Daniel Onofrei

Active exterior cloaking for the 2D Laplace and Helmholtz equations.

A new cloaking method is presented for 2D quasistatics and the 2D Helmholtz equation that we speculate extends to other linear wave equations. For 2D quasistatics it is proven how a single active exterior cloaking device can be used to shield an object from surrounding fields, yet produce very small scattered fields. The problem is reduced to finding a polynomial which is close to 1 in a disk and close to 0 in another disk, and such a polynomial is constructed. For the 2D Helmholtz equation it is numerically shown that three exterior cloaking devices placed around the object suffice to hide it.

Transformation Elastodynamics and Active Exterior Acoustic Cloaking

This chapter consists of three parts. In the first part we recall the elastodynamic equations under coordinate transformations. The idea is to use coordinate transformations to manipulate waves propagating in an elastic material. Then we study the effect of transformations on a mass-spring network model. The transformed networks can be realized with “torque springs”, which are introduced here and are springs with a force proportional to the displacement in a direction other than the direction of the spring terminals. Possible homogenizations of the transformed networks are presented, with potential applications to cloaking. In the second and third parts we present cloaking methods that are based on cancelling an incident field using active devices which are exterior to the cloaked region and that do not generate significant fields far away from the devices. In the second part, the exterior cloaking problem for the Laplace equation is reformulated as the problem of polynomial approximation of functions. An explicit solution is given that allows cloaking of larger objects at a fixed distance from the cloaking device, compared to previous explicit solutions. In the third part we consider the active exterior cloaking problem for the Helmholtz equation in 3D. Our method uses the Green’s formula and an addition theorem for spherical outgoing waves to design devices that mimic the effect of the single and double layer potentials in Green’s formula.

Complete Characterization and Synthesis of the Response Function of Elastodynamic Networks

We cloak a region from a known incident wave by surrounding the region with three or more devices that cancel out the field in the cloaked region without significantly radiating waves. Since very little waves reach scatterers within the cloaked region, the scattered field is small and the scatterers are for all practical purposes undetectable. The devices are multipolar point sources that can be determined from Greens formula and an addition theorem for Hankel functions. The cloaking devices are exterior to the cloaked region.The response function of a network of springs and masses, an elastodynamic network, is the matrix valued function W(ω), depending on the frequency ω, mapping the displacements of some accessible or terminal nodes to the net forces at the terminals. We give necessary and sufficient conditions for a given function W(ω) to be the response function of an elastodynamic network, assuming there is no damping. In particular we construct an elastodynamic network that can mimic a suitable response in the frequency or time domain. Our characterization is valid for networks in three dimensions and also for planar networks, which are networks where all the elements, displacements and forces are in a plane. The network we design can fit within an arbitrarily small neighborhood of the convex hull of the terminal nodes, provided the springs and masses occupy an arbitrarily small volume. Additionally, we prove stability of the network response to small changes in the spring constants and/or addition of springs with small spring constants.

On the active manipulation of fields and applications: I. The quasistatic case

Following the ideas proposed by Guevara-Vasquez et al (2009 Phys. Rev. Lett. 103; Opt. Express 17 14800–5) on active exterior cloaking, we present here a systematic integral equation method to generate suitable quasistatic fields for cloaking, illusions and energy focusing (with given accuracy) in multiple regions of interest. In the quasistatic regime, the central issue is to design appropriate source functions for the Laplace equation so that the resulting solution will satisfy the required properties. We show the existence and non-uniqueness of solutions to the problem and study the physically relevant unique L2-minimal energy solution. We also provide some numerical evidence on the feasibility of the proposed approach.

Γ-convergence for a fault model with slip-weakening friction and periodic barriers

We consider a three-dimensional elastic body with a plane fault under a slip-weakening friction. The fault has e-periodically distributed holes, called (small-scale) barriers. This problem arises in the modeling of the earthquake nucleation on a large-scale fault. In each e-square of the e-lattice on the fault plane, the friction contact is considered outside an open set T∈ (small-scale barrier) of size r∈ 0, then the fault behaves as a fault under a slip-dependent friction. The slip weakening rate of the equivalent fault is smaller than the undisturbed fault. Since the limit slip-weakening rate may be negative, a slip-hardening effect can also be expected. iii) if the barriers are too small (i.e. c = 0), then the presence of the barriers does not affect the friction law on the limit fault.

ASYMPTOTICS OF A SPECTRAL PROBLEM ASSOCIATED WITH THE NEUMANN SIEVE

In this paper, we analyze the asymptotic behavior of a Stekloff spectral problem associated with the Neumann Sieve model, i.e. a three-dimensional set Ω, cut by a hyperplane Σ where each of the two-dimensional holes, ∊-periodically distributed on Σ, have diameter r∊. Depending on the asymptotic behavior of the ratios we find the limit problem of the ∊ spectral problem and prove that the sequences , formed by the nth eigenvalue of the ∊ problem, converge to λn, the nth eigenvalue of the limit problem, for any n ∈ N. We also prove the weak convergence, on a subsequence, of the associated sequence of eigenvectors , to an eigenvector associated with λn. When λn is a simple eigenvalue, we show that the entire sequence of the eigenvectors converges. As a consequence, similar results hold for the spectrum of the DtN map associated to this model.

Engineering anisotropy to amplify a long-wavelength field without a limit

We present a simple design of circular or spherical shells capable of amplifying a long-wavelength or static field. This design makes use of only two isotropic materials and is optimally restricted to the prescribed geometric and materials constraint. Furthermore, it is shown that the amplification factor of the structure can be made arbitrarily large as the ratio of the radius of the inner sphere to that of the outer sphere decreases to zero. It is anticipated that the presented design will be useful for high-gain antennae for telecommunications, magnets generating strong, local uniform fields and thermoelectric devices harvesting thermal energy, among other applications.

Energy Accumulation in a Functionally Graded Spatial-Temporal Checkerboard

This letter extends the analysis of wave propagation in transmission lines with LC-parameters varying in space and time and the related effect of energy accumulation emerging from the concept of dynamic materials. We consider a practically important scenario of functionally graded checkerboard in space and time, i.e., the assembly combined of two dielectrics with material property transition zones applied instead of sharp interfaces. It is shown that the energy accumulation in traveling waves is preserved for certain ranges of material and geometric parameters.

Sensitivity analysis for active control of the Helmholtz equation

Abstract In previous works we considered the Helmholtz equation with fixed frequency k outside a discrete set of resonant frequencies, where it is implied that, given a source region D a ⊂ R d ( d = 2 , 3 ‾ ) and u 0 , a solution of the homogeneous scalar Helmholtz equation in a set containing the control region D c ⊂ R d , there exists an infinite class of boundary data on ∂ D a so that the radiating solution to the corresponding exterior scalar Helmholtz problem in R d ∖ D a will closely approximate u 0 in D c . Moreover, it will have vanishingly small values beyond a certain large enough “far-field” radius R. In this paper we study the minimal energy solution of the above problem (e.g. the solution obtained by using Tikhonov regularization with the Morozov discrepancy principle) and perform a detailed sensitivity analysis. In this regard we discuss the stability of the minimal energy solution with respect to measurement errors as well as the feasibility of the active scheme (power budget and accuracy) depending on: the mutual distances between the antenna, control region and far field radius R; value of the regularization parameter; frequency; location of the source.

Anomalous Localized Resonance Phenomena in the Nonmagnetic, Finite-Frequency Regime

The phenomenon of anomalous localized resonance (ALR) is observed at the interface between materials with positive and negative material parameters and is characterized by the fact that, when a given source is placed near the interface, the electric and magnetic fields start to have very fast and large oscillations around the interface as the absorption in the materials becomes very small while they remain smooth and regular away from the interface. In this paper, we discuss the phenomenon of anomalous localized resonance (ALR) in the context of an infinite slab of homogeneous, nonmagnetic material (