We analyze the time harmonic Maxwell equations in a complex three-dimensional geometry. The scatterer $\Omega\subset\mathbb{R}^3$ contains a periodic pattern of small wire structures of high conductivity, and the single element has the shape of a split ring. We rigorously derive effective equations for the scatterer and provide formulas for the effective permittivity and permeability. The latter turns out to be frequency dependent and has a negative real part for appropriate parameter values. This magnetic activity is the key feature of a left-handed metamaterial.

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Homogenization of Maxwell's Equations in a Split Ring Geometry

The two main versions of the boundary-value problem of magnetostatics (the one with h normal to the wall, the one with a tangential magnetic field) are studied in connection with a deep underlying mathematical structure: the existence of a decomposition of the space of square summable vector fields in five closed subspaces. This structure encompasses classical Helmholtz decomposition in its different guises. It is best conceived in terms of differential forms, but, for pedagogical reasons, this point of view has been left aside in the present text, which only makes use of classical notions of applied functional analysis.

Magnetostatic problems in multiply connected regions: some properties of the curl operator

We analyze the time harmonic Maxwell’s equations in a complex geometry. The scatterer Ω ⊂ R contains a periodic pattern of small wire structures of high conductivity, the single element has the shape of a split ring. We rigorously derive effective equations for the scatterer and provide formulas for the effective permittivity and permeability. The latter turns out to be frequency dependent and has a negative real part for appropriate parameter values. This magnetic activity is the key feature of a left-handed meta-material.

Homogenization of Maxwell’s equations with split rings

As a fundamental component of an overall program in modeling smart material damping devices, we consider inactive host material models for moderate to highly filled rubbers undergoing uniaxial tensile deformations. Beginning from a neo-Hookean strain energy function formulation for nonlinear extension, we develop general constitutive models for both quasi-static and dynamic deformations of a viscoelastic rod. The constitutive laws are nonlinear and contain hysteresis through a Boltzmann superposition integral term. The resulting integropartial differential equations models are shown to be equivalent to the usual Lagrangian dynamic distributed parameter models coupled with linear ordinary differential equations for internal variables (internal strains). Comprehensive well-posedness results (existence, uniqueness and continuous dependence) are summarized in a discussion of theoretical aspects of the systems.The models are validated with experiments designed and carried out explicitly for this study. In part...

Modeling of Nonlinear Hysteresis in Elastomers under Uniaxial Tension

We present a theoretical and numerical analysis of liquid surface waves (LSWs) localized at the boundary of a phononic crystal consisting of split-ring resonators (SRRs). We first derive the homogenized parameters of the fluid-filled structure using a three-scale asymptotic expansion in the linearized Navier-Stokes equations. In the limit when the wavelength of the LSW is much larger than the typical heterogeneity size of the phononic crystal, we show that it behaves as an artificial fluid with an anisotropic effective shear modulus and a dispersive effective-mass density. We then analyze dispersion diagrams associated with LSW propagating within an infinite array of SRR, for which eigensolutions are sought in the form of Floquet-Bloch waves. The main emphasis is given to the study of localized modes within such a periodic fluid-filled structure and to the control of low-frequency stop bands associated with resonances of SRRs. Considering a macrocell, we are able to compute the dispersion of LSW supported by a semi-infinite phononic crystal of SRRs. We find that the dispersion of this evanescent mode nearly sits within the first stop band of the doubly periodic structure. We further discover that it is linked to the frequency at which the effective-mass density of the homogenized phononic crystal becomes negative. We demonstrate that this surface mode displays the hallmarks of all-angle negative refraction and it leads to a superlensing effect. Last, we note that our homogenization results for the velocity potential can be applied mutatis mutandis to designs of electromagnetic and acoustic superlenses for transverse electric waves propagating in arrays of infinite conducting SRRs and antiplane shear waves in arrays of cracks shaped as SRRs.

Negative refraction, surface modes, and superlensing effect via homogenization near resonances for a finite array of split-ring resonators

In this paper we study a homogenization problem for a time periodic boundary value problem concerning the quasi-stationary Maxwell equations with a non linear monotone magne tic characteristic. The main features of the problem are related to the vanishing of the conductivity inside each period so that the type of the equations is rapidly oscillating. The unknowns are a vector potential and a scalar potential. We show that the first one converges to zero up to terms of second order, while the second one converges to the solution of a suitable homogenized stationary equation (with time as a parameter). We show moreover that when the magnetic characteristic is linear and symmetric the second order terms in the asymptotic expansion of the vector potential can be identified and related to the time derivative of the limit scalar potential.

Homogenization of the nonlinear quasistationary maxwell equations with degenerated coefficients

On the quasi-stationary Maxwell equations with monotone characteristics in a multiply connected domain

We study the homogenization of some boundary value time periodic problem for the quasistationary Maxwell equations in non linear laminated materials. The type of the equations is rapidly oscillating since the conductivity vani shes in the dielectric layers. We prove convergence of suita ble potentials to the solutions of limit equations for which homogenized characteristics are computed.

Angelo Negro

Papers

Homogenization of the nonlinear quasistationary maxwell equations with degenerated coefficients

On the quasi-stationary Maxwell equations with monotone characteristics in a multiply connected domain

Homogenization of the quasistationary Maxwell equations in nonlinear laminated materials

Homogenization of a non linear quasi-stationary Maxwell problem