# Finite Element Method to Model Electromagnetic Systems in Low Frequency

1. Edition March 2024

320 Pages, Hardcover*Wiley & Sons Ltd*

**978-1-78630-811-5**

Numerical modeling now plays a central role in the design and study of electromagnetic systems. In the field of devices operating in low frequency, it is the finite element method that has come to the fore in recent decades. Today, it is widely used by engineers and researchers in industry, as well as in research centers.

This book describes in detail all the steps required to discretize Maxwell's equations using the finite element method. This involves progressing from the basic equations in the continuous domain to equations in the discrete domain that are solved by a computer. This approach is carried out with a constant focus on maintaining a link between physics, i.e. the properties of electromagnetic fields, and numerical analysis. Numerous academic examples, which are used throughout the various stages of model construction, help to clarify the developments.

Chapter 1 Equations of Electromagnetism 1

1.1 Maxwell's equations 1

1.2 Behavior laws of materials 2

1.2.1 General case 2

1.2.2 Simplified forms 3

1.3 Interface between two media and boundary conditions 8

1.3.1 Continuity conditions between two media 9

1.3.2 Boundary conditions 12

1.4 Integral forms: fundamental theorems 13

1.4.1 Faraday's law 13

1.4.2 Ampère's law 14

1.4.3 Law of conservation of the magnetic flux 15

1.4.4 Gauss' law 16

1.5 Various forms of Maxwell's equations 17

1.5.1 Electrostatics 17

1.5.2 Electrokinetics 19

1.5.3 Magnetostatics 20

1.5.4 Magnetodynamics 22

Chapter 2 Function Spaces 25

2.1 Introduction 25

2.2 Spaces of differential operators 25

2.2.1 Definitions 25

2.2.2 Function spaces of grad, curl, div 26

2.2.3 Kernel of vector operators 27

2.2.4 Image spaces of operators 27

2.3 Studied topologies 29

2.3.1 Connected and disconnected domain 29

2.3.2 Simply connected and not simply connected domain 29

2.3.3 Contractible and non-contractible domain 30

2.3.4 Properties of function spaces 31

2.4 Relations between vector subspaces 31

2.4.1 Orthogonality of function spaces 31

2.4.2 Analysis of function subspaces 33

2.4.3 Organization of function spaces 39

2.5 Vector fields defined by a vector operator 40

2.5.1 Infinite number of solutions 41

2.5.2 Gauge conditions 42

2.6 Structure of function spaces 44

2.6.1 Adjoint operators 44

2.6.2 Tonti diagram 46

Chapter 3 Maxwell's Equations: Potential Formulations 49

3.1 Introduction 49

3.2 Consideration of source terms 49

3.2.1 Global source quantities imposed on the boundaries 50

3.2.2 Source quantities inside the domain 58

3.2.3 Examples of the calculation of support fields 62

3.3 Electrostatics 69

3.3.1 Source terms imposed on the boundary of the domain 69

3.3.2 Internal electrode 80

3.3.3 Tonti diagram 90

3.4 Electrokinetics 91

3.4.1 Elementary geometry 91

3.4.2 Multisource case 102

3.4.3 Tonti diagram 107

3.5 Magnetostatics 107

3.5.1 Studied problems 107

3.5.2 Scalar potential Phi formulation 108

3.5.3 Vector potential A formulation 121

3.5.4 Summarizing tables 129

3.5.5 Tonti diagram 131

3.6 Magnetodynamics 131

3.6.1 Imposed electric quantities 134

3.6.2 Imposed magnetic quantities 148

3.6.3 Summarizing tables 162

3.6.4 Tonti diagram 166

Chapter 4 Formulations in the Discrete Domain 169

4.1 Introduction 169

4.2 Weighted residual method: weak form of Maxwell's equations 170

4.2.1 Methodology 170

4.2.2 Weak form of the equations of electrostatics 173

4.2.3 Weak form of the equations of electrokinetics 179

4.2.4 Weak form of the equations of magnetostatics 183

4.2.5 Weak form of the equations of magnetodynamics 188

4.2.6 Synthesis of results 200

4.3 Finite element discretization 201

4.3.1 The need for discretization 201

4.3.2 Approximation functions 203

4.3.3 Discretization of vector operators 211

4.3.4 Discretization of physical quantities and associated fields 226

4.3.5 Taking into account homogeneous boundary conditions 228

4.3.6 Gauge conditions in the discrete domain 231

4.3.7 Discretization of support fields and associated potentials 240

4.4 Discretization of weak formulations 244

4.4.1 Notations 244

4.4.2 Ritz-Galerkin method 245

4.4.3 Electrostatics 248

4.4.4 Electrokinetics 260

4.4.5 Magnetostatics 269

4.4.6 Magnetodynamics 281

References 295

Index 297

Stéphane Clénet is Professor at ENSAM and researcher at the group L2EP in the field of computational electromagnetics in France.