Direct and inverse acoustic scattering problems involving smart obstacles

Abstract

Direct and inverse acoustic scattering problems involving smart obstacles are proposed and some ideas to study them are suggested. A smart obstacle is an obstacle that when hit by an incoming acoustic wave reacts circulating on its boundary a pressure current, that is a quantity dimensionally given by a pressure divided by a time, in order to generate a scattered wave that pursues a preassigned goal. In our models (see [6][7][8][9][10]12]) the smart obstacle pursues one of the following goals: i) to be undetectable, ii) to appear with a shape and/or acoustic boundary impedance different from its actual ones, iii) to appear with a shape and/or acoustic boundary impedance and in a location in space different from its actual ones. That is, in the first case the smart obstacle tries to be furtive, in the second case it tries to be masked that is it tries to appear as another obstacle that we call the mask and finally in the third case it tries to appear as another obstacle in a location in space different from its actual one. We refer to this last apparent obstacle as the ghost. The direct scattering problem considered is the following: given the incoming acoustic field, the obstacle, its acoustic impedance and its goal formulate an adequate mathematical model for the problems previously considered and find the optimal strategy to pursue the assigned goal within the proposed model. The inverse scattering problem considered is the following: given the knowledge of several far fields generated by the smart obstacle when hit by known incident acoustic fields it reacts with the optimal strategy and the knowledge of the goal pursued by the obstacle find the obstacle (i. e. find the shape, acoustic impedance and spatial location of the obstacle). For simplicity in this paper we limit our attention to the case of the obstacle that tries to be masked when the incoming acoustic field is time harmonic. Moreover in the inverse problem we assume that the acoustic boundary impedance of the obstacle and of the mask are known. In this case the direct scattering problem is translated in a constrained optimization problem and its solution is characterized as the solution of a set of auxiliary equations. The inverse scattering problem is translated in a two steps optimization procedure. Finally in a test case the inverse problem is solved numerically.

A method to solve an acoustic inverse scattering problem involving smart obstacles

“…A pressure current is a quantity whose physical dimension is pressure divided by time. A mathematical model of the acoustic time-dependent and time-harmonic direct scattering problems involving smart obstacles has been suggested recently (see [4][5][6][7][8][9][10][11]) and consists in the formulation of an optimal control problem for the wave equation in the time-dependent case that reduces, in the time-harmonic case, to a constrained optimization problem with the constraints given by an exterior boundary value problem for the Helmholtz equation. The pressure current is the control variable of the optimal control problem or the independent variable of the optimization problem respectively that must be determined.…”

“…The work presented here is only an attempt of studying these inverse problems: previous work includes [4,6] and further investigations will be presented elsewhere.…”

“…This dificulty is overcome defining a vector Herglotz wave function and treating appropriately the boundary conditions. In section 2 we formulate the time-harmonic masking direct scattering problem as a constrained optimization problem and we give the first-order optimality conditions associated to the corresponding Lagrangian functional (see [4] for further details). Moreover we recall some relations satisfied by the scattered field and the control function useful to formulate the masking inverse problem.…”

“…The assumption M ⊆ can be substituted with the assumption M ∩ = ∅ if we require that the masking effect takes place in IR 3 \ ∈ where ∈ is a set such that ⊂ ∈ , M ⊂ ∈ (see [4,5]). We call this last problem the ghost obstacle problem.…”

“…Under some hypotheses arguing as in [4] the first-order optimality condition of the constrained optimization problem [equations (2), (5), (16), (17)] can be derived using the Lagrangian functional associated to the optimization problem. This condition is given by the following boundary value problem for two coupled Helmholtz equations in the unknowns…”