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Three-Dimensional Axisymmetric Cloak Based on the Cancellationof Acoustic Scattering from a Sphere

L. Sanchis,1 V. M. Garca-Chocano,2 R. Llopis-Pontiveros,1 A. Climente,2

J. Martnez-Pastor,1 F. Cervera,2 and J. Sanchez-Dehesa2,*1

UMDO (Unidad Asociada al CSIC-IMM), Instituto de Ciencia de Materiales,

Universidad de Valencia, P.O. Box 22085, 46071 Valencia, Spain

2Wave Phenomena Group, Universitat Politecnica de Valencia, Camino de Vera s.n. (Edificio 7F), ES-46022 Valencia, Spain(Received 5 August 2012; revised manuscript received 27 November 2012; published 20 March 2013)

This Letter presents the design, fabrication, and experimental characterization of a directional three-

dimensional acoustic cloak for airborne sound. The cloak consists of 60 concentric acoustically rigid tori

surrounding the cloaked object, a sphere of radius 4 cm. The major radii and positions of the tori along the

symmetry axis are determined using the condition of complete cancellation of the acoustic field scattered

from the sphere. They are obtained through an optimization technique that combines genetic algorithm

and simulated annealing. The scattering cross section of the sphere with the cloak, which is the magnitude

that is minimized, is calculated using the method of fundamental solutions. The low-loss fabricated cloak

shows a reduction of the 90% of the sphere scattering cross section at the frequency of 8.55 kHz.

DOI: 10.1103/PhysRevLett.110.124301 PACS numbers: 43.20.+g, 43.28.+h, 43.58.+z

Acoustic cloaking is a phenomenon that renders objectsundetectable to sound waves [1,2]. Cloaking shells wereinitially proposed in two dimensions (2D) using acousticmetafluids with anisotropic inertia and scalar bulk mod-ulus [2] or isotropic inertia and anisotropic modulus [3].Afterwards, the proposal was extended to three dimensions(3D) using metafluids in spherical shells [4,5]. Acousticcloaking based on the scattering cancellation has also beenproposed for an elastic sphere covered by a thin elasticisotropic shell [6]. A hybrid method involving a trans-formation acoustic conformal map with a scattering can-cellation has also been suggested to reduce the scattering

cross section of objects in an aqueous environment [7].Despite the numerous works reporting proposals for mak-ing cloaks physically realizable [8,9], only the experimentsby Zhang et al. [10] and Popa et al. [11] claim the dem-onstration 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 also receivingincreasing interest due to its potential application as vibra-tion isolators. For example, the cloaking of bending wavestheoretically proposed by Farhat et al. [12] was followedby its experimental demonstration by Stenger et al. [13].

In order to design devices that perform a prescribedoperationlike acoustic cloakingthere is a powerfulmethod called inverse design which has been employedto develop a broad variety of devices in optics [1418] andin acoustics [19,20]. In particular, photonic cloaks basedon inverse design have been reported [18,21]. A directionalacoustic cloak has been experimentally realized in 2D byGarca-Chocano et al. [20].

This Letter reports the design, construction, and exp-erimental demonstration of a directional axisymmetric

acoustic cloak in 3D. It is based on a specific distributionof concentric tori with respect to the sound propagationdirection, which crosses the center of the cloaked objectan 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 an optimization toolthat combines the genetic algorithm [22] and the simulatedannealing [23]. The scattering cross section is the quantitythat is minimized in the optimization procedure, and itsvalue is obtained using the method of the fundamentalsolutions (MFS) [24,25]. It is demonstrated that the

designed cloak cancels the field scattered by the baresphere giving a perfect reconstruction of the incident soundwave front. The cloaking performance of the designedcloak has been proven with a prototype constructed witha 3D printer.

The chosen radius of the sphere is R0 4 cm and theprefixed operational frequency of the cloak has been selectedwith the condition R0, which for airborne sound corre-sponds to the linear frequency cb=R0 8:62 kHz,where cb 344:8 m=sec is the phase velocity of air atnormal room temperature (T 22) and normal atmos-pheric pressure (P0 0:751 m Hg) [26]. A set of 60 acous-tically rigid tori with equal transversal section with radiirt 0:267 cm has been employed as the cloaking shell.

The scattering problem defined by the sphere and the torihas been solved by the MFS, which is specially suitablefor axisymmetric scatterers [25]. In what follows we give abrief account of its application to calculate the scatteringcross section.

Let us consider an incident plane wave that travels alongthe z axis, Pincz; ! eikz, where k is the wave number.The angular frequency, ! 2 , is implicitly assumed

PRL 110, 124301 (2013) P H Y S I C A L R E V I E W L E T T E R Sweek ending

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in the temporal dependence and will be omitted in therest of the Letter for simplification. Within the MFS thetotal scattered pressure can be expressed in cylindricalcoordinates as

Pscatt; z XN

j1

jGaxis; z; 0

j; z0

j; (1)

where N is the total number of virtual ring sources locatedat positions (0j, z0

j) with amplitudes j, and Gaxis is the free

space axisymmetric Greens function of the correspondingHelmholtz equation [25]; i.e.,

Gaxis; z; 0

j; z0

j 41Z2

0

eikRj

Rjd; (2)

where Rj is the distance between thej source and the point

(, z) where the field is calculated [27]; i.e., R2j 2

02j z z0

j2 20j cos, with

0 .

Nsample points (i, zi) are appropriately selected on thesurface of the scatterers where we impose the condition

that they are acoustically rigid; i.e., the component ofthe velocity normal to the surface must be zero,j@Ptotal=@nji;zi 0, with ~n indicating the normal to the

surface at the given sample point. In this manner we arriveat the following linear system of coupled equations:

@Pinc

@n

i;zi

XN

j1

j

@Gaxis

@n

i;zi

0; i 1;. . .;N; (3)

where the coefficients j are the unknowns.

Once the multiple scattering problem is solved and the coefficients are obtained, the scattering cross section bythe sphere with the cloak c is obtained using the optical

theorem [27],

c! 2

kIm

XNj1

jeikzj

: (4)

In order to get the distribution of tori that conceal thecentral sphere at the prefixed frequency !0, we choose asthe fitness function

F 1 c!0

sph!0; (5)

where sph!0 is the cross section of the bare sphere.

After optimization, c

!0

takes its minimum value.Therefore, F 1 defines the condition of complete acous-tic concealment.

The 3D scattering algorithm described above isemployed 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 each

torus along the axis are calculated making use of twooptimization methods, the genetic algorithm and simulatedannealing. In brief, the binary-coded genetic algorithmoperates on a population of candidate structures in orderto produce the new ones with better performance in aniterative process inspired by Darwinian evolution [22]. Onthe other hand, the simulated annealing is an optimizationmethod inspired by a process of slowly cooling a molten

metal in order to obtain a uniform crystalline structure witha state of minimum energy [23].

Figure 1(a) shows an artistic representation of the spheresurrounded by the tori obtained after optimization. It isobserved that the spatial distribution of the tori is notuniform along the z axis and has a mirror symmetry planepassing through the origin of coordinates, z 0. The fitnessfunction of this optimum structure has a value F 0:978.

The calculated field maps for the sphere with and with-out the cloak are depicted in Fig. 2, where it is observedthat the shadowing produced by the rigid sphere is almostcompletely restored when it is surrounded by the set of tori.Let us remark that the cloaking performance described in

Fig. 2(b) can be understood as a destructive interferenceeffect between the waves scattered by the sphere and thecloak, respectively. For the case of the spherewith the cloak,a strong accumulation of field pressure near the spheresurface is seen. This effect represents a resonant modethat appears as a consequence of the multiple scatteringinterference phenomenon. The robustness of the cloaking

FIG. 1 (color online). (a) Schematic representation of the3D designed cloak. The cloak consists of 60 concentric torisurrounding the central spherical object. (b) Photograph of thecloak taken after its fabrication. It is made in plastic with acentral sphere of radius 4 cm. The total length of the sample(object cloak) along the z axis is 17 cm. (c),(d) Photographstaken inside the anechoic room.


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performance as a function of various effects is described in

the Supplemental Material [27].

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 of the

sphere with the cloak after its fabrication is shown inFig. 1(b), where it is noticeable how the tori are linked

by thin beams in order to keep fixed their calculated

positions. The material employed in the construction ofthe prototype is acoustically rigid in the range of frequen-

cies explored.The sphere with and without the cloak, respectively, was

separately characterized in an anechoic chamber withdimensions 8 6 3 m3. The incident sound is excitedusing a column speaker aligned with the axis of the cloak

[see Figs. 1(c) and 1(d)]. Two linear stages are used tomove a B&K 4958 microphone that performs a 2D sweep

inside the chamber. The robot scans an area behind the

samples of20 20 cm2 in the horizontal XZplane with aspatial resolution of 0.5 cm in each direction. This area isrepresented in Fig. 2 by the rectangle defined with dashed

lines. An area with the same dimensions and spatial reso-

lution is scanned in the vertical YZ plane. A chirp consist-ing of a frequency sweep between 7.5 and 9.5 kHz has been

used as an excitation signal. It is emitted at each spatialposition of the moving robot until the pressure is acquiredby the microphone. The received signal is processedthrough a fast Fourier transform which provides its ampli-tude and phase. In order to reduce the presence of randomnoise, several emissions are performed and the acquiredresponses are averaged.

A series of measurements around the prefixed opera-

tional frequency were performed in order to quantify thescattering reduction of the sphere with the cloak. Figure 3shows the pressure maps measured at 8.55 kHz, whichis the resulting frequency at which the averaged visibilityof the object with cloak takes the minimum value. Thisoptimum frequency is slightly different to 8.62 kHz, whichwas the frequency prefixed for cancellation. The smalldiscrepancy comes from the fact that the sound speedinside the anechoic chamber is 344:17 m=sec , correspon-ding to an average temperature of 21:63 C with humidityof 54.7% [26].

Figure 3 shows the real part of the total pressure mea-sured in the area of exploration belonging to the horizontal

plane (left-hand panels) and vertical plane (right-handpanels). The spectra taken in free space show some wig-gling in the wave fronts that was originated from unwanted

FIG. 3 (color online). Pressure maps measured at 8.55 kHz, theoperational frequency of the fabricated cloak. Maps on the left(right) represent the real part of the total pressure on thehorizontal (vertical) plane. (a) Free space, (b) bare sphere,(c) sphere cloak.

FIG. 2 (color online). (a) Map of the total pressure (real part inarbitrary units) obtained from the scattering of an incident planewave with frequency 8.62 kHz with a rigid sphere with radius4 cm. (b) Map corresponding to the sphere surrounded by thedesigned cloak. Both maps are calculated using a multiple

scattering approach based on the method of fundamental solu-tions. The rectangle defined by the dashed lines represents thearea scanned by the experimental setup.


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reflections in the free-echo chamber. These reflections aredue to the high working frequency of the designed cloaktogether with the presence of the components of theexperimental setup; some of them have sizes of the orderof the wavelength employed in the experiments (a few cm).The spectra of the bare sphere in Fig. 3(b) clearly show theshadowing by the diffracted sound. This shadowing issubstantially smaller than that observed for a 2D cylinder

[20]. Finally, the spectra with the cloak [see Fig. 3(c)] showthat the plane wave fronts are reconstructed but with thelimitations already observed in the free space spectra. It isimportant to remark that all the maps are represented withthe same color scale, which means that no appreciablelosses can be associated with the performance of the cloak.

The averaged visibility of an object is a parameteremployed to characterize the cloaking performance. It is afrequency dependent function that is defined as [10]

! 1

N

Xj

jPmax;jj jPmin;jj

jPmax;jj jPmin;jj; (6)

where Pmax;j and Pmin;j are the maximum and minimumpeak values along a given wave front j, and N is the totalnumber of wave fronts explored (see Fig. 3).

The averaged visibility can be compared to the traditionalmeasurement 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 c over allthe wave fronts on the transmitted side and compared withthat of the bare sphere sph. The results obtained for both the

horizontal and the vertical planes are depicted. The visibilityfor the free space 0 when there is neither sphere nor cloakis also plotted and is employed as a reference in our setup.

The measured visibilities (symbols) are compared with those(lines) derived from simulations using the method of funda-mental solutions. Note that the reference visibility (blacktriangles) that is the visibility for a propagating plane wavein free space, which should be zero, is spoiled by the pres-ence of unwanted scattering events inside the chamber.However, it is observed that the data follow fairly well thetrend of the numerical simulations. At 8.55 kHz we canread the visibility of 0.25 for the bare sphere whereas thevisibility of the cloaked sphere is reduced to 0.10, a valuevery close to 0.07, which is the one measured in free space.This result confirms the strong reduction of the shadowingobserved in the pressure map plot in Fig. 3(c).

The lower panels in Fig. 4 show a comparison betweenthe experimentally determined ratio c=sph (symbols)

and the ratio between cross sections c=sph (black line).

The last one is obtained from numerical simulationsemploying the method of fundamental solutions [27].The visibilities are taken with respect to the referencevisibility for a fair comparison. It is observed that the ratioc=sph effectively gives a good representation of =sphfor the frequencies near the one employed as the target in

the cancellation procedure. For smaller or higher frequen-cies the ratio c=sph does not necessarily account for the

strong perturbations produced by the backscattered field.These panels demonstrate the 3D performance of ourcloak. A 90% of maximum reduction of the sphere crosssection is experimentally obtained at 8.55 kHz. The cloakbandwidth is 120 Hz. This bandwidth has been establishedusing the convention [10] that, for any frequency enclosed

by the bandwidth, a scattering reduction larger than 70%should be measured [27].

In summary, we have demonstrated a low-loss 3D cloakthat can reduce the visibility of the hidden object for airb-orne sound waves. It has been obtained from the conditionof the total cancellation of the waves scattered by a sphereat a prefixed frequency. This result can be considered as the

acoustic analogue of the recently proposed cloaking struc-tures based on scattering cancellation of electromagneticwaves [28]. The fabricated cloak consists of 60 acousti-cally rigid tori that at 8.55 kHz suppresses up to 90% ofthe waves scattered by the sphere. Though the cloak is

FIG. 4 (color online). Upper panels: Plot of the averagedvisibility () as a function of the frequency. The experimentalresults with and without the cloak are marked by red circles andblue squares, respectively. The reference visibility (0) whenthere is no object is marked by black triangles. The linesrepresent the simulations. Lower panels: The purple diamondsplot the ratio between the averaged visibility of the sphere withthe cloak (c) and without cloak (sph). Both are shifted by the

reference visibility. The black lines represent the ratio betweenthe scattering cross sections of the sphere with and withoutcloak, c=sph, obtained from simulations. Full (empty) sym-

bols define data on the horizontal (vertical) plane.


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directional and narrow banded, there are several possibil-ities for its improvement using additional parameters. Forexample, a broadband operation could be made by impos-ing the cancellation over several frequencies. Designing anomnidirectional cloak could also be possible by perform-ing a multiobjective optimization [16]. Moreover, cloaksfor underwater operation could also be feasible by includ-ing the elasticity of the tori employed in their design.

This work is partially supported by the SpanishMinisterio de Economa y Competitividad underContracts No. TEC2010-19751, No. TEC2011-29120-C05-01, and No. CSD2008-00066 (CONSOLIDERProgram), and by the U.S. Office of Naval Research. Theauthors acknowledge the Centro de Tecnologas Fsicasat the UPV for technical help during data acquisition.We also acknowledge the computing facilities providedby the Universidad de Valencia.

*Corresponding author.

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Supplementary Information

L. Sanchis, V. M. Garca-Chocano, R. Llopis-Pontveros, A. Climente, J. Martnez-

Pastor, F. Cervera and J. Snchez-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 withan 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 (theZ-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 planez=0 (see Figure S1 where M=60).

Let us consider an incident plane wave propagating along thez-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 (S1)

where 2

1

0

( , , , ) 4jikR

axis j j

j

eG z z d

R

is the free space axisymmetric Greens

function for the Helmholtz equation. Polar angles, and , are omitted because of thesymmetry, k=2/,Nis the total number of ring-sources located at points ( , )j jz with

amplitudes j, and the distance between source and sample points is

cos2222jjjj

zzR , where .


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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, on the surface of the scatters and imposing

Neumanns boundary conditions to the total field, scattincscatt PPP , on the surface;that is

0,

,

ii z

total

n

P

(S2)

where n

is the normal to the surface at the selected point. We obtain a linear system of

Nequations for the unknowns j.

N,in

G

n

P

iiii z

N

j

axisj

z

inc

,10,1,

(S3)

For each torus m=1,,M centered at ( , )m mz with minor radius Rt we select Nt field-

points and Nt source-points. The coordinates of the field-points are


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( , ) ( ( ), ( ))mn mn m t n m t n

z R Cos z R Sin , and the ones for source points are

( , ) ( ( ), ( ))mn mn m t t n m t t nz R Cos z R Sin with 1 0,..., 2tN and t being a

real parameter such that 0 1t . On the other hand, the coordinates of the Ns fieldand source points for the sphere (with radius Rs) are, respectively,

( , ) ( ( ), ( ))n n s n s nz R Cos R Sin and ( , ) ( ( ), ( ))n n s s n s s nz R Cos R Sin with1

/ 2,..., / 2sN

and 0 1s . The parameters t and s are chosen by

minimizing the residual error function [1]:

N

i z

N

j

axisj

inc

st

ii

n

G

n

P

NE

1 ),(1

1,

(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 sourcepoints in the geometry of the whole system. Note that all the source points are out of thedomain 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 t=0.726 and s=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 for thewhole system formed by the object with the cloak. The cancellation of defines thetotal 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 modeledby the scattered amplitude functionf(, ) where and are the spherical coordinates

of the observation point and r the distance to the origin:r

efrPikr

scatt ),(),,( .

For the case of axisymmetric bodies where the incoming wave is aligned with the

symmetry axis, which we chose as the z-axis:r

efrPikr

scatt )(),( .

If we locate the observation point on the z-axis far from the origin (=0, =0,

z>>) we can obtain ikzscatto zePf )0( , and can be calculated using the opticaltheorem, )0(Im4 of

k . Therefore,

ikzscattzePk

Im4 , (S5)

where Pscatt

is the scattered field generated by the N tori located at positions ( , )j j

z

along the z-axis; i.e.,

N

j j

ikR

j

N

j

jjaxisj

scatt d

R

ezzGzP

j

1

2

01 4

1,;,0,0

, (S6)


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where the distances 2 2( )j j jR z z do not depend on , the polar angle in the

plane perpendicular to the z-axis a passing through the pointz 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, thesesimplifications, the expression for the scattering cross section can be cast as:

1

2Im j

Nikz

j

j

ek

(S7)

The minimization of the scattering cross section is the condition employed in the

optimization algorithm to obtain the positions of the tori along thez-axis and their major

radius. Consequently, the fitness function is

001

sph

cF , (S8)

withsph (0) being the cross section of the bare sphere calculated at the prefixedoperational frequency of the cloak (0=28.62 kHz). Therefore a value F=1 meanscomplete concealment of the sphere by the cloak.

The analytical expression obtained for using the MFS is an extremely fast algorithmthat 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

/sphere 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 crosssection of the sphere is practically cancelled due to the cloak.

Fig. S2.- The ratio ()/sphere() 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 c/sph has aminimum 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.


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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 naturalselection 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


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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) =R/2, (b) =R, and (c) =2R. It is observedthat, 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) =R/2, (b) =R, and (c) =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.


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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 wehave 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).

PRL Lorenzo 3DCloakGA 2013

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