Energy methods are used extensively in the formulation of discrete system models. They simplify the systematic integration of diverse kinematic schemes, and are well suited for characterizing complex energy domain coupling effects. Continuum mechanics models are by contrast normally based on partial differential equation descriptions of the physical system. The research presented here develops a new Hamiltonian method for the simulation of distributed parameter electromagnetic and thermo-electromagnetic systems. It expands the application of current system dynamics modeling techniques, to encompass complex distributed parameter electromagnetic systems. / text

Ce travail de thèse constitue une contribution originale et importante à la compréhension de la transformation d’espace et ouvre la voie au design de nouvelles structures éléctromagnétiques. Le couplage entre cette technique innovante et les métamatériaux a permis la réalisation de prototypes aux propriétés uniques. C’est ainsi que nous avons pu concevoir une cape d’invisibilité polygonale, un adapteur de modes ou encore une antenne directive ou isotrope. La fabrication de notre antenne très directive par cette méthode est le seul prototype dans la littérature qui allie le contrôle de la permittivité et la perméabilité à partir de résonnateurs électriques et magnétiques. Ce contrôle ultime de la lumière à partir d’une ingénierie de l’espace trouvera son utilité dans la recherche fondamentale mais aussi pour les ingénieurs et dévellopeurs recherchant plus de précision dans leur conception de dispositifs électromagnétiques. / This phD work is an original and important contribution to the understanding of transformation optics and paves the way to the design of new electromagnetic structures. The coupling between this innovative technique and metamaterials has led to prototypes with unique properties. We have thus developed an invisibility polygonal cloak, an electromagnetic taper, a directional antenna and isotropic source. The realization of our high-directive antenna with this method is the only prototype in the literature that combines controlled variations of the permittivity and permeability from electric and magnetic resonators. The ultimate control of light from an engineering space will find its usefulness in fundamental research but also for engineers and developers who are looking for more precision in the design of electromagnetic devices.

<p>The Discontinuous Galerkin time domain (DGTD) method is promising in modeling of realistic multiscale electromagnetic systems. This method defines the basic concept for implementing the communication between multiple domains with different scales.</p><p>Constructing a DGTD system consists of several careful choices: (a) governing equations; (b) element shape and corresponding basis functions for the spatial discretization of each subdomain; (c) numerical fluxes onto interfaces to bond all subdomains together; and (d) time stepping scheme based on properties of a discretized</p><p>system. This work present the advances in each one of these steps.</p><p> </p><p>First, a unified framework based on the theory of differential forms and the finite element method is used to analyze the discretization of the Maxwell's equations. Based on this study, field intensities (<bold>E</bold> and <bold>H</bold>) are associated to 1-forms and curl-conforming basis functions; flux densities (<bold>D</bold> and <bold>B</bold>) are associated to 2-forms and divergence-conforming basis functions; and the constitutive relations are defined by Hodge operators.</p><p>A different approach is the study of numerical dispersion. Semidiscrete analysis is the traditional method, but for high order elements modal analysis is prefered. From these analyses, we conclude that a correct discretization of fields belonging to different p-form (e.g., <bold>E</bold> and <bold>B</bold>) uses basis functions with same order of interpolation; however, different order of interpolation must be used if two fields belong to the same p-form (e.g., <bold>E</bold> and <bold>H</bold>). An alternative method to evaluate numerical dispersion based on evaluation of dispersive Hodge operators is also presented. Both dispersion analyses are equivalent and reveal same fundamental results. Eigenvalues, eigenvector and transient results are studied to verify accuracy and computational costs of different schemes. </p><p>Two different approaches are used for implementing the DG Method. The first is based on <bold>E</bold> and <bold>H</bold> fields, which use curl-conforming basis functions with different order of interpolation. In this case, the Riemman solver shows the best performance to treat interfaces between subdomains. A new spectral prismatic element, useful for modeling of layer structures, is also implemented for this approach. Furthermore, a new efficient and very accurate time integration method for sequential subdomains is implemented.</p><p>The second approach for solving multidomain cases is based on <bold>E</bold> and <bold>B</bold> fields, which use curl- and divergence-conforming basis functions, respectively, with same order of interpolation. In this way, higher accuracy and lower memory consumption are obtained with respect to the first approach based on <bold>E</bold> and <bold>H</bold> fields. The centered flux is used to treat interfaces with non-conforming meshes, and both explicit Runge-Kutta method and implicit Crank-Nicholson method are implemented for time integration. </p><p>Numerical examples and realistic cases are presented to verify that the proposed methods are non-spurious and efficient DGTD schemes.</p> / Dissertation