Active manipulation of fields modeled by the Helmholtz equation

Abstract

In this paper we extend the results proposed in [36] and study the problem of active control in the context of a scalar Helmholtz equation. Given a source region Da and {v 0 , v 1 ,. .. , vn}, a set of solutions of the homogeneous scalar Helmholtz equation in n mutually disjoint "control" regions {D 0 , D 1 ,. .. , Dn} of R 2 or R 3 , respectively, the main objective of this paper is to characterize the necessary boundary data on ∂Da so that the solution to the corresponding exterior scalar Helmholtz problem will closely approximate v i in D i , respectively, for each i ∈ {0,. .. , n}. Building up on the previous ideas presented in [36] we show the existence of a class of solutions to the problem, provide numerical support of the results in 2D and 3D and discuss the existence of a minimal energy solution and its stability. 1. Introduction. During recent years, there has been a growing interest in the development of feasible strategies for the control of acoustic and electromagnetic fields with one possible application being the construction of robust schemes for sonar or radar cloaking. One main approach controls fields in the regions of interest by changing the material properties of the medium in certain surrounding regions [4, 5, 8, 13, 14, 15, 39] (and the references therein). Several alternative techniques are proposed in the literature (other than transformation optics strategies) such as: plasmonic designs (see [2] and the references therein), strategies based on anomalous resonance phenomena (see [31, 32, 33]), conformal mapping techniques (see [24, 25]), and complementary media strategies (see [23]). In this paper, we will study an approximate control problem for the exterior scalar Helmholtz equation, i.e., we will characterize the boundary control functions on the source ∂D a so that we achieve Keywords and phrases. Active manipulation, Helmholtz equation, layer potentials, integral equation, active exterior cloaking. The AFOSR supported this work under the 2013 YIP Award FA9550-13-1-0078.

On the synthesis of acoustic sources with controllable near fields

“…The paper is organized as follows: In Section 2 we present the theoretical results in two parts: first, in Subsection 2.1 we first briefly recall the theoretical results of [22] for acoustic control in free space and then, in Subsection 2.2 we discuss their extension to the problem of underwater acoustic control in the context of a constant depth homogeneous ocean environments. In Section 3, we build up on our previous results in [24] and discuss the L 2 -Tikhonov regularization with Morozov discrepancy numerical approximation for the acoustic control problem in 3D and (assuming the superposition principle) without loosing the generality present numerical simulations in the three important situations listed above: first, in Subsection 3.1 we present the synthesis of an acoustic source approximating a prescribed plane wave in a give near field sub-region; then in Subsection 3.2 we present the synthesis of an acoustic source with a null in a given sub-region of its near field and approximating an outgoing plane wave in a disjoint near field sub-region; and finally, in Subsection 3.3 we synthesize a very weak acoustic radiator (almost non-radiating source (ANR)) approximating, in a sub-region of its near field, a given backward propagating (propagating towards the source) plane wave.…”

“…Based on these onsiderations it is concluded in [26] that the double layer and the single layer operators associated with the Green's function G have the same compactness properties and satisfy the same jump relations as the classical layer potentials associate to Φ. Thus, by using this together with a few elementary technical adjustments it can be proved that the results presented in [22] will extend to this case, i.e., normal velocities (or pressures) given by (2.9) (or (2.10)) will generate acoustic fields described by double layer potentials associated to G and satisfying (2.6) and (2.7). Moreover, we make the observation that the expressions (2.9), (2.10) can be used in computations since the Green's function G is computed explicitly in [26].…”

“…but, it is elementary to see how the results of [22] can be extended to obtain solutions of (2.2), (2.3) represented as single layer potentials or a linear combination between double layer and single layer potentials.…”

“…Similarly as in the 2D case treated in [24], the 3D L 2 -optimization scheme for problem (2.2), (2.3) is based on Tikhonov regularization with Morozov discrepancy principle. In this context, as in [22], [24] regularity results and the well posedness of the interior and exterior acoustic boundary value problem (recall that k was chosen non resonant) imply that in order to achieve approximate smooth controls in D 1 and D 2 it will be sufficient to have approximate L 2 controls on the boundaries of two slightly larger sets, W…”

Active Radar Cross Section Reduction of an Object Using Microstrip Antennas

“…In these works, cancellation of both electrostatic fields has been shown, as well as time‐varying scalar fields in two and three dimensions by using integral representation theorems. More recently, in Onofrei (, ), Onofrei proposed a unified integral equation method for the construction of a class of approximate solutions for the problem of scalar field manipulation using scalar sources. A sensitivity study for this design in the context of scalar field scattering cancellation was proposed in Norris et al () and Hubenthal and Onofrei ().…”