Modified Fundamental Solutions for the Scattering of Elastic Waves by a Cavity

Bencheikh, L.

doi:10.1093/qjmam/43.1.57

Cited by 14 publications

(7 citation statements)

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“…We begin our work by expanding the pointsource incident field in terms of cylindrical Navier eigenvectors [14]: In particular, we consider the scattering problem in which the scatterer is a circular disc of radius R, with Dirichlet boundary condition.…”

Section: The Exact Green Function For An Elastic Rigid Circular Discmentioning

confidence: 99%

On the scattering of two-dimensional elastic point sources and related near-field inverse problems for small discs

Athanasiadis

Pelekanos

Sevroglou

et al. 2009

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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The problem of scattering of a point-generated elastic dyadic field by a bounded obstacle or a penetrable body in two dimensions is considered. The direct scattering problem for each case is formulated in a dyadic form. For two point sources, dyadic far-field pattern generators are defined and general scattering theorems and mixed scattering relations are presented. The direct scattering problem for a rigid circular disc is considered, and the exact Green function and the elastic far-field patterns of the radiating solution in the form of infinite series are obtained. Under the low-frequency assumption, approximations for the longitudinal and transverse far-field patterns of the scattered field are obtained, in addition to an asymptotic expansion for the corresponding scattering cross-section. A simple inversion scheme that locates the radius and the position of a rigid circular disc, which is based on a closed-form approximation of the scattered field at the location of the incident point source, is proposed.

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Section: The Exact Green Function For An Elastic Rigid Circular Discmentioning

confidence: 99%

On the scattering of two-dimensional elastic point sources and related near-field inverse problems for small discs

Athanasiadis

Pelekanos

Sevroglou

et al. 2009

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…Let us recall the formulation of the boundary value problem as it is stated in [1]: Determine a function & x for P e D, satisfying 1. Elastodynamic equations of motion in D k- 2 .V(y.u sc (P))-K- 2 .Vx(Vxn sc (P))+u sc (P) = 0, PzD.…”

Section: Basic Formulationmentioning

confidence: 99%

“…In two recent works [1][2], we considered the problem of scattering of time-harmonic stress waves by an infinite cylindrical cavity of arbitrary smooth cross-section, in an otherwise unbounded, homogenous, isotropic, linearly elastic solid. This problem was solved using the boundary integral equation (B.I.E.)…”

Section: Introductionmentioning

confidence: 99%

Low frequency scattering of elastic waves by a cavity using a matched asymptotic expansion method

Bencheikh

1995

J. Aust. Math. Soc. Series B, Appl. Math

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This work deals with low-frequency asymptotic solutions using the method of matched asymptotic expansions. It is based on two papers by Buchwald [3] and Buchwald and Tran Cong [4] who studied the diffraction of elastic waves by a small circular cavity and a small elliptic cavity, respectively, in an otherwise unbounded domain. Here we clarify and systematize some aspects of their work and extend it to the diffraction of elastic waves by a small cylindrical cavity with a hypotrochoidal boundary. Results for the case of an incident P-wave are compared, in the special case of an elliptic boundary, with the results from the numerical solution of the boundary integral equation method.

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“…Argyropoulos et al presented the modified Green function technique for the investigation of the exterior Dirichlet problem in linear elasticity. Bencheikh gave the modification for an elastic cavity. For elasticity and thermoelasticity, Natroshvili and Jentsch introduced a very explicit and simple approach, with a proper combination of potentials, to overcome the disadvantage of nonuniqueness and consequently to reduce to uniquely solvable integral equations.…”

Section: Introductionmentioning

confidence: 99%

The modified Green function technique for the exterior Dirichlet problem in linear thermoelasticity

Argyropoulos

Argyropoulou

Kiriaki

2018

Math Methods in App Sciences

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In this work, the modified Green function technique for the exterior Dirichlet problem in linear thermoelasticity is presented. Expressing the solution of the problem as a double‐layer potential of an unknown density, we form the associated boundary integral equation that describes the problem. Exploiting that the discrete spectrum of the irregular values of the associated integral equation is identified with the spectrum of eigenvalues of the corresponding interior homogeneous Neumann problem for the transverse part of the elastic displacement field, we introduce a modification of the fundamental solution of the elastic field. We establish the sufficient conditions that the coefficients of the modification must satisfy to overcome the problem of nonuniqueness for the thermoelastic problem.

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Modified Fundamental Solutions for the Scattering of Elastic Waves by a Cavity

Cited by 14 publications

References 0 publications

On the scattering of two-dimensional elastic point sources and related near-field inverse problems for small discs

On the scattering of two-dimensional elastic point sources and related near-field inverse problems for small discs

Low frequency scattering of elastic waves by a cavity using a matched asymptotic expansion method

The modified Green function technique for the exterior Dirichlet problem in linear thermoelasticity

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