Scuola di Ingegneria Industriale e dell’Informazione

Corso di Laurea Magistrale in Ingegneria Elettrica

Master of Science in Electrical Engineering

PIEZOELECTRIC ENERGY HARVESTING IN AN ACOUSTIC METAMATERIAL CAVITY

Supervisor: Ing. Alessandro FERRERO

Co-Supervisor: Ing. Giovanni DOTELLI

Master Thesis of:

Letizia Chisari Personal code: 858295

Academic Year 2016 - 2017

PIEZOELECTRIC ENERGY HARVESTING IN A DOUBLY COILED-UP ACOUSTIC METAMATERIAL CAVITY

An investigation for optimal position of the piezoelectric bimorph plate

inside the metamaterial cavity.

By

Letizia Chisari

Supervisor Ing. Alessandro FERRERO

Co-Supervisor

Ing. Giovanni DOTELLI

Industrial and Information Engineering School Department of Electronics, Information and Bioengineering

Division of Electrical Engineering POLITECNICO DI MILANO Milano, Italia, April 2018

V

“Everything is determined … by forces over which we have no control. It is determined for the insects as well as for the stars. Human beings, vegetables, or cosmic dust – we all dance to a mysterious tune, intoned in the distance by an invisible piper.” – Albert Einstein

VI

Aknowledgments

I would like to thank my advisor, Ing. Alessandro Ferrero, for providing me

the wonderful opportunity to finish my master degree working on this

exciting project. His unwavering support, invaluable guidance and

suggestions in exploration this research greatly appreciated.

At the same time, many thanks to the chemical department members, Ing.

Giovanni Dotelli, Dr. Salvatore Latorrata and Dr. Andres Ricardo Leon

Garzon, for their invaluable suggestions and help.

Finally, I would like to thank my family for their constant encouragement

and belief in me during this course. I would also like to extend my thanks to

all my friends who kept me in good spirits during my stay here.

Letizia Chisari

Milan, Italy

Thursday, April 19, 2018

VII

Abstract

Piezoelectric energy harvesting in a doubly coiled-up acoustic

metamaterial cavity.

An investigation for optimal position of the piezoelectric bimorph plate

inside the metamaterial cavity.

This work represents a quasi-experimental research on acoustic

metamaterials. It describes a new class of materials with unusual

properties that can be engineered using existing materials for usual

properties. It investigates a particular engineered application in the energy

harvesting field. This work specifically explains the physical influence of

the piezoelectric bimorph plate placed inside a metamaterial cavity on the

sound pressure level from the theoretical point of view. Then simulations

based on this system have been done with COMSOL Multiphysics to predict

the optimal position of the piezoelectric bimorph plate inside a double-

walled metamaterial cavity and to detect its physical influence on

amplification efficiency and resonant behavior of the cavity. Results will be

showed and discussed. Moreover, this work describes applications of

acoustic metamaterials inside industrial, biomedical, civil and automotive

contexts.

KEY WORDS: acoustic metamaterial, space-coiling metamaterial, acoustic energy harvesting, piezoelectric bimorph plate, sound waves, sound pressure level gain, material unusual properties, local resonance

VIII

General Index

1 Introduction 1.1 Background of the thesis

Acoustic Metamaterials

Energy Harvesting

1.2 Purpose of the thesis

Space-coiling Metamaterials

Acoustic Energy Harvesting

2 Modeling and Analysis 2.1 Unit Cell and Multi-Cells system

2.2 The Acoustic Metamaterial Cavity

2.3 Piezoelectric Bimorph Plate

3 Results and Discussion 3.1 Unit Cell Effective Parameters calculation

3.2 Doubly coiled-up Acoustic Metamaterial Cavity amplification

3.3 Piezoelectric Bimorph Plate influence

4 Summary 4.2 Conclusion and Outlook

4.3 References

IX

Index of Figures

Figure 1 | Illustration of wave propagation effects.

Figure 2 | Parameter space for mass density ρ and bulk modulus K. Figure 3 | Notion of quantal meta-surface inspired from analogue-to-

digital conversion and image compression60.

Figure 4 | Conceptual illustration of transformation acoustics.

Figure 5 | Conceptual examples of active acoustic metamaterial designs. Figure 6 | Functionality and possible applications of active metamaterials. Figure 7 | Space-coiling metamaterials.

Figure 8 | Classification of energy harvesting approaches, based on

vibration sources.

Figure 9 | Metastructure which consists of a double-walled slab. Figure 10 | Unit cell.

Figure 11 | Schematic illustration of multilevel modeling of computer-

aided micromechanics for metamaterial development.

Figure 12 | Method for retrieving effective material properties. Figure 13 | Unit cell geometry.

Figure 14 a | Half wavelength resonator (open outlet), compared with b |

Quarter wavelength resonator (closed outlet).

Figure 15 | A series of LC resonator.

Figure 16 | Cross-sectional view of two two acoustic metamaterial slabs

that are separated by a subwavelength air gap.

Figure 17 | Theoretical prediction of sound pressure amplification.

Figure 18 | Bimorph plates are connected in parallel; arrows show the

poling direction.

Figure 19 | Piezoelectric bimorph plate structure with a tip mass.

Figure 20 | An array of two piezoelectric bimorph plates connected in

parallel.

Figure 21 | Unit cell effective parameters calculation.

Figure 22 | Theoretical prediction of sound pressure amplification. a |

Calculated sound pressure within metamaterial cavity, compared with b |

calculated values obtained using effective medium theory.

X

Figure 23 | a | Graph of sound pressure level of the effective

subwavelength-scale metastructure obtained using a cut line 2D. b | Graph

of sound pressure level of the final geometry obtained using a cut line 3D.

Figure 24 | a | Higher effective index subwavelength-scale metastructure.

b | Graph of sound pressure level of the 2D subwavelength-scale

metastructure obtained using a cut line 2D.

Figure 25 | Cross-sectional view of the spatial SPL gain distribution inside

the metastructure.

Figure 26 | a | A 3D rendering of Memoli’s brick. b | Cross-sections of 16

selected bricks and the corresponding phase maps at normal incidence.

Figure 27 | Configuration of the proposed piezoelectric bimorph plate. a |

The piezoelectric bimorph measures with a tip mass whose piezoelectric

layers are connected in parallel. b | Tip mass measures.

Figure 28 | 1st bending mode - (FP resonant mode).

Figure 29 | 1st twisting mode.

Figure 30 | 2nd bending mode.

Figure 31 | SPL graphs of the metamaterial cavity. a | without

piezoelectric bimorph plate. b | within piezoelectric element.

Figure 32 | SPL graphs of the metamaterial cavity. a | when an array of

horizontally oriented piezoelectric bimorph plates in placed inside. b |

when an array of vertically oriented piezoelectric bimorph plates in placed

inside.

XI

Index of Tables

Table 1 | Analogy between acoustic and electromagnetic variables and

material characteristics.

Table 2 | Characteristic properties of the three-dimensional acoustic

metastructure.

INTRODUCTION

- 1 -

Chapter 1

Introduction

1.1 Background of the thesis

Acoustic Metamaterials

Energy Harvesting

1.2 Purpose of the thesis

Space-coiling Metamaterials

Acoustic Energy Harvesting

INTRODUCTION

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1.1 Background of the thesis

Acoustics is the branch of science that studies the propagation of sound and

vibrational waves. Acoustic wave science studies the propagation of matter

oscillation through an elastic medium such as air or water and therefore

explains energy transfer through the medium. While the movement of

oscillating materials is limited through its equilibrium position, vibrational

waves can propagate in a long distance and can be reflected, refracted,

attenuated or, more generally, manipulated by the medium. According to

the oscillation frequency, acoustic waves have been classified to different

fields that cover the audio (or sonic waves, 20 𝐻𝑧 − 20 𝑘𝐻𝑧 frequency

range), ultrasonic waves (with frequencies greater than 20 𝑘𝐻𝑧) and

infrasonic waves (0 − 20 𝐻𝑧 frequency range), or seismic waves at much

larger scale which are waves of energy travelling through the Earth’s layer.

Audible acoustic waves are ubiquitous in our everyday experience: they

form the basis of verbal human communication, and the combination of

pitch and rhythm transforms sound vibrations into music. Waves with

frequencies beyond the limit of human audibility are used in many

ultrasonic imaging devices for medicine and industry. However, acoustic

waves are not always easy to control. Audible sound waves spread with

modest attenuation through air and are often able to penetrate thick

barriers with ease. Electronic devices are able to amplify and manipulate

sound signals, but only after they are converted to electronic form. New

tools to control these waves as they propagate, in the form of new artificial

materials, are extremely desirable. Materials have been used to control

wave propagation for centuries, and optics is a prime example. By precisely

shaping lenses, it is possible to make various optical devices for focusing

and manipulating light. In nature, this strategy is demonstrated by, for

example, the lenses in animal eyes, which are used to manipulate light,

and by the melon organ that Cetaceans use to focus sound waves for

underwater echolocation. These organs use relatively simple materials to

achieve control of wave propagation.

INTRODUCTION

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Particle velocity and sound pressure are both quantitative attributes of a

sound wave. As the wave moves out and away from its source, air

molecules bounce back and forth, jostled by others nearer the sound

source, and small variations in ambient air pressure are produced. It is

interesting to note that most large animals have evolved specialized

pressure transducers (a.k.a ears or tympannae) by which they detect and

extract information from these pressure fluctuations. On the other hand

many small animals, such as ants or fruit flies, without tympanal ears,

apparently do not perceive the pressure component of sound, but instead

have specialized movement receptors (usually small hairs on body or

antennae) that detect sound particle velocity, the oscillations of air

molecules in a sound field.

INTRODUCTION

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Acoustic Metamaterials

By designing and engineering artificial materials with more complex

properties, unprecedented functionalities can be obtained. The science of

designing and engineering such materials is the subject of the field of

metamaterials (DF 1), and the subset of this field, in which the aim is the

control of acoustic waves, is acoustic metamaterials. Metamaterials are

artificial structures, typically periodic (but not necessarily so), composed of

small meta-atoms that, in the bulk, behave like a continuous material with

unconventional effective properties. This includes the generation of band

gaps, which are frequency ranges with high levels of wave attenuation. In

the context of acoustics, these band gaps can be tuned to occur at low

frequencies where the acoustic wavelength is large compared to the

material, and where the performance of traditional passive noise control

treatments is limited. Therefore, such acoustic metamaterials have been

shown to offer a significant performance advantage, however, due to their

resonant behaviour, the band gaps tend to occur over a relatively narrow

frequency range. A significant increase in performance can be achieved by

incorporating active control elements into acoustic metamaterials and a

significant enhancement in the transmission loss is achieved too, as

Cummer et al. explain in their recent study1.

DF 1 | Metamaterials The term metamaterial is now broadly applied to engineered materials, usually composites, in which an internal structure is used to induce effective properties in the artificial material that are substantially different from those found in its components. The term originated from the field of electromagnetic materials, in which metamaterials were engineered to control light and radio wave propagation, and is used specifically to indicate materials composed of conducting structures that, by generating controlled electric and magnetic dipole responses to applied fields, result in a negative refractive index10. This property is not found in any known natural material. The term metamaterial is not very precisely defined, but a good working definition is: a material with ‘on-demand’ effective properties13, without the constraints imposed by what nature provides. For acoustic metamaterials, the goal is to create a structural building block that, when assembled into a larger sample, exhibits the desired values of the key effective

INTRODUCTION

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parameters — the mass density 𝜌 and the bulk modulus 𝐾 — as discussed in DF 2.

These two parameters are analogous to the electromagnetic parameters, permittivity ε and permeability 𝜇, as can be seen in the following expression of the refractive index 𝑛 and the impedance 𝑍.

𝑛 = √𝜌

𝐾 (acoustics) 𝑛 = √𝜀𝜇 (electromagnetism)

𝑍 = √𝜌𝐾 (acoustics) 𝑍 = √𝜇/𝜀 (electromagnetism)

The most common approach to constructing acoustic metamaterials is based on the use of structures whose interaction with acoustic waves is dominated by the internal behaviour of a single unit cell of a periodic structure, often referred to as a meta-atom. To make this internal meta-atom response dominant, the size of the meta-atom generally needs to be much smaller (about ten or more times smaller) than the smallest acoustic wavelength that is being manipulated. By contrast, in so-called phononic (for sound) or photonic (for light) crystals, unusual wave behaviour is created via the mutual interaction (multiple scattering) of unit cells whose dimensions are typically about half of the operating wavelength (although recent work14 has shown how local and multiple scattering responses can be combined in a single structure to achieve interesting effects, blurring the line between these different classes of artificial media). This subwavelength constraint ensures that the metamaterial behaves like a real material in the sense that the material response is not affected by the shape or boundaries of the sample. This equivalence will not hold for periodic materials in the phononic crystal regime, in which long-range interactions and spatial dispersion dominate the response. Instead, when the material response is determined by the local meta-atom response, effective bulk-material properties can be defined and estimated from simulations or measurements of very small samples. The fact that the effective parameters of a metamaterial composed of thousands or millions of meta-atoms can be determined using simple and efficient methods is one of the most powerful aspects of the metamaterial approach to artificial material design.

The science of waves propagating in periodic structures goes back

decades2; however, our modern appreciation of the use of engineered

structures to control wave properties began with photonic3 and phononic4

crystals. Research in this area rapidly expanded with the understanding

that relatively simple, but subwavelength, building blocks can be

assembled into structures that are similar to continuous materials, yet have

unusual wave properties that differ substantially from those of

INTRODUCTION

- 6 -

conventional media. In acoustics, the first artificial metamaterial5 used

rubber-coated spheres to create locally resonant and deeply subwavelength

structures that responded to incident acoustic waves. An assembly of these

meta-atoms into a bulk metamaterial exhibited peculiar, but useful,

acoustic properties.

By careful designing and engineering the parameters of the meta-atom

structures such as shape, geometry, size or orientation, fascinating

functionalities beyond the capability of conventional materials can be

realized. The concept of metamaterials was first proposed by Veselago6 in

1968 for electromagnetic waves, he predicted that a medium with

simultaneous negative permittivity and negative permeability were shown

to have a negative refractive index. But this negative index medium

remained as an academic curiosity for almost thirty years, it needed to

wait for around 30 years for the next step when Pendry reported artificial

designs with effectively negative permeability and permittivity in 19997,8.

Metamaterials with negative refractive index was first experimentally

demonstrated at GHz frequency by Smith and Shelby9,10 and have since

been a subject of numerous studies in a wide variety of wave-matter

interaction.

Building on this work, and taking inspiration from developments in

electromagnetism11, the field of acoustic metamaterials has focused on

developing artificial structures that are capable of controlling the

propagation of sound in new ways, made possible by the creation of

unusual material properties. These efforts have been successful on many

fronts. For instance, it is now possible to design acoustic metamaterials

that can acoustically conceal an object, acting as cloaks of ‘inaudibility’.

Also, acoustic metamaterials with a negative refractive index can be

designed to bend sound the ‘wrong’ way when insonified by a loudspeaker,

enabling new ways of focusing and shaping sound fields. Over the past

15 years, the field of acoustic metamaterials has branched out in many

directions, and it has been shown that acoustic waves can be manipulated

and controlled in ways not previously imagined. The two waves:

electromagnetic and acoustic are certainly different. In electromagnetism,

both electric and magnetic fields are transverse wave. Acoustic wave is

longitudinal wave; the parameters used to describe the wave are pressure

INTRODUCTION

- 7 -

𝑝 and particle velocity 𝑣𝑝. However, the two wave systems have common

physical concepts as wavevector, wave impedance and power flow.

Moreover, in a two-dimensional (2D) case, when there is only one

polarization mode, the electromagnetic wave has scalar wave formulation.

Therefore, the two sets of equations for acoustic and electromagnetic

waves in isotropic media are dual of each other by the replacement as

shown in Table 1.1 and this isomorphism holds for anisotropic medium as

well. Table1.1 presents the analogy between acoustic and transverse

magnetic field in 2D under harmonic excitation.

The fundamental physics properties related to the novel metamaterial

based applications16 include Interference, Diffraction, Absorption,

Scattering, Polarization, Dispersion, Reflection, Refraction, and

Transmission. These materials are targeted for breakthroughs in energy

harvesting, miniaturization of communication antenna, medical and

security imaging, and defense stealth applications.

INTRODUCTION

- 8 -

The critical destructive interference, constructive interference and

scattering effects are shown in Figure 1:

Figure 2 | Illustration of wave propagation effects: a| constructive interference, b| destructive interference and c | scattering effects.

Concept of metamaterials expands beyond electromagnetics15. Acoustics

metamaterial is analogues to magnetic metamaterial, where in density and

elastic stiffness, wave propagation parameters are engineered for unusual

properties. The fabrication methods used are conventional fabrication

methods that are extended for metamaterial fabrication, including 3D

manufacturing.

Metamaterials with negative parameters

Sound-wave propagation is controlled by the mass density and the bulk

modulus of a material (DF 2). In conventional media, both of these

parameters are positive and cannot be easily altered because they are

directly associated with the chemical composition and the microstructure

of the material, so the bonding structures of the constituted atoms.

However, if metamaterials are constructed using resonant subwavelength

meta-atoms structures that behaves like a continuous material in the bulk

and that enhance sound–matter interaction, then it is possible to engineer

the wave properties to obtain values of the effective acoustic-material

parameters that are not observed in nature.

When an atom is deviated from the equilibrium state, it will be pulled back

to the balance position by a central force explained by Newton’s second

law 𝐹 = 𝑚�̈� . Although the mass of an atom must be always positive,

negative effective mass density can be achieved in a periodic structure

comprising of artificial meta-atoms near its resonant frequency.

INTRODUCTION

- 9 -

While effective bulk modulus can reach a negative value when the external

force oscillates near the resonant frequency.

Researchers have explained in previous publications that either effective

mass density or effective bulk modulus of acoustic parameters can be

negative near resonant frequency of a periodic artificial structure and then

a fully opaque acoustic material is possible. However, an inverse effect in

which sound wave energy propagates instead of attenuation will occur

when both these two parameters are negative simultaneously. As it is

explained by Lee et al.20 In a mechanical system, a dipole resonance is

related to the effective mass density because the resonance vibrates along

a certain direction, resulting in the inertial response and oscillating like a

spring-mass system5,17,18,19.

A monopole resonance, however, vibrates in all directions associated with a

compressive or expansive motion which functions like the change of volume

of Helmholtz resonator and is thus related to the effective bulk

modulus18,21,22. Therefore, to realize double negative parameters, two

resonance symmetries including dipole and monopole resonances must be

exploited.

DF 2 | Acoustics principles and material parameters Acoustics is the science of vibrational wave propagation in fluids such as air or water, including the familiar audio frequency waves in air that we know as sound. For the purposes of controlling sound propagation with acoustic metamaterials, a key step is the identification of the material parameters that control wave propagation. Linear acoustics describes small pressure fluctuations that form a travelling wave of low intensity. One defining equation of acoustics comes from Newton’s second law (𝐹 = 𝑚�̈�) and connects the acoustic particle

perturbation velocity 𝑣𝑝 to the acoustic pressure 𝑝 as

𝜌𝜕𝑣𝑝

𝜕𝑡= −∇𝑝 (1)

Here the scaling constant is the fluid mass density 𝜌, which is one of the two critical constants that control acoustic wave propagation. To connect the motion of a non-viscous and stationary (not flowing) fluid with its compression and expansion, we express the conservation of mass through the continuity equation.

INTRODUCTION

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Assuming that acoustic wave propagation can be regarded as isentropic (adiabatic and reversible with constant entropy), which means that thermal processes can be neglected, the continuity equation is

𝜕𝑝

𝜕𝑡+ 𝐾∇ ∙ 𝑣𝑝 = 0 (2)

Here the scaling constant is the bulk modulus 𝐾, which is essentially the compressional stiffness of the fluid (medium’s resistance to external uniform compression), and which is the second of the two critical material constants. When these two equations are combined into a single equation for the pressure 𝑝, the scalar wave equation emerges

𝜕2𝑝

𝜕𝑡2=

𝐾

𝜌∇2𝑝 (3)

The acoustic wave velocity, which controls changes in wave direction at

interfaces, is thus given explicitly by 𝑐 = √(𝐾/𝜌). It can also be shown that the acoustic wave impedance, which controls wave reflection and transmission amplitudes at interfaces, and which is defined as the ratio of

pressure to fluid velocity in the wave, is 𝑍 = 𝑝/𝑣𝑝 = √𝐾𝜌. Therefore, the

fluid mass density 𝜌 and bulk modulus 𝐾 are the two fluid parameters that control the propagation of acoustic waves. Consequently, these are the parameters that we wish to control when designing metamaterial structures. Although there are some fundamental differences between acoustic and electromagnetic waves (such as their longitudinal and transverse natures, respectively), the two acoustic parameters are in many ways analogous to the two parameters that control electromagnetic wave propagation, the electric permittivity and the magnetic permeability. This is why the field of acoustic metamaterials has been able to borrow concepts so successfully from electromagnetic metamaterials.

One of the most unusual regimes for acoustic metamaterials arises when

the real parts of the effective mass density and bulk modulus are negative

in the same frequency range1. This regime is analogous to negative-index

metamaterials for electromagnetic waves. These materials, developed in

the early 2000s, use metallic structures that generate out-of-phase

(negative) electric and magnetic dipole responses to incident

electromagnetic fields, leading to a negative phase velocity and a negative

index of refraction23,24.

INTRODUCTION

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Examples in which artificial materials are engineered to have parameters

with negative or near-zero values are illustrated (Fig.2). These media

enable metamaterials designers to construct devices with surprising

effects, such as energy flow in the direction opposite to that of the wave

vector or sound propagation without phase variations. Such materials allow

for the guiding and focusing of acoustic signals. Acoustic metamaterials

were initially created for use in sound-attenuating applications5. The first

acoustic meta-atoms were spherical metal cores coated with a soft rubber

shell packed to a simple-cubic lattice in a host material, which could

exhibit a Mie-type resonance frequency — which describes the scattering of

an electromagnetic plane wave by a homogeneous sphere — far below the

wavelength-scale Bragg resonance frequency of the lattice5,25–27. Depending

on the underlying mechanical motion in such resonances, negative effective

values of the mass density and of the bulk modulus can be obtained. In the

context of spherical and cylindrical scatterers, monopolar modes give rise

to a resonant response of the bulk modulus, whereas the dipolar modes

create resonances in the mass density28. Numerical simulations of rubber

spheres suspended in water, which have been experimentally verified29,

show that these modes can coexist, leading to a band in parameter space

characterized by a negative index of refraction29,30. Other architectures for

acoustic metamaterials involve segments of pipes and resonators in the

form of open and closed cavities. In 1922, a seminal paper by G. W.

Stewart31 that discusses lumped acoustic elements for filter applications

characterized these structures as simple oscillators.

However, these elements were not used to form artificial media until 2006,

when metamaterials composed of a waveguide loaded with an array of

coupled Helmholtz resonators were constructed. Helmholtz resonators are

closed cavities connected to a waveguide via a narrow channel (Fig. 2b). At

their collective resonance frequency, a low-frequency stopband is formed,

the origin of which can be traced back to the negative effective bulk

modulus 𝐾 — which occurs when a parcel of fluid compresses under

dynamic stretching — of the loaded waveguide32. This is an example of a

locally resonant acoustic metamaterial. Altering the volume of the cavity

results in a change in its resonance frequency.

INTRODUCTION

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Thus, attaching a series of open side-branches to the waveguide produces

resonators with very low resonance frequency, and sound waves are

entirely reflected up to the frequency at which the sign of the bulk

modulus changes33. Designing an entire panel of these open side-branches

creates a so-called ‘acoustic double fishnet’ structure that could sustains

this attenuation band for a wide range of frequencies and angles, and that

could provide acoustic shielding to block or exploit environmental

noise34,35.

Insight into the nature of acoustic responses facilitates additional

metamaterial design approaches. If a fluid segment accelerates out of

phase with respect to

the acoustic driving

force, then a negative

mass density is possible,

as implied by equation

(2) in DF 2. This

acoustic response can

be created using

membranes fixed at the

rims of a tube or an

array of holes17,36,37

(Fig. 2c). Furthermore,

changing the size of the

membranes or loading

them with a mass makes

it possible to alter the

resonance over a

spectrally extended

range. If either the

effective bulk modulus

𝐾 or the mass density 𝜌 are negative, then fully opaque materials with

purely imaginary phase velocities are possible. However, in a similar

manner to the coexistence of monopolar and dipolar bubble resonances,

composing a structure of Helmholtz and membrane units for which 𝜌 and 𝐾

are simultaneously negative (Fig. 2d) creates a band in which energy can

Figure 2 | Parameter space for mass density ρ and bulk modulus K. a|For all known natural materials, the acoustic constitutive parameters are strictly positive (K>0 and ρ>0). b|Metamaterials with K<0 and ρ>0 can be obtained with ope and closed cavity resonators. c|Metamaterials with K>0 and ρ<0 are typically membranes or coated-bead structures. d|Space-coiling or coupled filter-element structures give rise to double-negative (K<0 and ρ<0) metamaterials.

INTRODUCTION

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propagate instead of attenuate, as happens when only one of these

parameters is negative.

Meta-surfaces and quantal meta-surfaces

The quest to enhance wave–matter interactions and to manipulate waves

using the smallest possible amount of space has led to the exploration of

acoustic meta-surfaces. Meta-surfaces belong to the family of wavefront-

shaping devices with thicknesses much smaller than the wavelength. In

acoustics, their building units are, for example, coiled elements, Helmholtz

resonators or resonant membranes that enable wave steering and focusing

through designs based on spatial phase gradients54-56. Strictly speaking,

meta-surfaces are monolayer materials that are able to impart an arbitrary

phase and amplitude modulation to the impinging wave, and constitute an

alternative to bulky crystals, whose performance may be hindered in some

cases by material losses. These ultrathin materials are able to support

curious effects, such as scattering of waves with anomalous reflection and

refraction angles57,58. Akin to the principle of a graded-index lens, a

properly designed meta-surface can also act as an ultra-thin, planar

acoustic lens whose focal length and position is engineered through the in-

plane phase profile59. Because of these effects, suitably designed meta-

surfaces with a 2π phase span could potentially generate unconventional

wave steering abilities. Memoli, G. et al.60 developed the notion of quantal

meta-surfaces to demonstrate a different metamaterial concept, based on

the use of a small set of pre-manufactured 3D unit cells, termed

metamaterial bricks, which can be assembled into 2D structures on-

demand. The bricks become, in isolation, the building blocks of an

assembly, encoding prerequisite phase delays. This operation is a form of

analogue-to-digital conversion (Fig. 3): the desired acoustic pressure field

is sampled at a certain distance from a meta-surface and used as input for

acoustic holography, leading to a phase distribution that gets quantized in

the spatial and phase domains, whose values are then mapped into a series

of pre-manufactured metamaterial bricks. Starting from a limited set of

unique bricks, they used a discrete wavelet transform based method to

synthesize the meta-surface needed in a given application, optimizing the

number of bricks needed.

INTRODUCTION

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This simple, yet powerful, concept simplifies the design of acoustic devices

and systems, and lays the foundations for realization of spatial sound

modulators (SSMs).

Figure 3 | Notion of quantal meta-surface inspired from analogue-to-digital conversion and image compression60. a| Quantization of an analogue phase distribution with a uniform 2π-span and a fixed spatial resolution. b| Lossy and c| Lossless compressions of figure (a) using wavelet transforms, with and without thresholding, respectively.

INTRODUCTION

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Transformation acoustics and cloaking

The development of materials with unusual constitutive acoustic

parameters has led to new ways to model the flow of sound. One of the

most powerful tools that can be used to design materials to control sound,

including those with the ability to hide or cloak objects from sound is the

concept of transformation acoustics. Often in metamaterials, ideas emerge

from concepts that originate in electromagnetism and optics. The

coordinate-transformation invariance of the Maxwell equations for

electromagnetism implies that any coordinate-transformation-based

deformation of electromagnetic fields, such as stretching and squeezing,

can be physically created with the right distribution of the electromagnetic

material properties71. The material properties that are required to obtain

such effects are usually complicated and difficult to implement, but the

generality of the concept implies that even complex deformations of

electromagnetic fields, such as those required for cloaking71, can in

principle be obtained using the right materials; such deformations have

been experimentally demonstrated at radio72,73 and optical frequencies74.

This concept of transformation optics raised the question of whether

similar manipulation can be applied to other types of waves, in particular

acoustic waves75, where it would have many potential applications. This

question was ultimately answered, first in two29 and then in three76

dimensions, by showing that the equations of linear acoustics take the

same form as certain equations governing electromagnetic waves. In three

dimensions, the analogous equations are those of electric current and

conductivity, which have been shown to be transformation invariant77.

Interestingly, and in contrast to electromagnetism, transformation

acoustics theory is not independent of the velocity of the background

fluid78, although at low flow speeds the effect is quite modest. A further

theoretical step forward in the field of transformation acoustics was the

finding that there are available degrees of freedom79,80 offering a wide

range of acoustic material properties that can realize a specific coordinate

transformation, instead of the one-to-one mapping available in

electromagnetism80,81.

Collectively, these findings show that the transformation-based design

approach (Fig. 4) can be used to design devices that are capable of

INTRODUCTION

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manipulating acoustic waves in very complex ways, provided that some

unusual acoustic material properties can be realized.

Inertial metafluids are generally composed of meta-atoms in which solid

inclusions are surrounded by a host fluid82,35. Experimental studies have

recently explored acoustic-metamaterial implementations of inertial

metafluids and have shown that they are relatively simple to make. For

example, a rotationally asymmetric arrangement of simple scatterers in a

fluid naturally homogenizes to yield an anisotropic effective mass density83.

By exploiting a previous result84, it was also shown how thin alternating

layers of fluids can, in principle, be assembled to yield the inhomogeneity

and strong anisotropy needed to create an acoustic cloak85,86. More

physically realizable structures made of elongated rigid scatterers

surrounded by a background fluid were shown in simulation82,35 and

experiment87 to behave as a fluid in which the anisotropy in the effective

dynamic mass density is tunable.

Figure 4 | Conceptual illustration of transformation acoustics. a | An acoustic wave propagates through a simple medium with known acoustic material properties. b | The acoustic wave is deformed in a finite region via a coordinate transformation that stretches or twist the underlying coordinate grid. c | Through the mechanics of transformation acoustics, one can determine the acoustic material properties that will deform the acoustic wave in precisely the way that coordinate transformation did.

INTRODUCTION

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Anisotropic Acoustic Metamaterials

Recently, a new design paradigm called conformal mapping and coordinate

transformation has inspired a series of key explorations to manipulate,

store and control the flow of energy, in form of either sound, elastic waves

or light radiation. In electromagnetism, because of the coordinate

invariance of Maxwell’s equations, the space for light can be bent in almost

arbitrary ways by providing a desired spatial distribution of electric

permittivity ε and magnetic permeability μ.88,89 Similar design approach can

be applied to acoustic waves by a engineered space with desired

distribution of effective density and compressibility90,91,75. A set of novel

optical/acoustic devices were proposed based on transformation

optics92,93,94,95; they usually call for complicated medium with anisotropic

and spatially varying material parameter tensor to accomplish the desired

functionality. Therefore, the 2D isotropic transmission line model is

extended in this section to build an anisotropic acoustic metamaterial

which promise potential application for a myriad of fascinating devices

based on coordinate transformation.

Active Acoustic Metamaterials

Low frequencies have long wavelengths, which means treatments have to

be large to perturb or absorb the wavefronts. Active control technologies

offer the possibility of bass absorption or diffuse reflections from relatively

shallow surfaces, as well as a capability for variable acoustics. An example

application for active absorption is the control of modes in small rooms.

The cost and difficulties of implementation are considerable, however, and

this is one reason why this technology has not been more widely applied.

Active absorption has much in common with active noise control; indeed, in

many ways, it is the same concept, just reorganized by a slightly different

philosophy. Olson and May carried out pioneering experiments, and they

suggested an active noise control method based on interference5. In their

method, an electroacoustic feedback loop was used to drive the acoustic

pressure to 0 near an error microphone places close to a secondary

loudspeaker. More sophisticated systems alter the surface impedance of

the control loudspeaker towards a desired target value. They may be

configured as feedforward or feedback devices and are often constructed

INTRODUCTION

- 18 -

around single channel, filtered-x least mean square (LMS) adaptive filter

algorithms. A more effective method places resistive material in front of

the control surface (loudspeaker) to gain energy dissipation. The active

system then maximizes the particle velocity through the material.

A significant challenge in noise control engineering is achieving a high level

of performance, or noise attenuation, within low size and weight

constraints168. This challenge becomes particularly demanding at low

frequencies where the acoustic wavelength is large and the required

dimensions of traditional passive noise control treatments become

impractical for many applications. An alternative to increasing the size of

the acoustic treatment is to increase its mass but this method has clear

practical restrictions in many applications. One alternative method of

controlling low frequency noise that has been extensively investigated, and

has relatively recently become a practically viable solution in a number of

sectors, is active noise control. In the acoustic domain this technology is

based on generating a secondary sound field using control sources that

destructively interferes with the primary, unwanted sound field. This

technology has been applied in the automotive, maritime and aerospace

sectors, as well as quite extensively in consumer audio applications. In

addition to the benefits afforded by active control technology in terms of

the size and weight requirements, there is also a significant benefit in that

the system can adapt to changes in the unwanted noise source.

In more recent years an alternative approach to achieving noise control at

low frequencies has emerged based on the principles of electromagnetic

metamaterials successively developed in the acoustic field. These so-called

acoustic metamaterials use an engineered sub-wavelength structure to

achieve significant levels of noise control. In particular, such materials

might use a structure consisting of periodically arranged locally resonant

elements to introduce band gaps, which are spectral regions in which wave

propagation is forbidden. When an array of identical resonators is

employed, the bandwidth over which significant attenuation is achieved is

somewhat limited, due to the resonant nature of the band gaps. The

bandwidth over which attenuation can be achieved can be extended by

tuning the resonators in the array to multiple frequencies, however, this

method requires a large number of resonators to be employed.

INTRODUCTION

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To overcome the limitations of passive acoustic metamaterials, more

recently attention has been focused on introducing active control

technologies into acoustic metamaterials. In fact, passivity, linearity and

time-invariance impose fundamental bounds on the available choices of

acoustic parameters113,114.

Active unit cells for metamaterials with unusual acoustic properties have

been considered in several designs. The term active is used quite generally

to indicate inclusions that can provide energy to the impinging wave and

feedback to the acoustic system, that can be controlled or that are

externally biased. The most common elements used in active meta-atoms

are active transducers, micro- or nano-electromechanical systems,

piezoelectric materials and electrically loaded acoustic elements (Fig. 5).

These architectures have enabled reconfigurability and real-time

tunability, among other features63,115–129. Figure 5a shows an example of a

metamaterial composed of an array of masses with variable mechanical

connectivity, whose effective material properties can be tuned in real time

with properly controlled piezoelectric discs128 (Fig. 5b). Piezoelectric

materials provide an ideal platform to tune and control the acoustic

properties of a metamaterial in a compact way, because they respond

strongly to electrical signals and can be controlled with relatively simple

electronics118,119 (Fig. 5c). Piezoelectric effects may also be exploited in

semiconductor substrates124, and these materials may be used to provide

effective acoustic gain, that is, to amplify the acoustic wave as it

propagates through them. A similar route to acoustic gain and active

control of the acoustic properties of a metamaterial may be provided by

loading loudspeakers, which, similar to piezoelectrics, convert airborne

acoustic waves into electric signals and vice versa, using electronic

circuitry (Fig. 5d). These strategies have been successfully used, for

instance, to obtain controllable acoustic gain and loss in a lumped element

configuration112,118.

INTRODUCTION

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Parity–time-symmetric acoustic metamaterials

One particularly interesting subclass of active metamaterials, briefly

mentioned above in the context of cloaking, is that in which active

elements pumping energy into the incoming wave are paired with their

time-reversed images, which correspond to absorbing elements. Such a

combination of elements satisfies a balanced loss–gain condition that has

been shown to provide unusual acoustic responses. This field of research

has stemmed from theoretical research in the area of quantum mechanics,

where it was shown that a special class of Hamiltonians that commute with

the parity–time (PT) operator can support real energy eigenvalues even

though they are non-Hermitian130,131,132.

Recently, PT symmetry has become relevant to the field of acoustic

metamaterials, in which gain and active components are readily available.

By pairing a resonant acoustic sensor, which absorbs a substantial portion

of the impinging energy, with its time-reversed image (Fig. 6a), under

proper conditions it is possible to realize a system that can absorb the

incoming wave without creating shadows or reflections.

Figure 5 | Conceptual examples of active

acoustic metamaterial designs. a,b | A periodic

array of masses connected to the substrate

through piezoelectric discs. c | A reconfigurable

metamaterial based on piezoelectric membranes

controlled by electronics. d | Electrically loaded

loudspeakers. R, resistance; L, inductance.

INTRODUCTION

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The functionality of this PT-symmetric system is based on eigenmodal

resonances, and therefore does not require external feedback or control.

Figure 6 | Functionality and possible applications of active metamaterials. a | An invisible

acoustic sensor based on parity–time-symmetric metamaterials. b | Reciprocity in sound propagation

implies that, after reversing source and sensor, the transmission is the same. Transmitted (𝑆1, 𝑆2) and

received (𝑃1, 𝑃2) signals are related by 𝑆1𝑃2 = 𝑆2𝑃1. c | A basic nonlinear non-reciprocal system for

free space isolation composed of a frequency-selective surface (FSS) and a nonlinear material for

second-harmonic generation (SHG), which converts an incoming wave at frequency f0 to 2f0. d | An

acoustic radially symmetric resonant cavity connected to three waveguides (labelled). e | The same

structure becomes a non-reciprocal device, an acoustic circulator, as the filling fluid is moved with

moderate rotation velocity. p, pressure.

INTRODUCTION

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Non-reciprocal acoustic metamaterials

Another area of research in the field of active acoustic metamaterials is

that of non-reciprocal metamaterials, in which the breaking of time-

reversal symmetry as well as one-way propagation and isolation are

allowed. In conventional media, sound travels symmetrically in the sense

that if it is possible to transmit a signal from A to B, then it is usually

possible to transmit it with the same intensity from B to A (Fig. 6b). This

symmetry, known as reciprocity, is a fundamental property of many wave

phenomena and is attributable to the fact that wave propagation in

conventional media is time-reversible. Figure 6d shows a basic power

splitter for airborne acoustic waves, formed by a radially symmetric cavity

connected to three waveguides. At resonance, an input sound at port 1

splits equally between the output ports 2 and 3. The device is reciprocal

and, therefore, the same transmission levels are expected when sound is

input at each port. In this system, a very large isolation of over 40 dB was

realized for airborne acoustic waves upon filling the subwavelength

acoustic ring cavity with a rotating fluid (Fig. 6e). Because the filling fluid

had a modest velocity, resonant transmission was shown to be strongly

asymmetric, and the acoustic waves impinging at port 1 were routed to

port 2, isolating port 3. The non-reciprocal circulation of sound was

provided here by the fluid motion; therefore, exciting the same structure

from port 2 would provide strong transmission to port 3, breaking the

symmetry in transmission as sketched in Figure 6b. Although it is

interesting to see how such a basic active component can modify the way

sound propagates, mechanical motion of the filling material is not always

convenient or practical.

A basic scheme involves asymmetric frequency conversion and suitable

filtering, which may be achieved in its simplest form by combining a

nonlinear medium with a frequency-selective mirror (Fig. 6c): for example,

using a phononic crystal that filters the fundamental frequency but not the

second harmonic. When excited from the side of the filter at the

fundamental frequency, the structure is highly reflective; however, if

excited from the opposite side, then the nonlinear medium converts most

of the impinging energy to the second harmonic, which tunnels unaltered

through the frequency-selective mirror, breaking reciprocity.

INTRODUCTION

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Recently, some of the described non-reciprocal elements and other related

components have been properly embedded in periodic lattices to realize

topologically non-trivial band diagrams. As discussed above, the careful

tailoring of frequency, wave vector and phase propagation has led to

advances in the areas of phononic crystals and metamaterials.

Acoustic metamaterials have exploited the use of nonlinearities associated

with sound propagation in several ways other than by breaking non-

reciprocity. Tunability and reconfigurability are other desirable

characteristics for acoustic metamaterials, which could be enabled by

active unit cells with feedback and control. These characteristics could be

especially useful when combined with self-control and the ability to learn

to adapt to changes in the background. Active self-reconfiguring

metamaterials and smart materials might be employed for several

applications, such as camouflaging and advanced imaging.

INTRODUCTION

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Energy Harvesting Energy harvesting is the process of obtaining electrical energy from the

surrounding vibratory mechanical systems through an energy conversion

method using smart structures, like, piezoelectric, electrostatic

materials169. Recent advancements in low power electronic gadgets, micro

electro mechanical systems, and wireless sensors have significantly

increased local power demand. In order to circumvent the energy demand;

to allow limitless power supply, and to avoid chemical waste from

conventional batteries, low power local energy harvesters are proposed for

harvesting energy from different ambient energy sources.

Ambient energy harvesting is also known as energy scavenging or power

harvesting, and it is the process where energy is obtained from the

environment. A variety of techniques are available for energy scavenging,

including solar and wind powers, ocean waves, piezoelectricity,

thermoelectricity, and physical motions. For example, some systems

convert random motions, including ocean waves, into useful electrical

energy that can be used by oceanographic monitoring wireless sensor nodes

for autonomous surveillance. Ambient energy sources are classified as

energy reservoirs, power distribution methods, or power-scavenging

methods, which may enable portable or wireless systems to be completely

battery independent and self-sustaining.

INTRODUCTION

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Mechanical Vibrations Indoor operating environments may have reliable and constant mechanical

vibration sources for ambient energy scavenging. Vibration energy

harvesting devices can be either electromechanical or piezoelectric.

Electromechanical harvesting devices, however, are more commonly

researched and used. Roundy, Wright, and Rabaey96,97,98 reported that

energy withdrawal from vibrations could be based on the movement of a

spring-mounted mass relative to its support frame. Mechanical acceleration

is produced by vibrations that, in turn, cause the mass component to move

and oscillate. This relative dislocation causes opposing frictional and

damping forces to be applied against the mass, thereby reducing and

eventually extinguishing the oscillations. The damping force energy can be

converted into electrical energy via an electric field (electrostatic),

magnetic field (electromagnetic), or strain on a piezoelectric material.

Piezoelectric

By straining a piezoelectric material, it is possible to alter mechanical

energy into electrical energy166. Strain or deformation of a piezoelectric

material causes charge separation across the device, producing an electric

field and consequently a voltage drop proportional to the stress applied.

The oscillating system is typically a cantilever beam structure with a mass

at the unattached end of the lever, which provides higher strain for a given

input force ( Roundy & Wright, 2004 ). The voltage produced varies with

time and strain, effectively producing an irregular AC signal on the

average. Piezoelectric energy conversion produces relatively higher voltage

and power density levels than the electromagnetic system. Moreover,

piezoelectricity has the ability of some elements, such as crystals and some

types of ceramics, to generate an electric potential from a mechanical

stress99. This process takes the form of separation of electric charge within

a crystal lattice. If the piezoelectric material is not short circuited, the

applied mechanical stress induces a voltage across the material. There are

many applications based on piezoelectric materials.

INTRODUCTION

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1.2 Purpose of the Thesis

Space-coiling Metamaterials

Very recent examples of acoustic meta-surfaces include the use of

labyrinthine structures61, helical structures62, space-coiling63,64 multi-

slits65,66 and Helmholtz resonators67-70. This thesis is focused on space-

coiling metamaterials, known as a subset of double negative parameters in

acoustic metamaterials, they have recently drawn great of interest for the

exploration of extraordinary constitutive acoustic parameters38-43. The

concept is first proposed by Liang and Li38 and the corresponding design as a

single curled unit is represented in Figure 7a. Instead of using local

resonance structures such as membranes or Helmholtz resonators which are

suitable only for narrow frequency range devices, the negative refractive

index is achieved over a broad range of frequency simply by coiling the

space inside the meta-surface and prism as can be seen in Figure 7b. The

structure consists of thin plates arranged in periodic channels. In Figure 7a,

the zigzag arrows on the left-hand side denote a path of waves in the

second quadrant inside curled channels and X-shaped blue region on the

right-hand side shows a simple view of the path of the waves through the

curled channels. Through the dispersion relation derived by Floquet–Bloch

theory, which provides a strategy to analyse the behaviour of systems with

a periodic structure, unusual properties such as negative, higher and zero

refractive index could be indeed realized to satisfy the dispersion relation.

Negative and higher index are obtained below the band-gap, whereas zero

refractive index are obtained at nearly one point of frequency range which

is exactly a band-gap frequency. In fact, each curled unit cell deliberately

leads to propagate the air flow in curled channels and elongate the path of

air flow. Therefore, the phase delay occurs along the elongated path,

resulting in high refractive index. If a phase change is given with a negative

value, then the negative refractive index can be obtained. Also, zero

refractive index can be realized by squeezing waves inside the meta-

surface at a specific frequency, which shows a high transmission (Fig. 7c).

INTRODUCTION

- 27 -

This kind of symmetric geometry could be designed easily not only for two-

dimensions but also for three-dimensions through the 3D printing

technique.

Figure 7 | Space-coiling metamaterials. a | Scheme of a single curled unit cell (left-hand side). It consists of thin plates (length L, thickness d) arranging in channels of width d and lattice constant a. The zigzag arrows denote a path of waves in the second quadrant inside curled channels. X-shaped blue region shows a simple view of the path of the waves through the curled channels (right-hand side). b | Pressure field of the space-coiling metamaterials (left-hand side) and the effective medium (right-hand side) which has same conditions without coiling. It shows both are well matched and the negative refractive index is obtained. c | Pressure field for the cases of a hard solid phase (above) and coiling metamaterials surrounding a hard plate (below). High transmission with no reflection by coiling metamaterials is obtained.

The energy propagates with a negative refractive index, which causes

energy to flow in the direction opposite to that of the wave44. This counter-

intuitive effect forces an incident wave impinging on such a structure to

refract in the opposite way compared to what happens with natural

materials, enabling new ways of controlling sound waves. Several other

metamaterial-based approaches for realizing unusual acoustic refraction

have been demonstrated. By coiling up space with labyrinthine structures,

the sound propagation phase is delayed such that band folding with

negative dispersion (𝜌 < 0 and 𝐾 < 0) is compressed towards the long-

wavelength regime38,41,42. This approach has the advantage of creating

negative refraction with a relatively simple metamaterial structure.

Another strategy to obtain negative refraction relies on stacking several

holey plates to form an anisotropic structure with hyperbolic dispersion.

INTRODUCTION

- 28 -

Owing to the hyperbolic shape of the dispersion contours, refraction of

sound can take place at negative angles for almost any direction of incident

sound45,46.

Finally, an interesting regime in which the effective mass density is close to

zero has recently been explored and tested for advanced phase control and

super-squeezing of sound waves in narrow channels47,48. Such media

transmit sound waves with no distortion or phase change across the entire

length of the material and enable new sound imaging and detection

modalities. Most of the acoustic metamaterial designs described above

make use of periodic structures. The same is true for the overwhelming

majority of acoustic (and electromagnetic) metamaterials, primarily for

ease of fabrication. However, given that the concept of acoustic

metamaterials is based on the local, internal mechanical response of the

structure (DF 2), there is no reason why metamaterials cannot be made

from aperiodic architectures, provided the average number of inclusions

per unit volume remains quite uniform on the scale of a wavelength49-51.

Implementing all these different acoustic metamaterial designs requires

techniques to compute the effective acoustic properties of a given

structure. Such techniques have been developed to describe composite

materials52,53 and are suitable for many types of artificial media, providing

the valuable possibility of efficiently describing the material response in

terms of its effective mass density and bulk modulus.

INTRODUCTION

- 29 -

Acoustic Energy Harvesting

Energy harvesting technology has received considerable attention for its

promising applications in areas such as wireless sensor networks and

microelectronic devices100.For more than a decade, tremendous efforts

have been made in the harvesting of environmental energy sources

including light, thermal101 and mechanical energies102,103. In recent years,

acoustic or sound energy, which is an abundant, clean, and viable energy

resource despite being mostly wasted, has been of great interest in the

area of energy harvesting104,105.

Piezoelectric materials have received tremendous interest in energy

harvesting technology due to its unique ability to capitalize the ambient

vibrations to generate electric potential. Their crystalline configuration

allows the material to convert mechanical strain energy into electrical

potential, and vice versa. There are various approaches in vibration based

energy scavenging where piezoelectric materials are employed as the

energy conversion medium.

Energy harvesters utilize the ability of smart materials (e.g. piezoelectric,

electrostatic) to generate electric potential in response to the external

mechanical deformation100,102,113,114. Additionally, the lifespans of the

embedded batteries are limited and shorter compared to the operational

life of the host electronic devices. In many occasions, there placements or

recharging of the batteries are unproductive and at times impossible.

Battery replacements or recharging the portable electronics can be

tedious, since the batteries can die without any indication. In this digital

world, the maintenance-free wireless sensors are employed in very remote

and complex territories; for example, sensors on the civil structures (e.g.

bridge, building or aircraft) for structural health monitoring, or the use of

global positioning system tracking devices in the forests. At instances

where the battery is fully dead, it can be very expensive to replace the

battery. Sometimes sensors are integrated within a device or embedded

inside the structures, such as civil infrastructure, making it almost

impossible to replace the batteries. An energy harvesting device can be a

utilitarian alternative to the batteries. There are many areas where the

INTRODUCTION

- 30 -

harvester can generate continuous power from the ambient or the

structural vibration. Overall, the key motivation of the energy harvesting

research is to introduce the self-powered wireless electronic systems in

order to alleviate the extended power demand and eliminate the

maintenance, replacement, and the chemical waste from the old

batteries106. Energy harvesting technology is principally driven by the

deformation of the host structure due to the mechanical or the acoustic

vibrations. Typically, smart structures are embedded in the host structure

to convert the strain energy of the host structure, due to the deformation,

into the electrical potential. The converted electrical energy can either be

used to power the electronic devices directly or to store the energy in to a

battery/capacitor for later use on demand. Various types of smart

materials, like, piezoelectrics, piezoresistives, and magnetostrictives often

are used in the vibration-based energy harvesting devices.

Piezoelectric materials can be of three forms: ceramic type, polymeric

type or composite type.

The piezoelectric material exhibits two types of piezoelectric effect: direct

and converse. The direct piezoelectric effect defines the piezoelectric

material’s capacity to transform the mechanical strain into the electrical

energy while the converse effect describes the capacity to transform the

applied electrical potential into the mechanical strain energy. The direct

piezoelectric effect is responsible for the material’s ability to function as a

sensor and the converse piezoelectric effect is accountable for its ability to

function as an actuator. Thus, the piezoelectric material can be employed

both as a sensor (using direct piezoelectric effect) and an actuator (using

converse piezoelectric effect).

INTRODUCTION

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The coupled electromechanical behaviour of the piezoelectric materials

can be represented by two constitutive equations107, as follows

1) Direct piezoelectric effect: 𝐷𝑖 = 𝑒𝑗𝑗

𝜎 𝐸𝑗 + 𝑑𝑖𝑚𝑑 𝜎𝑚

2) Converse piezoelectric effect: 𝜀𝑘 = 𝑑𝑗𝑘

𝑐 𝐸𝑗 + 𝑆𝑘𝑚𝐸 𝜎𝑚

where 𝐷𝑖 represents the dielectric displacement in 𝑁𝑚/𝑉 or 𝐶/𝑚2, 𝜀𝑘 is the

strain vector, 𝐸𝑗is the applied electric field vector in 𝑉/𝑚, and 𝜎𝑚 is the

stress vector in 𝑁/𝑚2.

𝑑𝑖𝑚𝑑 and 𝑑𝑗𝑘

𝑐 are the piezoelectric coefficients in 𝑚/𝑉 or 𝐶/𝑁, 𝑒𝑗𝑗𝜎 is the

dielectric permittivity in 𝑁/𝑉2 or 𝐹/𝑚, and 𝑆𝑘𝑚𝐸 is the elastic compliance

matrix in 𝑚2/𝑁.

Figure 8 | Classification of energy harvesting approaches, based on vibration sources.

In recent years, numerous articles have reviewed the vibration based

energy harvesting technology with different means. Kim et al.108 discussed

various vibrational energy harvesting devices, discussing the energy

conversion mediums and the respective mathematical models. Sodano et

al.109 reviewed various aspects (e.g. source vibration, device efficiency,

power storage, circuitry and the damping effects) of conventional energy

harvesting models and also discussed the future goals of the piezoelectric

INTRODUCTION

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energy scavenging. Bi-stable energy harvesting systems were briefly

discussed in several articles110,111. Traditional unit cell scavengers harvest

power at a distinct frequency however, the bi-stable systems can allow

multi-frequency harvesting. A detailed review of multi-frequency and

broadband vibration energy scavenging was presented by Zhu et al.112.

Only recently, the acoustic metamaterials have been brought into the

domain of energy harvesting because of their exceptional ability to create

local resonance in a structure. Acoustic metamaterials are traditionally

used for filtering acoustic or electromagnetic waves by introducing negative

effective material properties. Chen et al.115 made an attempt to discuss

the acoustic metamaterial based energy harvesting approaches presented

in various studies.

Based on the vibration sources, energy harvesting approaches can be

classified into two major categories: intermittent and continuous (Fig.8).

Continuous sources represent the models where the host structure is

characterized by a continuous vibration at a specific frequency or band of

few frequencies, such as with machine vibration. Conversely, intermittent

sources do not rely on a set of input frequencies, but instead, depends on

the availability of the host structure deformation. The resulting power

generation is created as a result of these deformations, as in the case of a

device harvesting from irregular footsteps of a pedestrian. One major

difference between a continuous and an intermittent source, is in their

operating principal. While resonance phenomenon is the key to generate

maximum power using the continuous source, intermittent source uses pure

bending mode to harvest energy. Intermittent vibration usually comes from

living and environmental sources. The motion of the human body parts have

been the most commonly investigated living source of energy for

harvesting, however, a few attempts were made also to transform the

animal motions into the electrical power. For environmental sources, wind

and water flow are the most common forms. Continuous vibration operates

using resonance phenomena. At resonance frequency, maximum power

output can be recorded due to amplified deflection in the host structure. At

an off-resonance frequency, power output is significantly lower compared

to the resonance frequency. However, some nonlinear systems have

recently been proposed for broadband energy harvesting, where the system

INTRODUCTION

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exhibits resonance response at wider band of frequencies. Another popular

approach for broadband energy harvesting is the use of linear generator

arrays. Two types of resonance behaviours are adopted in the domain of

vibration based energy harvesting: structural resonance and local

resonance. While in the structural resonance, the whole host structure

experiences resonance behaviour, in local resonance, only a part of the

host structure exhibits resonance behaviour. Typically, the models adopting

the structural resonance are designed for high frequency energy harvesting

since a smaller geometrical configuration can be allowed. However, large

geometrical configuration is essential for low frequency applications. Low

frequency energy scavenging with sub-wavelength scale geometry is

possible by utilizing the local resonance phenomenon.

The cantilever beam is the most popular and widely used model for the

power harvesting that uses the structural resonance to harvest the energy.

Additionally, during last years a few articles have presented the energy

scavenging possibilities using the plate type harvesters112. Phononic and

sonic crystals are traditionally used for filtering acoustic waves. However,

very recently, both phononic and sonic crystals have been introduced to

the domain of energy harvesting for low frequency power generation while

maintaining the sub-wavelength scale geometry, employing their unique

ability to exhibit local resonance phenomena.

Continuous sources can be adopted for harvesting energy using resonance

phenomenon to generate maximum power when the excitation frequency

matches with the natural frequency of the host structure. Two types of

resonance phenomena (structural resonance and local resonance) have

been introduced in vibration based energy scavenging.

In recent years, metamaterials and metastructures, have been introduced

for harvesting energy. Metamaterials can be used to harvest energy by

implementing the local resonance phenomenon in electromagnetism, optics

and acoustics fields. Unlike structural resonance, in local resonance, wave

energy localizes inside the metamaterial and can be converted into the

electrical potential by placing a smart material close to the resonating

point. Metamaterials are typically a composite structure of multiple

materials or geometry. Initially, two types of acoustic metamaterials were

available: phononic crystal (PC) and sonic crystal (SC). While PC acoustic

INTRODUCTION

- 34 -

metamaterial uses the geometrical liberty to introduce the local

resonance, SCs achieve the similar effect by using multi-phase materials

with different mechanical properties. Since the local resonance frequency

of the SC metamaterials principally depends on the material properties and

the geometrical arrangement, rather than the dimension of the structure,

it is possible to scale down the harvesting devices or keep the geometry of

the harvesters to the sub-wavelength scale to access the low frequency

energy harvesting116-118.

-PC based energy harvesting: As discussed earlier, the PCs has the ability to

forbid the elastic waves from propagating at specific frequencies, called

the band gaps119. At the band gaps, the wave energy is localized in the

structure and can be harvested as the electrical energy, when the PCs are

interfaced with an energy conversion mechanism. In 2009, Gonella et al. 120

presented the mechanism between the acoustic bandgaps and piezoelectric

energy harvesting. Per the dispersion curves applying Bloch-Floquet

boundary condition, the periodic arrangement of the PCs can be used to

obtain the frequency band gaps. Resulting energy localization allows a high

level of strain at the local region, which is ideal for the effective power

harvesting. The efficiency of the mechanism of the power conversion is

comparatively very small with the conversional energy harvesters at the

lower frequencies. Thus, these devices are typically designed to operate at

the higher frequencies (>∼1KHz). To design a low frequency energy

scavenger, PC based metamaterials can be introduced by creating point

defects and exploiting the physics of resonant cavity121. The unit cells in

the PC’s may be arranged periodically as per the requirement of the stop

band. The defect or the cavity in a PC can be created by removing one or

more unit cells from the arrangement. At the resonance frequency,

acoustic energy localizes in that cavity of the PCs where Wu et al. proposed

harvesting the acoustic energy using a polyvinylidenefluride (PVDF) film122-

124. The efficiency of the PC based energy harvesters is significantly higher

compared to the conventional harvesters employed before. One additional

advantage of the PCs is that such structures can provide multiple close

frequency band gaps. Hence, energy harvesting at multiple frequencies is

possible. However, one major limitation is that the central frequency of

the bandgap depends on the lattice constant of the PCs. The higher the

INTRODUCTION

- 35 -

lattice constant, the lower the central frequency, which may limit the

implementation of the PC based energy harvesters for low-frequency

applications, considering the size requirement121.

-SC based energy harvesting: SCs, first proposed by Liu et al.125 in 2000, are

composite structures consisting of soft and stiff components. In SC, the soft

material is used as a host matrix to house the heavy mass, known as the

resonator. Ideally, SCs resemble a spring-mass mechanism in a mass-in-

mass system. Like PCs, SCs are conventionally used to stop the acoustic

wave propagation at a particular frequency125 and as a result, low

frequency stop band filters are designed using mass-in-mass systems126.

Since the filtered wave energy at the local resonance frequencies is

trapped inside the soft constituent of the SC as dynamic strain energy127, it

is possible to recover the same energy by strategically embedding smart

materials118. A maximum power can be harvested when local resonance of

the embedded mass, strains the soft composite matrix and thus the

piezoelectric material. Very limited attempts were made to model the

energy harvesters based on SCs. Zhang et al.127 introduced a SC structural

unit consisting of a square mass connected to a square frame by four

convolute folded beams. A US patent128 invented by McCoy et al., reported

a metamaterial-based vibration energy harvesting device which included a

housing element encapsulating a multiplicity of oscillators,

mechanical/electrical energy converters, and internal circuitry. The

significant advantage of this invention was its ability to harvest a significant

percentage of the total available energy in a vibrating structure. However,

these are yet to be proved experimentally. However, an acoustic energy

harvesting system usually consists of an acoustic resonating component and

an energy conversion component with either piezoelectric or

electromagnetic materials. For energy harvesting using acoustic resonators,

two types of investigations are typically carried out by means of:

conventional acoustic resonators such as the Helmholtz resonator130-133 and

quarter-wave resonator134,135, and artificial acoustic structures such as

phononic crystals and metamaterials136-140. Horowitz et al.130 first proposed

a microelectromechanical acoustic energy harvester for which a Helmholtz

resonator was used with a piezoelectric diaphragm. Li et al.134,135

developed a low-frequency acoustic energy harvester by using piezoelectric

INTRODUCTION

- 36 -

beam arrays in a quarter-wave resonator. Regarding the acoustic

metamaterials, Wu et al.137 investigated an acoustic energy harvester using

the effect of wave localization in the cavity of a sonic crystal.

Cui et al.139 demonstrated a sound energy harvester using an acoustic

grating that consists of periodic metal slits and a thin flat metal screen.

More recently, Li et al.140 proposed a membrane-type metamaterial that is

capable of both energy harvesting and sound insulation. Although a number

of acoustic energy harvesting systems can be found in the literature, many

issues have yet to be resolved such as enhancing efficiency, enabling

miniaturization, and operating at a broadband frequency bandwidth.

Especially, to harvest the sound energy in a low frequency acoustic

environment, the conventional acoustic resonators have intrinsic limitations

of size and volume. For instance, while a typical Helmholtz resonator

requires a large cavity with a long and narrow neck, a quarter-wave

resonator should have a long tube for low frequency applications. Even in

the metamaterial-based acoustic energy harvesting, most literature have

mainly reported the energy harvesting results in the high frequency range

above tens of kilohertz. Based on this observation, and on the emerging

interest in miniaturizing acoustic metamaterials for sensor applications, the

aim of this work is to investigate the optimal position of piezoelectric

elements placed inside a doubly coiled-up acoustic metastructure

presented in 2017 by Kyung Ho Sun et al.129. Firstly, a single piezoelectric

bimorph plate has been considered, secondly two different arrangements of

an array of piezoelectric plates have been introduced inside the

metamaterial cavity and will be discussed. It can be expected that the

multiple arrays of piezoelectric plates inside the acoustic metamaterial

cavity will extract more electric energy from the sound wave. Results will

be showed, discussing the interference over the sound pressure level gain

inside the metamaterial cavity, due the presence of the piezoelectric

bimorph plates and the effect on the acoustic resonant behavior of the

cavity itself. Furthermore, the proposed system could be improved both for

sensing application incorporating broadband sound energy harvesting by

optimizing the acoustic metastructure and for frequency detection

application.

INTRODUCTION

- 37 -

In a recently developed acoustic metamaterial cavity129, which consists of

double-walled metastructure, it was found that strong sound amplification

within a very small gap could be obtained by the Fabry–Perot resonance

mechanism.

This corrugated structure with an artificial ‘zigzag’ path, can provide an

acoustic metamaterial cavity at the subwavelength scale due to the

extraordinarily high refractive index38,144. Motivated by such investigation,

in this study, I propose a detection of the physical influence of the

piezoelectric plates above the acoustic amplification efficiency inside the

metamaterial cavity. Initially, the effective material properties have been

retrieved, secondly subwavelength-scale unit-cell has been designed in

COMSOL Multiphysics considering the target resonance frequency of 640 Hz

inside the metamaterial cavity of a multicell system, then various acoustic

pressure simulations inside the final 3D geometry have been considered.

The resonant frequency of the piezoelectric bimorph plates has to be tuned

at the target low frequency of 640Hz. When a piezoelectric plate is

positioned inside such metamaterial cavity, high energy conversion could

be realized by the resonance frequency matching between the acoustic

metamaterial cavity and the piezoelectric bimorph plate129.

Lastly, two different arrangements of an array of piezoelectric plates have

been made, showing their influence on resonant behavior of the cavity but

which potentially might extract more electric energy from the sound wave

which is impinging on the double-walled metastructure.

MODELING AND ANALYSIS

- 38 -

Chapter 2

Modeling and Analysis 2.1 Unit Cell and Multi-cells system

2.2 The acoustic Metamaterial Cavity

2.3 Piezoelectric Bimorph Plate and Arrays

MODELING AND ANALYSIS

- 39 -

2.1 Unit-cell and Multi-cells system The focus of this chapter is on the modeling methods for predicting the effective properties53 and performance of metacomposites for novel application development, specifically for energy harvesting. Numerical models in COMSOL Multiphysics were developed to investigate the macro behavior using homogenized macro and heterogeneous microstructure based models. Computer aided Micro Mechanical models leverage the actual micro structure to predict the macro performance. The homogenized model uses the effective properties for macro performance evaluation of metacomposites. The macro micro modeling method provides many benefits for metacomposites performance evaluation at the cost of complexity, time and resources. The micro mechanical model allows engineering the constituents to get the desired effective properties. The details of the model development and the simulation method are reported16. A theoretical description of the proposed sound energy harvesting system

has been made considering the final geometry showed in Figure 9.

The basic system consists of

an acoustic metastructure

and a piezoelectric bimorph

plate installed inside the

metastructure. The acoustic

metamaterial cavity is

formed between two coiled-

up structures, and the

interior is filled with air.

To build up an acoustic

metamaterial cavity

operating at low frequency

in this investigation,

subwavelength-scale zigzag

elements are introduced to elongate the acoustic path, as shown in Figure

10.

Figure 9 | Metastructure, which consists of a double-walled slab. The acoustic metamaterial cavity is formed inside.

MODELING AND ANALYSIS

- 40 -

DF 3| Unit cell A basic element of metamaterial design

is a unit-cell of identical building blocks, like the crystal

lattice. These unit cell interaction with electromagnetic,

acoustics or other waves manifests into macro performance with unusual

properties. The wave interaction effects with the unit-cell are critical and

the constituents of the unit-cell can be engineered to interact for unusual

properties.

-The benefits of computer aided micro mechanics

The wave propagation inside the material is due to effective interaction of

wave with the constituents, when waves move through the material they

respond to the material as a whole, as if it were a homogeneous substance.

This behavior is leveraged in two step processes for novel application

development by, first macro application development with effective

properties and secondly engineering constituents for effective property.

The optimization methodology available in COSMOL can also be leveraged

for finding the optimal constituent configuration and properties as an

inverse material design problem. The benefits of computer aided micro

mechanics are virtual cost effective new product development, reduction

in actual experimentation, faster product development, and virtual

optimization for multiple properties145. Conventional experimentation-

based methods do not leverage the essential physics of composites for

simulation-based material development. The advantages of computer-aided

micromechanics16 are rooted in physics-based model from atomic to

molecular level. This methodology enables to design new material system

with unusual and novel properties by leveraging existing materials, as

synthesis of new materials are relatively expensive. In order to develop

unique material, we need to engineer constituent properties, interface

behavior, and the microstructure or morphology. The macro properties are

emerging from the constituent property and interaction. The response of a

composite structure initiates from atomic level to molecular to morphology

to constituents to macro properties. Tailoring these parameters through

Figure 10 | Unit cell

MODELING AND ANALYSIS

- 41 -

simulations can help to design material with better properties by factoring

the constituent effects and interaction. The multilevel modeling mythology

can enable digital design of filler based materials. The schematic

illustration of multilevel modeling of computer-aided micromechanics for

metamaterial development is shown in Figure 11.

Inverse material design is another material

design opportunity, which helps to design

materials to solve for a specific industrial

problem. Exploration of microstructure or

morphology by experiments alone will take

more time. The virtual exploration is a

successful approach for faster material

design beyond the experimental limitation.

Figure 11 | Schematic illustration of multilevel modeling of computer-aided micromechanics for metamaterial development.

MODELING AND ANALYSIS

- 42 -

Numerical Model development

In this paragraph the acoustics metamaterials are considered. Hence, the

governing equation related to acoustics simulations are given below. The

following two Maxwell equations governs the interaction of electromagnetic

field with materials and relate the time variations of one field to spatial

variation of the other.

∇̅ × �̅� = 𝜎�̅� + 𝜀𝜕�̅�

𝜕𝑡

∇̅ × �̅� = −𝜇𝜕�̅�

𝜕𝑥

Where,

�̅�, electric field vector

�̅�, magnetic field vector

𝜎, conductivity

𝜀, permittivity

𝜇, permeability

Maxwell’s equation reduced into the wave equation, is used for wave

propagation investigations, as given below,

(∇2 −𝑛2

𝑐2

𝜕2

𝜕𝑡2) 𝛹 = 0

Where, n is refractive index, 𝑐 is the velocity of light in vacuum and (n2/c2 )

= . Further, refers to the electrical permittivity and refers to the

magnetic permeability. Parallelism between electromagnetism and

acoustics can be made, as was said in the previous chapter.

The acoustics wave propagation in the medium is handled by the wave

propagation equation.

MODELING AND ANALYSIS

- 43 -

Acoustic waves in a lossless medium are governed by the following

inhomogeneous Helmholtz equation:

1

𝜌0𝑐𝑠2

𝜕2𝑝

𝜕𝑡2+ ∇ (−

1

𝜌0(∇𝑝 − 𝑞)) = 𝑄

Where, ρ0 in kg/m3, refers to the density and cs in m/s is the speed of

sound, p in (N/m2), is the differential pressure and Q in 1/s2 is the source.

The pressure acoustics interface in COMSOL Multiphysics is used for

acoustical wave propagation simulations for acoustical metamaterial

applications. A model in COMSOL was obtained in 2D and 3D with

appropriate material properties, boundary conditions and mesh parameters

were used for performance. The constituent material properties and

effective properties of unit-cell models were also validated. In the next

chapter results will be discussed. The integration features were used to

estimate and compare the overall performance. The frequency dependent

effects were also considered.53

Retrieval Method for the Calculation of Effective Properties

A retrieval scattering method to extract effective properties from

reflection and transmission coefficients has been used, presented by Zhang

et al.53 This retrieval method can be used to analyze various acoustic

metamaterials. The development of these acoustic metamaterials has led

to groundbreaking demonstrations of negative acoustic properties. These

negative effective properties manifest when the appropriate resonances in

the metamaterial are strong enough so that the scattered field prevails

over the background incident field. However, since the acoustic wavelength

is much longer than the lattice constant of locally resonant acoustic

metamaterials, the scattering in an average sense is considered and

macroscopic effective properties are assigned to the metamaterial.

Effective properties can provide an accurate and simple description of

wave interaction with the associated metamaterial. Reflection and

transmission coefficients used for retrieving effective properties can be

determined also experimentally from measurements. Basically Zhang et al.

extended a method for retrieving effective material properties of

MODELING AND ANALYSIS

- 44 -

electromagnetic materials146,147 to acoustic metamaterials. The word

effective is used to signify that this is the density experienced by the

acoustic waves rather than the more normal definition of mass divided by

volume. The bulk modulus is the ratio of the pressure applied to a material

to the resultant fractional change in volume it undergoes. In this retrieval

method, the effective refractive index 𝑛𝑒 and impedance 𝜉𝑒 are obtained

from reflection (R) and transmission (T) coefficients for a plane wave

normally incident on a slab. The effective mass density and sound speed

are then calculated from 𝑛𝑒 and 𝜉𝑒.

A scheme illustrating the retrieval method is shown in Figure 12.

The metamaterial is replaced by a

homogeneous fluid slab of material

which provides the same amplitude

and phase of reflection and

transmission coefficients. Then

effective properties are obtained by

using an inverse technique.

The pressure of a plane wave propagating in a direction r is as follows148:

𝑝(𝑡, 𝑟) = 𝐴𝑒𝑖(𝜔𝑡−𝑘𝑟) = 𝐴𝑒𝑖(𝜔𝑡−𝑘𝑥𝑥−𝑘𝑦𝑦−𝑘𝑧𝑧), where 𝑘 = {𝑘𝑥, 𝑘𝑦, 𝑘𝑧} is the

wavenumber, with 𝑘𝑥 being the component in the x direction, 𝑘2 = |𝑘|2 =

𝑘𝑥2 + 𝑘𝑥

2 + 𝑘𝑦2 + 𝑘𝑧

2; A is a constant related to the magnitude of the wave;

𝑟 = {𝑥, 𝑦, 𝑧} is the location of the observation point; t is time; and 𝜔 =

2𝜋𝑓 = 𝑘𝑐 is the angular frequency, where f is the frequency and c is the

speed of sound.

Let us consider the reflection R and transmission T coefficients for a plane

wave incident on an acoustic layer with density ρ2 and sound speed c2

placed between two different media with densities ρ1, ρ3 and sound speeds

c1, c3 149 :

𝑅 = (𝑍1 + 𝑍2)(𝑍2 − 𝑍3)𝑒−2𝑖𝛷 + (𝑍1 − 𝑍2)(𝑍2 + 𝑍3)

(𝑍1 + 𝑍2)(𝑍2 − 𝑍3)𝑒−2𝑖𝛷 + (𝑍1 − 𝑍2)(𝑍2 − 𝑍3), (1)

𝑇 = 4𝑍1𝑍2

(𝑍1 − 𝑍2)(𝑍2 − 𝑍3)𝑒𝑖𝛷 + (𝑍1 + 𝑍2)(𝑍2 + 𝑍3). (2)

Figure 12 | Method for retrieving effective material properties. The metamaterial is replaced by a homogeneous fluid slab, providing same properties retrieved before.

MODELING AND ANALYSIS

- 45 -

In these equations, 𝑍𝑖 = 𝜌𝑖𝑐𝑖 (𝑖 = 1,2) is the acoustic impedance, ϑi is the

angle between the wave vector and layer normal, 𝛷 = 2𝜋𝑓ℎ/𝑐2 the phase

change across the metamaterial slab, f the frequency of the acoustic wave,

and ℎ the slab thickness (Fig. 13).

For the simplified case of a plane wave normally incident on a slab with

identical medium (air) on both sides, the reflection and transmission

coefficients reduce to:

𝑅 =𝑍2

2 − 𝑍12

𝑍12 + 𝑍2

2 − 2𝑖𝑍1 𝑍2cot 𝛷, (3)

𝑇 =1 + 𝑅

cos 𝛷 +𝑍2 𝑖 sin 𝛷

𝑍1 , (4)

Introducing 𝑚 = 𝜌2/𝜌1, 𝑛 = 𝑐1/𝑐2, 𝑘 = 𝜔/𝑐1, and 𝜉 =𝜌2𝑐2

𝜌1𝑐1=

𝑍2

𝑍1, we obtain

𝑅 = tan(𝑛𝑘ℎ)(

1𝜉

− 𝜉)𝑖

2 − tan(𝑛𝑘ℎ) (1𝜉

+ 𝜉)𝑖, (5)

𝑇 = 2

cos(𝑛𝑘ℎ) [2 − tan(𝑛𝑘ℎ)(1𝜉

+ 𝜉)𝑖]. (6)

These formulas are identical to the ones obtained for the electromagnetic

field146,150.

By inverting last two equations, the effective acoustic impedance 𝜉𝑒 and

effective refractive index 𝑛𝑒 of the acoustic metamaterial slabs are given

by the following:

𝑛𝑒 =±𝑐𝑜𝑠−1(

12𝑇

[1 − (𝑅2 − 𝑇2)])

𝑘ℎ+

2𝜋𝑚

𝑘ℎ , (7)

𝜉𝑒 = ±√(1 + 𝑅)2 − 𝑇2

(1 − 𝑅)2 − 𝑇2, (8)

where m is the branch number of 𝑐𝑜𝑠−1 function.

MODELING AND ANALYSIS

- 46 -

As can be seen both the effective impedance and refractive index are

complex functions of complex variables. Mathematically, any combination

of signs in (7) and (8) and any m result in the same values for reflection and

transmission coefficients. This problem of selecting the branch number m

can be circumvented by determining the effective parameters (ρe, Ke) of a

minimum thickness metamaterial, for which m is zero.

In this work, the effective acoustic parameters have been estimated by

adopting the conditions to satisfy a passive acoustic medium, such

as 𝑅𝑒 (𝜉𝑒) > 0 and 𝐼𝑚 (𝑛𝑒) > 0. The effective refractive index and

impedance for the geometry shown in Figure 13 were calculated by Finite

Element Analysis (FEA), integrating the total acoustic pressure field on the

inlet side of the waves and on the outlet and their values will be given in

the next session.

In Figure 13, the zigzag metamaterial

slab is replaced by a slab of a

homogeneous material. Alternatively

the effective refractive index can be

intuitively estimated by the following

simple relation for the geometrical

dimensions: 𝑛𝑒 ∝𝑙

ℎ , where l is the

length of the zigzag path and h the

slab thickness.

Physically, multiple elongated zigzag paths provide a high refractive index

ne due to the low effective sound velocity per unit thickness of a slab.

Therefore, the extension of the zigzag path of the proposed metamaterial

slab in the direction of wave propagation yields the strong confinement of

the sound energy inside the acoustic metamaterial cavity.

Figure 13 | Unit cell geometry. h is the slab thickness, l the zigzag path length.

INLET

OUTLET

MODELING AND ANALYSIS

- 47 -

When 𝑅𝑒 (𝜉) or 𝐼𝑚 (𝑛) is close to zero, errors in measurement or calculation

of reflection and transmission coefficients may cause incorrect

combinations of signs in Eqs. (7) and (8)129. This would create

discontinuities in n and Z with frequency. To overcome this problem, last

two equations can be rewritten in the form:

𝜉 =𝑟

1 − 2𝑅 + 𝑅2 − 𝑇2 , 𝑛 =

−𝑖 log 𝑥 + 2𝜋𝑚

𝑘𝑑 , (9)

where

𝑟 = ∓√(𝑅2 − 𝑇2 − 1)2 − 4𝑇2 , 𝑥 = (1 − 𝑅2 + 𝑇2 + 𝑟)

2𝑇 . (10)

Solving the right-hand side of the expression for r, and selecting whichever

of the two roots yields a positive solution for 𝑅𝑒 (𝜉). This value of r, if used

in the expression for x eliminates its ambiguity in the expression for n. This

procedure provides consistent results and allows one to avoid nonphysical

solutions due to incorrect selection of second sign in Equations (7) and (8).

However, the determination of 𝑅𝑒 (𝑛) from Equation (5) is complicated by

the fact of choosing the proper value of m. Moreover, for thick

metamaterials, the solutions of n for different values of m can lie close to

each other.

Another important issue in obtaining the effective material properties is

the determination of positions of metamaterial boundaries. The phase of

reflected and transmitted waves should be measured at the surfaces of the

metamaterial, which are not well defined. The importance of boundary

positions is illustrated by Zhang et al. using an example53.

The presented method was adapted as a computer code for retrieval

properties of a slab with known properties. The reconstructed values of c

and ρ will be discussed in the next chapter. They show excellent agreement

with values obtained experimentally by other authors. Average absolute

error was equal to 10−6 when reflection and transmission coefficients were

obtained from finite element modeling data (FEM)151.

MODELING AND ANALYSIS

- 48 -

2.2 The acoustic Metamaterial cavity

The Acoustic Metamaterial Cavity (without pzt)

In this section, we describe how the acoustic pressure is amplified in the

acoustic metamaterial cavity and then used to harvest electric energy

introducing a piezoelectric bimorph plate inside the cavity itself for energy

scavenging applications. For acoustical performance of the acoustic

metamaterial cavity a small frequency range was considered to perform the

simulation from 450 Hz to 750 Hz, with Δf = 10 Hz. The plane wave

radiation boundary condition is adopted on both inlet and outlet to

eliminate the reflected waves and an incident pressure field is applied of 2

Pa at the inlet only along the y-axis characterized by longitudinal waves.

The choice of boundary conditions will be specified in the next chapter.

To obtain the strong sound amplification in the acoustic metamaterial

cavity, two mechanisms based on the Fabry–Perot resonance are used as

follows; (1) the compression of sound wave by the effect of high refractive

index provided by multiple elongated zigzag paths, due to the low effective

sound velocity per unit thickness of a slab and (2) the formation of a half-

wavelength resonator due to the double walled structures.

In fact, another important mechanism in obtaining high sound pressure

level (SPL) is the formation of

a half- wavelength resonator

in the cavity in which a

standing wave is compressed

inside a small effective volume

cavity. A complete transmission of

the incident wave can be observed

at the fundamental resonant

frequency (Fabry-Perot resonance) in this metastructure, due to the

absence of the cut-off frequency unlike electromagnetic waves152. In this

situation, the sound intensity 𝐼𝑠𝑙𝑎𝑏 = 𝑃𝑠𝑙𝑎𝑏𝑣𝑝 , where 𝑃𝑠𝑙𝑎𝑏 is the sound

pressure of the slab and 𝑣𝑝 is the particle velocity, high sound pressure

amplification inside the metamaterial cavity can be attained, because the

particle velocity inside the cavity is very slow.

Figure 14 a | Half wavelength resonator (open outlet),

compared with b | Quarter wavelength resonator (closed outlet).

MODELING AND ANALYSIS

- 49 -

At the fundamental resonant frequency, therefore, the doubly coiled-up

structures essentially create a half-wavelength resonator in which a node

with the lowest particle velocity is located at the center of the cavity as

will be showed in the results discussed in next session.

In addition, a simple design guideline to the acoustic metamaterial cavity

can be provided through the circuit analysis. In terms of a lumped acoustic

system, the metastructure with the acoustic cavity can be considered as a

series of LC resonator (Fig. 15).

Two metamaterial slabs and the acoustic cavity

sandwiched in between are regarded as an acoustic

inductance element (i.e. acoustic mass) and acoustic

capacitance element (i.e. acoustic compliance),

respectively.

The values of the effective inductance 𝐿𝑒 and capacitance 𝐶𝑒 can be found

from 153

𝐿𝑒 ∝2𝜌𝑎𝑖𝑟𝑙

𝑠 𝑎𝑛𝑑 𝐶𝑒 ∝

𝑎𝑔

𝜌𝑎𝑖𝑟𝑐𝑎𝑖𝑟2 (9)

where s is the channel width, a is the width of the periodic

unit cell and g is the cavity size as shown in Figure 16. The

symbols 𝜌𝑎𝑖𝑟 and 𝑐𝑎𝑖𝑟 denote the density and speed of the

air, respectively. The resonant frequency 𝜔0 and its

bandwidth B can be determined by

𝜔0 ≈ √1

𝐿𝑒𝐶𝑒 𝑎𝑛𝑑 𝐵 ≈ √

𝐶𝑒

𝐿𝑒 (10)

Figure 15 | A series of LC resonator.

Figure 16 | Cross-sectional view of two metamaterial slabs, separated by a subwavelength air gap.

MODELING AND ANALYSIS

- 50 -

From equations (9) and (10), one can find that the elongated zigzag path

(large l value) and narrow channel (small s value) lead to lower resonant

frequency with narrow frequency bandwidth, due to the large value of

effective inductance. A longer cavity (large g value) also leads to lower

resonant frequency with broad frequency bandwidth.

Finally, to design a metamaterial cavity for a desired resonant frequency

and a bandwidth, the size of a unit cell (a), the cavity size (g) and the

length of the zigzag path (l) should be synthetically adjusted, considering

equations (7)–(10).

Recently Song, K et al. design and experimentally demonstrate an acoustic

metamaterial localization cavity which is used for sound pressure level

(SPL) gain using doubly coiled-up space like structures. With a

subwavelength cavity that is 1/10th of the wavelength of the incident

acoustic wave, the SPL gain of an external acoustic signal inside an ultra-

small cavity consisting of 13 dB can be obtained at a fundamental

frequency of 990 Hz143. Sun et al. increased a sound pressure level (SPL) by

up to ∼16dB at a resonant frequency of 600 Hz129.

In the next chapter the amplification results obtained from the Fabry-Perot

resonance of the cavity, performed using FEM simulations will be showed.

To confine the acoustic energy into a small volume, two parallel slabs

composed of periodic corrugated structures has been used. This structure

provides an effective sound path, resulting in a high effective refractive

index 𝑛𝑒 and effective impedance 𝜉𝑒, that provide highly efficient sound

entrapment and miniaturization of the device, respectively.

The acoustic metastructure that has been designed for this study is shown

in Figure 9. This sample has dimensions of Lx (130 mm) × Ly (2h+g) × Lz (100

mm). The subwavelength-scale metastructure sizes are: h = 11 mm, a = 10

mm, w = 7 mm, t = 1 mm and g = 30mm. The localization cavity is created

using two acoustic metamaterial slabs that are separated by a

subwavelength air gap, g, to form a cavity. The stiff corrugated structures

form an artificial zigzag path along the direction of wave propagation,

effectively creating a coiling-up space38,142,154.

MODELING AND ANALYSIS

- 51 -

I have used COMSOL Multiphysics, a finite element software package, to

theoretically predict the SPL

using the following physical

parameters for the structure,

density = 2.7 ⨯ 103 𝑘𝑔/𝑚3,

Poisson’s ratio = 0.35, and

Young’s modulus = 70 ⨯ 109 𝑃𝑎 , to simulate aluminium, and air is used as

the working fluid. The subwavelength-scale metastructure has been

designed with a resonant frequency tuned at the target low frequency of

640 Hz following the study carried out by Sun et al. At this frequency, the

wavelength is approximately 50 times the periodicity and 11 times the total

length of the structure in the direction of wave propagation. I’ve

theoretically predicted the SPL amplification rate by a subwavelength

cavity that is formed by two effective medium slabs for which the

refractive index and the impedance have been calculated by the numerical

scattering method. At full transmission, the sound intensity inside the

cavity, 𝐼𝑐𝑎𝑣, is identical to the incident sound intensity, 𝐼0. This result

indicates that the energy inside the cavity is not changed by the presence

of the acoustic metamaterial cavity. The sound intensity is the product of

the sound pressure, P, and the particle velocity, 𝑣𝑝; thus, the sound

pressure, P, inside the zigzag metamaterial, 𝑃𝑐𝑎𝑣, can be expressed as

𝑃𝑐𝑎𝑣 = 𝐼0

𝑣𝑝⁄ . This equation shows that for equal sound intensities, the SPL

inside the cavity becomes a function of 𝑣𝑝 only. Thus, the extraordinary

amplification of the SPL originated from the low particle velocity within

the cavity. That is, under these conditions, a standing wave forms inside

the metamaterial cavity, and at the fundamental resonance frequency, a

displacement node forms at the centre, which is the point of maximum

pressure variation. Therefore, this configuration can form a first harmonic

inside a highly miniaturized cavity, thereby ‘‘focusing’’ the pressure field

inside the subwavelength gap. Note that the amplification of the SPL in the

gap is controlled by 𝜉𝑒 and 𝑛𝑒 because 𝑣𝑝 depends on the reflection

coefficients of the two walls, which in turn are also determined by 𝜉𝑒 and

𝑛𝑒. At the fundamental resonant frequency, the SPL in the metamaterial

Figure 17 | Theoretical prediction of sound pressure amplification.

MODELING AND ANALYSIS

- 52 -

cavity may be influenced by the piezoelectric plates presence. This effect

will be discussed in the next section.

However, strong sound confinement inside the air gap can be achieved by

high impedance discontinuities between the metamaterial and air. Thus, as

the path length is decreased, the resulting SPLs are also lowered because

of the decrease in 𝜉𝑒 and 𝑛𝑒, as shown in Song K. et al. study, who

analysed the sound amplification for three different samples obtained by

calculating the SPL and the particle velocity across the acoustic cavity

when normal incident sound waves impinged on the cavity.

MODELING AND ANALYSIS

- 53 -

2.3 Piezoelectric bimorph plate

The Piezoelectric Energy Harvester:

In the proposed geometry, the confined sound energy with the amplified

pressure inside the cavity is converted to electrical energy through the

acoustically driven mechanical vibrations of a piezoelectric material. To

convert the acoustic energy to the electrical one, firstly a piezoelectric

bimorph plate with a tip mass is considered in this thesis, where both ends

are fixed with respect to the structure, then an array of piezoelectric

plates will be considered.

The piezoelectric layers on the top and bottom of the aluminium substrate

are electrically connected in parallel as shown in Figure 18.

From a theoretical point of view the electro-mechanically coupled

governing equations of a piezoelectric bimorph plate can be expressed

based on the Rayleigh–Ritz method as155

𝑀𝑒𝑞ü𝑦(𝑡) + 𝐶𝑒𝑞ů𝑦(𝑡) + 𝐾𝑒𝑞𝑢𝑦(𝑡) + 𝛼𝑉0(𝑡) = 𝑓𝑦(𝑡) (11)

𝛼 ∙ 𝑢𝑦(𝑡) − 𝐶𝑝 ∙ 𝑉0(𝑡) = −𝑄𝑒(𝑡) (12) Where 𝑄𝑒(𝑡) is the generated charge at the electrodes, 𝑉0(𝑡) is the output

voltage, 𝐶𝑝 is the capacitance of the piezoelectric layers at constant strain,

𝛼 is the electromechanical coefficient, 𝑢𝑦(𝑡) denotes the displacement of

the tip mass in the transverse direction, and 𝑓𝑦(𝑡) is the applied force due

to the acoustic pressure. The symbols, 𝑀𝑒𝑞, 𝐶𝑒𝑞 , and 𝐾𝑒𝑞 denote the

equivalent mass, damping, and short-circuit stiffness respectively,

evaluated at the fundamental bending mode.

Figure 18 | Bimorph plates are connected in parallel; arrows show the poling direction.

MODELING AND ANALYSIS

- 54 -

Although the proposed energy harvesting system in is an acoustic-

structural-piezoelectric coupled system, equations (11) and (12) do not

represent governing equations fully coupled analysis (for a more rigorous

treatment, see156). However, through equations (11) and (12) the electrical

output is expected to be obtained by piezoelectricity because the applied

force 𝑓𝑦(𝑡), produces mechanical strain. This thesis does not pursue the

fully coupled analysis for design of the piezoelectric bimorph plate inside

the cavity; instead a simple vibration mode analysis of a piezoelectric

bimorph plate alone will be performed for tuning the resonant frequency to

that of the acoustic metamaterial cavity.

For the piezoelectric material, PZT 5H is employed because of its high

energy density. PZT 5H is

characterized by mass density of 7500

𝐾𝑔/𝑚3, Poisson ratio of 0.33 and

Young’s modulus 70 𝐺𝑃𝑎. Brass is used

as proof mass, with Young’s modulus

of 110 𝐺𝑃𝑎 and mass density of

8470 𝐾𝑔/𝑚3.

While aluminium is used for the substrate, with Young’s modulus of 70GPa

and mass density of 2700 𝐾𝑔/𝑚3. The electrical connection of the

piezoelectric plates has been shown in Figure 19.

Figure 19 | Piezoelectric bimorph plate structure with a tip mass.

MODELING AND ANALYSIS

- 55 -

Array of Piezoelectric Bimorph Plates

An array of two piezoelectric bimorph plates in represented below (Fig.20),

to show their electrical connection. The piezoelectric

bimorph plates are connected in parallel. Different

simulations have been performed after having

characterized the SPL gain inside the metastructure.

The physical influence of the piezoelectric bimorph

plate on the sound pressure amplification inside the

metamaterial cavity has been considered. The

piezoelectric element is seen as an obstacle for the

plane wave radiation field imposed externally. Firstly, a single

piezoelectric bimorph plate has been introduced at the centre of the

metamaterial cavity, secondly two different configurations have been

considered inserting three piezoelectric bimorph plates aligned centrally

along the horizontal axis and lastly along the vertical axis. Simulations’

results will be showed in the next section, highlighting the piezoelectric

elements influence over the resonant behaviour of the acoustic

metamaterial cavity.

Figure 20 | An array of two piezoelectric bimorph plates connected in parallel.

RESULTS AND DISCUSSION

- 56 -

Chapter 3

Results and Discussion 3.1 Unit Cell Effective Parameter calculation 3.2 Doubly coiled-up Acoustic Metamaterial Cavity amplification

3.3 Piezoelectric Bimorph Pate influence

RESULTS AND DISCUSSION

- 57 -

3.1 Unit Cell effective parameter calculation

In order to calculate the refractive index 𝑛𝑒 and 𝜉𝑒, the retrieval scattering

method has been used, based on the reflectance and transmission

coefficients, which can be obtained by COMSOL Multiphysics 5.3, a finite

element software package. These coefficients can be calculated on a single

unit cell (Fig. 21) by using normally incident soundwaves53. In order to

obtain reflection and transmission coefficients an acoustic field has to be

integrated at both inlet and outlet boundaries of the acoustic wave path.

Air has been used for the acoustic wave path and aluminium for the slits of

the unit cell. The pressure acoustic, frequency domain interface in COMSOL

has been adopted. To generate the plane wave incident field the radiation

boundary condition has been used, with a pressure amplitude of 2 Pa. The

periodic arrangement of the unit cells in the final geometry yields possible

to apply a Floquet periodic condition. By inverting the scattering

coefficients for calculating 𝑛𝑒 and 𝜉𝑒 by using constitutive conditions of

passive acoustic medium: 𝑅𝑒 (𝜉𝑒) > 0 and 𝐼𝑚 (𝑛𝑒) > 0. The importance of

the determination of positions of metamaterial boundaries is explained by

Sun et al.53, cause the phase of reflected and transmitted waves should be

measured at the surfaces of the

metamaterial, which are not well

defined. The importance of boundary

positions is illustrated using a

metamaterial constructed from hollow

silicone rubber cylinders immersed in

water. The weight function Ψ provides

a frequency averaged measure of the

difference in effective properties

between the two slabs as a function of the displacements of effective

boundaries position of the metamaterial from the cylinder surface,

considering the total number of frequencies.

From equations (7) and (8), showed in the previous section the designed

metamaterial slab provides the effective acoustic parameters with 𝑛𝑒 =

4.01 and 𝜉𝑒 = 9.38, thus yielding a high refractive index and high

Figure 21 | Unit cell effective parameters calculation.

RESULTS AND DISCUSSION

- 58 -

impedance, respectively (in this work, 𝜉𝑒 and 𝑛𝑒 are normalized to the

background fluid which is air).

In addition, the effective density 𝜌𝑒 and effective velocity 𝑐𝑒 of the

metamaterial slab can be obtained by the following relations129

𝜌𝑒 = (𝑛𝑒 ∗ 𝜉𝑒) ∗ 𝜌𝑎𝑖𝑟

𝑐𝑒 =𝑐𝑎𝑖𝑟

𝑛𝑒⁄

It should be noted that the imaginary part of density changes sign with

frequency129. This should not be interpreted as changing from attenuation

to gain, as that is dependent on the imaginary part of the sound speed. The

sign change in the imaginary part of the effective density simply coincides

with a change in the phase of reflected and transmitted acoustic waves.

Finite element simulations150 confirm this fact.

RESULTS AND DISCUSSION

- 59 -

3.2 Doubly coiled-up acoustic Metamaterial Cavity amplification

Numerical simulations were carried out using COMSOL Multiphysics 5.3

software package. In this investigation, the subwavelength-scale

metastructure has been designed (Fig. 21) with a resonant frequency tuned

at the target low frequency of 640 Hz.

The zigzag metamaterial slab is replaced by a slab of a homogeneous

material, characterized by the proprieties retrieved before. The sound

amplification was obtained by calculating the sound pressure level (SPL)

and the particle velocity (𝑣𝑝) across the acoustic cavity when normal

incident sound waves impinged on the cavity. The acoustic metamaterial

was periodic; thus, periodic boundary conditions have been used for all of

the calculations. Numerical simulations are provided to evaluate the sound

amplification performance of the final geometry in 3D in the low frequency

range from 450 Hz to 750 Hz, with Δf = 10 Hz.

Figure 22 | Theoretical prediction of sound pressure amplification. a | Calculated sound pressure within metamaterial cavity, compared with b | calculated values obtained using effective medium theory.

RESULTS AND DISCUSSION

- 60 -

Characteristics of the Acoustic Metastructure

In this paragraph, the characteristic performance of the empty three-

dimensional acoustic metastructure are investigated.

These properties are shown in Table 2, and used in numerical simulations.

Metamaterial slab (aluminium)

Density

2700 𝑘𝑔/𝑚3

Young’s modulus

70 𝐺𝑃𝑎

Poisson’s ratio

0.33

Medium (air at 293.15 K)

Density (𝜌𝑎𝑖𝑟) Sound speed (𝑐𝑎𝑖𝑟)

1.203 𝑘𝑔/𝑚3 343𝑚/𝑠

Effective medium slab

Density(𝜌𝑒) Sound speed (𝑐𝑒)

45.25 𝑘𝑔/𝑚3 85.54 𝑚/𝑠

From equations (7) and (8), showed in the previous section the designed

metamaterial slab provides the effective acoustic parameters with 𝑛𝑒 =

4.01 and 𝜉𝑒 = 9.38, thus yielding a high refractive index and high

impedance, respectively (in this work, 𝜉𝑒 and 𝑛𝑒 are normalized to the

background fluid which is air).

The calculated properties of the effective medium slab are found in Table

2.

Next, I predicted the sound pressure amplification of the acoustic cavity

formed in a three-dimensional FEA, a rigid wall boundary condition is

applied at all of the planes parallel to the direction of the incident wave

propagation. Here, damping effects are also considered because the

damping mechanism of thermal-viscous losses is inevitable for sound

propagation inside the narrow zigzag channels157. The amplitudes of

transmission peaks are measured to be slightly less than unity due to the

damping effect. Physically, the transmission peaks are attributed to the

existence of Fabry-Perot resonance modes inside the high-indexed

metamaterial cavity. It needs to be mentioned that the Fabry–Perot

resonance imposes a substantial restriction on high transmission in

broadband. To overcome such a problem, helical-structured metamaterial

Table 2 | Characteristic properties of the three-dimensional acoustic metastructure.

RESULTS AND DISCUSSION

- 61 -

has been designed62 with progressive lead, which has much less acoustic

impedance mismatch with air than

the one with constant lead.

Then, the SPL gain of the final 3D

metastructure is calculated and

measured along y-axis, at the

position of 𝐿𝑥

2⁄ ,𝐿𝑧

2⁄ (Fig. 23b)

using a cut line 3D, obtaining a good

agreement with the results obtained

for the subwavelength-scale

effective metastructure (Fig.

23a) using a cut line 2D.

Figure 23 shows a numerical

comparison of the SPL gain for a

metastructure with the cavity size of 𝑔 = 30 𝑚𝑚 .

As shown, the three-dimensional FEA simulation results made on the final

structure are in good agreement with those obtained for the subwavelength

metastructure. Note that, as the cavity size is increased, the SPL gain

decreases129. Nonetheless, the increased cavity size causes the confining of

the acoustic waves with a lower resonant frequency inside the cavity. It

can be observed that the acoustic metamaterial cavity with 𝑔 = 30 𝑚𝑚

yields the maximum SPL gain of 14 𝑑𝐵 at around 640 𝐻𝑧.

Figure 23 | a | Graph of sound pressure level of the effective subwavelength-scale metastructure obtained using a cut line 2D. b | Graph of sound pressure level of the final geometry obtained using a cut line 3D.

a

b

RESULTS AND DISCUSSION

- 62 -

An acoustic metamaterial unit cell with a higher effective refractive index

has been designed by increasing the length of the narrowest channel (𝑙), as

can be seen in Figure 24a and b. It should be noted an higher SPL gain of

16 𝑑𝐵 at around 420 𝐻𝑧 .

Note that the wavelength (𝜆 ~ 536 𝑚𝑚) at 640 𝐻𝑧 is approximately 11

times the width of the metastructure 𝐿𝑦 = 52 𝑚𝑚 thereby realizing a

compact acoustic energy harvesting system operating at low frequencies.

Figure 24 | a | Higher effective index subwavelength-scale metastructure. b | Graph of sound pressure level of the 2D subwavelength-scale metastructure obtained using a cut line 2D.

a b

RESULTS AND DISCUSSION

- 63 -

Figure 25 shows a cross-sectional view of the spatial SPL gain distribution

with 𝑔 = 30 mm at 640Hz.

The metastructure can be

observed to behave similar to

a half-wavelength resonator

with the maximum SPL gain at

the centre of the cavity,

although it yields an almost

uniform SPL gain inside the

entire space of the

metamaterial cavity. Pressure

is increased due to reduced

particle velocity (𝑃 = 𝐼𝑣𝑝

⁄ ).

This amplification behaviour can be explained by the high impedance

discontinuities between the air and the metamaterial slabs.

The acoustic path-coiling metamaterial was then proven to support high

transmission close to the Fabry-Pérot resonant frequency with

subwavelength cell size. The proposed system could be improved to

incorporate broadband sound energy harvesting by optimizing the acoustic

metastructure. By choosing suitable channel width and path length, the

unit cell can cover a complete 2π phase change across the structure. To

ensure precise phase control over the 2π range, the path length has to be

changed for different unit cells.

However, such resonance-based acoustic metamaterial unit cells suffer

from a narrow bandwidth, as high transmission only exists near the

resonant frequencies. When the frequency of the incident wave is off

resonance, the impedance mismatch between the narrow channel and the

surrounding background medium at the entrance leads to high reflection. In

order to circumvent this problem, impedance matching layers (IMLs) have

been introduced in a helical cell design to allow better impedance

matching158. Cummer et al. developed a design approach for a set of

labyrinthine metamaterial unit cells of different indices with shared,

controllable working frequency and bandwidth. The additional phase shift

Figure 25 | Cross-sectional view of the spatial SPL gain distribution inside the metastructure.

RESULTS AND DISCUSSION

- 64 -

induced by each unit cell is measured by the time delay of the signal at the

microphones, which are attached at inner sides of the waveguide used in

their experimental setup.

On the other hand, Memoli et al.60 developed the notion of quantal meta-

surfaces to demonstrate a different metamaterial concept, based on the

use of a small set of pre-manufactured 3D unit cells, termed metamaterial

bricks, which can be assembled into 2D structures on-demand using a

compression algorithm based on the discrete wavelet transform (DWT). The

bricks become, in isolation, the building blocks of an assembly, encoding

prerequisite phase delays.

It is of growing interest the investigation of broadband metamaterials,

potentially via the use of multilayer or 3D metamaterials that enables

multiple resonances to be cascaded to extend the bandwidth response.

Emphasis will be on relatively constant constitutive parameters across the

operating frequency band.

Figure 26 | a | A 3D rendering of Memoli’s brick. b | Cross-sections of 16 selected bricks and the corresponding phase maps at normal incidence.

RESULTS AND DISCUSSION

- 65 -

3.3 Piezoelectric Bimorph Pate influence Finally, when piezoelectric plates are positioned inside such metamaterial

cavity and the same amplification simulation is performed, I will show that

the piezoelectric element presence influences the resonant behaviour of

the acoustic metamaterial cavity.

Piezoelectric Bimorph Plate

The confined sound energy with the amplified pressure inside the cavity is

converted to electrical energy through the acoustically driven mechanical

vibrations of the piezoelectric plate. To convert the acoustic energy to the

electrical one, a piezoelectric bimorph plate with a brass tip mass is

considered in this study, with the fundamental frequency of about 640 Hz,

as shown in Figure 27. The use of a tip mass affects the structural integrity

and durability of the harvesters while the increased effective surface area

designs exhibit torsion (causing voltage cancellation effect) in the very

important first vibration mode. The proof mass is stepped to minimize the

zero stain area in the attachment to the piezoelectric material.

Both ends of the bimorph plate

are simulated to be fixed to the

supporting beams of the

structure applying a fixed

constraint on external

boundaries.

The piezoelectric layers on the top and bottom of the aluminium substrate

have to be electrically connected in parallel as shown in the second

chapter.

Figure 27 | Configuration of the proposed piezoelectric bimorph plate. a | The piezoelectric bimorph measures with a tip mass whose piezoelectric layers are connected in parallel. b | Tip mass measures.

a b

RESULTS AND DISCUSSION

- 66 -

A simple vibration mode analysis of the piezoelectric structure alone will

be performed for tuning the resonant frequency to that of the acoustic

metamaterial cavity using the solid mechanics module for piezoelectric

devices in COMSOL Multiphysics.

The electrical output is expected

to be obtained by piezoelectricity

because the applied force

𝑓𝑦(𝑡) , which is calculated by

integrating the pressure

distribution over the entire

surface of the piezoelectric

bimorph plate with a proper

assumed mode, produces

mechanical strains.

The first three vibration modes of the proposed piezoelectric bimorph plate

are shown in figures. To estimate the energy conversion efficiency of the

acoustic metamaterial cavity, the voltage gain (G) has to be calculated by

using the relation of 𝐺 = 20 log10 (𝑉𝑚

𝑉𝑜⁄ ), where 𝑉𝑚 and 𝑉𝑜 represent the

output voltage of the piezoelectric bimorph plate with and without the

cavity, respectively. The result should be almost identical to the SPL gain

of ∼14dB measured inside the cavity at 640Hz. It also indicates that the

resonant frequency of the acoustic metamaterial cavity with 𝑔 = 30 𝑚𝑚 is

approximately equal to the fundamental frequency of the piezoelectric

Figure 29 | 1st twisting mode.

Figure 28 | 1st bending mode - (FP resonant mode).

Figure 30 | 2nd bending mode.

RESULTS AND DISCUSSION

- 67 -

bimorph plate. K H Sun et al.129 experimentally validated their results

obtained through simulations, calculating the output power by 𝑉𝑚

2

2𝑅⁄ .

They reached the maximum output power of 0.345 μW at an optimal

resistance of 20 kΩ, which is about 40 times higher than that of the

piezoelectric bimorph plate without the acoustic metamaterial cavity.

In this work, an amplification simulation of the 3D metastructure is

performed showing piezoelectric plates influence over the resonant

behaviour of the acoustic metamaterial cavity, as shown in Figure 31a and

b.

As can be seen from the

graphs, the SPL gained

inside the empty

metamaterial cavity

(Fig. 31a) is around 14

dB, when a

piezoelectric bimorph

plate is positioned at

the centre of the

metastructure (Fig.

31b) it does not change

significantly.

It seems that the

amplification performance

of the metastructure is

not influenced by the

presence of the

piezoelectric element.

It represents physically an

obstacle for the plane

waves of the incident

pressure field of amplitude

2 𝑃𝑎, as can be seen from

discontinuities in the

a

b

Figure 31 | SPL graphs of the metamaterial cavity. a | without piezoelectric bimorph plate. b | within piezoelectric element.

RESULTS AND DISCUSSION

- 68 -

graph but it does not prevent to reach the amplification inside the

metamaterial cavity. Moreover, it can be noted that the SPL gain is almost

the same in both cases but in presence of the piezoelectric element it is

reached at a resonant frequency of the metamaterial cavity slightly higher

with the respect to the first case.

Higher resonant frequency means a lower refractive index so, can be noted

a slight influence over the FP resonant mode of the acoustic metamaterial

cavity. This behaviour can be showed with a couple of examples, inserting

an array of piezoelectric bimorph plates disposed horizontally (case a) and

vertically (case b), centred in the acoustic metamaterial cavity. In this

work, will be analysed the piezoelectric plates influence of amplification

performance inside the final 3D structure. An amplification simulation is

performed with the same features explained before and same boundary

conditions.

In Figure 32a is showed the SPL gain inside the metamaterial cavity when

an array of horizontally oriented piezoelectric bimorph plates is

introduced.

a

RESULTS AND DISCUSSION

- 69 -

In Figure 32b is

showed the SPL

in case of

vertical array.

If these graphs are compared to one in Figure 31a, can be seen that there

is a smooth influence due to the presence of the piezoelectric plates over

the resonant behaviour of the acoustic metamaterial cavity. Moreover, has

been noted that in presence of the piezoelectric array the resonant

frequency of the metamaterial cavity increases about 8Hz. It can represent

a lower refractive index compared with the previous case, which means a

change in the FP resonant mode. So, the piezoelectric element position

inside the metamaterial cavity is extremely important and has to be

considered to avoid a mismatch between the resonant frequency of the

cavity and the fundamental frequency of the piezoelectric element. In

addition, in this thesis work has been studied the trend of the SPL along the

x-coordinate and z-coordinate in both cases for all eigenfrequencies. It can

be expected that the vertical arrangement of the piezoelectric bimorph

plates places centrally in the cavity, guarantees higher output voltage

compared with the horizontal array but the presence of the array itself

inside the metamaterial cavity determine a slight shift in the position of

the maximum SPL, due to the smooth change in resonant frequency of the

Figure 32 | SPL graphs of the metamaterial cavity. a | when an array of horizontally oriented piezoelectric bimorph plates in placed inside. b | when an array of vertically oriented piezoelectric bimorph plates in placed inside.

b

RESULTS AND DISCUSSION

- 70 -

cavity. Specifically: −40 𝑚𝑚 in the 𝑥 direction, −5mm in the 𝑦 direction

and −20mm in the 𝑧 direction.

If the fundamental frequency of every piezoelectric element matches the

resonant frequency of the metamaterial cavity, the output power of the

supplementary piezoelectric bimorph plates is expected to be lower with

respect to the one obtained by tuning piezoelectric elements with different

frequencies in this metastructure, both in vertical and horizontal arrays

because the maximum transmission for each resonant frequency exhibits in

a specific point inside the metamaterial cavity. To obtain a better setup of

the final 3D geometry, multiple piezoelectric bimorph plates have to be

tuned at different fundamental frequencies depending on the

subwavelength design of the metamaterial structure, enabling multiple

resonances to be cascaded to extend the bandwidth response. On the other

hand, could be possible to exploit the narrow frequency bandwidth

transmitted by the actually designed 3D structure, even if the transmission

is maximum for the FP resonance only.

However, if the piezoelectric bimorph plates in the array will be tuned to

different frequencies, their total output power is expected to be higher

both for vertical or horizonal arrays, due to the possibility of engineering

the metastructure through the subwavelength design of the cells, with

different path length (𝑙) and width of the coiled channels (𝑠) thus enabling

multiple resonances.

RESULTS AND DISCUSSION

- 71 -

Overall, the energy scavenging of the complete system using a

piezoelectric array is expected to be higher with respect to the previously

analysed case with a single piezoelectric bimorph plate, due to the

connection itself even if a smooth influence over the sound amplification

performance is detected. The connection could be of series or parallel

type. A voltage difference is built up across the top and bottom electrodes

of each piezoelectric bimorph plate. For a parallel connected bimorph,

electric charges with the same sign are generated on the top and bottom

electrodes. Electric voltage is built up between the surface electrodes and

the interface electrode of the bimorph. With the same external force, the

electric charges generated by a parallel bimorph is twice the value

generated by a series bimorph. However, the generated electric voltage in

the parallel bimorph is half the value produced by a series bimorph, since

the capacitance of the parallel bimorph is four times that of the series

bimorph:

𝑉𝑝𝑎𝑟 = 𝑄𝑝𝑎𝑟

𝐶𝑝𝑎𝑟=

2𝑄𝑠𝑒𝑟

4𝐶𝑠𝑒𝑟=

1

2𝑉𝑠𝑒𝑟

Moreover, the piezoelectric array series connection expanded the excited

frequency bandwidth in low frequency range. To achieve an expanded

frequency bandwidth, each piezoelectric element should be tuned to a

different optimal frequency. If a harvester includes piezoelectric elements

with different optimal frequencies which are connected in parallel, then

only the element that generates the higher voltage powers the load, while

the other elements do not. This is because these parallel elements reduce

the overall load connected to the operating element. The harvester with

three bimorphs of identical optimal frequency can be used either if the

excitation acoustic pressure amplitude is small or if it is required to

generate higher voltage but the choice depends on the application.

SUMMARY

- 72 -

Chapter 4

Summary

4.1 Conclusions and Outlook

4.2 References

SUMMARY

- 73 -

4.1 Conclusions and Outlook

The ability to fully control the behaviour of classical waves (e.g.,

electromagnetic and acoustic waves) has long been desired and is at

present a highly active research area. Metamaterial-based energy

harvesting is challenging but promising field of research area that has

currently emerged as a ‘hot-spot’ of active research. The rapid

development of this field suggests that metamaterials will be an alternative

method over conventional energy harvesting methods.

Commonly, metamaterials are of most use to those working in

seismology164, underwater acoustics, or ultrasonics. On the other hand,

because of the growing interest in maintenance-free wireless sensing

applications, acoustic metamaterials are going to be employed for remote

monitoring. Cause of the growing interest in enabling maintainance-free

wireless sensing applications, acoustic metamaterials is a promising area.

Wireless sensor network (WSN) is an emerging application for low power

energy scavenging165. Studies performed to power WSN’s were

acknowledged by Wan et al.166.

In acoustics, various unit cell topologies have been proposed to achieve a

homogenized effective index to control the local transmitted or reflected

phase shift, such as labyrinthine cells, spiral cells and helical cells to name

a few. Metamaterials are structures made up of subwavelength unit cells,

which allow the global acoustic properties of a material to be altered by

changing the subwavelength design. However, the efficiency of phase shift

devices is fundamentally restricted by the scattering into unwanted

directions, which hinders their use for some applications. The origin of the

problem is attributed to the local reflection produced by the individual unit

cells and, to enable better performance, there is a fundamental limitation

that originates in the impedance mismatch between incident and refracted

waves163. Many approaches have been applied to improve the transmission

of the unit cells, such as making helical cells62,162, tapered spiral cells159,

changing the geometry of cell apertures160,60, or filling the channel with

light materials161. By appropriate choice of components, these materials

can appear to defy the laws of nature, producing negative densities and

other strange properties, to ensure the control of not only the phase

SUMMARY

- 74 -

gradient along the metasurfaces but also the impedance matching between

the incident and the desired scattered waves.

But methods to allow efficient cost-effective manufacture of

metamaterials need developing and allowing even miniaturization.

The structures that are used to experimentally demonstrate ideas are not

always well suited to real-world applications because of their size,

mechanical robustness and manufacturability. New designs and fabrication

techniques will be needed to enable the production of acoustic

metamaterials for practical use. The recent rapid increase in additive

manufacturing, or 3D printing1,60, has been very beneficial to the field,

because the materials used and the range of available sizes, are well suited

for metamaterials that manipulate audio-frequency airborne sound. Passive

and fixed structures have remarkable acoustic properties and

performances, but, with active structures, even greater performance can

be achieved. This is partly due to the relatively slow timescale (of the

order of a millisecond) of human-perceived audio frequencies, during which

considerable processing and actuation can be performed. The research

reviewed here has demonstrated how extreme manipulation of sound can

be achieved in ways that are not possible with conventional or passive

materials, but at the cost of increased complexity that must be carefully

considered for any practical applications. New ideas in the context of

acoustic meta materials, beyond those reviewed here, will certainly

emerge in the coming years S. A. Cummer et al. said1, driven by the range

of applications in which the ability to manipulate sound in new ways would

prove useful — consumer audio, ultrasound imaging, underwater acoustics

and sonar, and architectural acoustics such as air-born acoustics, sound

recording and sound detection for sensing application, to name just a few.

It has to be anticipated that design optimization will be a key component

of the transition from proof-of-concept experiments to acoustic

metamaterials that have application-specific properties.

SUMMARY

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In this thesis the proposed harvester has shown the ability to harvest

energy at a specific frequency from a unit cell model. Upon numerically,

analytically and also experimentally having validated this approach, a

broadband energy scavenger with a multi-cell model would be further

designed with a systematic variation of geometrical properties such as the

length of the curled paths 𝑙 and the width 𝑤 of the channels.

Wave front manipulations with acoustic metamaterials are dominated by

the phase control of the reflected or transmitted waves. Inspired by the

generalized Snell law initiated in the field of optics, which describes the

refraction modalities of a light beam in the transition between two means

with different refractive index167. The coiling up space concept has been

used to realize acoustic metamaterials for a predefined phase profile for

desired wave tailoring. The geometrical configuration of an energy

harvesting meta-device will need to be managed depending on its operation

site. However, power output and operating frequency of the conventional

energy harvesting devices are greatly dependent on its geometry. Hence,

future research needs to concentrate on these issues to optimize the

related parameters before the devices can be used in any commercial

operations.

SUMMARY

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