The factorization method for scatterers with different physical properties

Abstract

The scattering of time-harmonic acoustic plane waves by a mixed type scatterer is considered. Such a scatterer is given as the union of several components with different physical properties. Some of them are impenetrable obstacles with Dirichlet or impedance boundary conditions, while the others are penetrable inhomogeneous media with compact support. This paper is concerned with modifications of the factorization method for the following two basic cases, one is the scattering by a priori separated sound-soft and soundhard obstacles, the other one is the scattering by a scatterer with impenetrable (Dirichlet) and penetrable components. The other cases can be dealt with similarly. Finally, some numerical experiments are presented to demonstrate the feasibility and effectiveness of the modified factorization methods.

A modification of the factorization method for scatterers with different physical properties

“…Finally, we give the following functional analytic theorem behind the factorization method. The proof is completely analogous to previous works, for example, theorem 2.15 in Kirsch and Grinberg, theorem 2.1 in Lechleiter, and theorem 2.1 in Liu …”

“…In section 4 of Kirsch and Liu, numerical examples are given to compare modification method (which uses the artificial far‐field operator corresponding to an inner domain) with previous method numerically, where we find that the modification method provides numerically a better reconstruction than previous one. Therefore, we expect that even in a scatterer consisting of two objects with different physical properties, our modification method (which uses several artificial far‐field operators) would also provide a better reconstruction than previous ones such as Kirsch and Liu and Liu …”

“…For recent works discussing such a scatterer, we refer to other studies. () We remark that these works have to assume that the wave number of the incident wave is not an eigenvalue of the Laplacian on impenetrable obstacles included in a scatterer. Our aim of this paper is to eliminate this restriction by developing the idea of Kirsch and Liu …”

“… denotes the unit normal vector at x ∈ ∂ Ω 2 . We refer to theorem 7.15 in McLean for the well posedness of problems to and refer to Kirsch and Grinberg and Liu for the factorization method in this case.…”

“…Let us compare our works (Theorems 1.2 and 1.4) with previous works from the mathematical point of view of a priori assumptions. For Theorem 1.2, we refer to theorem 2.5 of Liu, 12 and for Theorem 1.4, we refer to theorem 3.9(b) of Kirsch and Liu. 11 These previous works also gave the characterization of Ω 1 by assuming the existence of outer domain B 2 of Ω 2 and that the wave number k 2 is not an eigenvalue on an obstacle, while, in our work, we can choose arbitrary wave number k > 0 by introducing extra artificial domains such as B 1 , B 2 , and B 3 , which are not so difficult topological assumptions.…”