Finite Element Method for Electromagnetic Modeling

Gérard Meunier received his PhD from the INP Grenoble University in 1981. He joined the CNRS in 1982 in the Grenoble Electrical Engineering Research Center (G2Elab) where he is currently CNRS Research Director. His research is devoted to the numerical modeling of electromagnetic phenomena.

• Chapter 1. Introduction to Nodal Finite Elements. 1Jean-Louis COULOMB1.1. Introduction 11.1.1. The finite element method 11.2. The 1D finite element method 21.2.1. A simple electrostatics problem 21.2.2. Differential approach 31.2.3. Variational approach 41.2.4. First-order finite elements 61.2.5. Second-order finite elements 91.3. The finite element method in two dimensions 101.3.1. The problem of the condenser with square section 101.3.2. Differential approach 121.3.3. Variational approach 141.3.4. Meshing in first-order triangular finite elements 151.3.5. Finite element interpolation 171.3.6. Construction of the system of equations by the Ritz method 191.3.7. Calculation of the matrix coefficients 211.3.8. Analysis of the results 251.3.9. Dual formations, framing and convergence 421.3.10. Resolution of the nonlinear problems 441.3.11. Alternative to the variational method: the weighted residues method 451.4. The reference elements 471.4.1. Linear reference elements 481.4.2. Surface reference elements 491.4.3. Volume reference elements 521.4.4. Properties of the shape functions 531.4.5. Transformation from reference coordinates to domain coordinates 541.4.6. Approximation of the physical variable 561.4.7. Numerical integrations on the reference elements 601.4.8. Local Jacobian derivative method 631.5. Conclusion 661.6. References 66Chapter 2. Static Formulations: Electrostatic, Electrokinetic, Magnetostatics 69Patrick DULAR and Francis PIRIOU2.1. Problems to solve 702.1.1. Maxwell’s equations 702.1.2. Behavior laws of materials 712.1.3. Boundary conditions 712.1.4. Complete static models 742.1.5. The formulations in potentials 752.2. Function spaces in the fields and weak formulations 822.2.1. Integral expressions: introduction 822.2.2. Definitions of function spaces 822.2.3. Tonti diagram: synthesis scheme of a problem 842.2.4. Weak formulations 862.3. Discretization of function spaces and weak formulations 912.3.1. Finite elements 912.3.2. Sequence of discrete spaces 932.3.3. Gauge conditions and source terms in discrete spaces 1062.3.4. Weak discrete formulations 1092.3.5. Expression of global variables 1142.4. References 115Chapter 3. Magnetodynamic Formulations 117Zhuoxiang REN and Frédéric BOUILLAULT3.1. Introduction 1173.2. Electric formulations 1193.2.1. Formulation in electric field 1193.2.2. Formulation in combined potentials a - 1203.2.3. Comparison of the formulations in field and in combined potentials 1213.3. Magnetic formulations 1233.3.1. Formulation in magnetic field 1233.3.2. Formulation in combined potentials t - ɸ 1243.3.3. Numerical example 1253.4. Hybrid formulation 1273.5. Electric and magnetic formulation complementarities 1283.5.1. Complementary features 1283.5.2. Concerning the energy bounds 1293.5.3. Numerical example 1293.6. Conclusion 1333.7. References 134Chapter 4. Mixed Finite Element Methods in Electromagnetism 139Bernard BANDELIER and Françoise RIOUX-DAMIDAU4.1. Introduction 1394.2. Mixed formulations in magnetostatics 1404.2.1. Magnetic induction oriented formulation 1414.2.2. Formulation oriented magnetic field 1444.2.3. Formulation in induction and field 1464.2.4. Alternate case 1474.3. Energy approach: minimization problems, searching for a saddle-point 1474.3.1. Minimization of a functional calculus related to energy 1474.3.2. Variational principle of magnetic energy 1494.3.3. Searching for a saddle-point 1514.3.4. Functional calculus related to the constitutive relationship 1544.4. Hybrid formulations 1544.4.1. Magnetic induction oriented hybrid formulation 1544.4.2. Hybrid formulation oriented magnetic field 1564.4.3. Mixed hybrid method 1574.5. Compatibility of approximation spaces – inf-sup condition 1574.5.1. Mixed magnetic induction oriented formulation 1584.5.2. Mixed formulation oriented magnetic field 1604.5.3. General case 1604.6. Mixed finite elements, Whitney elements 1614.6.1. Magnetic induction oriented formulation 1624.6.2. Magnetic field oriented formulation 1634.7. Mixed formulations in magnetodynamics 1644.7.1. Magnetic field oriented formulation 1644.7.2. Formulation oriented electric field 1674.8. Solving techniques 1674.8.1. Penalization methods 1684.8.2. Algorithm using the Schur complement 1714.9. References 174Chapter 5. Behavior Laws of Materials 177Frédéric BOUILLAULT, Afef KEDOUS-LEBOUC, Gérard MEUNIER, Florence OSSART and Francis PIRIOU5.1. Introduction 1775.2. Behavior law of ferromagnetic materials 1785.2.1. Definitions 1785.2.2. Hysteresis and anisotropy 1795.2.3. Classificiation of models dealing with the behavior law 1805.3. Implementation of nonlinear behavior models 1835.3.1. Newton method 1835.3.2. Fixed point method 1875.3.3. Particular case of a behavior with hysteresis 1915.4. Modeling of magnetic sheets 1925.4.1. Some words about magnetic sheets 1925.4.2. Example of stress in the electric machines 1925.4.3. Anisotropy of sheets with oriented grains 1945.4.4. Hysteresis and dynamic behavior under uniaxial stress 2005.4.5. Determination of iron losses in electric machines: nonlinear isotropic finite element modeling and calculation of the losses a posteriori 2095.4.6. Conclusion 2155.5. Modeling of permanent magnets 2165.5.1. Introduction. 2165.5.2. Magnets obtained by powder metallurgy 2165.5.3. Study of linear anisotropic behavior 2185.5.4. Study of nonlinear behavior 2205.5.5. Implementation of the model in finite element software 2235.5.6. Validation: the experiment by Joel Chavanne 2245.5.7. Conductive magnet subjected to an AC field 2255.6. Modeling of superconductors 2265.6.1. Introduction 2265.6.2. Behavior of superconductors 2275.6.3. Modeling of electric behavior of superconductors 2305.6.4. Particular case of the Bean model 2325.6.5. Examples of modeling 2375.7. Conclusion 2405.8. References 241Chapter 6. Modeling on Thin and Line Regions 245Christophe GUÉRIN6.1. Introduction 2456.2. Different special elements and their interest 2456.3. Method for taking into account thin regions without potential jump 2496.4. Method for taking into account thin regions with potential jump 2506.4.1. Analytical integration method 2516.4.2. Numerical integration method 2526.5. Method for taking thin regions into account 2556.6. Thin and line regions in magnetostatics 2566.6.1. Thin and line regions in magnetic scalar potential formulations 2566.6.2. Thin and line regions in magnetic vector potential formulations 2576.7. Thin and line regions in magnetoharmonics 2576.7.1. Solid conducting regions presenting a strong skin effect 2586.7.2. Thin conducting regions 2656.8. Thin regions in electrostatic problems: “electric harmonic problems” and electric conduction problems 2726.9. Thin thermal regions 2726.10. References 273Chapter 7. Coupling with Circuit Equations 277Gérard MEUNIER, Yvan LEFEVRE, Patrick LOMBARD and Yann LE FLOCH7.1. Introduction 2777.2. Review of the various methods of setting up electric circuit equations 2787.2.1. Circuit equations with nodal potentials 2787.2.2. Circuit equations with mesh currents 2797.2.3. Circuit equations with time integrated nodal potentials 2807.2.4. Formulation of circuit equations in the form of state equations 2817.2.5. Conclusion on the methods of setting up electric equations 2837.3. Different types of coupling 2847.3.1. Indirect coupling 2857.3.2. Integro-differential formulation 2857.3.3. Simultaneous resolution 2857.3.4. Conclusion 2857.4. Establishment of the “current-voltage” relations 2867.4.1. Insulated massive conductor with two ends: basic assumptions and preliminary relations 2867.4.2. Current-voltage relations using the magnetic vector potential 2877.4.3. Current-voltage relations using magnetic induction 2887.4.4. Wound conductors 2907.4.5. Losses in the wound conductors 2917.5. Establishment of the coupled field and circuit equations 2927.5.1. Coupling with a vector potential formulation in 2D 2927.5.2. Coupling with a vector potential formulation in 3D 3037.5.3. Coupling with a scalar potential formulation in 3D 3107.6. General conclusion 3177.7. References 318Chapter 8. Modeling of Motion: Accounting for Movement in the Modeling of Magnetic Phenomena 321Vincent LECONTE8.1. Introduction 3218.2. Formulation of an electromagnetic problem with motion 3228.2.1. Definition of motion 3228.2.2. Maxwell equations and motion 3258.2.3. Formulations in potentials 3298.2.4. Eulerian approach 3358.2.5. Lagrangian approach 3388.2.6. Example application 3428.3. Methods for taking the movement into account 3468.3.1. Introduction. 3468.3.2. Methods for rotating machines 3468.3.3. Coupling methods without meshing and with the finite element method 3488.3.4. Coupling of boundary integrals with the finite element method 3508.3.5. Automatic remeshing methods for large distortions 3558.4. Conclusion 3628.5. References 363Chapter 9. Symmetric Components and Numerical Modeling 369Jacques LOBRY, Eric NENS and Christian BROCHE9.1. Introduction 3699.2. Representation of group theory 3719.2.1. Finite groups 3719.2.2. Symmetric functions and irreducible representations 3749.2.3. Orthogonal decomposition of a function 3789.2.4. Symmetries and vector fields 3799.3. Poisson’s problem and geometric symmetries 3849.3.1. Differential and integral formulations 3849.3.2. Numerical processing 3879.4. Applications 3889.4.1. 2D magnetostatics 3889.4.2. 3D magnetodynamics 3949.5. Conclusions and future work 4039.6. References 404Chapter 10. Magneto-thermal Coupling 405Mouloud FÉLIACHI and Javad FOULADGAR10.1. Introduction 40510.2. Magneto-thermal phenomena and fundamental equations 40610.2.1. Electromagentism 40610.2.2. Thermal 40810.2.3. Flow 40810.3. Behavior laws and couplings 40910.3.1. Electrmagnetic phenomena 40910.3.2. Thermal phenomena 40910.3.3. Flow phenomena 40910.4. Resolution methods 40910.4.1. Numerical methods 40910.4.2. Semi-analytical methods 41010.4.3. Analytical-numerical methods 41110.4.4. Magneto-thermal coupling models 41110.5. Heating of a moving work piece 41310.6. Induction plasma 41710.6.1. Introduction 41710.6.2. Inductive plasma installation 41810.6.3. Mathematical models 41810.6.4. Results 42610.6.5. Conclusion 42710.7. References 428Chapter 11. Magneto-mechanical Modeling 431Yvan LEFEVRE and Gilbert REYNE11.1. Introduction 43111.2. Modeling of coupled magneto-mechancial phenomena 43211.2.1. Modeling of mechanical structure 43311.2.2. Coupled magneto-mechanical modeling 43711.2.3. Conclusion 44211.3. Numerical modeling of electromechancial conversion in conventional actuators 44211.3.1. General simulation procedure 44311.3.2. Global magnetic force calculation method 44411.3.3. Conclusion 44711.4. Numerical modeling of electromagnetic vibrations 44711.4.1. Magnetostriction vs. magnetic forces 44711.4.2. Procedure for simulating vibrations of magnetic origin 44911.4.3. Magnetic forces density 44911.4.4. Case of rotating machine teeth 45211.4.5. Magnetic response modeling 45311.4.6. Model superposition method 45511.4.7. Conclusion 45811.5. Modeling strongly coupled phenomena 45911.5.1. Weak coupling and strong coupling from a physical viewpoint 45911.5.2. Weak coupling or strong coupling problem from a numerical modeling analysis 46011.5.3. Weak coupling and intelligent use of software tools 46111.5.4. Displacement and deformation of a magnetic system 46311.5.5. Structural modeling based on magnetostrictive materials 46511.5.6. Electromagnetic induction launchers 46911.6. Conclusion 47011.7. References 471Chapter 12. Magnetohydrodynamics: Modeling of a Kinematic Dynamo 477Franck PLUNIAN and Philippe MASSÉ12.1. Introduction 47712.1.1. Generalities 47712.1.2. Maxwell’s equations and Ohm’s law 48112.1.3. The induction equation 48212.1.4. The dimensionless equation 48312.2. Modeling the induction equation using finite elements 48512.2.1. Potential (A,ɸ) quadric-vector formulation 48512.2.2. 2D1/2 quadri-vector potential formulation 48812.3. Some simulation examples 49112.3.1. Screw dynamo (Ponomarenko dynamo) 49112.3.2. Two-scale dynamo without walls (Roberts dynamo) 49512.3.3. Two-scale dynamo with walls 49812.3.4. A dynamo at the industrial scale 50212.4. Modeling of the dynamic problem 50312.5. References 504Chapter 13. Mesh Generation 509Yves DU TERRAIL COUVAT, François-Xavier ZGAINSKI and Yves MARÉCHAL13.1. Introduction 50913.2. General definition 51013.3. A short history 51213.4. Mesh algorithms 51213.4.1. The basic algorithms 51213.4.2. General mesh algorithms 51813.5. Mesh regularization 52613.5.1. Regularization by displacement of nodes 52613.5.2. Regularization by bubbles 52813.5.3. Adaptation of nodes population 53013.5.4. Insertion in meshing algorithms 53013.5.5. Value of bubble regularization 53113.6. Mesh processer and modeling environment 53313.6.1. Some typical criteria 53313.6.2. Electromagnetism and meshing constraints 53413.7. Conclusion 54113.8. References 541Chapter 14. Optimization 547Jean-Louis COULOMB14.1. Introduction 54714.1.1. Optimization: who, why, how? 54714.1.2. Optimization by numerical simulation: is this reasonable? 54814.1.3. Optimization by numerical simulation: difficulties 54914.1.4. Numerical design of experiments (DOE) method: an elegant solution 54914.1.5. Sensitivity analysis: an “added value” accessible by simulation 55014.1.6. Organization of this chapter 55114.2. Optimization methods 55114.2.1. Optimization problems: some definitions 55114.2.2. Optimization problems without constraints 55314.2.3. Constrained optimization problems 55914.2.4. Multi-objective optimization 56014.3. Design of experiments (DOE) method 56214.3.1. The direct control of the simulation tool by an optimization algorithm: principle and disadvantages 56214.3.2. The response surface: an approximation enabling indirect optimization 56314.3.3. DOE method: a short history 56514.3.4. DOE method: a simple example 56514.4. Response surfaces 57214.4.1. Basic principles 57214.4.2. Polynomial surfaces of degree 1 without interaction: simple but sometimes useful 57314.4.3. Polynomial surfaces of degree 1 with interactions: quite useful for screening 57314.4.4. Polynomial surfaces of degree 2: a first approach for nonlinearities 57414.4.5. Response surfaces of degrees 1 and 2: interests and limits 57614.4.6. Response surfaces by combination of radial functions 57614.4.7. Response surfaces using diffuse elements 57714.4.8. Adaptive response surfaces 57914.5. Sensitivity analysis 57914.5.1. Finite difference method 57914.5.2. Method for local derivation of the Jacobian matrix 58014.5.3. Steadiness of state variables: steadiness of state equations 58114.5.4. Sensitivity of the objective function: the adjoint state method 58314.5.5. Higher order derivative 58314.6. A complete example of optimization 58414.6.1. The problem of optimization 58414.6.2. Determination of the influential parameters by the DOE method 58514.6.3. Approximation of the objective function by a response surface 58714.6.4. Search for the optimum on the response surface 58714.6.5. Verification of the solution by simulation 58714.7. Conclusion 58814.8. References 588List of Authors 595Index 599