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Comment poser et résoudre les équations de Maxwell ? On présente ici les outils nécessaires : formulations faibles, éléments d’arêtes, méthodes intégrales, en insistant sur la modélisation, c’est-à-dire la « mise en équations » de situations concrètes : il y a en effet, pour chaque catégorie de problèmes en électrotechnique un modèle mathématique approprié, qu’il faut savoir dériver des équations fondamentales. Depuis sa rédaction, ce cours a été enseigné dans plusieurs formations doctorales européennes, destinées à des ingénieurs en électrotechnique et à des étudiants en analyse numérique.
Comment poser et résoudre les équations de Maxwell ? On présente ici les outils nécessaires : formulations faibles, éléments d’arêtes, méthodes intégrales, en insistant sur la modélisation, c’est-à-dire la « mise en équations » de situations concrètes : il y a en effet, pour chaque catégorie de problèmes en électrotechnique (chauffage par induction, machines et moteurs électriques, cavités résonnantes, etc.), un modèle mathématique approprié, qu’il faut savoir dériver des équations fondamentales. Pour chacun des modèles étudiés dans ce livre (« Maxwell complet », « conduction », « ondes en milieu borné », etc.) on donne la formulation appropriée et la version « discrétisée », c’est-à-dire exploitable sur un ordinateur, que la méthode des éléments finis permet d’en déduire.
Written by specialists of modeling in electromagnetism, this book provides a comprehensive review of the finite element method for low frequency applications. Fundamentals of the method as well as new advances in the field are described in detail. Chapters 1 to 4 present general 2D and 3D static and dynamic formulations by the use of scalar and vector unknowns and adapted interpolations for the fields (nodal, edge, face or volume). Chapter 5 is dedicated to the presentation of different macroscopic behavior laws of materials and their implementation in a finite element context: anisotropy and hysteretic properties for magnetic sheets, iron losses, non-linear permanent magnets and superconductors. More specific formulations are then proposed: the modeling of thin regions when finite elements become misfit (Chapter 6), infinite domains by using geometrical transformations (Chapter 7), the coupling of 2D and 3D formulations with circuit equations (Chapter 8), taking into account the movement, particularly in the presence of Eddy currents (Chapter 9) and an original approach for the treatment of geometrical symmetries when the sources are not symmetric (Chapter 10). Chapters 11 to 13 are devoted to coupled problems: magneto-thermal coupling for induction heating, magneto-mechanical coupling by introducing the notion of strong and weak coupling and magneto-hydrodynamical coupling focusing on electromagnetic instabilities in fluid conductors. Chapter 14 presents different meshing methods in the context of electromagnetism (presence of air) and introduces self-adaptive mesh refinement procedures. Optimization techniques are then covered in Chapter 15, with the adaptation of deterministic and probabilistic methods to the numerical finite element environment. Chapter 16 presents a variational approach of electromagnetism, showing how Maxwell equations are derived from thermodynamic principles.
Magnetic Materials and 3D Finite Element Modeling explores material characterization and finite element modeling (FEM) applications. This book relates to electromagnetic analysis based on Maxwell’s equations and application of the finite element (FE) method to low frequency devices. A great source for senior undergraduate and graduate students in electromagnetics, it also supports industry professionals working in magnetics, electromagnetics, ferromagnetic materials science and electrical engineering. The authors present current concepts on ferromagnetic material characterizations and losses. They provide introductory material; highlight basic electromagnetics, present experimental and numerical modeling related to losses and focus on FEM applied to 3D applications. They also explain various formulations, and discuss numerical codes. • Furnishes algorithms in computational language • Summarizes concepts related to the FE method • Uses classical algebra to present the method, making it easily accessible to engineers Written in an easy-to-understand tutorial format, the text begins with a short presentation of Maxwell’s equations, discusses the generation mechanism of iron losses, and introduces their static and dynamic components. It then demonstrates simplified models for the hysteresis phenomena under alternating magnetic fields. The book also focuses on the Preisach and Jiles–Atherton models, discusses vector hysterisis modeling, introduces the FE technique, and presents nodal and edge elements applied to 3D FE formulation connected to the hysteretic phenomena. The book discusses the concept of source-field for magnetostatic cases, magnetodynamic fields, eddy currents, and anisotropy. It also explores the need for more sophisticated coding, and presents techniques for solving linear systems generated by the FE cases while considering advantages and drawbacks.
This book concerns the analysis and design of induction heating of poor electrical conduction materials. Some innovating applications such as inductive plasma installation or transformers, thermo inductive non-destructive testing and carbon-reinforced composite materials heating are studied. Analytical, semi-analytical and numerical models are combined to obtain the best modeling technique for each case. Each model has been tested with experimental results and validated. The principal aspects of a computational package to solve these kinds of coupled problems are described. In the first chapter, the mathematical tools for coupled electromagnetic and thermal phenomena are introduced. In Chapter 2, these tools are used to analyze a radio frequency inductive plasma installation. The third chapter describes the methodology of designing a low frequency plasma transformer. Chapter 4 studies the feasibility of the thermo inductive technique for non-destructive testing and the final chapter is dedicated to the use of induction heating in the lifecycle of carbon-reinforced composite materials. Contents 1. Thermal and Electromagnetic Coupling, Javad Fouladgar, Didier Trichet and Brahim Ramdane. 2. Simplified Model of a Radiofrequency Inductive Thermal Plasma Installation, Javad Fouladgar and Jean-Pierre Ploteau. 3. Design Methodology of A Very Low-Frequency Plasma Transformer, Javad Fouladgar and Souri Mohamed Mimoune. 4. Non Destructive Testing by Thermo-Inductive Method, Javad Fouladgar, Brahim Ramdane, Didier Trichet and Tayeb Saidi. 5. Induction Heating of Composite Materials, Javad Fouladgar, Didier Trichet, Samir Bensaid and Guillaume Wasselynck
This book provides the reader with basic tools to solve problems of electromagnetism in their natural functional frameworks thanks to modern mathematical methods: integral surface methods, and also semigroups, variational methods, etc., well adapted to a numerical approach.As examples of applications of these tools and concepts, we solve several fundamental problems of electromagnetism, stationary or time-dependent: scattering of an incident wave by an obstacle, bounded or not, by gratings; wave propagation in a waveguide, with junctions and cascades. We hope that mathematical notions will allow a better understanding of modelization in electromagnetism and emphasize the essential features related to the geometry and nature of materials.
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.
In this introductory textbook, a revised and extended version of well-known lectures by L. Hörmander from 1986, four chapters are devoted to weak solutions of systems of conservation laws. Apart from that the book only studies classical solutions. Two chapters concern the existence of global solutions or estimates of the lifespan for solutions of nonlinear perturbations of the wave or Klein-Gordon equation with small initial data. Four chapters are devoted to microanalysis of the singularities of the solutions. This part assumes some familiarity with pseudodifferential operators which are standard in the theory of linear differential operators, but the extension to the more exotic classes of opertors needed in the nonlinear theory is presented in complete detail.
Multiphysics Modeling: Numerical Methods and Engineering Applications: Tsinghua University Press Computational Mechanics Series describes the basic principles and methods for multiphysics modeling, covering related areas of physics such as structure mechanics, fluid dynamics, heat transfer, electromagnetic field, and noise. The book provides the latest information on basic numerical methods, also considering coupled problems spanning fluid-solid interaction, thermal-stress coupling, fluid-solid-thermal coupling, electromagnetic solid thermal fluid coupling, and structure-noise coupling. Users will find a comprehensive book that covers background theory, algorithms, key technologies, and applications for each coupling method. Presents a wealth of multiphysics modeling methods, issues, and worked examples in a single volume Provides a go-to resource for coupling and multiphysics problems Covers the multiphysics details not touched upon in broader numerical methods references, including load transfer between physics, element level strong coupling, and interface strong coupling, amongst others Discusses practical applications throughout and tackles real-life multiphysics problems across areas such as automotive, aerospace, and biomedical engineering