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

Abstract

For a fluid-solid interaction problem with Lipschitz interface, we investigate the partial Fréchet differentiability of the solutions and the approximate far-fieldpattern with respect to solid material parameters. Differentiability is shown in standard Sobolev framework, and the derivatives are characterized as solutions to inhomogeneous fluid-solid transmission problems. To validate the accuracy of the characterization, we compare analytical values with numerical ones given by Interior Penalty Discontinuous Galerkin (IPDG) in a setting with circular obstacles. Our comparisons also show that IPDG gives results with high precision and incurs almost no effect of discretization error accumulation.

An effective numerical strategy for retrieving all characteristic parameters of an elastic scatterer from its FFP measurements

“…• The shape of the scatterer and its material properties are of different nature and scales. Their variations have different influence on the amplitude diffusion of the scattered field, as reported in [50,47]. Hence, the proposed approach addresses the scale issue by considering the non-dimensional formulation (see Section 1.3) when solving the forward scattering-type problems that occur at each Newton iteration.…”

“…Note that BVP has been extensively studied. Mathematical results pertaining to existence, uniqueness, and regularity of the solution can be found in [44,45,46,47,48,49], among other references.…”

“…The characteristic parameters of a given elastic scatterer Ω s are of different nature. Indeed, it has been reported in [50,47] that the sensitivity of the FFP given by (2) to changes on the shape Γ is in general quite different to the ones on the material parameters triplet (λ, µ, ρ s ). Such a variability can have, depending on the frequency regime, a dramatic impact on the scattering observability.…”

“…Such calculations must be performed efficiently and with a high accuracy level to ensure stability, fast convergence, and cost effectiveness of the proposed computational procedure. To this end, we exploit the mathematical results pertaining to the dependence of the operator F with respect to the domain [58,48] and to the material parameters [50,47]. The results established in [58] state that the operator F is continuously differentiable with respect to the domain Ω s .…”

“…The proof of the result assumes Ω s to be a Lipschitz continuous polygonal domain. Furthermore, the results established in [50,47] prove that F is also continuous differentiable with respect to the material parameters and provide a characterization of its corresponding derivatives. More specifically, let p i.e., corresponding to the configuration Ω s (q (n) ).…”