The Proper Orthogonal Decomposition (POD) combined with the (Discrete) Empirical Interpolation Method (DEIM) can be used to reduce the computation time of the solution of a FE model. However, it can lead to numerical instabilities. To increase the robustness, the POD_DEIM model must be constructed by preserving the structure of the full FE model. In this article, the structure preserving is applied for different potential formulations used to solve electromagnetic problems.

Proper Orthogonal Decomposition (POD) has been successfully used to reduce the size of the equation system and the computation time of linear Finite Element (FE) problems. With a nonlinear behavior law, the POD is not so efficient due to the computation cost of nonlinear entries of the full FE model. Then, the POD approach must be combined with an interpolation method of nonlinear terms to obtain an efficient reduced model. An interpolation method consists on the computation of a small number of nonlinear entries and on the interpolation of other terms. Different methods have been presented to select the set of nonlinear entries to be calculated. Then, the (Discrete) Empirical Interpolation method ((D)EIM) and the Best Points Interpolation Method (BPIM) have been developed. In this article, we propose to compare two reduced models based on the POD-(D)EIM and on the POD-BPIM in the case of nonlinear magnetostatics coupled with electric equation.

In this paper a method in 2-D frequency domain is presented to simulate a laminated iron core with a short-circuit between several magnetic sheets. The approach consists in coupling homogenization methods and finite element method. The defect is modeled with A* modified vector potential formulation and the rest of the structure with a homogenization method. The coupled method is applied to a lamination stack containing a short-circuit and compared to the reference, where the A* formulation is applied on the whole domain. Finally, a thermal modeling of lamination stack is presented to study the influence of an insulating defect.

Predetermination of losses and inductance values in the design phase, is necessary for the development of high-performance magnetic components for power electronics. Numerical modeling, based on the Finite Element Method (FEM) can be used to determine the characteristics of a particular component with a complex geometry in high frequency (HF). These models are very accurate but the computation time required is high compared to analytical models. The Model Order Reduction (MOR) methods can be applied to reduce the computation time while maintaining high accuracy. Nowadays, the Proper Orthogonal Decomposition (POD) is the most popular of MOR approaches. This technique has been applied to study problems in many elds of engineering. In this paper, the POD method is developed to solve magneto-harmonic problems in order to study a planar magnetic inductor.

In numerical computation, the finite element (FE) method associated with external electric circuits is often used to evaluate electromagnetic devices with voltage sources. To study the solution of the steady state, the computation time can be prohibitive due to a large transient state compared to the time step used to discretize the time domain. In this paper, a method based on Waveform Relaxation Method is developed in order to impose the steady state of the solution in the case of a nonlinear magnetostatic problem coupled with electric circuit equations.

Model order reduction (MOR) methods are more and more applied on many different fields of physics in order to reduce the number of unknowns and thus the computational time of large-scale systems. However, their application is quite recent in the field of computational electromagnetics. In the case of electrical machine, the numerical model has to take into account the nonlinear behaviour of ferromagnetic materials, motion of the rotor, circuit equations and mechanical coupling. In this context, we propose to apply the proper orthogonal decomposition combined with the (Discrete) empirical interpolation method in order to reduce the computation time required to study the start-up of an electrical machine until it reaches the steady state. An empirical offline/online approach based on electrical engineering is proposed in order to build an efficient reduced model accurate on the whole operating range. Finally, a 2D example of a synchronous machine is studied with a reduced model deduced from the proposed approach.

The stray load losses (SLLs) in electrical machines represent a non-negligible contribution of the total losses and a key point for an accurate evaluation of the energy efficiency of considered device. In this paper, one aspect of these SLLs, the end-region leakage fluxes and losses, is investigated and considered for the case of a high-power cage induction motor. The study is performed at locked rotor, no-load, and rated load conditions using a 3-D finite-element modeling approach. The influence of the leakage flux on the end-region conductive parts of the motor is analyzed together with the eddy current loss calculation. Finally, the SLLs are calculated and compared with the experimental measurements based on the IEEE standard 112-method B test.

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.

We present an optimization problem that requires the modeling of a multirate system composed of subsystems with different time constants. We use waveform relaxation method (WRM) in order to simulate such a system, but computation time can be penalizing in an optimization context. Thus, we apply output space mapping (OSM) that uses several models of the system to accelerate optimization. WRM is one of the models used in OSM.

To solve multi-physics problems, weak coupling of finite element calculations can be carried out: the subproblems of which the physical nature differs, are solved separately on their own meshes. In this case, Galerkin projection provides useful tool to ensure the transfer of physical fields between different meshes. In terms of implementation, the Galerkin projection system can be either accurately assembled over the intersection of two meshes or approximately integrated over the target mesh. This paper describes and compares these two implementation techniques for the Galerkin projection.

Coupled problems are made up of subproblems of which the physical nature differs. Using indirect coupling models, the subproblems are calculated separately on their own meshes to ensure precision. To obtain a precise solution for the total problem, it is important to ensure the transmission of information between the subproblems. In this paper, we present field projection methods on overlapping domains. In comparison to earlier works, the classical $L^{2}$ or $mathbf{L}^{2}$ projection theory is extended to $H(mathbf{grad})$, $mathbf{H}(mathbf{curl})$ and $mathbf{H}(div)$ to obtain increased projection accuracy for the distributional derivatives. A Petrov-Galerkin method is then presented to fill the test space using a bi-orthogonal basis, without losing the optimality of the result in comparison to the $L^{2}$ or $mathbf{L}^{2}$ Ritz-Galerkin method. Using the Petrov-Galerkin method and bi-orthogonal test functions, the projection is presented using a diagonal matrix. However, in the standard Ritz-Galerkin projections, a linear system must be solved.

Short circuits in turbo-generator stators can lead to a modification of the magnetic flux distribution that impacts the performances and, in some extreme cases, can lead to local damage of the iron core. This work presents the modeling of short circuited laminations in a stator yoke of a turbo-generator. A 3D finite element (FE) model, associated to a homogenization technique, is used to calculate the short circuit current. The results are compared with the experiment, especially for the electrical signature of the diagnosis test known as Electromagnetic Core imperfection detector (El Cid).

Overlapping elements can be used to connect non-conforming meshes in the finite-element method. This approach has been developed with the scalar potential formulation and used to solve magnetostatic problems but not in the case of the vector potential formulation. In this paper, we propose to introduce the overlapping element method in this second formulation

An approach based on the double Lagrange multipliers is developed in the finite element method in order to impose complex periodic or anti-periodic boundary conditions. The magnetostatic equations are solved using the vector or scalar potential formulations. In order to show the possibilities of the proposed approach, an example of application is studied and the results are discussed.

In this paper, we propose to introduce the single and double Lagrange multipliers approaches in the case of the finite element method (FEM). These approaches allow non-conforming meshes to be linked together. The methods introduced are developed in the case of a magnetostatic problem solved by the scalar potential formulation. An application is studied and the results obtained by both approaches are compared.

Coreplates in large generators may suffer from local short circuits. An accurate analysis is required to avoid these failures and detect them when occurring. The purpose of this paper is to develop a lamination stack model compliant with interlamination default analysis.

The accuracy of the magnetic field strength estimation in rotational single sheet tester is investigated with the finite element method and measurements. The incorporation of a vector hysteresis model, derived from the original scalar Jiles-Atherton model, in a finite element magnetic field code is performed. The application of this method leads to use the differential reluctivity tensor that arises naturally from the vector model. The effect of the shielding on a Rotational Single Sheet Tester is analyzed in order to improve the field homogeneity in the sample area.

For designers, calculation of local fluxes can be very useful. In the vector potential formulation, the local fluxes can be easily deduced. In the scalar potential formulation, the determination of these fluxes presents some difficulties. In this paper, we present three methods to compute a flux through any surface in the scalar potential formulation. These are compared with the one used in the vector potential formulation for two application examples.

In the field of computational electromagnetics, taking into account the rotation of a sub-domain is required to simulate certain devices such as electrical machines. Several methods have been proposed in the literature, but they remain quite difficult to implement. In this paper, we propose a sliding surface method based on a spatial Fourier interpolation in order to take into account any rotation angle with a very simple numerical implementation.

Harmonic Balance Finite Element method combined with homogenization method is used to model lamination stacks. The Harmonic Balance gives directly the steady-state solution and the homogenization method reduces the number of unknowns. The numerical model takes into account the nonlinear magnetic behavior and the electric conductivity. The results of the proposed method are compared with those obtained from a classic approach.

The Harmonic Balance combined with the finite element method enables to obtain the steady-state solution of electromagnetic problems. In this communication, we propose to apply this approach to study a laminated iron core submitted to a magnetic flux arising from a Pulse Width Modulation (PWM) voltage. The proposed numerical model takes into account the nonlinear magnetic behaviour, the electric conductivity and a short circuit between steel sheets.

In the field of computational electromagnetics, taking into account the rotation of a sub-domain is required to simulate certain devices such as electrical machines. We propose a sliding surface method based on a spatial Fourier interpolation in order to take into account any rotation angle with a very simple numerical implementation.

The Proper Orthogonal Decomposition combinedwith the (Discrete) Empirical Interpolation Method can be used to reduce the size of a numerical model. To conserve robustness, the reduced model must be constructed by preserving the structure of the full model. In this communication, an approach is proposed with potential formulations used to solve electromagnetic problems.

This paper presents a 3-D finite element method to determine the steady state operation for magnetostatic problems coupled with electrical circuit equations. For this purpose, the waveform relaxation method combined with the Newton method (WR-NM) is developed. This method is especially suitable to long transient problems. To show the efficiency of this approach a three-phase transformer is studied. The results show that the WR-NM becomes very efficiency when the transient state is more important than about 10 periods.

In this paper, the Harmonic Balance Method is used to obtain the steady-state solution of a laminated iron core with an imposed magnetic flux. The numerical model takes into account the nonlinear magnetic behaviour, the electric conductivity and a short circuit between steel sheets. The results of the Harmonic Balance Method are compared with those obtained from a classic approach.

In the domain of numerical computation, Model Order Reduction (MOR) methods are more and more applied in mechanics and have shown their efficiency in terms of computation time and memory requirements. In computational electromagnetics, research has started recently and the different methods available in the literature need to be compared in order to find the most efficient one. We propose to evaluate MOR approaches in order to solve linear magnetoquasistatic field problems. Therefore, the Proper Orthogonal Decomposition (POD), the Centroidal Voronoi Tessellation (CVT), the Proper Generalized Decomposition (PGD) and the Arnoldi-Krylov projection (AKP) are developed and compared.

Model Order Reduction (MOR) methods are an active research field in the numerical analysis domain. They are applied to many different areas in physics, especially in mechanics because they allow to dramatically reduce the computational time. MOR is quite recent in electromagnetics and needs still to be investigated. The Proper Orthogonal Decomposition (POD) is the most famous one and has already shown very promising results. However, the POD approach minimizes the error in the L2 sense on the whole domain and cannot be very accurate to calculate quantities of interest, like flux associated with a probe in region where the field is low. In this communication, we present the Balanced Proper Orthogonal Decomposition (BPOD) which extends the POD by taking account of probes in its model. The BPOD and POD approaches will be compared on a 3D linear magnetoquasistatic field problem.

In numerical computation, the finite element (FE) method associated with external electric circuits is often used to evaluate electromagnetism devices with voltage sources. To study the solution of the steady state, the computation time can be prohibitive due to a large transient state compared with the time step used to discretize the time domain. In this communication, the Waveform Relaxation Method is developed to impose the steady state of the solution in the case of magnetostatic problem coupled with electric cricuit equation.

Two model order reduction approaches are applied to reduce the computational time of a quasistatic finite element problem. Both methods are compared on an academic example.

The modeling of a multirate system -formed by components with heteroge- neous time constants- can be done using fixed-point method. This method allows a time- discretization of each subsystem with respect to its own time constant. In an optimization process, executing the loop of the fixed-point at each model evaluation can be time consum- ing. By adding one of the searched waveform of the system to the optimization variables, the loop can be avoided. This strategy is applied to the optimization of a transformer.

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.

For the optimization of a component from a multirate system, the article presents benefits of the joint use of several models for the optimization and of fixed-point strategy for the modeling. Thus, space mapping allows reducing time computation of an optimization process by limiting the number of evaluations of time consuming models. In the case of multirate problems, these models can be modeled by Waveform Relaxation Method to provide an additional time saving.

Usually, coupled problems (e.g. magneto-mechanical or magneto-thermal problems) with weak interactions can be decomposed into several sub-problems and solved separately. However, in the area of finite element computation, depending on the nature of the sub-problem to solve, the used meshes generally differ. In this case, transfers of fields are required to take into account the interaction between the sub-problems. Given a previously-computed field, in order to obtain on the target mesh its approximate image, the Galerkin projection enjoys several advantages over interpolation, especially in terms of precision. This method consists of minimizing the norm of the interpolation error in the chosen target space. The accuracy of the solution is ensured by the quasi-best property of the Galerkin method. Nevertheless, in practice the general implementation of the Galerkin projection has been proved challenging. As the source and target fields are discretized on different meshes, the accurate assembly of the projection system necessitates the numeric integrating over an intermediate mesh. In practice, such an intermediate mesh can be calculated by means of intersecting the source and target meshes. However, the use of intersection requires heavy work on programming and high computation cost (P.E. Farrell and J.R. Maddison, ”Conservative interpolation between volume meshes by local Galerkin projection”, in Comput. Methods Appl. Mech Engrg, Vol. 200, pp 89-100, 2009). This proposed paper discusses the possibility to avoid mesh intersections, by using an approximate integrating algorithm. Instead of integrating over the lowest common mesh, the integrals are computed over the target mesh. Over the quadrature points of targets elements, the source solution is interpolated and approximately integrated. This is less accurate than integrating over intersections. However, the accuracy can be improved using high-oder quadrature rules. Through analytical field distributions such as skin effect current densities, we have compared this approximate method to the intersection method, on node, edge and facet Whitney elements. The robustness of the approximate method will also be studied using reciprocal projection between fine and coarse meshes. In our examples of projection on edge elements, the computation time can be reduced by a factor up to one hundred. Meanwhile, the lost accuracy is less than 0.6%. The integration of physical constraints in the projection formulation will be shown to improve coarse-to-fine-mesh projected solutions.

In the communication, the WRM and the POD techniques are applied to study a transformer supplied by a power converter. The transformer is modeled by Finite Element Method. A reduced model of the transformer provided by the POD tech- nique is used to couple the power converter and the transformer using the WRM technique.

Short-circuits in turbo-generator stators can lead to a modification of the magnetic flux distribution that impact the performances and, in some extreme cases, to local damage of the iron core. This work presents the modeling of short-circuited laminations in a stator yoke of a turbo-generator. A 3D finite element (FE) model, associated to a homogenization technique, is used to calculate the short-circuit current. The results are compared with the experiment, especially for the electrical signature of the diagnosis test known as Electromagnetic Core imperfection detector (El Cid).

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.

This paper deals with the modelling of a lamination stack, crossed by a conducting building bar, in the presence of an inter-lamination short-circuit. Under a timevarying magnetic flux, an induced current loop flows through both the building bar and the fault. To reduce computational times, a homogenization technique is used to model the lamination stack. Eddy current losses and total magnetic energy values are compared for cases with and without homogenization for 50 Hz and 100 Hz.

In this paper, we propose to compare two types of approximation of the vector potential used to solve a magnetostatic problem in the case of the domain decompostion. The single and Lagrange multipliers approaches and the Mortar method are considered in this context.

This paper presents a simple model to simulate the currents in a transformer, especially during the transient state. The model is based on a reluctance circuit using a finite element (FE) tool for identifying the global behaviour laws of the transformer iron core. A three-phase three-limb transformer is modelled to test the proposed approach.

In this paper we compare different approaches for the lamination stack homogenization with a short-circuit due to an electrical fault between several sheets. The presented techniques are based on the determination of a complex equivalent magnetic permeability for the homogenized domain. The eddy current losses are calculated in the homogenized lamination stack and in the fault. These losses are compared with those obtained from a reference system where the lamination is modeled with its real geometry.

This paper deals with the identification of a new high power starter-alternator system, using both: a Finite Element Method (FEM) modeling and an elementary experimental vector control. The drive is composed of a synchronous 7-phase claw-pole machine supplied with a low voltage / high current Voltage Source Inverter (VSI). This structure needs specific approaches to plan its electrical and mechanical behaviors and to identify the parameters needed for control purpose. At first, a Finite Element Method (FEM) modeling of the machine is presented. It is used for the predetermination of the electromotive forces and of the torque. Experimental results are in good accordance with numerical results. In a second part, resistive and inductive parameters of the drive are determined by an original experimental approach that takes into account each component of the drive: the battery, the VSI and the machine.

The analysis of micro-electro-mechanical systems (MEMS) requires to take into account both electric and mechanical phenomena. An iterative FEM approach, which involves the computation of the electrostatic field and the displacement of the moving part, is presented. The maximum displacement allows computing the pull-in voltage.

For the experimental seven-phase machine studied in this paper, parameters necessary for the control such as inductances and electromotive force (EMF) are sensitive to harmonics. A conventional analytical method in which only the first harmonic is taken into account does not give good results. Besides, the machine is an axial-flux one and has two asymmetrical rotors: a 3D-FEM is then necessary. Comparisons between predeterminations and experimental results show sufficient accuracy to achieve a control model.

Specific actuators are now widely used. The design and study of such a converter must be carried out using specific models. This paper deals with the study of a permanent magnet planar actuator with Multiple Degrees of Freedom (MdoF). A 3D permeance network model is used to design the actuator and study its performance. The 3D-FEM is also used to investigate more accurately the prototype behavior. A prototype of the actuator has been built. Results, given by both models are shown and compared with those obtained experimentally.

Nous proposons une approche de couplage entre une formulation éléments finis spectrale et une approche d’homogénéisation afin de modéliser un empilement de tôles. L’approche spectrale permet d’obtenir directement le régime permanent, l’homogénéisation permet de réduire le nombre d’inconnues et la formulation classique nous permet de prendre en compte les effets de bord des tôles. Notre approche prend en compte une loi de comportement magnétique non linéaire et une conductivité électrique dans les tôles. Les résultats de notre approche de couplage sont comparés aux résultats d’une approche classique de type éléments finis pas à pas dans le temps.

La prédétermination, dans la phase de dimensionnement, des pertes et des valeurs d’inductances est nécessaire à l’élaboration de composants magnétiques performants pour l’électronique de puissance. La modélisation numérique peut être utilisée pour déterminer les caractéristiques propres d’un composant notamment lorsque sa géométrie est "complexe" ou que la fréquence d’utilisation est élevée. Ces modèles sont très précis mais le temps de simulation nécessaire est important par rapport à des modèles analytiques plus simplistes. Les méthodes de réduction de modèles, type POD ou autres, permettent de diminuer le nombre de ces simulations numériques tout en conservant une grande précision.

Une méthode de projection de champs entre deux maillages non coïncidents basée sur une approche énergétique est développée. Cette approche est proposée en magnétostatique pour les deux formulations en potentiel classiquement utilisées. Un exemple académique permet de montrer les possibilités.

To compute the magneto-elastic problems, a chaining of III. CALCUL DES FORCES finite element calculations is realized. The coupling term is magnetic Diverses m�thodes peuvent �tre envisag�es pour le calcul force. To compute it and realize the chaining, various methods are ana- des forces : m�thode des travaux virtuels, forces de Lorentz lyzed. The approaches are then compared to an academic example with [1, 2]. Ces diff�rentes m�thodes sont th�oriquement �quiva- analytical solution.