Resonance modes for exterior vibro-acoustic problems, application to adielectric elastomer loudspeaker
Emil GARNELL(1) ,Olivier DOARÉ(1) ,Corinne Rouby(1)(1) IMSIA, ENSTA ParisTech, 828 Boulevard des Maréchaux, 91120 Palaiseau, France. {emil.garnell, olivier.doare,corinne.rouby}@ensta-paristech.fr

Abstract
Modal expansion techniques are efficient order reduction tools for bounded problems, but their use for exterior
problems is less investigated. In the present article a simple vibrating structure embedded in a heavy exterior
acoustic fluid is studied. Using a coupled finite element model, with perfectly matched layers (PMLs) to insure
the Sommerfeld boundary condition, we show that the coupled fluid-structure modes can be obtained by solving
a linear eigenvalue problem. The full basis comprising the fluid loaded structural modes and numerical PML
modes is then used to expand the structural displacement and the radiated acoustic pressure, providing good
accuracy. It is shown that the displacement potential appears to be a more natural variable than the acousticpressure to describe the fluid state. This modal approach is then applied in a real case where a dielectric
elastomer loudspeaker is studied, and validated by measurements.
Keywords: Dielectric elastomer, Vibro-acoustic coupling, Perfectly Matched Layers

1

INTRODUCTION

Modal approaches are common model order reduction tools for bounded problems, and are broadly used for
all types of dynamical systems. When part of the problem is open, like for exterior acoustic radiation, modal
approaches are less investigated. Hein et al. [4] used Perfectly Matched Layers (PMLs) to compute the resonances of open acoustical systems. They were able to retrieve the trapped modes (non-damped modes) that may
exist in open systems, but also found so-called leaky modes (modes which are damped because of radiation).
However, the obtained modal basis is not used to reconstruct the radiated acoustic field.
Modal superposition methods for the calculation of exterior radiation have been used for example in [8], whereresonance modes are used to solve an exterior acoustic problem consisting of an open cavity in an exterior fluid.
A FEM/BEM coupled model has been reduced using resonance modes to compute the radiated sound pressure
by Peters et al. [11]. In the field of optics and electrodynamics, the use of resonance modes to describe
open systems has been studied in a larger extent [7], mainly for laser applications. The modal approach ispresented as an efficient method, especially for its easy physical interpretation compared to direct frequency
domain approaches.
To the author’s knowledge, few publications present acoustic radiation calculations of vibro-acoustic systems
using model order reduction techniques based on resonance modes. In order to introduce the proposed method,a simple vibro-acoustic system is considered, consisting in a tensioned membrane embedded in a infinite plane,
and immersed in an acoustic fluid. A fully coupled finite element model is set-up, using PMLs to implement theradiation boundary condition. A linear eigenvalue problem is obtained, and solved using a standard eigenvalue
solver. The resulting modal basis is used to reconstruct the radiated acoustic field, and compared to semianalytical results.
To demonstrate the applicability of the method, a real system is then analysed using this modal approach. We
consider an dielectric elastomer membrane, which is an active material made of a thin silicone film (200 µm)
sandwiched between two compliant electrodes. This material is capable of large deformations (up to 100%)
when a high voltage is applied between the electrodes [10]. In the studied configuration this membrane isinflated over a closed cavity, and radiates sound when an audio signal is applied [1, 5].

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2

MODAL EXPANSION FOR RADIATION OF FLUID LOADED STRUCTURES

2.1 Description of the studied system
In order to investigate modal methods for exterior radiation of resonant structures, a simple axi-symmetric system is considered, consisting of a flat tensioned membrane embedded in an infinite baffle (see Fig. 1). The
speed of sound is denoted cF , the fluid density ρF , the speed of transversal waves in the membrane cS , themembrane density ρS , and the membrane thickness h.

PML

Symmetry axis

Membrane

Figure 1. Schematics of the simplified system: a membrane is embedded in an infinite baffle.
The non-dimensional equations governing the dynamics of the system are:∂ξ∂ξ−
+ P = Pforce on Σ ,∂t 2 ∂ r2∂P− α∆P = 0 in Ω ,∂t 2∂ξ(r = 0) = 0 ,∂r∂ξ∇ P · n = β 2 on Σ ,∂tξ (r = 1) = 0 ,(1)∇ P · n = 0 on Σn ,(2)

where Ω is the whole acoustical domain, ξ the membrane displacement along z, P the acoustic pressure, α =ρF /ρS , and β = c2F /c2S . The time is scaled by h/cS , all lengths by h, and the pressures by ρS c2S . The Pforce (r,t)
is an external pressure applied to the membrane. In the following the time dependence of all fields is eiωt .
In the rest of the present study, the fluid state is described by the displacement potential Q = P/ω 2 instead of
the pressure P. This choice will be justified later. The governing equations using Q instead of P read [9]:∂ξ+ ω 2 Q = Pforce on Σ ,∂ r2
ω 2 Q + α∆Q = 0 in Ω ,− ω 2ξ −∂ξ
(r = 0) = 0 ,∂r∇ Q · n = −β ξ on Σ ,ξ (r = 1) = 0 ,(3)∇ Q · n = 0 on Σn .

(4)

A Sommerfeld radiation boundary condition must be added on Σint . This condition is implemented using Perfectly Matched Layers (PMLs), by introducing the following complex change of variables:RRr − i rrint σr (s)ds for r > rintz − i zzint σz (s)ds for z > zint
r̃ =
,
z̃ =
,
r
otherwisezotherwise
where σr (r) = σ0 (r − rint )2 and σz (z) = σ0 (z − zint )2 are the attenuation functions, and σ0 is an attenuation parameter that should be adjusted. The changes of variables imply the following changes of the partial derivatives:
∂
1
∂
1 ∂
=
=
,
∂ r̃ 1 − iσr (r) ∂ r γr (r) ∂ r
where the γ functions are defined as:
1 − iσr (r) for r > rint
γr (r) =,1
otherwise

∂
1
∂
1 ∂
=
=
,
∂ z̃ 1 − iσz (z) ∂ z γz (z) ∂ z

γz (z) =1 − iσz (z)1for z > zint.
otherwise

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The weak form of the structural and acoustic equations is obtained by multiplying Eq. (4) and Eq. (3) by test
functions Q̂ and ξˆ , integrating by parts, and using the change of variables r̃ and z̃:

Z
Z
Z 
γz ∂ Q ∂ Q̂ γr ∂ Q ∂ Q̂r̃drdz + αβ ξ Q̂r̃dr = 0 ,(5)−ω 2 QQ̂γr γz r̃drdz + α+
γz ∂ z ∂ zΣΩΩ γr ∂ r ∂ r
ZZZZ∂ ξ ∂ ξˆ−ω 2 ξ ξˆ rdr +rdr + ω 2 Qξˆ rdr = Pforce ξˆ rdr .
(6)ΣΣ ∂r ∂rΣΣ
2.2 Numerical solving
The weak form obtained in previous section is discretized using finite elements. This is performed in the
FreeFem++ software [3], which provides interpolation routines that allow to compute integrals on a common
line of functions defined on different meshes. This allows us to build the total mass and stiffness matrices:     MF0αKF αβ R
Q
0
−ω 2
+
=
⇔
(−ω 2 M + K)X = F .
(7)
0
KS
ξ
FS
−R† MS
where MF and KF are the mass and stiffness matrices of the fluid, MS and KS those of the structure, R the
coupling matrix, R† the hermitian transpose of R, and FS the force vector on the structure. There is no acoustic
source.
The matrix system Eq. (7) can be solved by various methods. The first option is to invert the system for all
frequencies of interest. This will be referred to as the FEM method:Q(ω)X(ω) == (−ω 2 M + K)−1 F(ω) .
(8)
ξ (ω)
Modal methods can also be used, even if the problem involves exterior radiation. Indeed, as the reader may havenoticed, the mass and stiffness matrices are frequency independent so a linear eigenvalue problem is obtained.
This happens because frequency independent PMLs have been chosen, and because no viscous losses are accounted for. Structural losses are modeled a posteriori using Rayleigh damping. The mass and stiffness matrices
are not symmetric, so left and right modeshapes are computed: (−ωn2 M + K)ψnR = 0 and ψnL (−ωn2 M + K) = 0.
If all eigenvalues are of order one, which is the case here, the following bi-orthogonality relations hold:ψ L Mψ R = diag(µn ) and ψ L Kψ R = diag(κn ), where ψ R and ψ L are the matrices containing the right and leftmodeshapes respectively, and µn and κn the modal mass and stiffness of mode n. The total displacement canthen be expanded on the right modeshapes as X = ψ R c, where c is the vector of modal amplitudes. The modal
amplitudes are easily obtained as:
ψnL F(ω).
(9)
cn (ω) =µn (ωn2 − ω 2 )This finally yields:

 

N
Q(ω)
P(ω)/ω 2
X(ω) === ∑ cn (ω)ψnR .
(10)
ξ (ω)
ξ (ω)
n=1
The displacement of the membrane and the acoustic pressure calculated using Eq. (10) will be referred to as the
Modal results. The convergence of this method with the number of modes N taken into account is of primaryinterest, and is studied in section 2.3.
A last option to compute the radiated field is to use the Rayleigh integral. Indeed, as the membrane is flat andembedded in a infinite plane, the radiated pressure can be computed by:P(xxr , ω) = −ω 2 β

Z

ξ (ω)
Σ

e−ik|xxs −xxr |dS(xxs ) ,2π|xxs − x r |(11)where the element source is located at xs and the receiver at xr , and the membrane displacement ξ is computedby Eq. (8). This method for computing the radiated pressure is called FEM Rayleigh.

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Table 1. Parameters used in all numerical tests
α

β


100

1

1
0

Pforceif r < 0.5
otherwise

σ0

rint

zint

dPML

100

1.5

0.7

0.5

2.3 Results
The membrane is meshed by 50 Lagrangian P2 elements along its radius, and the total mesh is shown in
Fig. 2(a). The parameters used in the numerical tests are given in Table 1. These parameters correspond to
typical values of the dielectric elastomer loudspeaker studied in section 3. The parameter α is much larger thanone, meaning that the first modes of the membrane occur at low frequencies for acoustics (the acoustical wavelength is much larger than the structural wavelength). The parameter β is of order 1, so the fluid loading effecton the membrane is not negligible. The PML absorption parameter σ0 has been adjusted so that reflections on
the PML are as small as possible in the frequency range of interest. The PML efficiency is analysed in Fig. 2
where the FEM and FEM Rayleigh solutions are compared to each other. They both yield the same results.
The FEM Rayleigh method implements directly the free-field radiation, so if the FEM method gives the same
results as the Rayleigh method, it means that no reflections occur on the PML surface Σint .
(a)

(b)
60
FEM
FEM Rayleigh

P (dB)

50

40

30

20
100

101

102

(-)

Figure 2. (a) Mesh of the membrane and surrounding fluid. The yellow area is the PML, and the black dot thereceiver location. (b) Acoustic pressure at the receiver location.
The first 300 modes (ordered by increasing modulus of the eigenfrequency) are computed, and the eigenfrequencies are plotted in the complex plane in Fig. 3, together with the modal loss factors. This figure clearlyshows that there are two categories of modes: the resonant modes of the fluid loaded membrane (which wecall membrane modes), and a series of so-called PML modes. The PML modes arise from reflections inside
the PMLs. The pressure in these modes is very large inside the PMLs and smaller in the physical part of thedomain (see Fig. 4). As the PML is largely damped, PML modes have a high modal loss factor. We thus distinguish PML modes from membrane modes using a simple threshold on the modal loss factor (see Fig. 3(b)).
It has also been checked that the membrane modes are independent of the PML parameters, and that the PML
modes are not.
The radiated acoustic pressure is calculated using the FEM and the Modal methods. The convergence of themodal method to the reference FEM calculation when the number of modes is increased is studied in Fig. 5.
Three subsets of modes are considered: the first 30 membrane modes among the first 300 modes, the first
100 modes including PML modes, and the first 300 modes including PML modes. Figure 5 shows that the

4679

(a)

(b)
150

105

Modal loss factor (-)

Imag( )All modes
Membrane modes

100

50

0
0

20

40

60

80

Real( )

100

120

140

All modes
Membrane modes
Threshold

100

10-5

0

50

100

150

200

250

300

Mode number

Figure 3. (a) First 300 eigenfrequencies in the complex plane. (b) Modal loss factors of the first 300 modes.

Figure 4. First 12 coupled modes. The black line is the displacement of the membrane, and the colour in the
acoustical domain is the real part of the pressure of the mode. Here modes 1, 2, 4, 5, 8 and 10 are membranemodes, the rest are PML modes.
membrane displacement is correctly described by a small number of modes. What is more, if PML modes are
removed from the modal expansion (only membrane modes are kept), the membrane displacement is still very
well predicted. This suggests that the set of membrane fluid-loaded resonance modes forms a complete basis
for the membrane vibrations.
On the other hand, if only membrane modes are used to compute the acoustic radiation, a poor estimation of
the radiated pressure is obtained. This means that the basis of fluid loaded membrane modes is not a complete
basis to describe exterior acoustics. If the PML modes are included, the Modal solution converges to the FEM
solution. With 300 modes, the Modal expansion gives exactly the same solution as the FEM calculation. Thismeans that the full basis including PML modes seems to form a complete basis for exterior radiation.

4680

(a)

(b)
60

60

50

20
0
-20

P (dB)

(dB)

40

FEM
30 membrane modes
100 modes
300 modes

40
FEM
30 membrane modes
100 modes
300 modes

30

20

-40
100

101

(-)

102

100

101

102

(-)

Figure 5. Convergence of the Modal solution to the FEM solution. (a) Displacement of the membrane at radiusr = 0.5. (b) Pressure at the receiver location.
Lalanne et al. [7] investigated extensively the use of modal methods to study open resonators, but for opticalapplications. They also coupled PMLs to finite elements, and concluded that in the general case PML modes
should be included in the modal expansion to yield accurate results. The same is observed here: we need toinclude PML modes to get an accurate prediction of the acoustic response.
If the fluid were described by the pressure P rather than the displacement potential Q, the modal expansion of
the pressure would have been of the following form:


N
P(ω)
= ∑ dn (ω)φnR ,
(12)
ξ (ω)
n=1
where dn are the modal amplitudes, and φnR the right modeshapes. Comparing Eqs. (10) and (12) shows that if
the pressure is used as the main variable, P and ξ have the same frequency dependence, whereas it is not thecase in Eq. (10). The convergence of the modal solution to the FEM solution for the acoustic pressure is thenmuch worse, as shown in Fig. 6. Interestingly, if the static contribution of higher order modes is included in
Eq. (12) (see [12] for example), the frequency dependence of the modal amplitudes in Eq. (12) is corrected andEq. (10) is retrieved. To conclude, the displacement potential appears to be a more efficient variable to describe
the fluid state for coupled fluid/structure radiation problems.

3

APPLICATION TO THE SOUND RADIATION OF A DIELECTRIC ELASTOMERLOUDSPEAKER

3.1 Description of the studied system
In this section the modal method for exterior sound radiation presented in a simplified case in section 2 isapplied to estimate the pressure radiated by a dielectric elastomer loudspeaker. This loudspeaker consists of an
inflated silicone membrane coated on both sides with conductive grease, see Fig. 7(a).
When a voltage is applied to the electrodes, the membrane area increases [10]. This effect can be used toradiate sound [5, 1, 6]. We built a prototype using this principle, and measured its sound radiation on axis at a
distance of 1 m in an anechoic chamber.
An electro-mechanical model of the membrane dynamics (similar to [13]) is built, and coupled to acoustics
inside and outside the cavity. The details of the coupled model are too heavy to be presented here, but will be

4681

(b)
60

60

50

50

P (dB)P (dB)

(a)

40

30

40

30

FEM
Modal P
Modal Q

20

FEM
Modal P
Modal Q

20
100

101

102

100

(-)

101

102

(-)

Figure 6. Pressure at the receiver location, calculation with the FEM method, and with modal methods. Modal
P is the modal solution described in Eq. (12), and Modal Q is the solution given in Eq. (10). (a) 100 modes
included in the modal expansion. (b) 300 modes.
(b)
PML
Siliconemembrane
Conductive
greasePressurized
cavitySymmetry axis

90

P / W s (dB ref 2e-5 Pa/1V2 )(a)Measured
Modal KH

80
70
60
50
40
30
20
102

103

Frequency (Hz)Figure 7. (a) Schematics of the dielectric elastomer loudspeaker. (b) Frequency response function between theradiated pressure on axis at 1m, and the excitation signal Ws .
published soon [2]. The coupled resonance modes of the system are computed, and used as a basis to calculate
the acoustical radiation of the loudspeaker, when it is excited by the electro-static force created by the charged
electrodes.
The acoustic mesh around the loudspeaker is small, to limit the number of degrees of freedom. The modal
method therefore only allows to express the acoustic pressure close to the membrane, which is then propagatedto the far field using a Kirshoff-Helmhotlz (KH) integral. Here the first 500 modes (including PML modes) areused in the modal expansion.
3.2 Results
The transfer function between the excitation signal Ws and the radiated acoustic pressure on axis at 1m isplotted in Fig. 7(b). The match of the resonance frequencies is really good, and the peak amplitudes also
fit rather well. What is more, the two anti-resonances and resonances around 1100 Hz and 2200 Hz, which4682

correspond to standing waves in the cavity are reproduced by the coupled model. This is a pure effect ofstrong vibro-acoustic coupling, demonstrating that a fully coupled model is mandatory to correctly describe thebehaviour of this loudspeaker.

4

CONCLUSION

In the present study the use of fluid-loaded resonance modes for the calculation of exterior acoustic radiation
has been investigated. It has been shown that an alternative formulation where the displacement potential is usedinstead of the pressure in the fluid domain allows a much better convergence of the modal expansion. The modal
expansion on the resonance modes has then been used to compute the acoustic radiation of a dielectric elastomer
loudspeaker, demonstrating its potential application on a real system. This model method is especially well
suited for modal force optimizations, for example when the shape of the electrodes of the dielectric elastomer
membrane is optimized to improve the spectral balance. Indeed, the system then remains almost identical, so
the modes only need to be computed once.

ACKNOWLEDGEMENTS
The authors acknowledge the support of the French National Research Agency (ANR) within the project SMArT(ANR-15-CE08-0007-02).

REFERENCES[1] J. W. Fox and N. C. Goulbourne. Electric field-induced surface transformations and experimental dynamic characteristics of dielectric elastomer membranes. Journal of the Mechanics and Physics of Solids,
57(8):1417–1435, Aug. 2009.
[2] E. Garnell, C. Rouby, and O. Doaré. Dynamics and sound radiation of a dielectric elastomer membrane.
Submitted for journal publication, Apr. 2019.
[3] F. Hecht. New development in freefem++. Journal of Numerical Mathematics, 20(3-4):251–266, 2013.
[4] S. Hein, T. Hohage, and W. Koch. On resonances in open systems. Journal of Fluid Mechanics, 506:255–
284, May 2004.
[5] R. Heydt, R. Pelrine, J. Joseph, J. Eckerle, and R. Kornbluh. Acoustical performance of an electrostrictive
polymer film loudspeaker. The Journal of the Acoustical Society of America, 107(2):833–839, 2000.
[6] N. Hosoya, H. Masuda, and S. Maeda. Balloon dielectric elastomer actuator speaker. Applied Acoustics,
148:238–245, May 2019.
[7] P. Lalanne, W. Yan, K. Vynck, C. Sauvan, and J.-P. Hugonin. Light Interaction with Photonic and Plasmonic Resonances. Laser & Photonics Reviews, 12(5):1700113, May 2018.
[8] L. Moheit and S. Marburg. Normal Modes and Modal Reduction in Exterior Acoustics. Journal of Theoretical and Computational Acoustics, 26(03):1850029, Sept. 2018.
[9] R. Ohayon and C. Soize. Structural Acoustics and Vibration: Mechanical Models, Variational Formulations
and Discretization. Academic Press, San Diego, 1998. OCLC: 247803545.
[10] R. Pelrine, R. Kornbluh, Q. Pei, and J. Joseph. High-Speed Electrically Actuated Elastomers with Strain
Greater Than 100%. Science, 287(5454):836–839, Feb. 2000.
[11] H. Peters, N. Kessissoglou, and S. Marburg. Modal decomposition of exterior acoustic-structure interaction.
The Journal of the Acoustical Society of America, 133(5):2668–2677, May 2013.
[12] J.-F. Sigrist. Fluid-Structure Interaction: An Introduction to Finite Element Coupling. John Wiley & Sons,
2015.
[13] J. Zhu, S. Cai, and Z. Suo. Resonant behavior of a membrane of a dielectric elastomer. International
Journal of Solids and Structures, 47(24):3254–3262, Dec. 2010.

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