl Phi/ finite element formulation. The vector potential is represented by means of edge elements and the scalar potential by nodal elements. The results obtained are compared with transient computation.