Abstract
In this paper, the interior elastic direct and inverse scattering problem of time-harmonic waves for a non-penetrable partially coated obstacle placed in a homogeneous and isotropic medium is studied. The scattering problem is formulated via the Navier equation, considering incident circular waves due to point-source fields, where the corresponding scattered data are measured on a closed curve inside the obstacle. Our model, from the mathematical point of view, is described by a mixed boundary value problem in which the scattered field satisfies mixed Dirichlet-Robin boundary conditions on the Lipschitz boundary of the obstacle. Using a variational equation method in an appropriate Sobolev space setting, uniqueness and existence results as well as stability ones are established. The corresponding inverse problem is also studied, and using some specific auxiliary integral operators an appropriate modified factorisation method is given. In addition, an inversion algorithm for shape recovering of the partially coated obstacle is presented and proved. Last but not least, useful remarks and conclusions concerning the direct scattering problem and its linchpin with the corresponding inverse one are given.
Highlights
The first problems have the following interpretation: If the total field is the superposition of an incident field uinc and the scattered field usct, the direct scattering problem is to determine the scattered field from the knowledge of the incident wave field and the governing differential equation of the wave motion
Our partially coated obstacle will be referred to as the scatterer, which consists of an inhomogeneity to the propagation of a given time-harmonic elastic wave field; our incident wave is a point-source field located at a point inside the scatterer
In contrast to traditional approaches applied in inverse scattering theory, which deal with iterative techniques such as regularised Newton methods [9,10,11] or conjugate gradient methods, our study focuses on a non-iterative method
Summary
Let B ⊂ R2 denote a closed bounded and connected domain with Lipschitz boundary ∂B ≡ Γ := Γ D ∪ Π ∪ Γ I , where Γ D (D : stands for the Dirichlet boundary condition) and Γ I (I: stands for the Robin boundary condition) are disjoint, relatively open subsets of Γ having Π as their common boundary in Γ. We state the direct scattering problem which is described by the following mixed boundary value problem: For a given elastic incident point-source urinc. Throughout this paper c > 0 is the surface impedance being a constant, λ, μ, are the Lamé coefficients as before and ρ(r) denotes the mass density, given by r ∈ R2 \ B ρ0 , ρ(r) =. In what follows in this paper, we assume C to be a Lipschitz closed curve inside B and. We define the following interior mixed boundary value problem (IMBVP):. The above IMBVP will be expressed in a variational sense and its well-posedness will be exploited
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
