Wave mechanics in media pinned at Bravais lattice points

The propagation of waves through microstructured media with periodically arranged inclusions has applications in many areas of physics and engineering, stretching from photonic crystals through to seismic metamaterials. In the high-frequency regime, modelling such behaviour is complicated by multiple scattering of the resulting short waves between the inclusions. Our aim is to develop an asymptotic theory for modelling systems with arbitrarily-shaped inclusions located on general Bravais lattices. We then consider the limit of point-like inclusions, the advantage being that exact solutions can be obtained using Fourier methods, and go on to derive effective medium equations using asymptotic analysis. This approach allows us to explore the underlying reasons for dynamic anisotropy, localisation of waves, and other properties typical of such systems, and in particular their dependence upon geometry. Solutions of the effective medium equations are compared with the exact solutions, shedding further light on the underlying physics. We focus on examples that exhibit dynamic anisotropy as these demonstrate the capability of the asymptotic theory to pick up detailed qualitative and quantitative features

Bloch wave homogenization in a medium perforated by critical holes

Publicación ISIIn this Note, we use the Bloch wave method to study the asymptotic behavior of the solution of the Laplace equation in a periodically perforated domain, under a non-homogeneous Neumann condition on the boundary of the holes, as the hole size goes to zero more rapidly than the domain period. We prove that for a critical size, the non-homogeneous boundary condition generates an additional term in the homogenized problem, commonly referred to as 'the strange term' in the literature

WAVE PROPAGATION IN ELASTIC MEDIA WITH INTERNAL STRUCTURE. PERIODIC TRANSFORMATIONS AND CURVED BEAMS

In this thesis we will focus on the linear elastodynamic properties of complex materials. In Chapter 2, we review the linear theory of elasticity. This chapter provides a brief overview of the basic laws of elasticity theory for di_erent coordinate systems, that will facilitate the further development of our research. Dispersion properties of 2D periodic systems for di_erent lattices are reported in Chapter 3. The governing equations of out of plane wave and examples of coordinate transformations in Cartesian and cylindrical are considered in Chapter 4. Also the scattering problems by square and cylindrical uncloaked and cloaked holes are investigated analytically and numerically. In Chapter 5 a periodic transformation approach has been applied to the problem of out of plane shear wave propagation in an isotropic linear elastic material. The Chapter is organized as follows. In Section 5.1 we present initial and transformed equations of motion and corresponding boundary conditions, describing the periodic locally radial geometric transformation. In Section 5.2 we report the comparative analysis of dispersion properties and briey describe the applied multipole expansion method. In particular, we focus our attention on classical, overlapping and unfolding transformations by also performing a low-frequency, long wavelength homogenisation. In Section 5.3 we show several application including a transmission problems in a continuum and in a waveguide, the detection of defect modes and the design of the transformation for the existence of Dirac points. In Chapter 6, the mathematical model of a curved beam that is connected to two semi-in_nite straight beams is developed. Dispersion properties of curved beams are derived, characterized by three di_erent propagating regimes. By implementing the Transfer matrix approach, the reection and transmission coe_cients that depend on the curvature, frequency and total angle of the curved beam are determined. By analysing the e_ect of the curvature, frequency and total angle on energy ux, separation between high frequency/low curvature regime, where the incident wave is practically totally transmitted, and low frequency/high curvature regime where, in addition to reection there is a strong coupling between longitudinal and exural waves, are de_ned. Finally, general conclusions are given in the last chapter

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 corrector in the homogenization of a conductive-radiative heat transfer problem

International audienceThis paper focuses on the contribution of the so-called second order corrector in periodic homogenization applied to a conductive-radiative heat transfer problem. More precisely, heat is diffusing in a periodically perforated domain with a non-local boundary condition modelling the radiative transfer in each hole. If the source term is a periodically oscillating function (which is the case in our application to nuclear reactor physics), a strong gradient of the temperature takes place in each periodicity cell, corresponding to a large heat flux between the sources and the perforations. This effect cannot be taken into account by the homogenized model, neither by the first order corrector. We show that this local gradient effect can be reproduced if the second order corrector is added to the reconstructed solution