Sponsored by: 1

Designing of a simulation software for acoustical layout

and shape optimization of new materials for diffusers

in room acoustics and architecture

Dr.-Ing. Marco Norambuena, Prof. Dr.-Ing. Martin Ochmann, Willsingh Wilson;

Project duration: 08.06.2012 - 07.12.2013

1. Abstract

The present report describes the research on acoustic metamaterials carried out throughout this project. It is

of great interest, specially, in the private sector to offer new and cost-effective solution to solve difficult

problems. This need requires continuous transfer of knowledge and technology from academia to the private

sector in order to develop novel products. In this effort to provide the partner company with new resources,

metamaterials offer countless options for this purpose. The novel field of acoustic cloaking is studied in depth.

Several designs of cloaking devices are presented together with their advantages and disadvantages.

Performance among them is assessed through simulations of sound pressure field distribution and total

scattering cross section. Lastly, a beam-bending metamaterial is presented and evaluated.

2. Introduction

In the last 10 years physics has experienced the arrival and growth of a new research field. This new area,

called metamaterials, has attracted special interest due to its potential applications. The term metamaterial is a

broad definition that describes a group of engineered materials which provide certain properties not

commonly available in nature, such as negative refractive index, negative effective mass density, etc. Their

particular properties are generated by its internal physical structure rather than its chemical composition.

Although these materials were initially proposed and investigated in the field of electromagnetism, their

physical characteristics have allowed to expand their development to other areas such as optics, mechanics,

acoustics, etc. where also materials interact with propagating waves. The range where a metamaterial behaves

as such occurs where the internal structure of the material is much smaller than the wavelength of the

interacting wave.

The incidence of any type of wave on a rigid body, in this case acoustic waves, generates a reflected wave as a

result. The control of these reflected waves is particularly important since it allows to improve the acoustic

quality in many situations. For example, in architectural acoustics such control of the reflection it translates

into a better perception of music or speech. In industrial applications, the use of specially crafted materials as

the ones investigated here, allows to reduce the scattering generated by rigid bodies.

3. Results

Acoustic Cloaking

Cloaking refers to the ability of a metamaterial to make an object invisible. This idea was introduced by Pendry

et al. [Pen 06, Cum 06] in 2006 and since then on it has led to numerous papers. The principle of cloaking is

based on the form-invariance of Maxwell’s equations and a design technique called transformation optics [Che

09, Yan 09], which together define how to shape a material to achieve certain purpose, in this case a cloaking

device.

In 2007 Cummer et al. [Cum 07] presented one of the first acoustic cloaks, this was based on the experience

gained through previous work regarding electromagnetic cloak. It exploits the equivalence between acoustic

wave propagation in two dimensions and electromagnetic propagation in isotropic media. The duality of both

equations is given by the values shown below where

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Acoustics Electromagnetism

p Pressure Ez Electric field

vr,ϕ Velocity Hϕ,r Magnetic field

ρr,ϕ Mass density μϕ,r Permeability

λ Bulk modulus εz Permittivity

The result of this equivalence implies that is possible to create a cylindrical shell with a particular spatial

distribution of mass density and bulk modulus that will bend the trajectory of any incident wave around the

center of a rigid object with minimum scattering. The material spatial distribution required for such result is

given by

ρrρ0=

r

r− a,ρϕρ0=r− a

r,λλ 0

=(b− ab )2r

r− a

where r is the radius, a and b are the inner and outer radii of the cloak, and the quantities with subscript 0 are

those of the background medium. The effect of a cloaking device is shown in Figure 1. There, three cases are

depicted: (a) A point source radiating in free field, (b) A rigid cylinder generates a perturbation in the sound

field and (c) A metamaterial around the cylinder minimizes the perturbations.

(a) Point source in free field (b) Rigid scatterer (c) Cloaking around scatterer

Abb. 1 – Sound pressure field for three different conditions under cylindrical wave incidence.

A year later two new designs for acoustic cloaking were proposed by Cheng et al. [Cheg 08] and Chen et al.

[Che 08], these designs were based on a bi-layered structure that could potentially facilitate the actual

realization. The idea behind these designs was to eliminate the need of an anisotropic material distribution by

replacing this by a set of inhomogeneous isotropic layers that emulate the same effect. Both designs present a

rather good performance as show in Abb 2.

(a) Cheng’s cloak design (b) Chen’s cloak design

Abb. 2 – Sound pressure field generated by two different bi-layered

cloak designs under plane wave incidence.

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Abb. 3 - Total scattering cross section comparison of a rigid

cylinder with and without cloaking.

Solid Acoustic Cloaking

Although these proposed cloaking devices proved to greatly minimize the scattering and therefore provide a

significant performance, these designs assume that the constitutive materials of the cloak are fluids and any

practical cloaking would require some sort of solid components to at least sustain its structural integrity.

Since the beginning of research in this area it was pointed out that equations of motion for elastic media do not

share the form-invariance present in equations of electromagnetism and acoustics. This unfortunate

characteristic of elastic media increases the difficulty to create a solid cloak since it implies that coordinate

transformations cannot be implemented precisely as distribution of elastic properties. Despite this limitation,

in 2010 Urzhumov et al. [Urz 10] presented a solid cloaking and studied the influence of the shear modulus.

Urzhumov concluded that it was possible to implement a solid cloak under some restrictions. Although this

discretized model does not require an anisotropic material it imposes several new constrains regarding

thickness and filling fraction of the layers. Specifically, the ratio of the filling fraction between layers required is

extremely high, between 100-2000 times. This unfortunate fact makes difficult to test such design using a Finite

Element Method (FEM), which is the common tool used to predict the acoustic behavior in these type cases.

However, if the idealized, continuous and anisotropic model of the cloak is used, it is possible to simulate its

behavior successfully. Figure 4 shows some examples at different frequencies. The simulations there show

similar results compared to the images in Figure 1. The main difference is given by the incident wave, which

now consists of a plane wave whereas before a cylindrical wave was used.

510 [Hz] 1000 [Hz] 2570 [Hz]

Without

cloak

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With cloak

Abb. 4 - Sound pressure field around a rigid body without and with a cloaking device for different frequencies

of incidence.

In 2008, Li [Li 08] introduced a new concept of cloaking where the hidden region is mapped into a line or a

plane. This characteristic make it to be known as Carpet cloak. The advantage of this design is that none of the

required parameters have a singularity and, moreover, all their components are isotropic. In the same way as

before, the original carpet cloak was developed in the field of electromagnetics and then extended into

acoustics [Zhu 10, Zha 12, Pop 11, Zig 12]. The simplicity of this design offers the advantage that can be easily

manufactured without the complexity that an anisotropic material imposes, where material parameters not

only depend on the spatial position but also on the direction of wave propagation.

As many metamaterials, the carpet cloak, consists of an array of unit-cells with a given distribution. First, the

parameters of the unit cell must be defined according with the cloaking requirements. Once the parameters

estimation is done [Fok 07], the array is created on top of the region that wants to be cloaked. A diagram of the

carpet cloak geometry is shown in Figure 5. Given the small size of the numerous perforations used, it is

tempting to assume that viscous losses will play an important role in the effect created by the cloak. However,

it was proved that such losses do not influence the resulting sound field in any special way, i.e. the cloaking

performance remains constant whether the losses are considered or not. The only noticeable effect is a slight

reduction of the local sound pressure.

Unit cell Array of cells (perforated plate)

Array of plates

Abb. 5 – Structure diagram of a carpet cloak.

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The broadband performance of the carpet cloak was assessed through simulations for several frequencies, as

shown in Figure 6. In a similar way, the response of the cloak at different angles of incidence was simulated.

The results, shown in Figure 7, present a consistent performance irrespective of the angle of incidence.

1715 [Hz] 3430 [Hz] 6860 [Hz]

Abb. 6 – Scattered pressure with and without carpet cloak for different frequencies.

0 [deg] 30 [deg] 45 [deg] 60 [deg]

Abb. 7 – Scattered sound pressure with and without carpet cloak for different angles of incidence relative to

the ground.

Beam-bending metamaterial

The idea behind this type of materials is to be able bend a normally incident wave front to a given angle. This

metamaterial is so-called Beam-Bending [Pop 09] or Beam-shifter. Although there are several beam-bending

designs found in the literature [Akl 12, Lia 12], the characteristic of the one presented here is that the exit

angle is modified whereas in all other cases the exit angle is identical to the incident angle and the beam-

bending takes place only inside the material itself. The change in the trajectory angle increases the travel

distance of a wave and thereby increases the energy losses when the metamaterial is combined with a classic

acoustic absorbent material. Commonly this task at low frequencies is highly inefficient, when only classic

absorbing materials are used, since the large wavelengths in this spectral range requires the implementation of

large size absorbers.

Simulations showed that, as expected, an increase in the thickness of the metamaterial results in an increase in

the exit angle. With a reasonable thickness of 0.6λ, an angle of 30 [deg] is obtained, Figure 8 depicts this case.

Beside this, a particularly interest behavior occurs when the material is exposed to different angles of

incidence. Irrespective of the angle of incidence, the difference between incident and exit angle is always

constant. This is clearly visible if the incident angle is kept constant and the metamaterial is rotated, the exit

angle does not change. Figure 9 shows the result of the radiation pattern when only the metamaterial is

rotated. As shown there, the main lobe of the pressure points to approximately 30 [deg] in every case.

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(a) Traveling wave passing through the metamaterial of

0.6 λ thickness. λ is the wavelength.

(b) Pressure field generated by a gaussian beam

impinging on the metamaterial with a rotation of 22

[deg]. Main lobe of the exit angle situated at 30

[deg].

Abb. 8 – Sound pressure simulations of a beam-bending metamaterial

Abb. 9 – Polar diagram of the exit angle for different

rotation angles of the metamaterial. A thickness of 0.6λ

generates an approx. angle of 30 [deg].

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4. Summary

As stated at the beginning, one of the main objectives of this project was not only to study deeply the novel

field of acoustic metamaterials but also serve as a bridge between this new area of science and the partner

company wax GmbH. Specifically, this consisted on the introduction of acoustic metamaterials, proposition

metamaterials designs and evaluation of their performance; Focusing always in bringing together the scientific

research close to a real implementation scenario. Throughout the project, special emphasis was placed on the

development of feasible solution to create an acoustic cloaking metamaterial. Such metamaterial is able to

create a unique behavior that is not possible to reproduce with classic materials. This behavior can be

considered as a special case of uniformed diffusion. The simulations conducted proved that it is possible to

hide an acoustically rigid object inside the cloak and minimize almost completely any perturbation in the sound

field. Moreover, the acoustic cloaking provides a broadband effectiveness. It was important to notice that

there are fundamental differences between fluid and solid cloaks. As part of the solid cloak designs, the

described carpet cloak presents itself as a viable option from the manufacturing point of view. Alike as before,

this design generates a reasonable performance under broadband excitation.

5. References

[Pen 06] J. Pendry, D. Schurig and D. Smith: Controlling electromagnetic fields. Science, 2006. DOI:

10.1126/science.1125907.

[Cum 06] S. Cummer, B. Popa, D. Schurig, D. Smith, J. Pendry: Full-wave simulations of electromagnetic

cloaking structures. Physical Review E, 2006. DOI: 10.1103/PhysRevE.74.036621.

[Che 09] H. Chen: Transformation optics in orthogonal coordinates. Journal of Optics A, 2009.

DOI:10.1088/1464-4258/11/7/075102

[Yan 09] M. Yan, W. Yan, M. Qiu: Invisibility Cloaking by Coordinate Transformation. Progress in Optics,

Elsevier, 2009. DOI: 10.1016/S0079-6638(08)00006-1.

[Cum 07] S. Cummer and D. Schurig: One path to acoustic cloaking. New Journal of Physics, 2007. DOI:

10.1088/1367-2630/9/3/045.

[Cheg 08] Y. Cheng, F. Yang, J. Xu and X. Liu: A multilayer structured acoustic cloak with homogeneous

isotropic materials. Applied Physics Letters, 2008. DOI: 10.1063/1.2903500.

[Che 08] H. Chen, T. Yang, X. Luo and H. Ma: Impedance-Matched Reduced Acoustic Cloaking with

Realizable Mass and Its Layered Design. Chinese Physics Letters, 2008. DOI: 10.1088/0256-

307X/25/10/049.

[Urz 10] Y. Urzhumov, F. Ghezzo, J. Hunt and D. Smith: Acoustic cloaking transformations from

attainable material properties. New Journal of Physics, 2010. DOI: 10.1088/1367-

2630/12/7/073014.

[Li 08] J. Li and J. Pendry: Hiding under the Carpet: A New Strategy for Cloaking. Physical Review

Letters, 2008. DOI: 10.1103/PhysRevLett.101.203901.

[Zuh 10] W. Zhu, C. Ding and X. Zhao: A numerical method for designing acoustic cloak with

homogeneous metamaterials. Applied physics letters 97, 131902, 2010. DOI:

10.1063/1.3492851.

[Zha 12] X. Zhang, X. Ni, M. Lu and Y. Chen: A feasible approach to achieve acoustic carpet cloak in air.

Physics Letters A, 376, 2012. DOI: 10.1016/j.physleta.2011.10.059.

[Pop 11] B. Popa, L. Zigoneanu and S. Cummer: Experimental Acoustic Ground Cloak in Air. Physical

Review Letters, 106, 253901, 2011. DOI: 10.1103/PhysRevLett.106.253901.

[Zig 12] L. Zigoneanu, B. Popa and S. Cummer: Sound Manipulation with Acoustic Metamaterials.

Proceedings of Internoise 2012/ASME NCAD meeting August 19-22, 2012. IN12_1277.

[Fok 07] V. Fokin, M. Ambati, C. Sun and X. Zhang: Method for retrieving effective properties of locally

resonant acoustic metamaterials. Physical Review B, 76, 144302, 2007. DOI:

10.1103/PhysRevB.76.144302.

[Pop 09] B. Popa and S. Cummer: Design and characterization of broadband acoustic composite

metamaterials. Physical review B, 80, 174303, 2009. DOI: 10.1103/PhysRevB.80.174303.

[Akl 12] W. Akl and A. Baz: A technique for physical realization of anisotropic density matrices with

application to acoustic beam shifters. Journal of applied physics 111, 024907, 2012. DOI:

10.1063/1.3678634.

[Lia 12] Z. liang and J. Li: Extreme Acoustic Metamaterial by Coiling Up Space. Physical Review Letters

108, 114301, 2012. DOI: 10.1103/PhysRevLett.108.114301

Sponsored by: 8

6. Contact details

Prof. Dr.-Ing. Martin Ochmann

Beuth Hochschule für Technik Berlin

Fachbereich II

Luxemburger Str. 10, 13353 Berlin

Tel. +49 30 4504 -2931

E-Mail: ochmann@beuth-hochschule.de

web: projekt.beuth-hochschule.de/ca/

Co-operation partner

wax GmbH

Dr. Oliver Hüfner, Timo Krämer and Willsingh Wilson

Nürnberger Str. 18

D 10789 Berlin

Tel. +49 (0)30 609 898-660

Fax: +49 (0)30 609 898-669

E-Mail: mail@w-a-x.com

www.w-a-x.com