Sonic Crystals: Fundamentals, characterization and ...

Sonic Crystals: Fundamentals, characterization and experimental

techniques A. Ceb recos

1 L a u m , L e M a n s U n i v e r s i t é , C N R S , A v. O . M e s s i a e n , 7 2 0 8 5 , L e M a n s

L A U M : J . P. G r o b y, V. R o m e r o - G a r c í a

U P V: N . J i m é n e z , V. S á n c h e z - M o r c i l l o , L . M . G a r c í a - R a f f i

U C B : M . H u s s e i n , D . K r a t t i g e r

Col lab orators in t h i s work:

Outline

• Part I ◦ Introduction to Sonic Crystals

◦ Origins and fundamentals

◦ Bandgaps. Dispersion relation.

◦ Group and phase velocities.

◦ Dispersion.

◦ EFC

◦ Methods for Dispersion Relation calculation

◦ Plane Wave expansion

◦ Finite-elements for band structure calculation in time-domain

• Part II ◦ Experimental techniques for the characterization of Sonic Crystals

◦ Dispersion relation using Space-time to frequency-wavevector transformation

◦ Methodology

◦ 2D Experimental relation using SLatCow

◦ Deconvolution method for analysis of reflected fields by SCs

◦ Methodology

◦ Practical application: Sonic Crystals for noise reduction at the launch pad

EXPERIMENTAL TECHNIQUES FOR THE CHARACTERIZATION OF SONIC CRYSTALS 2

A crystal is a solid material whose constituents, such as atoms, molecules or ions, are arranged in a highly ordered microscopic structure, forming a lattice that extends in all directions

Photonic crystal

E. Yablonovitch. PRL, 58, 2059, (1987).

Engineer photonic density of states to

control the spontaneous emission of

materials

embedded in the photonic crystal.

S. John. PRL, 58, 2486, (1987).

Using photonic crystals to affect localisation

and control of light.

Phononic crystal

Sonic crystal

Periodic distribution of solid scatterers in a solid host

medium

M. M. Sigalas and E. N. Economou. JSV. 158, 377. (1992)

M. S. Kushwaha et al,. PRL, 71, 2022. (1993)

Particular case in which the host medium is a fluid.

R. Martínez-Sala. Nature, 378. 241. (1995).

M. S. Kushwaha. APL, 70, 3218. (1997)

J. V. Sánchez-Pérez, PRL, 80, 5325. (1998)

E. Yablonovitch et al. PRL, 67, 2295, (1991).

Definition and origins

General definition: (Solid state physics)

Artificial (or even natural) materials whose physical properties are periodic functions of the space (1D, 2D, 3D)

Optical properties (electromagnetic waves) Dielectric constant (refractive index)

Elastic properties (elastic and acoustic waves) Density and elastic constants (speed of sound)

The study of photonic, phononic and sonic crystals makes use of the same concepts and theories developed in quantum mechanics for electron motion:

Direct and reciprocal lattices, Brillouin zone, Bragg interferences, Bloch periodicity, band structures…

ORIGINS AND FUNDAMENTALS

Direct Space

•

• There is a unique one-dimensional (1D) periodic system. • Five two-dimensional (2D). • Fourteen three-dimensional (3D) different lattices

• Unit cell and lattice constant

• Filling fraction

• Scatterer shape

Geometrical parameters


2D Square lattice

•

Reciprocal Space

The reciprocal space (k-space) is used to study the wave propagation characteristics in periodic structures (Dispersion relation, bandgaps, propagation direction):


•

Reciprocal Space

The study of the band structure can be limited to the Irreducible Brillouin zone. For the case of a square lattice the IBZ is reduced to a triangle.

The scatterer should accomplish the same symmetries in the direct space


Direct and Reciprocal Space

• Direct space • Reciprocal space

Propagation pressure fields, Vibrational modes Band structures, Equifrequency contours, surfaces


Origins and interpretation of the band gaps: Bragg interferences

J. D. Joannopoulos (Photonic crystals: Molding the flow of

light)

Illustration of Evanescent behaviour, 1D

E. Yablonovitch,. Scientific American,

285. (6). Dec. 2001, pp. 47-55

Bragg interferences

BANDGAPS. DISPERSION RELATION

Evidence of evanescent modes inside the BG


V. Romero-García et al,. Appl. Phys. Lett., 96, 124102, (2010)

V. Romero-García et al,. J. Appl. Phys. 108, 044907, (2010)

Analytical, numerical and experimental results

Origins of the band gaps

Plane wave propagating in a 1D homogeneous medium


• w(k) in homogeneous medium




• Periodicity in real space periodicity in reciprocal space




• Periodic w(k) repetitions of period 2pi/a (replicated bands due to periodicity)


Plane wave propagating in a 1D periodic medium


• Periodic variation of the physical properties of the medium




• Creation of bandgaps and dispersion



• Band gap frequency and dependence of its width


2D Dispersion relation



• w(k) in 2D is now a surface

• Periodicity in x and y directions. Main directions of simmetry in reciprocal space




• Dispersion relation along the main simmetry points of the reciprocal space




2D dispersion relation


• Periodic variation of the physical properties. Creation of pseudo gaps for low ff


2D dispersión relation


• Increasing ff and/or impedance contrast will eventually create a full band gap


•

Phase and group velocities

DISPERSION. GROUP AND PHASE VELOCITIES

Group velocity and phase velocity


Source: Institute of Sound and Vibration Research. University of Southampton

• Negative group velocity: Certain frequencies in the second band




• Higher phase velocity: Dispersive part of first band


• Zero group velocity: Localized wave



Understanding dispersion

• Group velocity

• Phase velocity

25



N = 200 layers

Rigid boundary conditions

Gaussian pulse at wn = 0.4


• Group velocity

• Phase velocity

26



N = 200 layers


Gaussian pulse at wn = 0.7


• Group velocity

• Phase velocity

27



N = 200 layers


Gaussian pulse at wn = 1

Equifrequency contours


Generally considering infinite media by applying periodic boundary conditions (Bloch theory) PERIODIC : INFINITE MEDIUM

• Transfer matrix method (TMM)

• Plane wave expansion (PWE)

• Multiple scattering method (MST)

• Finite element method (FEM)

• Finite difference in time domain(FDTD)

• Finite element in time domain (FETD)

CHARACTERISTICS

• Dimensionality: 1D,2D,3D (TMM 1D, PWE, FEM, FDTD all, etc.)

• Handling different media (limitations in PWE solid-fluid except rigid inclusions in air or holes in solid matrix )

• Handling geometry (structure factor in PWE, meshing in FEM or FDTD)

• Steady-state or time dependent

29 METHODS FOR DISPERSION RELATION CALCULATION

Methods for dispersion relation calculation

PWE. Eigenvalue problem

Wave equation (inhomogeneous medium)

Considering a periodic medium

Fourier series

expansion The solution of a wave equation with periodic

potential can be written in the form:

Bloch’s theorem

PLANE WAVE EXPANSION

• Plane wave expansion (PWE)


Structure factor

Consider that the periodic structure is made of two materials, A and B, being B the host

material

Other shapes Square Cylinder

V. Romero-García. JPD, 40. 305108.

(2013)

R. Wang et al, JAP, 90, 2001

Hexagonal, rectangular, elliptic

• Model the shape of the scatterer:

Kushwaha et al., PRB, 49. (1994)




• Finite element in time-domain for elastic band structure calculation (FETD) ◦ Time-dependent Bloch periodicity

◦ Excitation required

• Continuum Equation of motion:

Finite element method in time-domain

Strong form general elastodynamic problem

Space Discretization

A. Cebrecos et al. The finite-element method for elastic band structure calculation. Computer Physics Communications. Under review

FINITE-ELEMENTS FOR BAND STRUCTURE CALCULATION IN TIME-DOMAIN

Unit-cell finite-element model

• Weak form:

Direct stiffness method Element to global

Bloch theory (BC’s) Option: Eigenvalue problem or time-integration (forced term)

Weighting and shape functions


Unit-cell time-domain simulation

• Time integration method:

◦ Center difference Newmark scheme (Explicit) ◦ Computationally efficient

◦ Less storage compared to implicit methods

◦ Conditionally stable

CFL



• Transient excitation ◦ Ricker wavelet

• Calculation of frequency band structure



• Transient excitation ◦ Ricker wavelet

• Calculation of frequency band structure


Numerical examples. Bloch Modeshapes


Dispersion relation using Space-time to frequency-wavevector transformation

PART I I

39

• Space-Time Fourier Transformation

1D Dispersion relation recovery: Methodology

40 DISPERSION RELATION USING SPACE-TIME TO FREQUENCY-WAVEVECTOR TRANSFORMATION

1D periodic medium

• Spatial sampling

1D Dispersion relation recovery

41

1D periodic medium

DISPERSION RELATION USING SPACE-TIME TO FREQUENCY-WAVEVECTOR TRANSFORMATION

• What is the right spatial sampling measuring periodic structures?

• Using analogy from temporal signals:

• From Nyquist sampling theorem, considering periodicity in reciprocal space:

• Finally, consider a SC of a given length:


42

Lattice constant

N° unit cells

For smaller dx, greater N

N° UNIT CELLS DEFINES RESOLUTION in k-space



• Influence of SC size, undersampling and oversampling space


Number of Unit cells Undersampling Oversampling


• 2D Acoustic metamaterial ◦ Quarter-wave resonators as scatterers (Lossless)

◦ 50 x 50 unit cells

◦ 50 measurement points in center line

◦ Point source at the center

44

Angular components in all directions

Bands overlapping

Dispersion relation incomplete!



• 2D Acoustic metamaterial ◦ Quarter-wave resonators as scatterers (Lossless)


◦ 50 measurement points in center line

◦ Plane wave excitation

45

New angular components created due to scattering

Dispersion relation incomplete!



• Spatial “filtering” in the reciprocal space (k-space)

1. 1 measurement point per scatterer

2. Transformation to k-space (2D)

3. Selection of main symmetry directions for frequencies of interest

46

Isofrequency contours




• Results using a point source





• Results using a plane wave

Experimental dispersion relation measurement

• SLaTCow method: Complex dispersion relation

49

A. Geslain et al., J. Appl. Phys. 120,135107,

(2016) Wednesday morning at SAPEM

• Set of parameters defining theoretically the propagatio: amplitude, phase, real and imaginary part of the wave vector

• Optimization minimizing the difference between

and

Analysis for every frequency


Experimental dispersion relation measurement

• SLaTCow method: Complex dispersion relation

50

• Set of parameters defining theoretically the propagatio: amplitude, phase, real and imaginary part of the wave vector

• Optimization minimizing the difference between

and

Analysis for every frequency


Experimental dispersion relation

51


◦ 1 measurement point per scatterer

Numerical results Experiments



52





53




Deconvolution method for analysis of reflected fields by SCs

PART I I

54

Experimental evaluation of a SC Impulse Response

• Context of the problem: ◦ Study feasibility of Sonic Crystals in reducing impact of the backward

reflected field emitted by a sound source in a simplified scaled model of a launch pad (Proof of concept)

• Requisites (Greatly simplified problem): ◦ Linear regime

◦ Static source

◦ Broadband frequency study

◦ Scaled model in water (ultrasonic regime)

• Challenge: ◦ Analysis of reflected field by a SC

◦ Insertion loss in reflection

◦ Diffusion coefficient

ESA – ITI Type A project: Sonic Crystals for Noise Reduction at the Launch Pad

DECONVOLUTION METHOD FOR ANALYSIS OF REFLECTED FIELDS BY SCS

Experimental evaluation of a SC Impulse Response

ESA – ITI Type A project: Sonic Crystals for Noise Reduction at the Launch Pad

• Context of the problem: ◦ Study feasibility of Sonic Crystals in reducing impact of the backward

reflected field emitted by a sound source in a simplified scaled model of a launch pad (Proof of concept)

• Requisites:

◦ Linear regime

◦ Static source

◦ Broadband frequency study

◦ Scaled model in water (ultrasonic regime)

• Challenge: ◦ Analysis of reflected field by a SC

◦ Insertion loss in reflection

◦ Diffusion coefficient

Sound source

Reflector: • Sonic crystal • Rigid reflector

Ground surface

Areas of interest


•

Linear and time invariant system: Theory

Transfer function H(f):

• Widely used in experiments using SC’s

• Calculation of reflection, transmission and absorption coefficients

• Insertion Loss in attenuation devices


•

Linear and time invariant system: Theory

Impulse response h(t):

• Room acoustics: Acoustic quality parameters

• Weakly nonlinear system identification

• Incident and reflected pressure field

Farina A, Fausti P. Journal of Sound

and Vibration 2000;232(1):213-29

A. Novak, et al., IEEE Transactions on. Vol. 63(8), pp. 2044-2051. (2014)

Farina, A. Audio Engineering Society Convention 108. Audio Engineering Society, 2000.


• Input signal: Logarithmic sine sweep:

• Quasi-ideal case 1:

Impulse response deconvolution: Practical example


• Input signal: Logarithmic Sine sweep:

• Quasi-ideal case 2:

Impulse response deconvolution: Practical example


• Signal parameters

Experimental setup

Computer. Labview for measurement

control

DAQ-PXI Digital Signal Generator

Digital oscilloscope RF Amplifier

3D Motorized axis

Hydrophone

Piezoelectric transducer



• Real case: ◦ System response + reflections

Impulse response deconvolution: Example



• Real case: ◦ System response + reflections

Impulse response deconvolution: Example


• Characteristics of the SCs ◦ Four different samples (2D SCs)

◦ Square shape scatterers

◦ Square and triangular lattice

◦ Low and filling fraction

◦ 3D Printed ◦ Selective laser melting

Samples used in the experiments


• Experimental results: Incident field f = 700 kHz

Second Band

Incident and reflected field separation

f = 350 kHz

First Band f = 500 kHz

Band Gap


f = 700 kHz

High Band

• Reflected field. Square lattice High filling fraction

Incident and reflected field separation

f = 350 kHz

Low Band f = 500 kHz

Band Gap


Experimental wave propagation. Convolution signal on demand

67

• Input: Sinusoidal pulse (Narrow bandwidth, band gap)


68

Experimental wave propagation

• Input: Sinusoidal pulse (Narrow bandwidth, 2° Band)


Insertion loss results

69

• Modified Insertion loss to study reflection

• IL along the space integrated in frequency

Reduction of the reflected field depending on the reflection angle


Insertion loss results

70

• Modified Insertion Loss to study reflection

• IL in frequency bands integrated in ROI

Reduction of the reflected field due to diffusion Higher in propagative bands Lower in the BG


• Near to far field transformation

◦ Projection of the pressure in near to field to infinity ◦ Esentially it is a spatial Fourier transform

◦ Angular information

Difussion coefficient

71

T. J. Cox, P. D'antonio. Acoustic absorbers and diffusers: theory, design and application. Crc Press, 2009.

Reflection from a flat rigid surface


• Diffusion coefficient

• Normalized diffussion coeficient

• Quantification of the type of reflection

◦ Specular

◦ Diffuse

Difussion coefficient

72

ISO 17497-2:2012 Acoustics -- Sound-scattering properties of surfaces -- Part 2: Measurement of the directional diffusion coefficient in a free field


• Far field results



• Quantification of the type of reflection

◦ Specular

◦ Diffuse

Difussion coefficient results

73 DECONVOLUTION METHOD FOR ANALYSIS OF REFLECTED FIELDS BY SCS

Sonic Crystals: Fundamentals, characterization and experimental

techniques A. Ceb recos

1 L a u m , L e M a n s U n i v e r s i t é , C N R S , A v. O . M e s s i a e n , 7 2 0 8 5 , L e M a n s

L A U M : J . P. G r o b y, V. R o m e r o - G a r c í a

U P V: N . J i m é n e z , V. S á n c h e z - M o r c i l l o , L . M . G a r c í a - R a f f i

U C B : M . H u s s e i n , D . K r a t t i g e r

Col lab orators in t h i s work:

Positive, zero and negative diffraction.

• Focusing of waves using finite SC’s:

◦ Curvature of the wave front

◦ Character of the incident wave

◦ Plane wave

◦ Point source

◦ Sound beam (Gaussian beam) ◦ Width of the source

D=2a

D=8a


Isofrequency contours & Focusing regimes

• Focusing of waves using finite SC’s:

– Interplay between beam and periodic media

• Band structure and Isofrequency contours (PWE)

• Spatial spectrum of the incident beam

a = 5,25 mm r = 0,8 mm

D=2a

D=8a

Extended BZ

Source angular spectrum

Isofrequency contour

Band structure


• Spatial dispersion completely parabolic

Results

Accumulated phase shift

Focusing distance

D=8a

D=2a


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