The Effect of a Thin Coating on the Scattering of a Time-Harmonic Wave for the Helmholtz Equation

Bendali, Abderrahmane; Lemrabet, Keddour

doi:10.1137/s0036139995281822

Cited by 107 publications

(175 citation statements)

References 12 publications

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“…i- 6 and for The figure 10 gives the rate of convergence of u h -u when computed over an arbitrary mesh of the type given m figure 5, v/hils figure 11 represents the rate of convergence of p^ -p e for the energy norm || . || w £ .…”

Section: Numerical Resultsmentioning

confidence: 99%

“…However, it is the study of an electric current scattering in a plate bordered by a highly conductive rod which, in steady-state, constitutes the most typical example modeled by (1). This model is also a good approximation of wave scattering problems by a scatterer covered with a penetrable thin shell, provided we limit ourselves to the main part of the operator of the partial differential équation set on Q + (see [15] and [6]). …”

Section: Physical Motivationmentioning

confidence: 99%

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Numerical approximation of stiff transmission problems by mixed finite element methods

Capatina-Papaghiuc¹,

Raynaud²

1998

ESAIM: M2AN

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Section: Numerical Resultsmentioning

confidence: 99%

Section: Physical Motivationmentioning

confidence: 99%

Numerical approximation of stiff transmission problems by mixed finite element methods

Capatina-Papaghiuc¹,

Raynaud²

1998

ESAIM: M2AN

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“…Thus, one would like to replace the sheet by an internal interface, where Generalised Impedance Boundary conditions (GIBCs) [2], [3] model the effect of the sheet. We will derive higher order GIBCs by asymptotic expansions [4] for smooth sheets and constant conductivity c(x). Even so we are looking for the solution for a sheet of a particular thickness d, we replace (1) by a family of problems (see Fig.…”

Section: Outline Of the Problemmentioning

confidence: 99%

Asymptotic expansion of highly conductive thin sheets

Schmidt

Tordeux

2007

Proc Appl Math and Mech

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Sensitive measurement and control equipment are protected from disturbing electromagnetic fields by thin shielding sheets. Alternatively to discretisation of the sheets, the electromagnetic fields are modeled only in the surrounding of the layer taking them into account with the so called Generalised Impedance Boundary Conditions.We study the shielding effect by means of the model problem of a diffusion equation with additional dissipation in the curved thin sheet. We use the asymptotic expansion techniques to derive a limit problem, when the thickness of the sheet ε tends to zero, as well as the models for contribution to the solution of higher order in ε. These problems are posed in limit area of vanishing ε with condition for the jump of the solution and it's normal derivative, which avoid to mesh the computational domain, even locally, at the scale of ε.We derive the problems for arbitrary order and show their existence and uniqueness. Numerical experiments for the problems up to second order show the asymptotic convergence of the solution of right order in mean of the thickness parameter ε.Copyright line will be provided by the publisher Outline of the problemThe diffusion-reaction equation in 2Dis a model problem for the magnetoquasistatic Eddy-current model with a thin conducting sheet [1], whereas c(x) is vanishing besides the thin sheet Ω C of constant thickness d. Domains with very thin sheets are hardly to mesh by todays grid generators. Thus, one would like to replace the sheet by an internal interface, where Generalised Impedance Boundary conditions (GIBCs)[2], [3] model the effect of the sheet. We will derive higher order GIBCs by asymptotic expansions [4] for smooth sheets and constant conductivity c(x). Even so we are looking for the solution for a sheet of a particular thickness d, we replace (1) by a family of problems (see Fig. 1), ordered by the thickness parameter εWe consider these equations for all ε < ε 0 , such that the support of f is outside the sheet Ω ε int . We choose in this model a conductivity scaled like 1 /ε. To solve for a particular thickness d, one select c 0 = c d, by what the problems (2) and (1) coincide for ε = d. Due to this scaling the limit solution for ε → 0 is non-trivial, meaning that neither the effect of the sheet disappear nor the sheet gets a perfect conductor. The external solution u ε ext (x) in the non-conducting subdomain Ω ε ext and the internal solution u ε int (x) in Ω ε int are related by Dirichlet and Neumann transmission conditions. Asymptotic expansionsExternal functions In the external domain Ω ε ext we are looking for an asymptotic of the solution with the formInternal functions We introduce local coordinates (t, s) in the sheet, where t is the tangential coordinate along the midline Γ m and s ∈ [−ε/2, ε/2] is the coordinate in thickness direction. Furthermore, let S = s/ε the stretched coordinate, the parameter domain of (t, S) is called Ω and the internal solution is U ε int (t, S) := u ε int (t, s). We are looking for an asymptotic of the soluti...

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“…involving second order tangential derivatives of the displacement field). These conditions appear frequently as the result of an asymptotic analysis to derive impedance conditions for thin layer problems (see [17,6,22]). The first order approximation for the Laplace-Neumann transmission problem with a thin layer is…”

Section: Introductionmentioning

confidence: 99%

“…The same type of transmission conditions appears in the work [28] of Marigo and Pideri in the context of linear elasticity in dimension three, where the authors study the effects of inclusions of a second material under the assumptions that the defects are coplanar. Finally, in the context of wall laws for rough boundary, Bresch and Milisic obtained in [12] a second order wall law which is written in the form 6) where β and γ are solutions of the two cells problem and have opposite signs. In a first work on the scalar case [9], we studied the existence and uniqueness for the non-coercive Ventcel problem.…”

Section: Introductionmentioning

confidence: 99%

Artificial conditions for the linear elasticity equations

Bonnaillie‐Noël

Dambrine

Hérau³

et al. 2014

Math. Comp.

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In this paper, we consider the equations of linear elasticity in an exterior domain. We exhibit artificial boundary conditions on a circle, which lead to a non-coercive second order boundary value problem. In the particular case of an axisymmetric geometry, explicit computations can be performed in Fourier series proving the wellposedness except for a countable set of parameters. A perturbation argument allows to consider near-circular domains. We complete the analysis by some numerical simulations.

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The Effect of a Thin Coating on the Scattering of a Time-Harmonic Wave for the Helmholtz Equation

Cited by 107 publications

References 12 publications

Numerical approximation of stiff transmission problems by mixed finite element methods

Numerical approximation of stiff transmission problems by mixed finite element methods

Asymptotic expansion of highly conductive thin sheets

Artificial conditions for the linear elasticity equations

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