Direct Elastic Imaging of a Small Inclusion

Ammari, Habib; Calmon, Pierre; Яковлева, Екатерина

doi:10.1137/070696076

Cited by 56 publications

(46 citation statements)

References 14 publications

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“…In 3d, the elastic moment tensor M for a unit ball-inclusion x has also been computed in [11]. The topological derivative (4.5) for a ball-inclusion is:…”

Section: Resultsmentioning

confidence: 99%

Damage and fracture evolution in brittle materials by shape optimization methods

Allaire¹,

Jouve²,

Goethem³

2011

Journal of Computational Physics

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a b s t r a c tThis paper is devoted to a numerical implementation of the Francfort-Marigo model of damage evolution in brittle materials. This quasi-static model is based, at each time step, on the minimization of a total energy which is the sum of an elastic energy and a Griffith-type dissipated energy. Such a minimization is carried over all geometric mixtures of the two, healthy and damaged, elastic phases, respecting an irreversibility constraint. Numerically, we consider a situation where two well-separated phases coexist, and model their interface by a level set function that is transported according to the shape derivative of the minimized total energy. In the context of interface variations (Hadamard method) and using a steepest descent algorithm, we compute local minimizers of this quasi-static damage model. Initially, the damaged zone is nucleated by using the so-called topological derivative. We show that, when the damaged phase is very weak, our numerical method is able to predict crack propagation, including kinking and branching. Several numerical examples in 2d and 3d are discussed.

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“…In 3d, the elastic moment tensor M for a unit ball-inclusion x has also been computed in [11]. The topological derivative (4.5) for a ball-inclusion is:…”

Section: Resultsmentioning

confidence: 99%

Damage and fracture evolution in brittle materials by shape optimization methods

Allaire¹,

Jouve²,

Goethem³

2011

Journal of Computational Physics

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“…Regarding the Lamé system, we cite the works , , , where, as we just mentioned, the asymptotics are given in terms of the size of the scatterers only. In these works, the authors considered transmission problems and showed that the corresponding moment tensors are in general anisotropic.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

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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“…The concept of topological sensitivity (TS), since its inception in the context of structural shape optimization, [9,10] has been generalized and applied to deal with inverse scattering problems in acoustics, [7,[11][12][13][14][15][16][17] electromagnetism, [18][19][20][21][22][23] and elastodynamics. [24][25][26][27][28][29][30][31][32][33] In the reconstruction approach, the TS indicator function is defined as the sensitivity of a given cost functional (that would commonly be used as a platform for non-linear minimization) with respect to the nucleation of an infinitesimal obstacle at a prescribed sampling point in the reference (background) medium. Accordingly, the support of hidden obstacles is exposed through the spatial distribution of topological sensitivity, namely the regions where TS attains pronounced negative values.…”

Section: Introductionmentioning

confidence: 99%

Inverse acoustic scattering by solid obstacles: topological sensitivity and its preliminary application

Yuan

Bracq

Lin

2015

Inverse Problems in Science and Engineering

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A topological sensitivity approach is developed for the imaging of penetrable solid obstacles embedded in a fluid background medium by way of inverse acoustic scattering. To this end, an asymptotic form for the scattered field caused by a nucleating spherical solid obstacle in the reference fluid medium is derived within the boundary integral equation framework, where the required limiting behaviour of the acceleration field inside the perturbation is established based on solutions of two simplified fluid-solid interaction problems. With this result, the equivalence of acoustic scattering by fluid and solid scatterers in terms of asymptotic behaviour is observed and numerically validated. The direct and adjoint-field topological sensitivity expressions for transient and time-harmonic excitations are obtained accordingly. The utility of the proposed method as a tool for preliminary obstacle reconstruction and bulk modulus characterization is illustrated through numerical examples and an exploratory experimental study. On the basis of adjusted topological sensitivity formulas for fluid reference domains containing pre-existing solid inhomogeneities, an iterative 3D obstacle reconstruction approach is established and its performance with synthetic data is demonstrated.

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Direct Elastic Imaging of a Small Inclusion

Cited by 56 publications

References 14 publications

Damage and fracture evolution in brittle materials by shape optimization methods

Damage and fracture evolution in brittle materials by shape optimization methods

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

Inverse acoustic scattering by solid obstacles: topological sensitivity and its preliminary application

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