On the Justification of the Foldy--Lax Approximation for the Acoustic Scattering by Small Rigid Bodies of Arbitrary Shapes

Challa, Durga Prasad; Sini, Mourad

doi:10.1137/130919313

Cited by 35 publications

(65 citation statements)

References 36 publications

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“…Second, using formulas of the type and , one can solve the inverse problems which consists of localizing the centers,

z_{m}

, of the obstacles from the far‐field measurements and also estimating their sizes from the computed capacitances

C_{m}

, see , , , for instance.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

“…The goal of the present work is to extend the results in to the Lamé system and derive the error of the approximation explicitly in terms of the whole denseness of the scatterers, i.e.

M, a

and d .…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

“…For any functions

f, g

defined on

\partial D_{ε}

and

\partial B

respectively, we define

{(f)}^{\land} (ξ) : = \hat{f} (ξ) : = f (ε ξ + z) and {(g)}^{\lor} (x) : = \overset{̌}{g} (x) : = g (\frac{x - z}{ε}) .

Let T 1 and T 2 be an orthonormal basis for the tangent plane to

\partial D_{ε}

at x and let

\partial \partial T = \sum_{l = 1}^{2} \partial \partial T_{p} T_{p}

, denote the tangential derivative on

\partial D_{ε}

. Then the space

H^{1} (\partial D_{ε})

is defined as

H^{1} (\partial D_{ε}) : = \{ϕ \in L^{2} (\partial D_{ε}); \partial ϕ \partial T \in L^{2} (\partial D_{ε})\} .

We have the following lemma from . Lemma Suppose

0 < ε \leq 1

and

D_{ε} : = ε B + z \subset {double-struckR}^{n}

. Then for every

ψ \in L^{2} (\partial D_{ε})

and...…”

Section: Proof Of Theoremmentioning

confidence: 99%

See 2 more Smart Citations

The Foldy-Lax approximation of the scattered waves by many small bodies for the Lamé system

Challa

Sini

2015

Math. Nachr.

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We are concerned with the linearized, isotropic and homogeneous elastic scattering problem by many small rigid obstacles of arbitrary, Lipschitz regular, shapes in 3D case. We prove that there exists two constant a0 and c0, depending only on the Lipschitz character of the obstacles, such that under the conditions a ≤ a0 and≤ c0 on the number M of the obstacles, their maximum diameter a and the minimum distance between them d, the corresponding Foldy-Lax approximation of the farfields is valid. In addition, we provide the error of this approximation explicitly in terms of the three parameters M, a and d. These approximations can be used, in particular, in the identification problems (i.e. inverse problems) and in the design problems (i.e. effective medium theory).

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“…Second, using formulas of the type and , one can solve the inverse problems which consists of localizing the centers,

z_{m}

, of the obstacles from the far‐field measurements and also estimating their sizes from the computed capacitances

C_{m}

, see , , , for instance.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

“…The goal of the present work is to extend the results in to the Lamé system and derive the error of the approximation explicitly in terms of the whole denseness of the scatterers, i.e.

M, a

and d .…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

“…For any functions

f, g

defined on

\partial D_{ε}

and

\partial B

respectively, we define

{(f)}^{\land} (ξ) : = \hat{f} (ξ) : = f (ε ξ + z) and {(g)}^{\lor} (x) : = \overset{̌}{g} (x) : = g (\frac{x - z}{ε}) .

Let T 1 and T 2 be an orthonormal basis for the tangent plane to

\partial D_{ε}

at x and let

\partial \partial T = \sum_{l = 1}^{2} \partial \partial T_{p} T_{p}

, denote the tangential derivative on

\partial D_{ε}

. Then the space

H^{1} (\partial D_{ε})

is defined as

H^{1} (\partial D_{ε}) : = \{ϕ \in L^{2} (\partial D_{ε}); \partial ϕ \partial T \in L^{2} (\partial D_{ε})\} .

We have the following lemma from . Lemma Suppose

0 < ε \leq 1

and

D_{ε} : = ε B + z \subset {double-struckR}^{n}

. Then for every

ψ \in L^{2} (\partial D_{ε})

and...…”

Section: Proof Of Theoremmentioning

confidence: 99%

See 1 more Smart Citation

The Foldy-Lax approximation of the scattered waves by many small bodies for the Lamé system

Challa

Sini

2015

Math. Nachr.

Self Cite

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“…Geuzaine et al [22] provided an open source finite element solver based on domain decomposition methods; the open source solver software available online, is suitable for solution of high frequency time-harmonic electromagnetic wave problems. Challa and Sini [23] studied the acoustic MS problem by many small rigid scatterers of arbitrary shapes using the Foldy-Lax approximation; they solved the inverse problem of scattering by the small obstacles. Thierry et al [24] proposed a new efficient Matlab toolbox, so-called 'μ-diff' suitable for a large class of 2D MS problems with any deterministic or random distribution of cylinders.…”

Section: Background Reviewmentioning

confidence: 99%

Acoustic multiple scattering using recursive algorithms

Amirkulova

Norris

2015

Journal of Computational Physics

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“…In the present work, this last condition is the only one we (naturally) impose on t. Hence, compared to [15], we allow here the small scatterers to be as close as we want since t can be as large as we want, i.e. we cover the mesoscale regime.…”

mentioning

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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On the Justification of the Foldy--Lax Approximation for the Acoustic Scattering by Small Rigid Bodies of Arbitrary Shapes

Cited by 35 publications

References 36 publications

The Foldy-Lax approximation of the scattered waves by many small bodies for the Lamé system

The Foldy-Lax approximation of the scattered waves by many small bodies for the Lamé system

Acoustic multiple scattering using recursive algorithms

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

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