Electromagnetic Modeling by Finite Element Methods
Murr 5. McLyman 8. Poularikas 9. Interconnected Dynamical Systems, Raymond A.
DeCarto and Richard Saeks Jacquot Jones McLyman Engelmann Andreas Electromagnetic Compossibility, Heinz M. Schlicke Gottling Taylor Multivariable Control: An Introduction, P. Sinha Edstrom, Jr. Green way, and William P. Kelly Browne, Jr. Kit Sum Emery Mahmoud, Mohamed F. Hassan, and Mohamed G. Darwish Fathi and Cedric V.
Armstrong Sponsored by Ontario Centre for Microelectronics Perry Tzafestas Buchanan Dexter Miniet Fuqua Kussy and Jack L. Warren Salmon Protective Relaying: Principles and Applications, J. Lewis Blackburn Powell Chalam Hnatek Swartzlander, Jr. Bellanger Levinson Alford Kheir Rigling Sakis Meliopoulos Signal Processing Handbook, edited by C. Chen Richard Stillwell Garzia and Mario R. Burke Regalia Handbook of Electric Motors, edited by Richard H. Engelmann and William H.
Middendorf Power-Switching Converters, Simon S. Ang Spatial Electric Load Forecasting, H. Lee Willis Gieras and Mitchell Wing Garzon Radio Receiver Design, Robert C. Dixon Slade Phadke LaCombe Pilot Protective Relaying, edited by Walter A. Elmore Weston New students in this area will find a didactical approach for a first contact with the FEM including some codes and many examples. For researchers and teachers having experience in the area, this book presents advanced topics related to their works as well as useful text for classes.
Our text focuses on three complementary issues. The first is related to a didactical approach of EM equations and the application of the FEM to electromagnetic classical cases. The second one is the coupling of EM equations with other phenomena that exist in electromagnetic structures, such as external electrical and electronic circuits, movement and mechanical equations, vibration analysis, heating, eddy currents, and nonlinearity.
The final issue is the analysis of electrical and magnetic losses, including hysteresis, eddy currents and anomalous losses. This book is intended primarily for graduate students but what must be pointed out is that more and more undergraduate students have been introduced to this area and this is the reason why efforts have been made to use a very didactical approach to the subjects presented in the book. Coupling and losses, advanced topics of the book, have been the objects of a great deal of scientific research in the last two decades and many related technical papers have been published in periodicals and at conferences.
In spite of being active research topics, the content we have chosen is based on well-proven techniques. These may be applied without general restrictions. More classically, FEM is more commonly presented for mechanics and we consider that this brief review of EM is appropriate here. The goal is not to analyze this method very deeply many books with this purpose are available but to bring out the most important aspects of the FEM for EM analysis. It is a concise chapter in which virtually all the FE concepts are introduced and it is clearly shown how they should be linked in order to implement a computational code.
Thermal equations are also included in this chapter. In this chapter much of our experience and advanced research work are extensively described. The formulation reaches advanced phenomena as in, for instance, linking EM devices to converters, whose topology is not known "a priori". It means that the dynamic behavior of the converter is taken into account considering the switching of thyristors, diodes, etc.
Eddy current phenomenon is also treated in "thick" conductors. In this chapter, methods for discretizing airgaps and for simulating the physical displacement are presented. In the final part of this chapter a method based on 2D simulations to take into account the skew effect in rotating machines is proposed.
Many different and commonly employed methods are presented and compared. Also, results on vibrational behavior of EM structures coupling mechanical equations with EM ones are presented. Advanced studies on eddy current, anomalous and hysteresis losses are described. In our text we present modeling for hysteresis and its implementation in a FEM code, using, as indicated above, proven methods. We hope that the book will provide reliable and useful information for students and researchers dealing with EM problems.
Finally, we would like to express our sincere gratitude to many colleagues and friends who helped us to develop the works presented in this book. Without their support it would have been impossible to publish it. We would like specially to thank Dr. Rioux Univ. Paris VI , our thesis advisors, who gave us the scientific background for our research and professional life; Prof.
Ida Univ. Cardoso USP and their teams for continual collaboration and technical support; and Prof. Kost T. Cottbus for his cooperation and technical exchanges. Joao Pedro A. Contents Preface 1. Mathematical Preliminaries 1. Introduction 1. The Vector Notation 1. Vector Derivation 1. The Nabla V Operato 1. Definition of the Gradient, Divergence, and Rotational 1. The Gradient 1. Example of Gradient 1. The Divergence 1. Definition of Flu 1. The Divergence Theorem 1. The Conservative Flux 1.
Example of Divergence 1. The Rotational 1. Circulation of a Vector 1. Stokes' Theorem 1. Example of Rotational 1. Second-Order Operators 1. Application of Operators to More than One Function 1. Expressions in Cylindrical and Spherical Coordinates 2. Introduction 2. The EM Quantities 2. The Electric Field Intensity E 2. The Magnetic Field Intensity 2. The Surface Current Density J 2. Volume Charge Density p 2. The Electric Conductivity 2. Local Form of the Equations 2. The Anisotropy 2. The Approximation to Maxwell's Equations 2.
The Integral Form of Maxwell's Equations 2. Electrostatic Fields 2. The Ele 2. The Electric Field 2. Force on an Electr 2. Nonconservative Fields: Electromotive Force 2. Refraction of the Electric Field 2. Dielectric Strength 2. Magnetostatic Fields 2. Maxwell's Equations in Magnetostatics 2. The Biot-Savart Law 2. Magnetic Field Refraction 2. Energy in the Magnetic Field 2. Magnetic Materials 2. Diamagnetic Materials 2. Paramagnetic Materials 2. Inductance and Mutual Inductance 2. Definition of Inductance 2. Energy in a Linear Syste 2. Magnetodynamic Fields 2.
Maxwell's E namic Field 2. The Equation for H 2. The Equation forB 2. The Equation forE 2. The Equation for J 2. Solution of the Equations 3. Brief Presentation of the Finite Element Method 3. Introduction 3. The Galerki 3. The Establishment of the Physical Equation 3. The First-Order Triangle 3.
Application of the Weighted Residual Metho 3. Application of the Finite Element Method and Solution 3. The Boundary Conditions 3. A First-Order Finite El 3. Example for Use of the Finite Element Program 3. Generalization of the Finite Element Method 3. High-Order Finite Elements: General 3. High-Order Finite Elements: Notation 3. High-Order Finite Elements: Implementation 3. Continuity of Finite Elements 3. Polynomial Basis 3. Transformation of Quantities - the Jacobian 3. Evaluation of the Integrals 3. Numerical Integration 3. Some 2D Finite Elements 3. First-Order Triangular Element 3.
Second-Order Triangular Element 3. Quadrilateral Bi-linear Element 3. Quadrilateral Quadratic Element 3. Coupling Different Finite Elements 3. Coupling Different Types of Finite Elements 3. Calculation of Some Terms in the Field Equation 3. The Stiffness Matri 3. Evaluation of the Second Term in Eq. Evaluation of the Third Term in Eq. Evaluation of the Source Term 3. The Problem to Be Solved 3.
The Discretized Domain 3. The Finite Element Program 4. Introduction 4. Some Static Cases 4. Electrostatic Fields: Dielectric Materials 4. Stationary Currents: Conducting Mater 4. Magnetic Fields: Scalar Potential 4. The Magnetic Field: Vector Pote 4. The Electric Vector Potential 4.
Application to 2D Eddy Current Problem 4. First-Order Element in Local Coordinates 4. The Complex Vector Potential Equation 4. Structures with Moving Parts 4. Axi-Symmetric Application 4. Non-linear Applications 4. Method of Successive Approximation 4. The Newton-Raphson Method 4. Geometric Repetition of Domains 4. Periodicit 4. Anti-Perio 4. Thermal Problems 4. Thermal Conduction 4. Convection Transmission 4.
Radiation 4. FE Implementation 4. Voltage-Fed Electromagnetic Devices 4. Static Examples 4. Calculation of Electrostatic Fields 4. Calculation of Static Currents 4. Calculation of the Magnetic Field - Scalar Potential 4. Calculation of the Magnetic Field - Vector Potentia 4. Dynamic Examples 4. Eddy Currents: Time Discretizatio 4. Moving Conducting Piece in Front of an Electromagnet 4. Thermal Case: Heating by Eddy Currents 5. Coupling of Field and Electrical Circuit Equations 5. Introduction 5. Formulation Using the Magnetic Vector Potential 5.
The Formulation in Two Dimensions 5. Equations for Conductors 5. Thick Conductors 5. Thin Conductors 5. Equations for the Whole Domain 5. The Finite Element Method 5. Equations for Different Conductor Configurations 5. Thick Conductors Connections 5. Series Connection 5. Parallel Connection 5. Independent Voltage Sources 5. Star Connection with Neutral 5. Polygon Connection 5. Star Connection without Neutral Wir 5. Reduced Equations of Electromagnetic Devices 5. Circuit Topology Concepts 5. Determination of Matrices G] to G6 5.
Example 5. Discre 5. Examples 5. Sim 5. A Didactical Example 5. Three-Phase Induction Moto 5. Massive Conductors in Series Connection 5. Movement Modeling for Electrical Machines 6. Introduction 6. Met 6. Methods with Discretized Airgaps 6. The Macro-Element 6. The Moving Band 6.
Examples 6. Thre 6. Permanent Magnet Motor 7. Interaction Between Electromagnetic and Mechanical Forces Introduction 7. Methods Based on Direct Formulations 7. Method of the Magnetic Co-Energy Variation 7. The Maxwell Stress Tensor Method 7. The Method Proposed by Arkkio 7. Examples of Torque Calculation 7. Methods Based on the Force Density 7. Preliminary Considerations 7. Equivalent Sources Formulation 7. Equivalent Currents 7. Equivalent Magnetic Char 7. Other Equivalent Source Dist 7. Formulation Based on the Energy Derivation 7. Comparison Among the Different Methods 7.
Magnetic Force Calculation 7. Mechanical Calculation 7. Calculation of the Natural Response 7. Example of Vibration Calculatio 7. Iron Losses 8. Introduction 8. Eddy Curre 8. Hysteresis 8. Anomalous or Excess Losses 8.
Total Iron Losses 8. Example 8. The Jiles-Atherton Model 8. The JA Equations 8. Determination of the Parameters from Experimental Hysteresi Loops 8. Numerical Algorithm 8. The Inverse Jiles-Atherton Model 8. The Inverse JA Metho 8. Including Iron Losses in 8. Hysteresis Modeling by Means of a Differential Reluctivit 8. Introduction In this chapter we review a few ideas from vector algebra and calculus, which are used extensively in future chapters.
We assume that operations like integration and differentiation as well as the bases for elementary vector calculus are known. This chapter is written in a concise fashion, and therefore, only those subjects directly applicable to this work are included. Readers wishing to expand on material introduced here can do so by consulting specialized books on the subject. It should be emphasized that we favor the geometrical interpretation rather than complete, rigorous mathematical exposition. We look with particular interest at the ideas of gradient, divergence, and rotational as well as at the divergence and Stokes' theorems.
These notions are of fundamental importance for the understanding of electromagnetic fields in terms of Maxwell's equations. The latter are presented in local or point form in this work. The Vector Notation Many physical quantities posses an intrinsic vector character. Examples are velocity, acceleration, and force, with which we associate a direction in space. Other quantities, like mass and time, lack this quality. These are scalar quantities. Another important concept is the vector field. A force, applied to a point of a body is a vector; however, the velocity of a gas inside a tube is a vector defined throughout a region i.
In the latter case, we have a vector field. We use this concept extensively since many of the electromagnetic quantities electric and magnetic fields, for example are vector fields. The Nabla V Operator First, we recall that a scalar function may depend on more than one variable. The nabla is a mathematical operator to which, by itself, we cannot associate any geometrical meaning. It is the interaction of the nabla operator with other quantities that gives it geometric significance. Figure 1. The gradient is orthogonal to a constant potential surface. Defining the vector dM.
Hence, for all differential displacements M and M1 on this surface, we can write dU — 0. From Eq. Assume now that the displacement of M to M" is in the direction of increasing U, as shown in Figure 1. From the foregoing arguments we conclude that grad U is a vector, perpendicular to a surface on which U is constant and that it points to the direction of increasing U. Example off Gradient Given a function r, as the distance of a point M x,y,z from the origin 0 0,0,0 , determine the gradient of this function.
Geometrical representation of the gradient.https://pleanabnaginea.ml/map25.php
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The magnitude of the gradient is gradr y Figure 1. Definition of the direction of gradr. Therefore, grad r points to the direction of increasing r, or towards spheres with radii larger than r, as was indicated formally above. Definition of Flux Consider a point M in the vector field A, as well as a differential surface ds at this point, as in Figure 1.
We choose a point N such that the vector MN is perpendicular to ds. Definition of normal unit vector to a surface ds. The flux is maximum when A and ds are parallel, or, when A is perpendicularly incident on the surface ds. Since ds is a vector, it possesses three components which represent the projections of the vector on the three planes of the system Oxyz see Figure 1. Components of the vector ds. A closed surface in Cartesian coordinates: definition of divergence.
In the case of a closed surface S, n always points to the outward direction of the volume enclosed by the surface S. On the upper surface the normal to the surface is positive, and the component Az of the vector A, is augmented by dAz. Using the same rationale on the other two pairs of parallel surfaces, yields the expression dx dy y dA. Conservative Flux Consider a tube of flux, such that the field of vectors A defines a volume in which the vectors are tangent to the lateral walls, as shown in Figure 1. S3 is the lateral surface of the tube section.
A tube of flux. Because Sj and S2 are arbitrary surfaces, they can be approximated geometrically. It is clear that if S2 tends to Si and at the same time A2 tends to A i, the sum above tends to zero. Since the flux entering the tube is equal to the flux leaving it, we conclude that the flux through the closed surface, in this case, is zero. This leads to the conclusion that the flux is conservative the flux in the tube is conserved.
From the discussion here we conclude that when the flux is conservative, the divergence of the field is zero. A radial vector field. Directions of A and ds for the field in Figure 1.
A circumferential vector field. Example of Divergence Consider a radial vector field as shown in Figure 1. To calculate the flux of the vector A through a spherical shell of radius R, we note that ds and A are collinear and in the same direction as in Figure 1. Examining the transverse cross-section Figure 1. Observing these two examples, we note that in the first case, vectors of the field are literally divergent at point M and, therefore, divergence of the field is different than zero.
In the second example, vectors of the field do not depart but rather suggest rotation or a vortex, divergence. In this case the divergence of the vector is zero. If A and dl are parallel, as in Figure 1. In the first case, the vector A circulates along the contour L, while in the second case, where A is perpendicular to the contour, it does not circulate along the contour L. Definition of circulation of a vector A along contour L.
Maximum circulation. Zero circulation. Assuming that dl is equal to dM in Eq. If the contour L is closed, or, if Q and P coincide, the circulation of A is zero. It remains to be verified if all vectors can be identified by a gradient. Writing A. This theorem will show an important relation for this vector field A. Assume that there is a closed contour defining a surface. This surface is divided into small areas s. We examine one infinitely small rectangle, with sides dx, dy, parallel to the axes of the coordinate system Ox and Oy as in Figure 1.
This is done separately for each section of the contour. A small surface defined by a closed contour. Therefore, we conclude that the flux of a vector A through the open surface S is equal to the circulation of the vector A along the contour L, encircling the surface. A vector field A with constant magnitude at distance R from point M. Example of Rotational Consider the field of vectors A, as indicated in Figure 1.
Defining a surface 5" such that it is enclosed by the circle of radius R, we can calculate the circulation of A along the contour L. Because the vector rot A is perpendicular to A by definition of the cross product , we observe that the geometrical positions of A and rot A are as shown in Figure 1. Another example is shown in Figure 1. Geometric representation of the rotA Figure 1.
In this particular vector field, there is no rotation or vortex. On the other hand, the field is divergent. The possible combinations are: divgradU 1. Note that, for example, rot divA , cannot exist by definition since the rotational operates only on vectors, while divA is, by definition, a scalar. We now calculate, as an example, the operator div gradU.
Finally, Eq. Written in an appropriate form, they describe the phenomena of diffusion of fields, either electromagnetic electric or magnetic fields , or mechanical diffusion of heat, flow of fluids, etc. Cylindrical coordinate system, its relation to the Cartesian system, and components of vector A. Expressions in Cylindrical and Spherical Coordinates The expressions below give the principal operators in cylindrical and spherical coordinates, applied to a vector function A and a scalar function U. Spherical coordinates R,0, f 1 dAz Figure 1. Spherical coordinate system, its relation to the Cartesian system, and components of a vector A.
Introduction In this chapter a brief presentation of Maxwell's equations is done, assuming that the bases of Dedromagnetics EM are already known. The main idea here is to recall the principal concepts necessary to the work developed afterwards. Besides, it is interesting to define and establish the notation of the main physical quantities that will be used in the following chapters. Electromagnetics EM can be described by the equations of Maxwell and the constitutive relations.
The theory of EM took a long time to be established, and it can be understood by the fact that the EM quantities are "abstract" or, in other words, can not be "seen" or "touched" contrarily to most others, such as mechanical and thermal quantities. Actually, the majority of the EM phenomena were established by other scientists before Maxwell, such as Ampere , Gauss , Faraday , Lenz among others. However, there was some incompatibility on the formulation and Maxwell , by introducing an additional term in to the Ampere's law, could synthesize the EM in four equations.
The genius of this man brought the EM to a very simple formalism, kept mainly by only four equations. The physical possibility of this group of equations along with constitutive ones is so high that very distinct phenomena like p.
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In fact, when, p. In this work, we will be interested in low-frequency phenomena and the contents of the chapters are mainly focused on this part of the EM. As a matter of fact, Maxwell's equations do not make distinction between low and high frequency, as already mentioned above, but bringing them to practical applications, it is possible to adapt them to these two situations.
Moreover, when describing low frequency problems, generally, the Maxwell's equations can be divided into two groups: electrostatics and magnetics and, an important aspect: they can be treated independently. The diagram below Fig. EM division for physical applications thus, in each block of this diagram, the four Maxwell's equations are adapted to the corresponding physical situation. The EM Quantities Maxwell's equations are a set of partial differential equations in space and time applied to electromagnetic quantities. We attribute properties of principles or postulates to Maxwell's equations.
E Q Figure 2. Electric field due to a charge Q, or an equivalent charge distribution. To do so we assume here that the notions of electric charge and electric current are known. The Electric Field Intensity E An electric charge or an assembly of electric charges Q, stationary in space, has the property of creating an electric quantity in space, called an electric field intensity E as shown in Figure 2.
The electric field intensity is a vector quantity and obeys the rules of vector fields. The manner in which the electric field intensity E is calculated will be shown in subsequent sections. In this case, a magnetic field intensity H is generated as shown in Figure 2. A moving charge and the magnetic field intensity it generates. A moving charge or charges lead to the idea of the electric current. This is the ultimate result of the vector field H, whose calculation we will present shortly.
If this movement of charges occurs in a conducting wire as it is the case in a majority of real situations , the electric field intensity is practically nonexistent since the electrons move between vacant positions in the atoms of the conducting material and the net sum of charges is essentially zero. Later on we will see that variations in electric field intensities also generate magnetic field intensities. Suppose that, by external means, we create a magnetic field intensity H in both materials and that H is constant throughout the cross-section S.
Two materials of different permeabilities maintain different magnetic flux densities for the same field intensity. We obtain Note that the larger is the permeability of the medium, the larger are the magnetic flux density and the flux passing through its cross-section S.
In other words, B is called "magnetic flux density" or "induction" since this quantity expresses the capacity to induce flux within a medium. As in the example above, a high flux density is associated with a high permeability J,. Using the literal meaning of the terms "induction" and "permeability," we can say that a large flux is "induced" in a medium and that the medium is highly "permeable" to flux. D is also called "electric induction" and it plays an important role in Gauss' theorem, as will be presented soon. There are, in spite of similarities with magnetic quantities, a few salient differences between them.
The first difference is the fact that 8 varies little between materials, in contrast to the permeability fJ,. In useful dielectric materials 8 varies no more than by a factor of while the variation in J, can often attain factors of 10 or more. Straight conductor Figure 2. Conductor dimensions 2. We define a unit vector u perpendicular to the surface S. The mean average surface current density crossing the area S is given by If we assume that the surface S is small, the current density J can be considered to be constant over the surface.
In many cases, J varies throughout the cross-section. The Electric Conductivity a In general, when analyzing electric field problems we distinguish between two types of materials: dielectric or insulating materials and conducting materials. Insulating materials are characterized by their permittivity 8 and their dielectric strength which will be discussed later. Conducting materials are characterized by their conductivity a. The latter expresses the material's capacity to conduct electric current. We have seen earlier that s and we state that the electric field in this case is where V is the electric potential difference on this section of the conductor.
It defines a quantity at any point in space. They describe the relations between field quantities based on the electric and magnetic properties of materials 8 , [i , and G. Local Form of the Equations Maxwell's four equations are as follows: 2. Applying the divergence on both sides of Eq. This is significant in that the flux of the vector or, similarly, the conduction current is conservative. In other words, the current entering a given volume is equal to the current leaving the volume. In fact, in practically all electromagnetic devices, the current injected into the device is equal to the current leaving it.
An Efficient Vector Finite Element Method for Nonlinear Electromagnetic Modeling
When this does not happen, there is an accumulation of charges in the device, or a certain amount of charge is extracted from the device. This is shown schematically in Figure 2. The negative sign of the expression Figure 2. Accumulation of charges in a volume due to nonzero divergence of the current density. Now, we will analyze, under local form, the Maxwell equations. We assume first the situation in Figure 2. As we have seen in the previous section, H and J are connected by a rotation or curl relationship.
The geometric relation between these quantities is demonstrated in Figure 2. The flux of the vector J is the conduction current. Figure 2. Relation between conduction current density and magnetic field intensity. To understand this we can say that the magnetic flux entering a volume is equal to the magnetic flux leaving the volume. This relation corresponds to a condition which allows understanding of the field behavior and serves, in various cases, as an additional mean for determining the magnetic field intensity.
However, Eq. The geometrical situation connecting these quantities is shown in Figure 2. Assuming that B increases as it comes out of the plane of Figure 2. Relation between the time derivative of the magnetic flux density and the electric field intensity. We can easily imagine a volume in which there is a difference between the electric fluxes entering and leaving the volume. This situation is shown in Figure 2.
The flux traversing the volume is outward-oriented. D and p are related through the divergence, according to the relations shown in Chapter 1. The geometrical relation between the two quantities is shown in Figure 2. The flux of the vector D traversing the surface that encloses the volume V of the sphere is nonzero. The nature of the electric flux. Two types of anisotropic materials.
The material on the left has grain-oriented structure while the one on the right is made of thin insulated sheets. The Anisotropy It is possible to apply Maxwell's equations in various situations and in combinations of different materials. However, instead of discussing all possible applications, we prefer to present the equations through a general situation. For this purpose it is necessary to introduce the concept of magnetic anisotropy. Consider a material whose magnetic permeability is dominant in a certain direction.
One such material is a sheet of iron with grain-oriented structure or thin plates made of sheet metal which form, for example, the core of a transformer, as in Figure 2. It is reasonable to assume that in both cases, the magnetic flux flows with more ease in the direction Ox. In the first case, this is due to the orientation of the grains and in the second due to the presence of small gaps between the layers of sheet metal. In this case, there is an angle between H and B. We conclude that the relation where J, is a scalar, is not general since it does not satisfy the cases above.
Because of this, we introduce the concept of the "permeability tensor" denoted by ki. While in the example above, Bx depended only on Hx, now it may depend on all three components of H. In general, if the tensor jj, is not a diagonal tensor, we can write a more complex relation as Besides the concept of anisotropy, which complicates the study of magnetic materials, we introduce another phenomenon, frequently encountered in electromagnetic devices.
Finite Element Modeling of scattered electromagnetic waves for stroke analysis.
In these devices, the magnetic permeability is not constant but depends on the particular value of H in the magnetic material in question. This phenomenon is called "non-linearity. For practical purposes, it is often useful to simplify these equations based on the conditions of operation. Obviously for static cases this term is zero. If it is not the case we notice that this term, appearing in 2. At first look, one can imagine that as 2. That is the link between 2.
We can consider the following physical situation. In Fig. V can depend on time, and the resulting E too. Of course, the electrical field E created is calculated using the equation 2. If in our study domain we are proceeding with magnetic calculations, in the equation 2. For this reason we can neglect parasitic capacitances in wires.
We cannot neglect this term if, in the domain under consideration, we have large capacitors and large electric fields, or if the involved frequencies are very high. In these cases, the displacement current can be large. Magnetodynamic Formulations, Z. Ren, F. Bandelier, F. Behavior Laws of Materials, F. Bouillault, A. Kedous-Lebouc, G. Meunier, F. Ossart, F. Modeling on Thin and Line Regions, C. Coupling with Circuit Equations, G.
Meunier, Y. Lefevre, P. Lombard, Y. Le Floch. Symmetric Components and Numerical Modeling, J. Lobry, E. Nens, C. Magneto-thermal Coupling, M. Magneto-mechanical Modeling, Y.