## 2/ Designs of cloaks with gradient index soils using buried columns and/or forest of trees

29 September 2016 *( maj : 5 October 2016 )*, by

Conformal platonics is an emerging topic of transformation elastodynamics using conformal mappings in order to ease the design of structured soils for civil engineering applications: one should first simplify the design of structured plates so as to prevent anisotropy in the metamaterial’s parameters. One way to do so is to extend the concept of conformal optics to platonics: the former was proposed in (Leonhardt, 2006) and is valid in the geometrical optics limit (when the wavelength is much smaller than the diffracting obstacle).

**Figure**- Design of the Maxwell fisheye for Rayleigh waves; Stereographic projection of a virtual sphere (a thin shell) on a semi-infinite substrate. Numerical simulation (SimSonic3D) of a Lamb wave emitted by a point source (lower left) Gaussian pulse centered at 10 kHz frequency with 200µs duration that is converted into a plane wave (lower right) when passing through a half Maxwell fisheye lens. Note that by reciprocity, an incoming plane wave from the right would be focused at an image point on the left hand side of the half Maxwell fisheye lens. This makes possible lots of interesting effects such as invisibility cloaks combining Luneburg or Maxwell’s fisheye lenses (Colombi et al., 2016).

Although constrained to certain frequency range, this approach only requires a spatially varying refractive index. As a first paradigm of conformal platonics applied to geophysics, (Colombi et al. 2016) proposed four Luneburg lenses arranged in a checkerboard fashion to focus Rayleigh waves thanks to specific trajectories that lead not only to four images of an incoming plane wave, but that also create an exclusion zone, thereby making a seismic cloak. This work has laid the foundations for a whole new class of earthquake protections when we consider a combination of Luneburg lenses. Other, more realistic cloaking systems, such as using Maxwell’s fisheyes as shown in

WP4.1 Fig. 1, shall be further investigated during the ANR project. In Fig. 1, a stereographic projection of a sphere onto a flat plane is conformal, meaning that it preserves angles but neither distances nor areas. Using such a conformal mapping, Leonhardt (2006) designed some super antennas in optics.

An example of conformal optics based super antenna for Lamb waves is shown in Fig. 1 for a spatially varying plate density like:

where Rmax is the virtual sphere radius, and ρ(r) the heterogeneous density which in practice is achieved here with 10 isotropic homogeneous rings that have a specific density d given by the relation (Eq. 1).

Such a conformal geophysics route to Rayleigh wave control has the advantage of avoiding artificial anisotropy, but would only work in the geometrical optics limit. We will investigate theoretically and numerically such conformal seismic cloak designs for soils with buried columns of concrete and/or forest of trees in the frequency ranges of interest for civil engineering applications.