Eddy current canonical problems (with applications to nondestructive evaluation) (original) (raw)
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The book, having eight chapters, offers one semester course (with four credit hours per week) on diverse topics usually included in the mathematics syllabus of the first year Engineering class of most Universities. The first chapter deals with the introductory topics in 2-dimensional coordinate geometry: straight lines and conic sections. A brief discussion of theory of equations is given in the second chapter. Vectors (mostly in 3-dimensional space) are introduced in the third chapter. Vector algebra and products of vectors are included. The next chapter deals with the matrices and determinants. Matrix algebra including multiplication of matrices is given. Further concepts such as rank of a matrix etc. are avoided. A comprehensive course on convergence of infinite series forms the subject mater of Chapter 5. Presuming that the students already had a first course on calculus only main concepts and results of differentiation and integration of functions of single variable have been given in the chapter 6. Power series, Maclaurin’s and Taylor’s expansions are explained in detail in the same chapter. Two Eulerian integrals, usually called Beta and Gamma functions, are also introduced and some properties of Gamma functions are given. Applications of vectors to geometry dealing with the vector equations of straight lines and planes are given in the Chapter 7. The last chapter deals with the partial derivation of functions of more than one variable. Both vector and scalar functions are considered and the vector differential operator of the first order is introduced. The total derivative of functions and famous Euler’s theorem on homogeneous functions are explained. Besides several worked out exercises (called Examples), Chapters 5, 7 and 8 are also supplemented by Problem-sets of unsolved exercises with necessary hints to the challenging ones. Tutorial sheets and Test Papers containing model questions and a short bibliography of the topics are provided. The alphabetical index added at the end makes the access to the contents faster. Chapters are divided into Sections, which are numbered chapter-wise. The discussion within the Sections is presented in the form of Definitions, Theorems, Corollaries, Notes and (solved) Examples. These subtitles within the Sections are numbered in decimal pattern. For instance, the equation number (c.s.e) refers to the eth equation in the sth section of Chapter c. When the number c coincides with the chapter at hand it is dropped. Adequate references to the previously appeared results are made in the text and unnecessary repetitions are avoided.
On the analysis of axis-symmetric eddy-currents
2017
This work is primarily concerned with the numerical simulation of linear time-harmonic electromagnetic field problems and to a lesser extent with some applications of these simulations to non-destructive testing. Chapter 1 provides a broad overview of the work and also pauses briefly to discuss the original inspiration of the project; eddy-current non-destructive testing. Chapter 2 looks in some depth at the underlying mathematical structure of the problem. With the aid of Tonti diagrams we systematically develop equations for the potential functions. The equations for the potentials as they originally stand do not have unique solutions. In order that there be a single unique solution to the problem gauge constraints must be enforced. One particular approach to enforcing gauge constraints, that of augmenting the operator, is treated in some depth and generality. In Chapter 3 having determined the equations governing the flow of eddy-currents and having determined that, given suitabl...
Ingenieria Y Ciencia Ing Cienc, 2013
The eddy current model is obtained from Maxwell's equations by neglecting the displacement currents in the Ampère-Maxwell's law. The so-called "A, V − A potential formulation" is nowadays one of the most accepted formulations to solve the eddy current equations numerically, and Bíró & Valli have recently provided its well-posedness and convergence analysis for the time-harmonic eddy current problem. The aim of this paper is to extend the analysis performed by Bíró & Valli to the general transient eddy current model. We provide a backward-Euler fully-discrete approximation based on nodal finite elements and we show that the resulting discrete variational problem is well posed. Furthermore, error estimates that prove optimal convergence are settled.
Mathematical Methods for Engineers and Scientists 1
Dr. K. T. Tang, 2005
broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication Law of September 9, 1965, in its current version, and permission for use must always be obtained is concerned, specifically the rights of of illustrations, recitation, translation, reprinting, reuse of this publication or parts thereof is permitted only under the provisions of the German Copyright A E The use of general descriptive names, registered names, trademarks, etc. in this publication does not SPIN: 11576396 57/3100/SPi Typesetting by the author and SPi using a Springer LT X macro package protective laws and regulations and therefore free for general use. imply, even in the absence of a specific statement, that such names are exempt from the relevant from Springer. Violations are liable for prosecution under the German Copyright Law.
Numerical solution of eddy current problems in bounded domains using realistic boundary conditions
The aim of this paper is to analyze a finite element method to solve the low-frequency harmonic Maxwell equations in a bounded domain containing conductors and dielectrics, and using realistic boundary conditions in that they can be easily measured. These equations provide a model for the so-called eddy currents. The problem is formulated in terms of the magnetic field. This formulation is discretized by using Nédélec edge finite elements on a tetrahedral mesh. Error estimates are easily obtained when the curl-free condition is imposed explicitly on the elements in the dielectric domain.
Numerical analysis of the electric field formulation of an eddy current problem
Comptes Rendus Mathematique, 2003
In this paper we analyze a finite element method for the numerical solution of an eddy Version franç aise abrégée On s'intéresseà un domaine conducteur borné qui est traversé par un courant alternatif de fréquence angulaire ω. Dans ce cas le modèle est constitué deséquations (1)-(3). Dans [3] et [4] nous avonsétudié ce problème avec des conditions aux limites réalistes d'un point de vue pratique. Plus précisement, associéà la décomposition de la frontière donnéeà la Section 2 nous considérons les conditions aux limites (4)-(8) où les seules données sont les intensités du courant, I n , n = 1, . . . N . Ce problème aété analysé dans [4] Note présentée par First name NAME S0764-4442(00)0????-?/FLA c 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés.
Mathematical Methods for Engineers
2017
The present book is based on lectures given by the author to students of various colleges studying mathematics. In designing this course the author tried to select the most important mathematical facts and present them so that the reader could acquire the necessary mathematical conception and apply mathematics to other branches. Therefore in most cases we did not give rigorous formal proofs of the theorems. The rigorousness of a proof often fails to be fruitful and therefore it is usually ignored in practical applications. The book can be of use to readers of various professions dealing with applications of mathematics in their current work. The subject matter is presented in a very systematic and logical manner. It contains material which you will find of great use, not only in the technica1 courses you have yet to take, but also in your profession after graduation, as long as you deal with the analytical aspects of your field. This book consists of eight chapters. Vector Analysis....
Numerical analysis of electric field formulations of the eddy current model
In this paper we analyze a finite element method for the numerical solution of an eddy Version franç aise abrégée On s'intéresseà un domaine conducteur borné qui est traversé par un courant alternatif de fréquence angulaire ω. Dans ce cas le modèle est constitué deséquations (1)-(3). Dans [3] et [4] nous avonsétudié ce problème avec des conditions aux limites réalistes d'un point de vue pratique. Plus précisement, associéà la décomposition de la frontière donnéeà la Section 2 nous considérons les conditions aux limites (4)-(8) où les seules données sont les intensités du courant, I n , n = 1, . . . N . Ce problème aété analysé dans [4] Note présentée par First name NAME S0764-4442(00)0????-?/FLA c 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés.