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R - PMMH

Transformational acoustics and homogenization
in structured plates
Sébastien Guenneau
Institut Fresnel, CNRS,
Aix-Marseille Université,
13013 Marseille, France.
work in collaboration with:
Younes Achaoui, Ronald Aznavourian, André Diatta, Stefan Enoch (Institut Fresnel)
Stéphane Brûlé, Ménard soil dynamic laboratory (Lyon, France)
Philippe Roux (ISTerre, Grenoble, France)
Marc Dubois, Gautier Lefebvre, Ros-Kiri Ing, Emmanuel Bossy,
Geoffroy Lerosey, Patrick Sebbah (Institut Langevin, Paris, France)
Tryfon Antonakakis, Andrea Colombi, Richard Craster (Imperial College London, UK)
Bogdan Ungureanu (Technical University of Iasi, Romania)
Nicolas Vandenberghe (IRPHE, Marseille, France)
Cargèse summer school " Wave propagation in complex media", 2015, August 17th - 28th
1
Outline of the talk:
Principle of transformational physics
From electromagnetic to acoustic Metamaterials
Platonic Metamaterials
Seismic metamaterials
Conclusion and perpectives.
2
Principle of transformational
physics
S. Guenneau: Cargèse summer school " Wave propagation in complex media", 2015, August 17th - 28th
3
A simple change of coordinates…
Original space:
Let us start with a PDE with a scalar coefficient m
(isotropic heterogeneous medium)
(1)
x’ = p(x,y,z,t)
Transformed space:
y’ = q(x,y,z,t)
z’ = r(x,y,z,t)
After coordinate change, we get a modified equation
The PDE now has a matrix coefficient M
(anisotropic heterogeneous medium)
(2)
…can have big consequences!
4
First example: 3D Maxwell’s equation in frequency domain
Original space:
Let us start with the time-harmonic vector Maxwell
curl curl equation in vacuum
(3)
x’ = p(x,y,z)
y’ = q(x,y,z)
Transformed space:
z’ = r(x,y,z)
After coordinate change, Maxwell’s equation describes
an anisotropic heterogeneous medium (Nicolet et al.,
1994; Ward & Pendry, 1996, Greenleaf et al. 2003,
Zolla et al. 2005, Pendry et al. 2006 etc.)
 The PDE now has matrix valued spatially varying
(in general dispersive) permittivity and permeability!
(4)
5
Second example: 2D Helmholtz’s equation
Let us start with the scalar Helmholtz equation
(for transverse electromagnetic wave, pressure
wave, anti-plane shear wave, water wave, etc.)
Original space:
(5)
x’ = p(x,y)
Transformed space:
y’ = q(x,y)
After coordinate change, Helmholtz’s equation has
both a matrix coefficient A and a scalar coefficient b
(6)
We shall derive the transformed equation on black board
So far, so good, the transformed equation has the same structure as the original one!
6
Paradigm of Transformational physics: Cloaking in optics
(Pendry, Schurig and Smith, Science 2006)
Consider a map
r’=R1+ r(R2-R1)/R2
from a disk r<R 2
onto a ring R1<r’<R2, this leads to two sets of parameters:
in Hz polarization
(7)
in Ez polarization
Permittivity and permeability describe a transformed medium called metamaterial inside the
ring (invisibility cloak). Hz and Ez the longitudinal magnetic and electric fields.
R1
R2
Point source in
free space
Point source near
a scattering object
Point source near object
surrounded by the cloak
7
Paradigm of Transformational physics: Cloaking in Ez polarization
(Schurig et al., Science 2006)
One may argue a metamaterial would be hard to make in practice, as it requires artificial
anisotropy and magnetism, and one can choose between two types of material parameters:
Pendry, Schurig
and Smith’s
choice for experiments
R1
r’=R1+ r(R2-R1)/R2
R2
Ez polarization
Imperial College/Duke cloak
6 MONTHS
Practical implementation
Concept of invisibility cloak
(Schurig et al., Science November 2006)
(Pendry et al., Science June 2006)
8
Paradigm of Transformational physics: cloaking in Hz polarization
(Kanté et al., PRB 2009)
Following Pendry’s work, Kanté, Germain and de Lustrac proposed a cloak in the other
polarization!
B. Kanté, Romain and
de Lustrac’s choice
for experiments
IEF cloak in Hz polarization
Numerics versus experiments at 11 GHz
9
Can we avoid material anisotry?
Transformational optics versus conformal optics
Cloaking via artificial anisotropy
(cloaking for Maxwell)
Pendry, Schurig, Smith, 2006
(Imperial College & Duke)
Cloaking via gradient index
(cloaking in ray optics limit)
One goal
Transformation Optics (TO)
Leonhardt, 2006
(Saint-Andrews)
Conformal Optics (CO)
Two ways to do it
BTW: a combination of TO and CO makes invisibility carpets (best of two worlds)
10
From electromagnetic to
acoustic metamaterials
11
Plethora of experimental results for
electromagnetic waves
Imperial/Duke 2006
Duke 2009
Duke/Imperial 2006
Berkeley 2009
Cornell 2009
KIT 2010
and many more since 2010
Remark: c)-f) are invisibility carpets as proposed by Li and Pendry, PRL 2008.
One way to see it is that carpets combine TO & CO.
12
From microwave to water wave cloaks:
Pendry’s
cloak mimicks an effective tensor of permeability
The waterwave cloak mimicks an effective tensor of viscosity
Microwave cloak @ 8.5 GHz Anisotropy guides waves
(split ring resonators)
Water wave cloak (5-15Hz)
(no resonators)
Institut Fresnel, Liverpool University 2008
Duke University, Imperial College 2006
Cloaking via artificial anisotropy is a
multiphysics concept!
13
Examples of manufactured carpets for optics and water waves
Telecommunications (1.55 mm)
Plasmonics (700-900 nm)
Scherrer et al. PRB 2013 (with group of de Fornel in Dijon and Lippens iand Vanbesien n Lille)
Linear surface water waves (0.5-2 m)
(with group of B. Molin & O. Kimoun in IRPHE)
Dupont, Kimmoun, Molin, Guenneau, Enoch, arXiv 2011 & PRE 2015
17m
See also Berraquero, Maurel, Petitjean, Pagneux, PRE 2013 & Porter
and Newman JFM 2014 for cloaking with varying depth
Kadic et al., Opt. Exp. 2010,
Kadic et al. Plasmonics 2012
(Barcelona, Marseille)
14
Experiments on microwaves and acoustics with a waterwave cloak
Cloaking between 3-8 GHz
Cloaking between 2-7 kHz
Benchmark:
Fang’s group
(PRL 2011)
Simulation (IF)
Measure 3.6 Hz
(Institut Fresnel)
Computation
3D SER (CST)
Redha Abdeddaim
(Institut Fresnel)
Measure pressure field
(Fang, MIT) @ 5kHz
@50 kHz
speaker
Xu et al. Scientific Reports 2015
Microphone
15
Experimental proof of acoustic cloaking @5kHz (Fang’s group MIT)
Xu et al. Scientific Reports 2015
Experimental data (MIT) for
glass bottle cloaked (5 kHz)
Computation 2D
SER (COMSOL)
Mohamed Farhat
(KAUST)
Conclusion: The same cloak works for transverse electric waves (3-8 GHz),
pressure acoustic waves (2-7KHz) and water waves (5-15 Hz)
16
The case of transformed elastodynamics
Modified Willis equations versus transformed Navier equations
Original space:
Let us start with the time-harmonic tensor Navier
equation in isotropic homogeneous elastic medium
described by a rank 4 elasticty tensor C and a density
(8)
x’ = p(x,y,z) such that u’=JT u
with J jacobian
Transformed space:
y’ = q(x,y,z) matrix of transform
z’ = r(x,y,z)
After coordinate change, Navier’s equation does not
retain its form in general (Milton, Briane, Willis 2006)
 The PDE now has additional rank 3 tensors D and E
However, take J=Id, then one gets (Brun, Guenneau,
Movchan, 2009; Shuvalov and Norris, 2011):
(9)
(10)
BUT C’ijkl different from C’ijlk (Cossérat)
17
Numerical results for a 2D elastic (Cosserat) cloak
for coupled pressure and shear waves in frequency and time domains
2D elastic cloak in frequency domain
Guenneau, Brun, Movchan
Institut Fresnel/Liv. Univ. (2009)
2D elastic cloak in time domain
Komatitsch, Cristini, Xie (LMA/CNRS):
Convergence problem with Specfem when
wave reaches infinite speed!
Heterogeneous scalar density:
Elasticity tensor without the
minor symmetries:
For other elastic cloaks see Milton, Briane, Willis NJP 2006, Parnell and Norris APL 2012 etc.
2013: First mechanical pentamode cloak (static)
KIT (Weneger’s group) Nature Comm. 2013
Inspired by theoretical work of Norris (PRSA 2008)
Remaining challenge: mechanical cloak in dynamic regime
19
Proposal for a 3D elastodynamic Cosserat cloak
Transformed Navier equation for
u’=JT u with J=Id:
with density
C’ has 21 non-zero entries
in spherical basis
Diatta & Guenneau APL 2014
20
3D elastodynamic cloak for coupled pressure and shear waves
Cloak @ 3Hz with r1=2m, r2=4m dressing a void of radius 2m.
Norm of elastic displacement for a spherical void
(left), no void (middle) and void+cloak (right).
3 components of strain tensor for a spherical void
(top), no void (middle) and void+cloak (bottom).
Diatta & Guenneau APL 2014
Remark: The elasticity tensor has 34=81 heterogeneous entries in Cartesian coordinates.
21
The case of transformed elastodynamics for plates
Navier equation does not behave nicely under geometric transform, what
about the Kirchhoff-Love equation for thin plates?
Original space:
Let us start with the biharmonic equation for flexural
waves in thin elastic plates with E the Young modulus,
r the density, h the thickness and D the plate rigidity
(11)
x’ = p(x,y,z)
Transformed space:
y’ = q(x,y,z)
After coordinate change, the biharmonic equation reads
the same (Farhat, Guenneau, Enoch and Movchan, 2009)
but one now has new Young modulus, density, thickness
and flexural plate rigidity parameters:
(12)
22
Platonic metamaterials: Lamb
waves in structured plates
23
Effective plate model with a beam lattice
Bernoulli-Euler equation (in four beams in a unit cell):
General solution (16 coefficients):
l
h
b
Floquet-Bloch conditions:
Floquet-Bloch conditions:
l
Continuity of displacement at node:
Rigid body motion at node:
Angular and linear momenta at node:
Farhat et al. APL 96, 081909, 2010
24
Focusing Lamb waves in a perforated plate
(platonic crystal)
15
Frequency (kHz)
Dubois et al., APL 103, 071915, 2013
Farhat et al. APL 96, 081909, 2010
Negative
group
velocity
10
duraluminium plate (500*300*1mm)
5
0
X
M

Bande acoustique théorique et numérique
Collaboration Langevin/Fresnel (ANR PLATON)
M. Dubois , E. Bossy & P. Sebbah
M. Farhat, S. Enoch & S. Guenneau
Lensing @10KHz
Numerical evidence of cloaking in a perforated plate
Biharmonic equation in homogeneous phase
with
D
And zero bending moments and shearing stress at boundaries
no cloak
cloak 100
cloak 200
(top) and 0.36 (bottom)
Farhat, Guenneau, Enoch, PRB85, 020301 (2012)
26
Experiments in plates: Lensing and cloaking
With group Patrick Sebbah (Langevin, Paris)
With Nicolas Vandenberghe
(IRPHE, Marseille)
Réfraction
négative
FDTD simulation (using SIMSONIC)
performed by Ronald Aznavourian
FDTD simulation (using SIMSONIC: 3D full elasticity
790
freeware developed by Emmanuel
Bossy, Langevin)
Frequency (kHz)
15
10
5
0
X
M

@10000 Hertz
@400 Hertz
27
High frequency homogenization theory of pinned plates
(Antonakakis & Craster, PRSA 2012)
FEM
HFH
@W
=6.58
FEM
@W
=6.58
HFH
=
(hyperbolic metamaterial)
Antonakakis, Craster & Guenneau, EPL 2014
28
High Frequency Homogenization versus experiments in pinned plates
Hyperbolic metamaterial plate (T11*T22<0)
HFH gives @M
T11=27.05
T22=-15.12
FEM
HFH
Experiments
Anisotropic metamaterial plate (T11<<T22)
HFH gives @M
T11=0.90
T22=467.33
M
G
X
M
FEM
Experiments
Lefebvre, Achaoui, Ing, Antonakakis, Craster,
Guenneau, Sebbah (in preparation)
29
Maxwell’s fisheye lens in plates of varying thickness
(Dubois, Lefevbre et al. APL 2015) platonics
FDTD simulations (3D elasticity)
optics
(D + n2) u=0
rh/D proportional to n4
Snapshot @155 ms
0.01
Snapshot @605 ms
0.01
0.008
0.008
0.006
0.006
0.004
0.004
0.002
0.002
0
0
-0.002
-0.002
-0.004
-0.004
-0.006
-0.006
-0.008
-0.008
-0.01
-0.01
Lefebvre, Achaoui, Ing, Guenneau, Sebbah (APL 2015)
30
Cloaking Lamb waves with forest of trees
on plates (varying effective thickness)
(Colombi, Roux et al. JASA 2015)
31
Seismic metamaterials: Moulding
surface elastic waves in structured soils
32
MAIN CONCEPT : CHANGE THE PROPERTIES OF THE INITIAL SOIL
TECHNOLOGIES : OUR ALTERNATIVE = GROUND IMPROVEMENT
Soil cross section
Z
Densification
Soil
Ground improvement without admixture
Inclusions
Reinforcement with rigid inclusions
(steel, concrete) or boreholes (air)
Metamaterial ?
33
Selected
design:
a seismic
shield
via platonic
stop band
DESCRIPTION
OF
A FULL
SCALE
SEISMIC
TEST
Shielding effect – Source (50 Hz)
2D plate Model with Floquet-Bloch
Conditions – solved with Comsol
Soil parameters (Ménard):
Vs ~ 150 m/s, E=100 Mpa, n=0.3
Density soil r= 1500 kg/m3
Density bore holes r= 1.2 kg/m3
Grid spacing 1.73 m
Holes’ diameter : 0.32 m
Example of lensing effect
Partial stop band around 50Hz: A source at frequency 50 Hz should be reflected
by a finite array of 30 boreholes
34
First experiment on seismic metamaterial (Ménard, Grenoble, August 2012)
DESCRIPTION OF A FULL SCALE SEISMIC TEST
1.73 m
Length : 5 m
Depth : 2 m
35
Construction site (Ménard, Grenoble, August 2012)
MAKING COLUMNS
setup
with low
seismic
source, boreholes, and velocimeters
RESULTS Experimental
: J2-J1 -2m
E/A1
normed
37
RESULTS
: J2-J1
-2melastic
E/A1energy
low (Ménard,
normed
Map
of measured
Grenoble, Summer 2012)
Map of J2-J1 where J2 is the
elastic energy with boreholes
and J1 without boreholes
Map of J2/J1 where J2 is the
elastic energy with boreholes
and J1 without boreholes
0.2<J2/J1 <1 in blue area i.e.
up to 5 times less elastic NRJ
Black rectangles: sensors; white discs: boreholes
38
Word ofOF
caution
on seismic
shields:
both periodicity
DESCRIPTION
A FULL
SCALE
SEISMIC
TEST and
overall shape of seismic metamaterial matter…
Concentration effect – Source (50 Hz)
2D Model – Comsol Conditions de
passage sur les bords des trous
(formulation faible, cf. eq. 1)
Energy field
Soil :
Vs ~ 150 m/s, E=100 Mpa, n=0.3
Density soil r= 1500 kg/m3
Density bore holes r= 1.2 kg/m3
Grid spacing 1.73 m
Example of lensing effect
avec
 The ring-type structure leads to a concentration effect!
39
Second experiment on seismic metamaterial (Ménard, Saint-Priest, September 2012)
DESCRIPTION OF A FULL SCALE SEISMIC TEST
1.73 m
Depth : 2 m
Length : 5 m
Holes’ diameter : 2 m, pitch: 6m
40
Seismic flat lens (Saint Priest, September 2012)
experiment
d
@5 Hz (source at d/2)
experiment
simulation
simulation
Hz
50
40
3
30
0
20
An approximate plate simulation suggests
negative refraction around 3-7Hz
10
@5 Hz (source at d)
M

X
M
Brûlé, Enoch, Guenneau (under preparation)
41
Seismic flat lens (experiment of Saint Priest, September 2012)
Frequency domain
Experiment (4.5Hz)
Time domain
d
simulation
@4.5 Hz (source at d)
one `image’ inside the lens after 2.178s
one `image’ at the exit of the lens after 2.270s
Brûlé, Enoch, Guenneau (under preparation)
42
Conclusion and perspectives
43
Theoretical and experimental results in plates
Collaboration with group of Patrick Sebbah
@ Langevin, Paris: Experiments on lensing
of flexural waves (Pendry-Veselago flat
lens and Maxwell fisheye) ANR PLATON
Collaboration with group of Nicolas Vandenberghe
@ IRPHE, Marseille: Experiment on cloaking of
flexural waves through concentric layers of holes
FEM
HFH
Collaboration with group of Richard Craster
@ Imperial, London: high-frequency
homogenization of pinned plates
(experiments with group of P. Sebbah)
44
Experimental achievements on seismic metamaterials
Shielding effect around 50 Hz (Grenoble)
Focusing effect around 5 Hz (St Priest)
Brûlé, Javelaud, Enoch, Guenneau, PRL 2014
source
Partnership with Ménard company (Dynamic soil laboratory of Stéphane Brûlé)
Reminiscent of Work Sang-Hoon Kim and Mukunda P. Das
on seismic waveguide metamaterials, Mod. Phys. B 2012
& work of ETZ Zurich’s group, Ext. Mech. Lett 2015
45
Perspectives on seismic metamaterials
Seismic cloak in sedimentary soils
Shielding effects with a forest of trees
Institut Fresnel, La Recherche, 2012&2014
Work in progress on simulations and experiment
with Ménard company, the group of Philippe Roux at
ISTerre (Grenoble) and the group of Richard Craster
at Imperial College London
Objective:
Seismic cloak:
~100m in diameter
Frequencies:
1-15 Hz
Experiments led by the group of Philippe Roux
at ISTerre (Grenoble). Computations made
by Andrea Colombi at Imperial College London
46
Thanks for your attention!
Ongoing work in ERC Team ANAMORPHISM:
Dr. Younes Achaoui: numerics on seismic metamaterials
Dr. André Diatta: theory and numerics on elastic cloaking
Dr. Guillaume Dupont: experiments & numerics on water waves
Mr. Ronald Aznavourian: FDTD and morphing

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