Fréchet differentiability of the elasto‐acoustic scattered field with respect to Lipschitz domains

Barucq, Hélène; Djellouli, Rabïa; Estecahandy, Elodie

doi:10.1002/mma.3444

Cited by 6 publications

(14 citation statements)

References 55 publications

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“…Note that most references assume at least C 2 regularity, cf. [6,2], with the exception of [5] for acoustic scattering with Lipschitz obstacles, and [14,15,16] for acoustic-elastic transmission (in fact OP ∞ ) with polygonal-shaped obstacles, and the aforementioned [11,8] for elasticity on bounded Lipschitz domain. Under Lipschitz assumption, the auxiliary problems can contain singular boundary or interface terms that do not fit in the canonical boundary Sobolev spaces, and to give sense to the problem is not trivial (cf.…”

Section: Introductionmentioning

confidence: 99%

“…Remark 5. Although [14,15,16] also study the auxiliary problems in PDE form, they use implicit function theorem to show differentiability, while we work directly with the solution operator and use standard analysis 4 . Other approaches to show Fréchet differentiability can be e.g.…”

Section: Introductionmentioning

confidence: 99%

“…Water has the shortest wavelength, due to its speed c f inferior to both the primary speed c P and the shear speed c S in steel as indicated in Table 1. 16 They are uniformly convergent on compact subsets.…”

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confidence: 99%

“…While the true analytical values are infinite sums 16 , those for numerical purposes are truncated versions with N = 2 κ b + 1 modes (a standard choice cf. [13]).…”

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confidence: 99%

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Characterization of partial derivatives with respect to material parameters in a fluid–solid interaction problem

Azpiroz

Barucq

Djellouli

et al. 2018

Journal of Mathematical Analysis and Applications

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For a fluid-solid interaction problem with Lipschitz interface, we investigate the partial Fréchet differentiability of the solutions and the approximate far-fieldpattern with respect to solid material parameters. Differentiability is shown in standard Sobolev framework, and the derivatives are characterized as solutions to inhomogeneous fluid-solid transmission problems. To validate the accuracy of the characterization, we compare analytical values with numerical ones given by Interior Penalty Discontinuous Galerkin (IPDG) in a setting with circular obstacles. Our comparisons also show that IPDG gives results with high precision and incurs almost no effect of discretization error accumulation.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

“…While the true analytical values are infinite sums 16 , those for numerical purposes are truncated versions with N = 2 κ b + 1 modes (a standard choice cf. [13]).…”

mentioning

confidence: 99%

See 2 more Smart Citations

Characterization of partial derivatives with respect to material parameters in a fluid–solid interaction problem

Azpiroz

Barucq

Djellouli

et al. 2018

Journal of Mathematical Analysis and Applications

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“…The case of electromagnetic scattering problems has been also studied by several authors and results can be found in [36,21,9,10,30], among other references. The case of elasto-acoustic problems was recently partially addressed in [7]. It was proved that the elasto-acoustic scattered field and its corresponding far-field pattern are continuously Fréchet differentiable with respect to the elastic domain.…”

mentioning

confidence: 99%

Mathematical Determination of the Fréchet Derivative with Respect to the Domain for a Fluid-Structure Scattering Problem: Case of Polygonal-Shaped Domains

Barucq¹,

Djellouli²,

Estecahandy³

et al. 2018

SIAM J. Math. Anal.

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Abstract. The characterization of the Fréchet derivative of the elasto-acoustic scattered field with respect to Lipschitz continuous polygonal domains is established. The considered class of domains is of practical interest since two-dimensional scatterers are always transformed into polygonalshaped domains when employing finite element methods for solving direct and inverse scattering problems. The obtained result indicates that the Fréchet derivative with respect to the scatterer of the scattered field is the solution of the same elasto-acoustic scattering problem but with additional right-hand side terms in the transmission conditions across the fluid-structure interface. This characterization has the potential to advance the state-of-the-art of the solution of inverse obstacle problems. 1. Introduction. One of the basic inverse scattering problems in scattering theory is the determination of the shape of the scatterer using some measured scattered far-field patterns [8]. This model problem, called inverse obstacle problem (IOP), is relevant to numerous real-world applications including radar and sonar detection, geophysical exploration, structural design, medical imaging, and atmospheric studies. In spite of their apparent simple formulations, IOPs are very challenging problems from both mathematical and computational viewpoints. The difficulties in studying and/or solving IOPs are mainly due to their nonlinear and severely ill-posed nature [8]. Nevertheless, given the applied nature of these problems and their prevalence in sciences and engineering, IOPs have been subject to extensive studies leading to a tremendous growth within the last three decades with an emphasis on the development of computational methods (see, for example, [11,14], and the references therein).

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An effective numerical strategy for retrieving all characteristic parameters of an elastic scatterer from its FFP measurements

Azpiroz

Barucq

Diaz

et al. 2020

Journal of Computational Physics

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A new computational strategy is proposed for determining all elastic scatterer characteristics including the shape, the material properties (Lamé coefficients and density), and the location from the knowledge of farfield pattern (FFP) measurements. The proposed numerical approach is a multi-stage procedure in which a carefully designed regularized iterative method plays a central role. The adopted approach is critical for recognizing that the different nature and scales of the sought-after parameters as well as the frequency regime have different effects on the scattering observability. Identification results for two-dimensional elastic configurations highlight the performance of the designed solution methodology.

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Fréchet differentiability of the elasto‐acoustic scattered field with respect to Lipschitz domains

Cited by 6 publications

References 55 publications

Characterization of partial derivatives with respect to material parameters in a fluid–solid interaction problem

Characterization of partial derivatives with respect to material parameters in a fluid–solid interaction problem

Mathematical Determination of the Fréchet Derivative with Respect to the Domain for a Fluid-Structure Scattering Problem: Case of Polygonal-Shaped Domains

An effective numerical strategy for retrieving all characteristic parameters of an elastic scatterer from its FFP measurements

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