Solution Of Eddy Current Problems Research Articles

Low-rank compression techniques in integral methods for eddy currents problems

Volume integral methods for the solution of eddy current problems are very appealing in practice since only the conducting regions need to be meshed. However, they require the assembly and storage of a dense stiffness matrix. With the objective of cutting down assembly time and memory occupation, low-rank approximation techniques like the Adaptive Cross Approximation (ACA) have been considered a major breakthrough. Recently, the VINCO framework has been introduced to reduce significantly memory occupation and computational time thanks to a novel factorization of the dense stiffness matrix. The aim of this paper is introducing a new matrix compression technique enabled by the VINCO framework. We compare the performance of VINCO framework approaches with state-of-the-art alternatives in terms of memory occupation, computational time and accuracy by solving benchmark eddy current problems at increasing mesh sizes; the comparisons are carried out using both direct and iterative solvers. The results clearly indicate that the so-called VINCO-FAIME approach which exploits the Fast Multipole Method (FMM) has the best performance.

Computation of Eigenvalues and Eigenfunctions in the Solution of Eddy Current Problems

The solution of the eigenvalue problem in bounded domains with planar and cylindrical stratification is a necessary preliminary task for the construction of modal solutions to canonical problems with discontinuities. The computation of the complex eigenvalue spectrum must be very accurate since losing or misplacing one of the thereto linked modes will have an important impact on the field solution. The approach followed in a number of previous works is to construct the corresponding transcendental equation and locate its roots in the complex plane using the Newton-Raphson method or Cauchy-integral-based techniques. Nevertheless, this approach is cumbersome, and its numerical stability decreases dramatically with the number of layers. An alternative, approach consists in the numerical evaluation of the matrix eigenvalues for the weak formulation for the respective 1D Sturm-Liouville problem using linear algebra tools. An arbitrary number of layers can thus be easily and robustly treated, with continuous material gradients being a limiting case. Although this approach is often used in high frequency studies involving wave propagation, this is the first time that has been used for the induction problem arising in an eddy current inspection situation. The developed method is implemented in Matlab and is used to deal with the following problems: magnetic material with a hole, a magnetic cylinder, and a magnetic ring. In all the conducted tests, the results are obtained in a very short time, without missing a single eigenvalue.

Cyclic Symmetry in Volume Integral Formulations for Eddy Currents: Cohomology Computation and Gauging

This contribution addresses the solution of eddy-current problems by means of a volume integral formulation based on the electric vector potential on a computational domain that exhibits a cyclic symmetry. Even if grids discretizing the domain are typically composed of tetrahedral or hexahedral elements, the proposed approach also works for general polyhedral meshes, such as those ones obtained by subgridding . In this article, an algorithm to compute a set of suitable cohomology generators needed when the conductors are not simply connected is introduced first. Besides being purely combinatorial, with linear-time worst case complexity and suitable with polyhedral meshes, it reuses a code that computes generators for triangular surface meshes, with obvious advantages concerning the implementation effort. Second, the formulation and the algorithm for cohomology computation are tweaked to be able to solve eddy-current problems with cyclic symmetry reserving specific attention to the construction of suitable tree–cotree decomposition for the problem gauging.

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.

Analytical Solutions of Eddy-Current Problems in a Finite Length Cylinder

Magnetic field and eddy currents in a cylinder of finite length are calculated by separation of variables. The magnetic field outside the cylinder or inside the bore of the hollow cylinder and shell is expressed in terms of Bessel functions. Both axial and transverse applied fields are considered for the solid and hollow cylinders. The equations for the vector potential components are transformed in one-dimensional equations along the radial coordinate with the consequent integration by the method of variation of parameters. The equation for the scalar electric potential when required is also integrated analytically. Expressions for the magnetic moment and loss are derived. An alternative analytical solution in terms of scalar magnetic potential is derived for the finite length thin shells. All formulas are validated by the comparison with the solutions by finite–element and finite-difference methods.

Iterative Solution of Eddy Current Problems on Polyhedral Meshes

We present an effective technique to solve eddy current problems on the conductors of arbitrary topology by an iterative geometric formulation suitable for an arbitrary polyhedral mesh. The formulation is based on the reformulation of the volume integral formulation in a form suitable to be solved by fixed point iterations. Given that nowadays, the computation of Biot–Savart fields can be performed efficiently on graphics processing units, the right-hand side of the systems is constructed at each fixed point iteration without computing the inductance matrix, which is fully populated. The proposed technique is useful in eddy current problems at industrial frequency where meshing the complement of conductors is hard, thus representing a sound alternative to integral formulations.

Element-free galerkin method in eddy-current problems with ferromagnetic media

This paper focuses on the application of the element-free Galerkin method to the solution of eddy-current problems in ferromagnetic media. The method is first validated by comparison with an analytical linear solution, discussing the role of the numerical parameters that affect the solution accuracy. Then, the application is extended to nonlinear and hysteretic media, adopting the fixed point iterative scheme and the classical Preisach model. The proposed method is also tailored for nonhomogenous structures. The merits and drawbacks with respect to a finite-element approach are discussed for all the considered problems.

Eddy current problems in nonlinear media by the element-free Galerkin method

The element-free Galerkin method is applied to the solution of eddy current problems in nonlinear and hysteretic media. The method is first validated by comparison with an analytical linear solution and successively applied to different nonlinear problems discussing merits and drawbacks with respect to a Finite Element approach.

Numerical solution of eddy current problems in bounded domains using realistic boundary conditions

The aim of this paper is to analyze a finite element method to solve the low-frequency harmonic Maxwell equations in a bounded domain containing conductors and dielectrics, and using realistic boundary conditions in that they can be easily measured. These equations provide a model for the so-called eddy currents. The problem is formulated in terms of the magnetic field. This formulation is discretized by using Nédélec edge finite elements on a tetrahedral mesh. Error estimates are easily obtained when the curl-free condition is imposed explicitly on the elements in the dielectric domain.A multivalued magnetic scalar potential is introduced then to impose this curl-free condition. The discrete counterpart of this formulation leads to an important saving in computational effort. Problems related to the topology are also considered, more precisely, the possibility of having a non-simply connected dielectric domain is taken into account. Finally, the method is applied to solve two three-dimensional model problems: a test with a known analytical solution and the computation of the electromagnetic field in a metallurgical arc furnace.

Numerical solution of eddy current problems in ferromagnetic bodies travelling in a transverse magnetic field

AbstractEddy currents are investigated in a ferromagnetic bar travelling in a transverse magnetic field. Such an open boundary field problem is analysed by a hybrid approach based on Galerkin finite element formulation coupled with a separation of variables. A steady state is considered, introducing time‐periodic boundary conditions. The resultant system of non‐linear equations is solved by an iterative procedure based on Brouwer's fixed‐point theorem. Numerical results are presented for a bar of circular cross‐section made of cast steel or cast iron. Selected examples of the field distribution and characteristics of eddy‐current power losses are enclosed in graphic form. Copyright © 2003 John Wiley & Sons, Ltd.

Finite-element solution of eddy-current problems in unbounded domains by means of the hybrid FEM-DBCI method

A hybrid method, called finite-element method-Dirichlet boundary-condition iteration (FEM-DBCI), is described for the finite-element solution of time-harmonic eddy current problems in three-dimensional unbounded domains. The procedure couples a differential equation for the interior problem in terms of the electric field with an integral equation for the exterior one, which expresses the Dirichlet condition on the truncation boundary. The global algebraic system is efficiently solved in an iterative way. FEM-DBCI leads to shorter solving times and not much higher memory requirements with respect to the FEM-boundary-element method.

Influence of surface charges in the finite volume modeling of conductive regions for 3-D eddy-current computation

The finite volume discretization of massive conductors for the calculation of eddy-current distribution causes fictitious surface charges at the boundary between adjacent volume elements having different current values. In this paper, the effects of these surface charges on the numerical solution of three-dimensional eddy-current problems are evidenced. Furthermore, comparing solutions of eddy-current problems in the presence and absence of surface charges, an empirical criterion is derived for the mesh refinements of the current inside massive volumes.

Three dimensional finite elements modeling of superconductors

In this paper we present a variational formulation for the solution of eddy current problems in presence of superconductors. A suitable formulation of the macroscopic constitutive law is used so that an unconstrained minimization has to be performed. The electromagnetic problem is solved using an integral formulation, thus allowing us to discretize only the conducting regions. The method, although tested against a 1D analytical solution, is implemented for fully 3D geometries.

Solution of magnetic field and eddy current problem induced by rotating magnetic poles (abstract)

The magnetic field and eddy current problems induced by rotating permanent magnet poles occur in electromagnetic dampers, magnetic couplings, and many other devices. Whereas numerical techniques, for example, finite element methods can be exploited to study various features of these problems, such as heat generation and drag torque development, etc., the analytical solution is always of interest to the designers since it helps them to gain the insight into the interdependence of the parameters involved and provides an efficient tool for designing. Some of the previous work showed that the solution of the eddy current problem due to the linearly moving magnet poles can give satisfactory approximation for the eddy current problem due to rotating fields. However, in many practical cases, especially when the number of magnet poles is small, there is significant effect of flux focusing due to the geometry. The above approximation can therefore lead to marked errors in the theoretical predictions of the device performance. Bernot et al. recently described an analytical solution in a polar coordinate system where the radial field is excited by a time-varying source. A discussion of an analytical solution of the magnetic field and eddy current problems induced by moving magnet poles in radial field machines will be given in this article. The theoretical predictions obtained from this method is compared with the results obtained from finite element calculations. The validity of the method is also checked by the comparison of the theoretical predictions and the measurements from a test machine. It is shown that the introduced solution leads to a significant improvement in the air gap field prediction as compared with the results obtained from the analytical solution that models the eddy current problems induced by linearly moving magnet poles.

Three-dimensional finite-element formulation for problems involving time-varying fields, relative motion, and magnetic saturation

A numerical method for the solution of eddy current problems which involve magnetic saturation, relative motion and time-varying field is presented. The model is based on the finite-element method and the A-ψ-φ formulation is employed. The moving conductor is modelled using three components of magnetic vector potential, whereas the rest of the problem domain is modelled using either of two scalar potentials. The model is verified by means of experimental measurements made on a test problem in which the three-dimensional effects are clearly evident.

A new approach to the solution of eddy current problem with discontinuous electrical conductivity

Presents a novel method for use in finite elements to solve the eddy current problem caused by discontinuous electrical conductivity. The modified vector potential is directly used in this method to solve the problem that the modified vector potential formulation cannot be used when discontinuous conductivity occurs. This method also solves the problem that the airside normal component of the modified vector potential on the interface between the air region and the conductor cannot be determined if the conductivity of the air region is ideally taken as zero. Being applied to a model of one conductor in the air region, it leads to an interface condition which is commonly used for this conductor. The method is further verified when the result of a connected-conductor model is compared with that of a hybrid method.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Numerically stable finite element methods for the Galerkin solution of eddy current problems

Two sources of error that contribute to the instability of Galerkin solutions of vector eddy-current problems are identified. The first results from a magnification of the error in modeling or integrating the solenoidal forcing function. The second arises from contamination of the deterministic solutions by spurious modes. To obtain stable finite-element solutions of vector eddy-current problems, it is therefore necessary to model the forcing function accurately and to use vector basis functions that do not produce spurious modes. Two such sets of stable basis functions are found to be standard Lagrangian elements on a common C/sup 1/ mesh and tangential vector elements on an arbitrary mesh.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Complementary variational finite-element solution of Eddy current problems using the field variables

A variational finite element method, based upon a functional related to the classical Lagrange density and originally embodied into a program intended for high frequency problems with loss, has been tested on a simple skin-effect eddy current problem. Either of the dual variables E or H may be used subject to complementary boundary constraints. The results so far have been worked in terms of the H variable and show good agreement with the theoretical skin-effect result.

Approximate treatment of nonlinear eddy current problems

This work presents a new systematic approach for the computation of the fundamental frequency eddy current losses in a magnetic material. It is based on the use of the time averaged value of the permeability over a complete cycle. This depends on the magnetization characteristics of the material and the time variation of the source. Curves depicting the dependence of this weighted permeability (complex in the presence of hysteresis) on the magnetic field are given for different magnetic materials. The practical application of this concept for the solution of eddy current problems is demonstrated for one dimensional multilayer problems. A combination of the micro circuit and transfer function representations is used, with the former restricted to the nonlinear layers. An iterative procedure, in which a continuously increasing part of the magnetic material is represented by its equivalent circuit, has been developed. Numerical results are presented for cylindrical geometries and are compared with those obtained by experiment, time domain analysis, and using a value of μ corresponding to a given value of H.

Use of numerical analysis of nonlinear eddy current problems by finite elements in the determination of parameters of electrical machines with solid iron rotors

This paper reports on the application of the concept of effective permeability using the finite element technique, to the solution of eddy current problems, and the determination of damping resistances as well as the characteristics of the solid iron rotors in rotating machinery during transients. The formulation is based on the concept of magnetic vector potential in phasor form, coupled with effective permeability which is related to the average magnetic energy throughout an a.c. cycle. The technique is applied to the solid iron rotors of turbogenerators for the determination of equivalent damping resistances, etc., during asynchronous operation. Sample eddy current loss results compare favorably with experimental test data. Instantaneous change in damping resistances during machine transients are obtained and reported on here based on this technique.