In this series of two articles, we consider the propagation of a time harmonic wave in a medium made of the junction of a half-space (containing possibly scatterers) with a thin slot. The Neumann boundary condition is considered along the boundary on the propagation domain, which authorizes the propagation of the wave inside the slot, even if the width of the slot is very small. We perform a complete asymptotic expansion of the solution of this problem with respect to the small parameter ε/λ, the ratio between the width of the slot, and the wavelength. We use the method of matched asymptopic expansions which allows us to describe the solution in terms of asymptotic series whose terms are characterized as the solutions of (coupled) boundary value problems posed in simple geometrical domains, independent of ε/λ: the (perturbed) half-space, the half-line, a junction zone. In this first article, we derive and analyze, from the mathematical point of view, these boundary value problems. The second one will be devoted to establishing error estimates for truncated series.

Abstract.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.Mathematics Subject Classification.

The presence of small inclusions modifies the solution of the Laplace equation posed in a reference domain Ω 0 . This question has been deeply studied for a single inclusion or well separated inclusions. We investigate the case where the distance between the holes tends to zero but remains large with respect to their characteristic size. We first consider two perfectly insulated inclusions. In this configuration we give a complete multiscale asymptotic expansion of the solution to the Laplace equation. We also address the situation of a single inclusion close to a singular perturbation of the boundary ∂Ω 0 . We also present numerical experiments implementing a multiscale superposition method based on our first order expansion.

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We derive and analyse models which reduce conducting sheets of a small thickness ε in two dimensions to an interface and approximate their shielding behaviour by conditions on this interface. For this we consider a model problem with a conductivity scaled reciprocal to the thickness ε, which leads to a nontrivial limit solution for ε → 0. The functions of the expansion are defined hierarchically, i.e. order by order. Our analysis shows that for smooth sheets the models are well defined for any order and have optimal convergence meaning that the H 1 -modelling error for an expansion with N terms is bounded by O(ε N +1 ) in the exterior of the sheet and by O(ε N +1/2 ) in its interior. We explicitly specify the models of order zero, one and two. Numerical experiments for sheets with varying curvature validate the theoretical results. Mathematics Subject Classification IntroductionMany electric devices contain very thin conducting parts either for electromagnetic shielding [13,16], or as casings, tank walls [9,25] or supply lines [5]. The large aspect ratio of these sheets of about few millimetres or centimetres to metres or hundreds of micrometres to centimetres and the high conductivity causes variations in thickness direction in much smaller scales than in the longitudinal directions. Their discretisation by the finite element method (FEM) is challenging when the thickness ε of the thin sheets is considerably smaller than the size of neighbouring parts for three reasons. First, domains with such thin sheets are difficult to mesh by most mesh generators. Secondly, a discretisation on meshes with cell sizes of different magnitudes can lead to ill-conditioned matrices, and thirdly, meshes of good quality may also contain cells around the sheet with sizes comparable to the sheet thickness which leads to a high number of additional degrees of freedom. By reducing the thin sheet to an interface and by approximating its effect by conditions on this interface, a highly accurate modelling with standard discretisation schemes like the FEM is possible.The so called impedance boundary conditions (IBCs), first proposed by Shchukin [29] and Leontovich [19], are traditionally used for replacing solid conductors, where the domain is artificially confined, by an approximate boundary condition [1][2][3]8,11,15,28]. This technique is proved to be accurate for smooth sheets and can be readily implemented.However, in the context of thin conducting sheets this technique of Shchukin and Leontovich has been seldom applied. Interface conditions for thin sheets are often based on a tensor product ansatz of a set of simple functions in thickness direction and functions defined on the interface. The simplest approaches assume no variation in thickness direction, which leads to a surface quantity [5,22]. Using two functions in thickness direction Krähenbühl and Muller [18] derived a relation between the mean value of the tangential component of the electric or magnetic field on the interfaces of the sheet and the jump of the magnetic or elec...

We study the exterior derivative as a symmetric unbounded operator on square integrable 1-forms on a 3D bounded domain D. We aim to identify boundary conditions that render this operator self-adjoint. By the symplectic version of the Glazman-Krein-Naimark theorem this amounts to identifying complete Lagrangian subspaces of the trace space of H(curl, D) equipped with a symplectic pairing arising from the ∧-product of 1-forms on ∂D. Substantially generalizing earlier results, we characterize Lagrangian subspaces associated with closed and co-closed traces. In the case of non-trivial topology of the domain, different contributions from co-homology spaces also distinguish different self-adjoint extension. Finally, all self-adjoint extensions discussed in the paper are shown to possess a discrete point spectrum, and their relationship with curl curl-operators is discussed.

Abstract.We are concerned with a 2D time harmonic wave propagation problem in a medium including a thin slot whose thickness ε is small with respect to the wavelength. In a previous article, we derived formally an asymptotic expansion of the solution with respect to ε using the method of matched asymptotic expansions. We also proved the existence and uniqueness of the terms of the asymptotics. In this paper, we complete the mathematical justification of our work by deriving optimal error estimates between the exact solutions and truncated expansions at any order.Mathematics Subject Classification. 35J05, 34E05, 78A45, 78A50.

This paper is devoted to the mathematical justification of the usual models predicting the effective reflection and transmission of an acoustic wave by a low porosity multiperforated plate. Some previous intuitive approximations require that the wavelength be large compared with the spacing separating two neighboring apertures. In particular, we show that this basic assumption is not mandatory. Actually, it is enough to assume that this distance is less than a half-wavelength. The main tools used are the method of matched asymptotic expansions and lattice sums for the Helmholtz equations. Some numerical experiments illustrate the theoretical derivations.1. Introduction. Perforated plates are widely used in engineering systems, either as resistive sheets for acoustic liners [24] or for cooling purposes at the walls of the combustion chambers of jet engines [19]. As these plates generally contain tens of thousands of holes for realistic geometries, the direct determination of the effect of the perforations is out of reach in numerical simulations. As a result, the effective acoustic behavior of such plates is usually reproduced by designing suitable semi-analytic models.Most of these models are based on the calculation of the Rayleigh conductivity of a hole, which relates the net fluctuating volume flux through the perforation to the fluctuating pressure drop across it (cf., e.g., [32,12]). Such an approach has several advantages. The most valuable of them is the possibility of coupling models taking into account the absorption of acoustic energy by viscosity at the orifices with the usual equations governing the propagation of acoustic waves elsewhere. The most frequently used model relies on the expression for the conductivity of a single circular hole in an infinitely thin wall derived by Howe [10]. In this model, a bias flow crosses the orifice. The limiting case, when the velocity of the bias flow goes to zero, is reduced to the very classical approach, earlier derived by Rayleigh [32]. In [35], one can find a clear derivation of the Rayleigh conductivity model for circular perforations in an infinitely thin plate, although formal and implicit as regards the method of matched asymptotic expansion. Howe's model was later extended to a hole in a thick plate [11,15]. These models are widely used in the literature for plates of low porosity