e Kiriaki

The linear sampling method for the transmission problem in three-dimensional linear elasticity

In this paper the sampling method for the shape reconstruction of a penetrable scatterer in three-dimensional linear elasticity is examined. We formulate the governing differential equations of the problem in dyadic form in order to acquire a symmetric and uniform representation for the underlying elastic fields. The corresponding far-field operator is defined in the appropriate space setting. We establish the interior transmission problem in the weak sense and consider the case where the nonhomogeneous boundary data are generated by a dyadic source point located in the interior domain. Assuming that the inclusion has absorbing behaviour, we prove the existence and uniqueness of the weak solution of the interior transmission problem. In this framework the main theorem for the shape reconstruction for the transmission case is established. As for the cases of the rigid body and the cavity an approximate far-field equation is derived with the known dyadic Green function term with the source point an interior point of the inclusion. The inversion scheme which is proposed is based on the unboundedness for the solution of an equation of the first kind. More precisely, the support of the body can be found by noting that the solution of the integral equation is not bounded as the point of the location of the fundamental solution approaches the boundary of the scatterer from interior points.

The factorization method in inverse elastic scattering from penetrable bodies

The present work is concerned with the extension of the factorization method to the inverse elastic scattering problem by penetrable isotropic bodies embedded in an isotropic host environment for time-harmonic plane wave incidence. Although the former method has been successfully employed for the shape reconstruction problem in the field of elastodynamic scattering by rigid bodies or cavities, no corresponding results have been recorded, so far, for the very interesting (both from a theoretical and a practical point of view) case of isotropic elastic inclusions. This paper aims at closing this gap by developing the theoretical framework which is necessary for the application of the factorization method to the inverse transmission problem in elastodynamics. As in the previous works referring to the particular reconstruction method, the main outcome is the construction of a binary criterion which determines whether a given point is inside or outside the scattering obstacle by using only the spectral data of the far-field operator.

The far-field equations in linear elasticity : An inversion scheme

In this paper the far-field equations in linear elasticity for the rigid body and the cavity are considered. The direct scattering problem is formulated as a dyadic one. This imbedding of the vector problem for the displacement field into a dyadic field is enforced by the dyadic nature of the free space Greens function. Assuming that the incident field is produced by a superposition of plane dyadic incident waves it is proved that the scattered field is also expressed as the superposition of the corresponding scattered fields. A pair of integral equations of the first kind which hold independently of the boundary conditions are constructed in the far-field region. The properties of the Herglotz functions are used to derive solvability conditions and to build approximate far-field equations. Having this theoretical framework, approximate far-field equations are derived for a specific incidence which generates as far-field patterns simple known functions. An inversion scheme is proposed based on the unboundedness for the solutions of these approximate “far-field equations” and the support of the body is found by noting that the solutions of the integral equations are not bounded as the point of the location of the fundamental solution approaches the boundary of the scatterer from interioir points. It is also pointed that it is sufficient to recover the support of the body if only one approximate “far-field equation” is used. The case of the rigid sphere is considered to illuminate the unboundedness property on the boundary.

Scattering theorems for complete dyadic fields

Abstract An incident plane dyadic field is scattered by a body, on the surface of which the boundary conditions correspond to vanishing displacements, to vanishing traction, or to elastic transmission. Both longitudinal and transverse waves with all possible polarizations are embodied in the complete incident dyadic wave. We exhibit the most general reciprocity and scattering theorems for this dyadic scattering problem and we show how to recover all related known theorems as special cases. The nine scalar relations, coming out of the component analysis of each dyadic theorem, can be used as a basis for constructing many more new results for acoustic, electromagnetic and elastic scattering problems.

The linear sampling method for non-absorbing penetrable elastic bodies

In this paper the linear sampling method for the shape reconstruction of a penetrable non-dissipative scatterer in two-dimensional linear elasticity is examined. We formulate the governing differential equations of the problem in dyadic form in order to acquire a symmetric and uniform representation for the underlying elastic fields. The corresponding far-field operator is defined in the appropriate space setting. Assuming that the inclusion has non-absorbing behaviour, we consider this problem as a degenerate case of a non-dissipative anisotropic inclusion. Results for the existence and uniqueness of the weak solution of the interior transmission problem are obtained. In this framework the main theorem for the shape reconstruction for the transmission case is established. As in the previous works referring to the linear sampling method in acoustics and linear elasticity, the inversion scheme which is proposed is based on the unboundedness of the solution of an equation of the first kind having as the known term the far-field of the free-space Green dyadic, generated by a source inside the inclusion approaching the boundary. Numerical results are presented for different inclusion geometries, assuring the simple and efficient implementation of the algorithm, using synthetic data derived from the boundary element method.

On the scattering amplitudes for elastic waves

Reciprocity and scattering theorems for the normalized spherical scattering amplitude for elastic waves are obtained for the case of a rigid scatterer, a cavity and a penetrable scattering region. Depending on the polarization of the two incident waves reciprocity relations of the radial-radial, radial-angular, and angular-angular type are established. Radial and angular scattering theorems, expressing the corresponding scattering amplitudes via integrals of the amplitudes over all directions of observation, as well as their special forms for scatterers with inversion symmetry are also provided. As a consequence of the stated scattering theorems the scattering cross-section for either a longitudinal, or a transverse incident wave is expressed through the forward value of the radial, or the angular amplitude, correspondingly. All the known relative theorems for acoustic scattering are trivially recovered from their elastic counterparts.

Computation of light scattering by axisymmetric nonspherical particles and comparison with experimental results

A laboratory prototype of a novel experimental apparatus for the analysis of spherical and axisymmetric nonspherical particles in liquid suspensions has been developed. This apparatus determines shape, volume, and refractive index, and this is the main difference of this apparatus from commercially available particle analyzers. Characterization is based on the scattering of a monochromatic laser beam by particles [which can be inorganic, organic, or biological (such as red blood cells and bacteria)] and on the strong relation between the light-scattering pattern and the morphology and the volume, shape, and refractive index of the particles. To keep things relatively simple, first we focus attention on axisymmetrical particles, in which case hydrodynamic alignment can be used to simplify signal gathering and processing. Fast and reliable characterization is achieved by comparison of certain properly selected characteristics of the scattered-light pattern with the corresponding theoretical values, which are readily derived from theoretical data and are stored in a look-up table. The data in this table were generated with a powerful boundary-element method, which can solve the direct scattering problem for virtually arbitrary shapes. A specially developed fast pattern-recognition technique makes possible the on-line characterization of axisymmetric particles. Successful results with red blood cells and bacteria are presented.

The inverse scattering problem for a rigid ellipsoid in linear elasticity

The inverse scattering problem for the rigid ellipsoid in linear elasticity is examined. It has been proved that six measurements of the first-order imaginary coefficient of the radial scattering amplitude in the low-frequency region are necessary in order to evaluate the semi-axes of the ellipsoid and to fix the position of the principal axes of the ellipsoid. Numerical results and degenerate cases of the ellipsoidal geometry are also presented.

On the uniqueness of the inverse elastic scattering problem for periodic structures

In this paper, Schiffers theorem of inverse elastic scattering theory for periodic rigid structures is formulated. The examination of this uniqueness problem requires the study of the spectral properties of an infinite periodic layer, defined by a pair of candidate scattering curves. It is proved that, knowledge of the scattered field on a straight line above the corrugations for P-type one-direction incidence and for an interval of wavenumbers, is sufficient for the unique determination of the scatterers curve. Moreover, Schiffers result holds for a finite set of wavenumbers, if a priori information about the height of the curve is available.

On the condition number of integral equations in linear elasticity using the modified Green's function

In this work the modified Greens function technique for an exterior Dirichlet and Neumann problem in linear elasticity is investigated. We introduce a modification of the fundamental solution in order to remove the lack of uniqueness for the solution of the boundary integral equations describing the problems, and to simultaneously minimise their condition number. In view of this procedure the cases of the sphere and perturbations of the sphere are examined. Numerical results that demonstrate the effect of increasing the number of coefficients in the modification on the optimal condition number are also presented.