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Publication# Challenges in Computational Electromagnetics

Résumé

To meet strict requirements of the information society technologies, antennas and circuit elements are becoming increasingly complex. Frequently, their electromagnetic (EM) properties cannot be anymore expressed in closed-form analytical expressions mainly because of the multitude of irregular geometries found in actual devices. Therefore, accurate and efficient (in terms of computational time and memory) electromagnetic models coupled with the robust optimization techniques, are needed in order to be able to predict and optimize the behavior of the innovative antennas in complex environments. The contribution of this thesis consists in the development and improvement of accurate electromagnetic modeling and optimization algorithms for an ubiquitous class of antennas, the planar printed antennas. The approach most commonly used to model and analyze this type of structures is the Integral Equation (IE) technique numerically solved using the Method of Moments (MoM). From the computational point of view, the main challenge is to develop techniques for efficient numerical evaluation of spatial-domain Green's functions, which are commonly expressed in terms of the well-known Sommerfeld integrals (SIs), i.e., semi-infinite range integrals with Bessel function kernels. Generally, the analytical solution of the SIs is not available, and their numerical evaluation is notoriously difficult and time-consuming because the integrands are both oscillatory and slowly decaying, and might possess singularities on and/or near the integration path. Due to the key role that SIs play in many EM problems, the development of fast and accurate techniques for their evaluation is of paramount relevance. This problem is studied in detail and several efficient methods are developed. Finally, the applicability of one of these methods, namely the Weighted Averages (WA) technique, is extended to the challenging case appearing in many practical EM problems: the evaluation of semi-infinite integrals involving products of Bessel functions. However, the development of effective analysis codes is only one aspect. At least equally important is the availability of reliable optimization techniques for an adequate design of antennas. For that purpose, the Particle Swarm Optimization (PSO) algorithm is introduced in the context of our analysis codes. Moreover, the innovative hybrid version of the PSO algorithm, called the Tournament Selection PSO, has been proposed with the aim of even further improving convergence performances of the classical PSO algorithm. Detailed theoretical description of this socially inspired evolutionary algorithm is given in the thesis. Finally, the characteristics of both algorithms are compared throughout several EM optimization problems.

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Analyse numérique

L’analyse numérique est une discipline à l'interface des mathématiques et de l'informatique. Elle s’intéresse tant aux fondements qu’à la mise en pratique des méthodes permettant de résoudre, par des

Antenne radioélectrique

thumb|Antenne rideau HF de télécommunication.
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thumb|Montage d'une antenne de station terrienne au Nicaragua.
thumb|upright=1.8|Un diagramme animé d'une an

Arnold Sommerfeld

Arnold Johannes Wilhelm Sommerfeld ( à Königsberg, Royaume de Prusse – à Munich, Allemagne) est un physicien théoricien allemand.
Biographie
Il étudia les mathématiques et les sciences na

Printed circuits in bounded media encompass a wide range of practical structures such as discontinuities in waveguides, planar circuits embedded in shielded multilayered media or even two-dimensional printed periodic structures. The Electromagnetic (EM) modeling of printed circuits in layered bounded media is performed via an Integral Equation (IE) technique. Green's functions (GFs) are specially defined to satisfy both the Boundary Conditions (BCs) imposed by the layered media and by the transverse boundary enclosing the entire structure. Finally, a system of IEs on the equivalent sources can be solved numerically by means of the Method of Moments (MoM). Each of the problems enumerated above has already been solved by other authors using IE-MoM techniques. Nevertheless, our formulation introduces a unified approach applicable to all the aforementioned problems. Due to the symmetry presented by a bounded layered media, the GF problem can be reduced into a two-dimensional transverse boundary problem and a one-dimensional transmission line problem in the normal direction. Both problems can be treated independently. This thesis proposes and fully develops an efficient technique that encompasses different laterally bounded multilayered problems with a seamless transition between them. The method is based on a modal representation of the transverse boundary problem and on the expansion of the equivalent surface currents by zero-curl & constant-charge Basis Functions (BFs). It offers a unified and versatile approach that, on one hand eliminates redundancy in the formulation and on the other hand simplifies each particular problem to the evaluation of constant coefficients or basic line integrals. Analytical solutions can be found for the combination of linear subsectional basis functions in rectangular and circular Perfect Electric Conductor (PEC) boundaries as well as for periodic lattices. This thesis then solves the problem of transmission line model in the longitudinal direction by proposing an efficient algorithm that guarantees numerical stability under a variety of known critical conditions where other already known formulations fail. In addition, it introduces alternate equivalent expressions of this formulation that allow new interpretations of the problem. Due to its practical interest, the method is applied for the EM modeling of multilayered boxed printed circuits. This motivated the implementation of a dedicated software tool for the efficient analysis of these topologies including losses. Extensive numerical experiments have been carried out to assess the validity of the aforementioned theory and some properties of test-structures (losses, mesh, etc).

The present thesis deals with the electromagnetic modeling, design and practical implementation of a planar antenna for the reception of satellite broadcasting services from user terminals on board automotive platforms. This antenna is intended to address the market of low cost consumer applications. As a consequence, stringent structural and performance-to-price ratio requirements have to be imposed. The antenna has been conceived as a low profile phased array, on a multilayer planar technology, with fully electronic beam steering and polarization tracking capabilities. The success of this approach strongly relies on the ability of designing highly sophisticated planar multilayered radiators that will act as array elements providing both adequate performance and enough geometrical flexibility to match the constraints dictated by the imposed array topology and structure. The main subject of this thesis is the basic building block of such an antenna, the so-called Elementary Radiating Cell. This cell not only comprises the bare radiators but also the passive circuits (hybrids, power combiners, long via-holes through the multilayered substrates) able to connect the radiating elements to the array beamforming feeding network. The implementation of this Elementary Radiating Cell must be compatible with the available multilayer technologies and with the selected array lattice. This severely limits the available volume and poses some trade-off problems, whose solution has been one of the most challenging efforts in this thesis. This thesis has been carried out in the framework of a joint project between the European Space Agency, several key industrial partners in the sector of satellite R&D and consumer applications, led by IMST-Germany, and the EPFL Laboratoire d'électromagnétisme et d'Acoustique (EPFL-LEMA). The project has demonstrated the feasibility of the electronically steered phased array antenna concept in the development of a new generation of compact satellite terminals for the automotive market.

Surface Mixed Potential Integral Equation (MPIE) formulations together with the Method of Moments (MoM) are widely used to solve electromagnetic problems. An accurate evaluation of the Green functions (GF) associated to the integral equation and of the coupling integrals needed to fill the MoM matrix are the cornerstone steps in the implementation of integral equation algorithms. This thesis is mainly focused on these two topics. The main intended application of our MPIE-MoM formulation is the analysis of enclosed structures, with the shield being materialized by rectangular cavities with perfect conducting (PEC) walls. GFs for rectangular cavities constitute a classic research topic, where there is still a lot of room for improvements. In this area, three main original results are presented in this thesis. Firstly, the exponential convergence of the modal series is ensured via a sophisticated coordinate permutation method. In second place, a study which allows setting the relationship between cavity resonances, excited modes and GF components' singularities, is fully developed. Finally, a novel hybrid method, to compute the GF static part is introduced. This method combines in a new original way both, the modal and image expansions of the cavity GFs. The discretization of the MPIE via the method of moments leads to a matrix equation. In the Galerkin version of the MoM, the matrix elements are given by four-dimensional integrals over source and observer surface domains of the GFs multiplied by some basis and test functions. These so-called coupling integrals invoke the integration of the GF singularity, which in the MPIE case is of the weak type (1/R). The accurate integration of this singularity is a very challenging topic, which has been tackled following many different strategies. Here, the closed analytical expressions of the 4D integral over rectangular domains of this singularity are presented. The problem related to the integration of the GF singularity on arbitrary shaped domains is solved through a hybrid numerical-analytical technique based on an original integral transformation and using by the first time double exponential (DE) numerical integration rules. The thesis concludes with several numerical examples and benchmarks of practical interest. They ascertain the validity of strategies, concepts and results of this thesis and they strongly hint to the development of future competitive computer tools.