Mathematical Determination of the Fréchet Derivative with Respect to the Domain for a Fluid-Structure Scattering Problem: Case of Polygonal-Shaped Domains

Abstract

Abstract. The characterization of the Fréchet derivative of the elasto-acoustic scattered field with respect to Lipschitz continuous polygonal domains is established. The considered class of domains is of practical interest since two-dimensional scatterers are always transformed into polygonalshaped domains when employing finite element methods for solving direct and inverse scattering problems. The obtained result indicates that the Fréchet derivative with respect to the scatterer of the scattered field is the solution of the same elasto-acoustic scattering problem but with additional right-hand side terms in the transmission conditions across the fluid-structure interface. This characterization has the potential to advance the state-of-the-art of the solution of inverse obstacle problems. 1. Introduction. One of the basic inverse scattering problems in scattering theory is the determination of the shape of the scatterer using some measured scattered far-field patterns [8]. This model problem, called inverse obstacle problem (IOP), is relevant to numerous real-world applications including radar and sonar detection, geophysical exploration, structural design, medical imaging, and atmospheric studies. In spite of their apparent simple formulations, IOPs are very challenging problems from both mathematical and computational viewpoints. The difficulties in studying and/or solving IOPs are mainly due to their nonlinear and severely ill-posed nature [8]. Nevertheless, given the applied nature of these problems and their prevalence in sciences and engineering, IOPs have been subject to extensive studies leading to a tremendous growth within the last three decades with an emphasis on the development of computational methods (see, for example, [11,14], and the references therein).

Characterization of partial derivatives with respect to material parameters in a fluid–solid interaction problem

“…That for T 1 follows additionally from the boundedness of the curvature of Σ. We next verify the second property in (14). For φ ∈ H 1/2 (Σ), we have…”

“…Lemma 1. The following choices of B satisfy properties (14): B = T when Σ is spherical (or circular), B = T 0 for general Σ, or B = T 1 when Σ is convex with bounded positive curvature.…”

“…Under Lipschitz assumption, the auxiliary problems can contain singular boundary or interface terms that do not fit in the canonical boundary Sobolev spaces, and to give sense to the problem is not trivial (cf. [14] and further discussion in subsection 2.3). Furthermore, for fluidsolid problems, existence of solutions is not always guaranteed due to the existence of solid resonance modes called Jones frequencies (defined in (26)).…”

“…Note that most references assume at least C 2 regularity, cf. [6,2], with the exception of [5] for acoustic scattering with Lipschitz obstacles, and [14,15,16] for acoustic-elastic transmission (in fact OP ∞ ) with polygonal-shaped obstacles, and the aforementioned [11,8] for elasticity on bounded Lipschitz domain. Under Lipschitz assumption, the auxiliary problems can contain singular boundary or interface terms that do not fit in the canonical boundary Sobolev spaces, and to give sense to the problem is not trivial (cf.…”

“…Remark 5. Although [14,15,16] also study the auxiliary problems in PDE form, they use implicit function theorem to show differentiability, while we work directly with the solution operator and use standard analysis 4 . Other approaches to show Fréchet differentiability can be e.g.…”