Antonios Charalambopoulos

The effect of boundary conditions on guided wave propagation in two-dimensional models of healing bone.

Guided wave propagation has recently drawn significant interest in the ultrasonic characterization of bone. In this work, we present a two-dimensional computational study of ultrasound propagation in healing bones aiming at monitoring the fracture healing process. In particular, we address the effect of fluid loading boundary conditions on the characteristics of guided wave propagation, using both time and time-frequency (t-f) signal analysis techniques, for three study cases. In the first case, the bone was assumed immersed in blood which occupied the semi-infinite spaces of the upper and lower surfaces of the plate. In the second case, the bone model was assumed to have the upper surface loaded by a 2mm thick layer of blood and the lower surface loaded by a semi-infinite fluid with properties close to those of bone marrow. The third case, involves a three-layer model in which the upper surface of the plate was again loaded by a layer of blood, whereas the lower surface was loaded by a 2mm layer of a fluid which simulated bone marrow. The callus tissue was modeled as an inhomogeneous material and fracture healing was simulated as a three-stage process. The results clearly indicate that the application of realistic boundary conditions has a significant effect on the dispersion of guided waves when compared to simplified models in which the bones surfaces are assumed free.

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 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.

A linear sampling method for the inverse transmission problem in near-field elastodynamics

Elastic-wave shape reconstruction of buried penetrable scatterers from near-field surface measurements is examined within the framework of the linear sampling method. The proposed inversion scheme is based on a linear integral equation of the first kind whose solution becomes unbounded as the (trial) source point of the reference Greens function approaches the boundary of an elastic scatterer from its interior. We provide a comprehensive theoretical setting to establish (i) the necessary transmission problems for near-field elastodynamics and (ii) solvability properties of the postulated linear equation in the context of penetrable obstacles. A set of numerical results with simply and multiply connected elastic scatterers is included to illustrate the performance of the reconstruction technique.

On the Interior Transmission Problem in Nondissipative, Inhomogeneous, Anisotropic Elasticity

In this paper, the interior transmission problem for the non absorbing, anisotropic and inhomogeneous elasticity is investigated. The direct scattering problem for the penetrable inhomogeneous, anisotropic and nondissipative scatterer is first studied and the existence and uniqueness of its solution are established. In the sequel, the interior transmission problem in its classical and weak form is presented and suitable variational formulations of it are settled. Finally, it is proved that the interior transmission eigenvalues constitute a discrete set.

On the dyadic scattering problem in three-dimensional gradient elasticity: an analytic approach

The investigation of the direct scattering problem of an elastic dyadic incident field from a spherical inclusion, is the main outcome of this work, in the case where the scatterer and the host environment dispose microstructure. The framework of the method is based on the implication of Mindlins gradient theory. The development of the method is fully analytic and gives successively several byproducts, which are indispensable for the solution of the scattering problem but constitute also independent results of their own theoretical and practical value. So the numerable set of Navier eigendyadics is constructed, which is proved to be a basis for every dyadic field obeying the dynamic gradient elasticity equation. This permits the construction of a useful spectral representation for every gradient elasticity field. Furthermore, the set of dyadic spherical harmonics is built, which stands for the extension of the well-known spherical vector harmonics to the dyadic realm. Every dyadic field restricted on the unit sphere can be expanded in terms of these spherical dyadic harmonics. The orthogonality relations of these functions are determined in close form and this is the prerequisite for the fully analytic treatment of the boundary conditions involving the scattering problem under consideration.

The linear sampling method for the transmission problem in 2D anisotropic elasticity

In the present work, the problem of reconstructing the shape of two-dimensional elastic anisotropic inclusions embedded in isotropic media is investigated within the framework of the linear sampling method. It is well known that the latter approach has been extensively used as an inverse solver in acoustic, electromagnetic and elastic scattering problems dealing with isotropic media and only recently in anisotropic acoustics and electromagnetics. The work at hand aims at contributing to the extension of the linear sampling method to anisotropic elastic inverse scattering. As in the previous works referring to the aforementioned reconstruction method, the proposed inversion scheme is based on the unboundedness of the solution of a linear integral equation of the first kind. Numerical results are also presented for several inclusion geometries and a system thereof exhibiting the applicability of the method.

On isovector fields and similarity solutions of generalized dynamic thermoelasticity

The purpose of the present work is the investigation of the isovector fields as well as the similarity solutions of the PDEs describing the generalized dynamic thermoelasticity. The adopted methodology and solution techniques belong totally to the analytical realm, while special treatment of the reduced differential equations resulting from similarity solution adoption has been realized.

Inverse scattering for an acoustically soft scatterer in the low-frequency region

Abstract In this work, a new method is presented for the determination of the shape of a scatterer in R 3 when it is a priori known that a scatterers surface can be represented as a power series of Cartesian coordinates. The necessary data for the application of the method are provided by the knowledge of the scattering amplitude as an analytic function of the wave number k in the low-frequency region.

An analytic algorithm for shape reconstruction from low-frequency moments

In the present work, a novel method, concerning the solution of the inverse scattering problem, is developed and implemented, in the realm of low-frequency acoustics. The method is based on the suitable exploitation of the low-frequency moments, which are the structural pieces of the far-field pattern. The stimulus for the present method has been offered by a recent accomplishment permitting the extraction of the moments from the far-field pattern via a systematic, direct, and stable manner. The aim of the method is to reconstruct polynomial scatterers and to approximate general scatterers by polynomial surfaces. This is accomplished via the formulation of suitable objective functionals involving the unknown coefficients of the Cartesian representation of the sought polynomial surface along with the low-frequency moments. These functionals are constructed by forcing the target polynomial surface to comply with the moments extracted from real data. The minimization of these functionals provides the optimiz...