On the optimization of a piezoelectric speaker for hearing aid

Engineering Optimization IV – Rodrigues et al. (Eds)
© 2015 Taylor & Francis Group, London, ISBN 978-1-138-02725-1
On the optimization of a piezoelectric speaker for hearing aid application
through multi-physical FE models
G.C. Martins, P.R. Nunes & J.A. Cordioli
Department of Mechanical Engineering, Federal University of Santa Catarina, Florianópolis (SC), Brazil
ABSTRACT: The use of piezoelectric materials in hearing aid speakers, also called receivers, presents technical
and economic advantages such as reducing the number of parts of the system and its manufacturing cost. However,
the performance of such systems is still not competitive when compared to traditional electrodynamic speakers. In
order to achieve an appropriate performance, one option is to apply optimization techniques to these systems, and
this is the main aim of this work.The analysis of the vibro-acoustic performance of a piezoelectric speaker involves
the construction of multi-physical models, and the first part of the work concerns the development of a multiphysical numerical model of a miniaturized speaker prototype by using the Finite Element Method (FEM). The
multi-physical model is composed of piezoelectric, structural and acoustic coupled FE models. It is important
to mention that the acoustic model of the small cavities of the speaker accounts for thermal and viscous effects
on the acoustic propagation. These effects are included in a simplified form to reduce the computational cost of
the multi-physical FE model solution. After the FE model presentation, two different speaker designs are then
proposed and optimized. The methods of Genetic Algorithm and Nelder-Mead (simplex) methods are applied
sequentially to optimize the designs. The final designs obtained display similar performance to commercially
available speakers for hearing aids.
1
INTRODUCTION
Current speakers of hearing aids are electrodynamic
system, being composed of electromagnetic, mechanical and acoustical components. The electromagnetic
component is usually a complex element involving
many parts that are assembled with great precision.
The replacement of the electromagnetic component by
a piezoelectric component can be quite advantageous,
since the piezoelectric components would need fewer
parts to fulfill the function of the electromagnetic components. Therefore, the application of piezoelectric
components in speakers for hearing aids can bring both
technical (durability, consumption, etc) and economic
advantages.
The modeling of miniaturized piezoelectric transducers, like speakers and microphones for hearing
aids, involves multi-physical models with strong coupling between the involved physics. In the literature,
some analytical models of these components can
be found as in Lotton et al. (1999) and Gazengel
et al. (2011). However, these analytical models usually
involve very simple geometries or are very simplified. These characteristics restrict the use of such
models as a design tool for miniaturized piezoelectric transducers. The aim of this paper is to develop
a numerical multi-physical model of a hearing aid
piezoelectric speaker, and use it as a design tool to
apply optimization procedures. The numerical multiphysical model was implemented by using the Finite
Element Method (FEM) via the commercial software
Comsol AB (2012) together with lumped parameter
models.
In what follows, two designs of hearing aid speakers were proposed. These designs were modeled and
optimized to improve their performances. The optimization procedure were performed using Genetic
Algorithm (GA) and Nelder-Mead (NM) methods
which are already implemented in commercial software MATLAB (MathWorks 2012) used in this
work.
The paper starts with a presentation of the multiphysical model equations where the piezoelectric,
structural and acoustical model equations are briefly
presented with their coupling terms. Then, the hearing
aid piezoelectric speaker model is presented, followed
by the speaker design optimization and a comparison
of the results with commercial loudspeakers.
2
MULTI-PHYSICAL MODEL EQUATIONS
To properly model a hearing aid speaker model, it is
necessary to account for the piezoelectric, structural
and acoustic behavior of this system. In the following
sections, the equations of each of these phenomena
will be reviewed, while the FE model formulation are
all implemented in the commercial software Comsol
AB (2012) which was used in this work.
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2.1
Piezoelectric model equations
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The electro-mechanic response of a piezoelectric body
of volume and regular boundary surface ∂, is
governed by the mechanical, dynamic and electrostatic equilibrium equations (Benjeddou 2000). These
equilibrium equations, considering harmonic perturbations, can be written as
where u, fb , ρ, qb and D, are the mechanical displacement vector, mechanical body force vector, mass
density, electric body charge and electric displacement
vector, respectively, while [σ] is the Cauchy stress tensor. The piezoelectric boundary ∂, could be subject
to either essential or natural boundary conditions, or a
combination of them, as:
•
Mechanical boundary conditions:
•
Electrical boundary conditions:
(FLNS) equations, as presented by Kampinga et al.
(2008). The description of the FLNS FE formulation is
well presented in Kampinga (2010), and it has already
been implemented in Comsol AB (2012), where it is
possible to couple this model with either structural and
piezoelectric models.
However, in the optimization process, the application of the FLNS FE model is very inconvenient due
to its high computational cost. Therefore, a simplified
viscothermal acoustic model called Low Reduced Frequency (LRF) model (Tijdeman 1975, Beltman 1998)
is used as presented below.
2.3.1 Low reduced frequency model
This simplified model uses the standard acoustic partial differential equation with lumped functions to represent the viscothermal effects in the acoustic models.
These lumped functions are achieved from analytical
solutions and can be placed in the form of standard
acoustic differential equation as
where
where n is the boundary outward unit normal vector.
Equation (1) and the boundary conditions in equations (2) and (3) state the problem in strong form
which must be continuously satisfied at all points of the
domain and boundary ∂. By applying variational
calculus to the equation (1), the weak formulation of
the equilibrium equations is obtained, as shown in
Benjeddou (2000). Finally, the Finite Element Method
can be applied based on the weak formulation to solve
this equations.
2.2
Structural model equation
The equation to model the mechanic response of
structure is the same presented in equation (1a). The
structural model equation could be subject to the same
mechanical boundary condition presented in the equations (2). The development of the weak formulation
and FEM for the structural model equation is well
described in many books as in Cook et al. (2001).
The coupling of the piezoelectric and the structural
models are made by assuming displacement continuity
on the interface of these models.
2.3 Viscothermal acoustic model equations
In miniaturized acoustic devices it is common to find
small cavities, and this feature brings viscothermal
effects into acoustic propagation. These effects are due
to the viscous friction and thermal diffusion which are
usually neglected in standard acoustic models.
Viscothermal wave propagation is a subproblem of
fluid dynamics which, under the continuum assumption, can be modeled by the Full Linear Navier-Stokes
The lumped functions B(s) and D(s, Pr) are dimensionless, and its parameters s and Pr are the shear wave
number and Prandtl number, respectively (Tijdeman
1975, Beltman 1998). These lumped functions are
achieved by taking the average of analytical solutions
which are found in (Beltman 1998). Although these
functions can be obtained only in simple systems,
they were applied in the FE models by assuming each
acoustic subsystems as tubes or layers.
The same boundary conditions of the standard
acoustic equation can be applied to Equation 4, and its
coupling with structural (or piezoelectric) models is
made the same way as the standard acoustic-structure
coupling.
3 THE HEARING AID PIEZOELECTRIC
SPEAKER MODEL
The performance of a hearing aid speaker, is usually
evaluated by coupling it to a standard microphone coupler which simulate the ear canal impedance and provides an approximation of the incident sound presure at
the ear drum. This acoustic coupler has small diameter
tubes and a 1/2 diameter cavity with volume of 2 cm3 ,
which is the average volume of an adult ear channel
(Dillon 2001). Therefore, a model that aims in evaluating the performance of a hearing aid speaker needs
to consider this acoustic coupler and the microphone
acoustic surface impedance.
An overview scheme of the speaker performance
analysis model can be visualized in Figure 1. The
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Figure 1. Overview scheme of the speaker performance
analysis model.
speaker system is modeled by FE coupled piezoelectric, structural and viscothermal acoustic models. The
speaker FE model design is inspired on the experimentally validated design presented in a previous study
(Martins et al. 2012) which is an axis-symmetric twodimensional FE model. This model is described in
section 3.2.
The acoustic coupler is represented by a lumped
parameter model to reduce the computational cost of
the analysis, as presented in the following section.
3.1 The acoustic coupler lumped parameter model
The acoustic coupler was considered as three coupled
acoustic tubes as shown in Figure 1. By means of the
transfer matrix method, it is possible to characterize
an acoustic tube as a matrix [M ] so that
Figure 2. Multi-physical speaker FE model set-up.
2 and 3 as circular layers. Simply supported mechanical BC was applied to edges of the structural domain
as shown in Figure 2.
The procedure of coupling the FE speaker model
and the acoustic lumped parameter model described
in Kampinga (2010) requires the solution of the FE
model twice to get its transfer matrix. As the optimization process also needs to solve FE model many
times, a different approach for the coupling of the FE
and the lumped model was used. In this approach, the
impedance BC (Z) was applied considering the acoustic coupler lumped matrix [M ]c described in section
3.1 and the impedance of the 1/2 microphone, Zmic
given in Kjaer (1996), as
where
where p2 and V2 are the average acoustic pressure and
average volume velocity at one end of the tube, respectively; p1 and V1 are the average acoustic pressure and
average volume velocity at the other end of the tube,
respectively.
The transfer matrix for a tube can be obtained from
Munjal (1987), and the entire acoustic coupler transfer
matrix could be written as
After the FE model solution, the pressure obtained
by the microphone (pout ) considering the lumped
model is given by
In the next section, this matrix is used to calculate the impedance applied to the FE model and the
pressure obtained at the microphone surface.
where pFE is the average pressure obtained on the FE
model surface where Z is applied.
3.2 FE model
3.3 The approach validation
Figure 2 presents the domains and boundary conditions (BC) applied to the speaker FE model. As
mentioned before, the acoustic domains were modeled with the LRF model to account for viscothermal
effects. Therefore, equation (4) with the LRF lumped
functions B(s) and D(s, Pr) was used considering
Acoustic domain 1 as a tube and Acoustic domains
In the previous paper (Martins et al. 2012), the FE
piezoelectric speaker model was experimentally validated through a prototype designed with dimensions
larger than hearing aid speaker designs. This prototype
was modeled with the approach presented above, and
the results were compared with experimental result
and two commercial hearing aid speakers (Knowles)
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Figure 3. Comparison of the SPL of prototype’s model, its
experimental results and the Knowles hearing aid speakers
(ϕin = 1 Volt).
measured with the same acoustic coupler, as shown
in Figure 3. The dimensions of the prototype and the
experimental set-up can be found in Martins et al.
(2012).
As can be seen in Figure 3, the model showed good
agreement with the experimental result. It can be also
noted that the prototype has lower performance than
the commercial loudspeakers at frequencies below
6000 Hz. This frequency range is very important
because this is the frequency range of speech. Therefore, it is important for hearing aid systems that the
speaker has higher performance in this frequency
range.
Figure 4. Piezoelectric speaker designs for optimization.
Table 1.
4 THE SPEAKER DESIGN OPTIMIZATION
The main objective of this paper is to generate designs
of piezoelectric hearing aid speakers with improved
performance. The design of a hearing aid speaker is
very difficult because of the multi-physical parameters
involved. So, it needs an experienced designer who
knows the sensibility of each parameter to obtain a
speaker with an acceptable performance. Instead of
doing an exhaustive analysis of many speaker designs,
an optimization procedure was employed for two basic
designs proposed in next section.
4.1
Hearing aid speaker basic designs
Two designs inspired on the prototype validated in
Martins et al. (2012) were created, as shown in
Figure 4. The parameters Xi are a group of mechanical, acoustic and piezoelectric properties considered
as design variables in the optimization process. The
material-dependent variables (X1 and X2 ) are shown
in Table 1.
In Design 1, the parameters X1 and X2 are denoting materials of the structural and piezoelectric
domains, respectively. In Design 2, both domains are
piezoelectric, and these parameters are denoting two
piezoelectric materials.
Materials related by X1 and X2 design parameters.
Index
Structural
Piezoelectric
1
2
3
4
Steel
Copper
Aluminum
Acrylic
PZT-5A
Barium Titanate
Aluminum Nitride
PVDF
4.2 Optimization problem
An important speaker performance parameter is the
frequency response function (FRF) relating the pressure measured by the microphone (pout ) and the electric potential applied to the speaker (ϕin ). The FRF is
evaluated in the model by taking the average sound
pressure level (SPL) at the microphone surface position for a unit input spectra of ϕin at the speaker
electrode surface.
The objective of the speaker model optimization is
to improve the SPL spectrum in the frequency range up
to 8 kHz. Ideally, a similar level to that displayed by the
commercial hearing aid speakers would be obtained.
To achieve this goal, the fitness function was written as
320
Table 2.
process.
Range of design parameters in the optimization
Table 3. Main input parameters applied in the MATLAB
optimization algorithms.
X
Xl
Xu
GA parameters
X1
X2
X3
X4
X5
X6
X7
X8
1
1
1500 [µm]
1 [µm]
10 [µm]
10 [µm]
1 [µm]
10 [µm]
4
4
3000 [µm]
1000 [µm]
2000 [µm]
4500 [µm]
1000 [µm]
1000 [µm]
Parameter
Design 1
Design 2
PopulationSize
EliteCount
CrossoverFraction
StallGenLimit
TolFun
16
1
0.7
50
1e-6
14
1
0.7
50
1e-6
Parameter
Design 1
Design 2
MaxIter
MaxFunEvals
TolFun
TolX
500
1600
1e-4
1e-4
500
1400
1e-4
1e-4
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NM parameters
The SPL function is dependent on f and X which are
the frequencies and design parameters vectors, respectively, and the fitness function take the minimum value
of the SPL in the frequency range of interest. The components of X are dependent on the speaker designs
described above.
The optimization problem applied in this paper
could be written as
The optimization of the speaker designs were performed in two sequential steps as:
1. Optimize with GA method and get the optimized
design XGA ;
2. Optimize with NM method with Xinit = XGA .
with
l
u
where X and X denote the lower and upper bounds on
X , respectively, and their values are shown in Table 2.
As X1 and X2 are non-continuous parameters, they have
been applied as an integer parameter that relates an
index for each group of material properties.
The GA and NM methods were applied using
the optimization toolbox in commercial software
MATLAB (MathWorks 2012) by the “ga.m” and
“fminsearch.m” algorithms, respectively. The main
input parameters applied in these optimization algorithms are shown in Table 3 while the remaining
parameters were kept at standard values found in
(MathWorks 2012).
4.3 Optimization procedure
The optimization methods applied in this work were
the Genetic Algorithm (GA) method and the NelderMead (NM) method. These methods do not take in
account the gradients of the objective and the constraints functions, which are very complicated to
obtain in the case of a multi-physical FE model.
Therefore, the implementation of these methods in
the present study is simpler since the FE commercial software is only used to evaluate the fitness
function and the optimization algorithms take it to
operate in the design improvement, i.e., maximization
or minimization of the fitness function.
The GA method is an evolutionary optimization method which, by giving an initial group of
design parameters X (initial population), improves the
designs by doing selection, crossing and mutation over
the population (Haupt & Haupt 2004). The advantage
of this method is that it allows non-stagnation in a
local minimum design and improves the possibility of
reaching a point close to a global optimum design.
The NM method is a local optimizer which, from
the initial design Xinit , constructs a initial simplex
and improves the design by operations of reflection,
expansion and contraction of the simplex (Rao 2009).
4.4 Results
Table 4 shows the optimized parameters obtained in the
optimization procedure. It shows that some parameters
were equal and next to the minimum constraint such as
X4 and X7 which denote the thickness of piezoelectric
and structural layers, respectively. These features were
expected because they increase the range of motion of
the diaphragm and, consequently, the SPL. This effect
also justifies the values of X3 and X8 which are near to
the maximum constraints.
Figure 5 shows a comparison of the optimized
designs with a commercial speaker and the validated
prototype (Martins et al. 2012). It can be seen that
both designs have similar performance, with their
SPL spectrum close to the commercial speaker and
much higher than the validated prototype speaker.
Comparing with validated prototype, which has larger
diameter than optimized designs, the minimum SPL
has increased more than 20 dB for most frequencies
below 5 kHz.
In fact, the SPL spectrum of optimized designs
have higher levels than commercial speaker in some
frequencies as can be seen in Figure 5. Design 1 SPL
321
Table 4.
designs.
validation. The optimized designs showed better performance than the validated prototype and similar SPL
levels to a commercial hearing aid speaker. Although
the optimized designs were not experimentally validated, the optimization procedure has been showed to
be an appropriate tool to improve piezoelectric hearing
aid speaker designs.
Optimized design parameters for the proposed
X
Design 1
Design 2
X1
X2
X3
X4
X5
X6
X7
X8
4 (Acrylic)
1 (PZT-5A)
2893 [µm]
1 [µm]
70 [µm]
4060 [µm]
111 [µm]
10 [µm]
4 (PVDF)
1 (PZT-5A)
2957 [µm]
1 [µm]
216 [µm]
2886 [µm]
96 [µm]
N/A
ACKNOWLEDGMENTS
The authors would like to thank Olavo Silva and
Guillaume Barrault for discussions and advices and
the financial support provided by FINEP, CNPq
and Acstica Amplivox Ltda.
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REFERENCES
Figure 5. SPL of optimized designs compared with the
Knowles and validated prototype speakers (ϕin = 1 Volt).
spectrum is flatter than the commercial speaker, which
is a desired feature for a hearing aid speaker. Both
designs have also higher SPL levels than the commercial speaker in frequencies above 5 kHz, which
could be a sound quality improvement for hearing aid
systems.
The optimized designs were not experimentally validated, however, the optimization procedure applied
here showed that it is possible to produce hearing
aid speaker based on the piezoelectric effect with
performance similar to electrodynamic hearing aid
speakers.
5
DISCUSSIONS AND CONCLUSIONS
This paper has presented a multi-physical model to
analyze piezoelectric speaker designs. The model was
validated with experimental results of a previous
paper. Then, an optimization procedure was applied
to improve the performance of two initial designs
inspired by the prototype used in the experimental
Beltman, W.M. (1998). Viscothermal wave propagation
including acousto-elastic interaction. Ph. D. thesis,
University of Twente, Enschede (the Netherlands).
Benjeddou, A. (2000). Advances in piezoelectric finite element modeling of adaptive structural elements: a survey.
Computers & Structures 76, 347–363.
Comsol AB (2012). Comsol Multiphysics user’s guide
(4.3 ed.).
Cook, R., D. Malkus, M. Plesha, & R. Witt (2001). Concepts
and Applications of Finite Element Analysis. Wiley.
Dillon, H. (2001). Hearing aids. Thieme.
Gazengel, B., P. Hamery, P. Lotton, & A. Ritty (2011). A dome
shaped pvdf loudspeaker model. Acta Acustica United
with Acustica 97, 800–808.
Haupt, R. & S. Haupt (2004). Practical genetic algorithms.
John Wiley & Sons.
Kampinga, W. R. (2010, June). Viscothermal acoustics using
finite elements : analysis tools for engineers. Ph. D. thesis,
Enschede.
Kampinga, W. R., Y. H. Wijnant, & A. de Boer (2008).
A finite element for viscothermal wave propagation. In
Proceedings of ISMA 2008, pp. 4271–4278.
Kjaer, B. (1996). Technical Documentation: Microphone
handbook Vol. 1: Theory. Naerum, Denmark.
Lotton, P., M. Bruneau, Z. Skvor, & A. Bruneau (1999).
A model to describe the behavior of a laterally radiating
piezoelectric loudspeaker. Applied Acoustics 58.
Martins, G., J. Cordioli, & R. Jordan (2012). Numerical simulation of a piezoelectric loudspeaker including viscothermal effects for hearing aid applications. In INTER-NOISE
and NOISE-CON Congress and Conference Proceedings,
Volume 2012, pp. 8506–8516. Institute of Noise Control
Engineering.
MathWorks (2012). MATLAB R2012b User’s Guide.
Munjal, M. (1987). Acoustics of ducts and mufflers with
application to exhaust and ventilation system design.
Wiley New York (NY) et al.
Rao, S. (2009). Engineering optimization: theory and practice. Wiley.
Tijdeman, H. (1975). On the propagation of sound waves
in cylindrical tubes. Journal of Sound and Vibration 39,
1–33.
322