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Three-dimensional Axisymmetric Cloak Based on the Cancellation of Acoustic
Scattering from a Sphere
L. Sanchis1, V. M. Garcıa-Chocano2, R. Llopis-Pontiveros1, A.
Climente2, J. Martınez-Pastor1, F. Cervera2, and J. Sanchez-Dehesa2∗1UMDO (Unidad Asociada al CSIC-IMM), Instituto de Ciencia de Materiales,
Universidad de Valencia, P.O. Box 22085, 46071 Valencia, Spain. and2Wave Phenomena Group, Universitat Politecnica de Valencia,Camino de Vera s.n. (Edificio 7F), ES-46022 Valencia, Spain.
(Dated: November 27, 2012)
This Letter presents the design, fabrication and experimental characterization of a directionalthree-dimensional acoustic cloak for airborne sound. The cloak consists of 60 concentric acousticallyrigid tori surrounding the cloaked object, a sphere of radius 4 cm. The major radii and positions ofthe tori along the symmetry axis are determined using the condition of complete cancellation of theacoustic field scattered from the sphere. They are obtained through an optimization technique thatcombines genetic algorithm and simulated annealing. The scattering cross section of the sphere withthe cloak, which is the magnitude that is minimized, is calculated using the method of fundamentalsolutions. The low-loss fabricated cloak shows a reduction of the 90% of the sphere scattering crosssection at the frequency of 8.55 kHz.
PACS numbers: 43.20.+g, 43.28.+h, 43.58.+z
Keywords: acoustic cloaking, scattering cancellation, genetic algorithm, simulated annealing
Acoustic cloaking is a phenomenon that renders ob-jects undetectable to sound waves [1, 2]. The physicalmechanism behind the cloaking is the guiding of soundwaves around the cloaked object thanks to the specificproperties of the fluid-like shell surrounding the object.Cloaking shells were initially proposed in two dimen-sions (2D) using acoustic metafluids with anisotropic in-ertia and scalar bulk modulus[2] or isotropic inertia andanisotropic modulus[3]. Afterwards, the proposal was ex-tended to three dimensions (3D) using metafluids in thespherical shells [4, 5]. Acoustic cloaking based on thescattering cancellation has been also proposed for an elas-tic sphere covered by a thin elastic isotropic shell[6]. Thisapproach follows a similar proposal in the electromag-netic (EM) domain where the scattering cancellation isachieved using plasmonic and metamaterials covers[7, 8].An hybrid method involving a transformation acousticconformal map with a scattering cancellation has beenalso suggested to reduce the scattering cross section ofobjects in aqueous environment[9]. Despite of the numer-ous works reporting proposals for making cloaks physi-cally realizable [10, 11], only the experiments by Zhanget al. [12] and by Popa et al. [13] claim for the demon-stration of acoustic cloaking. No practical realizations of3D cloaks have been reported so far though, in principle,they can be engineered following the same recipes.
The cloaking of mechanical waves is receiving increas-ing interest due to its potential application as vibrationisolators. The cloaking of bending waves theoreticallyproposed by Farhat et al.[14] was followed by its experi-mental demonstration by Stenger et al.[15].
In order to design devices that perform a prescribedoperation - like acoustic cloaking - there is a powerfulmethod called ”inverse design” which has been employed
z
x( c )
( a ) ( b )
( d )
( a )
y
z
z
FIG. 1: (a) Schematic representation of the 3D designedcloak. The cloak consists of 60 concentric tori surrounding thecentral spherical object. (b) Photograph of the cloak takenafter its fabrication. It is made in plastic with a central sphereof radius 4 cm. The total length of the sample (object+cloak)along the z−axis is 17 cm. (c)-(d) Photographs taken insidethe anechoic room.
to develop a broad variety of devices either in optics[16–19, 23] or in acoustics [20, 21]. In particular, pho-tonic cloaks based on inverse design have been reported[22, 23]. A directional acoustic cloak has been experi-mentally realized in 2D by Garcıa-Chocano et al. [21].2D Acoustic cloaks based on topology optimization havebeen also designed for airborne sound without experi-mental verification [24].This Letter reports the design, construction and ex-
perimental demonstration of a directional axisymmetricacoustic cloak in 3D. It is based on a specific distribution
2
of concentric tori with respect to the sound propagationdirection, which crosses the center of the cloaked object- an acoustically rigid sphere. The positions and majorradii of the tori surrounding the sphere are determinedsuch that they cancel the field scattered by the sphere.Those parameters are obtained using a optimization toolthat combines the genetic algorithm[25] and the simu-lated annealing[26]. The scattering cross section is thequantity that is minimized in the optimization procedure,and its value is obtained using the method of the funda-mental solutions (MFS)[27, 28]. This method offers a fastsolution of the scattering problem at each optimizationstep, providing the optimum structure after a reasonablecomputing time. It is demonstrated that the designedcloak cancels the field scattered by the bare sphere givinga perfect reconstruction of the incident sound wavefront.The cloaking performance of the designed cloak has beenproven with a prototype constructed with a 3D printer.The chosen radius of the sphere is R0 =4 cm and
the prefixed operational frequency of the cloak has beenselected with the condition λ = R0, which for air-borne sound corresponds to the linear frequency ν =cb/R0 =8.62 kHz, where cb =344.8 m/sec is the phasevelocity of air at normal room temperature (T =22o)and normal atmospheric pressure (P0 =0.751 m Hg) [29].A set of 60 acoustically rigid tori with equal transver-sal section with radii rt =0.267 cm has been employedas the cloaking shell. The dimensions of the sphere andthe common (minor) radius of tori were imposed by thefabrication limitations of the 3D printer.The scattering problem defined by the sphere and the
tori has been solved by the MFS, which is specially suit-able for axisymmetric scatterers [28]. In what follows wegive a brief account of its application to calculate thescattering cross section.Let us consider an incident plane wave that travels
along the z−axis, P inc(z;ω) = eikz , where k is thewavenumber. The angular frequency, ω =2π × ν, is im-plicitly assumed in the temporal dependence and willbe omitted in the rest of the article for simplification.Within the MFS the total scattered pressure can be ex-pressed in cylindrical coordinates as
P scatt(ρ, z) =
N∑
j=1
αjGaxis(ρ, z; ρ′
j, z′
j), (1)
where N is the total number of virtual ring-sources lo-cated at positions (ρ′j , z
′
j) with amplitudes αj , and Gaxis
is the free space axisymmetric Green’s function of thecorresponding Helmholtz equation [28]; i.e.,
Gaxis(ρ, z; ρ′
j, z′
j) = (4π)−1
∫ 2π
0
eikRj
Rj
dφ, (2)
where Rj is the distance between the j−source and thepoint (ρ, z) where the field is calculated [30]; i.e., R2
j =
ρ2 + ρ′2j + (z − z′j)2 − 2ρρ′jcosφ, with φ = θ′ − θ
N sample points (ρi, zi) are appropriately selected onthe surface of the scatterers where we impose the con-dition that they are acoustically rigid; i.e., the compo-nent of the velocity normal to the surface must be zero,∣
∣∂P total/∂n∣
∣
(ρi,zi)=0 with ~n indicating the normal to
the surface at the given sample point. In this manner wearrive to the following linear system of coupled equations
[
∂P inc
∂n
]
(ρi,zi)
+
N∑
j=1
αj
[
∂Gaxis
∂n
]
(ρi,zi)
= 0, i = 1, . . . , N
(3)where the coefficients αj are the unknowns.Once the multiple scattering problem is solved and the
α−coefficients are obtained, the scattering cross sectionby the sphere with the cloak, σc, is obtained using theoptical theorem [30],
σc(ω) =2π
kIm
N∑
j=1
αje−ikzj
, (4)
In order to get the distribution of tori that conceal thecentral sphere at the prefixed frequency ω0, we choose asthe fitness function
F = 1−σc(ω0)
σsph(ω0), (5)
where σsph(ω0) is the cross section of the bare sphere. Af-ter optimization σc(ω0) takes its minimum value. There-fore, F = 1 defines the condition of complete acousticconcealment.The 3D scattering algorithm described above is em-
ployed as the direct solver providing a fast and accuratecalculation of the fitness function (see Eq. 5) of bothstructures, the bare sphere and the sphere with the cloak.The optimization procedure starts with the tori homo-
geneously distributed along the z−axis with their planesperpendicular to it. The dimension and position of eachtorus along the axis are calculated making use of twooptimization methods, the genetic algorithm (GA) andsimulated annealing (SA). In brief, the binary-coded GAoperates on a population of candidate structures in or-der to produce the new ones with better performance inan iterative process inspired by Darwinian evolution[25].On the other hand, the SA is an optimization methodinspired by a process of slowly cooling a molten metalin order to obtain a uniform crystalline structure with astate of minimum energy[26].Figure 1(a) shows an artistic representation of the
sphere surrounded by the tori obtained after optimiza-tion. It is observed that the spatial distribution of thetori is not uniform along the z−axis and has a mirrorsymmetry plane passing through the origin of coordi-nates, z =0. The fitness function of this optimum struc-ture has a value F =0.978.
3
-10
0
10
x (c
m)
-2.0 -1.0 0 1.0 2.0
(a)
-10 0 10 20 30
-10
0
10
x (c
m)
z (cm)
(b)
FIG. 2: (a) Map of the total pressure (real part in arbitraryunits) obtained from the scattering of an incident plane wavewith frequency 8.62 kHz with a rigid sphere with radius 4cm. (b) Map corresponding to the sphere surrounded by thedesigned cloak. Both maps are calculated using a multiplescattering approach based on the method of fundamental so-lutions. The rectangle defined by the dashed lines representsthe area scanned by the experimental setup.
The calculated field maps for the sphere with and with-out the cloak are depicted in Fig. 2, where it is ob-served that the shadowing produced by the rigid sphereis almost completely restored when it is surrounded bythe set of tori. Let us remark that the cloaking per-formance described in Fig. 2(b) can be understood asa destructive interference effect between the waves scat-tered by the sphere and the cloak, respectively. For thecase of the sphere with the cloak it is seen a strongaccumulation of field pressure near the sphere surface.This effect represent a resonant mode that appears as aconsequence of the multiple scattering interference phe-nomenon. The robustness of the cloaking performanceas a function of various effects is described in the supple-mentary information[30]
To support experimentally the previous findings, asphere of radius 4 cm and the sphere with the cloak werefabricated using a commercial 3D printer. A photo ofthe sphere with the cloak after its fabrication is shown inFig. 1(b), where it is noticeable how the tori are linkedby thin beams in order to keep fixed their calculated po-
sitions. The material employed in the construction of theprototype is acoustically rigid in the range of frequenciesexplored.
The sphere with and without the cloak, respectively,were separately characterized in an anechoic chamberwith dimensions 8×6×3 m3. The incident sound is ex-cited using a column speaker aligned with the axis of thecloak (see Figs. 1(c)-(d)). Two linear stages are used tomove a B&K 4958 microphone that performs a 2D sweepinside the chamber. The robot scans an area behind thesamples of 0.20×0.20 m2 in the horizontal XZ−planewith a spatial resolution of 0.5 cm in each direction. Thisarea is represented in Fig. 2 by the rectangle defined withdashed lines. An area with the same dimensions and spa-tial resolution is scanned in the vertical Y Z−plane. Achirp consisting of a frequency sweep between 7.5 and9.5 kHz has been used as an excitation signal. It is emit-ted at each spatial position of the moving robot until thepressure is acquired by the microphone. The received sig-nal is processed through a fast Fourier transform whichprovides its amplitude and phase. In order to reducethe presence of random noise several emissions are per-formed and the acquired responses are averaged. Tem-perature and humidity sensors have been incorporated insuch a manner that these parameters, which determinethe sound velocity, are continuously recorded by the con-trol software.
A series of measurements around the prefixed opera-tional frequency were performed in order to quantify thescattering reduction of the sphere with the cloak. Figure3 shows the pressure maps measured at 8.55 kHz, whichis the resulting frequency at which the averaged visibil-ity of the object with cloak takes the minimum value.This optimum frequency is slightly different to 8.62 kHz,which was the frequency prefixed for cancellation. Thesmall discrepancy comes from the fact that the soundspeed inside the anechoic chamber is 344.17 m/sec, cor-responding to an average temperature of 21.63oC withhumidity of 54.7% [29].
Figure 3 shows the real part of the total pressure mea-sured in the area of exploration belonging to the horizon-tal plane (left panels) and vertical plane (right panels).The spectra taken in free space show some wiggling inthe wavefronts that was originated from unwanted re-flections in the free-echo chamber. These reflections aredue to the high working frequency of the designed cloaktogether with the presence of the components of the ex-perimental setup; some of them have sizes of the order ofthe wavelength employed in the experiments (a few cms).The spectra of the bare sphere in Fig. 3(b) clearly showthe shadowing by the diffracted sound. This shadow-ing is substantially smaller than that observed for a 2Dcylinder[21]. Finally, the spectra with the cloak (see Fig.3(c)) show that the plane wavefronts are reconstructedbut with the limitations already observed in the free-space spectra. It is important to remark that all the
4
-10
-5
0
5
10
x (c
m)
-10
-5
0
5
10
y (c
m)
-10
-5
0
5
10
x (c
m)
-10
-5
0
5
10 y
(cm
)
15 20 25 30 35-10
-5
0
5
10
x (c
m)
z (cm)15 20 25 30 35
-10
-5
0
5
10( c ) Sphere + Cloak
( b ) Bare sphere
y (c
m)
z (cm)
( a ) Free space
FIG. 3: (Color online) Pressure maps measured at 8.55 kHz,the operational frequency of the fabricated cloak. Maps onthe left (right) represent the real part of the total pressure onthe horizontal (vertical) plane.
maps are represented with the same color scale, whichmeans that no appreciable losses can be associated tothe performance of the cloak.The averaged visibility of an object, γ, is a parameter
employed to characterize the cloaking performance. It isa frequency dependent function that is defined as[12]:
γ(ω) =1
N
∑
j
|Pmax,j | − |Pmin,j |
|Pmax,j |+ |Pmin,j |, (6)
where Pmax,j and Pmin,j are the maximum and minimumpeak values along a given wavefront j.The visibility can be compared to the traditional mea-
surement of scattering cross-section, but performed forthe convenience of the field measurement available in ourexperimental setup. The upper panels in Fig. 4 plot theaverage visibility of the cloaked sphere, γc, over all thewavefronts on the transmitted side and compared withthat of the bare sphere, γsph. The results obtained forboth the horizontal and vertical planes are depicted. Thevisibility for the free space, γ0, when there is neithersphere nor cloak is also plot and is employed as a refer-ence in our setup. The measured visibilities (symbols)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Ave
rage
d vi
sibi
lity
()
Horizontal XZ-plane
VerticalYZ-plane
8.2 8.4 8.6 8.8 9.00
1
2
3
(c -
0) / (
sph -
0)Frequency (kHz)
c /
sph
8.2 8.4 8.6 8.8 9.0
Frequency (kHz)
FIG. 4: (Color online) Upper panels: Plot of the averagedvisibility (γ) as a function of the frequency. The experimentalresults with and without the cloak are marked by red circlesand blue squares, respectively. The reference visibility (γ0)when there is no object is marked by black triangles. Thelines represent the simulations. Lower panels: The purplediamonds plot the ratio between the averaged visibility of thesphere with the cloak (γc) and without cloak (γsph. Both areshifted by the reference visibility. The black lines representthe ratio between the scattering cross sections of the spherewith and without cloak, σc/σsph, obtained from simulations.Full (empty) symbols define data on the horizontal (vertical)plane.
are compared with those (lines) derived from simulationsusing the method of fundamental solutions. Note thatthe reference visibility (black triangles) that is the visi-bility for a propagating plane wave in free space, whichshould be zero, is spoiled by the presence of unwantedscattering events inside the chamber. This effect is worsein the vertical plane since we have to add a vertical armto attach the stage that moves the microphone along theY direction. However, it is observed that the data followsfairly well the trend of the numerical simulations. At 8.55kHz we can read the visibility of 0.25 for the bare spherewhereas the visibility of the cloaked sphere is reduced to0.10, a value very close to 0.07 that is the one measuredin free space. This result confirms the strong reductionof the shadowing observed in the pressure map plot inFig. 3(c).
Lower panels in Fig. 4 shows a comparison between the
5
experimentally determined ratio γc/γsph (symbols) andthe ratio between cross-sections σc/σsph (black line). Thelast one is obtained from numerical simulations employ-ing the method of fundamental solutions [30]. The visi-bilities are taken with respect to the reference visibilityfor a fair comparison. It is observed that the ratio γc/γspheffectively gives a good representation of σ/σsph for thefrequencies near the one employed as the target in thecancellation procedure. For smaller or higher frequen-cies the ratio γc/γsph does not necessarily account forthe strong perturbations produced by the backscatteredfield. These panels demonstrate the 3D performance ofour cloak. A 90% of maximum reduction of the spherecross section is experimentally obtained at 8.55 kHz. Thecloak bandwidth is 120 Hz. This bandwidth has been es-tablished using the convention[12] that, for any frequencyenclosed by the bandwidth, a scattering reduction largerthan a 70% should be measured[30].In summary, we have demonstrated a low loss 3D cloak
that can reduce the visibility of the hidden object for air-borne sound waves. It has been obtained from the con-dition of the total cancellation of the waves scattered bya sphere at a prefixed frequency. This result can be con-sidered as the acoustic analogue of the recently proposedcloaking structures based on scattering cancellation ofEM waves[7, 8, 31]. The fabricated cloak consists of 60acoustically rigid tori that at 8.55 kHz suppresses up toa 90% of the waves scattered by the sphere. Though thecloak is directional and narrow banded, there are severalpossibilities for its improvement using additional parame-ters. For example, a broadband operation could be madeby imposing the cancellation over several frequencies[24].Designing an omnidirectional cloak could be also possibleby performing a multi-objective optimization[18] and us-ing set of tori orientated along different axis. Moreover,cloaks for underwater operation could be also feasible byincluding the elasticity of the tori employed in their de-sign.This work is partially supported by the Spanish Min-
isterio de Economıa y Competitividad under ContractsTEC2010-19751, TEC2011-29120-C05-01, and CSD2008-00066 (CONSOLIDER Program), and by the U.S. Officeof Naval Research. The authors acknowledge the ”Cen-tro de Tecnologıas Fısicas” at the UPV for technical helpduring data acquisition. We also acknowledge the com-puting facilities provided by the Universidad de Valencia.
∗ Electronic address: [email protected][1] G. W. Milton, M. Briane and J. R. Willis, New J. Phys.
8, 248 (2006).[2] S. A. Cummer and D. Schurig, New J. Phys. 9, 45 (2007).[3] A. N. Norris, J. Acoust. Soc. Am. 125, 839 (2009).
[4] H.Y. Chen and C.T. Chan, Appl. Phys. Lett. 91, 183518(2007)
[5] S. A. Cummer, B. I. Popa, D. Schurig, D. Smith, J.Pendry, M. Rahm, and A. Starr, Phys. Rev. Lett. 100,024301 (2008).
[6] M. D. Guild, A. Alu, and M. R. Habermann, J. Acoust.Soc. Am. 129, 1355 (2011).
[7] B. Edwards, A. Alu, M. G. Silveirinha, and N. Engheta,Phys. Rev. Lett. 103, 153901 (2009).
[8] D. Rainwater, A. Kerkhoff, K. Melin, J.C. Soric, G.Moreno, and A. Alu, New J. Phys. 14, 013054 (2012).
[9] T. P. Martin and G. J. Orris, Appl. Phys. Lett. 100,033056 (2012).
[10] D. Torrent and J. Sanchez-Dehesa, New J. Phys. 10,063015 (2008).
[11] Y. Cheng, F. Yang, J.Y. Xu, X.J. Liu, Appl. Phys. Lett.92 151913 (2008).
[12] S. Zhang, C. Xia, and N. Fang, Phys. Rev. Lett. 106,024301 (2011).
[13] B.-I. Popa, L. Zigoneanu, and S. A. Cummer, Phys. Rev.Lett. 106, 0253901 (2011).
[14] M. Farhat, S. Guenneau, and S. Enoch, Phys. Rev. Lett.103 024301 (2009).
[15] N. Stenger, M. Wilhelm, and M. Wegener, Phys. Rev.Lett. 108 014301 (2012).
[16] L. Sanchis, A. Hakansson, D. Lopez-Zanon, J. Bravo-Abad, and J. Sanchez-Dehesa, Appl. Phys. Lett. 84, 4460(2004).
[17] S. Preble, M. Lipson and H. Lipson, Appl. Phys. Lett.86, 061111 (2005).
[18] A. Hakansson, H. T. Miyazaki and J. Sanchez-Dehesa,Phys. Rev. Lett. 96, 153902 (2006).
[19] L. Sanchis, M.J. Cryan, J. Pozo, I. J. Craddock, and J.G.Rarity, Phys. Rev. B, 76 045118 (2007).
[20] A. Hakansson, F. Cervera and J. Sanchez-Dehesa, Appl.Phys. Lett. 86, 054102 (2005).
[21] V. M. Garcıa-Chocano, L. Sanchis, A. Dıaz-Rubio, J.Martınez-Pastor, F. Cervera, R. Llopis-Pontiveros, andJ. Sanchez-Dehesa, Appl. Phys. Lett. 99, 074102 (2011).
[22] A. Hakansson, Opt. Express 15, 4328 (2007).[23] J. Andkajaer and O. Sigmund, Appl. Phys. Lett. 98,
021112 (2011).[24] J. Andkajaer and O. Sigmund, In Metamaterials’2011:
The Fifth International Congress on Advanced Electrom-ganetic Materials in MIcrowave and Optics, Barcelona,Spain, October 10-13.
[25] D.E. Goldberg, Genetic Algorithms in Search, Optimiza-tion and Learning (Addison Wesley, Reading, MA, 1989).
[26] S. Kirkpatrick, C. D. Gelatt, Jr., M.P. Vecchi, Science,220 4598, (1983).
[27] A. F. Seybert, B. Soenarko, F. J. Rizzo, and D. J. Shippy,J. Acoust. Soc. Am. 80, 1241 (1986).
[28] A. Karageorghis and G. Fairweather, J. Acoust. Soc. Am.104, 3212 (1998).
[29] Leo L. Beranek, Acoustics (Acoustical Soc. of America,New York, 1996) p. 10.
[30] See Supplementary Material for a thorough discussion onthe mathematical procedure.
[31] G. W. Milton, and N.-A. P. Nicorivici, Proc. R. Soc. A.462, 3027 (2006).
1
Supplementary Information L. Sanchis, V. M. García-Chocano, R. Llopis-Pontíveros, A. Climente, J. Martínez-Pastor, F. Cervera and J. Sánchez-Dehesa Three-Dimensional Axisymmetric Cloak Based on the Cancellation of Acoustic Scattering from a Sphere Movies A direct visualization of the cloaking performance, like the field snapshots shown in Fig. 2, is presented as an animation. The sequential advancement of the phase fronts gives an account of the propagation of the measured sound wave in air with and without the cloak. Movie S1: Calculated pressure field pattern as in Fig. 2 (a) of the bare rigid sphere with an incident plane wave with frequency 8.62 kHz Movie S2: Calculated pressure field pattern as in Fig. 2 (b) of the sphere with the cloak an incident plane wave with frequency 8.62 kHz Multiple scattering approach: the Method of Fundamental Solutions
In order to obtain the pressure field at any arbitrary point (x, y, z) in the 3D space we have employed the Method of Fundamental Solutions (MFS) [1], which is especially suitable for axisymmetric scatters [2]. This method is based on a proper distribution of parallel ring sources with a common symmetry axis (the Z-axis) in the exterior domain in such a way that the boundary conditions are met. These boundaries are defined by the system formed by a central sphere and M surrounding tori with symmetry axis Z and specular symmetry with respect to the plane z=0 (see Figure S1 where M=60).
Let us consider an incident plane wave propagating along the z-axis, ikzinc eP = . This impinging wave over the set of tori produces a total scattered pressure, which in cylindrical coordinates is given by
( ) ( ) , ,;,,1
∑=
′′=N
jjjaxisj
scatt zzGzP ρραρ (S1)
where ( )2
1
0
( , , , ) 4jikR
axis j jj
eG z z dR
π
ρ ρ π φ−′ ′ ′= ∫ is the free space axisymmetric Green´s
function for the Helmholtz equation. Polar angles, θ and θ’, are omitted because of the symmetry, k=2π/λ, N is the total number of ring-sources located at points ( , )j jzρ′ ′ with amplitudes αj, and the distance between source and sample points is
( ) ( )φρρρρ ′′−′−+′+= cos2222jjjj zzR , where θθφ −′=′ .
2
Figure S1.- Positions of source points (red circles) and sample points (hollow circles) in the method of fundamental solutions. The sample points are always on the surface of the sphere and the tori, respectively. By choosing N sample points ( )ii z,ρ on the surface of the scatters and imposing Neumann’s boundary conditions to the total field, scattincscatt PPP += , on the surface; that is
( ) 0,
,
=
∂
∂
ii z
total
nP
ρ (S2)
where n is the normal to the surface at the selected point. We obtain a linear system of N equations for the unknowns αj.
( ) ( )
N,in
Gn
P
iiii z
N
j
axisj
z
inc
,1 0,1,
==
∂+
∂
∂ ∑= ρρ
α (S3)
For each torus m=1,…,M centered at ( , )m mzρ with minor radius Rt we select Nt field-points and Nt source-points. The coordinates of the field-points are
3
( , ) ( ( ), ( ))mn mn m t n m t nz R Cos z R Sinρ ρ ϑ ϑ= + + , and the ones for source points are ( , ) ( ( ), ( ))mn mn m t t n m t t nz R Cos z R Sinρ ρ η ϑ η ϑ′ ′ = + + with 1 0,..., 2
tNϑ ϑ π= = and ηt being a real parameter such that 0 1tη< < . On the other hand, the coordinates of the Ns field and source points for the sphere (with radius Rs) are, respectively, ( , ) ( ( ), ( ))n n s n s nz R Cos R Sinρ ϑ ϑ= and ( , ) ( ( ), ( ))n n s s n s s nz R Cos R Sinρ η ϑ η ϑ′ ′ = with
1 / 2,..., / 2sNϑ π ϑ π= − = and 0 1sη< < . The parameters ηt and ηs are chosen by
minimizing the residual error function [1]:
( ) ∑ ∑= =
∂
∂+
∂∂
=N
i z
N
j
axisj
inc
st
iin
Gn
PN
E1 ),(1
1,ρ
αηη (S4)
The following relation holds: N= M × Nt + Ns. The values employed for M, Nt and Ns were 60, 15 and 40, respectively. Figure S1 shows the distribution of sample and source points in the geometry of the whole system. Note that all the source points are out of the domain space (inside the tori and sphere). The integrals of the derivative of the Green function are evaluated numerically using the trapezoidal adaptive rule with relative tolerance 0.001. The values obtained from minimization were ηt=0.726 and ηs=0.8, which are kept constant in the rest of the process. Fitness parameter: the Scattering Cross Section
The quantity representing concealing is the scattering cross section σ for the
whole system formed by the object with the cloak. The cancellation of σ defines the total concealment of the object by the cloak. It is well known that the scattered field at large distances can be represented with a spherical wave from a point source modeled by the scattered amplitude function f (θ, φ) where θ and φ are the spherical coordinates of the observation point and r the distance to the origin: r
efrPikrscatt ),(),,( ϕθθϕ = .
For the case of axisymmetric bodies where the incoming wave is aligned with the symmetry axis, which we chose as the z-axis: r
efrPikrscatt )(),( θθ = .
If we locate the observation point on the z-axis far from the origin (θ=0º, ρ=0, z>>λ) we can obtain ikzscatto zePf =)0( , and σ can be calculated using the optical
theorem, ( ) [ ])0(Im4 ofkπσ = . Therefore,
( ) ( ) [ ]ikzscatt zePk Im4πωσ = , (S5)
where Pscatt is the scattered field generated by the N tori located at positions ( , )j jzρ′ ′ along the z-axis; i.e.,
( ) ( ) ∑ ∫∑==
′=′′=
N
j j
ikR
j
N
jjjaxisj
scatt dR
ezzGzPj
1
2
01 4
1,;,0,0π
φαπ
ρα , (S6)
4
where the distances 2 2( )j j jR z zρ′ ′= + − do not depend on φ′ , the polar angle in the plane perpendicular to the z-axis a passing through the point z´j
We employed the expression for the scattered field into (S5) and assume that with z → +∞, Rj → z for amplitudes and Rj → z – zj for the phases. After, these simplifications, the expression for the scattering cross section can be cast as:
1
2 Im jN
ikzj
je
kπσ α −
=
=
∑ (S7)
The minimization of the scattering cross section is the condition employed in the optimization algorithm to obtain the positions of the tori along the z-axis and their major radius. Consequently, the fitness function is
( )( )0
01ωσ
ωσ
sph
cF −= , (S8)
with σsph (ω0) being the cross section of the bare sphere calculated at the prefixed operational frequency of the cloak (ω0=2π×8.62 kHz). Therefore a value F=1 means complete concealment of the sphere by the cloak. The analytical expression obtained for σ using the MFS is an extremely fast algorithm that solves the scattering problem. It has been embedded in the optimization procedure to obtain the optimum structure in a controllable time. Figure S2(a) shows the ratio σ/σsphere calculated as a function of the frequency for the cloak structure resulting from the minimization. Note that at the prefixed frequency 8.62 kHz the scattering cross section of the sphere is practically cancelled due to the cloak.
Fig. S2.- The ratio σ(ω)/σsphere(ω) calculated by the method of fundamental solutions. (a) As a function of the frequency. (b) The ratio as a function on the incident angle at the frequency 8.62 kHz. The horizontal dashed line is a guide for the eye. Figure S2b represents the robustness of the cloak at the frequency where σc/σsph has a minimum value. This frequency is 8.62 kHz and defines the working frequency of the cloak. As a convention we consider that an increase of a 30% of this ratio defines the bandwidth of the cloak.
5
Optimization procedure: Results and discussion
The optimization tool integrates a fast direct solver, which has been described above, and a combination of two optimization methods: The Genetic Algorithm [3] (GA) and the Simulated Annealing [4] (SA) methods. In brief, GAs and SA are adaptive heuristic search algorithms inspired by the Darwinian evolutionary theory of natural selection and the process of cooling melted solids in order to obtain a uniform crystalline structure with an optimum energy state. They operate on a population of candidate structures to obtain a solution that maximizes the fitness function, which constitutes a metric of the performance of each candidate solution. In this work we join these two concepts to obtain 3D axisymmetric cloaking formed by an arrangement of rigid concentric tori surrounding the cloaked object.
Each displacement of the 30 algorithm-guided tori out of the 60 that form the cloaking shell (because of symmetry with respect to the z=0 plane), which initially are homogeneously distributed in the annular region R2 <ρ0
2+z2 < (2R)2, is described by:
0
0 z
Tjz z Tj
ρρ ρ= +
= + (S9)
where (ρ0, z0) are the coordinates at equilibrium while (jρ, jz) are integers taking values from –7 to 7, and T is a control parameter called temperature.
In order to avoid inconsistencies, the optimization algorithm automatically assigns a null value to the fitness of the individuals that have one o more tori overlapped. For a fixed temperature, we obtain (152)30=3.67×1070 possibilities that are coded with binary strings of 240 elements. Such a huge size of the search space is prohibitive even for efficient stochastic optimization routines like GAs. In order to work out the problem we have performed a sequence of short GA optimizations called annealing steps where the initial state, with certain probability, is the best of the precedent GA optimization state while gradually has been decreased the T parameter. If the difference between correlative GA optimized fitness values is ∆E, the probability to a change of state is proportional to e∆E/T. The behavior differentiating this algorithm from the hill-climbing algorithms is the fact that not all the times the best configurations are accepted as the new GA initial states. This annealing step process is repeated until the temperature takes a null value. In this way, the algorithm here utilized is a combination of GA and SA optimization methods capable of dealing with such huge space search. (Also, with the GA-SA algorithm the scatters can migrate freely and they are not constrained to move around fixed point as would happen if only one GA run were used). For the GA parameters we have considered a population of 70 individuals, a mutation probability of 0.01 and a maximum number of generations of 60. Regarding the SA parameters an initial temperature of 0.0009 and 100 annealing steps have been considered.
6
Fig. S3.- Convergence of the optimization procedure with the number of function calls.
Figure S4 shows the pressure maps and the corresponding distribution of tori for three different propagating wavelengths; (a) λ=R/2, (b) λ=R, and (c) λ=2R. It is observed that, for a fixed number of 60 tori, the optimization procedure get a better convergence as smaller is the propagating wavelength. For larger wavelengths the optimization aggregates the tori in clusters to enhance the scattering by the incident wave. In order to improve the fitness we should employ large number of cylinders and/or increase their radii.
Figure S4.- pressure maps and cloaks resulting from three different impinging wavelengths: (a) λ=R/2, (b) λ=R, and (c) λ=2R. The black lines define the level zero, which is used as guide for the eye. As a final observation, we should point out that the performance of the cloak rapidly degrades when the sphere becomes an ellipsoidal body. The position of the tori must be optimized to restore the fitness function.
7
Parallel Computing Despite the efficient multiple scattering algorithm that has embedded simplifications associated to the symmetry of the problem, a powerful parallel computer has been required to carry out the computations in a reasonable time frame. For this, we have used server Altix Ultraviolet 1000 with 174 cores and 2TB of RAM. On this system we have used 36 Nehalem-EX cores at 2,66GHz to execute our FORTRAN code with OpenMP extensions for parallelism. Solving the multiple scattering problem of the sphere with a given configuration of the tori takes 33 seconds in one core. Figure S3 shows that the structure with the optimum fitness parameter was obtained after 420000 calls to the direct solver. The CPU time required to run the complete Fig. S3 using the 36 cores were about 5 days. References: [1] G. Fairweather, F. J. Rizzo, D. J. Shippy, The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math., 9, 69 (1998). [2] A. F. Seybert, B. Soenarko, F.J. Rizzo, and D.J. Shipp, “A special integral equation formulation for acoustic radiation and scattering for axisymmetric bodies and boundary conditions”. J. Acoust. Soc. Am., 80, 1241 (1986) [3] D.E. Goldberg, Genetic Algorithms in Search, Optimization and Learning (Addison Wesley, Reading, MA, 1989). [4] S. Kirkpatrick, C. D. Gelatt, Jr., M.P. Vecchi, “Optimization by simulate annealing”. Science 220, 4598 (1983).
