Metamaterials and high-contrast homogenisation: analysis, numerics and application
Date: Tuesday 14 June 2011
Time: 11.30-15.45
Venue: Cardiff School of Mathematics, Room M/0.40
The workshop will focus on a topic of intense interest, situated on the interface between pure and applied mathematics. It was discovered recently through experiments that composite materials with a high degree of contrast between constituents can possess remarkable properties, such as negative refraction and 'cloaking'. The speakers all work in this area and will bring in different angles at it: analytical, numerical and applied. The meeting will be hosted by the Cardiff Analysis group, whose members are involved in the related research. Due to the overarching nature of the proposed topic, the workshop will attract a wide range of analysts and applied mathematicians, mainly from but not limited to, Wales (Cardiff, Swansea, Aberystwyth), Bath, Oxford, London (UCL, Imperial College, King's College), Liverpool.
The planned events for the day are as follows:
Capacitary problems in elasticity with torsion effects (11.30 - 12.30)
Speaker: Michel Bellieud (University of Perpignan).
Abstract: We analyze a two-phase hyperelastic medium wherein a periodic set of stiff parallel fibres of small cross section is embedded in a softer matrix. This task is set in the context of a simplified model of small-deformation nonlinear elasticity embracing the Norton-Hoff model of viscoplastic flows. We show that when the rate of growth $p$ of the stored-energy function of the matrix is smaller than or equal to 2, a concentration of strain energy possibly emerges in a small region of space containing the fibres. This extra contribution depends on the density of the sections of the fibres with respect to some specific notion of capacity adjusted for elasticity. This phenomenon is caused by the discrepancy between the averaged effective displacement of the fibres and that of the matrix and, if $p<2$, by the possible rotations of the fibres with respect to their longitudinal axes. This rotating behavior may generate in parallel the storage of torsional energy within the fibres.
Lunch Break (12:30 - 13:30)
Buffet will be provided
Transformation plasmonics: carpets, cloaks, wormholes and perfect lenses (13:40 - 14:40)
Speaker: Sebastien Guenneau (Fresnel Institute, Marseille).
Abstract: There is currently a keen interest in transformational optics, whereby light trajectories follow unprecedented routes associated with geodesics in transformed coordinates. Electromagnetic paradigms include invisibility cloaks and carpets, described by anisotropic heterogeneous material parameters, as well as perfect lenses with a negative refractive index. We shall investigate these for a class of electromagnetic surface waves propagating at metal interfaces: Surface Plasmon Polaritons. Analytical formulae will be illustrated by three-dimensional finite element computations. Finally, a plasmonic counterpart of an Einstein-Rosen's solution to the equations of general relativity will be introduced: Plasmonic wormholes.
Numerical methods for high-contrast problems and their analysis (14:45 - 15:45)
Speaker: Robert Scheichl (University of Bath).
Abstract: The modelling and simulation of heterogeneous/composite materials with high contrast material properties and very small scale variation is of great interest in many application areas from material science and optics to porous media flow. Unless the material properties are varying (or assumed to vary) periodically, numerical computation is essential. Due to the small scales introduced by the variation this is a very challenging task (especially for high contrast and in 3D). In this talk we will describe (in the context of elliptic PDEs) some promising approaches to design and analyse robust numerical methods for such problems that exhibit a complexity comparable to standard methods for the respective homogeneous problem. In the course of this discussion we
will highlight the importance of weighted Poincare-type inequalities and provide necessary and sufficient conditions for the Poincare constant to
be uniformly bounded independent of the contrast.