Solvability of the integrodifferential equation of Eshelby's equivalent inclusion method

Gintides, Drossos; Kiriaki, K.

doi:10.1093/qjmam/hbu025

Cited by 7 publications

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“…This work also extends results in recent investigations on Eshelby's equivalent inclusion method [3], in particular [4,9] (addressing the existence of solutions for Eshelby's method in the case of ellipsoidal inhomogeneities) and [5] (where the solvability of the singular VIE for isotropic inhomogeneity and background materials is established using Mikhlin's theory of singular integral operators [11]). …”

supporting

confidence: 75%

“…, as done recently for studying the equivalent inclusion problem in [5]. To this aim, we first note that the background elasticity tensor admits the decomposition…”

Section: 2mentioning

confidence: 99%

“…The solvability of the elastostatic SVIE is established in [5] for the less-general case of isotropic inhomogeneity and background materials, by explicitly computing the symbolic determinant of the singular integral operator and invoking Mikhlin's theory of singular integral operators [11]. Moreover, solvability results for the special case of ellipsoidal inhomogeneities are given in [4,9].…”

Section: 2mentioning

confidence: 99%

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A modified volume integral equation for anisotropic elastic or conducting inhomogeneities: Unconditional solvability by Neumann series

Bonnet¹

2017

J. Integral Equations Applications

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ABSTRACT. This work addresses the solvability and solution of volume integrodifferential equations (VIEs) associated with 3D free-space transmission problems (FSTPs) involving elastic or conductive inhomogeneities. A modified version of the singular volume integral equation (SVIE) associated with the VIE is introduced and shown to be of second kind involving a contraction operator, i.e. solvable by Neumann series, implying the wellposedness of the initial VIE. Then, the solvability of VIEs for frequency-domain FSTPs (modelling the scattering of waves by compactly-supported inhomogeneities) follows by a compact perturbation argument. This approach extends work by Potthast (1999) on 2D electromagnetic problems (transverse-electric polarization conditions) involving orthotropic inhomogeneities in a isotropic background, and contains recent results on the solvability of Eshelby's equivalent inclusion problem as special cases. The proposed modified SVIE is also useful for iterative solution methods, as Neumannn series converge (i) unconditionally for static problems and (ii) on some inhomogeneity configurations for which divergence occurs with the usual SVIE for wave scattering problems.

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supporting

confidence: 75%

“…, as done recently for studying the equivalent inclusion problem in [5]. To this aim, we first note that the background elasticity tensor admits the decomposition…”

Section: 2mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

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A modified volume integral equation for anisotropic elastic or conducting inhomogeneities: Unconditional solvability by Neumann series

Bonnet¹

2017

J. Integral Equations Applications

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“…where can then be recast in either of two equivalent VIE forms (a) and (b), depending on whether the main unknown is chosen to be the restriction to [22] (which focuses on the solvability of the related integral equation (6.1) for σ := ΔC : ε[u B ], assuming isotropic materials) to general anisotropic materials. Its proof is given in Section 6.…”

Section: Definition and Vie Formulationmentioning

confidence: 99%

Higher order topological derivatives for three-dimensional anisotropic elasticity

Bonnet

Cornaggia

2017

ESAIM: M2AN

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This article concerns an extension of the topological derivative concept for 3D elasticity problems involving elastic inhomogeneities, whereby an objective function J is expanded in powers of the characteristic size a of a single small inhomogeneity. The O(a 6) approximation of J is derived and justified for an inhomogeneity of given location, shape and elastic properties embedded in a 3D solid of arbitrary shape and elastic properties; the background and the inhomogeneity materials may both be anisotropic. The generalization to multiple small inhomogeneities is concisely described. Computational issues, and examples of objective functions commonly used in solid mechanics, are discussed.

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“…Remarks on the important special case of isotropic background properties are given in Section 4, where our definition of a radiating solution is in particular shown to be equivalent to the Sommerfeld-Kupradze radiation conditions [25]. Finally, unique solvability for anisotropic elastostatics is obtained as a by-product in Section 5, this time without restrictions on anisotropy; this result is closely related to that of [16] on the equivalentinclusion method. Some auxiliary proofs are finally given in Section 6 1.1.…”

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confidence: 95%

Solvability of a volume integral equation formulation for anisotropic elastodynamic scattering

Bonnet¹

2016

J. Integral Equations Applications

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This article investigates the solvability of volume integral equations arising in elastodynamic scattering by penetrable obstacles. The elasticity tensor and mass density are allowed to be smoothly heterogeneous inside the obstacle and may be discontinuous across the background-obstacle interface, the background elastic material being homogeneous. Both materials may be anisotropic, within certainme limitations for the background medium. The volume integral equation associated with this problem is first derived, relying on known properties of the background fundamental tensor. To avoid difficulties associated with existing radiation conditions for anisotropic elastic media, we also propose a definition of the radiating character of transmission solutions. The unique solvability of the volume integral equation (and of the scattering problem) is established. For the important special case of isotropic background properties, our definition of a radiating solution is found to be equivalent to the Sommerfeld-Kupradze radiation conditions. Moreover, solvability for anisotropic elastostatics, directly related to known results on the equivalent inclusion method, is recovered as a by-product. 1. Introduction. Volume integral equations, also known as Lippmann-Schwinger integral equations, arise naturally when considering the scattering of waves by penetrable inhomogeneities embedded in a homogeneous background medium, for which a fundamental solution is known. They have been developed and used in various areas of physics such as electromagnetism and optics [3, Chap. 2] [8, Chap. 9], acoustics [28] [9, Chap. 8], or elastodynamics [34, 35, 39], since a long time. If the penetrable object has homogeneous properties, scattering may alternatively be modelled using coupled surface integral equations, see e.g. [13] for elastodynamics. Variational formulations, combined with appropriate handling of the solution behavior at infinity, can also be applied to such scattering problems.Volume integral equations have a geometrical support restricted to the spatial region where material properties differ from the background. This feature makes them useful e.g. for deriving asymptotic or homogenized models involving inhomogeneities of low contrast or vanishing size [2,38]. Moreover, as they provide a direct mathematical link between unknown inhomogeneities and remote measurements, they are also convenient for medium imaging inverse problems [5], for instance providing a foundation for contrast source inversion methods [32] or allowing rather explicit expressions of far-field patterns.In contrast with the vast existing literature on the mathematical aspects of boundary integral equations and their application to scattering by impenetrable obstacles characterized by Dirichlet, Neumann or impedant boundary conditions, comparatively few studies are available regarding the mathematical properties of volume integral equations. The well-posedness of volume integral formulations for various electromagnetic scattering problems is addressed in [33] f...

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Solvability of the integrodifferential equation of Eshelby's equivalent inclusion method

Cited by 7 publications

References 0 publications

A modified volume integral equation for anisotropic elastic or conducting inhomogeneities: Unconditional solvability by Neumann series

A modified volume integral equation for anisotropic elastic or conducting inhomogeneities: Unconditional solvability by Neumann series

Higher order topological derivatives for three-dimensional anisotropic elasticity

Solvability of a volume integral equation formulation for anisotropic elastodynamic scattering

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