On Well-Posedness of Scattering Problems in a Kirchhoff--Love Infinite Plate

Abstract

We address scattering problems for impenetrable obstacles in an infinite elastic Kirchhoff-Love two-dimensional plate. The analysis is restricted to the purely bending case and the time-harmonic regime. Considering four types of boundary conditions on the obstacle, well-posedness for those problems is proved with the help of a variational approach: (i) for any wave number k when the plate is clamped, simply supported or roller supported; (ii) for any k except a discrete set when the plate is free (this set is finite for convex obstacles).

The Linear Sampling Method for Kirchhoff-Love infinite plates

“…The impenetrable obstacle D ⊂ R 2 is supposed to be a bounded open domain of class C 3 which is either characterized by a Dirichlet or a Neumann boundary condition. More precisely, by using the notations of [2], in particular Ω = R 2 \ D, the scattered field v s satisfies in the unbounded domain Ω the problem (1) Here k > 0 is the wave number, u i is an incident field which satisfies ∆ 2 u i −k 4 u i = 0 in a domain including D, B r is the open ball centered at 0 and of radius r, n is the outward normal to B r and s is the measure on ∂B r . The reader will refer to [2] for a short justification of how the system (1) is derived.…”

“…More precisely, by using the notations of [2], in particular Ω = R 2 \ D, the scattered field v s satisfies in the unbounded domain Ω the problem (1) Here k > 0 is the wave number, u i is an incident field which satisfies ∆ 2 u i −k 4 u i = 0 in a domain including D, B r is the open ball centered at 0 and of radius r, n is the outward normal to B r and s is the measure on ∂B r . The reader will refer to [2] for a short justification of how the system (1) is derived. Roughly speaking, the first line describes the motion of the plate in the frequency domain, the second one characterizes the boundary conditions on the boundary of the obstacle while the third one is the radiation condition, which specifies that only outgoing scattering waves are admissible.…”

“…Roughly speaking, the first line describes the motion of the plate in the frequency domain, the second one characterizes the boundary conditions on the boundary of the obstacle while the third one is the radiation condition, which specifies that only outgoing scattering waves are admissible. It is shown in [2] that the classical Sommerfeld condition for the Helmholtz equation is also valid for the bilaplacian case. In order to specify the surface differential operators B 1 and B 2 , we need to introduce some notations.…”

“…In other words, the Dirichlet boundary condition corresponds to the clamped plate while the Neumann boundary condition corresponds to the free plate. Let us now consider the bounded domain Ω R = Ω ∩ B R , where R > 0 is such that B R contains the obstacle D. It is proved in [2] that the problem (1) is equivalent to the following problem set in the bounded domain Ω R : find u s ∈ H 2 (Ω R ) such that…”

“…Here, H 1 m denotes the Hankel function of the first kind and of order m. Wellposedness of the forward problem (1) is the main purpose of [2]. The proof consists of a Fredholm analysis of the equivalent problem (2).…”