Topological derivative for the inverse scattering of elastic waves

Guzina, Bojan B.; Bonnet, Marc

doi:10.1093/qjmam/57.2.161

Cited by 105 publications

(152 citation statements)

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“…The leading contribution to J(a) in the small-crack limit has been found in [4], on the basis of identity (18) truncated to first order in (v a , q a ) (i.e. without the last two integrals), to be given by…”

Section: Previous Results On Topological Sensitivity For Crack Problemsmentioning

confidence: 99%

“…To incorporate the effect of the leading contribution as a → 0 of the quadratic terms v 2 a and q 2 a in the asymptotic analysis, an expansion of J(a) must, in view of (18) and (22), be performed to order O(a 4 ) at least. As (18) involves integrals over the vanishing crack D a , the position vectorξ ∈ D a is scaled for this purpose according to:…”

Section: Derivation Of Expansion Of J(a): Methodology and Notationmentioning

confidence: 99%

“…As (18) involves integrals over the vanishing crack D a , the position vectorξ ∈ D a is scaled for this purpose according to:…”

Section: Derivation Of Expansion Of J(a): Methodology and Notationmentioning

confidence: 99%

“…Topological sensitivity is concerned with quantifying the perturbation of an objective function J with respect to the nucleation of a small object D a (z) of characteristic linear size a and specified location z, as a function of z. Denoting by J(a; z) the value achieved by J when D a (z) is the only perturbation to an otherwise known reference medium, then in 2-D situations with Neumann or transmission conditions on ∂D a (z) the topological derivative T 2 (z) appears through an expansion of the form J(a; z) = J(0) + a 2 T 2 (z) + o(a 2 ). Subsequent investigations have also established the usefulness of the topological sensitivity as a preliminary sampling tool for defect identification problems, providing estimates of location, size and number of a set of sought defects [6,10,15,18,19,21,22]. This approach has also been extended by formulating higher-order expansions of J(a; z).…”

Section: Introductionmentioning

confidence: 99%

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Fast identification of cracks using higher-order topological sensitivity for 2-D potential problems

Bonnet¹

2011

Engineering Analysis with Boundary Elements

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To cite this version:Marc Bonnet. Fast identification of cracks using higher-order topological sensitivity for 2-D potential problems. Engineering Analysis with Boundary Elements, Elsevier, 2011, 35, pp.223-235 Abstract This article concerns an extension of the topological sensitivity (TS) concept for 2D potential problems involving insulated cracks, whereby a misfit functional J is expanded in powers of the characteristic size a of a crack. Going beyond the standard TS, which evaluates (in the present context) the leading O(a 2 ) approximation of J, the higher-order TS established here for a small crack of arbitrarily given location and shape embedded in a 2-D region of arbitrary shape and conductivity yields the O(a 4 ) approximation of J. Simpler and more explicit versions of this formulation are obtained for a centrally-symmetric crack and a straight crack. A simple approximate global procedure for crack identification, based on minimizing the O(a 4 ) expansion of J over a dense search grid, is proposed and demonstrated on a synthetic numerical example. BIE formulations are prominently used in both the mathematical treatment leading to the O(a 4 ) approximation of J and the subsequent numerical experiments.

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Section: Previous Results On Topological Sensitivity For Crack Problemsmentioning

confidence: 99%

Section: Derivation Of Expansion Of J(a): Methodology and Notationmentioning

confidence: 99%

“…As (18) involves integrals over the vanishing crack D a , the position vectorξ ∈ D a is scaled for this purpose according to:…”

Section: Derivation Of Expansion Of J(a): Methodology and Notationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Fast identification of cracks using higher-order topological sensitivity for 2-D potential problems

Bonnet¹

2011

Engineering Analysis with Boundary Elements

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“…the curse of dimensionality), leading to the use of simplified (i.e., less generally applicable) parameterizations of the unknown property field. In contrast, gradient-based methods are not only substantially more computationally efficient overall, but are also not as affected by the curse of dimensionality, particularly if using an adjoint approach or something similar to calculate the gradients [11,12,13]. Therefore, gradient-based methods are capable of converging to a solution estimate with relative computational efficiency, even with a generally applicable high-dimensional finite-element type parameterization of the unknown material property field.…”

Section: Introductionmentioning

confidence: 99%

A generalized computationally efficient inverse characterization approach combining direct inversion solution initialization with gradient-based optimization

Wang

Brigham

2016

Comput Mech

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The nal publication is available at Springer via https://doi.org/10.1007/s00466-016-1362-3Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. AbstractA computationally efficient gradient-based optimization approach for inverse material characterization from incomplete system response measurements that can utilize a generally applicable parameterization (e.g., finite elementtype parameterization) is presented and evaluated. The key to this inverse characterization algorithm is the use of a direct inversion strategy with Gappy proper orthogonal decomposition (POD) response field estimation to initialize the inverse solution estimate prior to gradient-based optimization. Gappy POD is used to estimate the complete (i.e., all components over the entire spatial domain) system response field from incomplete (e.g., partial spatial distribution) measurements obtained from some type of system testing along with some amount of a priori information regarding the potential distribution of the unknown material property. The estimated complete system response is used within a physics-based direct inversion procedure with a finite element-type parameterization to estimate the spatial distribution of the desired unknown material property with minimal computational expense. Then, this estimated spatial distribution of the unknown material property is used to initialize a gradient-based optimization approach, which * Corresponding author Email addresses: mengyu_wang@meei.harvard.edu (Mengyu Wang), john.brigham@durham.ac.uk (John C. Brigham) Preprint submitted to Computational MechanicsNovember 23, 2016 uses the adjoint method for computationally efficient gradient calculations, to produce the final estimate of the material property distribution. The three-step ((1) Gappy POD, (2) direct inversion, and (3) gradient-based optimization) inverse characterization approach is evaluated through simulated test problems based on the characterization of elastic modulus distributions with localized variations (e.g., inclusions) within simple structures. Overall, this inverse characterization approach is shown to efficiently and consistently provide accurate inverse characterization estimates for material property distributions from incomplete response field measurements. Moreover, the solution procedure is shown to be capable of extrapolating significantly beyond the initial assumptions regarding the potential nature of the unknown material property distribution.

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Sounding of finite solid bodies by way of topological derivative

Bonnet

Guzina

2004

Numerical Meth Engineering

135

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This paper is concerned with an application of the concept of topological derivative to elastic-wave imaging of finite solid bodies containing cavities. Building on the approach originally proposed in the (elastostatic) theory of shape optimization, the topological derivative, which quantifies the sensitivity of a featured cost functional due to the creation of an infinitesimal hole in the cavity-free (reference) body, is used as a void indicator through an assembly of sampling points where it attains negative values. The computation of topological derivative is shown to involve an elastodynamic solution to a set of supplementary boundary-value problems for the reference body, which are here formulated as boundary integral equations. For a comprehensive treatment of the subject, formulas for topological sensitivity are obtained using three alternative methodologies, namely (i) direct differentiation approach, (ii) adjoint field method, and (iii) limiting form of the shape sensitivity analysis. The competing techniques are further shown to lead to distinct computational procedures. Methodologies (i) and (ii) are implemented within a BEM-based platform and validated against an analytical solution. A set of numerical results is included to illustrate the utility of topological derivative for 3D elastic-wave sounding of solid bodies; an approach that may perform best when used as a pre-conditioning tool for more accurate, gradient-based imaging algorithms. Despite the fact that the formulation and results presented in this investigation are established on the basis of a boundary integral solution, the proposed methodology is readily applicable to other computational platforms such as the finite element and finite difference techniques

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Topological derivative for the inverse scattering of elastic waves

Cited by 105 publications

References 3 publications

Fast identification of cracks using higher-order topological sensitivity for 2-D potential problems

Fast identification of cracks using higher-order topological sensitivity for 2-D potential problems

A generalized computationally efficient inverse characterization approach combining direct inversion solution initialization with gradient-based optimization

Sounding of finite solid bodies by way of topological derivative

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