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Periodic structures based on coupled Helmholtz resonators
for broadband noise suppression
Mariia Krasikova1,2, Aleksandra Pavliuk2, Sergey Krasikov2, Anton Melnikov1,
Yuri Baloshin2, David A. Powell3, Steffen Marburg1, Andrey Bogdanov2,4
1Chair of Vibroacoustics of Vehicles and Machines, Technical University of Munich, Garching b. M¨unchen 85748, Germany
2School of Physics and Engineering, ITMO University, St. Petersburg 197101, Russia,
Email: mariia.krasikova@metalab.ifmo.ru
3School of Engineering and Information Technology, University of New South Wales, Canberra, ACT 2610, Australia
4Harbin Engineering University, Harbin 150001, Heilongjiang , Peoples R China
Introduction
Acoustic metamaterials and phononic crystals have been
receiving a lot of attention during the last decade as
they provide novel routes for the control and manipu-
lation of acoustic fields [1, 2]. One of the particular di-
rections of studies in this area is dedicated to the de-
velopment of noise-insulating systems that may be si-
multaneously thin, lightweight, and ventilated [3, 4]. In
this work, we consider the structure based on strongly
coupled Helmholtz resonators. In Ref. [5] it was demon-
strated that such structures can be characterized by not
only broad stop-bands but also period-dependent venti-
lation and partial optical transparency. We develop this
concept further to improve the noise-insulating proper-
ties of these structures via tuning the distances between
the resonators resulting in a merging and broadening of
stop-bands.
System Description
The considered system consists of Helmholtz resonators
made of pipes with slits carved along their axes [see
Fig. 1(a)]. Under the assumption that the length of
pipes is infinite, the system can be considered as two-
dimensional. It is also assumed that the system is finite
along the direction of plane wave propagation but in-
finitely periodic along the other direction [see Fig. 1(b)].
The period along the y-axis is ay= 120 mm. All of the
resonators are the same and characterized by the outer
radius R= 53 mm, the inner radius r= 48 mm, and the
width of slit w= 40 mm. The thickness of the considered
structure is considered to be 3 pairs of the resonators,
such that initially the distances between the centers of
the resonators is 120 mm. The resonators are made of
polylactic acid (PLA). The choice of the parameters is
based on Ref. [5].
Numerical computations are provided in COMSOL Mul-
tiphysics where the incident wave with the amplitude
p0= 1 Pa is modelled via background pressure field and
Floquet conditions are used for the periodic boundaries.
The area in front and behind the structure is supple-
mented by perfectly matched layers to emulate free space.
The transmission spectra are calculated as
T= 20 log10 p/p0,(1)
where pis the root mean square pressure calculated at
three arbitrary sampling points located behind the struc-
ture.
The optimization procedure was based on the genetic al-
gorithms [6] such that the cost function was an inverse of
the linear combination of the spectrally averaged trans-
mission coefficient and maximal transmission coefficient
within the range 200 – 2100 Hz:
C= (a1Tavg +a2Tmax)−1,(2)
such that various combinations of a1and a2coefficients
were used during the calculations. The purpose of the
algorithm was to minimize the cost function via adjust-
ment of the distances within the pairs of the resonators.
Selection of the parents and crossover procedures were
implemented via the roulette wheel selection and uniform
crossover techniques, respectively.
(a) Resonator: (b) Semi-infinite structure:
periodic condi�on
θ1
w
r
R
d1d2
air
d3
x
y
Figure 1: Schematic geometry of the considered structures. (a) Single Helmholtz resonator with circular cross-section. The
resonator is characterized by the outer radius R= 53 mm, the inner radius r= 48 mm and the slit width w= 40 mm. (b)
Semi-infinite structure which has finite thickness along the x-axis and is periodic along y-axis with the period ay= 120 mm.
The incident plane wave propagates along the x-axis.
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120 mm
360 mm
Ini�al structure:
(c)
Op�mized structure:
(d)
(e)
(a) (b)
x
y
kx
ky
ΓX
M
kx
ky
ΓX
M
Figure 2: Schematic representation of (a) the initial and (b) the optimized semi-infinite structures. (c) Transmission spectra
for the initial (blue line), optimized (red line) are shown. The green line corresponds to the structure consisting of four pairs,
such that there are two pairs with d= 360 mm and two pairs with d= 120 mm. The corresponding band diagram for unit
cells with (d) d= 120 mm and (e) d= 360 mm. The unit cell height in both cases is ay= 120 mm, while the width is 420 and
240 mm, respectively.
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Results
First of all, the initial structure was modified via the
removement of two resonators in such a way that the
resulting structure consists of two pairs of resonators
with the distances d1= 360 mm and d2= 120 mm
between their centers [see Fig. 2(a)]. Despite the fact
that the modified system consists of lower number of res-
onators, its noise-insulation is better since the peak near
1100 Hz is suppressed and two stop-bands are merged
into a single one [see Fig. 2(b)]. This result can be ex-
plained by the consideration of two equivalent infinite
systems consisting of the unit cells with d1= 360 mm
and d2= 120 mm. The corresponding band structures
are shown in Figs. 2(c) and 2(d) from which it is clear
that the modes of one system lie in the band-gap regions
of the other system. Hence, the resonances corresponding
to the eigenmodes are not pronounced for the structure
with the finite thickness. Obviously, the increase of the
number of pairs will result in a more pronounced stop-
band covering the broad region from 180 to 2000 Hz as
Fig. 2(b) demonstrate where the spectra of the structure
consisting of two pairs of each type is shown (4 pairs in
total).
Hence, since the distances between the resonators are
crucial for enhancement of noise-insulation properties of
the considered structures it is reasonable to provide tun-
ing of these parameters. For that we utilize genetic op-
timization algorithm described in the System Descrip-
tion section. As initially, the system consists of 3 pairs
of resonators but the distances between the resonators
within each pair are varied. Transmission spectrum for
one of the most beneficial designs with d1= 109 mm,
d2= 117 mm and d3= 344 mm is shown in Fig. 3. As
previously, the resonance at 1100 Hz is suppressed and
the spectrum is characterized by the broad stop-band.
Basically, the spectrum of the optimized structure al-
most coincide with the initial spectra except the region
around 1100 Hz. Hence, while the overall thickness of the
optimized structure is larger, the amount of the material
and its mass is the same and its noise insulation proper-
ties are better. It is also should be noted that such type
of a structure in which one of the parameters changes
gradually is and example of the so-called chirped struc-
tures [7–9]. Despite the fact that here the band struc-
tures are not provided, the underlying mechanism of the
broadening of the stop-band is also related to overlaps
between the eigenmodes of a system with one unit cell
with the band-gaps of the two other systems with other
unit cells. It is also should be noted that the distance
between the resonators is not the only parameter which
allows to find such an overlap of the band structures.
However, this is one of the parameters affecting the lo-
cal coupling strength of the resonators and change of its
value results in a significant shift of the eigenmodes.
Conclusion
To conclude, we have demonstrated that noise-insulating
properties of semi-infinite periodic structures based on
strongly coupled Helmholtz resonators can be improved
via combination of unit cell with different distances be-
tween the resonators. The improvement is based on the
combination of band-gaps corresponding to different unit
cells which for the case of systems with finite thickness
leads to broadening of stop-bands. In particular, for the
structure optimized via genetic algorithm the distances
within the pairs of the resonators are d1= 109 mm,
d2= 117 mm and d3= 344 mm. The transmission spec-
tra of this structure is characterized by the pronounced
stop-band covering the range from 200 to 2000 with
the maximal value of approximately −20 dB and spec-
trally averaged transmission coefficient within this range
Tavg =−42 dB. The obtained results demonstrate that
periodic systems made of strongly coupled resonators are
promising for the implementation of noise-insulating sys-
tems.
Figure 3: Transmission spectra for the initial structure and the structure optimized via genetic algorithm. Both structures
consist of three pairs of strongly coupled resonators. For the initial structure the distance between the centers of the resonators
within pairs is d1=d2=d3= 120 mm while for the optimized structure d1= 109 mm, d2= 117 mm and d3= 344 mm. All
other parameters are the same.
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Acknowledgements
M.K., A.P., S.K and A.B. acknowledge the Ministry of
Science and Higher Education of the Russian Federation
(Project No. 075-15-2022-1120).
References
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