Dans cette thèse, nous nous intéressons à la modélisation mathématique des hétérogénéités de longueurs caractéristiques beaucoup plus petites que la longueur d'ondes. La thèse consiste en deux parties. La partie théorique est dédiée à l'obtention d'un développement asymptotique raccordé: la solution est décrite à l'aide d'un développement de champ proche au voisinage de l'obstacle et par un développement de champ lointain hors de ce voisinage. Le développement de champ lointain met en jeu des solutions singulières de l'équation des ondes tandis que le champ proche lui est régi par un modèle quasi-statique. Ces deux développements sont alors raccordés dans une zone intermédiaire dite de raccord. Nous obtenons alors des estimations d'erreurs permettant de rendre rigoureux ce développement asymptotique formel. La deuxième partie est numérique. Elle décrit à la fois la méthode de Galerkine discontinue, une méthode de raffinement de maillage espace-temps et propose une discrétisation des modèles asymptotiques obtenues précédemment. Elle est illustrée par un certain nombre de tests numériques. / In this thesis, we focus our attention on the modeling of heterogeneities which are smaller than the wavelength. The document is decomposed into two parts : a theoretical one and a numerical one. In the first part, we derive a matched asymptotic expansion composed of a far-field expansion and a near-field expansion. The terms of the far-field expansion are singular solutions of the wave equation whereas the terms of the near-field expansion satisfy quasistatic problems. These expansions are matched in an intermediate region. We justify mathematically this theory by proving error estimates. In the second part, we describe the Discontinuous Galerkin method, a local time stepping method and the implementation of the matched asymptotic method. Numerical simulations illustrate these results.

In this thesis, we study the phenomenon of thermally induced polarisation using a combination of theory and computer simulation. Molecules of sufficiently low symmetry exhibit thermo-molecular orientation when subjected to a temperature gradient, leading to considerable electrostatic fields in polar liquids. Here, we first use non-equilibrium molecular dynamics simulations to study this interesting effect numerically. To this end, we propose an integration algorithm to impose a constant heat flux in simulations and show that it greatly improves energy conservation compared to a previous algorithm. We next investigate the thermal polarisation of water and find that truncation of electrostatic interactions can lead to severe artefacts, such as the wrong sign of polarisation and an overestimation of the electric field. We further show that the quadrupole-moment contribution to the electric field is significant and responsible for an inversion of its sign. To facilitate the theoretical description of electrostatic interactions, we propose a new dipolar model fluid as a perturbation of a Stockmayer fluid. Using this modified Stockmayer model, we provide numerical evidence for the recently proposed phenomenon of thermally induced monopoles. We show that the electrostatic field generated by a pair of heated/cooled colloidal particles immersed in such a solvent can be trivially described by two Coulomb charges. Finally, we propose a mean-field theory to predict the thermo-polarisation effect exhibited by our model fluid theoretically, and demonstrate near quantitative agreement with simulation results.

In this work, a new method for mass determination of single low-charged particles in the sub-picogram regime is developed. It opens applications to chemical physics and surface science via determination of growth rates. The method combines the well-known electrodynamic quadrupole ion trap in a UHV-chamber and fourier transformation of scattered light. The achieved mass resolution of down to $10^{-4}$ at 100 fg mass on a time scale of ten seconds allows a resolution of a few percent of the mass of an adsorbed monolayer and to determine growth rates down to one molecule per second on a time scale of one day. The observation of temperature dependent sticking coefficients results in the measures of the energy of an adsorption barrier. Observation of discrete steps in the rate gives information about the density of molecules in an ordered layer. Temperature dependent desorption data gives the binding energy. The dependence of these observables on the controllable curvature and charge of the substrate's surface is measurable. The first part of this dissertation consists of a description of the common theory of the quadrupole ion trap with the completion of not widely known, newly introduced, contributions to the trapping potential. These contributions lead to systematic shifts in the mass determination. In particular the influence of the inhomogenity of the electrical field, that is used for compensating the gravitational force, is investigated analytically and corroborated experimentally. It is assumed, that the particle's finite size effects in a further shift. In the experimental part initial demonstrative measurements are presented: the time-resolved adsorption of fullerene, anthracene and NO on silica spheres with 500nm diameter has been measured at room temperature. In addition the secondary electron yield of in-situ prepared particles during irradiation with monoenergetic electrons has been determined by analyzing the distribution of change of the number of elementary charges by single events of charging.

Two gas-phase ion-molecule reaction systems are presented here based on measurements done in a temperature variable 22-pole trapping machine. In the first case, the proton affinity of methane is determined based on a new technique for measuring the equilibrium constant of the HCO2+ + CH4 <=> CH5+ + CO2 reaction. The second case reports to the (Ar + N2 )+ reaction system, with reaction rate temperature dependencies measurements made both in the forward and reverse direction with different and complementary methods. The temperature variable 22-pole trapping machine allows one to determine equilibrium constants and reaction rate coefficients over a wide range of temperatures. The coupling of an effusive beam to the setup overcomes the problem of neutral gas wall condensation and extends the temperature range measurements beyond condensation point. The introduction (Chapter 1) gives a short overview about the rf technology and parallel experimental techniques developed in order to better characterize and understand the several mechanisms related to ion-molecule reactions. It also focuses some aspects of reaction rate temperature dependencies determination as well as thermodynamical equilibrium in laboratory environment. A short description of the setup and experimental methods are presented in Chapter 2. Based on equilibrium constant measurements, Chapter 3 is dedicated to the proton affinity of methane. This concept has applications on several fields such as atmospheric and combustion modelling, or testing empirical and ab initio theories for electronic structures. The (Ar − N2 )+ system presented in Chapter 4, is known for being a good case study for inferring the role of vibrational excitation in reaction dynamics and to the existence of non-adiabatic coupling. The experimental results here presented for the N2+ + Ar reaction demonstrate that it is possible to avoid parallel reactions with first vibrational excited state of nitrogen (N2 (ν = 1)). On the other hand, the reverse reaction experiments confirm the existence of a minimum of the reaction rate in the 30 to 300 K range, due to the existence of two reaction channels. The question of the high rate coefficient towards lower temperatures being related to the N2 rotational ground state population is raised. A summary and outlook are presented in Chapter 5, where some new possible paths of investigation are pointed out.

High-frequency wave phenomena is observed in many physical settings, most notably in acoustics, electromagnetics, and elasticity. In all of these fields, numerical simulation and modeling of the forward propagation problem is important to the design and analysis of many systems; a few examples which rely on these computations are the development of metamaterial technologies and geophysical prospecting for natural resources. There are two modes of modeling the forward problem: the frequency domain and the time domain. As the title states, this work is concerned with the former regime. The difficulties of solving the high-frequency wave propagation problem accurately lies in the large number of degrees of freedom required. Conventional wisdom in the computational electromagnetics commmunity suggests that about 10 degrees of freedom per wavelength be used in each coordinate direction to resolve each oscillation. If K is the width of the domain in wavelengths, the number of unknowns N grows at least by O(K^2) for surface discretizations and O(K^3) for volume discretizations in 3D. The memory requirements and asymptotic complexity estimates of direct algorithms such as the multifrontal method are too costly for such problems. Thus, iterative solvers must be used. In this dissertation, I will present fast algorithms which, in conjunction with GMRES, allow the solution of the forward problem in O(N) or O(N log N) time. / text

Η παρούσα διδακτορική διατριβή αποτελεί μια συμβολή στην ανάλυση του ηλεκτρικού πεδίου στο περιβάλλον εναέριων γραμμών απλού και διπλού κυκλώματος. Αναπτύσσονται αναλυτικές σχέσεις για τον ακριβή υπολογισμό της έντασης του ηλεκτρικού πεδίου καθώς και απλούστερες προσεγγιστικές σχέσεις. Η προτεινόμενη μέθοδος υπολογισμού προέκυψε από τον συνδυασμό της ανάλυσης των φορτίων των αγωγών σε συμμετρικές συνιστώσες, της περιγραφής της έντασης του ηλεκτρικού πεδίου με διπλούς μιγαδικούς αριθμούς και της χρήσης του αναπτύγματος πολυπόλων για την έκφραση της έντασης του ηλεκτρικού πεδίου. Με τους διπλούς μιγαδικούς αριθμούς επιτυγχάνεται η παράσταση με μιγαδικούς αριθμούς τόσο των χρονικά ημιτονοειδώς μεταβαλλόμενων μεγεθών (τάσεων και φορτίων) όσο και των γεωμετρικών αποστάσεων. Το ανάπτυγμα πολυπόλων είναι μια σειρά απείρων όρων, όπου ο κάθε επόμενος όρος είναι αντιστρόφως ανάλογος της αυξανόμενης δύναμης της απόστασης. Αναπτύσσονται επίσης ακριβείς αναλυτικές σχέσεις, καθώς και απλούστερες προσεγγιστικές σχέσεις, για τον υπολογισμό της μαγνητικής επαγωγής γραμμών διπλού κυκλώματος. Για την ανάπτυξη των σχέσεων αυτών συνδυάζεται η χρήση των διπλών μιγαδικών αριθμών με το ανάπτυγμα πολυπόλων για την έκφραση της μαγνητικής επαγωγής. / This thesis is a contribution to the analysis of the electric field in the vicinity of single-circuit and double-circuit overhead power lines. Analytical formulas are developed for the accurate calculation of the intensity of the electric field as well as simpler approximate formulas. The proposed calculation method has arisen from the combination of the analysis of the conductors’ charges in their symmetrical components, the description of the intensity of the electric field using double complex numbers and the use of multipole expansions to express the intensity of the electric field. The use of double complex numbers allows the simultaneous representation of sinusoidal varying quantities (voltages and charges) and geometrical distances with complex numbers. The multipole expansion is a series of infinite terms, where each term is inversely proportional to an increasing power of the distance. Accurate analytical formulas are also developed as well as simpler approximate formulas for the calculation of the magnetic flux density produced by double-circuit lines. For the development of those formulas the use of double complex numbers is combined with the multipole expansion to express the magnetic flux density.

In this work, a new method for mass determination of single low-charged particles in the sub-picogram regime is developed. It opens applications to chemical physics and surface science via determination of growth rates. The method combines the well-known electrodynamic quadrupole ion trap in a UHV-chamber and fourier transformation of scattered light. The achieved mass resolution of down to $10^{-4}$ at 100 fg mass on a time scale of ten seconds allows a resolution of a few percent of the mass of an adsorbed monolayer and to determine growth rates down to one molecule per second on a time scale of one day. The observation of temperature dependent sticking coefficients results in the measures of the energy of an adsorption barrier. Observation of discrete steps in the rate gives information about the density of molecules in an ordered layer. Temperature dependent desorption data gives the binding energy. The dependence of these observables on the controllable curvature and charge of the substrate's surface is measurable. The first part of this dissertation consists of a description of the common theory of the quadrupole ion trap with the completion of not widely known, newly introduced, contributions to the trapping potential. These contributions lead to systematic shifts in the mass determination. In particular the influence of the inhomogenity of the electrical field, that is used for compensating the gravitational force, is investigated analytically and corroborated experimentally. It is assumed, that the particle's finite size effects in a further shift. In the experimental part initial demonstrative measurements are presented: the time-resolved adsorption of fullerene, anthracene and NO on silica spheres with 500nm diameter has been measured at room temperature. In addition the secondary electron yield of in-situ prepared particles during irradiation with monoenergetic electrons has been determined by analyzing the distribution of change of the number of elementary charges by single events of charging.

Two gas-phase ion-molecule reaction systems are presented here based on measurements done in a temperature variable 22-pole trapping machine. In the first case, the proton affinity of methane is determined based on a new technique for measuring the equilibrium constant of the HCO2+ + CH4 <=> CH5+ + CO2 reaction. The second case reports to the (Ar + N2 )+ reaction system, with reaction rate temperature dependencies measurements made both in the forward and reverse direction with different and complementary methods. The temperature variable 22-pole trapping machine allows one to determine equilibrium constants and reaction rate coefficients over a wide range of temperatures. The coupling of an effusive beam to the setup overcomes the problem of neutral gas wall condensation and extends the temperature range measurements beyond condensation point. The introduction (Chapter 1) gives a short overview about the rf technology and parallel experimental techniques developed in order to better characterize and understand the several mechanisms related to ion-molecule reactions. It also focuses some aspects of reaction rate temperature dependencies determination as well as thermodynamical equilibrium in laboratory environment. A short description of the setup and experimental methods are presented in Chapter 2. Based on equilibrium constant measurements, Chapter 3 is dedicated to the proton affinity of methane. This concept has applications on several fields such as atmospheric and combustion modelling, or testing empirical and ab initio theories for electronic structures. The (Ar − N2 )+ system presented in Chapter 4, is known for being a good case study for inferring the role of vibrational excitation in reaction dynamics and to the existence of non-adiabatic coupling. The experimental results here presented for the N2+ + Ar reaction demonstrate that it is possible to avoid parallel reactions with first vibrational excited state of nitrogen (N2 (ν = 1)). On the other hand, the reverse reaction experiments confirm the existence of a minimum of the reaction rate in the 30 to 300 K range, due to the existence of two reaction channels. The question of the high rate coefficient towards lower temperatures being related to the N2 rotational ground state population is raised. A summary and outlook are presented in Chapter 5, where some new possible paths of investigation are pointed out.