Effect of point defects in a two-dimensional phononic crystal on the

Effect of point defects in a two-dimensional phononic crystal on the

KONRAD GRUSZKA, SEBASTIAN GARUS, JUSTYNA GARUS, KATARZYNA BŁOCH,
MARCIN NABIAŁEK
Effect of point defects in a two-dimensional
phononic crystal on the reemission
of acoustic wave
INTRODUCTION
FDTD METHOD
Phononic crystals (PC) in a rather broad sense are synthetic
materials with periodically changing acoustic properties of the
medium (eg. density and speed of propagation of the acoustic
wave) and have been recently intensively investigated due to the
wide application range as well as interesting physical properties of
these systems [1]. In spite of considerable simplicity of the
construction of such materials they enable the construction of
frequency filters that will allow to suppress undesired frequencies.
In some cases, with proper distribution of repetitive elements
making up the crystal, PC allow to create sound barriers in any
band of acustic spectrum. Certain frequency ranges which are
subject to strong damping, i.e. when the phonons do not propagate
through the crystal, are called phononic band gaps.
Significant impact on the parameters of phononic crystals has
location and the distance between the periodic components, which
by the appropriate arrangement are allowing to modify the center
frequency of the pass-filter, and thus adjust the position of band
gap [2]. If the distance between these components are comparable
to the acoustic wavelength of the incident wave, and the top layer
of PC will be uniform across the width, the acoustic wave will
leave the crystal plane with very good convergence even at
considerable large distances from the sound source. This allows
the construction of the so-called. directional speakers and opens
wide possibilities of commercial use in many applications ranging
from medicine up to military applications. Figure 1 shows the
typical structure of two-dimensional PC.
FDTD method (finite-difference time-domain method) is method
known for many years which allows to simulate the propagation of
waves of different types (e.g. electromagnetic) in any medium [3]
and is widely used in a number of experiments relating to the
acoustic simulations [4÷6].
We start from the first-order differential equations that describe
the propagation of sound waves in the medium:

p ( xˆ , t )    uˆ
t
d
 0  r uˆ ( xˆ , t )  p( xˆ , t )
dt

(1)
(2)
2
2
where: p ( xˆ , t ) is the pressure field F / m  kg /( m  sec ),
uˆ ( xˆ , t ) is vector velocity, m / s,  0 is the density of the medium,
 r is a relative density with respect to 0 ,  is a compressibility
of medium   1 /( 0   r  c 2 ), c – the speed of sound in the
medium.
Differentiating after time and space above equations, we get the
following two formulas, describing the propagation of the socalled. Yee cell [7]:
p n 1 / 2 ( i, j,k )  p n 1 / 2 ( i, j,k )
t  0  r c 2
 u xn ( i  1 / 2 , j,k )  u xn ( i  1 / 2 , j,k )
x
t  0  r c 2

 u xn ( i, j  1 / 2 ,k )  u xn ( i, j  1 / 2 ,k )
y


(3)
t  0  r c 2
 u xn ( i, j,k  1 / 2 )  u xn ( i, j,k  1 / 2 )
z
u n 1 / 2 ( i, j,k )  u n 1 / 2 ( i, j,k )
t
  p n ( i,1 j,k )  p n ( i, j,k )
 r ( i  1 / 2 , j,k )  0   x 
t

  p n ( i, j  1,k )  p n ( i, j,k )
 r ( i, j  1 / 2 ,k )  0   y 
t

  p n ( i, j,k  1 )  p n ( i, j,k )
 r ( i  1 / 2 , j,k  1 / 2 )  0   z 

Fig. 1. Typical phononic crystal structure 1 – first material; 2 – second
material: d – diameter of periodic element, a, b – distances between
element centers
Rys. 1. Typowa struktura kryształu fononicznego 1 – materiał pierwszy,
2 – materiał drugi: d – średnica części periodycznej, a, b – odległości
między środkami elementów
Dr inż. Konrad Gruszka ([email protected]), mgr inż. Sebastian Garus, mgr
inż. Justyna Garus, dr Katarzyna Błoch, dr hab. prof. PCz. Marcin Nabiałek –
Wydział Inżynierii Procesowej, Materiałowej i Fizyki Stosowanej, Instytut Fizyki,
Politechnika Częstochowska
(4)
For the above equations, n is a step in the two-dimensional
space {p, u} formed by pressure field p ( xˆ , t ) and the velocity field
uˆ ( xˆ , t ) . Grid formed by the given field is a square grid, so
Δx = Δy. The spatial intervals Δx, Δy and time interwal Δt must
take into account the so-called Courant stability criterion, assessed
using Von Neumann's criteria, so it will not lead to a simulation
breakdown [8]:
ct 
1
2
(1 / x)  (1 / y ) 2
(5)
In the two-dimensional FDTD simulation carried out in the
field of acoustics, both the particle velocity components and the
scalar sound pressure values are arranged around the Yee cell, as
shown in Figure 2 [8].
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Figure 5 shows the ways of the distribution of vacancies in the
simulated crystals. Initial structure (a) was modified by removing
the sequence of (b) – eight rods, (d) – sixteen rods, (e) – one rod
from layer 11, (f) – a single rod from the surface layer, (g) – the
two rods from the surface layer (thus increasing the depth of
a defect), and (h) – the three rods from the surface layer. Structure
(c) represents a subgroup of the four structures in which 5 rods
were removed from the lyer 3, 10, 16 and 20 counting from the
sound source (in Figure 5 from the top).
Fig. 2. Acoustic Yee cell used in FDTD algorithm [8]
Rys. 2. Akustyczna komórka Yee użyta w algorytmie FDTD [8]
SIMULATION PARAMETERS
Simulation enviroment
Numerical simulation was performed using the self written
algorithm implemented in C++ using the equations (3) i (4), and
using the Yee cell presented in Figure 2. As in the simulation
lossless medium was used (phonons of acoustic wave does not
change their energy with traveled distance), at the edges of the
simulation several damping layers PML (Perfectly Matched
Layers) were applied to eliminate the interference of the reflected
sound wave, with the wave emitted from the source. This layer
had a thickness of 8 cells. PML layer is placed on the edges of the
simulation, since the acoustic wave is reflected from the edges,
just like if it pass to another medium. The frequency of source is
chosen in such a way that the wavelength is comparable with the
distances between the elements forming the PC, and for all
simulations it was fixed to 1.5 kHz. The amplitude of the source
was set to 120 dB. The main medium in which the sound
propagate is air, and PC-forming elements are made of rectangular
copper bar with a thickness of 1 mm. Simulation space dimensions
were 100×250 mm. The data on the sound pressure were collected
from the opposite side of the source (230 mm from the beginning
of the simulation). Figure 3 shows a schematic of the simulation.
Fig. 3. FDTD simulation scheme: 1 – sound source, 2 – phononic
crystall (PC), 3 – microphones
Rys. 3. Schemat symulacji FDTD: 1 – źródło dźwięku, 2 – krzyształ
fononiczny, 3 – mikrofony
Parameters of the model phononic crystal
Phononic crystal was built from 22 layers consisting of
respectively 21 and 22 alternating copper rods with dimensions of
1 mm, which was obtained by the superimposition of the two
simple networks where suitable vectors were: a = 3 mm and b = 2
mm. The initial system of phononic crystal is presented in Figure
4. Such a choice of structure allowed to achieve a high symmetry
of the system, and further study of the possible impact of
vacancies on the acoustic wave after passing through the PC. In
other structures, some rods were removed from the inner layers of
PC, as well as from its surface.
Fig. 4. Basic phononic crystal cell: 1 – first simple primitive, 2 –
second primitive network, a, b – translation vectors
Rys. 4. Podstawowa komórka kryształu fononicznego: 1 – pierwsza
podtsawowa sieć, 2 – druga podstawowa sieć, a, b – wektory translacyjne
STUDIES
The phononic crystal structures
investigated in this work
Research were carried out on 10 different structures containing
defects in the form of vacancies, depending on their position, and
on the number of vacancies in the layer or in the whole crystal
volume in relation to the initial structure.
Fig. 5. Vacancies placement in PC. Black color marks removed rods
Rys. 5. Sposób rozmieszczenia wakansów w krzyształ fononiczny.
Czarnym kolorem zaznaczono usunięte pręty
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Results
Figure 6 shows the simulation space for the the initial phononic
crystal structure (Fig. 5a) after n = 100 000 steps (t = 100 ms). In
the initial period of the simulation (when step count n < 45 000)
harmonics of the fundamental frequency appearing, and due to the
interference effects caused by the propagation in the PC,
simulation is not stable. Studies have shown that for the simulation
of the investigated systems, stability is achieved after about 45 000
steps throughout the range space under consideration. Acoustic
wave before PC front propagates circularly from the source. After
propagation through the PC obtained wave was flat, stable
throughout the whole range of the simulation, showing only slight
variations in the sound pressure level at ±1 dB. Pressure graph
collected from 84 points placed perpendicular to the incident wave
front at the time of 100 ms was included in the Figures 7 and 8.
As expected, the signal is symmetrical over the entire width of
the test. After passing through the PC, the intensity of the acoustic
wave falls significantly from the original level of 120 dB to 45 dB,
while maintaining a good homogeneity.
Fig. 7. Pressure for initial structure
Rys. 7. Ciśnienie w dla struktury wyjściowej
Defects in the PC volume
When one vacancy is placed in the the PC (Fig. 5e) it has
practically no effect on the acoustic wave, and the sound pressure
chart is the same as that for the initial structure. Increasing the
number of defects within the volume of the crystal structure to 8
shows Figure 9. Such a number of randomly distributed defects
has a negligible effect on the measured SPL levels. Increasing the
number of vacancies to 16 also has little effect on the shape of the
recorded pressure level sound curve. It can be concluded that
a small percentage (about 4%) of vacancies placed inside the PC
does not play a significant role in the formation of a flat,
symmetrical acoustic wave.
Fig. 8. Sonic pressure level for initial structure
Rys. 8. Ciśnienie akustyczne dla struktury wyjściowej
Defects on the PC surface
Placing a single defect on the surface of PC according to Figure 5f
also gave a similar result as a small amount of vacancies within
the crystal volume. Changes in the shape of the pressure curve was
obtained only in the case of enhanced ("deep") defect of the
surface defect (Fig. 5g). Sound pressure curve for this case is
given on Figure 10.
Fig. 9. Sonic pressure level for structure with 8 vacancies
Rys. 9. Ciśnienie akustyczne dla struktury z 8 wakansami
Fig. 6. Simulation space. Intensity of the color represent acoustic pressure
Rys. 6. Przestrzeń symulacji. Jasność koloru reprezentuje ciśnienie akustyczne
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Near the center of the simulation (the center of symmetry of the
acoustic wave), the sound pressure curve collapses, showing
a noticeable decrease in the amplitude. Increasing the number of
surface defects to 3, while maintaining a single depth only slightly
changes the shape of the curve of acoustic pressure, which is
shown in Figure 11. In this case the curve response is much flatter
and its left side diverges from that symmetry shape. This
corresponds to the situation shown in Figure 5h, where it can be
seen, that the left side of the surface defect density is higher.
Influence of the position of a fixed amount
of defects on acoustic wave propagation
Figure 12 shows the curves obtained for the four substructures
shown in Figure 5c. If the point defects are present in layers
distant from the surface layer, from which there is a secondary
emission of the acoustic wave, the effect of these defects is small.
The same distribution of vacancies in the structure of PC, but in
a layer located closer to the surface layer (layer 10 from the source
of the acoustic wave) is associated with a slightly higher sound
pressure deformation curve, and as it approaches to the surface,
layer defects influence is increasing. Figure 12b is a close-up area
of the curves in the range from 40 dB to 47 dB, in which the
difference in the sound pressure levels are exposed. As can be
seen, as the distance from acoustic waves source is increasing,
sound pressure extremes are deepening symmetrically. Because in
this case the relative position of point defects does not change (and
also their total number), the effect on the acoustic wave leaving
Fig. 12. Sonic pressure level for subgrup of four structures with 5
defects in: 1 – third, 2 – tenth, 3 – sixteenth and 4 – twenty layer
Rys. 12. SPL [dB] dla podgrupy czterech struktur z 5 defektami
w warstwach: 1 – trzeciej, 2 – dziesiątej, 3 – szesnastej i 4 – dwudziestej
the phononic crystal is caused only by the distance from the
surface layer, from which reemission occurs.
CONCLUSIONS
Some types of defects in the form of vacancies have a significant
impact on the reemission of acoustic wave from phononic crystal.
A small number of randomly distributed point defects in the
phononic crystal volume, has no significant effect on the acoustic
wave. A significant influence on the sound pressure curve was
achieved by placing the vacancies on the surface of PC, and
intensify of this effect was achieved by modifying the depth of
surface defects. In addition, there is a rather complex relationship
between distance of occurrence of equally placed, fixed number of
vacancies within the crystal volume, and shape of the resulting
acoustic wave leaving the phononic crystal.
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Rys. 11. Ciśnienie akustyczne dla struktury z trzema defektami
umieszczonymi na powierzchni kryształu fononicznego
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