Reciprocity principle and crack identification

Andrieux, Stéphane; Abda, Amel Ben; Bui, Huy Duong

doi:10.1088/0266-5611/15/1/010

Cited by 93 publications

(117 citation statements)

References 5 publications

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“…The asymptotic form of integral equation (12) as a → 0 is now sought. For this purpose, and following customary practice for such asymptotic analyses, scaled coordinatesξ are introduced so that…”

Section: Preliminariesmentioning

confidence: 99%

“…Non-iterative approaches relevant to elastodynamic crack identification notably include methods based on the concepts of linear sampling [22,27,46], reciprocity gap [12,14,21], or topological derivative, the latter being considered herein. Initially introduced in connection with structural topology optimization [31,51], the concept of topological derivative (TD) has, since then, also been applied as a convenient, computationally economical means for qualitative flaw identification, see e.g.…”

Section: Introductionmentioning

confidence: 99%

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Qualitative identification of cracks using 3D transient elastodynamic topological derivative: Formulation and FE implementation

Bellis

Bonnet

2013

Computer Methods in Applied Mechanics and Engineering

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To cite this version:Cédric Bellis, Marc Bonnet. Abstract A time-domain topological derivative (TD) approach is developed for transient elasticwave imaging of buried cracks. The TD, which quantifies the sensitivity of the misfit cost functional to the creation at a specified location of an infinitesimal trial crack, is expressed in terms of the time convolution of the free field and an adjoint field as a function of that specified location and of the trial crack shape. Following previous studies on cavity identification in similar conditions, the TD field is here considered as a natural and computationally efficient approach for defining a crack location indicator function. This study emphasizes the implementation and exploitation of TD fields using the standard displacement-based FEM, a straightforward exploitation of the relevant sensitivity formulation established here. Results on several numerical experiments on 3D elastodynamic and acoustic configurations are reported and discussed, allowing to assess and highlight many features of the proposed TD-based fast qualitative crack identification, including its ability to identify multiple cracks and its robustness against data noise.

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Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Qualitative identification of cracks using 3D transient elastodynamic topological derivative: Formulation and FE implementation

Bellis

Bonnet

2013

Computer Methods in Applied Mechanics and Engineering

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“…New topics in mechanical tomography have recently been the subjects of several works. For example, solutions to crack inverse problems in two and three dimensions are known in elasticity [Andrieux and Ben Abda 1992;Andrieux et al 1999;Bui et al 2005] and in viscoelasticity , in statics as well as in dynamics, under the assumption of small frequencies. In elastodynamics, solutions of inverse crack problems are obtained in [Bui et al 2005] where the solution to an earthquake Keywords: nonlinear inverse problem, inclusion geometry, antiplane problem.…”

Section: Introductionmentioning

confidence: 99%

On the mystery of Calderón’s formula for the geometry of an inclusion in elastic materials

Bui

2011

J. Mech. Mater. Struct.

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I write this paper to pay homage to Marie-Louise Steele and in honor of Charles R. Steele. I have had the pleasure and the honor to serve their journals IJSS and JoMMS, with George Herrmann. They have made Solids & Structures and now Material Sciences a subject of nobility to all of us.We consider the nonlinear inverse problem of determining an inclusion in an elastic body, in antiplane shear loading. The perturbation of the shear modulus due to the inclusion was determined by Calderón (1980) in the case of a small amplitude of perturbation. For the general nonlinear case, the problem is decomposed into two linear problems: a source inverse problem, which determines the geometry of the inclusion, and a Volterra integral equation of the first kind for determining the amplitude. In this paper, we deal only with the determination of the inclusion geometry in the two-dimensional problem. We derive a simple formula for determining the inclusion geometry. This formula enables us to investigate the mystery of Calderón's solution for the linearized perturbation h 0 , raised by Isaacson and Isaacson (1986), in the case of axisymmetry. By using a series method for numerical analysis, they found that the supports of the perturbation, in the linearized theory and the nonlinear theory in the axisymmetric case, are practically the same. We elucidate the mystery by discovering that both theories give exactly the same support of the perturbation, supp(h 0 ) ≡ supp(h), for the general case of geometry and loadings. Then, we discuss an application of the geometry method to locate an inclusion and solve the source inverse problem, which gives an indication of the amplitude of the perturbation.

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“…In medecine, tomography techniques using mechanical loads such as antiplane shear loading on life tissue, considered as a viscoelastic medium, have been worked out for Kelvin-Voigt's viscoelasticity (Catheline et al [12], Muller et al [18]) and for pure elasticity. In the elastic case, solutions to crack inverse problems in 2D and 3D are already known, see Andrieux and Ben Abda [4], Andrieux et al [5], Bui et al [8]. It is thus important to know if viscoelastic inverse problems can be studied by using classical correspondence between viscoelasticity and elasticity.…”

Section: Introductionmentioning

confidence: 99%

On a nonlinear inverse problem in viscoelasticity

Bui¹,

Chaillat²

2009

Vietnam J. Mech.

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Abstract. We consider an inverse problem for determining an inhomogeneity in a viscoelastic body of the Zener type, using Cauchy boundary data, under cyclic loads at low frequency. We show that the inverse problem reduces to the one for the Helmholtz equation and to the same nonlinear Calderon equation given for the harmonic case. A method of solution is proposed which consists in two steps : solution of a source inverse problem, then solution of a linear Volterra integral equation.

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Reciprocity principle and crack identification

Cited by 93 publications

References 5 publications

Qualitative identification of cracks using 3D transient elastodynamic topological derivative: Formulation and FE implementation

Qualitative identification of cracks using 3D transient elastodynamic topological derivative: Formulation and FE implementation

On the mystery of Calderón’s formula for the geometry of an inclusion in elastic materials

On a nonlinear inverse problem in viscoelasticity

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