The aim of this paper is to present comparisons between numerical simulations and experimental me... more The aim of this paper is to present comparisons between numerical simulations and experimental measurements of the multipaction effect. After a brief description of this phenomenon and a presentation of the experiment to be modeled, we exposed the numerical techniques and specific methodology that we implemented in our code DEGAS. This methodology tries to provide an efficient numerical tool for the analysis and the understanding of the multipaction phenomenon. Some comparisons are shown that point out a good agreement between experimental and numerical results. The final aim of this work would be to give a help in the designing of R.F. component free from multipaction effect.
L'identification de milieux elastiques 1d a partir de donnees de sismiques marines, consiste ... more L'identification de milieux elastiques 1d a partir de donnees de sismiques marines, consiste a determiner les distributions des vitesses de propagation des ondes et de densite, parametres caracteristiques d'un sous-sol. Notre etude se compose de trois parties. Dans un premier temps, on presente et analyse la methode utilisee. On definit l'operateur de modelisation qui, a une distribution de parametres, associes la restriction a un plan, de la solution d'une equation de propagation a deux dimensions. Pour integrer ce systeme d'equations couplees acoustique-elastodynamique, une classe de schemas numeriques est etudiee. On s'interesse plus particulierement aux proprietes de stabilite et de dispersion numerique. Le probleme inverse est ensuite formule comme un probleme de moindres carres. De premieres experiences numeriques illustrent la faisabilite de cette technique. Dans une deuxieme partie, nous etudions la linearisation de l'operateur de modelisation. L&...
In this chapter, we specifically study the approximate models that we derived from Maxwell’s equa... more In this chapter, we specifically study the approximate models that we derived from Maxwell’s equations. We refer to Chap. 1 for the models, and we rely on the mathematical tools introduced in Chaps. 2, 3 and 4. Unless otherwise specified, we consider complex-valued function spaces. Constants that are independent of the data, but that may depend on the domain or on the parameters defining the model, are generically denoted by C, C0, C1, etc. We provide incremental proofs for the well-posedness of the static, quasi-static and Darwin models, in the sense that solving the quasi-static models relies on the solution of static problems, whereas solving the Darwin models relies on the solution of static and quasi-static problems.
In this chapter, we analyse the coupled models introduced in Sect. 1.3, namely the Vlasov–Poisson... more In this chapter, we analyse the coupled models introduced in Sect. 1.3, namely the Vlasov–Poisson, Vlasov–Maxwell and magnetohydrodynamics (MHD) systems. They are basic models of charged particle, plasma and conducting fluid physics. We present the useful mathematical tools, and a variety of existence and uniqueness results for several types of solution.
We first introduce basic notions on Banach and Hilbert spaces. Afterwards, we recall some well-kn... more We first introduce basic notions on Banach and Hilbert spaces. Afterwards, we recall some well-known results, which help prove the well-posedness of the various sets of equations we study throughout this book.
In this chapter, we devote our attention to establishing mathematical properties concerning the e... more In this chapter, we devote our attention to establishing mathematical properties concerning the electromagnetic fields that are governed by the time-dependent Maxwell equations. For that, we investigate a number of physical properties of the electromagnetic fields exhibited in Chap. 1, using the mathematical tools introduced in Chaps. 2, 3 and 4. We focus mainly on four items.
To measure data and solutions spatially, we recall a number of useful definitions and results on ... more To measure data and solutions spatially, we recall a number of useful definitions and results on Lebesgue and standard Sobolev spaces. Then, we introduce more specialized Sobolev spaces, which are better suited to measuring solutions to electromagnetics problems, in particular, the divergence and the curl of fields. This also allows one to measure their trace at interfaces between two media, or on the boundary. Last, we construct ad hoc function spaces, adapted to the study of time- and space-dependent electromagnetic fields.
Computational Methods in Applied Mathematics, 2022
Even today, solving numerically the time-dependent Vlasov–Maxwell equations is a challenging issu... more Even today, solving numerically the time-dependent Vlasov–Maxwell equations is a challenging issue, and developing simpler but accurate approximate models is still worthwhile. Here, we propose a new family of paraxial asymptotic models that approximates the Vlasov–Maxwell system of equations. We introduce parameters in our models that allow us to handle relativistic cases, much slower beams or even non-relativistic cases. These models are derived by introducing a small parameter and provide static or quasi-static approximate equations that are 𝑛-th order accurate; 𝑛 may be chosen as required. Practically, one can select a model by determining the regime one is interested in and choosing the degree of accuracy needed.
This chapter is devoted to an alternative, second-order formulation of the Maxwell’s equations. W... more This chapter is devoted to an alternative, second-order formulation of the Maxwell’s equations. We rigorously justify the process we outlined in Sect. 1.5.3. This new formulation is especially relevant for computational applications, as it admits several variational formulations, which can be simulated by versatile finite element methods. Our attention will be focused on three issues: equivalence of the second-order equations with the original, first-order equations studied in Chap. 5, the well-posedness of the new formulation and the regularity of its solution, as we did in that chapter. We also study how to take into account the conditions on the divergence of the fields, incorporating them explicitly at some point in the variational formulations. To these ends, we shall again rely on the mathematical tools introduced in Chaps. 2, 3 and 4, as well as on the specific properties of the spaces of electromagnetic fields introduced in Chap. 6.
An explicit k-dependency in W^m,p error estimates is proved to derive two probabilistic laws whic... more An explicit k-dependency in W^m,p error estimates is proved to derive two probabilistic laws which evaluate the relative accuracy between Lagrange finite elements P_k_1 and P_k_2, (k_1 < k_2). Thanks to these estimates we show a weak asymptotic relation in D'(R) between these probabilistic laws when k_2-k_1 goes to infinity. In this case, we get that P_k_2 finite element is surely more accurate than P_k_1, for any value of the mesh size h. This brings a complementary perspective regarding the classical way of comparing two finite elements, which is usually limited to the asymptotic rate of convergence, when h goes to zero. Moreover, we also get new insights which highlight cases such that P_k_1 finite element is more likely accurate than P_k_2 one, for a range of specific values of h.
The aim of this paper is to present comparisons between numerical simulations and experimental me... more The aim of this paper is to present comparisons between numerical simulations and experimental measurements of the multipaction effect. After a brief description of this phenomenon and a presentation of the experiment to be modeled, we exposed the numerical techniques and specific methodology that we implemented in our code DEGAS. This methodology tries to provide an efficient numerical tool for the analysis and the understanding of the multipaction phenomenon. Some comparisons are shown that point out a good agreement between experimental and numerical results. The final aim of this work would be to give a help in the designing of R.F. component free from multipaction effect.
L'identification de milieux elastiques 1d a partir de donnees de sismiques marines, consiste ... more L'identification de milieux elastiques 1d a partir de donnees de sismiques marines, consiste a determiner les distributions des vitesses de propagation des ondes et de densite, parametres caracteristiques d'un sous-sol. Notre etude se compose de trois parties. Dans un premier temps, on presente et analyse la methode utilisee. On definit l'operateur de modelisation qui, a une distribution de parametres, associes la restriction a un plan, de la solution d'une equation de propagation a deux dimensions. Pour integrer ce systeme d'equations couplees acoustique-elastodynamique, une classe de schemas numeriques est etudiee. On s'interesse plus particulierement aux proprietes de stabilite et de dispersion numerique. Le probleme inverse est ensuite formule comme un probleme de moindres carres. De premieres experiences numeriques illustrent la faisabilite de cette technique. Dans une deuxieme partie, nous etudions la linearisation de l'operateur de modelisation. L&...
In this chapter, we specifically study the approximate models that we derived from Maxwell’s equa... more In this chapter, we specifically study the approximate models that we derived from Maxwell’s equations. We refer to Chap. 1 for the models, and we rely on the mathematical tools introduced in Chaps. 2, 3 and 4. Unless otherwise specified, we consider complex-valued function spaces. Constants that are independent of the data, but that may depend on the domain or on the parameters defining the model, are generically denoted by C, C0, C1, etc. We provide incremental proofs for the well-posedness of the static, quasi-static and Darwin models, in the sense that solving the quasi-static models relies on the solution of static problems, whereas solving the Darwin models relies on the solution of static and quasi-static problems.
In this chapter, we analyse the coupled models introduced in Sect. 1.3, namely the Vlasov–Poisson... more In this chapter, we analyse the coupled models introduced in Sect. 1.3, namely the Vlasov–Poisson, Vlasov–Maxwell and magnetohydrodynamics (MHD) systems. They are basic models of charged particle, plasma and conducting fluid physics. We present the useful mathematical tools, and a variety of existence and uniqueness results for several types of solution.
We first introduce basic notions on Banach and Hilbert spaces. Afterwards, we recall some well-kn... more We first introduce basic notions on Banach and Hilbert spaces. Afterwards, we recall some well-known results, which help prove the well-posedness of the various sets of equations we study throughout this book.
In this chapter, we devote our attention to establishing mathematical properties concerning the e... more In this chapter, we devote our attention to establishing mathematical properties concerning the electromagnetic fields that are governed by the time-dependent Maxwell equations. For that, we investigate a number of physical properties of the electromagnetic fields exhibited in Chap. 1, using the mathematical tools introduced in Chaps. 2, 3 and 4. We focus mainly on four items.
To measure data and solutions spatially, we recall a number of useful definitions and results on ... more To measure data and solutions spatially, we recall a number of useful definitions and results on Lebesgue and standard Sobolev spaces. Then, we introduce more specialized Sobolev spaces, which are better suited to measuring solutions to electromagnetics problems, in particular, the divergence and the curl of fields. This also allows one to measure their trace at interfaces between two media, or on the boundary. Last, we construct ad hoc function spaces, adapted to the study of time- and space-dependent electromagnetic fields.
Computational Methods in Applied Mathematics, 2022
Even today, solving numerically the time-dependent Vlasov–Maxwell equations is a challenging issu... more Even today, solving numerically the time-dependent Vlasov–Maxwell equations is a challenging issue, and developing simpler but accurate approximate models is still worthwhile. Here, we propose a new family of paraxial asymptotic models that approximates the Vlasov–Maxwell system of equations. We introduce parameters in our models that allow us to handle relativistic cases, much slower beams or even non-relativistic cases. These models are derived by introducing a small parameter and provide static or quasi-static approximate equations that are 𝑛-th order accurate; 𝑛 may be chosen as required. Practically, one can select a model by determining the regime one is interested in and choosing the degree of accuracy needed.
This chapter is devoted to an alternative, second-order formulation of the Maxwell’s equations. W... more This chapter is devoted to an alternative, second-order formulation of the Maxwell’s equations. We rigorously justify the process we outlined in Sect. 1.5.3. This new formulation is especially relevant for computational applications, as it admits several variational formulations, which can be simulated by versatile finite element methods. Our attention will be focused on three issues: equivalence of the second-order equations with the original, first-order equations studied in Chap. 5, the well-posedness of the new formulation and the regularity of its solution, as we did in that chapter. We also study how to take into account the conditions on the divergence of the fields, incorporating them explicitly at some point in the variational formulations. To these ends, we shall again rely on the mathematical tools introduced in Chaps. 2, 3 and 4, as well as on the specific properties of the spaces of electromagnetic fields introduced in Chap. 6.
An explicit k-dependency in W^m,p error estimates is proved to derive two probabilistic laws whic... more An explicit k-dependency in W^m,p error estimates is proved to derive two probabilistic laws which evaluate the relative accuracy between Lagrange finite elements P_k_1 and P_k_2, (k_1 < k_2). Thanks to these estimates we show a weak asymptotic relation in D'(R) between these probabilistic laws when k_2-k_1 goes to infinity. In this case, we get that P_k_2 finite element is surely more accurate than P_k_1, for any value of the mesh size h. This brings a complementary perspective regarding the classical way of comparing two finite elements, which is usually limited to the asymptotic rate of convergence, when h goes to zero. Moreover, we also get new insights which highlight cases such that P_k_1 finite element is more likely accurate than P_k_2 one, for a range of specific values of h.
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