Eddy Current Problems Research Articles

A Mixed Multiscale FEM for the Eddy-Current Problem With T, Φ–Φ in Laminated Conducting Media

A novel mixed multiscale finite-element method for the eddy-current problem is presented to avoid the necessity of modeling each laminate of the core of electrical devices. The method is based on a current vector potential $T$ and a reduced magnetic scalar potential (RMSP) $\Phi $ and copes with the 3-D problems. The edge effect is considered. Material properties are assumed to be linear. Hence, the method is developed for the frequency domain. External currents are represented by the Biot–Savart field serving as excitation. The planes of symmetry are exploited. Numerical simulations are presented, showing excellent accuracy at minimal computational costs.

MSFEM and MOR to Minimize the Computational Costs of Nonlinear Eddy-Current Problems in Laminated Iron Cores

MSFEM and MOR to Minimize the Computational Costs of Nonlinear Eddy-Current Problems in Laminated Iron Cores

3-D BEM Formulations for Eddy-Current Problems With Multiply-Connected Domains and Circuit Coupling

Quasi-static linear problems can be solved efficiently with Boundary Element Method (BEM). This method is based on surface integral equations dealing with equivalent magnetic and electric surface current densities. Many works have shown the potentiality of BEM especially for the modeling of non-destructive testing devices. In this paper, after selecting formulations enabling the modeling of multiply-connected regions, an original coupling is proposed in order to take into account external electric circuit in the problem.

A Novel h–φ Approach for Solving Eddy–Current Problems in Multiply Connected Regions

A novel h-φ approach for solving 3-D time-harmonic eddy current problems is presented. It makes it possible to limit the number of degrees of freedom required for the discretization such as the T-Ω method, while overcoming topological issues related to it when multiply connected domains are considered. Global basis functions, needed for representing magnetic field in the insulating region, are obtained by a fast iterative solver. The computation of thick cuts by high-complexity computational topology tools, typically required by the T-Ω method, is thus avoided. The final matrix system turns out to be symmetric and full-rank unlike the more classical A-A method, which requires gauging of magnetic vector potential to ensure uniqueness. Numerical tests show that the proposed method is accurate and the field problem solution is obtained in a reasonable computational time even for 3-D models with millions of mesh elements.

FEM and BEM Implementations of a High Order Surface Impedance Boundary Condition for Three-Dimensional Eddy Current Problems

We implement a three-dimensional formulation for eddy current problems based on the reduced magnetic scalar potential enforcing a high order surface impedance boundary condition (SIBC) which takes into account the curvatures of the conductor surface. Based on perturbation theory, the formulation reduces to three Laplace boundary value problems with Neumann boundary conditions and hence can be easily implemented in any existing Finite Element Method (FEM) or Boundary Element Method (BEM) code for the Laplace equation. The appropriate choice of the small parameter in the perturbation approach correctly represents the order of accuracy of the SIBC. The validation is carried out by comparison with full FEM solutions of a canonical test problem and of a more realistic example of a non-destructive testing probe. The validity of the extension of a high order SIBC to lower frequencies is verified and the fields can be obtained at any frequency in the range of interest once the formulation is solved only once.

Measurements of Thickness for Metallic Plates With Co-Axial Holes Using a Novel Analytical Method With the Modified Integration Range

The existence of the hole on a plate affects the calculation of eddy current problems. Consequently, the accuracy for the prediction of the material properties decreases if the effect of the hole is not taken into account. In this paper, a novel analytical method based on the modified integration range is proposed which can address the presence of the hole. Due to the presence of the hole, the conventional Dodd-Deeds analytical solution cannot be used to calculate the inductance change. Therefore, a revised upper integration limit is introduced to replace the original limit - ∞ when using the co-axially air-core electromagnetic sensor. With the presence of the hole, the magnitude of the received signal reduces, and the peak frequency feature changes. The analytical method is validated by measured and numerical simulation results. It is found that the upper limit is related to the radius of the open hole. With the new technique, the thickness of sample plates with holes can be estimated based on the peak frequency feature.

Mathematical and Numerical Study of a Three-Dimensional Inverse Eddy Current Problem

We study an inverse problem associated with an eddy current model. We first address the ill-posedness of the inverse problem by proving the compactness of the forward map with respect to the conductivity and the nonuniqueness of the recovery process. Then by virtue of nonradiating source conceptions, we establish a regularity result for the tangential trace of the true solution on the boundary, which is necessary to justify our subsequent mathematical formulation. After that, we formulate the inverse problem as a constrained optimization problem with an appropriate regularization and prove the existence and stability of the regularized minimizers. To facilitate the numerical solution of the nonlinear nonconvex constrained optimization, we introduce a feasible Lagrangian and its discrete variant. Then the gradient of the objective functional is derived using the adjoint technique. By means of the gradient, a nonlinear conjugate gradient method is formulated for solving the optimization system, and a Sobolev gradient is incorporated to accelerate the iterative process. Numerical examples are provided to demonstrate the feasibility of the proposed algorithm.

不可压缩黏性流体的二维Navier-Stokes方程的间断有限元模拟

The incompressible Navier-Stokes equations are composed of the conservation law and the diffusion and constrained development equations. To test the numerical method, based on the unstructured grid, a discontinuous Galerkin scheme was established. The numerical results of the eddy current problem for different viscosity coefficients υ were discussed. The simulation results show that, the method has high precision and can solve the incompressible viscous fluid problem with moving interface, which makes the simulation boundary layer, the shear layer and the complex vortex solution be very effective, and the shock structure can be successfully extended to the numerical simulation of complex phenomena.

A boundary and finite element coupling for a magnetically nonlinear eddy current problem

The aim of this paper is to provide a mathematical and numerical analysis for a FEM-BEM coupling approximation of a magnetically nonlinear eddy current formulation by using FEM only on the conducting domain, and by imposing the integral conditions on its boundary. The nonlinear relationship between flux density and the magnetic field intensity is given by a physical parameter called magnetic reluctivity, which is assumed to depend on the Euclidean norm of the magnetic induction in the conducting domain. We use the nonlinear monotone operator theory for parabolic equations to show that the continuous formulation obtained for the coupling is a well-posed problem. Furthermore, we use Nédélec edge elements, standard nodal finite elements, and a backward-Euler time scheme, to obtain a fully discrete formulation and to prove quasi-optimal error estimates.

A Posteriori Error Estimation for the p-Curl Problem

We derive a posteriori error estimates for a semidiscrete finite element approximation of a nonlinear eddy current problem arising from applied superconductivity, known as the p-curl problem. In particular, we show the reliability for nonconforming Nédélec elements based on a residual-type argument and a Helmholtz--Weyl decomposition of W^p_0(curl;Omega). As a consequence, we are also able to derive an a posteriori error estimate for a quantity of interest called the AC loss. The nonlinearity for this form of Maxwell's equation is an analogue of the one found in the p-Laplacian. It is handled without linearizing around the approximate solution. The nonconformity is dealt with by adapting error decomposition techniques of Carstensen, Hu, and Orlando. Geometric nonconformities also appear because the continuous problem is defined over a bounded C^1,1 domain, while the discrete problem is formulated over a weaker polyhedral domain. The semidiscrete formulation studied in this paper is often encountered in commercial codes and is shown to be well-posed. The paper concludes with numerical results confirming the reliability of the a posteriori error estimate.

Application of n-Fold Rotational Symmetries to Eddy Currents Integral Model in the Time Domain

Group Theory is a powerful “tool” which has been extensively exploited to reduce the computational cost in the numerical analysis of several physical problems, including computational electromagnetics. This article proposes the extension of the application of group theory to an integral formulation of the eddy current problems in the presence of n-fold rotational symmetries. The proposed approach is applied to the eddy currents computation in toroidal fusion devices, where rotational symmetries play a key role.

H-$\Phi$ Field Formulation With Lumped Sources and Unbounded Domains

We consider a H-Φ field formulation to solve 3-D frequency-domain eddy-current problems. This formulation uses vector and scalar tetrahedral finite elements within, respectively, the conductive and nonconductive domains. It can handle multiply-connected regions and eliminates the need to compute the source current density and source magnetic field before the actual simulation. We propose three ways to couple finite elements with the multiple multipole program (MMP) and solve this H-Φ variational form on an unbounded domain. The MMP is a method that uses exact solutions of the homogeneous equations as basis functions (the so-called “multipoles”). The desired behavior at infinity is given by the chosen multipoles-this eliminates the need for artificially truncating the computational domain. Interface conditions between the finite-element method (FEM) and MMP domains allow to express the coupled FEM-MMP problem.

Residual-Type APosterioriError Estimates for 3-D Low-Frequency Stable Maxwell Formulations in Both Frequency and Time Domains

In this article, residual-type a posteriori error estimates developed in our previous work for magnetostatic and eddy-current problems are extended to low-frequency (LF) Maxwell problems using A/φ formulation, where both inductive and capacitive effects can be handled simultaneously. Classical low-order finite-element basis (LOFEB) as well as high-order finite-element basis (HOFEB) of the edge and nodal types are adopted in numerical examples to evaluate the performance of the proposed estimators.

Numerical analysis of a penalty approach for the solution of a transient eddy current problem

The aim of this paper is to propose and analyze a numerical method to solve transient eddy current problems formulated in terms of the magnetic field intensity. Space discretization is based on Nédélec edge elements, while a backward Euler scheme is used for time discretization; the curl-free constraint in the dielectric domain is imposed by means of a penalty strategy. Convergence of the penalized problem as the penalty parameter goes to zero is proved for the continuous and the discrete problems, for the latter uniformly in the discretization parameters. Optimal order error estimates for the convergence of the discrete penalized problem with respect to the penalty and the discretization parameters are also proved. Finally, some numerical tests are reported to assess the performance of this approach.

Multigrid‐reduction‐in‐time for Eddy Current problems

AbstractParallel‐in‐time methods have shown success for reducing the simulation time of many time‐dependent problems. Here, we consider applying the multigrid‐reduction‐in‐time (MGRIT) algorithm to a voltage‐driven eddy current model problem.

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.

Locating complex eigenvalues for analytical eddy-current models used to detect flaws

PurposeDiscrete eigenvalues occur in eddy current problems in which the solution domain was truncated on its edge. In case of conductive material with a hole, the eigenvalues are complex numbers. Their computation consists of finding complex roots of a complex function that satisfies the electromagnetic interface conditions. The purpose of this paper is to present a method of computing complex eigenvalues that are roots of such a function.Design/methodology/approachThe proposed approach involves precise determination of regions in which the roots are found and applying sets of initial points, as well as the Cauchy argument principle to calculate them.FindingsThe elaborated algorithm was implemented in Matlab and the obtained results were verified using Newton’s method and the fsolve procedure. Both in the case of magnetic and nonmagnetic materials, such a solution was the only one that did not skip any of the eigenvalues, obtaining the results in the shortest time.Originality/valueThe paper presents a new effective method of locating complex eigenvalues for analytical solutions of eddy current problems containing a conductive material with a hole.

Massively Parallel Computation for 3-D Nonlinear Finite Edge Element Problem With Transmission Line Decoupling Technique

Transmission line method (TLM) has been used in 2-D scalar finite-element (FE) analysis due to its parallelism and constant admittance matrix. In this paper, the TLM is extended for the 3-D nonlinear vector FE problem that is more widely used for electromagnetic apparatus in practice. TLM is specially adapted for tetrahedron edge elements to calculate quasi-static electromagnetic field distribution for eddy current problems, and a dummy scalar gauge is applied to make the reduced magnetic vector FE formulation full-ranked and uniquely solvable by TLM. For each element, the nonlinearity is separated by transmission lines and only local small-scale Newton–Rapson iteration is needed, which is suitable for massive parallelization due to the independence between different elements. The TLM is implemented on a many-core GPU for a nonlinear FE power inductor case study, and the comparison of the results with a commercial FE software shows over 50 times speedup with a relative error of less than 2%.

Homogenization of Multiscale Eddy Current Problem by Localized Orthogonal Decomposition Method

A homogenization approach for the solution of multiscale eddy current problem is proposed. The method is based on the subspace decomposition and it involves a coarse space and a nested fine space. The homogenized problem is posed in the coarse space with the help of a projection operator acting between the coarse space and a space of rapidly oscillating functions. A Helmholtz decomposition is applied to treat the null-space of the curl operator so that the projection can be locally calculated. The results are illustrated in a 2-D numerical example.

Coupling Nonlinear Magneto‐Thermal Problems in Frequency Domain with the Mechanical Field

AbstractThis paper investigates a computationally efficient strategy for simulating induction heating processes, using a multiharmonic ansatz and harmonic balancing for the solution of the nonlinear magnetic eddy current problem in frequency domain. In order to increase the flexibility of the meshing procedure and decrease the number of degrees of freedom, a methodology for nonconforming interfaces and Nédéléc edge elements is presented. The verification of the proposed method is carried out by comparison to the conform (conforming mesh) solution. Furthermore, the results from the eddy current problem are used to simulate the temperature distrubution in a thin steel sheet and compute the resulting stresses.