Justine Gontier
email: gontier.justine@gmail.com

Carlos Hervas-Garcia
email: carlos.hervas-garcia@isae.fr

Joel Bordeneuve-Guibe
email: joel.bordeneuve-guibe@isae.fr

Christine Espinosa
email: christine.espinosa@isae.fr

CANSAT : MULTIPHYSICAL EXPERIMENTAL DESIGN OF A PROBE COME BACK MISSION

Earth reentry and recovering of a small space probe (typical of a students' project satellite) set questions related to guidance and control. If the probe is asked to reach a target point with constraints on the landing precision, then other questions related to the foil and to the complete system performance arise. In this frame, the aim of the present work is to design, to build and to operate a decelerator system that is automatically controlled by the satellite probe in the objective to reach an a priori chosen landing point with a required precision. This multiphysics project involves at the same time design and fabrication of the decelerator (parafoil and suspension lines), flight mechanics of the deformable decelerator, guidance and control development, system engineering of the probe development. The project is divided in two similar but distinct projects. The first one called "small probe" (33 cL / up to 350 g, International Class) aims to improve the complete system global performance with the goal of precise landing and in-flight missions, for a low altitude release configuration. The "small probe" students' project aims the French CanSat challenge, an international competition organised in France by CNES and Planète Sciences Association. The second one is to launch a "big probe" (1 L / up to 1 kg) from a 1/25th scale replica of the Soyouz Rocket developed by a partner student's team of Samara State Aerospace University (Russia). The latter case aims to test previous developments in more complex and severe release and descent conditions.

I. INTRODUCTION

Earth re-entries of small space sample payloads [START_REF] Benton | Development of precision parafoil flight at high alitude for sample return applications[END_REF][START_REF] Ramus | Characterizing an experimental decelerator for delivering Nano-sat payloads to planetary surfaces[END_REF] or recovering of high altitude scientific analysis devices are the typical kinds of return application under consideration (Figure 1). After atmosphere re-entry of the former both kinds of project need to develop both a reliable automatic guidance and control system, and a stable decelerator system that can be controlled by the probe. 1 : opening and self-stabilization, stable descent, approach and final landing phase. Reliable guidance and control systems are necessary to give the system autonomy to self-compute position and distance to the target point, and give the control order to the governing actuator. In the case of a small parafoil decelerator systems (small mass of payload and small foil), the extreme thinness and deformability and the foil make it sensitive to external wind amplitude and changes. As a consequence the foil does not keep a permanent shape making difficult its control through the suspension lines. Due to its small weight, the system furthermore suffers high displacements and rotations modifications (and sometimes looping!) between two sequenced instants of the control system [START_REF] Archen | Cansat[END_REF].

Stability of the foil for different sizes has previously been studied by our student team through laboratory wind tunnel experiments [START_REF] Ben Yamin | Caractérisation de voilures souples pour concours Cansat[END_REF] and complete systems drop tests from a balloon [START_REF] Castellon | CanSat[END_REF]. Numerical simulations of the fluidstructure interaction between the parafoil and the air during the stable descent phase have also been studied by partner students' internships [START_REF] Loukota | Fluid Structure Interaction of a Parafoil[END_REF] . For the global project further investigations it was shown [START_REF] Pierré | CanSat: multiphysics experimental design of a probe come back mission[END_REF] that the control algorithm had to be improved for low altitude drop tests of a small probe under balloon, and that the complete system had to be improved for higher altitude launch. As a secondary objective, it is aimed to challenge our both projects during the French International CanSat championship in August 2013.

The purpose of the present paper is to explain the development strategies and results of the two similar projects called "small probe" and "big probe". The small probe is an International Class CanSat device (33 cl up to 350 gr) dropped from a captive balloon. A special solar balloon system has been developed to allow rapid testing and to respect French students operation regulations. The big probe is an Open Class CanSat device (1 L up to 1 kg) launched from a 1/25 th scale replica of the Soyouz Rocket designed, build and operated by a partner student team of the Samara State Aerospace University (Russia). The CanSat Challenge consists in the conception of an autonomous device to be dropped from a captive balloon at 150 meters high. This device is required to realize 3 technical and scientific missions and respect specifications. Indeed, the volume is typical of a soda can, that is 330 mL, and its mass must be less than or equal to 350 grams.

II. THE CANSAT CHALLENGE REQUIREMENTS

The principal mandatory missions are precise landing, and atmospheric air environment measurements (pressure, temperature, hydrothermal state). Measured data must be transmitted through a real time connection to a ground station. The third mission is up to the team proposal.

II.I. ATMOSPHERIC AIR MEASUREMENTS

The CanSat has to be able to make some measures of atmospheric pressure, hygrometry, and temperature. The values of the different parameters are sent in live to the ground station to be analysed, the data transmission are assured by two antennas, an emitter on-board antenna and a receiving one on the ground.

The recording device, coupled with the real time transmission, enables a record of the flight necessary for a data post-treatment. All the data during the flight are stored on a SD card in order to get the results even if there is a transmission problem during the flight.

II.II. PRECISION LANDING

The objective is to reach a target on the ground. This target is positioned in less than 75 meters from the position of the launch of the CanSat. The exact placement is done by GPS coordinates just before the launch. The goal of this mission is to land as near as possible from the target thanks to a piloting sail or a mechanical system of terrestrial transport after parachuting.

II.III. FREE MISSION

The recording device, coupled with the live date transmission, enables a record of the flight necessary for a data post-treatment. The video gives information about the stability of the CanSat that could be used to optimise the CanSat's answers to the request of the servo-motor.

All the data during the flight are stored on a SD card in order to get the results even if there is a transmission problem during the flight. Different free missions: record in flight paraglide opening and stability, record ground vision to help or complete GPS for the ultimate landing phase. The video gives information about the stability of the CanSat that could be used to optimise the CanSat's answers to the request of the servo-motor.

Other missions aimed to test the effect of the probe configuration under the foil, either horizontal or vertical, on the global system performance and stability.

III.II. PROJECT MANAGEMENT AND TEAMS

This CanSat project is divided in 3 teams: the prototype International Class, the project Soyouz-CanSat and the development of a test system, described on Figure 4.

IV. CONCEPTION OF THE CANSAT : TECHNICAL DEVELOPMENTS IV.I. THE COMPONENTS

The CanSat is composed by different components to be supplied, to complete its missions. Its on-board electronic system is controlled by a calculator mini Arduino pro, likened to C++ programming software, and supplied by a lithium polymer battery.

The atmospheric sounding is realised by a pressuretemperature captor. The position of the CanSat is given by a GPS EM406A, and the sail is piloted by two servomotors. The data are sent from an XBee emitter (Figure 5) on the CanSat and an XBee receptor on the ground station (Figure 6). The recording is done by a camera which could record 45 minutes. In order to have the cap followed by the CanSat, the use of an inertial power station has been considered but not realised yet.

IV.II. THE ELECTRONIC CARD

The electronic card allows the connexion between the different components and their alimentation. It is a "simple face" type and it is realised like other classical electric circuit with copper, light-sensitive resin, epoxy support and protector film. For the conception of the card, we have used the software CAO Eagle.

IV.III. THE STRUCTURE

The structure has been realized thanks to the software Catia * in such a way that the limited volume of the CanSat could be optimised. 9). This kind of sail has the advantage to be built easily and to be modified by being adjusted by the rigging [START_REF] Hiraki | development of autonomous flight systems with small parafoil[END_REF][START_REF] Bayada | Etude de la Pilotabilité d'un Parafoil[END_REF].

We use two NPW5 sails. The dimensions and the lengths of the riggings are calculated according to the lift/drag ratio, the mass of the CanSat and the progress speed. This speed has to be at least 5 m/s et the mass is limited by 300 g. The lift/drag ratio is around 2. So, thanks to the computation software "TW NPW5", the sails have a span around 50 cm and a surface of 0,36 m 2 . For the piloting of the CanSat, two configurations have been thought, but no one is optimized. The question of the piloting is not yet completely solved [START_REF] Belnou | CanSat : Pilotage Guidage[END_REF].

Vertical configuration

In this configuration, the CanSat turns thanks to a unique servo-motor which attaches the sail. The gravity centre is moved from a referential linked to the sail. Thus, the probe turns. This configuration is difficult to pilot because the rig-gings are linked together to the pale of the servo-motor.

Horizontal configuration

For this configuration we wanted to reproduce an architecture based on the one used for parachute or paragliding. In this configuration too, the CanSat moves thanks to the moving of its gravity centre. The difference is that the sail is deformed like the one on a paragliding, pulling the riggings.

In this configuration,the use of two servo-motors, for each side of the sail, is necessary. An other difference is that with this configuration, only one rigging is piloting and the others are fixed to the structure. Nevertheless, in this position, the gravity centre is on the servo-motors axis and so the CanSat has not the wanted performance.

The piloting of the CanSat is still something which has to be improved.

V.II. GUIDANCE OF THE CANSAT

To fulfill to the precision landing mission, an algorithm has to be developed in two parts, the approach strategy and the finale approach.

The approach strategy

The direction calculated by the Arduino thanks to the GPS coordinates is different from the real direction taken by the CanSat (Figure 10). The GPS measures are taken every 2 seconds and the potential trajectory curve. The principle of the algorithm is to send a command to the motors every two GPS-periods. The command sent to the servo-motors is a proportional gain K multiplied by the direction error :

Motorcommand = K.ε Cap (1)
with

ε Cap = theoreticalCap -f ollowedCap (2)
The final approach This strategy consists in measure the distance between the CanSat and the target every GPS-period. As soon as this distance is under 5 metres † the Arduino sends the maximal command to the CanSat which spirals down and stays in the area of the target.If the CanSat is going out of the target area, due to the wind for example, the approach strategy is used again.

Even with those codes, the result is not always good. Indeed, the mass of the CanSat is very low and so it is very sensible to the wind.

V.III. ASSOCIATED CODE

In the aim to pilot the CanSat, a micro-controller Arduino V0023 is used for executing in loop the code in C language. All the wished actions on the different CanSat components are done by the Arduino. The GPS coordinates are recovered and then analyzed by the code which allows to give the commands to the servo-motors. In the same way, the temperature and the pressure datas are collected and analyzed to calculate the height of the CanSat. All those datas are saved in a SD card and sent directly to the ground thanks to the XBee.

The code is started thanks to an umbilical cable which is removed when the CanSat is launched.

VI. TESTS WITH A SOLAR BALLOON

VI.I. CREATION OF THE SOLAR BALLOON

In order to test the piloting guidance of the CanSat, in the conditions of the challenge, the use of a captive solar balloon of 4 meters of diameter is a good solution [START_REF] Chevalier | Développement d'un moyen d'essais[END_REF]. Indeed, thanks to the sun ray, the air inside the balloon becomes hotter than the air outside and the ballon begins to ascend. This ballon can ascend to 150 meters with 2 kilograms of load.

A tube in composite has been made to put the CanSat into and to launch the CanSat in a similar way to the challenge. † Distance determined in advance and equal to the GPS precision Nevertheless, this mode of tests is very sensible to the climate, like the temperature and the wind.

VI.II. TESTS OF THE CANSAT

Tests at 50 meters have been achieved. The sail of the CanSat opened well. The good response of the command showed that the piloting and the guidance of the CanSat were working. The precision of the piloting guidance could be tested. To do that, the CanSat would have to be launched at 100 meters.

VII.I. THE SOYOUZ-CANSAT PROJECT

One of the natural evolutions of the CanSat project is the Open Class CanSat or, in other words, the CanSat 1Kg. This evolution comes from the initiative of cooperation between ISAE and the Samara State Aerospace University in Russia, in an act of commemoration of the first french-russian space cooperation.Teams from both universities agreed to develop and integrate a new CanSat into a 1/25th Soyouz rocket replica to compete annually in Open Class Categories of CanSat competitions. This cooperation work requires the integration of the already existing CanSat technology into the launcher constraints. Besides, the redesign and implementation of new technology was necessary in order to respond to the new requirements.

VII.II. MISSIONS' DEFINITION AND NEW CHAL-

LENGES

The missions of the Open Class CanSat are the same as the ones of the International Class CanSat as far as precision landing and atmospheric air measurements are concerned. However, the "big probe" will be launched from the Soyouz rocket at an altitude between 700 and 1000 meters, hence the following adapted missions :

• Real-time telemetry: once the CanSat is ejected from the Soyouz rocket, it will have to transmit data to the ground station. Tough real-time sounding is not required by the competition basis, this achievement could make the difference from other projects. Therefore a great effort has been done to develop a communication subsystem powerful enough to perform real-time transmission from the very beginning of the CanSat flight.

• Comeback: the CanSat will not simply be dropped. One of the challenges of a CanSat project is to build a payload capable of, autonomously, reach a target previously set. For this task the already existing piloting and guidance algorithm has been refined and tuned for this new CanSat.

• Integration into the Soyouz: the launcher is meant to climb up to 1 km height and eject at that point the CanSat. It introduces constraints in weight, volume and acceleration forces that the CanSat has to meet. One of the main challenges of this project is the continuous cooperation and work under constraints in what constitutes an authentic system integration project.

VII.III. OPEN CLASS CANSAT DEVELOPMENT

Structure and parachute

The initial existing CanSat structure was no longer valid. The new CanSat must prove its strength to accelerations forces close to 12 g at take-off ‡ as well as strong decelerations at the time of deployment of the parachute or at the moment of impact with the ground. Accordingly, an important redesign and reinforce study of the thermoplastic structure was held.

Firstly, a finite elements study in SAMCEF § was carried out. The 3 most critical phases were modelled for its posterior study: the take-off, the parachute deployment and the impact with the ground. This study revealed the weakest points of the initial concept, designed in CATIA V. Next, this concept was modified in order to strengthen ‡ This acceleration is imposed by the launcher § Samcef is a finite elements computation software the most critic points in terms of load. The final model was achieved after an iterative process of analysis and redesign. Some results of the structural study and redesign work are shown in Figures 16 and13. Besides, a new adequate parachute had to be found to make possible the control of the flight. This parachute was chosen regarding its performance and manageability when attached to a 1 kg load. The parachute had been designed, built and tested entirely within the context of a final year work project of ISAE students [START_REF] Ben Yamin | Caractérisation de voilures souples pour concours Cansat[END_REF]. This year it has been re-adapted and tested in order to meet the competition requirements. • Microcontroller subsystem: The Arduino Pro Mini (based on the ATmega168 electronic board) is the brain and the hub of all the electronics CanSat. It is responsible for the execution of the piloting and guidance code as well as the management of information between the components and between the CanSat and the ground.

• Peripheral components subsystem: formed by the temperature, pressure and hygrometry sensors, the camera, the GPS module and the servo-motor. This set of components is aimed at either receiving data from the environment and send it to the Arduino or execute the Arduino's commands.

• Communication subsystem: Dedicated to make possible the real-time data link between the CanSat and the ground station when airborne, several improvements from previous CanSats were necessary. In addition to the Xbee wireless communication modules previously used, the antennae had been conceived and developed specifically for this project. Regarding the importance of this feature within the Open Class CansSat project, the next point of this article will be dedicated to this subsystem. All this subsystems are assembled in an electronic board with the aim of centralizing the functions around the microcontroller and ease the reparation or substitution in the case of an eventual failure in any subsystem.

Communication Subsystem

One of the most important involvements of the Open Class CanSat project is to adapt to the exigencies of the Russian launcher. Apart from the structural and mechanical constraints, the fact of being ejected at 1 Km height implies that, for the success of the missions, the communication must have a range greater than 1000 m.

Establishing a simple equation for the link budget be-tween the CanSat and the ground station it shows that:

P rec = λ 2 4πR 2 G Emax G Rmax P in (3) hence, R max = λ 4π G Emax G Rmax P in P rec (4)
Being restricted for the public band frequency between 2.4 and 2.5 GHz (and hence the wavelength), as well as for the power emitted (by the French electromagnetic emission regulations) and the power received (dependent on the receptor technology), the maximum range can only be increased by increasing either GEmax or GRmax of the onboard and reception antennae respectively or both. Thereby one of the main tasks of this project has been to think out a technological solution to this problem that allowed increasing the emission and reception gains while adapting to the volume, mass and shape constraints.

Amongst the limited technological solution, the patch antenna seemed to be the most appropriate. It is a similar technology to the one used in modern GPS receptors. Three different antenna models were theoretically conceived. Some of the principal pursuit features of these antennae were:

• Radiation pattern as roundest as possible, with the maximum gain increment within a beam of 120 o from the vertical axis of the CanSat.

• Resonance frequency of the antenna on the public band frequency together with an optimal adaptation of the transmission line to minimize the reflection losses.

• Circular polarization of the electromagnetic field so as not to depend of the CanSat orientation with respect to ground.

For its theoretical validation, a complete computational simulation in HFSS ¶ was carried out. Later, these antennae were fabricated and tested to verify and validate its technical specifications.

Piloting and guidance Strategy

At this point, the Open Class CanSat reused the existing piloting and guidance strategy for the CanSat vertical configuration. A few changes on the proportional gain and initial setup were necessary to adapt the already developed law to the 1kg CanSat. These changes showed to perform adequately. ¶ HFSS is a commercial electromagnetic simulation software Project management and cooperation work with the Russian Team

The participation in the CanSat competition together with the required continuous communication with the Russian Team was responsible for a demanding commitment of organization. A regular monthly call as well as daily mailing with them ensured the success of this cooperation. In addition a representative of the Open Class CanSat ISAE team went to the Russian facilities in Samara to finalize the integration of the interface Soyouz-CanSat.

Figure 1 :

 1 Figure 1: Flight mission illustration

Figure 2 :

 2 Figure 2: Comparison between real and numerical deflection

Figure 3 :

 3 Figure 3: The CanSat International Class compared to a can

  III. CONCEPTION OF THE CANSAT : PROJECT MANAGEMENT III.I. CONTEXT AND SUMMARY OF PREVIOUS WORKS Since 2009, our school, ISAE-ENSICA, has been represented in the CanSat France Competition through the former club Budstar. Since summer 2012, the former Budstar team has become an association called N6K'nSat. This year, we build two CanSat categories, an International Class and an Open Class. N6K'nSat was created to promote our study and work on CanSat, and more and more students have participated in the CanSat project. Indeed, ENSICA students have won CanSat France Competition every year since 2009 and one Spanish contest .

Figure 5 :

 5 Figure 5: XBee, transmitter in the CanSat

Figure 6 :

 6 Figure 6: XBee receiver on the ground, linked to the ground station

Figure 7 :

 7 Figure 7: Structure of the CanSat by the software Catia

Figure 4 :

 4 Figure 4: Organisation Chart

Figure 8 :

 8 Figure 8: The CanSat

Figure 9 :

 9 Figure 9: Nasa ParaWing 5 Sail

Figure 10 :

 10 Figure 10: Iterative Strategy with computation of the real course and comparison with the route to follow before turning

Figure 11 :

 11 Figure 11: Tests of the CanSat with the Solar Ballon

Figure 12 :

 12 Figure 12: Results of SAMCEF computation

Figure 14 :

 14 Figure 14: Sail used for the CanSat

Figure 15 :

 15 Figure 15: Patch antenna model

Figure 17 :

 17 Figure 17: Integration of the interface Soyouz-CanSat

ACKNOWLEDGEMENTS

This work has been partially funded by the following organisms: ISAE SUPAERO Association, CNES (French Space Agency), ISAE Direction Formation ENSICA. All the students and permanent people of ISAE who contributed to this project are sincerely thanked for their deep help. A special mention is due to the original ENSICA team Budstar Association for their friendly support.


Olivier Cherrier

Pierre Selva

Vale ´rie Pommier-Budinger

Fre ´de ´ric Lachaud

Joseph Morlier

Valerie Pommier-Budinger
email: valerie.budinger@isae.fr

Damage localization map using electromechanical impedance spectrums and inverse distance weighting interpolation: Experimental validation on thin composite structures

Keywords: Composite structures, damage localization map, electromechanical impedance, inverse distance weighting

Piezoelectric sensors are widely used for structure health monitoring technique. In particular, electromechanical impedance techniques give simple and low-cost solutions for detecting damage in composite structures. The purpose of the method proposed in this article is to generate a damage localization map based on both indicators computed from electromechanical impedance spectrums and inverse distance weighting interpolation. The weights for the interpolation have a physical sense and are computed according to an exponential law of the measured attenuation of acoustic waves. One of the main advantages of the method, so-called data-driven method, is that only experimental data are used as inputs for our algorithm. It does not rely on any model. The proposed method has been validated on both one-dimensional and two-dimensional composite structures.

Introduction

Structural health monitoring (SHM) is an emerging technology, dealing with the development and implementation of techniques and systems where monitoring, inspection and damage detection become an integral part of structures. It further merges with a variety of techniques that will provide information about the condition of a structure in terms of reliability and safety before the damage threatens the integrity of the structure. [START_REF] Farrar | An introduction to structural health monitoring[END_REF][START_REF] Doebling | A summary review of vibration-based damage identification methods[END_REF] In the paradigm of SHM, five major steps exist: (a) detection of damage in a structure, (b) localization of damage, (c) damage identification, (d) quantification of damage severity, and (e) prognostic of remaining service life of the structure. [START_REF] Farrar | Damage prognosis: the future of structural health monitoring[END_REF] SHM systems can be tuned to monitor damage such as fatigue cracks, corrosion, delamination, loose bolts, or impact damage either in real time or on demand. For aeronautical structures, a field where this problem has been quite studied, the components have to resist low-energy impacts caused by dropped tools, mishandling during assembly, and maintenance and in-service impacts by foreign objects such as stones or birds. In these impacts of relative low energy, a small indentation is often referred to as barely visible impact damage (BVID). Although not visually apparent, low-energy impact damage is found to be quite detrimental to the load carrying capabilities of composite structures, underscoring the need of reliable damage detection techniques for composite structures. [START_REF] Schubel | Impact and post impact behaviour of composite sandwich panels[END_REF][START_REF] Abrate | Impact on composite structures[END_REF][START_REF] Petit | Impact and compression after impact experimental study of a composite laminate with a cork thermal shield[END_REF] When visual inspection is needed, structural components can be extracted for maintenance. Successful nondestructive testing (NDT) techniques such as radiographic detection (X-ray) and ultrasound testing (C-Scan) exist but are impractical and/or expensive for large components and integrated vehicles. The major advantage of SHM techniques is their online implementation and their mixed global/local approach (network of sensors).

A promising technique, namely, vibration-based structural health monitoring (VBSHM), has been proposed to address the problem of globally detecting, characterizing, and, to a certain extent, localizing damage based on changes of modal parameters of a structure in operation. [START_REF] Cawley | The location of defects in structures from measurements of natural frequencies[END_REF][START_REF] Farrar | Vibration-based structural damage identification[END_REF] Any structure can be considered as a dynamic system with stiffness, mass, and damping. Once some damage emerges in the structures, the structural parameters will change, and the frequency response functions (FRFs) and modal parameters of the structural system will also change. This change of modal parameters can be taken as the signal of early damage occurrence in the structural system and especially in composite structures. [START_REF] Montalva˜o | A review of vibration-based structural health monitoring with special emphasis on composite materials[END_REF] Recently, some authors [START_REF] Shahdin | Correlating low energy impact damage with changes in modal parameters: a preliminary study on composite beams[END_REF][START_REF] Shahdin | Damage monitoring in sandwich beams by modal parameter shifts: a comparative study of burst random and sine dwell vibration testing[END_REF] evaluated experimentally the modal parameter changes due to impacts for several energies of impact. They also demonstrated the sensibility of damping changes to detect delamination in composite structures. Montalvao et al. [START_REF] Montalvao | A method for the localization of damage in a CFRP plate using damping[END_REF] have succeeded in creating damage localization maps, using three-dimensional (3D) laser vibrometer and an indicator containing information on damping changes and local stress field changes. Friedel et al. [START_REF] Frieden | Vibration-based characterization of impact induced delamination in composite plates using embedded FBG sensors and numerical modelling[END_REF] proposed a detailed numerical model including the damage pattern obtained from X-ray computed tomographic images. They demonstrated that most of the frequency changes can be explained by one type of damage (delamination) and thus linked the total delamination surface with the absorbed impact energy (affine relation). They built a homogenized damage model, including two damage factors, that allows predicting the change of natural frequencies for a known damage size.

An important research community tries to use local modal parameters such as high-resolution mode shapes (or derived data) to localize damage using wavelet analysis [START_REF] Morlier | Diagnosis of a portal frame using advanced signal processing of laser vibrometer data[END_REF][START_REF] Rucka | Application of continuous wavelet transform in vibration based damage detection method for beams and plates[END_REF][16][START_REF] Huang | Damage detection with spatially distributed 2D continuous wavelet transform[END_REF][START_REF] Loutridis | A twodimensional wavelet transform for detection of cracks in plates[END_REF][START_REF] Qiao | A 2-D continuous wavelet transform of mode shape data for damage detection of plate structures[END_REF][START_REF] Katunin | Damage identification in composite plates using two-dimensional B-spline wavelets[END_REF] or image processing. [START_REF] Morlier | Modeshapes recognition using Fourier descriptors: a simple SHM example[END_REF] Finally, another widespread trend is to use, instead of vibration data, highfrequency waves called Lamb waves. Piezoelectric wafer active transducers are used to generate and receive guided Lamb waves propagating in the structure. In the study by Yu and Giurgiutiu, 22 a virtual beam steering method and device used as embedded ultrasonic structural radar was implemented as a signal post-processing procedure. The arrays of piezoelectric transducers (lead zirconate titanate (PZT)) yielded good directionality within the full range of 360°and were able to detect damage anywhere in the entire plate. Ostachowicz et al. [START_REF] Ostachowicz | Damage localisation in plate-like structures based on PZT sensors[END_REF] also used elastic wave propagation phenomenon from two PZTs for damage detection and localization.

As a result of processing of the signals registered from the structure, special maps that indicated damage location on aluminum and composite plates were built. They extended their effort in the study by Malinowski et al. [START_REF] Malinowski | Damage detection potential of a triangular piezoelectric configuration[END_REF] and developed a triangular piezoelectric configuration and an algorithm that transfers information from the time domain to the spatial domain to localize damage on anisotropic material plate.

Classical SHM methods use model-driven data; it means that high-quality models for both undamaged (UD) (baseline) and damaged cases (delamination, crack, etc.) are required. Then, the damage is identified from a metric corresponding to the distance between experimental and numerical data (FRF), natural frequencies, mode shapes, etc.). [START_REF] Adams | Health monitoring of structural materials and components[END_REF][START_REF] Staszewski | Health monitoring of aerospace structures[END_REF][START_REF] Worden | The fundamental axioms of structural health monitoring[END_REF] The localization problem can be viewed as an inverse problem often solved using optimization and model updating process [START_REF] Moaveni | Damage identification of a composite beam using finite element model updating[END_REF][START_REF] Meo | Damage assessment on platelike structures using a global-local optimization approach[END_REF][START_REF] Niemann | Damage localization using experimental modal parameters and topology optimization[END_REF][START_REF] Shahdin | Correlating low energy impact damage with changes in modal parameters: diagnosis tools and FE validation[END_REF][START_REF] Viana | Combining experimental and computed data for effective SHM of critical structural components[END_REF] to minimize the error between experimental and numerical data.

In this article, our approach is mostly experimental. The purpose of our work is based on the following industrial needs: Low-cost instrumentation and easy connection; Simple criteria for detecting damage; Reliable visualization of damage location through signal processing.

To achieve the first point, especially for real-size industrial structures, SHM community often employs smart materials. In particular, the electromechanical impedance (EMI)-based SHM technique possesses distinct advantages such as the ability to detect incipient damage, use of nonintrusive piezoelectric transducers, and potentially low-cost applications. [START_REF] Park | Overview of piezoelectric impedance-based health monitoring and path forward[END_REF][START_REF] Lim | Structural identification and damage diagnosis using self-sensing piezo-impedance transducers[END_REF][START_REF] Yan | Monitoring interfacial defects in a composite beam using impedance signatures[END_REF][START_REF] Bhalla | Structural health monitoring by piezo-impedance transducers applications[END_REF][START_REF]Structural health monitoring using high-frequency electromechanical signatures[END_REF][START_REF] Bhalla | Ultra low-cost adaptations of electro-mechanical impedance technique for structural health monitoring[END_REF][START_REF] Giurgiutiu | Damage detection in thin plates and aerospace structures with the electro-mechanical impedance method[END_REF] A simple method to measure EMIs is the use of an impedance analyzer that allows you to receive information about many electrical quantities (e.g. voltage/current characteristics) as a function of the frequency. In this article, our low-cost SHM system consists of few PZTs (three PZTs for triangulation) that are bonded to the structure and can be plugged to the data acquisition hardware on demand.

For the second point, damage indicators (DIs) derived from the measured EMI are classically used to provide information about damage detection. [START_REF] Giurgiutiu | Damage detection in thin plates and aerospace structures with the electro-mechanical impedance method[END_REF] The earliest works on the EMI method are those published by Liang et al. in two articles. [START_REF] Liang | Coupled electromechanical analysis of adaptative systems-determination of the actuator power consumption and system energy transfer[END_REF][START_REF] Liang | An impedance method for dynamic analysis of active material system[END_REF] The first article presents a coupled electromechanical analysis of piezoelectric ceramic (PZT) actuators integrated in mechanical systems to determine the actuator power consumption and energy transfer in the electromechanical systems. The second article presents the essence of the method with usage of model of piezoelectric transducer bonded to 2-degree-of-freedom structure.

To achieve the last point and to get reliable localization maps, the analysis of DIs based on EMI is more difficult than the detection. In this article, our original method is based on acoustic wave attenuation and relies on the hypothesis that the interaction between the damaged zone and the sensor depends on the distance between them. Inverse distance weighting (IDW) interpolation will be used to express this hypothesis.

EMI for damage detection

Basics of EMI

The mechanical impedance of a structure, at the force application point, is defined as the ratio of the driving harmonic force to the resulting harmonic velocity at the same point. Thus, the mechanical impedance can be expressed as [START_REF] Bhalla | Structural health monitoring by piezo-impedance transducers applications[END_REF][START_REF] Giurgiutiu | Structural health monitoring with piezoelectric wafer active sensors[END_REF][START_REF] Bhalla | Structural health monitoring by piezo-impedance transducers part I: modeling[END_REF]

Z = R + jX = F v = F 0 e jvt v 0 e j vtÀ; ð Þ = F 0 v 0 e j; ð1Þ
where Z is the mechanical impedance, R is the resistance, X is the reactance, F is the force, F 0 is the initial force magnitude, n is the velocity, v 0 is the initial velocity, ; is the phase delay, and v is the angular frequency.

A continuous structure (such as plate or beam) can be considered as a mass-spring-damper system. Thus, the equivalent mechanical impedance is given by

Z eq = cm 2 v 2 c 2 + vm À k m À Á 2 + j mv c 2 À k m vm À k v À Á Â Ã c 2 + vm À k v À Á 2 ð2Þ
Damage in laminated composites (delamination due to impact, etc) causes locally an increase in damping and a decrease in stiffness. [START_REF] Cawley | The location of defects in structures from measurements of natural frequencies[END_REF][START_REF] Farrar | Vibration-based structural damage identification[END_REF][START_REF] Montalva˜o | A review of vibration-based structural health monitoring with special emphasis on composite materials[END_REF][START_REF] Shahdin | Correlating low energy impact damage with changes in modal parameters: a preliminary study on composite beams[END_REF] These changes induce a shift in the mechanical impedance. Figure 1 illustrates the effects of these changes on the mechanical impedance spectrum. Figure 1(a) gives an example of real and imaginary parts of a system with three values of the damping coefficient (c, 2c, and 4c). Figure 1(b) gives the real and imaginary parts of a system with three values of the stiffness (k, 0.98k, and 0.95k).

In recent years, the damage detection methods based on the EMI that is a function of the mechanical impedance of a structure have gained increased attention. The methods use small-size piezoelectric transducers intimately bonded to the structure under study. The measured EMI with a piezoelectric transducer is the result of a coupling between the transducer and the structure (Figure 2).

The EMI methods are mostly effective at ultrasonic frequencies, and they properly capture the changes in local dynamics due to incipient structural damage. In complex large structures, such changes are too small to affect the global dynamic behavior of the structure and hence cannot be readily detected by conventional lowfrequency vibration methods. Experimental demonstrations have shown that the real part of the highfrequency impedance spectrum is directly affected by the presence of damage or defects in the monitored structure. [START_REF] Giurgiutiu | Structural health monitoring with piezoelectric wafer active sensors[END_REF] Besides, a usual bandwidth for thin composite structures is 10-20 kHz.

For detecting damage, the variations of modal parameters have to be estimated with accuracy from the measurement of the EMI. The reliability of the estimation of modal parameters is strongly dependent on several parameters: the sampling frequency, the modal density in the selected bandwidth and also the level of the internal (structural) damping. The real part of the measured impedance is classically used in damage detection because of the simplicity of the process of peak detection.

Damage metrics

The development of suitable damage metrics and damage identification algorithms remains an open question in the practical application of EMI technique. The damage index compares the amplitudes of the two spectrums (damaged vs UD) and assigns a scalar value that serves as a metric for the damage analysis in the structure. [START_REF] Worden | The fundamental axioms of structural health monitoring[END_REF] In this article, two variation indices (so-called damage features in pattern recognition) are used. The first one is the root mean square deviation (RMSD) and the second is the mean frequency shift of modal peaks (Df mean ).

The RMSD is defined as

RMSD % ð Þ= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P N Real Z D ð Þ À Real(Z UD ) ½ 2 P N Real(Z UD ) ½ 2 v u u u u t 3100% ð3Þ
where Real(Z UD ) is the real part of impedance for the UD structure, Real(Z D ) is the real part of impedance for the damaged structure, and N is the number of samples.

The mean frequency shift is defined as

Df mean % ð Þ= P N n = 1 p f D n À f UD n f UD n N pks 3100% ð4Þ
where f UD n is the modal frequency of the UD structure for the mode n, f D n is the modal frequency of the damaged structure for the mode n, and N pks is the number of modal peaks in the studied frequency band.

These two variation indices can be used for damage detection by comparing their values to thresholds that are significant of damage presence. These thresholds have to be determined experimentally for each structure under study. In this article, we propose to use these indices in a damage localization process by propagating the local information extracted from each sensor to obtain a map of damage.

Damage localization map using acoustic wave attenuation

Acoustic attenuation

Acoustic attenuation is a measure of the energy loss of sound propagation in media and depends on several phenomena (diffraction, reflection, diffusion, and absorption). For laminated materials and unidirectional composite materials, acoustic wave propagation is strongly dependent on fiber spacing and orientation. [START_REF] Williams | Ultrasonic attenuation and velocity in AS3501-6 graphite/epoxy fiber composite[END_REF] The technique for measuring the acoustic attenuation coefficient is to evaluate the ratio between incident and transmitted wave levels with an exponential decrease in wave energy 44

a t a 0 = e Àm v ð Þx so m v ð Þ= À ln a t =a 0 ð Þ x ð5Þ
where m(v) is the acoustic attenuation coefficient function of the considered frequency, a 0 is the initial level of generated signal (incident wave), and a t is the received signal after propagation at distance x (transmitted wave). The coefficient used in the next part will be computed from the mean of five measured values. We also observed that for thin composite structures, this coefficient is strongly dependent on the first layer's orientation.

Damage localization map through acoustic attenuation

The main purpose of this article is to construct ''damage localization maps'' using few piezoelectric transducers. For example, for a two-dimensional (2D) structure like a plate, three transducers with unsymmetrical position are used to localize damage in the xy plane (Figure 3) using the principle of triangulation, which is a method of determining the position of an object by measuring its distance from other objects with known locations. In large structures, the boundary conditions introduce sudden changes in wave propagation, and the method requires to consider the structures as an assembly of several elementary substructures that will all be instrumented. For example, in an aeronautical fuselage, every stiffened panel (or set of panels depending on the accuracy of the damage localization process) is a substructure and needs to be instrumented by three PZTs for triangulation.

Classical triangulation can be conducted on time series for guided waves [START_REF] Flynn | Identifying scatter targets in 2D space using in situ phased-arrays for guided wave structural health monitoring[END_REF][START_REF] Kessler | A structural health monitoring software tool for optimization, diagnostics and prognostics[END_REF] or Lamb waves [START_REF] Chao | Quantitative evaluation of orientation-specific damage using elastic waves and probability-based diagnostic imaging[END_REF][START_REF] Su | Identification of damage using Lamb waves: from fundamentals to applications[END_REF] using signal processing tools to enhance the damage map and avoid false damage. The proposed method is based on an enhanced triangulation based on EMI spectrums and weighted DIs. It is inspired from classical IDW interpolation for 2D surface reconstruction of scatter points. IDW methods are based on the assumption that the interpolating surface should be influenced most by the nearby points and less by the more distant points. In the next chapter, our results from online (SHM) methods are compared to those of off-line (NDT) methods in order to prove the accuracy of our SHM-based method.

The proposed method for the construction of damage localization maps can be divided into four steps. Steps 1 and 2 are preliminary steps that are performed once before monitoring. Step 3 is performed continuously.

Step 4 is performed only if damage is detected in step 3.

Step 1: instrumentation. Depending on the size of structures, PZT transducers must be chosen large enough in order to give ''good'' EMI spectrums, that is to say with several modal peaks. They must be bonded with a rigid glue to get a good electromechanical coupling between the transducers and the structure. At least, they must also be bonded to optimal placements that result from a trade-off between requirements for triangulation and EMI measurements.

Step 2: map construction (meshing) using IDW for each piezoelectric transducer. The structure under study is meshed. Obviously, the precision of localization is directly dependent on the mesh size. Each mesh has at least the piezoelectric transducer size. Then, for each transducer denoted as i, every mesh j is weighted by acoustic attenuation A ij defined by

A ij = e Àm v ð Þx ij ð6Þ
where x ij is the distance between the transducer i and the center of each mesh j. Like in IDW methods, the acoustic attenuation is such that it decreases as the distance between the transducer and each scatter point increases.

This method enables to generate weighted maps with a ''physical'' sense. There are as many weighted maps as transducers.

Step 3: EMI measurements and damage metrics. EMI measurements should be performed regularly for monitoring. In this article, two EMI spectrums are measured for pre and post impact. Then, we can obtain RMSD and Df mean indices for each transducer using equations ( 3) and ( 4). These indices are then compared to thresholds to determine if damage occurs. If yes, a damage localization map is built using step 4 of the method.

Step 4: damage localization map computing. The DIs are computed using IDW interpolation and by taking the inverse in order to get a high value of the indicator in the damaged zone. For each mesh j, the DI based on RMSD index is computed by

DI j(RMSD) = X a i = 1 RMSD i A ij " # À1 ð7Þ
and the DI based on Df mean index is computed by

DI j (Df mean ) = X a i = 1 Df mean, i A ij " # À1 ð8Þ
where a is the number of transducers. For onedimensional (1D) structures, the number of transducers is at least 2, and for 2D structures, it is at least 3.

Plotting the values of a DI for each mesh then gives one damage location map that enables to evaluate the damage position on the structure. For 2D structures like thin plates, the same concept is carried out in two dimensions. Figure 5 gives an example of mesh of a 2D isotropic plate (Figure 5(a)) and of mesh using IDW interpolation with the same attenuation coefficient in any direction for a sensor located at one extremity.

Map examples

When the plates are made with unidirectional composite, the mean attenuation must be measured both in fiber direction and in direction transverse to the fibers. Figure 6 gives an example of mesh with IDW interpolation for a thin plate with a mean attenuation equal to 2.8e23 m 21 in the fiber direction and to 10e23 m 21 in the direction transverse to the fibers. The maps are computed for each transducer (three in the studied case) with these coefficients and for a first ply orientation of 45°.

Finally, Figure 7 gives an example of damage localization map. The map is interpolated at higher resolution (38) for a better visualization (Figure 7(a)). Then, only the zone with the highest probability is plotted to get a single dot and a clear position of damage (Figure 7(b)).

Experimental validation

Case 1: 1D structure (stiffener)

The 1D structure is a composite stiffener of an aircraft door (clamped at the extremities). It has the shape of an I-beam of mean dimensions 66 3 89 3 1045 mm. The beam web is composed of 16 plies of carbon/epoxy plus a peel ply of tissue (considered as equal to isotropic behavior). Two piezoelectric composite patches of type DuraAct Ò (PI ceramics Ò PIC255) of dimensions 50 3 30 3 0.5 mm are glued on to the beam web near each extremity of the beam with a structural glue of type 3M DP460 Ò (Figure 8).

A drop test machine is able to create an impact on a specimen with controlled measurements of the impact head (velocity and force) (Figure 9(a)). The chosen target energy is 35 J for each impact because it corresponds to the energy level for which the damage becomes significant but still remains a BVID. Resulting energy is computed from the measured velocity and the mass. Impact data are presented in Table 1. To test the proposed method of impact location, two positions of damage between the two transducers are investigated one after the other. Impacts are located on the flange of the beam (Figure 9(b)), and an example of damage is shown in Figure 9(c).

The stiffener web is meshed in equal parts (11 meshes of 95 mm length and 89 mm height). Preliminary tests demonstrate that the best EMI response (i.e. with detectable peaks) is between 8 and 16 kHz for the stiffener. In the beam web, the acoustic attenuation is measured in this frequency band, and the mean value of acoustic attenuation coefficient is 4e23 m 21 . Then, two maps, one for each transducer, can be computed with IDW interpolation and plotted (Figure 10).

Before and after damage generation, the real part of the EMI is measured (Figure 11). Df mean and RMSD indices are computed from EMI signatures. The variations of indices are presented in Table 2. Classically, indices for damage no. 2 are computed while considering the state with the first damage as the UD state (D1 replaces UD in equations ( 3) and (4) as reference).

In Figure 12(a) and (b), the probability of damage is plotted for RMSD. In Figure 12(c) and (d), the results for Df mean are highlighted. For the first damage, the location predicted by the RMSD index is good since the high-probability zone indicates the real zone of impact. For the second damage, the predicted location is less accurate. In comparison, the location predicted by the Df mean index is excellent for every impact.

Case 2: 2D structure (thin composite plates)

Six plates of dimensions 290 3 200 3 3 mm and composed of 12 plies of carbon/epoxy prepreg T700M21 with a stratification [45/-45/0/90/0/90]S for a total thickness of 3 mm are investigated in peripheral clamped configuration (Figure 13). Three PI Ò PZT piezoelectric ceramics of type PIC151 of dimensions 10 3 10 3 0.5 mm are bonded with structural glue type EPO-TEK Ò E4110 on each plate as shown in Figure 12.

In this part, six positions of damage are investigated. The chosen target energy is 20 J for each impact, and resulting energy is computed from the measured velocity and force. Impact data are presented in Table 3. In parallel, nondestructive tests (Ultrasonic control Cscans) are carried out to determine the exact position of every impact.

The structure is regularly meshed: 24 meshes of 48 mm length and 50 mm height. Preliminary tests show that the most sensitive EMI signatures are established for the bandwidth 10-20 kHz. The plates have anisotropic behavior, and the mean attenuation must be measured both in fiber direction and in a direction transverse to the fibers. The maps with IDW interpolation computed for each transducer correspond to the ones given as example in Figure 6. The real part of the EMI is then measured on every transducer (before and after impact). An example of the EMI signature for plate no. 1 is given in Figure 14. Using all EMI signatures, Df mean and RMSD indices can be computed for every plate (Tables 4 and5).

The damage localization maps are finally computed for both variation indices. The maps in Figures 15 and16 are overlaid for comparison with the Ultrasonic control C-scans. Both indices permit to give a good estimation of damage localization. Maximal errors are less than the damage size (typically around 20 mm), but in this case, RMSD is more reliable.

Conclusion

The proposed method permits to detect and localize a single isolated damage in composite structures. Piezoelectric transducers bonded to the structure enable to measure EMIs. The purpose of the method in this article is to generate a damage localization map based on both indicators computed from EMI spectrums and IDW interpolation. The weights for the interpolation have a physical sense and are computed according to an exponential law of the measured attenuation of acoustic waves. One of the main advantages of our method, so-called data-driven method, is that only experimental data are used as inputs for our algorithms. It does not rely on any model. The proposed method has been validated in both 1D and 2D composite structures. Future works will consist in evaluating damage severity through an updating process based on gradient optimization. These works will naturally lead to a comparison of our data-driven method to modeldriven method based on high-fidelity finite element (FE) models and supervised learning (probabilistic neural networks (PNN)). Other perspectives to improve the method are to consider more than three transducers for each substructure in order to carry out several damage localization analyses with sets of three transducers and then to compare the results of each set to avoid false damage detection and to increase the reliability.

Figure 1 .

 1 Figure 1. Simple damage model (related to pole shift): (a) example of real and imaginary parts of mechanical impedance versus frequency (a) for an increase in damping coefficients (c, 2c, and 4c) and (b) for a decrease in stiffness (k, 0.98k, and 0.95k).

Figure 2 .

 2 Figure 2. Schematic representation of piezoelectric actuator mounted on an unknown structure.

Figure 3 .

 3 Figure 3. 2D map reconstruction (so-called DI) using the propagation of the three measurements (green points in NDT results). The red area is the SHM damage localization, overlaid of the NDT results (green zone in the center of the plate).

Figure 4

 4 Figure 4 gives an example of mesh of a 1D structure (Figure4(a)) and of mesh using IDW interpolation for a sensor located at one extremity (Figure4(b)).

Figure 4 .

 4 Figure 4. Example of a 1D isotropic structure: (a) mesh in distance and (b) mesh with IDW interpolation computed by the exponential attenuation law for m = 40e24 mm 21 .

Figure 5 .

 5 Figure 5. Example of a 2D isotropic structure: (a) mesh in distance and (b) mesh with IDW interpolation computed by the exponential attenuation law for m = 10e23 mm 21 in all directions. 2D: two-dimensional; IDW: inverse distance weighting; PZT: lead zirconate titanate.

Figure 6 .

 6 Figure 6. Example of a 2D anisotropic structure-visualization of mesh with IDW interpolation for (a) transducer no. 1, (b) transducer no. 2, and (c) transducer no. 3. 2D: two-dimensional; IDW: inverse distance weighting.

Figure 9 .

 9 Figure 9. (a) Drop test machine, (b) location of the impacts on the composite stiffener, and (c) example of typical damage.

Figure 7 .

 7 Figure 7. Example of damage localization map for a composite plate (damage located on the top center): (a) Raw DI and (b) processed DI using a threshold that limits the map to higher probability. DI: damage indicator.

Figure 8 .

 8 Figure 8. Instrumented composite stiffener, two DuraAct Ò piezoelectric patches are bonded at the extremities (red zones).

Figure 11 .

 11 Figure 11. Experimental EMI signatures (between 8 and 16 kHz) for the two transducers on the studied stiffener for UD and damaged cases after a first impact energy of 35 J (D1 = 35 J) and after a second impact energy of 35 J (D2 = 35 J). UD: undamaged; EMI: electromechanical impedance.

Figure 10 .

 10 Figure 10. Visualization of mesh with IDW interpolation computed by the exponential attenuation law for m = 4e23 m 21 (a) for transducer no. 1 and (b) for transducer no. 2 in the stiffener case. IDW: inverse distance weighting.

Figure 12 .

 12 Figure 12. Damage localization maps for the two successive impacts (real in blue circle) for (a, b) RMSD index and for (c, d) Df mean index.

Figure 13 .

 13 Figure 13. Three piezoelectric transducers bonded to a composite plate in clamped configuration.

Figure 14 .

 14 Figure 14. Example of EMI signatures (between 10 and 20 kHz) for the three piezoelectric transducers on the plate no. 1 for UD and damaged cases after an impact energy of 20 J (D = 20 J). UD: undamaged; EMI: electromechanical impedance.

Figure 15 .

 15 Figure 15. Damage localization maps (Df mean index) overlaid on Ultrasonic C-scan maps for plate nos 1-6. The DI results (red zone) are compared to NDT test (green zone). NDT: nondestructive testing; DI: damage indicator.

Table 1 .

 1 Impact data for the composite stiffener.

		Maximum force (kN)	Mass (kg)	Velocity (m/s)	Target energy (J)	Resulting energy (J)
	Impact no. 1	14.5	4	4.25	35	36.2
	Impact no. 2	16.8	4	4.3	35	36.9

Table 2 .

 2 RMSD and Df mean indices for the composite stiffener and for the two transducers (pzt1 and pzt2).

		RMSD pzt1 (%)	RMSD pzt2 (%)	Df meanpzt1 (%)	Df meanpzt2 (%)
	Damage no. 1	3.600	8.800	0.180	0.250
	Damage no. 2	11.100	11.200	1.210	0.190

RMSD: root mean square deviation.

Table 3 .

 3 Impact data for all tested composite plates.

		Maximum force (N)	Mass (kg)	Velocity (m/s)	Target energy (J)	Resulted energy (J)
	Plate no. 1	6378.47	2.368	4.100	20	19.90
	Plate no. 2	6067.33	2.368	4.095	20	19.85
	Plate no. 3	6596.27	2.368	4.118	20	20.08
	Plate no. 4	5989.54	2.368	4.095	20	19.85
	Plate no. 5	5787.3	2.368	4.010	20	19.00
	Plate no. 6	6036.9	2.368	3.996	20	18.90

Table 5 .

 5 Df mean indices for all tested plates and for every PZT.

	Df	Plate no. 1	Plate no. 2	Plate no. 3	Plate no. 4	Plate no. 5	Plate no. 6
	PZT1	0.34%	0.20%	0.09%	0.36%	0.69%	0.33%
	PZT2	0.38%	0.22%	0.12%	0.33%	0.65%	0.50%
	PZT3	0.34%	0.21%	0.12%	0.43%	0.82%	0.51%

PZT: lead zirconate titanate.

Table 4 .

 4 RMSD indices for all tested plates and for every PZT.

	RMSD	Plate no. 1	Plate no. 2	Plate no. 3	Plate no. 4	Plate no. 5	Plate no. 6
	PZT1	12.99%	8.46%	4.66%	10.15%	14.71%	8.33%
	PZT2	13.50%	7.34%	4.35%	11.20%	13.96%	13.51%
	PZT3	15.34%	3.93%	10.36%	10.32%	17.76%	12.28%

RMSD: root mean square deviation; PZT: lead zirconate titanate.

PZT: lead zirconate titanate.

1D: one-dimensional; IDW: inverse distance weighting; PZT: lead zirconate titanate.

Acknowledgements

The authors thank N. Boudjemaa, LATECOERE, for his insightful comments.

Funding

This study received financial support from the Re´gion Midi-Pyre´ne´es, the DIRRECTE as well as the FEDER through EPICEA 2009 program related to the project titled ''SAPES COMPOSITES.''

Declaration of conflicting interests

The authors declare that there is no conflict of interest.