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Grain-oriented electrical steels (GOES) are widely used in the manufacturing of high efficiency energy conversion systems thanks to their excellent magnetic properties along the rolling direction. Nevertheless, on the one hand, GOES exhibit strong magnetic anisotropy, resulting in distinct magnetic properties regarding the direction of the applied magnetic field to the rolling direction. On the other hand, GOES undergoes changes in their magnetic properties due to the industrial manufacturing process and mechanical constraints during operation. In general, stress, and particularly compressive stress, deteriorates magnetic properties. This paper deals with the modeling of the magnetic behavior of GOES under compressive stress. The orientation distribution function (ODF) based approach recently applied to describe the B-H first magnetization curves of a conventional GOES without stress as well as under applied uniaxial mechanical tensile stress is extended to account for the effect of compressive stress, with different considerations of modeling. The magnetic properties and the sensitivity of the ODF-based model to the magnetization direction as well as to the applied compressive stress are analyzed. Oscillation issues inherent to the ODF-based model are also discussed.

In this work, we propose an a posteriori goal-oriented error estimator for the harmonic A-Phi 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.

Grain-oriented electrical steels (GOES) are extensively utilized in power transformers to enhance energy efficiency and reduce their size and weight. GOES are characterized by their strong magnetic anisotropy, leading to distinct magnetic characteristics (behavior law and iron losses) according to the direction and the intensity of the applied field with respect to the rolling direction (RD). The Orientation Distribution Function (ODF) based approach is a convenient method for describing the anisotropy of the GOES behavior law owing to its ease of identification and implementation. However, it exhibits a significant oscillation issue at low magnetic fields. To address this issue, a high-order ODF-based method has been proposed in the literature. In this article, the ODF-based model, initially stress-free, is extended to include the effects of mechanical tensile stress on the first magnetization curves of a conventional GOES. An ODF-based model considering both magnetic anisotropy and mechanical tensile stress is then proposed. The results are discussed and compared with experimental data.

The interest of non-destructive testing (NDT) in steel material characterization is growing especially for quality and life cycle control. The Magnetic Barkhausen Noise (MBN) analysis shows an extreme sensitivity to different parameters: steel grade, grain size, microstructure, magnetic domains wall structure, residual stress, surface condition, etc. It is very difficult to distinguish the individual effect of each parameter on the steel MBN signature. This paper attempts to address this problematic by fixing most of these parameters and varying only one parameter of interest (either austenitizing or tempering temperature) to assess the ability of the MBN technique to dissociate the mechanical properties of different martensitic steel grades. The MBN technique presented herein has proven to be able to distinguish high-performance martensitic steel grades when applied according to the proposed experimental protocol.

Grain-Oriented Electrical Steels (GOES) are widely used in transformers to increase energy efficiency while reducing the volume and weight of such transformers. These GOES are characterized by their strong anisotropy, which leads to different magnetic properties (behavior law and iron losses) depending on the strength and the direction of the applied magnetic field to the rolling direction (RD). To describe the anisotropy of the GOES behavior law, the Orientation Distribution Function (ODF) based approach shows interesting features regarding ease of identification and implementation. However, a significant oscillation issue in this model at low magnetic field exits. It is recently reported that a high-order ODF based method can improve the accuracy of the original model. But the high-order ODF based model implies an important number of experimental data according to different directions of the applied field to RD (different magnetization angles) to build a model with sufficient accuracy. In addition, the accuracy also depends on how the input magnetization angles are selected between the RD and the transverse direction (TD). In this article, an optimal algorithm for input magnetization angle selection is proposed for the high-order ODF based method. To validate the proposed algorithm, 13 magnetization angles have been measured. A comparative study has been conducted based on the original and the high-order ODF based models built with the proposed selected magnetization angles. It is shown that the proposed algorithm can reduce the number of experimental data needed for the ODF based model while providing an acceptable global error.

We present a unified framework for goal-oriented estimates for elliptic and parabolic problems that combines the dual-weighted residual method with equilibrated flux and potential reconstruction. These frameworks allow to analyze simultaneously different approximation schemes for the space discretization of the primal and the dual problems such as conforming or nonconforming finite element methods, discontinuous Galerkin methods, or the finite volume method. Our main contribution is twofold: first in a unified framework we prove the splitting of the error into a fully computable estimator η and a remainder, second this remainder is estimated by the product of the fully computable energy-based error estimators of the primal and dual problems. Some illustrative numerical examples that validate our theoretical results are finally presented.

The need for electrical machines is growing in a wide range of applications nowadays, such as for energy production, automotive, marine, and medical equipment, etc. The reliability, efficiency, performance and operational safety are always critical issues, which implies that the monitoring and analysis during the operation of the equipment need to be strengthened. In general, more sensors can give higher accuracy of the estimated physical quantity, but bring a more expensive cost. To address this issue, an algorithm principally based on the parameterized background data-weak (PBDW) method is proposed and adopted in this work to optimize both the number and the location of sensors. Besides, to further investigate the performance of the proposed algorithm, a prototype concerning an anisotropic magnetostatic problem is studied as an example for the placement of magnetic sensors. The numerical experiments shows that the proposed method can achieve very satisfactory results in different aspects.

Due to the advancement in the development of semiconductors used in the power converters, the printed circuit boards (PCBs) require an in-depth study of their electromagnetic behavior. To characterize the behavior of the PCBs, the Darwin model is employed, which can take into account all the coupled effects, namely resistive, inductive, and capacitive effects, at the intermediate frequencies. Nevertheless, the study of particular structures having a geometric dimension smaller than the others can create meshing difficulties. The modeling of thin structures by the finite element method requires the optimization of the mesh. To circumvent this issue, the shell elements for both node and edge elements are applied in this work. Finally, to validate the proposed approaches, two PCBs with different geometries are studied in both time and frequency domains, where the measurements for a single PCB are provided to compare with the numerical results.

Nondestructive testing (NDT) of industrial steel products is commonly used to detect defects such as cracks using among other methods, electromagnetic techniques. When properly calibrated, these NDT can be adapted to evaluate the mechanical properties of steels. It is especially of prime importance when rigorous quality control must be achieved for high performance steels. In this context, the proposed article deals with the investigation of the Magnetic Barkhausen Noise (MBN) to characterize the tensile properties of high mechanical performance martensitic steels.

A numerical issue arises when we extend the convolutional neural network (CNN) U-net to the anisotropic magnetostatic field computation. The output magnetic field has a significant gradient with respect to the input geometry parameter, which introduces inevitable errors in the training process to degrade the performance of deep learning (DL). To address this issue, the subset selection approach is utilized to divide the whole database into serval subsets, where the samples are assigned according to the gradient between the input and output. Then these subsets with different sample densities are combined into a global one. Taking the uniform dataset with the same sample size as a comparison, the influence of subset selection on DL is investigated by comparing the performance of CNN on different datasets. Numerical experiments illustrate that the adaptive subset selection can be employed to improve the accuracy and efficiency of the CNN network.

Convolutional neural networks (CNN) have shown great potentials and have been proven to be an effective tool for some image-based deep learning tasks in the field of computational electromagnetism (CEM). In this work, an energy-based physics-informed neural network (EPINN) is proposed for low-frequency electromagnetic computation. Two different physics-informed loss functions are designed. To help the network focus on the region of interest instead of computing the whole domain on average, the magnetic energy norm error loss function is proposed. Besides, the methodology of energy minimization is integrated into the CNN by introducing the magnetic energy error loss function. It is observed that the introduction of the physics-informed loss functions improved the accuracy of the network with the same architecture and database. Meanwhile, these changes also cause the network to be more sensitive to some hyperparameters and makes the training process oscillate or even diverge. To address this issue, the sensitivity of the network hyperparameters for both physics-informed loss functions are further investigated. Numerical experiments demonstrate that the proposed approaches have good accuracy and efficiency with fine-tuned hyperparameters. Furthermore, the post-test illustrates that the EPINN has excellent interpolation performance and can obtain good extrapolation results under certain restrictions.

The Darwin model, which simultaneously incorporates resistive, capacitive, and inductive effects but neglects the radiation one, has recently attracted more and more attention in the research area. For our industrial application needs, the finite-element (FE) system to solve derived from the Darwin model generally has a large size, which is beyond the capabilities of the direct solvers due to the memory limitation. In this work, a specially designed formulation amenable to an iterative solver is proposed for industrial applications. Moreover, a detailed comparison of different cases is carried out on two different examples.

The modeling of the capacitive phenomena, including the inductive effects becomes critical, especially in the case of a power converter with high switching frequencies, supplying an electrical device. At a low frequency, the electro-quasistatic (EQS) model is widely used to study the coupled resistive-capacitive effects, while the magneto-quasistatic (MQS) model is used to describe the coupled resistive-inductive effects. When the frequency increases, the Darwin model is preferred, which is able to capture the coupled resistive-capacitive-inductive effects by neglecting the radiation effects. In this work, we are interested in specifying the limits of these models, by investigating the influence of the frequency on the electromagnetic field distributions and the impedance of electromagnetic devices. Two different examples are carried out. For the first one, to validate the Darwin model, the measurement results are provided for comparison with the simulation results, which shows a good agreement. For the second one, the simulation results from three different models are compared, for both the local field distributions and the global impedances. It is shown that the EQS model can be used as an indicator to know at which frequency the Darwin model should be applied.

Deep learning (DL) has attracted more and more attention in computational electromagnetism. Particularly, the Convolutional Neural Network (CNN) is one of the most popular learning models in DL due to its excellent capacity for feature extraction and convergence. The efficiency of CNN mainly depends on how many training samples are needed to effectively converge the network. The sample preparation process often involves a lot of numerical computations, which can be very expensive and time-consuming. In this paper, based on the traditional DL network training procedure, two different approaches, namely adding smart training samples and reference samples, are proposed to help the DL network converge. The smart sample selection is based on a greedy algorithm, which can be applied for both training and reference samples. The influences of these two approaches on the CNN training process are investigated by an example of the coupled magneto-thermal computation applied to a transformer. Numerical results show that the two proposed approaches can significantly help the network to converge and improve the efficiency of the DL model.

The quality of a local physical quantity obtained by the numerical method such as the finite element method (FEM) attracts more and more attention in computational electromagnetism. Inspired by the idea of goal-oriented error estimate given for the Laplace problem, this work is devoted to a guaranteed a posteriori error estimate adapted for the quantity of interest (QOI) linked to magnetostatic problems, in particular, to the value of the magnetic flux density. The development is principally based on an equilibrated flux construction, which ensures fully computable estimators without any unknown constant. The main steps of the mathematical development are given in detail with the physical interpretation. An academic example using an analytical solution is considered to illustrate the performance of the approach, and a discussion about different aspects related to the practical point of view is proposed.

In time domain electromagnetic field problems that have a fast time response with sharp field change or need a long time transient process to get the steady-state solution, a large number of time step iterations are required using the classical time-stepping method. A reduced model based on Sinc interpolation is proposed in this work. The accuracy of the proposed method does not depend on the traditional time-domain discretization step but only the number of sampling points. Two numerical examples, namely an electro-quasistatic problem and a field-circuit coupled problem, are considered to illustrate the accuracy and performance of the proposed method. It is shown that only a few pre-computed samplings are sufficient to obtain a good performance in comparing with the traditional time-stepping method.

This paper aims to investigate the approach combine the deep learning (DL) and finite element method for the magneto-thermal coupled problem.

In this paper, 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 edge and nodal type are adopted in numerical examples to evaluate the performance of the proposed estimators.

In this work, the proper orthogonal decomposition (POD) is applied for parametric analysis in the gauged potential formulation of the Darwin model considering both capacitive and inductive effects. Due to the large contrast in material parameters, the resulted system matrix is ill-conditioned. Also, the condition number of the corresponding snapshot complex matrix is very huge. To improve the stability of the POD method, a structured-preserving strategy is considered and implemented for different unknown potentials, namely the magnetic vector potential A, the electric scalar potential φ, and the Lagrange multiplier p. Besides, a greedy algorithm is proposed to select the snapshots adaptively. Two numerical examples, including a parallel plate capacitor and a modified RLC device structure, are provided to illustrate the capability of proposed POD in model order reduction in frequency domain solvers.

The proper generalized decomposition (PGD) is ana priorimodel order reduction (MOR) method based on a variable-separatedexpression of the problem. Two iterative loops are needed in the PGD algorithm, namely the outer loop for enriching the reductionmodes progressively, and the inner loop for solving each mode by fixed point iterations. Setting the stopping criterion of these twoloops blindly can cause either the inaccuracy of the PGD or a waste of iterations. In this work, a special variable-separated PGDwith edge elements is proposed and implemented on a hexahedral mesh in magnetostatics. Also, an adaptive stopping criterion basedon dual formulations is applied to balance different error components, namely the discretization error, error for outer and innerloops of PGD. A numerical example is given to illustrate the proposed approach

When using magnetic vector potential (MVP)-based formulations for magnetostatic or eddy-current problems, either gauge conditions specifying the divergence of the MVP or tree gauging by eliminating redundant degrees of freedom of the MVP is usually imposed to ensure uniqueness of solutions. Explicit gauging of the MVP is not always necessary since classical iterative solvers can automatically and implicitly fix the gauge as long as the right-hand side vectors are consistent. Besides iterative solvers, implicit gauging is also observed when using state-of-the-art parallel sparse direct solvers (PSDSs), thanks to the built-in functions of handling null-spaces of either real symmetric positive semi-definite matrix systems or those complex symmetric systems from eddy-current problems. Both static and eddy-current examples are solved by PSDS to demonstrate results of local physical quantities or global quantities such as magnetic energy or joule losses. High-order edge/nodal elements are also considered in our numerical examples and it is observed that PSDS can also easily and correctly handle the delicate discrete null spaces.

Calculating the bounds of global energy is an important issue in computational electromagnetism, which can provide guaranteed results when extracting inductance parameters. In this paper, an improved equilibrated type a posteriori error estimate for open boundary magnetostatic problems is proposed. We derive our error estimator based on vector dual formulations, which can be efficiently solved using parallel sparse direct solvers. The new estimator can provide a sharp and guaranteed estimate of the finite-element spatial discretization error. Moreover, the computational cost is cheaper than using existing equilibrated error estimators. Numerical experiments are carried out to showcase the performance of our error estimator, including the modified TEAM workshop problem 13 and the benchmark IEEJ problem.

Dual formulations are accurate in use for computing energy-related global quantities such as inductance and providing upper and lower bounds of the unknown true values of these global parameters, which is not possible if using a single formulation. Since traditional dual formulations result in totally different algebraic matrix equations, people have to develop two different finite element programs and solve the resultant two algebraic equation systems respectively. In this work two practical dual formulations for open region magnetostatic problems, where the global finite element matrices are exactly the same, are adopted for extracting the winding inductances. Finite element formulation and implementation details are presented. Practical examples having complex windings are solved using the proposed methods to showcase the effectiveness and accuracy. High order FE basis functions are also used to enhance the solution accuracy. The proposed method is highly useful for medium-sized industrial applications by providing guaranteed inductance parameters.

In this paper, a symmetric field-circuit coupled finite element method (FEM) for low-frequency (LF) full-wave Maxwell problems using a magnetic vector potential (MVP) formulation is proposed. The resultant fully-discrete coefficient matrix is made symmetric for the first time by introducing the so-called source electric scalar potential (ESP) for solid conductors, where the terminal currents are converted from surface integrations of the current density vectors to volumetric integrations. Numerical examples, including a benchmark capacitor charging problem with external circuit connections, are solved and the numerical results match well with reference solutions. The proposed formulation is useful when analyzing electromagnetic fields with coupled inductive-capacitive effects and external circuit connections.

An improved starting point Newton method applied to 3-D scalar formulation in magnetostatics is proposed in this paper. Compared with the classical Newton method, the inexact-Newton and quasi-Newton methods are reported by testing on a benchmark problem as well as an industrial example. Remarkable convergence acceleration using the proposed strategy is observed, and thus, it significantly reduces the computational time.

In this paper, a novel potential formulation for low-frequency (LF) applications taking into account both inductive and capacitive effects but without considering wave propagation is proposed. Both time-domain and frequency-domain formulations are presented. The resultant fully discrete finite-element matrix is made symmetric by incorporating a gauge condition and also rewriting the current continuity equation. To improve numerical accuracy and computational efficiency, high-order mixed-edge elements and nodal elements are adopted to approximate the vector and scalar unknown variables together with high-order time-stepping schemes. Several numerical examples are solved to validate and showcase the accuracy of the proposed methods. The proposed formulations are stable in use for LF electromagnetic field computations by considering inductive and capacitive effects simultaneously, such as finding the resonant frequencies of wireless power transfer devices.

In this paper, we develop adaptive inexact versions of iterative algorithms applied to finite element discretizations of the linear Stokes problem. We base our developments on an equilibrated stress a posteriori error estimate distinguishing the different error components, namely the discretization error component, the (inner) algebraic solver error component, and possibly the outer algebraic solver error component for algorithms of the Uzawa type. We prove that our estimate gives a guaranteed upper bound on the total error, as well as a polynomial-degree-robust local efficiency, and this on each step of the employed iterative algorithm. Our adaptive algorithms stop the iterations when the corresponding error components do not have a significant influence on the total error. The developed framework covers all standard conforming and conforming stabilized finite element methods on simplicial and rectangular parallelepipeds meshes in two or three space dimensions and an arbitrary algebraic solver. Implementation into the FreeFem++ programming language is invoked and numerical examples showcase the performance of our a posteriori estimates and of the proposed adaptive strategies. As example, we choose here the unpreconditioned and preconditioned Uzawa algorithm and the preconditioned minimum residual algorithm, in combination with the Taylor–Hood discretization.

Source tensor projections are developed for the magneto-elastic coupled problems when magnetostriction-induced force and magnetic force are considered. Comparisons with classical force density projection are first performed on a simple example. Then, it is investigated on an application of a multilayer transformer core with the consideration of material anisotropy and multilayer inhomogeneity.

In this paper, we consider some potential formulations of electrostatic as well as time-harmonic eddy current problems with voltage or current excitation sources. The well-posedness of each formulation is first established. Then, the reliability of the corresponding residual-based a posteriori estimators is derived in the context of the finite element method approximation. Finally, the implementation in an industrial code is performed, and the obtained theoretical results are illustrated on an academic and on an industrial benchmark.

In this paper, we propose an a posteriori error estimator for the numerical approximation of a stochastic magnetostatic problem, whose solution depends on the spatial variable but also on a stochastic one. The spatial discretization is performed with finite elements and the stochastic one with a polynomial chaos expansion. As a consequence, the numerical error results from these two levels of discretization. In this paper, we propose an error estimator that takes into account these two sources of error, and which is evaluated from the residuals.

In the modeling of eddy current problems, potential formulations are widely used in recent days. In this paper, the results of residual-based a posteriori error estimators, which evaluate the discretization error in the finite-element computation, are extended to the case of several kinds of source terms for both A/φ and T/Ω harmonic formulations. The definitions of the estimators are given and some numerical examples are provided to show the behavior of the estimators.

A strategy of mesh adaptation in eddy current finite element modeling is developed from both residual and hierarchical error estimators. Wished distributions of element sizes of adapted meshes are determined from the element-wise local contributions to the estimators and define constraints for the mesh generator. Uniform distributions of the local error are searched.

Mesh-to-mesh field transfer arises frequently in finite element computations. Typical applications may concern remeshing, multigrid methods, domain decomposition and multi-physics problems. For electromagnetic fields, one of the essential constraints in such transfers is to conserve energetic quantities such as the magnetic energy and the joule heating. Within the framework of Galerkin projection on overlapping domains, we introduce the definition of energetic norms for electromagnetic fields. The corresponding formulations we propose, provide energy-conserving projection of electromagnetic fields between different meshes.

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.

This paper is devoted to the derivation of a Helmholtz decomposition of vector fields in the case ofmixed boundary conditions imposed on the boundary of the domain. This particular decomposition allows to obtain a residual a posteriori error estimator for the approximation ofmagnetostatic problems given in the so-called A-formulation, for which the reliability can be established. Numerical tests confirm the obtained theoretical predictions.

In finite element computations, the choice of the mesh is crucial to obtain accurate solutions. In order to evaluate the quality of the mesh, a posteriori error estimators can be used. In this paper, we develop residual-based error estimators for magnetostatic problems with both classical formulations in term of potentials used, as well as the equilibrated error estimator. We compare their behaviors on some numerical applications, to understand the interest of each of them in the remeshing process.

Purpose: In this paper, the aim is to propose a residual‐based error estimator to evaluate the numerical error induced by the computation of the electromagnetic systems using a finite element method in the case of the harmonic A‐φ formulation. Design/methodology/approach: The residual based error estimator used in this paper verifies the mathematical property of global and local error estimation (reliability and efficiency). Findings: This estimator used is based on the evaluation of quantities weakly verified in the case of harmonic A‐φ formulation. Originality/value: In this paper, it is shown that the proposed estimator, based on the mathematical developments, is hardness in the case of the typical applications.

For eddy current problems, potential formulations are widely used nowadays. In this paper, residual based a posteriori error estimators are introduced to evaluate the discretization error in the finite element calculation in both case of A /φ and T /Ω harmonic formulations. A device with eddy currents is studied in order to show the efficiency of the proposed estimators.

In this paper, we focus on an a posteriori residual-based error estimator for the T/Omega magnetodynamic harmonic formulation of the Maxwell system. Similarly to the A/Phi formulation, the weak continuous and discrete formulations are established, and the well-posedness of both of them is addressed. Some useful analytical tools are derived. Among them, an ad-hoc Helmholtz decomposition for the T/Omega case is derived, which allows to pertinently split the error. Consequently, an a posteriori error estimator is obtained, which is proven to be reliable and locally efficient. Finally, numerical tests confirm the theoretical results.

This paper is devoted to the derivation of an a posteriori residual-based error estimator for the A-Phi magnetodynamic harmonic formulation of the Maxwell system. The weak continuous and discrete formulations are established, and the well-posedness of both of them is addressed. Some useful analytical tools are derived. Among them, an ad hoc Helmholtz decomposition is proven, which allows to pertinently split the error. Consequently, an a posteriori error estimator is obtained, which is proven to be reliable and locally efficient. Finally, numerical tests confirm the theoretical results.

This paper aims to propose an Improved Starting Point (ISP-) Newton method applied to vector potential A formulation for circuit coupled magnetostatic problems. These problems are usually known to vary greatly during the time. This impacts the quality of the initial guess of the Newton method and thus increases significantly the computational cost. The Newton method with vector potential formulation A has been analyzed. Numerical examples show the performance of our proposed ISP-Newton method.

In this paper, a comparison between the model order reduction technique and deep learning technique is proposed, from the aspect of the surrogate model construction in computational electromagnetism. The merit and demerit of both approaches are discussed and compared via an academic application of magneto-thermal coupled analysis.

Electromagnetic non-destructive testing (NDT) of industrial steels is mostly employed to detect defects such as cracks or mechanical properties inhomogeneities. However, NDT can also be employed, when properly calibrated, to evaluate quantitatively these mechanical properties. It is especially of interest when rigorous quality testing must be achieved on finished steel-based products of which mechanical characteristics must be guaranteed. In that context, the proposed work deals with the evaluation of electromagnetic NDT to characterize the tensile properties of high mechanical performance steels.

The modeling of the capacitive phenomena including the inductive effects becomes critical, especially in the case of a power converter with high switching frequencies supplying an electrical device. To capture the coupled capacitive-inductive effects, the Darwin model is invoked in the frequency domain, which only neglects the radiation effects of the full Maxwell system. In this work, we are interested in the comparison of the different solvers, both iterative and direct, for ungauged and gauged finite-element (FE) systems for Darwin model. In addition, to handle the huge FE resulting matrix due to the industrial application, a novel formulation is proposed.

The Galerkin projection provides an useful tool to transfer electromagnetic fields between different meshes. Given an electromagnetic field calculated on the source mesh, the transfer to a different mesh can be employed for model-coupling, domain decomposition, remeshing, visualization and similar proposes. The Galerkin projection consists of calculating a target field which minimizes the interpolation error between two discretized fields. However, the $L^2$ Galerkin projection suffers from non-conservation of the electromagnetic energy. In this paper, we present an energetic approach for Galerkin projections.

The finite element computation of eddy current problems gives numerical error. This error cannot be calculated, but can only be estimated. Among all error estimators already developed, it is proposed to compare two proven estimators called residual and hierarchical error estimators.

For the eddy current problem, the potential formulations are widely used today. In this communication, residual based a posteriori error estimators are introduced to evaluate the discretization error in the finite element calculation in both case of A-Phi and T-Omega harmonic formulations. An example is carried out to show the behavior of our estimators.

In this paper a residual-based error estimator is proposed to evaluate the numerical error induced by the computation of the electromagnetic systems using a Finite Element Method in the case of the harmonic A-Phi formulation. This estimator is based on the evaluation of quantities weakly verified by the formulation. Furthermore as this estimator verifies the notions of reliability and efficiency it allows to estimate the qualities of local and global solutions. Two examples of electromagnetic systems with current density induced are used to show the efficiency of the proposed estimator.

In time-domain electromagnetic fields computation, numerical methods (such as Finite Element Method (FEM), Finite Integration Technique (FIT) [1-3] and etc.) have been applied. For the time- domain integration solution, explicit and implicit schemas have been widely used. In comparison with implicit methods, the explicit methods are easier to realize in terms of computation complexity, however, they are constrained by the stability condition. This condition may require a small time step and therefore a prohibitive computing time. Another possibility is to use an implicit schema which ensures the numerical stability. Unfortunately the implicit methods require equation solved at each time step [4]. As a consequence, despite of a free choice on time step, the computation time using the full implicit methods increases. In this paper, fixed-point explicit calculation is introduced to an implicit schema. This method combines the two advantages of implicit and explicit methods: no stability condition and no equation solving. The solver is then applied to a time-domain eddy current problem. Using orthogonal mesh cells and FIT, the mass-matrices in discrete formulations are diagonal. The fixed- point explicit method allows direct calculations without matrix inversion or decomposition.

In finite element computations, the choice of the mesh is crucial to obtain an accurate solution. In order to evaluate the quality of the mesh, a posteriori error estimators can be used. In this paper, we analyze and compare the residual and equilibrated error estimators for magnetostatic problems.

In this paper, a surrogate model base on deep learning (DL) is proposed to predict the hysteresis loops of ferro-magnetic materials. The databases of hysteresis loops were measured on the MPG200D Brockhaus equipment with an Epsteinframe. In order to reduce the experimental cost and accelerate measurements, a surrogate model based on the convolutional neural network (CNN) is proposed. First, presenting the measurement results in the form of 256× 256 × 1 images and extract the 6 most characteristic parameters, namely peak magnetic flux density, frequency, maximum magnetic field strength, remanence, coercivity, and the area of the hysteresis loop. All these physical parameters are taken as the label in the supervised DL process. These labe linformation are normalized to form a Gaussian distribution image of 256× 256 × 3 as the input, and the corresponding B-H curve is the output. Using image-to-image CNN U-net, once the network is effectively trained, the hysteresis loops under other excitation parameters can be predicted without further measurement. Our numerical examples show that the prediction results agree well with the measurement results. The sensitivity of CNN for hysteresis loops prediction with respect to the hyperparameters are investagted. A set of empirical hyperparameter configurations isput forward to guarantee an efficient convergence. This research shows that the proposed approach can be an efficient tool to predict the hysteresis loop of ferromagnetic materials under different circumstances, which can potentially contribute to nonlinear hysteresis FEM computation.

La simulation électromagnétique est un outil essentiel en génie électrique et joue un rôle crucial dans le développement des jumeaux numériques en tant que représentations virtuelles de systèmes physiques. Une solution de champ précise des modèles électromagnétiques est indispensable pour la conception et le diagnostic des dispositifs électromagnétiques, en particulier lorsqu’un comportement local détaillé est requis. Cependant, à mesure que les modèles électromagnétiques deviennent plus larges et plus complexes, les simulations précises et efficaces deviennent de plus en plus difficiles. Par conséquent, il est crucial de trouver un compromis entre précision et coût de calcul. Ce manuscrit décrit mes activités de recherche, menées principalement au sein de l’équipe OMN du laboratoire L2EP de l’Université de Lille. Il se compose de deux parties principales : l’étude du modèle de Darwin et l’investigation au Deep Learning. Dans la première partie, l’étude du modèle de Darwin est présentée. En négligeant les effets de rayonnement, le modèle de Darwin peut prendre en compte simultanément tous les effets résistifs, inductifs et capacitifs, montrant un grand potentiel dans les applications impliquant des composants électroniques de puissance modernes à haute fréquence. Nous présentons notre développement de formulations mathématiques et de techniques numériques associées, telles que l’estimation d’erreur a posteriori, la réduction d’ordre de modèle et la validation d’exemples industriels. Dans la deuxième partie, des travaux préliminaires d’investigation sur l’application du Deep Learning au calcul des champs électromagnétiques, sous l’hypothèse de peu de données, sont présentés. La motivation de cette partie était d’étudier si les techniques de Deep Learning peuvent aider à gérer les difficultés numériques qui se posent dans la construction de méta-modèles lors de l’application d’autres méthodes de réduction de modèle. Différentes approches sont proposées pour améliorer la précision et l’efficacité du modèle généré par la technique de Deep Learning. Enfin, des travaux de recherche en cours concernant l’estimation d’erreur pour la quantité d’intérêt et les outils numériques pour un placement optimal des capteurs sont présentés. Le manuscrit décrit également brièvement les perspectives de recherche futures.

Résumé Ce travail s’intéresse à la résolution numérique par éléments finis des équations de Maxwell en régime quasi-stationnaire et en formulations potentielles. L’objectif poursuivi consiste à développer des estimateurs d’erreur a posteriori résiduels, afin de contrôler l’erreur de discrétisation spatiale, dans le cadre d’applications en régime statique ou en régime dynamique harmonique.La première partie de cette thèse est composée de deux chapitres. Le premier est consacré à la modélisation des phénomènes physiques étudiés et à l’obtention des équations mathématiques en résultant. Dans le second, on présente les estimateurs a posteriori et leur intérêt dans le cadre de la mise en oeuvre de la méthode des éléments finis. On détaille notamment les notions de fiablité et d’efficacité d’un estimateur. La deuxième partie se décompose en trois chapitres. Le premier développe l’estimateur a posteriori dans le cas de la magnétostatique en formulation potentielle vecteur A. Les outils mathématiques nécessaires à l’étude sont en particulier détaillés. L’estimateur obtenu est alors validé sur quelques cas tests académiques. Le deuxième traite de l’estimateur a posteriori pour la formulation magnétodynamique en potentiel A/φ en régime harmonique. Un soin particulier est apporté pour générer une décomposition de Helmholtz ad hoc permettant d’obtenir la fiabilité de l’estimateur. Plusieurs configurations sont traitées en fonction de la position du domaine conducteur dans le domaine de calcul et des conditions aux limites associées. Un test numérique est ensuite effectué. Le troisième chapitre est consacré à l’estimateur d’erreur a posteriori pour la formulation T/Ω en régime harmonique pour le problème de la magnétodynamique, en supposant le domaine conducteur simplement connexe. Similairement à la formulation A/φ, une décomposition de Helmholtz est développée pour établir la fiabilité. Une validation numérique est proposée. Enfin, la troisième partie présente une batterie de tests numériques applicatifs et industriels permettant de tester les estimateurs développés dans des conditions réelles. Celle-ci se termine notamment par une application de EDF R&D ayant pour objet le contrôle non destructif par courant de Foucault de tubes générateurs de vapeur. abstract We are interested in resolving the Maxwell equations in the case of quasi-stationary and potential formulations when the finite element method is used. The aim of this work is to develop residual-based a posteriori estimators to control the spatial discretization error in magnetostatic and magnetodynamic problems. The first part is decomposed in two chapters. In the first one, the modeling of the physical phenomena involved are proposed and the mathematical equations are derived. Then, in the second one, the definition of the a posteriori estimators and their interest are presented in the context of the finite element method. The particular notions of reliability and efficiency of an estimator are presented. The second part can be decomposed into three chapters. In the first one, a residualbased a posteriori estimator for the vector potential formulation A in the case of magnetostatic problems is developed. Some necessary mathematical tools for the study are particularly detailed. The estimator is then validated by some academic tests. In the second chapter, a residual-based a posteriori estimator for the A/φ magnetodynamic harmonic formulation is developed. An ad-hoc Helmholtz decomposition is derived to obtain the reliability of the estimator. Several configurations are considered according to the position of the conductor domain in the computational domain as well as boundary conditions used. A numerical test is then performed. In the third chapter, a residual-based a posteriori estimator is derived for the T/Ω magnetodynamic harmonic formulation, when the conductor domain is simply connected. Similarly to the A/φ formulation, an ad-hoc Helmholtz decomposition is developed to establish the reliability. A numerical validation is proposed.Finally, in the third part, a set of numerical experiments and industrial applications are presented to evaluate our estimators. It ends with a particular application of EDF R&D focusing on the eddy current non-destructive evaluation of steam generator tubes.