Ευθέα και αντίστροφα προβλήματα σκέδασης ελαστικών κυμάτων σε ομογενές κατά τμήματα μέσο
Direct and inverse elastic scattering problems in a piecewise homogeneous medium
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Keywords
Δισδιάστατη γραμμική ελαστικότητα ; Εξίσωση Navier ; Μεικτά προβλήματα σκέδασης ; Τροποποιημένη μέθοδος παραγοντοποίησης ; Στοχαστικά προβλήματα συνοριακών τιμών ; Αναπτύγματα Wiener chaos ; Two-dimensional linear elasticity ; Navier equation ; Mixed boundary value scattering problems ; Stochastic boundary value problems ; Wiener chaos expansionsAbstract
In this thesis, we investigate boundary value problems involving Navier equation both in deterministic and stochastic invironment. Initially, we focus on the direct elastic scattering problem posed by a piecewise impedance obstacle. Specifically, we consider a
scatterer embedded in a homogeneous medium, which is piecewise coated with different impedance constants on distinct parts of its boundary. The uniqueness of the solution is established through the variational formulation of the problem, while the existence
and regularity properties of the solution are demonstrated using the boundary integral equation approach. Notably, our approach relies on representing the solution as a combination of single and double layer potentials, leading to existence results as well as to an
essential regularity result. Additionally, significant remarks and conclusions are provided to enhance the understanding of the problem.
Next, we address the direct and inverse scattering problem of time-harmonic elastic waves by an inhomogeneous medium, containing buried objects. Initially, we establish well-posedness for the direct scattering problem through a modified variational method within a suitable Sobolev space setting. We prove uniqueness, existence, and continuous dependence of the solution on the boundary data associated with the buried obstacles. Subsequently, we delve into the corresponding inverse problem, particularly exploring, via
the modified factorization method, for shape reconstruction and location of the support of the inhomogeneous medium. Our study also includes pertinent remarks and conclusions, focusing on the interconnection between the direct scattering problem and its inverse
counterpart in elastic media.
Finally, we study two stochastic boundary value problems arising in linear elasticity through a Wiener chaos expansion, in order to show the feasibility of the method and proceed in future paper to the study of a stochastic scattering problem in a piecewise homogeneous medium. Specifically, we establish an appropriate variational formulation for Navier equation with stochastic boundary data . The key idea is to reduce the stochastic problems into an infinite hierarchy of deterministic boundary value problems, where each problem is treated with an appropriate variational formulation. Subsequently, we establish well-posedness for this hierarchy of deterministic problems, establish the relevant connection to the stochastic problem, and employ uniqueness and existence arguments for the weighted Wiener chaos solution. Afterwards, we address a boundary value problem concerning the non homogeneous Navier equation where both the right hand side of the equation as well as the boundary data are given as Wiener chaos expansions, using similar reasoning, i.e the reduction of the problem into an infinite hierarchy of deterministic ones. As in the previous cases, valuable remarks and conclusions are included to further illuminate the topic at hand.