Scattering problems in elastodynamics

In electromagnetism, acoustics, and quantum mechanics, scattering problems can routinely be solved numerically by virtue of perfectly matched layers (PMLs) at simulation domain boundaries. Unfortunately, the same has not been possible for general elastodynamic wave problems in continuum mechanics. In this paper, we introduce a corresponding scattered-field formulation for the Navier equation. We derive PMLs based on complex-valued coordinate transformations leading to Cosserat elasticity-tensor distributions not obeying the minor symmetries. These layers are shown to work in two dimensions, for all polarizations, and all directions. By adaptative choice of the decay length, the deep subwavelength PMLs can be used all the way to the quasi-static regime. As demanding examples, we study the effectiveness of cylindrical elastodynamic cloaks of the Cosserat type and approximations thereof

Biharmonic Split Ring Resonator Metamaterial: Artificially dispersive effective density in thin periodically perforated plates

We present in this paper a theoretical and numerical analysis of bending waves localized on the boundary of a platonic crystal whose building blocks are split ring resonators (SRR). We first derive the homogenized parameters of the structured plate using a three-scale asymptotic expansion in the linearized biharmonic equation. In the limit when the wavelength of the bending wave is much larger than the typical heterogeneity size of the platonic crystal, we show that it behaves as an artificial plate with an anisotropic effective Young modulus and a dispersive effective mass density. We then analyze dispersion diagrams associated with bending waves propagating within an infinite array of SRR, for which eigen-solutions are sought in the form of Floquet-Bloch waves. We finally demonstrate that this structure displays the hallmarks of All-Angle-Negative-Refraction(AANR) and it leads to superlensing and ultrarefraction effects, interpreted thanks to our homogenization model as a consequence of negative and vanishing effective density, respectively.Comment: 17 pages, 6 figure

The homogenisation of Maxwell's equations with applications to photonic crystals and localised waveforms on metafilms

An asymptotic theory is developed to generate equations that model the global behaviour of electromagnetic waves in periodic photonic structures when the wavelength is not necessarily long relative to the periodic cell dimensions; potentially highly-oscillatory short-scale detail is encapsulated through integrated quantities. The theory we develop is then applied to two topical examples, the first being the case of aligned dielectric cylinders, which has great importance in the modelling of photonic crystal fibres. We then consider the propagation of waves in a structured metafilm, here chosen to be a planar array of dielectric spheres. At certain frequencies strongly directional dynamic anisotropy is observed, and the asymptotic theory is shown to capture the effect, giving highly accurate qualitative and quantitative results as well as providing interpretation for the underlying change from elliptic to hyperbolic behaviour

Homogenization Techniques for Periodic Structures

International audienceWe describe a selection of mathematical techniques and results that suggest interesting links between the theory of gratings and the theory of homogenization, including a brief introduction to the latter. We contrast the "classical" homogenization, which is well suited for the description of composites as we have known them since their advent until about a decade ago, and the "non-standard" approaches, high-frequency homogenization and high-contrast homogenization, developing in close relation to the study of photonic crystals and metamaterials, which exhibit properties unseen in conventional composite media, such as negative refraction allowing for super-lensing through a flat heterogeneous lens, and cloaking, which considerably reduces the scattering by finite size objects (invisibility) in certain frequency range. These novel electromagnetic paradigms have renewed the interest of physicists and applied mathematicians alike in the theory of gratings

Second-order homogenization of boundary and transmission conditions for one-dimensional waves in periodic media

International audienceWe consider the homogenized boundary and transmission conditions governing the mean-field approximations of 1D waves in finite periodic media within the framework of two-scale analysis. We establish the homogenization ansatz (up to the second order of approximation), for both types of problems, by obtaining the relevant boundary correctors and exposing the enriched boundary and transmission conditions as those of Robin type. Rigorous asymptotic analysis is performed for boundary conditions, while the applicability to transmission conditions is demonstrated via numerical simulations. Within this framework, we also propose an optimized second-order model of the homogenized wave equation for 1D periodic media, that follows more accurately the exact dispersion relationship and generally enhances the performance of second-order approximation. The proposed analysis is applied toward the long-wavelength approximation of waves in finite periodic bilaminates, subject to both boundary and transmission conditions. A set of numerical simulations is included to support the mathematical analysis and illustrate the effectiveness of the homogenization scheme