Application of the linear sampling method to identify cracks with impedance boundary conditions

Hassen, M. F. Ben; Boukari, Yosra; Haddar, Houssem

doi:10.1080/17415977.2012.686997

Cited by 28 publications

(58 citation statements)

References 8 publications

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“…The fracture is illuminated by a set of incident plane waves (1), taking k s such that the ratio between the shear wavelength and the arclength of Γ is λ s / = 0.7. Thus-induced scattered field is then measured in terms of the far-field pattern,ũ ∞ , given by (7)- (9). The spatial density of sensory data, for both illumination and sensing purposes, is given by N θ ×N φ = 25×12 directions given by the polar (θ j , j = 1, .…”

Section: Numerical Implementation and Resultsmentioning

confidence: 99%

“…In this vein, it is noteworthy that some anomaly-indicator functionals -such as those featured by the FM [34], (G)LSM [16,47] and TS [46] are largely insensitive to the (unknown) boundary condition of a hidden anomaly. For instance, [15,9] demonstrate that the LSM is successful in reconstructing electromagnetic obstacles and cracks regardless of their boundary condition. Recent advancements on the recovery of boundary (or interfacial) conditions are, on the other hand, mostly optimization-based as proposed in the context of acoustic and electromagnetic inverse scattering.…”

Section: Introductionmentioning

confidence: 99%

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A synoptic approach to the seismic sensing of heterogeneous fractures: From geometric reconstruction to interfacial characterization

Pourahmadian

Guzina

Haddar

2017

Computer Methods in Applied Mechanics and Engineering

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A non-iterative waveform sensing approach is proposed toward (i) geometric reconstruction of penetrable fractures, and (ii) quantitative identification of their heterogeneous contact condition by seismic i.e. elastic waves. To this end, the fracture support Γ (which may be non-planar and unconnected) is first recovered without prior knowledge of the interfacial condition by way of the recently established approaches to non-iterative waveform tomography of heterogeneous fractures, e.g. the methods of generalized linear sampling and topological sensitivity. Given suitable approximationΓ of the fracture geometry, the jump in the displacement field acrossΓ i.e. the fracture opening displacement (FOD) profile is computed from remote sensory data via a regularized inversion of the boundary integral representation mapping the FOD to remote observations of the scattered field. Thus obtained FOD is then used as input for solving the traction boundary integral equation onΓ for the unknown (linearized) contact parameters. In this study, linear and possibly dissipative interactions between the two faces of a fracture are parameterized in terms of a symmetric, complex-valued matrix K(ξ) collecting the normal, shear, and mixed-mode coefficients of specific stiffness. To facilitate the high-fidelity inversion for K(ξ), a 3-step regularization algorithm is devised to minimize the errors stemming from the inexact geometric reconstruction and FOD recovery. The performance of the inverse solution is illustrated by a set of numerical experiments where a cylindrical fracture, endowed with two example patterns of specific stiffness coefficients, is illuminated by plane waves and reconstructed in terms of its geometry and heterogeneous (dissipative) contact condition.

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Section: Numerical Implementation and Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A synoptic approach to the seismic sensing of heterogeneous fractures: From geometric reconstruction to interfacial characterization

Pourahmadian

Guzina

Haddar

2017

Computer Methods in Applied Mechanics and Engineering

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“…Next, suppose that there exists a ∈H 1/2 (Γ) 3 such that H * (a) = 0. In light of (19) and (21), it is apparent that H * is nothing else but the far-field operator stemming from the double-layer potential…”

Section: Key Properties For the Application Of Sampling Methodsmentioning

confidence: 99%

“…where Σ(ξ, y) is the (third-order) elastodynamic fundamental stress tensor given in Appendix B. By virtue of definition (19), vanishing far-field pattern of V (a) implies, by the Rellich Lemma and the unique continuation principle, that V (a) = 0 in R 3 \Γ. Owing to the fundamental jump property of double-layer potentials by which V = a, one obtains a = 0 which guarantees the injectivity of H * .…”

Section: Key Properties For the Application Of Sampling Methodsmentioning

confidence: 99%

Generalized linear sampling method for elastic-wave sensing of heterogeneous fractures

2017

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Abstract. A theoretical foundation is developed for active seismic reconstruction of fractures endowed with spatially-varying interfacial condition (e.g. partially-closed fractures, hydraulic fractures). The proposed indicator functional carries a superior localization property with no significant sensitivity to the fracture's contact condition, measurement errors, and illumination frequency. This is accomplished through the paradigm of the F -factorization technique and the recently developed Generalized Linear Sampling Method (GLSM) applied to elastodynamics. The direct scattering problem is formulated in the frequency domain where the fracture surface is illuminated by a set of incident plane waves, while monitoring the induced scattered field in the form of (elastic) far-field patterns. The analysis of the wellposedness of the forward problem leads to an admissibility condition on the fracture's (linearized) contact parameters. This in turn contributes toward establishing the applicability of the F -factorization method, and consequently aids the formulation of a convex GLSM cost functional whose minimizer can be computed without iterations. Such minimizer is then used to construct a robust fracture indicator function, whose performance is illustrated through a set of numerical experiments. For completeness, the results of the GLSM reconstruction are compared to those obtained by the classical linear sampling method (LSM).

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“…Our reconstruction method is a modified linear sampling method, adapted to our problem where we already know the interface Γ and only look for the delaminated part Γ 0 . The linear sampling method and factorization method have been used to reconstruct cracks or screens with various types of boundary conditions [8], [10], [12], [27] and [33] (see also the monographs [13] and [15]). Although numerically both the linear sampling method and factorization method provide similar reconstruction results, the factorization method is mathematically more satisfactory.…”

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confidence: 99%

Nondestructive Testing of the Delaminated Interface between Two Materials

Cakoni¹,

Teresa²,

Haddar³

et al. 2016

SIAM J. Appl. Math.

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We consider the problem of detecting if two materials that should be in contact have separated or delaminated. The goal is to find an acoustic technique to detect the delamination. We model the delamination as a thin opening between two materials of different acoustic properties, and using asymptotic techniques we derive a asymptotic model where the delaminated region is replaced by jump conditions on the acoustic field and flux. The asymptotic model has potential singularities due to the edges of the delaminated region, and we show that the forward problem is well posed for a large class of possible delaminations. We then design a special Linear Sampling Method (LSM) for detecting the shape of the delamination assuming that the background, undamaged, state is known. Finally we show, by numerical experiments, that our LSM can indeed determine the shape of delaminated regions.

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Application of the linear sampling method to identify cracks with impedance boundary conditions

Cited by 28 publications

References 8 publications

A synoptic approach to the seismic sensing of heterogeneous fractures: From geometric reconstruction to interfacial characterization

A synoptic approach to the seismic sensing of heterogeneous fractures: From geometric reconstruction to interfacial characterization

Generalized linear sampling method for elastic-wave sensing of heterogeneous fractures

Nondestructive Testing of the Delaminated Interface between Two Materials

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