V. Sevroglou

A boundary integral equations approach for mixed impedance problems in elasticity

Direct scattering problems for partially coated obstacles in linear elasticity lead to interior and exterior mixed impedance boundary value problems for the equations of steady-state elastic oscillations. We employ the potential method and reduce the mixed impedance problems to equivalent boundary pseudodifferential equations for arbitrary values of the oscillation parameter. We give a detailed analysis of the corresponding pseudodifferential equations which live on a proper submanifold of the boundary of the elastic body and establish uniqueness and existence results for the original mixed impedance problems for arbitrary values of the oscillation parameter; this is crucial in the study of inverse elastic scattering problems for partially coated obstacles. We also investigate regularity properties of solutions near the curves where the boundary conditions change and establish almost best Holder smoothness results.

An application of the reciprocity gap functional to inverse mixed impedance problems in elasticity

In this paper, a mixed impedance scattering problem for partially coated obstacles is studied. We formulate the direct scattering problem for the Navier equation and the mathematical setting for the description of the inverse problem. The scattered field satisfies mixed Dirichlet-impedance boundary conditions on the smooth boundary of the scatterer. A uniqueness theorem for the determination of the partially coated obstacle is proved. A method for the reconstruction of the shape of the obstacle is presented, for which no a priori information of the physical parameters of the scattering obstacle is needed. The inverse problem is treated using ideas from the linear sampling method and by the use of the reciprocity gap functional. Further, a characterization of the surface impedance is established.

The direct electromagnetic scattering problem by a mixed impedance screen in chiral media

In this article solvability results for the direct electromagnetic scattering problem for a mixed perfectly conducting-impedance screen in a chiral environment is studied. In particular, incident time-harmonic electromagnetic waves in a chiral medium upon a partially coated open surface Γ (the ‘screen’), that satisfies an impedance boundary condition on one side and a perfectly conducting boundary condition on the other side, are considered. We introduce the Beltrami fields, appropriate boundary integral relations for these fields are proved and via them a uniqueness result is established. A variational method in a suitable functional space setting is considered and using a Calderon type operator for the chiral case, existence for the scattering problem is established.

Point-Source Elastic Scattering by a Nested Piecewise Homogeneous Obstacle in an Elastic Environment:

A nested piecewise homogeneous elastic scatterer is embedded in a homogeneous elastic environment. The scatterer’s core may be rigid, cavity, Robin, or lossy penetrable. A 2D or 3D incident elastic field, generated by a point-source located in the homogeneous environment, impinges on the scatterer. The scattering problem is formulated in a dyadic form. The main purpose of this paper is to establish scattering relations for the elastic point-source excitation of a nested piecewise homogeneous scatterer. To this direction, we establish reciprocity principles and general scattering theorems relating the scattered fields with the corresponding far-field patterns. Furthermore, for a scatterer excited by a point-source and a plane wave, mixed scattering relations are derived. The optical theorem, relating the scattering cross-section with the field at the point-source’s location a is recovered as a corollary of the general scattering theorem. We present a detailed investigation for the 2D case and summarize the results for the 3D case, pointing out the main differences in the analysis.

A Mixed Impedance Scattering Problem for Partially Coated Obstacles in Two-Dimensional Linear Elasticity

In this paper the problem of scattering of time-harmonic elastic waves by a non-penetrable partially coated obstacle buried in a piecewise homogeneous medium is considered. We study the direct scattering problem as well as the inverse one. The direct problem for the Navier equation is formulated in a dyadic form and the issues of solvability due to uniqueness, existence and regularity are discussed. Uniqueness results for the corresponding inverse scattering problem are proved. In particular, the unique determination of the non-penetrable partially coated obstacle with its boundary condition as well as of the penetrable interface(s) between the layered media are established via the knowledge of the far-field pattern for elastic waves. The proof of the uniqueness is based on a mixed reciprocity relation and its connection between plane-waves and point-sources. The paper at hand deals with two-dimensional problems for a body consisting of two layers, but the obtained results also hold for multi-layered media; they are valid as well for the three-dimensional case.

Electromagnetic Scattering by a Chiral Impedance Screen

In this paper the solvability of the direct electromagnetic scattering problem by an impedance screen in a chiral environment is presented. Time-harmonic electromagnetic plane waves in a chiral medium are considered as incident fields. These propagating fields are scattered by an obstacle which is a partially coated open surface \(\Gamma \), well known as the “screen”. Uniqueness results are proved using appropriate relations for Beltrami fields, and in addition, existence results are established by using a variational method in suitable functional space setting.

Dyadic Elastic Scattering by Point Sources: Direct and Inverse Problems

This chapter is concerned with the scattering of elastic point sources by a bounded obstacle, as well as with a related near-field inverse problem for small scatterers. We consider the Dirichlet problem, where the displacement field is vanishing on the surface of the scatterer. A dyadic formulation for the aforementioned scattering problem is considered in order to gain the symmetry–compactness of the dyadic analysis [TAI94].