Transformation elastodynamics and active exterior acoustic cloaking

Abstract

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 analytic functions. An explicit solution is given that allows to cloak 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.

Submitted as a chapter for the volume "Acoustic metamaterials: Negative refraction, imaging, lensing and cloaking", Craster and Guenneau ed., Springer

References in corpus (13)

• Illusion optics: The optical transformation of an object into another object
• Design of Electromagnetic Cloaks and Concentrators Using Form-Invariant Coordinate Transformations of Maxwell's Equations
• General Relativity in Electrical Engineering
• Acoustic cloaking theory
• On the feasibility of pentamode mechanical metamaterials
• Achieving control of in-plane elastic waves
• Full-wave invisibility of active devices at all frequencies
• Superscatterer: Enhancement of scattering with complementary media
• New metamaterials with macroscopic behavior outside that of continuum elastodynamics
• Solutions in folded geometries, and associated cloaking due to anomalous resonance
• Mathematical analysis of the two dimensional active exterior cloaking in the quasistatic regime
• Exterior cloaking with active sources in two dimensional acoustics
• Realizable response matrices of multiterminal electrical, acoustic, and elastodynamic networks at a given frequency

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• Mathematical analysis of the two dimensional active exterior cloaking in the quasistatic regime
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• Sensitivity analysis for active control of the Helmholtz equation
• A unifying perspective on linear continuum equations prevalent in science. Part II: Canonical forms for time-harmonic equations
• Analytic Materials
• On the active manipulation of quasistatic fields and its applications
• Discrete Transformation Elasticity: An Approach to Design Lattice-based Polar Metamaterials
• Active cloaking of resonant coated inclusions for waves in membranes and Kirchhoff plates
• Active cloaking for clusters of pins in thin plates
• On the active manipulation of EM fields in open waveguides