Reconstruction of a Fully Anisotropic Elasticity Tensor from Knowledge of Displacement Fields

Bal, Guillaume; Monard, Francźois; Uhlmann, Günther

doi:10.1137/151005269

Cited by 24 publications

(29 citation statements)

References 36 publications

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“…In , uniqueness results are given for the determination of the shear modulus from a finite number of linearly independent displacement fields in two dimensions. The reconstruction of an anisotropic elasticity tensor from a finite number of displacement fields for the linear, stationary elasticity equation is the topic of . A comprehensive overview of various inverse problems in the field of elasticity is offer in the article .…”

Section: Introductionmentioning

confidence: 99%

On the linearization of identifying the stored energy function of a hyperelastic material from full knowledge of the displacement field

Seydel

Schuster

2016

Math Methods in App Sciences

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We compute a local linearization for the nonlinear, inverse problem of identifying the stored energy function of a hyperelastic material from the full knowledge of the displacement field. The displacement field is described as a solution of the nonlinear, dynamic, elastic wave equation, where the first Piola–Kirchhoff stress tensor is given as the gradient of the stored energy function. We assume that we have a dictionary at hand such that the energy function is given as a conic combination of the dictionary's elements. In that sense, the mathematical model of the direct problem is the nonlinear operator that maps the vector of expansion coefficients to the solution of the hyperelastic wave equation. In this article, we summarize some continuity results for this operator and deduce its Fréchet derivative as well as the adjoint of this derivative. Because the stored energy function encodes mechanical properties of the underlying, hyperelastic material, the considered inverse problem is of highest interest for structural health monitoring systems where defects are detected from boundary measurements of the displacement field. For solving the inverse problem iteratively by the Landweber method or Newton‐type methods, the knowledge of the Fréchet derivative and its adjoint is of utmost importance. Copyright © 2016 John Wiley & Sons, Ltd.

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Section: Introductionmentioning

confidence: 99%

On the linearization of identifying the stored energy function of a hyperelastic material from full knowledge of the displacement field

Seydel

Schuster

2016

Math Methods in App Sciences

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“…Upon viewing q as a four-vector q 0 q 1 q 2 q 3 T , a = a 1 e 1 + a 2 e 2 + a 3 e 3 and similarly for b, (14) takes the form of the following matrix-vector multiplication…”

Section: Evolving a Unit Quaternion Along A Curvementioning

confidence: 99%

“…Inverse conductivity problems share many similarities with inverse elasticity problems [8,27,14,23], and some of the current framework also applies there as well, see [14]. Other internal functionals for inverse conductivity may be considered, for instance current densities, see [22,38,36,35,12,11,34].…”

Section: Introductionmentioning

confidence: 97%

Imaging of isotropic and anisotropic conductivities from power densities in three dimensions

Monard

Rim

2018

Inverse Problems

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We present numerical reconstructions of anisotropic conductivity tensors in three dimensions, from knowledge of a finite family of power density functionals. Such a problem arises in the coupled-physics imaging modality Ultrasound Modulated Electrical Impedance Tomography for instance. We improve on the algorithms previously derived in [9,32] for both isotropic and anisotropic cases, and we address the well-known issue of vanishing determinants in particular. The algorithm is implemented and we provide numerical results that illustrate the improvements. 1 γ is uniformly elliptic if there exists a constant κ ≥ 1 such that κ −1 |ξ| 2 ≤ γ(x)ξ · ξ ≤ κ|ξ| 2 for every x ∈ X and ξ ∈ R 3 .

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“…The time-harmonic Lamé system from internal measurements was also considered in [8]. The reconstruction algorithms for fully anisotropic elasticity tensors from knowledge of a finite number of internal data were derived in [18]. In this section, we concern the linear isotropic elasticity setting.…”

Section: Transient Elastographymentioning

confidence: 99%

Applications of CGO solutions to coupled-physics inverse problems

Kocyigit¹,

Lai²,

Qiu³

et al. 2017

IPI

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This paper surveys inverse problems arising in several coupled-physics imaging modalities for both medical and geophysical purposes. These include Photoacoustic Tomography (PAT), Thermo-acoustic Tomography (TAT), Electro-Seismic Conversion, Transient Elastrography (TE) and Acousto-Electric Tomography (AET). These inverse problems typically consists of multiple inverse steps, each of which corresponds to one of the wave propagations involved. The review focus on those steps known as the inverse problems with internal data, in which the complex geometrical optics (CGO) solutions to the underlying equations turn out to be useful in showing the uniqueness and stability in determining the desired information.

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Reconstruction of a Fully Anisotropic Elasticity Tensor from Knowledge of Displacement Fields

Cited by 24 publications

References 36 publications

On the linearization of identifying the stored energy function of a hyperelastic material from full knowledge of the displacement field

On the linearization of identifying the stored energy function of a hyperelastic material from full knowledge of the displacement field

Imaging of isotropic and anisotropic conductivities from power densities in three dimensions

Applications of CGO solutions to coupled-physics inverse problems

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