Eddy Current Problems Research Articles

Arbitrary order spline representation of cohomology generators for isogeometric analysis of eddy current problems

Common formulations of the eddy current problem involve either vector or scalar potentials, each with its own advantages and disadvantages. An impasse arises when using scalar potential-based formulations in the presence of conductors with non-trivial topology. A remedy is to augment the approximation spaces with generators of the first cohomology group. Most existing algorithms for this require a special, e.g., hierarchical, finite element basis construction. Using insights from de Rham complex approximation with splines, we show that additional conditions are here unnecessary. Spanning tree techniques can be adapted to operate on a hexahedral mesh resulting from isomorphisms between spline spaces of differential forms and de Rham complexes on an auxiliary control mesh.

A Lagrangian approach for solving an axisymmetric thermo-electromagnetic problem. Application to time-varying geometry processes

The aim of this work is to introduce a thermo-electromagnetic model for calculating the temperature and the power dissipated in cylindrical pieces whose geometry varies with time and undergoes large deformations; the motion will be a known data. The work will be a first step towards building a complete thermo-electromagnetic-mechanical model suitable for simulating electrically assisted forming processes, which is the main motivation of the work. The electromagnetic model will be obtained from the time-harmonic eddy current problem with an in-plane current; the source will be given in terms of currents or voltages defined at some parts of the boundary. Finite element methods based on a Lagrangian weak formulation will be used for the numerical solution. This approach will avoid the need to compute and remesh the thermo-electromagnetic domain along the time. The numerical tools will be implemented in FEniCS and validated by using a suitable test also solved in Eulerian coordinates.

An Equilibrated Error Estimator for the 2-D/1-D MSFEM T-Formulation of the Eddy Current Problem

An Equilibrated Error Estimator for the 2-D/1-D MSFEM T-Formulation of the Eddy Current Problem

Numerical simulation of resistance furnaces by using distributed and lumped models

This work proposes a methodology that combines distributed and lumped models to simulate the current distribution within an indirect heat resistance furnace and, in particular, to calculate the current to be supplied for achieving a desired power output. The distributed model is a time-harmonic eddy current problem, which is solved numerically using the finite element method. The lumped model relies on calculating a reduced impedance associated with an equivalent circuit model. Numerical simulations and plant measurements demonstrate the effectiveness of this approach. The good correlation between the results indicates that this approximation is well-suited to support the design and improve the efficiency of the furnace in a short time.

An Iterative Method for the Inverse Eddy Current Problem with Total Variation Regularization

An Iterative Method for the Inverse Eddy Current Problem with Total Variation Regularization

MSFEM With MOR and DEIM to Solve Nonlinear Eddy Current Problems in Laminated Iron Cores

MSFEM With MOR and DEIM to Solve Nonlinear Eddy Current Problems in Laminated Iron Cores

Cauer Ladder Network With Constant Basis Functions for Eddy Current Problems Involving Conductor Movement

Cauer Ladder Network With Constant Basis Functions for Eddy Current Problems Involving Conductor Movement

Second-Order Approximation of Nonlinear Eddy-Current Problems by a Cauer Ladder Network

Second-Order Approximation of Nonlinear Eddy-Current Problems by a Cauer Ladder Network

Goal-oriented error estimation based on equilibrated flux reconstruction for the approximation of the harmonic formulations in eddy current problems

Abstract In this work, we propose an a posteriori goal-oriented error estimator for the harmonic $\textbf {A}$-$\varphi $ formulation arising in the modeling of eddy current problems, approximated by nonconforming finite element methods. It is based on the resolution of an adjoint problem associated with the initial one. For each of these two problems, a guaranteed equilibrated estimator is developed using some flux reconstructions. These fluxes also allow to obtain a goal-oriented error estimator that is fully computable and can be split in a principal part and a remainder one. Our theoretical results are illustrated by numerical experiments.

Normalized Nonlinear Impedance Boundary Condition in Anhysteretic Magnetic Material for Eddy Current Problems

Normalized Nonlinear Impedance Boundary Condition in Anhysteretic Magnetic Material for Eddy Current Problems

A hybrid spectral-spatial formulation for the calculation of the transient eddy-current response

A new approach for the solution of transient eddy-current problems involving pieces with one direction of symmetry is presented. The approach is applicable to pieces with translational or rotational symmetry. A Fourier series decomposition of the solution is introduced in the direction of symmetry, which converts the original 3D numerical problem into a series of independent 2D problems. The decomposition has significant benefits in terms of computational time and numerical noise reduction, and is inherently parallelisable.

Coupled electric circuits method applied to pulsed eddy current NDT problem

The present work concerns the study of the effect of variation of the thickness of 2D cylindrical test-piece and lift-off (the air gap between the sensor and the test-piece) on the sensor response in case of transient source current. The coupled circuit’s method employed is based on the mutual’s inductances calculation and with the association to Kirchhoff laws it yields to a transient algebraic equations system which is solved at each time step. The method is applied to the study of the pulsed eddy currents in cylindrical and axisymmetric device with electrical conductivity and geometrical dimensions variations.

A Posteriori Error Estimation for the Optimal Control of Time-Periodic Eddy Current Problems

Abstract This work presents the multiharmonic analysis and derivation of functional type a posteriori estimates of a distributed eddy current optimal control problem and its state equation in a time-periodic setting. The existence and uniqueness of the solution of a weak space-time variational formulation for the optimality system and the forward problem are proved by deriving inf-sup and sup-sup conditions. Using the inf-sup and sup-sup conditions, we derive guaranteed, sharp and fully computable bounds of the approximation error for the optimal control problem and the forward problem in the functional type a posteriori estimation framework. We present here the first computational results on the derived estimates.

Efficient asymptotic models for axisymmetric eddy current problems in linear ferromagnetic materials

The problem under consideration is that of time-harmonic eddy current problems in linear ferromagnetic materials surrounded by a dielectric medium with a smooth common interface. Assuming axisymmetric geometries and orthoradial axisymmetric data, we construct an efficient multiscale expansion for the orthoradial solution that provides reduced computational costs. We investigate numerically the accuracy of the approach using an analytical procedure and infinite cylinders as well. It results that the computation of two asymptotics is sufficient to ensure accurate solutions in the case of low frequencies.

Modelling the post‐buckling behaviour of steel sheets under induction heating

AbstractInduction heating has many engineering applications. To accurately describe the induction heating process of thin steel sheets one needs to account for their mechanical deformation. This might strongly affect the magnetic configuration since thin sheets are prone to thermal buckling, leading to large supercritical deformations. We suggest a modelling strategy combining a computationally highly efficient structural mechanical shell description for the thin sheet with a standard continuum model for eddy current and thermal problems through an iterative coupling algorithm. All benchmark problems are solved numerically using the finite element method.

Electromagnetic Shielding Technique for No-Insulation Superconducting Rotor Windings in Electrical Aircraft Propulsion

No-insulation (NI) high temperature superconductor (HTS) machine is a kind of synchronous semi-superconducting machine with high power density and enhanced thermal stability, which has a great potential for electric aircraft propulsion systems. However, NI HTS rotor windings in this machine always suffer the problem of eddy currents and losses induced by ripple magnetic fields in the synchronous machine environment, which can significantly reduce the efficiency and safety of the HTS machine. In this paper, an electromagnetic shielding technique is developed to minimize this induced eddy current and loss in NI HTS rotor windings. A numerical model is developed to analyse the effectiveness of this shielding technique, and experiments are performed to validate the model. Then, the effect of this electromagnetic shielding on the induced eddy current and loss of NI coil is studied using this model. Results show that electromagnetic shielding technique based on copper discs can effectively reduce the eddy current and loss on NI HTS coil at the operating temperature range of HTS machines, 20-30 K, thus it is promising to solve the problem of eddy current and loss of NI HTS machine design in electrical aircraft propulsion.

Reduced order model approaches for predicting the magnetic polarizability tensor for multiple parameters of interest

The magnetic polarizability tensor (MPT) is an economical characterisation of a conducting magnetic object, which can assist with identifying hidden targets in metal detection. The MPT’s coefficients depend on multiple parameters of interest including the object shape, size, electrical conductivity, magnetic permeability, and the frequency of excitation. The computation of the coefficients follow from post-processing an eddy current transmission problem solved numerically using high-order finite elements. To reduce the computational cost of constructing these characterisations for multiple different parameters, we compare three methods by which the MPT can be efficiently calculated for two-dimensional parameter sets, with different levels of code invasiveness. We compare, with numerical examples, a neural network regression of MPT eigenvalues with a projection-based reduced order model (ROM) and a neural network enhanced ROM (POD–NN) for predicting MPT coefficients.

A hybrid enrichment strategy for the efficient construction of reduced order models by the proper generalized decomposition

PurposeThe calculation of electromagnetic fields can involve many degrees of freedom (DOFs) to achieve accurate results. The DOFs are directly related to the computational effort of the simulation. The effort is decreased by using the proper generalized decomposition (PGD) and proper orthogonalized decomposition (POD). The purpose of this study is to combine the advantages of both methods. Therefore, a hybrid enrichment strategy is proposed and applied to different electromagnetic formulations.Design/methodology/approachThe POD is an a-priori method, which exploits the solution space by decomposing reference solutions of the field problem. The disadvantage of this method is given by the unknown number of solutions necessary to reconstruct an accurate field representation. The PGD is an a-priori approach, which does not rely on reference solutions, but require much more computational effort than the POD. A hybrid enrichment strategy is proposed, based on building a small POD model and using it as a starting point of the PGD enrichment process.FindingsThe hybrid enrichment process is able to accurately approximate the reference system with a smaller computational effort compared to POD and PGD models. The hybrid enrichment process can be combined with the magneto-dynamic T-Ω formulation and the magnetic vector potential formulation to solve eddy current or non-linear problems.Originality/valueThe PGD enrichment process is improved by exploiting a POD. A linear eddy current problem and a non-linear electrical machine simulation are analyzed in terms of accuracy and computational effort. Further the PGD-AV formulation is derived and compared to the PGD-T-Ω reduced order model.

Numerical analysis of a FEM based on a time-primitive of the electric field for solving a nonlinear transient eddy current problem

The aim of this paper is to analyze from a mathematical and a numerical point of view a formulation of a transient eddy current problem in terms of a time-primitive of the electric field in a bounded domain with ferromagnetic conductors. To this aim, we introduce a Lagrange multiplier to impose the free-divergence condition in the isolated domain. Thus, we obtain a nonlinear degenerate parabolic problem in mixed form and prove its well-posedness. Then, we propose a fully-discrete scheme by using an implicit Euler scheme for time-discretization and a finite element method based on edge and nodal elements for the spatial discretization. We prove quasi-optimal error estimates for the approximation and include some numerical examples to confirm the obtained theoretical results.

2-D Eddy Current Boundary Value Problems for Power Cables With Helicoidal Symmetry

Power cables have complex geometries in order to reduce their AC resistance. The cross-section of a cable consists of several conductors that are electrically insulated from each other to counteract the current displacement caused by the skin effect. Furthermore, the individual conductors are twisted over the cable's length. This geometry has a non-standard symmetry - a combination of translation and rotation. Exploiting this property allows formulating a dimensionally reduced boundary value problem. Dimension reduction is desirable, otherwise the electromagnetic modeling of these cables becomes impracticable due to tremendous computational efforts. We investigate 2D eddy current boundary value problems which still allow the analysis of 3D effects, such as the twisting of conductor layers.