# Andre DiattaCNRS. Institut Fresnel, Centrale, Aix-Marseille Université

Andre Diatta

Ph.D.

## About

49

Publications

8,509

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558

Citations

Citations since 2017

## Publications

Publications (49)

We discuss the classification of 2-solvable Frobenius Lie algebras. We show that they always split as a semidirect sum of a vector space V of dimension n and an n-dimensional maximal Abelian subalgebra (MASA) of the full space of endomorphisms of V. We supply a complete classification list of all 2-solvable Frobenius Lie algebras corresponding to n...

This work relates to three problems, the classification of maximal Abelian subalgebras (MASAs) of the Lie algebra of square matrices, the classification of 2-step solvable Frobenius Lie algebras and the Gerstenhaber's Theorem. Let M and N be two commuting square matrices of order n with entries in an algebraically closed field K. Then the associati...

We propose a design of cylindrical cloak for coupled in-plane shear waves consisting of concentric layers of sub-wavelength resonant stress-free inclusions shaped as Swiss rolls. The scaling factor between inclusions’ sizes is according to Pendry’s transform. Unlike the hitherto known situations, the present geometric transform starts from a Willis...

We outline some recent research advances on the control of elastic waves in thin and thick plates, that have occurred since the large scale experiment [S. Brûlé, Phys. Rev. Lett. 112 , 133901 (2014)] that demonstrated significant interaction of surface seismic waves with holes structuring sedimentary soils at the meter scale. We further investigate...

We propose a design of cylindrical elastic cloak for coupled in-plane shear waves consisting of concentric layers of sub-wavelength resonant stress-free inclusions shaped as swiss-rolls. The scaling factor between inclusions' sizes is according to Pendry's transform. Unlike the hitherto known situations, the present geometric transform starts from...

Steering waves in elastic solids is more demanding than steering waves in electromagnetism or acoustics. As a result, designing material distributions which are the counterpart of optical invisibility cloaks in elasticity poses a major challenge. Waves of all polarizations should be guided around an obstacle to emerge on the downstream side as thou...

Recent static experiments on twist effects in chiral three-dimensional mechanical metamaterials have been discussed in the context of micropolar Eringen continuum mechanics, which is a generalization of linear Cauchy elasticity. For cubic symmetry, Eringen elasticity comprises nine additional parameters with respect to linear Cauchy elasticity, of...

We outline some recent research advances on the control of elastic waves in thin and thick plates, that have occurred since the large scale experiment [Phys. Rev. Lett. 112, 133901, 2014] that demonstrated significant interaction of surface seismic waves with holes structuring sedimentary soils at the meter scale. We further investigate the seismic...

Recent static experiments on twist effects in chiral three-dimensional mechanical metamaterials have been discussed in the context of micropolar Eringen continuum mechanics, which is a generalization of Cauchy elasticity. For cubic symmetry, Eringen elasticity comprises nine additional parameters with respect to Cauchy elasticity, of which three di...

We make precise some results on the cloaking of displacement fields in linear elasticity. In the spirit of transformation media theory, we show that the transformed governing equations in Cosserat and Willis frameworks are equivalent to certain small defect problems for the usual Navier equations. We introduce a notion of nearly cloaking for elasti...

Viewed from the sky, the urban fabric pattern appears similar to the geometry of structured devices called metamaterials; these were developed by Physicists to interact with waves that have wavelengths in the range from nanometers to meters (from electromagnetic to seismic metamaterials). Visionary research in the late 1980s based on the interactio...

In [AIP Advances 6, 121707 (2016)], a soil structured with concrete columns distributed within two specially designed seismic cloaks thanks to a combination of transformational elastodynamics and effective medium theory was shown to detour Rayleigh waves of frequencies lower than 10 Hz around a cylindrical region. The aforementioned studies motivat...

We first review classical results on cloaking and mirage effects for electromagnetic waves. We then show that transformation optics allows the masking of objects or produces mirages in diffusive regimes. In order to achieve this, we consider the equation for diffusive photon density in transformed coordinates, which is valid for diffusive light in...

We explore interactions of elastic waves propagating in plates (with soil parameters) structured with concrete pillars buried in the soil. Pillars are 2 m in diameter, 30 m in depth and the plate is 50 m in thickness. We study the frequency range 5 to 10 Hz, for which Rayleigh wave wavelengths are smaller than the plate thickness. This frequency ra...

In this paper, we bring to the awareness of the scientific community and civil engineers, an important fact: the possible lack of wave protection of transformational elastic cloaks. To do so, we propose spherical cloaks described by a non-singular asymmetric elasticity tensor depending upon a small parameter η, that defines the softness of a region...

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 scatter...

A simple invisible structure made of two anisotropic layers is analyzed
theoretically in temporal regime. The frequency dispersion is introduced and
analytic expression of the transient part of the field is derived for large
times when the structure is illuminated by a causal excitation. This expression
shows that the limiting amplitude principle a...

Numerical simulations shed light on control of shear elastic wave propagation
in plates structured with inertial resonators. The structural element is
composed of a heavy core connected to the main freestanding plate through tiny
ligaments. It is shown that such a configuration exhibits a complete band gap
in the low frequency regime. As a byproduc...

We investigate the properties of principal elements of Frobenius Lie algebras, following the work of M. Gerstenhaber and A. Giaquinto. We prove that any Lie algebra with a left symmetric algebra structure can be embedded, in a natural way, as a subalgebra of some sl(m,K), for K= R or C. Hence, the work of Belavin and Drinfeld on solutions of the Cl...

We propose a cloak for coupled shear and pressure waves in solids. Its
elastic properties are deduced from a geometric transform that retains the form
of Navier equations. The spherical shell is made of an anisotropic and
heterogeneous medium described by an elasticity tensor C' (without the minor
symmetries) which has 21 non-zero spatially varying...

We discuss the concept of cloaking for elastic impedance tomography, in
which, we seek information on the elasticity tensor of an elastic medium from
the knowledge of measurements on its boundary. We derive some theoretical
results illustrated by some numerical simulations.

In this chapter, we review some recent developments in the field of photonics: cloaking, whereby an object becomes invisible to an observer, and mirages, whereby an object looks like another one (say, of a different shape). Such optical illusions are made possible thanks to the advent of Metamaterials, which are new kinds of composites designed usi...

We start by a review of the chronology of mathematical results on the Dirichlet-to-Neumann map
which paved the way toward the physics of transformational acoustics. We then rederive the
expression for the (anisotropic) density and bulkmodulus appearing in the pressure wave equation
written in the transformed coordinates. A spherical acoustic cloak...

This paper reviews some properties of lenses in curved and folded optical spaces. The point of the paper is to show some limitations of geometrical optics in the analysis of subwavelength focusing. We first provide a compre-hensive derivation for the equation of geodesics in curved optical spaces, which is a tool of choice to design metamaterials i...

We design non-singular cloaks enabling objects to scatter waves like objects with smaller size and very different shapes. We consider the Schrodinger equation which is valid e.g. in the contexts of geometrical and quantum optics. More precisely, we introduce a generalized non-singular transformation for star domains, and numerically demonstrate tha...

This paper extends the proposal of Li and Pendry [Phys. Rev. Lett. 101, 203901-4 (2008)] to invisibility carpets for infinite conducting planes and cylinders (or rigid planes and cylinders in the context of acoustic waves propagating in a compressible fluid). Carpets under consideration here do not touch the ground: they levitate in mid-air (or flo...

26 pages, 9 figures. Key words: Mathematical methods in physics; Mathematical Physics, electromagnetic theory; Metamaterials;Anisotropic optical materials; invisibility; cloak

We extend the design of radially symmetric three-dimensional invisibility cloaks through transformation optics [1] to cloaks with a surface of revolution. We derive the expression of the transformation matrix and show that one of its eigenvalues vanishes on the inner boundary of the cloaks, while the other two remain strictly positive and bounded....

In this paper, we consider the intensity surface of a 2D image, we study the evolution of the symmetry sets (and medial axes) of 1-parameter families of iso-intensity curves. This extends the investigation done on 1-parameter families of smooth plane curves (Bruce and Giblin, Giblin and Kimia, etc.) to the general case when the family of curves inc...

The point of the paper is to show some limitations of geometrical optics in the analysis of subwavelength focusing. We analyze the resolution of the image of a line source radiating in the Maxwell fisheye and the Veselago-Pendry slab lens. The former optical medium is deduced from the stereographic projection of a virtual sphere and displays a hete...

We derive the expression for the anisotropic heterogeneous matrices of permittivity and permeability associated with two-dimensional polygonal and star shaped cloaks. We numerically show using finite elements that the forward scattering worsens when we increase the number of sides in the latter cloaks, whereas it improves for the former ones. This...

This paper investigates the geometry of compact contact manifolds that are
uniformized by contact Lie groups, i.e., compact manifolds that are the
quotient of some Lie group G with a left invariant contact structure and a
uniform lattice subgroup. We re-examine Alexander's criteria for existence of
lattices on solvable Lie groups and apply them, al...

Let G be a Lie group, $T^*G$ its cotangent bundle with its natural Lie group structure obtained by performing a left trivialization of T^*G and endowing the resulting trivial bundle with the semi-direct product, using the coadjoint action of G on the dual space of its Lie algebra. We investigate the group of automorphisms of the Lie algebra of $T^*...

We investigate contact Lie groups having a left invariant Riemannian or pseudo-Riemannian metric with specific properties
such as being bi-invariant, flat, negatively curved, Einstein, etc. We classify some of such contact Lie groups and derive
some obstruction results to the existence of left invariant contact structures on Lie groups.

Mark all vertices on a curve evolving under a family of curves obtained by intersecting a smooth surface M with the 1-parameter family of planes parallel to the tangent plane of M at a point p. Those vertices trace out a set, called the vertex set through p. We take p to be a generic umbilic point on M and describe the perestroikas of the vertex se...

The investigation of 3D euclidean symmetry sets (SS) and medial axis is an important area, due in particular to their various important applications. The pre-symmetry set of a surface M in 3-space (resp. smooth closed curve in 2D) is the set of pairs of points which contribute to the symmetry set, that is, the closure of the set of pairs of distinc...

We prove that the level sets of a real C
s
function of two variables near a non-degenerate critical point are of class C
[s/2] and apply this to the study of planar sections of surfaces close to the singular section by the tangent plane at an elliptic or hyperbolic point, and in particular at an umbilic point. We go on to use the results to study s...

We investigate the behaviour of vertices and inflexions on 1-parameter families of curves on smooth surfaces in the 3-space, which include a singular member. In particular, we discuss the context where the curves evolve as sections of a smooth surface by parallel planes. More precisely we will trace the patterns of inflexions and vertices (maxima a...

A result from Gromov ensures the existence of a contact structure on any connected non-compact odd dimensional Lie group. But in general such structures are not invariant under left translations of the Lie group. The problem of finding which Lie groups admit a left invariant contact structure (contact Lie groups), is then still open. While Lie grou...

Let G be a Lie group with Lie algebra $ \Cal G: = T_\epsilon G$ and $T^*G = \Cal G^* \rtimes G$ its cotangent bundle considered as a Lie group, where G acts on $\Cal G^*$ via the coadjoint action. We show that there is a 1-1 correspondance between the skew-symmetric solutions $r\in \wedge^2 \Cal G$ of the Classical Yang-Baxter Equation in G, and th...

Étant donné (G, π) un groupe de Lie-Poisson et H un sous-groupe fermé connexe e G, notre principal résultat décrit les tenseurs de Poisson sur G/H pour lesquels l'action canonique de G est un morphisme de Poisson. Nous fournissons aussi une construction explicite des sous-algèbres lagrangiennes de l'algèbre de Lie double de (G, π). Cela permet de d...

Given a Poisson Lie group (G, π) and a closed and connected subgroup H of G, our main result gives the Poisson tensors on G/H for which the canonical action of G on G/H is a Poisson morphism. We also supply an explicit method to construct the Lagrangian subalgebras of the double Lie algebra of (G, π). Such a construction allows us to give a new pro...

## Projects

Project (1)