In this paper, we study a model for the magnetization in thin ferromagnetic films. It comes as a variational problem for $S^1$-valued maps $m^{\text{'}}$ (the magnetization) of two variables $x^{\text{'}}$: $E_{\epsilon }\left(m^{\text{'}}\right)=\epsilon \int {|{\nabla }^{\text{'}}·m^{\text{'}}|}^{2}d{x}^{\text{'}}+\frac{1}{2}\int {\left||{\nabla }^{\text{'}}{|}^{-1/2}{\nabla }^{\text{'}}·m^{\text{'}}\right|}^{2}d{x}^{\text{'}}$. We are interested in the behavior of minimizers as $\epsilon \to 0$. They are expected to be $S^1$-valued maps $m^{\text{'}}$ of vanishing distributional divergence ${\nabla }^{\text{'}}·m^{\text{'}}=0$, so that appropriate boundary conditions enforce line discontinuities. For finite $\epsilon >0$, these line discontinuities are approximated by smooth transition layers, the so-called Néel walls. Néel...

In this article, we derive a complete mathematical analysis of a coupled 1D-2D model for 2D wave propagation in media including thin slots. Our error estimates are illustrated by numerical results.

Asymptotics of solutions to Schrödinger equations with singular magnetic and electric potentials is investigated. By using a Almgren type monotonicity formula, separation of variables, and an iterative Brezis–Kato type procedure, we describe the exact behavior near the singularity of solutions to linear and semilinear (critical and subcritical) elliptic equations with an inverse square electric potential and a singular magnetic potential with a homogeneity of order −1.

This paper is devoted to the study of cloaking via anomalous localized resonance (CALR) in the two- and three-dimensional quasistatic regimes. CALR associated with negative index materials was discovered by Milton and Nicorovici [21] for constant plasmonic structures in the two-dimensional quasistatic regime. Two key features of this phenomenon are the localized resonance, i.e., the fields blow up in some regions and remain bounded in some others, and the connection between the localized resonance...

The purpose of this paper is to derive and study a new asymptotic model for the equilibrium state of a thin anisotropic piezoelectric plate in frictional contact with a rigid obstacle. In the asymptotic process, the thickness of the piezoelectric plate is driven to zero and the convergence of the unknowns is studied. This leads to two-dimensional Kirchhoff-Love plate equations, in which mechanical displacement and electric potential are partly decoupled. Based on this model numerical examples are presented...

We propose transmission conditions of order 1, 2 and 3 approximating the shielding behaviour of thin conducting curved sheets for the magneto-quasistatic eddy current model in 2D. This model reduction applies to sheets whose thicknesses ε are at the order of the skin depth or essentially smaller. The sheet has itself not to be resolved, only its midline is represented by an interface. The computation is directly in one step with almost no additional cost. We prove the well-posedness w.r.t. to...

We propose transmission conditions of order 1, 2 and 3 approximating the shielding behaviour of thin conducting curved sheets for the magneto-quasistatic eddy current model in 2D. This model reduction applies to sheets whose thicknesses ε are at the order of the skin depth or essentially smaller. The sheet has itself not to be resolved, only its midline is represented by an interface. The computation is directly in one step with almost no additional cost. We prove the well-posedness w.r.t. to...

A new system of integral equations for the exterior 2D time harmonic scattering problem is investigated. This system was first proposed by B. Després in [11]. Two new derivations of this system are given: one from elementary manipulations of classical equations, the other based on a minimization of a quadratic functional. Numerical issues are addressed to investigate the potential of the method.

The Asymptotic Numerical Method (ANM) is a family of algorithms, based on computation of truncated vectorial series, for path following problems [2]. In this paper, we present and discuss some techniques to define local parameterization [4, 6, 7] in the ANM. We give some numerical comparisons of pseudo arc-length parameterization and local parameterization on non-linear elastic shells problems

Nous étudions l’effet d’une couche mince rugueuse périodique déposée sur une structure semi-infinie, dans le contexte Helmholtz bi-dimensionnel. Formellement, nous obtenons des conditions de transmission équivalentes à l’ordre 1, par des techniques de type homogénéisation. Suivent alors la résolution du problème du milieu effectif éclairé par une onde plane, et le calcul de la fonction de Green effective ; le tout par analyse de Fourier. Dans un deuxième temps, nous considérons le problème de diffraction...

In this paper, we consider linear ordinary differential equations originating in electronic engineering, which exhibit exceedingly rapid oscillation. Moreover, the oscillation model is completely different from the familiar framework of asymptotic analysis of highly oscillatory integrals. Using a Bessel-function identity, we expand the oscillator into asymptotic series, and this allows us to extend Filon-type approach to this setting. The outcome is a time-stepping method that guarantees ...

La compréhension du passage des équations de la mécanique des fluides compressibles aux équations incompressibles a fait de grands progrès ces vingt dernières années. L’objectif de cet exposé est de présenter l’évolution des méthodes mathématiques mises en œuvre pour étudier ce passage à la limite, depuis les travaux de S. Klainerman et A. Majda dans les années quatre–vingts, jusqu’à ceux récents de G. Métivier et S. Schochet (pour les équations non isentropiques). Suivant les conditions initiales...

In this work, depending on the relation between the Deborah, the Reynolds and the aspect ratio numbers, we formally derived shallow-water type systems starting from a micro-macro description for non-Newtonian fluids in a thin domain governed by an elastic dumbbell type model with a slip boundary condition at the bottom. The result has been announced by the authors in [G. Narbona-Reina, D. Bresch, Numer. Math. and Advanced Appl. Springer Verlag (2010)] and in the present paper, we provide a self-contained...