Introduction

As calculation tools are reaching increasingly high performances, numerical modeling has developed significantly in all sectors of society. It can be used to predict the evolution of a given structure or device starting from an initial state, study physical phenomena by accessing quantities that are not measurable or develop virtual prototypes in order to improve a design process. Applied physics, and in particular low-frequency electromagnetism, which is the subject of this book, are no exceptions. Nowadays, high-performance simulation software is available for students, engineers and researchers. A prerequisite for making the best use of a tool, even in the field of computation, is obviously a good knowledge of its foundations and principles. In this context, it seemed interesting to propose a book that may grasp, under the best conditions, the path leading to building these numerical models.

The modeling of electromagnetic phenomena relies on two partial differential equations, known as Maxwell's equations:

These two equations should be completed by behavior laws that describe the reaction of media to electromagnetic fields, which are associated with physical phenomena such as dielectric polarization, electric conduction and ferromagnetism. Finally, for a proper formulation of the problem, boundary conditions should be added, for either a finite or infinite studied domain. Although it may appear simple, this problem, composed of several equations, has no analytical solution, except for the case of elementary geometry, with linear behavior laws.

As the exact solution to the problem is not available, there are two possibilities for reaching an approximation of this solution:

• Formulate hypotheses on the geometry, the behavior laws of the materials and the spatial distribution of electromagnetic fields. The objective is to make the analytical solution possible. This approach requires the "model builder" to have very deep, expert knowledge on the studied system, to be able to formulate the "right hypotheses". If the latter are not valid, there is a very high risk of reaching a low-quality solution, which is very far from the exact solution of the initial problem. Moreover, this approach is not always possible if complex phenomena, such as nonlinearities, are predominant.
• Or, reformulate the initial problem in discrete form, leading to a system of differential algebraic equations. An approximation of the exact solution is then obtained at the cost of a significant amount of computation, which can be readily processed by the computers that are available nowadays. This reformulation, requiring few or almost no hypotheses, is obtained by implementing numerical methods. In the field of electromagnetism, the most widespread such method is the finite element method.

This book focuses on the second approach, often referred to as "computational electromagnetics", providing a detailed description of the implementation of the finite element method in low-frequency electromagnetism. Our purpose is to explain the process starting from equations verified by electromagnetic fields in the continuous domain, in order to arrive at a system of equations that will be solved using a computer. This process, often called "discretization", will be conducted with a permanent concern for maintaining a link between physics, i.e. the properties of electromagnetic fields, and numerical analysis, through the finite element method.

Furthermore, this book is mainly addressed to students, engineers and researchers in the field of electrical engineering. They will be able to better understand the intricate details of (open-source or commercial) software that models the behavior of electromagnetic fields. They will thus have the possibility of better using these tools and therefore have a good knowledge of their limits. This book is also addressed to students, engineers and researchers in the field of numerical analysis who are interested in better understanding the links between numerical methods and physics in the field of electromagnetism.

Even though this book offers few pieces of information on numerical implementation, it provides all the elements required for understanding the theoretical foundations. It also allows us to conceive the link between physics and numerical methods and therefore between the applications and the software used.

The above-stated Maxwell's equations allow for the study of all electromagnetic phenomena. For certain low-frequency applications it is, however, possible to derive them in a "static" or "quasi-static" state. Under certain hypotheses, these simpler problems lead to solutions that are equal or very close to those that would have been obtained using the full Maxwell equation system. After discretization using a numerical method, they can be used to obtain smaller size systems of equations that are easier to solve due to their mathematical properties.

Approximations by problems under a static or quasi-static state are widely used in many domains such as power grids, electrical machines, power electronics and non-destructive testing. This book focuses in particular on three static problems, namely electrostatics, electrokinetics (when electric charges travel at constant speed, the fields do not depend on time) and magnetostatics. In the quasi-static state, Maxwell's equations can be written in the magnetoquasistatic form (more often referred to as "magnetodynamics") or in the electroquasistatic form. In this quasi-static case, our focus will be on magnetodynamics. On the contrary, electroquasistatic problems will not be considered, but the developments remain similar to those used in the case of magnetodynamics.

This book has four chapters, each corresponding to a stage of the process leading to the discretization of Maxwell's equations.

The objective of Chapter 1 is to formulate various problems in the static state and the magnetodynamic state, and then to solve them. For each problem, the equilibrium equations are written, as well as the behavior laws and the boundary conditions on the electromagnetic fields. A review of the properties of these fields also highlights their behavior at the interface between two media, and the nature of their integral forms. A key point of this chapter is the definition of electric and magnetic quantities, referred to as "source terms", which are at the origin of the creation of electromagnetic fields. These terms can be located inside the studied domain (electric charges, inductors, permanent magnets) or imposed on the boundary of the domain (electromotive or magnetomotive forces, current density or magnetic flux).

Chapter 2 is dedicated to the introduction of functional spaces associated with vector operators: gradient, curl and divergence. As these operators are used when writing the equations of static and quasi-static problems, they can be used to define the functional spaces to which various electromagnetic fields belong. An analysis is conducted on the properties of functional spaces and in particular on the images and kernels of the vector operators in relation to the topology of the studied domain. These properties lead quite naturally to the notion of scalar and vector potentials, widely used as intermediary for solving static and quasi-static problems, which will be introduced in Chapter 3. The notion of gauge is also presented, which imposes the uniqueness of a field when defining by a single vector operator. Gauge conditions will therefore be very useful to impose the uniqueness of potentials so that the problem is properly posed. They are also used in the construction of source terms.

Chapter 3 focuses on the potential-based formulations for static and magnetodynamic problems. In the case of static problems, the introduction of these potentials allows for the reduction of the number of unknowns, passing from two unknown fields to only one unknown potential. This potential can be a scalar or a vector quantity. For each problem, two formulations in terms of potential referred to as "scalar" or "vector" are obtained. In magnetodynamics, two potentials are used in close relation to those introduced for static problems. Two formulations known as "electric" and "magnetic" are then deduced.

These potentials are not necessarily introduced in a direct manner, requiring instead a reformulation of the source terms of the initial problem, located either inside or on the boundary of the domain. The first part of this chapter is dedicated to this reformulation. The number of sources is often limited, facilitating a focus on the essential, which is a systematic method for imposing source terms. However, as shown by the examples presented, the methodology is readily applicable to problems with a greater number of sources, using the superposition theorem (even though the behavior laws are not linear).

Chapter 4 is dedicated to the discretization of formulations of static and magnetodynamic problems. Successful completion of this discretization requires first of all finding the proper spaces of approximation within which the approximate solutions will be sought. These spaces must have a finite dimension for implementation on a computer. In the case of the finite element method, the spaces of approximation are defined from a mesh, which is obtained by splitting the studied domain into elements of simple shapes (tetrahedron, hexahedron, prism, etc.). A field is then perfectly defined by a vector, whose entries are...