Acoustic and Electromagnetic Equations

Nédélec, Jean-Claude

doi:10.1007/978-1-4757-4393-7

Cited by 718 publications

(583 citation statements)

References 15 publications

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“…Hence the total field U t := U i + U s satisfies the following exterior impedance problem of the acoustic waves where ν m is the outward unit normal vector of ∂D m , κ > 0 is the wave number and S.R.C stands for the Sommerfeld radiation condition. The scattering problem (1.1-1.3) is well posed in the Hölder or Sobolev spaces, see [18,19,27,29] for instance, in the case λ m ≥ 0. We will see that this condition can be relaxed to allow λ m to be negative under some conditions.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

Multiscale analysis of the acoustic scattering by many scatterers of impedance type

Challa

Sini

2016

Z. Angew. Math. Phys.

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We are concerned with the acoustic scattering problem, at a frequency κ, by many small obstacles of arbitrary shapes with impedance boundary condition. These scatterers are assumed to be included in a bounded domain Ω in R 3 which is embedded in an acoustic background characterized by an eventually locally varying index of refraction. The collection of the scatterers Dm, m = 1, ..., M is modeled by four parameters: their number M , their maximum radius a, their minimum distance d and the surface impedances λm, m = 1, ..., M . We consider the parameters M, d and λm's having the following scaling properties:≈ a t and λm := λm(a) = λm,0a −β , as a → 0, with non negative constants s, t and β and complex numbers λm,0's with eventually negative imaginary parts.We derive the asymptotic expansion of the farfields with explicit error estimate in terms of a, as a → 0. The dominant term is the Foldy-Lax field corresponding to the scattering by the point-like scatterers located at the centers zm's of the scatterers Dm's with λm|∂Dm| as the related scattering coefficients. This asymptotic expansion is justified under the following conditionsand the error of the approximation is C a 3−2β−s , as a → 0, where the positive constants a0, λ−, λ+ and C depend only on the a priori uniform bounds of the Lipschitz characters of the obstacles Dm's and the ones of M (a)a s and d(a) a t . We do not assume the periodicity in distributing the small scatterers. In addition, the scatterers can be arbitrary close since t can be arbitrary large, i.e. we can handle the mesoscale regime. Finally, for spherical scatterers, we can also allow the limit case β = 1 with a slightly better error of the approximation.

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Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

Multiscale analysis of the acoustic scattering by many scatterers of impedance type

Challa

Sini

2016

Z. Angew. Math. Phys.

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“…In a pre-processing step, we decompose (f Z ) k and (w Z ) k , for k = B dc − , B dc + , in the basis of vector spherical harmonics (see [16]) as follows, For instance, let us consider the rst problem Z = A. We are interested in the total force and torque exerted by the rst particles B dc − .…”

Section: The Interpolation Methods For Computing the Singular Eldsmentioning

confidence: 99%

“…The two errors are very close in the numerical computation. Hence, it is more convenient to use the rst de nition (16).…”

Section: Computation Of Correction Velocitiesmentioning

confidence: 99%

Numerical Determination of Truncation Orders in the Correction Method for Stokes Equations

Tran

Nguyen

2018

J. ADV. ENG. COMPUT.

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Abstract. In [15], the authors propose an accurate method, namely the correction method, for computing hydrodynamic interaction between very closed spherical particles in a Stokes uid. The accuracy of this method depends on two truncation parameters for approximating the Neumann to Dirichlet matrix and the velocity correction respectively. In this paper, we establish a numerical determination to estimate these parameters. We perform some numerical experiments to present our method.

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“…The following pseudodifferential operators are commonly seen; see [7]. The problem we are solving in this paper is posed as follows.…”

Section: The Problemmentioning

confidence: 99%

Solutions to Pseudodifferential Equations Using Spherical Radial Basis Functions

Pham¹,

Tran²

2009

Bull. Aust. Math. Soc.

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Spherical radial basis functions are used to define approximate solutions to pseudodifferential equations of negative order on the unit sphere. These equations arise from geodesy. The approximate solutions are found by the collocation method. A salient feature of our approach in this paper is a simple error analysis for the collocation method using the same argument as that for the Galerkin method.2000 Mathematics subject classification: primary 65N15, 65N30, 65N35.

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Acoustic and Electromagnetic Equations

Cited by 718 publications

References 15 publications

Multiscale analysis of the acoustic scattering by many scatterers of impedance type

Multiscale analysis of the acoustic scattering by many scatterers of impedance type

Numerical Determination of Truncation Orders in the Correction Method for Stokes Equations

Solutions to Pseudodifferential Equations Using Spherical Radial Basis Functions

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