Monday to Friday:
Monday 9th: Introduction, by J. Brégains
Tuesday 10th: DFs and Geometry, by R. Kotiuga
Wednesday 11th: Geometric Methods, by Ch. Geuzaine
Thursday 12th: Numerical Modelling In Optics, by A. Nicolet
Friday 13th: Cell Methods, by B. Auchmann
GEOMETRIC METHODS, by Ch. Geuzaine
The lecture will touch upon two recent developments for the numerical simulation of electromagnetic problems using finite element methods. In the first part, I will introduce domain decomposition techniques to tackle the open problem of solving large scale high-
DIFFERENTIAL FORMS AND GEOMETRY, by R. Kotiuga
1. An Informal Introduction by means of Differential Calculus. 2 Differential Forms and Variational Principles. 2.1 Simple Variational Principles. 2.1.1 The basic variational problems. 2.1.2 Direct varaitional methods and the FEM. 2.1.3 Symbols of differential operators and a general form of the multivariable E-
5.1 1; 2; n; 3; 4; ::: 5.2 From homotopy and homology to Algebraic Topology. 5.3 Revisiting the making cuts in 3-
1. A. Bossavit, Computational Electromagnetism: Variational Formulations, Complementarity, Edge Elements Academic Press, 1997.
2. H. Flanders, Differential Forms with Applications to the Physical Sciences Dover, 1989 reprint of the original Academic Press ed.
3. T. Frankel, The Geometry of Physics: An Introduction 2nd ed., Camb. U. Press, 2004.
4. P. Gross, P. R. Kotiuga, Electromagnetic Theory and Computation: A Topological Approach, MSRI Monograph Series, vol. 48, Camb, U. Press, 2004.
5. R. Weinstock Calculus of Variations: With Applications to Physics and Engineering, Dover Publications reprint of the 1952 McGraw-
6. J. Clerk Maxwell, A Treatise on Electricity and Magnetism, in two volumes, 3rd ed., Dover Publications, 1954.
INTRODUCTION, by J. Brégains
The mathemtatical language used throughout the course will be Differential Forms (DFs), which basically are covariant antisymmetric tensors. The introduction is aimed towards presenting the concept of DFs and seeing how they can be used as a mathematical tool complementary to the vector calculus. To this end, we depart from matrices, describe their use for representing vectors and tensors, and then extend such a utility to turn to the very definition of contravariant and covariant vectors and tensors. We then pass through the DFs mathematical field to finally exhibit its elegance and simplicity when applied to electromgnetics.
ELECTROMAGNETIC MODELLING IN OPTICS, by A. Nicolet
In this lecture, we will first review the mechanism of Maxwell's equation transformations and see how integral quantities are left invariants. This will be applied to the practical numerical modelling (using FEM) of various problems: unbounded geometries (in the static and quasi-
Extending this principle beyond continuous transformations allows us to design exotic optical devices such as the invisibility cloak. Another example of transformation optics devices are the super-
Another extremely useful application of transformation optics are the Perfectly Matched Layers, introduced in 1994 by Bérenger much before transformation optics. Nevertheless, harmonic PMLs are very nicely described as complex-
CELL METHODS, by B. Auchmann
The study of cell methods is useful for various reasons: because cell methods are implemented in commercial software, they are relatively easy to program from scratch, they suggest powerful and compact data structures, and they provide a geometric and intuitive understanding of important properties of the finite-