On the Existence and Uniqueness of a Solution to the Interior Transmission Problem for Piecewise-Homogeneous Solids

Abstract

The interior transmission problem (ITP), which plays a fundamental role in inverse scattering theories involving penetrable defects, is investigated within the framework of mechanical waves scattered by piecewise-homogeneous, elastic or viscoelastic obstacles in a likewise heterogeneous background solid. For generality, the obstacle is allowed to be multiply connected, having both penetrable components (inclusions) and impenetrable parts (cavities). A variational formulation is employed to establish sufficient conditions for the existence and uniqueness of a solution to the ITP, provided that the excitation frequency does not belong to (at most) countable spectrum of transmission eigenvalues. The featured sufficient conditions, expressed in terms of the mass density and elasticity parameters of the problem, represent an advancement over earlier works on the subject in that (i) they pose a precise, previously unavailable provision for the well-posedness of the ITP in situations when both the obstacle and the background solid are heterogeneous, and (ii) they are dimensionally consistent, i.e., invariant under the choice of physical units. For the case of a viscoelastic scatterer in an elastic solid it is further shown, consistent with earlier studies in acoustics, electromagnetism, and elasticity that the uniqueness of a solution to the ITP is maintained irrespective of the vibration frequency. When applied to the situation where both the scatterer and the background medium are viscoelastic, i.e., dissipative, on the other hand, the same type of analysis shows that the analogous claim of uniqueness does not hold. Physically, such anomalous behavior of the "viscoelastic-viscoelastic" case (that has eluded previous studies) has its origins in a lesser known fact that the homogeneous ITP is not mechanically insulated from its surroundings-a feature that is particularly cloaked in situations when either the background medium or the scatterer are dissipative. A set of numerical results, computed for ITP configurations that meet the sufficient conditions for the existence of a solution, is included to illustrate the problem. Consistent with the preceding analysis, the results indicate that the set of transmission values is indeed empty in the "elastic-

Integral Equation Methods for Electrostatics, Acoustics, and Electromagnetics in Smoothly Varying, Anisotropic Media

“…The class of coefficients which are piecewise smooth where the jumps occur on C 2 boundaries were studied in [12]. In [24], piecewise homogeneous objects were studied. It would be of great practical interest if the unique continuation property holds for piecewise smooth coefficients, whose jumps occur on piecewise C 2 boundaries, allowing our integral formulation to be applicable to domains with edges.…”

Nature of the transmission eigenvalue spectrum for elastic bodies

“…With reference to (1), it is further noted that c, C, c * and C * represent the bounds on the extreme eigenvalues of C and C * , computed with respect to double contraction with a second-order tensor. In the most general anisotropic case C and C * , which are endowed with major symmetry [5], may each have up to six distinct eigenvalues. Hereon, it is assumed that the two distributions of material properties are "non-intersecting" in the sense that either…”

“…A survey of particular issues and difficulties associated with the treatment of this non-traditional boundary-value problem can be found in [25]. Here one should mention that the early works on the ITP have primarily focused on the question of its well-posedness via either integral equation methods [20,23,32,39] or customized variational formulations [5,8,18,28] applied to Helmholtz, Maxwell, and Navier equations. These studies have in particular established a number of conditions, stated in terms of (contrast between) the physical properties of the scatterer and the background, under which the existence and uniqueness of a solution to the ITP can be ensured.…”

“…In this case, the aforementioned impediments plaguing the mathematical treatment of the ITP [25] are compounded by the tensorial structure of the elastic wave equation, which in particular features a fourth-order elastic tensor which may have up to six distinct eigenvalues in a general anisotropic case. The existing literature on the elastic ITP is relatively scarce [5,[18][19][20] and provides, for a limited number of physical configurations (specified in terms of the elasticity and mass density contrasts between the obstacle and the scatterer), sufficient conditions under which the problem is well-posed and has, at most, a countable set of eigenvalues.…”

Elastic transmission eigenvalues and their computation via the method of fundamental solutions

“…Interior transmission eigenvalues (ITEs) arise primarily in the study of inverse scattering theory, cf. [2,3,8]. Their major role comes along with conventional reconstruction algorithms for which incident wave frequencies of that specific order need to be excluded to fully justify the feasibility of qualitative methods for the target recovery process.…”