PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Bending waves focusing in arbitrary shaped plate-like structures:
application to spatial audio
Nassim BENBARA1; Marc REBILLAT1, Nazih MECHBAL1
1

ENSAM, CNRS, CNAM, HESAM Université, 151 Boulevard de l'Hôpital, 75013 Paris, France

ABSTRACT
Advanced audio applications are more and more demanding with respect to the visual impact of loudspeakers
while still requiring more channels for high quality spatial sound rendering. The use of arbitrary plate-likestructures driven by electromagnetic actuators or by piezoelectric elements appears as a promising solution
to tackle both issues. However, to meet spatial rendering audio constraints (omnidirectional piston-likesources), the generated bending waves must be focused to a certain extent within the host plate. Theoretically,
this means being able to invert the spatio-temporal wave propagation operator for the generated bending
waves to fit a given target shape. Several methods are here investigated to perform this task depending on
the available knowledge of wave propagation in the plate (theoretical, partial spatial and full spatial
knowledge). The various methods are presented in a unified theoretical framework and their performances
are compared in terms of sound radiation by means of two key performance indexes.
Keywords: Spatial vibration control, Bending wave focusing, Advanced signal processing

1. INTRODUCTION
In the past decades, several studies were performed on the sound restitution using an array ofloudspeakers. The main idea is to calibrate each loudspeaker of the array to reproduce with highfidelity the physics and the acoustics of the primary source (1). Historically, spatial rendering methods
such as stereophony (VBAP), binaural rendering or holophony (ambisonic or Wave Field Syn thesis(WFS)) are used with electromagnetic loudspeakers. There are many constraints imposed by suchmethods for them to work properly: a flat and omnidirectional response for the speaker, and several
piston-like actuators to avoid spatial aliasing and to cover the whole restitution area (2). Distributed
Mode Loudspeakers (DML) were introduced as a new kind of loudspeaker with a low visual profile(3). The idea is to produce sound waves by exciting the bending waves of a thin stiff plate trough anactuator. This technology was used to render spatial sound by the WFS Method (4,5). However, thistechnology suffers from some drawbacks because of the aliasing caus ed by joint panels and the modal
behaviour at low frequency of the plate which causes distortions of the signal and does not meetspatial audio requirements. Therefore, a new technology using larger plates called MultiActuator
Panels (MAP) or much larger called Large MultiActuator Panels (LaMAP) has been designed toovercome these limitations. The main idea is to excite the plate with several actuators driv en bydifferent signals to reproduce different sound sources (6). This technology was successfully used in
the context of the wave field synthesis to render spatial sound.(7). However, some drawbacks such as
the modal behavior at low frequencies and the interferences between sources causing sound distortions
and poor sound quality still exists with this technology.
Particularly in the automotive industry, using spatial sound rendering coupled with flat panel
loudspeaker will be very interesting. Indeed, it will allow to alert the users from several dangers andprovide high quality spatial sound. Moreover, this approach will decrease weight by replacing
electrodynamical loudspeakers with piezoelectric actuators and improve design purposes and visual
impact. However, the reconstitution zone is small, with a very limited number of canals and with very
complex operational conditions. Considering the previous observations, the focusing of the bending
1

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nassim.benbara@ensam.eu

waves on arbitrary plate-like interior parts of the car to create independent sound sourc e appears as
the best way to create an array for spatial sound rendering.
In the literature, there are various methods that allow to focus bending waves i n a media. The first
method called modal control (MC) is based on modal superposition (8). Indeed, it allows to focus a
monochromatic vibro-tactile peak on plates (9), or an audio source on a rectangular plate by computing
FIR filters of order 3 (10). A strategy of equalization was developed by the same authors (11). The
method is based on a simple approach and is sensitive to modeling error. The second method available
in the literature is based on the time reversal ( TR) principle, initially described by Fink et al. in (12).
It allows to focus a wave at a specific point in the space by learning partially the waves received at
the actuators positions and initiated at the focusing point. An improvement is available in (13)permitting to improve the TR in high damping medias. The third method studied in this paper is thespatio-temporal inverse filtering (STIF), initially studied by (14) and (15) to find a solution to thelack of robustness of the time reversal method. Firstly, the method consists in learning the propagationof the waves in the whole media, then, an inverse filter is created and used as input for each actuator.
Finally, another method based on the learning of the large band dynamic behavior of the structure,using the maximization of the contrast between the radiating zone and the whole media. It was initiatedfor room acoustics in (16), and applied on a plate structure in (17).
The objective of this study is to propose an approach enabling for the localization of bending waves
in a defined area of a geometrically complex structure and with arbitrary boundary conditions to makeit vibrate as a piston. First of all, this approach should be precise in terms of spatial focusing. Moreover,this method must operate in a wide frequency range and be suitable for the audio restitution. Finally,
the method should not rely on too much experimental measurements for practical reasons as theautomotive is targeted as the applicative field. These methods will be presented in a unified formalism
and compared numerically to highlight advantages and drawbacks of each method, in a structural andacoustics point of view.

2. PROBLEM STATEMENT
Let ܵ denote a plate-like structure of arbitrary shape and with arbitrary boundary conditions.
Assuming that this structure lies in the ሺ‫ݔ‬ǡ ‫ݕ‬ሻ plane, the positions of the N piezoelectric actuators
bonded on S are given by ሼሺ‫ݔ‬௡ ǡ ‫ݕ‬௡ ሻሽ௡‫א‬ሾଵǡேሿ and the out-of-plane displacement of the bending wave
velocity field is denoted ‫ݑ‬ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݐ‬ሻ.

Figure 1 – Arbitrary shaped-like structure with piezoelectric actuators and localized audio sources
The objective is now, for any input audio signal ܽሺ‫ݐ‬ሻ, to focus the bending wave field ‫ݑ‬ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݐ‬ሻ to
fit a given target shape ߶ሺ‫ݔ‬ǡ ‫ݕ‬ሻ using the ܰ actuators through ܰ FIR filters ‫ݎ‬௡ ሺ‫ݐ‬ሻ applied to theaudio input. Mathematically, this means that we would like to achieve:‫ݑݐ׊‬ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݐ‬ሻ  ൌ ߶ሺ‫ݔ‬ǡ ‫ݕ‬ሻܽሺ‫ݐ‬ሻ(1)Or equivalently in the frequency domain, with ܷሺ‫ݔ‬ǡ ‫ݕ‬ǡ ݂ሻ and ‫ܣ‬ሺ݂ሻ the Fourier transforms of thedisplacement and the audio signal respectively:‫ܷ݂׊‬ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ݂ሻ ൌ ߶ሺ‫ݔ‬ǡ ‫ݕ‬ሻ‫ܣ‬ሺ݂ሻǤ
(2)
Assuming the system is globally linear, the contributions ‫ݑ‬௡ ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݐ‬ሻ of all the ܰ actuators
generating the field ‫ݑ‬ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݐ‬ሻ sum up and can be written as:ேே‫ݑݐ׊‬ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݐ‬ሻ ൌ ෍ ‫ݑ‬௡ ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݐ‬ሻ ൌ ෍ ݄௡ ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݐ‬ሻ ‫ݎ כ‬௡ ሺ‫ݐ‬ሻ ‫ܽ כ‬ሺ‫ݐ‬ሻǡ
௡ୀଵ

(3)

௡ୀଵ

where “‫ ”כ‬stands for the convolution product and ݄௡ ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݐ‬ሻ is the spatiotemporal impulse response6321

of the ݊௧௛ actuator. Or equivalently in the frequency domain:ேே‫ܷ݂׊‬ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ݂ሻ ൌ  ෍ ܷ௡ ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ݂ሻ ൌ ෍ ‫ܪ‬௡ ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ݂ሻܴ௡ ሺ݂ሻ‫ܣ‬ሺ݂ሻǤ
௡ୀଵ

(4)

௡ୀଵ

By combining both expressions (2) and (4) of ܷሺ‫ݔ‬ǡ ‫ݕ‬ǡ ݂ሻ, the objective to achieve is then expressed in
the frequency domain and independent of the audio input signal ܽሺ‫ݐ‬ሻ as:ே‫݀݊݅ܨ‬ሼܴ௡ ሺ݂ሻሽ௡‫א‬ሾଵǡேሿ ‫݂׊ݐ݄ܽݐ݄ܿݑݏ‬ǡ ߶ሺ‫ݔ‬ǡ ‫ݕ‬ሻ ൌ  ෍ ‫ܪ‬௡ ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ݂ሻܴ௡ ሺ݂ሻǤ

(5)

௡ୀଵ

This means that the spatiotemporal propagation operators ‫ܪ‬௡ ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ݂ሻ associated with the ܰactuators needs to be inverted to design the ܰ FIR filters ܴ௡ ሺ݂ሻ able to focus the bending wave fieldon the target shape ߶ሺ‫ݔ‬ǡ ‫ݕ‬ሻ whatever the frequency ݂.

3. BENDING WAVES FOCUSING METHODS UNIFIED THEORETICALDESCRIPTIONIn this section, several methods able to in invert the spatiotemporal propagation operators‫ ݊ܪ‬ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ݂ሻ are presented. The methods differ depending on the available knowledge of wave
propagation in the plate they rely on (theoretical, partial spatial and full spatial knowledge). They are
all presented within a unified theoretical framework for comparison purpose.
Methods relying on a theoretical knowledge of ࡴ࢔ ሺ࢞ǡ ࢟ǡ ࢌሻ
The first class of methods presented here assumes that a theoretical knowledge of wave propagation
within the host structure is known. For bending wave, the out-of-plane displacement ‫ݑ‬ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݐ‬ሻ of thehost structure subject to the external point load ‫݌‬ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݐ‬ሻ can be expressed as (18):3.1‫׏ܦ‬ସ ‫ݑ‬ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݐ‬ሻ ൅ ߩ݄‫ݑ‬ሷ ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݐ‬ሻ ൅ ܿ‫ݑ‬ሶ ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݐ‬ሻ ൌ ‫݌‬ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݐ‬ሻ‫ ܦ‬ൌ‫݄ܧ‬ଷǤͳʹሺͳ െ ߥ ଶ ሻ(6)In the above equations, ‫ܧ‬, ߥ, and ߩ are the Young’s modulus, Poisson’s ratio, and density of thepanel respectively. Additionally, ܿ is the damping constant and ݄ is the panel thickness.
Let ሺߠ௠ ሺ‫ݔ‬ǡ ‫ݕ‬ሻǡ ݂௠ ሻ be the theoretical orthogonal modes associated with the structure under study
which can be computed analytically or through FEM models. The main idea is to decompose thedisplacement ‫ݑ‬ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݐ‬ሻ in the mode shapes basis, and reinjecting the displacement expression in the
movement equation (6). Next, thanks to the projection of the obtained equation on the ‫ ܯ‬first modes,
the propagation operator ‫ܪ‬௡ ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ݂ሻ can be obtained, as explained in (9,10).
Moreover, the next step consists in decomposing the target shape ߶ሺ‫ݔ‬ǡ ‫ݕ‬ሻ on the host structuremode shapes ߠ௠ ሺ‫ݔ‬ǡ ‫ݕ‬ሻ as:ெ߶ሺ‫ݔ‬ǡ ‫ݕ‬ሻ ൌ ෍ ߶௠ ߠ௠ ሺ‫ݔ‬ǡ ‫ݕ‬ሻ‫߶݄ݐ݅ݓ‬௠ ൌ ඵ ߶ሺ‫ݔ‬ǡ ‫ݕ‬ሻߠ௠ ሺ‫ݔ‬ǡ ‫ݕ‬ሻ݀‫ݕ݀ݔ‬Ǥ

(7)

ௌ

௠

It can be shown by combining (5) and (7) that the following equation can be written:ே‫݂׊‬ǡ ‫߶݉׊‬௠ ൌ ෍ ‫ܪ‬௡ ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ݂ሻܴ௡ ሺ݂ሻǡ
௡

(8)with the number of modes ‫ ܯ‬chosen equal to the number of actuators or superior (potential matrix
conditioning problem during the inversion). Moreover, it can be shown also in (10) that such equation
can be easily inverted and expressed in the discrete-time domain as follows:ܴ௡ ሺ‫ݖ‬ሻ ൌ ‫ܬ‬௡ ൅ ‫ܭ‬௡ ‫ି ݖ‬ଵ ൅ ‫ܮ‬௡ ‫ି ݖ‬ଶ Ǥ(9)Where ‫ܬ‬௡ , ‫ܭ‬௡ and ‫ܮ‬௡ can be expressed using the modal parameters (݂௠ ǡ ߠ௠ ሺǡ ሻሻ as well as thetarget shape parameters ߶௠ .
The method described above offers the advantage of not needing any prior experiments. In addition,
the computation cost is low because the FIR filter order is 3 and the method is not sensitive to noise.
However, the advantages listed before are available only for very simple structures . Additionally, for
an audio purpose the frequency bandwidth is in general large as well as the number of modes to beaddressed, which limits drastically the frequency band.

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3.2 Method relying on a partial knowledge of ࡴ࢔ ሺ࢞ǡ ࢟ǡ ࢌሻ
Let us now assume that a partial spatial knowledge of the propagation operator is available. Moreprecisely, it is assumed here that the spatiotemporal operator is known for an excitation located at one
spatial position ሺ‫ݔ‬௜ ǡ ‫ݕ‬௜ ሻ and measurements performed at actuator positions ൛ሺ‫ ݊ݔ‬ǡ ‫ ݊ݕ‬ሻൟ.
݊‫א‬ሾͳǡܰሿෝ ሺ‫ ݊ݔ‬ǡ ‫ ݊ݕ‬ǡ ݂ሻൟ
Mathematically, what is known in the frequency domain is ൛‫ܪ‬݊‫א‬ሾͳǣܰሿsuch that:෡ ሺ‫ݔ‬௡ ǡ ‫ݕ‬௡ ǡ ݂ሻ ൌ ܿ‫݆݊݋‬ሾ‫ܪ‬௡ ሺ‫ݔ‬௜ ǡ ‫ݕ‬௜ ǡ ݂ሻሿǡ(10)
‫ א ݊׊‬ሾͳǡ ܰሿ‫ܪ‬
according to the spatial reciprocity principle [8]. Thus, by choosing:෡ ሺ‫ݔ‬௡ ǡ ‫ݕ‬௡ ǡ ݂ሻሿǤ(11)
ܴ௡ ሺ݂ሻ ൌ ܿ‫݆݊݋‬ሾ‫ܪ‬One should be able to focus the bending waves around the location ሺ‫ݔ‬௜ ǡ ‫ݕ‬௜ ሻ. According to the theory,
this method is based on the time invariance principle of partial derivati ve equations in a non-dispersive
media. Moreover, this method is also based on the principle of spatial reciprocity of frequency
response functions. But if the media induces several losses, an iterative method was introduced by(13) and permits to converge to the STIF presented just below, but it this case is less flexible than the
STIF.
3.3 Method relying on a complete spatial knowledge of ࡴ࢔ ሺ࢞ǡ ࢟ǡ ࢌሻ
Let us now assume that a complete spatial knowledge of the propagation operator is available.
More precisely, it is assumed here that the spatiotemporal operator is known for any actuator ݊ ‫א‬ሾͳǡ ܰሿ and that measurements are performed over a complete grid of ‫ ܭ‬points on the structure for. Mathematically, what is known in the frequency domain ispositions ൛ሺ‫ ݇ݔ‬ǡ ‫ ݇ݕ‬ሻൟ݇‫א‬ሾͳǡ‫ܭ‬ሿ൛‫ ݊ܪ‬ሺ‫ ݇ݔ‬ǡ ‫ ݇ݕ‬ǡ ݂ሻൟ݊‫א‬ሾͳǣܰሿǡ݇‫א‬ሾͳǡ‫ܭ‬ሿ.

The problem to be solved can then be rewritten on the measurement spatial grid as:ே߶ሺ‫ݔ‬௞ ǡ ‫ݕ‬௞ ሻ ൌ ෍ ‫ܪ‬௡ ሺ‫ݔ‬௞ ǡ ‫ݕ‬௞ ǡ ݂ሻܴ௡ ሺ݂ሻǤ

(12)

௡ୀଵ

The resulting matrix can first be directly pseudo-inversed in the frequency range of interest by
keeping only large enough eigenvalues. This leads directly to t he FIR filters ܴ௡ ሺ݂ሻ:ܴሺ݂ሻ ൌ ‫ܪ‬ሺ݂ሻା Ȱǡ(13)
where “+” denotes the pseudo-inversion operator.
It is important to note that the pseudo-inversion step is sensitive to noise. Hence, during the pseudoinversion step, the lowest singular values are canceled to deal with those matrix conditioning issues.
More details are available in (15).
Firstly, the method is more robust because it allows to acquire information about the wave
propagation in the whole media. Additionally, the filters size is chosen by the user. Hence, the modeltakes into account the real structure and its complexity, with eventual losses, and the n otion of modal
density is not crucial. Finally, this method is very interesting for its ease of implementation. However,
it is necessary to realize several measures to learn the dynamic.

4. Numerical study of the investigated focusing methods
4.1 Structure under study
In order to test the previous methods and to compare their performances, a ribbed clamped
polypropylene plate was chosen, a material often used for car passenger garnishments. The dimensionsof the plate are given in Figure 1, with ribs of length 1 cm and thickness of 3 mm to approach as muchas possible the surfaces used in the automotive industry. Indeed, the ribs are stiffening the plate,
shifting the eigenmodes to higher frequencies and introducing localized modes around the ribs.
The material used is polypropylene, with a Young modulus of ‫ ܧ‬ൌ ͳǤͳ , a Poisson’s ratio ofߥ ൌ ͲǤ͵͵ and a density of ߩ ൌ ͻͻͲ ‫݃ܭ‬Ȁ݉ଷ . The dynamics of the structure with piezoelectric actuators
glued on it is computed thanks to a finite element model implemented on the software SDT (19) andan experimental modal analysis was performed jointly to identify modal dampings. The nine firstmode shapes are from 158 Hz to 707 Hz.The plate is modeled with shell elements (Figure 2). The
control algorithms are developed and implemented in Matlab with SDT jointly.

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Figure 2 – (left) Dimensions of the plate and (right) meshes of the studied plate and (c) ribbed plate.
The target shape consists of a rectangle measuring ሺͲǡͳͷ‫ܮ‬௫ Ǣ Ͳǡʹ‫ܮ‬௬ ሻ and centered at ሺͲǡ͵‫ܮ‬௫ Ǣ Ͳǡͷͷ‫ܮ‬௬ ሻ.
Moreover, the audio signal to be reconstructed is a sweep sine with a duration of 10 seconds, from100 Hz to 10 kHz, sampled at 40 kHz. In addition, a set of "NCE51" piezoelectric actuators developedby Noliac are used, with a diameter of 25 mm and a thickness of 1 mm. Indeed, there are 9 PZTs
between each rib placed quasi randomly, and equally spaced along y. The number of actuators andtheir disposition change the results. Here, an example with few actuators is studied, because in realconditions as in cars, only a small number of actuators is embedded for reasons of compactness andchannel addressing.
4.2 Sound radiation computation
Let ‫ݑ‬ሺ‫ݎ‬ௌ ǡ ‫ݐ‬ሻ be the out-of-plane displacement of the plate and ܷሺ‫ݎ‬ௌ ǡ ݂ሻ its Fourier transform, where‫ݎ‬ௌ ൌ ሺ‫ݔ‬ௌ ǡ ‫ݕ‬ௌ ሻ belongs to the plate. The sound radiation of a baffled plate in the fluid observation points‫ ݎ‬ൌ ሺ‫ݔ‬ெ ǡ ‫ݕ‬ெ ǡ ‫ݖ‬ெ ሻ is computed thanks to the Rayleigh integral (18), defined by:ܲሺ‫ݎ‬ǡ ݂ሻ ൌെ߱ଶ ߩ଴
ܷሺ‫ݎ‬ௌ ǡ ݂ሻ݁ ି௝௞ȁ௥ି௥ೄ ȁඵ݀ܵሺ‫ݎ‬ௌ ሻǡʹߨȁ‫ݎ̴ݎ‬ௌ ȁௌ

(14)

where ߩ଴ ൌ ͳǤʹ‫݃ܭ‬Ȁ݉ଷ the density of air, ܿ଴ ൌ ͵ͶͲ݉Ȁ‫ ݏ‬the sound speed in air.
The sound radiation is computed on a semi-circle centered on the target shape, at a radius of 50cm.
4.3 Indicators for the parametric study
The first indicator allows to calculate the smoothness of the directivity. Let ܲ௠௘௔௡ be the meanvalue of the pressure at the frequency ݂, on the semi-circle of length ‫ݎ‬௅ , the SPL mean error indicator
allows to compare the current pressure distribution with a smooth distribution, and is given by:ͳ ௥೘ೌೣ
ሺܲሺ‫ݎ‬ǡ ݂ሻ െ ܲ௠௘௔௡ ሻଶ ݀‫ݎ‬Ǥ߳ఏ ሺ݂ሻ ൌ ඨ න‫ݎ‬௅ ௥೘೔೙ (15)

Moreover, the second criterion permit to evaluate the flatness of the Sound Pressure Level (SPL),
on the semi-circle defined above. The SPL in dB is defined as:ȁܲሺ‫ݎ‬ǡ ݂ሻȁଶܵܲ‫ܮ‬ሺ݂ሻ ൌ ͳͲ ଵ଴ ቆ
ቇǡ
(16)
ଶ
ܲ௥௘௙
where, ܲ௥௘௙ ൌ ʹͲߤܲܽ is approximately the threshold of hearing at 1 kHz.
Let ܵܲ‫ܮ‬௠௘௔௡ be the mean SPL in the frequency range ȟ݂ ൌ ݂௠௔௫ െ ݂௠௜௡ , the SPL mean error
indicator is expressed as:߳ௌ௉௅ ሺ‫ݎ‬ሻ ൌ ඨͳ ௙೘ೌೣ
ሺܵܲ‫ܮ‬ሺ‫ݎ‬ǡ ݂ሻ െ ܵܲ‫ܮ‬௠௘௔௡ ሻଶ ݂݀Ǥනȟ݂ ௙೘೔೙

(17)

A smaller value for both indicators means a smoother response of the structure, when it increases, theresponses lost their flatness and the sound source loses quality.

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5. Numerical results
5.1 Structural results for the localization methods
The three methods described previously are applied numerically and compared to a case where nocontrol is imposed, with only the nearest actuator to the target shape actuated.

Figure 3 – Reconstructed shapes with (a) STIF, (b) TR, (c) MC and (d) without control.
The Figure 3 shows the normalized mean quadratic displacement between 100 Hz and 800 Hz. It
shows that the STIF method focuses the energy on the target area very efficiently (Figure 3a) comparedto the other methods. After 800 Hz, the method doesn’t work anymore. Additionally, the TR focuses
moderately the waves in the frequency band (Figure 3b) but does not maintain the target shape foreach frequency. Similarly, the MC localizes the energy alternatively next to the target shape andbetween ribs (Figure 3c), but does not work well since the mode shapes are not conventional flexuralmodes but a mixture of global and local modes. Finally, without control strategy, the energy is also
confined between the ribs near the target shape and shows better precision than MC and TR ( Figure3d). The initial learning of the dynamic behavior allows to have better focusing precision byconsidering the local modes in the dynamics, in a wider frequency band.
5.2 Sound radiation results for the localization methods
The directivity and SPL error in the frequency band [100Hz;10kHz] are plotted Figure 4 for eachmethod. Moreover, the overall directivity in [-90°;90°] are plotted versus the frequency in Figure 5.
Firstly, one can observe that the error of the STIF method is very low and increases abruptly above
1 dB after 800Hz. Additionally, the error shows four peaks at respectively 150 Hz, 270 Hz, 338 Hz,
and 440 Hz approximately. It corresponds to the first eigenfrequencies of the structure which
influences the target shape pattern and thus the sound radiation. It means that the directivity isomnidirectional until 500 Hz, except for modal frequencies, as shown in Figure 4a and Figure 5a.
Between 500 Hz and 800 Hz, there is an apparition of dips around [-90°;-45°] and [45°;90°], andabove 800 Hz, the structure radiates mainly in the middle of the plate. Hence, this behavior isconsistent with the precedent structural study, because below 800 Hz, the localized source behaves asa piston and above the plate doesn’t localize anymore vibrations. Moreover, the directivity error forthe other curves show the same trend, with an error increasing globally after 500 Hz. Indeed, the MC
produces also omnidirectional directivity pattern until 500 Hz approximately and after the soundradiation is concentrated in the middle as for the STIF, and the SPL increases with frequency (Figure5c). Additionally, when no control is implemented, the SPL shows several dips and peaks due to the6325modal behavior. Above 500 Hz, it doesn’t show any omnidirectional pattern (Figure 5d). Finally, theTR method gives a very poor SPL level compared to other methods (Figure 5b).

Figure 4 – a) Directivity and b) SPL error of the four reconstructed shapes between 100Hz and 10kHz.
The SPL error curves available in Figure 4b allow to understand the flatness of the SPL according
to the spatial position. The SPL error of the STIF method around 10 dB is fairly flat and thus showsthat the SPL is quite smooth. According to the precedent discussion, the MC exhibits an increasing
SPL from 50 dB to 150 dB and its error is very high and not flat (between 25 and 40 dB). The same
behavior is noticeable for the TR but with a decreasing and very low SPL level , and an error around
60 dB. This is due to the lack of robustness of the method in case of highly damped media . Finally,
without any control strategy, the error contains several oscillations of 5 dBs around 15 dB andconfirms the modal behavior. The STIF method achieves better performances in terms of directivity
and SPL response flatness than the other methods.

Figure 5 – Directivity vs Frequency for respectively a) STIF, b) TR, c) MC and d) without control.

6. CONCLUSIONS
To conclude, three focusing wave methods were tested and adapted for bending wave in the caseof sound reproduction. Moreover, the sound radiation was computed thanks to the Rayleigh integrals.
Within the automotive industry and spatial sound rendering, it is obvious that the STIF method would
be the preferred method. Indeed, even if the computation coast is higher, the method is easier toimplement and allows a better focusing in term of vibrational point of view. Additionally, it provides
the best directivity and flatness SPL response pattern. This will allow to achieve a better soundreproduction than with simple MAPs. Finally, several parameters as the number of actuators and theboundary conditions have been studied but not presented here, influence the localization results, andmust be considered to find the best compromise between those parameters and the aimed purpose.

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ACKNOWLEDGEMENTS
This work was financially supported by the French National Research Agency (ANR, contract
ANR-17-CE33-0004).

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