Gwénaël Toullelan
email: gwenael.toullelan@cea.fr

Sylvain Chatillon

Raphaële Raillon

Steve Mahaut

Sébastien Lonné

Souad Bannouf

Results of the 2016 UT Modeling Benchmark Proposed by the French Atomic Energy Commission (CEA) Obtained with Models Implemented in CIVA software

For several years, the World Federation of NDE Centers, WFNDEC, proposes benchmark studies in which simulated results (in either ultrasonic, X-rays or eddy current NDT configurations) obtained with various models are compared to experiments. This year the proposed UT benchmark proposed by CEA concerns inspection configurations with multi-skips echoes i.e. the incident beam undergoes several skips on the surface and bottom of the specimen before interacting with the defect. This technique is commonly used to inspect thin specimen and/or in case of limited access inspection. This technique relies on the use of T45° mode in order to avoid mode conversion and to facilitate the interpretation of the echoes. The inspections were carried out with two probes of different aperture working at 5MHz.

INTRODUCTION

The T45° multi-skips inspections have been carried out on two steel planar specimens containing one vertical backwall breaking flaw. To evaluate the influence of the beam divergence on the detectability after several skips, inspections were done with two probes working at the same frequency (5MHz) but with two different apertures. The inspections were performed using immersion technique rather than a contact one to ensure a better reproducibility.

This communication presents the results obtained with the models implemented in the CIVA software. In CIVA, the field radiated by the probe is computed by applying the so-called pencil-model [START_REF] Gengembre | Pencil method in elastodynamics: application to ultrasonic field computation[END_REF], the elastodynamic (for acoustics see [START_REF] Lu | Models Comparison for the scattering of an acoustic wave on immersed targets[END_REF], [START_REF] Lü | Numerical comparison of acoustic wedge models, with application to ultrasonic telemetry[END_REF]) PTD (Physical Theory of Diffraction ) system model [START_REF] Zernov | A refinement of the Kirchhoff approximation to the scattered elastic fields[END_REF], [START_REF] Darmon | A system model for ultrasonic NDT based on the Physical Theory of Diffraction (PTD)[END_REF] is then applied to predict the response of notches and the SOV (Separation Of Variables) model [START_REF] Darmon | Modelling of scattering of ultrasounds by flaws for NDT[END_REF] is used for the SDH responses used as reference. The comparison between simulated and experimental results are presented and discussed.

This work is done in the context of experimental validation [START_REF] Toullelan | Results of the 2015 UT Modeling Benchmark Obtained with Models Implemented in Civa[END_REF] of the ultrasonic module of CIVA which results are available on the EXTENDE website [8].

DESCRIPTION OF THE 2016 ULTRASONIC BENCHMARK PROPOSED BY CEA

Experimental measurements were carried out at CEA LIST. Acquisitions were performed using two immersion probes working at 5MHz, tilted in order to generate transversal waves at 45°. Two setups were considered, one for each probe (Figure 1). For setup 1, a Ø12.7mm probe was used to inspect a 12mm thick planar block while for the setup 2, a Ø6.35mm probe was used to inspect a 5mm thick planar block. This second setup aims at assessing the multi-skips echoes after a maximum number of skips. For each setup the probe was moved in two perpendicular directions over the notch. During the displacement of the probe, the multi-skips echoes waveforms received at each position were stored (experimental C-scans) and the maximum amplitude of each echo was also recorded. The reproducibility of the results has been checked and the confidence interval of the experimental data presented in this paper has been evaluated at +/-2dB.

An example of experimental C-scan obtained on setup 1 including the first 4 multi-skips echoes and the B-scan extracted above the defect center (red dotted line on the C-scan) are presented in Figure 1.

The echoes of highest amplitude (numbered 1 to 5 on the figure) correspond to paths for which the difference between the number of skips of the beam on the backwall of the block during the forward path (probe-> notch) and the number of skips during the return path (notch-> probe) is 1. These paths, corresponding to specular echoes, are represented in Figure 1 from echo number 1 corresponding to the T corner echo (1 skip of the beam on the backwall for the forward path and no skip during the return path) to echo number 4 (4 skips of the beam for the forward path and 3 skips for the return path).

In this communication, the overall amplitude of these echoes numbered 1 to n are compared on the figure (and referred to as "multi-skips echoes" n°1, n°2, n°3 ...).

The amplitude reference AREF for all the amplitudes given in dB is the maximum amplitude of the direct echo obtained from a side drilled hole of 2mm diameter located at 6mm depth for the setup 1 and at 4mm depth for the setup 2. The measured relative amplitude of the echo from a notch is given in dB: 20 log (A / AREF) -(G -GREF), where G and GREF are the gains in dB used during the measurement carried out respectively on the current notch and on the reference reflector.

SIMULATION PROCEDURES Input Parameters

Mock-ups: Acquisitions were carried out on two planar blocks, one of 12mm thick, and another one of 5mm thick. Both are made of steel and contain one artificial vertical backwall breaking notch of 2mm height. The material homogeneity was experimentally checked and the specimen material was modelled as isotropic and homogeneous.

Probes: the probes used were single element conventional immersion probes with a center frequency of 5MHz and diameters of 12.7mm and 6.35mm. The water path was 100mm for the 12.7mm probe and 25mm for the 6.35mm probe. The incident angle was applied in order to generate transversal waves at 45° in each specimen.

CIVA input signal: the probe input signal was a 5MHz synthetic signal generated by CIVA which parameters (bandwidth and phase) were precisely adjusted to ensure a good matching of the experimental and simulated wave forms of a T direct echo obtained in pulse echo mode on the reference amplitude SDH.

Echoes from the side drilled holes were calculated with the SOV model and the PTD model was used for the calculation of the notch multi-skips echoes [START_REF] Lu | Models Comparison for the scattering of an acoustic wave on immersed targets[END_REF].

Material Parameters Measurement

The longitudinal (L) and transversal (T) velocities were experimentally estimated using successive backwall echoes.

Setup 1: VL = 5950m.s-1 and VT = 3290m.s-1. The density was 7.8g.cm-3. Setup 2: VL = 5900m.s-1 and VT = 3220m.s-1. The density was 7.8g.cm-3. Attenuation was measured in two ways depending on the setup as following: Setup 1: Multi-skips echoes on the SDH located at 6mm depth in the 12mm thick mock-up were measured in order to evaluate the attenuation on the specimen (Figure 2 (a)). Ray-paths corresponding to the measured amplitudes are also presented in the figure. Results are represented against the sound-path in the specimen with a reference amplitude which is the SDH direct echo amplitude (Figure 3).

Setup 2: A mock-up made of the same steel grade than for the 5mm planar block and including SDHs from 4mm to 60mm depth was used to evaluate the attenuation (Figure 2 (b)). In this setup, amplitudes are presented according to the SDH depth with, as reference in amplitude, the one of the 4mm depth SDH (Figure 4).

For the both setups, attenuation was adjusted so that the simulated curve representing the SDH amplitude according to the ultrasonic path fits the experimental one (Figures 3 and4).

RESULTS FOR THE NOTCH IN THE SETUP 1

Figure 5 shows the experimental B-scan including the first 4 multi-skips echoes of the 2mm high notch but the evaluation had been done until multi-skip echo n°7. The bottom of the figure presents the superposition result of the experimental and simulated echodynamic curves and normalized A-scans (corresponding to the maximum amplitude of multi-skip echoes n°1 (corner echo) and n°3). This comparison shows the good prediction of the multi-skips echoes shape by CIVA.

Figure 6 presents a comparison between experimental and simulated multi-skips echoes amplitudes according to the path in the specimen. The evaluations were done for multi-skips echoes from n°1 to n°7 which corresponds to a path from 34mm to 440mm in the specimen. For this setup, the reference for the amplitude is the T direct echo of a Ø2mm SDH located at 6mm depth in the same mock-up.

A very good agreement between experimental and simulated multi-skips echo amplitudes is obtained with a maximum discrepancy of 2dB.

RESULTS FOR THE NOTCH IN THE SETUP 2

The same comparisons as in Figures 5 and6 are presented in Figures 7 and8 for the setup 2. The reference for the amplitude is the T direct echo of a Ø2mm SDH located at 4mm depth.

Experimental results show a linear decreasing of the amplitudes which is well reproduced by CIVA. The discrepancies observed between simulated and experimental multi-skip echoes amplitudes from n°1 to n°5 are less than 1dB.

However, the superposition between the experimental and simulated echodynamic curves shows differences after the multi-echo n°2. Indeed, after the second skip of the beam on the backwall, CIVA predicts much larger echoes. It was not observed in the setup 1, where a less divergent probe was used. The reason of this discrepancy is not yet explained but can be assigned to contributions of creeping and head waves due to the divergence of the probe. Analysis of these differences for the setup 2 is in progress.

The experimental and simulated A-scans for multi-skips echo n°1 and n°3 are in good agreement.

CONCLUSION

This communication presented the results obtained with the CIVA software of the 2016 UT Modeling Benchmark. Comparisons between simulation and experiments of echoes from SDH and vertical backwall breaking notches after several skips of the beam on the surface and the bottom of two steel planar blocks were presented. Two setups were considered, with two probes of different apertures and operating at 5MHz. Setup 1 considered a Ø12.7mm probe inspecting a 12mm thick planar specimen. Setup 2 considered a Ø6.35mm probe inspecting a 5mm thick planar specimen.

A very good agreement is obtained between experimental and simulated multi-skips echoes amplitudes for both setups. Discrepancies are less than 2dB in amplitude for all echoes. The shapes of the A-scans corresponding to the maximum amplitude of multi-skip echoes are also well predicted by CIVA.

Regarding echodynamic curve shapes, a good agreement is obtained for setup 1 while discrepancies are noted between the shapes of the echoes after the multi-echo n°2 for the smallest probe aperture (setup 2). The origin of this disagreement can be assigned to supplementary contributions (head waves) occurring because of the important divergence of the probe that was used, but hypothesis has to be confirmed by complementary studies.

FIGURE 1 .

 1 FIGURE 1. Description of the T45° multi-skips echoes inspection setup

FIGURE 2 .

 2 FIGURE 2. Attenuation measurement configurations. (a) Multi-skips echoes on a SDH for the configuration 1; (b) Direct echoes on SDHs located at different depths for the configuration 2

FIGURE 3 .

 3 FIGURE 3. Attenuation measurement for the setup 1. Experimental B-scans and superposition of measured and simulated multiskips echo amplitudes on the SDH located at 6mm depth

FIGURE 4 .

 4 FIGURE 4. Attenuation measurement for the setup 2. Experimental B-scans and superposition of measured and simulated direct echo amplitudes on SDHs against the SDH depth

FIGURE 5 .FIGURE 6 .

 56 FIGURE 5. Experimental B-scan (top); superposition of experimental and simulated echodynamic curves of the first four multiskips echoes for the setup 1; example of experimental and predicted A-scans

FIGURE 7 .FIGURE 8 .

 78 FIGURE 7. Experimental B-scan (top); superposition of experimental and simulated echodynamic curves of the first five multiskips echoes for the setup 2; example of experimental and predicted A-scans


𝑎 Sandra Le Coz
email: lecozsandra@yahoo.com

Aurélien Benoit-Lévy
email: aurelien.benoit-levy@cea.fr

Olivier A Martineau-Huynh

𝑏 𝑎

A neural network to classify GRAND radio time traces

Introduction

The sources of UHE cosmic rays are still a mystery. But UHE neutrinos are produced in the same sources than cosmic rays, with the advantage of being undeflected by magnetic fields. Their detection is then relevant to point to these UHE sources. The GRAND project [START_REF] Álvarez-Muñiz | The Giant Radio Array for Neutrino Detection (GRAND): Science and Design[END_REF] aims at the detection of the earth-skimming UHE tau neutrinos that interact in the rock via charge current, generating taus which then decay into the air, leading to hadronic air showers. To get a chance to detect neutrino events despite their low cross-section of interaction with matter, a huge area of detection is necessary. Antennas are a good option because they are cheap, and 1 km -2 antenna density is enough thanks to the large foot print of the very inclined air showers induced by neutrinos[]. Furthermore, for cost efficiency, antennas must be self-triggered, and thus be able to reject background signals to avoid the saturation of the acquisition. Yet, the trigger rate of background signals that were recorded by TREND [START_REF] Charrier | Autonomous radio detection of air showers with the TREND50 antenna array[END_REF], the seed-project for GRAND, was in average always above 10Hz, despite its remote location. For the sake of the future GRAND array, we have decided to use the TREND experimental data to study the potential of neural networks to reject background signals. Neural networks are indeed theoretically able to find the best combination of relevant features to perform the discrimination. TREND data are 5.120 𝜇s length time traces, digitized at 200MS/s, such as the ones in figure 1. A neural network would then be fed with 1024 features per input trace. The target this neural network would have to reach is 1 if the trace corresponds to an "air shower", and 0 otherwise. The output of the neural network would then be between 0 and 1, indicating the probability of the trace to have an air shower origin. In the next section, we will present the TREND experiment in more details, explain how we have selected traces to train and test our neural network, and how we were able to put « 0 » and « 1 » labels to these traces.

Training and test datasets

TREND was a self-triggered antennas array, consisting in 50 single-polarized, 50-100MHz, butterfly antennas, setup in XinJiang in China, and running between 2011 and 2014. TREND was triggered at the antenna level when the maximum absolute amplitude of a radio transient was at least 6 standard deviations beyond the baseline. TREND was then triggered at the array level if there was 5 or more antenna-level triggers in space-time coincidence. For the neural network trigger study, we used a set of 1.25 × 10 8 experimental coincidences, consisting of 9 × 10 8 radio time traces, corresponding to 80 days of acquisition. The main reason for which we decided to use experimental data rather than simulations is that neural networks might be very sensitive to differences between simulations and reality, that are sometimes hard to apprehend. Therefore, a neural network might work on simulations, but completely fail when put it into production. Furthermore, a lot of data from TREND were already available, for which the only critical point was to add labels. Actually, to add these "air shower" or "background" labels on the recorded time traces, we used the offline cuts that were defined in the final analysis of TREND, which was already published [START_REF] Charrier | Autonomous radio detection of air showers with the TREND50 antenna array[END_REF]. In this analysis, cuts were applied to reject : coincidences in time clusters, time traces with too long antenna signals, coincidences with a high 𝜒² of reconstruction, for which the source position is below 3 km, zenith angles above 80°, discontinuous trigger pattern at ground, and direction-time coincidences clusters. These cuts allowed to select 200 coincidences, consisting of 1495 radio time traces. Then, the aperture of TREND was computed from simulations of air showers and of the TREND detector response. About 370 recorded air showers were expected, out of which 160 would have been selected by the cuts. Figure 2 shows the distribution in zenith and azimuth of the selected experimental (black) and simulated (red) coincidences. The consistency between the two distributions indicates that the 200 experimental coincidences are mostly air showers, with a ∼ 20% contamination by background. Because contamination is higher for zenith angles above 70°a nd azimuth angles between 160°and 240°, we decided to reject these directions of the labelisation process. The 984 radio time traces consisting the remaining 139 coincidences were then labeled "air shower" while we labeled "background" the other 9 × 10 8 traces. Yet, using such an unbalanced dataset would have led the neural network to just ignore the "air shower" instances and to ouput « 0 » whatever the input. To obtain a class-balanced dataset while taking advantages of the almost unlimited pool of "background", we decided to put a weight of 2 for the 984 "air shower" instances and to randomly pick 984 × 2 = 1968 "background" instances. This dataset was split into a training set and a test set, with 80% and 20% fractions respectively.

Model

Convolutional Neural Network

A convolutional architecture has been chosen because it is far more efficient then a regular fully connected neural network for images recognition and related tasks. For each layer, there are ranges of neurons that share the same weights, because they are receptive to the same convolution pattern, but localized in different overlapping areas, so that they are not connected to all the inputs. The outputs of the neurons that share the same weights constitute a feature map. The neurons of the next layer are then receptive to patterns on the former-layer set of feature maps, also sharing weights, and being connected to limited scale inputs.

Training

Once the architecture is chosen, a neural network has to be trained to be able to make predictions. Training is the process of updating weights to minimize the loss function, which measures the discrepancy between the neural network output (the predictions) and the target to reach (the labels). The minimization method is the gradient descent: iteratively, the partial derivative of the loss function with respect to each weight is computed, and each weight is then updated in order to make the loss decrease (if the partial derivative is positive, weight is reduced, if it is negative, weight is increased). We chose "adam" for the gradient descent algorithm and a "cross entropy" loss function. For 2-classes classifications, cross entropy is just equivalent to -(1/𝑛) ln(1 -𝑒𝑟𝑟𝑜𝑟), where error is the absolute difference between the neural network output and the target. As the error tends towards 1, cross entropy therefore tends towards infinity, making it more relevant for classification tasks, compared to a "mean absolute error" or a "mean squared error" function.

Hyperparameters tuning and results

We used the Keras/TensorFlow libraries. We tried different first layer inputs: time traces, Fourier transforms, and combinations of both. Best inputs were FFT real and imaginary parts, put in a succession alongside frequency. Also, many hyperparameters combinations were tested: convolutional kernel sizes within [3,5,7,9,11,13,21,31,41,51...81], and number of filters per convolutional layer within [4,8,16,32]. We finally got the best performances with a neural network made of 3 conv-layers, with kernel size of 51, 8 filters for the first conv-layer and 16 filters for the second and third ones. Each of the conv-layer is followed by a maxpooling of size 2. A dense layer of 64 neurons bounds the last conv-layer to the final output, which is compelled within [0,1]. Dropout (60%) and L2 regularizations (2e-3) were adjusted to keep the lowest loss while avoiding overfitting over 80 training epochs (overfitting is when the loss on the training set is lower than the loss on the test set, and one epoch is when all the training instances has been used once). We chose the ReLU activation function because it is generally a good default, and padding was set to "same" not to loose information. We finally got a 83% accuracy for the training and test sets, the accuracy being defined as the number of well classified data divided by the total number of data in a set. Output of the neural network (prediction probability to be a shower) for all the traces of the test set, that are sorted by label on the x-axis ("background" first, "air shower" then). Right: Ratio of "air shower" traces (orange) and "background" traces (blue) that are predicted as "air shower", as a function of the prediction decision threshold. The test set is composed of 1968 × 0.2 distinct background traces and 984 × 0.2 distinct air shower traces (with a weight of 2 for the latter). Accuracy (dotted red) is 83% for a default decision threshold of 0.5.

The accuracy is here computed with the default value of 0.5 for the decision threshold (threshold relative to the neural network output to decide a trace is predicted to belong to class « 0 » or « 1 »). These results are shown on figures 3 and 4.

Inferences

Our aim was to simulate the discrimination performance of our neural network in production mode. To do that, we used our neural network to classify the whole 9 × 10 8 radio time traces of the 1.25 × 10 8 recorded coincidences. When the neural network output was above the decision threshold, we got a "Neural Network trigger" at the antenna level. If we had 5 or more antenna-level triggers in space-time coincidence, we got a "Neural Network trigger" at the array level, and the coincidence was considered as recorded by the neural network. The number of coincidences that may be recorded that way depends on how the traces of each coincidence are related to the others. If each individual trace in a coincidence has an identical probability 𝑝 to be classified as "air shower", then the probability of this air shower coincidence to be recorded is equal to 𝑝. If, on the contrary, the probabilities to be classified as "air shower" are completely independent for all the traces in a coincidence, the probability of an air shower coincidence to be recorded is ∼ 1binom.cdf(successes = 5, trials = 8, 𝑝) (because 8 is the average number of traces per recorded coincidence and we ask for 5 traces at least). Same consideration applies for background coincidences. Therefore, with a decision threshold set to 0.6 (where the accuracy is the highest), we expect the neural network trigger to preserve between 0.07% (87 500) and 18% (22.5 × 10 6 ) of the 1.25 × 10 8 coincidences that were recorded with the standard TREND trigger. As we have shown earlier, according to simulations, ∼ 370 of these 1.25 × 10 8 coincidences might be air showers. The neural network trigger is expected to preserve between 86% (318) and 91% (337) of the 370. It finally turned out that our "Neural Network trigger" recorded 3 × 10 6 coincidences (2.4%), meaning that traces in each background coincidences are correlated to some extent, from our neural network point of view. The intersection between that 3 × 10 6 and the 200 selected with the standard TREND analysis consist of 159 coincidences. One can note that this intersection was expected to contain at least between 120 and 126 coincidences, because 86% of the traces consisting 139 coincidences amongst the 200 have already been predicted as "air shower" by the neural network while training and test stages. For these 159 coincidences, the number of recorded traces per coincidence (multiplicity) has decreased by 7% because of the neural network trigger. That loss of multiplicity involves an average direction misreconstruction under 3% in zenith and under 1% in azimuth (see figure 5).

To complete the analysis of our neural network trigger performances, we applied a set of simplified offline cuts (rejection of time clusters, high 𝜒², near source position, zenith>80°, direction-time clusters) to the 3 × 10 6 coincidences that have triggered with the neural network, resulting in a selection of 225 coincidences. The angular distribution of these 225 coincidences is shown on figure 6. It exhibits a North-South pattern in azimuth, as expected for air showers at first order. For comparison, the figure also shows the distribution of the 200 coincidences that where selected by the TREND standard analysis, and the intersection between the two selected sets (108 coincidences). A summary of all the sets is shown on figure 7. As a conclusion to this section, our neural network has been able to improve trigger purity and to replace some offline cuts.

Conclusion

We studied the potential of deep learning techniques to reject significant parts of the massive amount of background signals that trigger antennas. For this study, we used a set of experimental data, consisting of 1.25 × 10 8 coincidences (9 × 10 8 radio time traces) that were recorded in 2012 by TREND, a self-triggered array of 50 single-polarized antennas, running in a remote valley of China. The standard analysis of this data had led to a selection of 200 coincidences. Simulations had shown that these 200 coincidences are mostly air showers, with a contamination estimated around 20%. For the purpose of this study, an angular selection was performed to increase the purity of this selection, leading to 139 coincidences, consisting of 984 radio times traces. These 984 radio time traces and a random selection amongst the much larger amount of background signals were used to feed a convolutional neural network. This neural network allowed to reject 82% of the background radio times traces while preserving 86% of the air shower radio time traces. At the array level (when combining traces in space-time correlations), the neural network allowed to reject 98% of the recorded coincidences, while preserving between 86 and 91% of the air shower coincidences. Because the training dataset was quite small, biased and not totally pure, the potential of the deep learning techniques was not fully probed. Furthermore, the antennas of the soon-tocome GRANDproto300 array [START_REF] Decoene | The GRANDProto300 experiment: a pathfinder with rich astroparticle and radioastronomy science case[END_REF] will for sure carry more information thanks to their 3 arms. There is thus plenty of room for improvements in the near future.

Figure 1 :

 1 Figure 1: TREND traces, recorded at 5ns sampling. Left was labeled "background", right was labeled "air shower".

Figure 2 :

 2 Figure 2: Zenith (left) and azimuth (right) distributions for the 200 selected experimental coincidences (black, plain line), and for the 160 air showers that would have been selected in that time according to simulations (red, dotted line). East is at 𝜙 = 0 • , North is at 𝜙 = 90 • .

Figure 3 :

 3 Figure 3: Loss (left) and accuracy (right) as a function of the number of training epochs (see text for definitions), for a training set composed of 1968 × 0.8 distinct background traces and 984 × 0.8 distinct air shower traces (with a weight of 2 for the latter).

Figure 4 :

 4 Figure4: Left: Output of the neural network (prediction probability to be a shower) for all the traces of the test set, that are sorted by label on the x-axis ("background" first, "air shower" then). Right: Ratio of "air shower" traces (orange) and "background" traces (blue) that are predicted as "air shower", as a function of the prediction decision threshold. The test set is composed of 1968 × 0.2 distinct background traces and 984 × 0.2 distinct air shower traces (with a weight of 2 for the latter). Accuracy (dotted red) is 83% for a default decision threshold of 0.5.

Figure 5 :

 5 Figure 5: Zenith (left) and azimuth (right) angles for the 159 coincidences that were both selected by the TREND standard analysis and recorded by the Neural Network trigger. Angular reconstruction differs because of a loss of multiplicity due to the Neural Network veto on traces.

Figure 6 :

 6 Figure 6: Zenith (left) and azimuth (right) distributions for selected coincidences (degrees). In blue with the Neural Network trigger and a set of offline cuts (225), in orange with the TREND standard trigger and selection (200). The intersection of the two selections is in green (108).

Figure 7 :

 7 Figure 7: Diagram of coincidences that have been recorded (grey circles) and that have been selected (sand circles) with the TREND standard analysis (darker) and with our Neural Network trigger (lighter).


domain: spi nlin

text:
Antoine Faugaret

Laurent Martin Witkowski

Yohann Duguet

Yann Fraigneau

Yann Fraigneau Boundary

Boundary condition influence on instabilities in a low Reynolds free surface rotating flow

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Boundary condition influence on instabilities in a low

Reynolds free surface rotating flow

A. Faugaret * ‡ , L. Martin Witkowski ‡ † , Y. Duguet ‡ and Y. Fraigneau ‡
The study of a rotating flow in one its most simple configuration (fixed cylindrical cavity, rotating bottom disk, free top surface) reveals a much more complicated physic than expected. While we tried to validate our experimental setup with both publication 1 and numerical result, we were able to obtain the right unstable modes, but at much lower Reynolds. After several investigation on possibles issue in our experience, we finally suspected free condition surface to deeply influence both base flow and instabilities.

We first used a condition introduced in 2 , that we call "Frozen condition", and that consists in setting the surface radial velocity to 0. But if this condition lowered the critical Reynolds, Stability analysis was no more able to predict the right mode. So we built a new top boundary condition as an α weighted sum of free and frozen conditions.

A search for critical Reynolds depending on α reveals a second unstable branch, with a lower critical Reynolds, and better visualization agreement 1(c) with instabilities that are experimentally observed 1(b).

Figure 1 :

 1 Figure 1: (a) Axial vorticity computed at Re = 95, with free surface condition. (b) Experimental visualization at Re ∼ 30. (c) Axial vorticity computed at Re = 53, with frozen surface condition


X Saloppé

& Pham

Virtual Reality (VR) in Forensic Psychology

Interests and issues in Research and Practice

VR in mental health

 propose a typology based on immersion, one of three central characteristics of VR with fidelity and patient depiction. The three levels of immersion: low, moderate and high, depend on the presence of devices in real world. Thereby, the use of joystick, mouse or even communication of instructions from the experimenter is categorized as low immersion in contrast to head-mounted display, considered as high immersion.

The use of this technology in mental health has increased for the last decades (Bogossian et al.,

VR in forensic research and practice

The use of VR in forensic research is sparse, despite its utility for diagnosis, risk assessment and therapy [START_REF] Fromberger | Virtual reality applications for diagnosis, risk assessment and therapy of child abusers[END_REF].

Psychiatric diagnoses are still largely based on clinical interviews or on self-questionnaires [START_REF] Van Bennekom | Virtual Reality Objectifies the Diagnosis of Psychiatric Disorders: A Literature Review[END_REF]. Semi-structured interviews allow the in-depth investigation of diagnoses through a clinical relationship. However, they present the disadvantages of requiring a prior training [START_REF] Lewis-Beck | The SAGE Encyclopedia of Social Science Research Methods[END_REF], and being sensitive to memory bias or to clinician's perception concerning the nature and severity of diagnosis [START_REF] Van Bennekom | Virtual Reality Objectifies the Diagnosis of Psychiatric Disorders: A Literature Review[END_REF]. Self-questionnaires present advantages: they are quick and convenient to administer, even in an expertise framework, and require less interpretation, few training, limited costs [START_REF] Lewis-Beck | The SAGE Encyclopedia of Social Science Research Methods[END_REF]. However, they should only be used when research objectives are simple and limited [START_REF] Lewis-Beck | The SAGE Encyclopedia of Social Science Research Methods[END_REF]. Moreover, clinical interviews and self-questionnaires are also limited by potential interviewees manipulation, social desirability and low ecological validity [START_REF] Van Bennekom | Virtual Reality Objectifies the Diagnosis of Psychiatric Disorders: A Literature Review[END_REF]. In order to overcome these limitations, new methodologies are needed.

VR brings both ethical and practical benefits to risk assessment. On the ethical side, a major pitfall of inaccurate risk assessment stems from the superficial environments (prison, forensic hospital) in which the offender is evaluated [START_REF] Fromberger | Virtual reality applications for diagnosis, risk assessment and therapy of child abusers[END_REF]. It would be unethical to confront a child sexual offender with a victim for the purpose of assessing its self-regulation abilities following a therapeutic program. But the use of VR allows us to assess abilities in close-to-real 3D environments without resorting to ethically questionable or even dangerous processes. On the practical side, VR also present the benefit of the variety of scenarii [START_REF] Meyer | Verhaltensbeobachtung von Kindesmissbrauchstätern in virtuellen Risikosituationen-eine Machbarkeitsstudie zu VR-gestützter Gefährlichkeitsprognostik[END_REF]). In combination with eye-tracking device, VR assesses realistic situations (Figure 1) without exposition to uncontrolled environment (Trottier et al., 2014). VR validates and specifies the assessment of paraphilic interests (Renaud et al., 2014), empathy (Figure 2; [START_REF] Hamilton-Giachritsis | Reducing risk and improving maternal perspective-taking and empathy using virtual embodiment[END_REF], or emotions decoding (Figure 3; Cigna et al., 2015;[START_REF] Seinfeld | Offenders become the victim in virtual reality: impact of changing perspective in domestic violence[END_REF].

The controlled virtual environment also allows to investigate the criminogenic needs (e.g.: (sexual) self-regulation, offense supportive cognitions, interpersonal functioning) highlighted as main therapeutic focus [START_REF] Ward | Risk management or goods promotion: The relationship between approach and avoidance goals in treatment for sex offenders[END_REF]. Furthermore, VR (Figure 4) better transposes skills developed in therapy to day-to-day life [START_REF] Joyal | Utilisation de la Réalité Virtuelle en neuropsychologie expérimentale et Clinique[END_REF]. Additionally, it has been showed that patients improve their test scoring (e.g.: WSCT) after cognitive remediation sessions but fail to transpose it (e.g.: cognitive flexibility) to real situations. In comparison to classical therapy, the VR-based therapy has shown better (g =.73-.88) or equal (g = -.007) results for treatment of anxiety, depression, phobia or even schizophrenia [START_REF] Carl | Virtual reality exposure therapy for anxiety and related disorders: A meta-analysis of randomized controlled trials[END_REF]Fodor et al., 2018;[START_REF] Rus-Calafell | A virtual reality-integrated program for improving social skills in patients with schizophrenia: A pilot study[END_REF][START_REF] Wechsler | Inferiority or Even Superiority of Virtual Reality Exposure Therapy in Phobias?-A Systematic Review and Quantitative Meta-Analysis on Randomized Controlled Trials Specifically Comparing the Efficacy of Virtual Reality Exposure to Gold Standard in vivo Exposure in Agoraphobia, Specific Phobia, and Social Phobia[END_REF]. However, meta-analysis or systematic review concerning the clinical efficiency of VR is still missing. (Trottier et al., 2014) Figure 2 -VR use for empathy enhancing (Hamilton-Giachistis et al., 2018) Figure 3 -Emotion recognition 3D stimuli (Cigna et al., 2015) Figure 4 -Virtual shopping environment (Parsons, 2012) Contact : luca.tiberi@umons.ac.be View publication stats

  Lack of standardized research frameworkUse of Augmented Reality (AR) as alternative

Figure 1 -

 1 Figure1-Combined used of VR and eye-tracking(Trottier et al., 2014)


L Ju

Fradkin ∨

A Kamta Djakou

C Prior

M Darmon

S Chatillon
email: sylvain.chatillon@cea.fr

P Calmon

The Alternative Kirchhoff Approximation in elastodynamics, with applications in ultrasonic non-destructive testing

Keywords: 2000 Mathematics subject classification: 74J05, 74J10, 74J15, 74J20, 74H10, 74H15, 74L99 Non-Destructive Testing, ultrasound, the high-frequency asymptotics, the Lamb's Green's tensor, the critical Gaussian beam

The Kirchhoff Approximation is widely used to describe scatter of elastodynamic waves. It simulates the scattered field as the convolution of the free-space Green's tensor with the geometrical elastodynamics approximation to the total field on the scatterer surface and therefore cannot be used to describe non-geometrical phenomena, such as head waves. The aim of this paper is to demonstrate that an alternative approximation, the convolution of the far-field asymptotics of the Lamb's Green's tensor with incident surface tractions has no such limitation. This is done by simulating scatter of a critical Gaussian beam of transverse motions from an infinite plane. The results are of interest in ultrasonic non-destructive testing.

Introduction

Non-Destructive Testing (ndt) is commonly used to assess size, location and orientation of flaws in industrial structures. The ultrasonic inspections offer the least expensive alternative. In nuclear industry, it is a legal requirement that ndt inspectors carry out inspection qualification to demonstrate that safety critical defects can be detected. As part of inspection qualification they routinely perform computer simulation of relevant configurations. If necessary this can be done with commercial finite-element packages, but when the speed of simulation is of essence, whenever possible, inspectors use packages such as CIVA [START_REF] Huet | Modeling of corner echo ultrasonic inspection with bulk and creeping waves, Ultrasonic Wave Propagation in Non Homogeneous Media[END_REF], which utilise approximate tools for simulating radiation, propagation and scattering of ultrasonic pulses in solids. One such tool is the elastodynamic Kirchhoff Approximation (ka) [START_REF] Darmon | Modelling of scattering of ultrasounds by flaws for NDT, Ultrasonic Wave Propagation in Non Homogeneous Media[END_REF], [START_REF] Schmerr | Fundamentals of ultrasonic non-destructive evaluation: A modeling approach[END_REF] that relies on the well-known free-space Green's tensor for a homogeneous and isotropic solid as well as the classical assumptions of the geometrical elastodynamics (GE): these are satisfied by large scatterers with locally straight edges, when both the incident wave front and scatterer surface are locally plane. The industrial ultrasonic probes operate at relatively high frequencies, so these assumptions usually hold.

The ka cannot be used to simulate non-geometric phenomena, such as head waves, which arise when the wave fronts or scatterers are curved or contain irregularities. One circumstance, in which such waves make a significant contribution arises when the probing beam of T (transverse) motions is incident on the plane surface at the critical angle. Then the reflected T beam and head wave both propagate in the same direction and may experience destructive interference. The accompanying effects are beam shifting (the Goos-Hänchen effect) [START_REF] Pott | Scattering of an acoustic Gaussian beam from a fluidsolid interface[END_REF] and a GoodierBishop type wave [START_REF] Zernov | Wedge diffraction of a critically incident Gaussian beam[END_REF].

In order to overcome this drawback we propose to simulate non-geometric aspects of scatter using an Alternative Kirchhoff Approximation (aka) that relies on (the highfrequency asymptotics) of the Lamb's Green's tensor instead of the free-space Green's tensor. The rationale behind the proposal is simple: the free-space Green's tensor does not contain any information on the boundary of the solid, while the Lamb's Green tensor describes the field radiated by a point source situated on such boundary (provided its is planar) and using its far-field asymptotics can lead a computer code that is still relatively fast. In this paper we demonstrate capabilities of the aka by simulating scatter of a critical Gaussian T beam by a planar scatterer. To be more precise, the incident beam is chosen to be generated by a CPS (Complex Point Source), which in the paraxial approximation behaves as a Gaussian beam. Such beams serve as a good model of beams generated by industrial ultrasonic transducers, are easy to treat analytically and have intensity dropping at the beam boundary so fast that the geometrical elastodynamics does not apply.

The article is organised as follows: the problem statement is offered in §2, followed by §3, where the incident CPS beam is described. The exact integral representation of the Lamb's Green's tensor as well as its high frequency asymptotics are introduced in §4 and appendices. The validation of aka is presented in §5.

Problem statement

We consider the displacement fields U exp[-i(ωt)kx] created in an homogeneous and isotropic solid half-space by a time-harmonic load, where i is the imaginary unit; x-point in space; t-time, ω-circular frequency; k-wave vector with amplitude k = ω/c; and c-wave speed. For simplicity of presentation, below the exponential factor exp(-iωt) is omitted but implied everywhere. We use bold symbols to denote vectors, and when in regular font, the same symbols denote the corresponding amplitudes.

Let the scatterer be the planar boundary of this half-space. Let us introduce the scatterer Cartesian coordinate system {e 1 , e 2 , e 3 }, with e 3 -the inner normal to this boundary. The corresponding Cartesian coordinates of any vector x are (x 1 , x 2 , x 3 ) and associated spherical polar coordinates are (s, φ, θ) (see Figure 1). Everywhere below the subscripts k, l, m, n = 1, 2, 3 refer to the the respective coordinates in the appopriate Cartesian system. Other subscripts and superscripts are used to indicate the field or scalar type. In particular, descriptors in, sc and tot relate to the incident, scattered and total fields, respectively.

Inside the half-space, let the incident and total (and hence scattered) displacement vector fields u(x) = (u m (x)) and the associated stress tensor fields σ(x) = (σ m (x)) satisfy the reduced equations of motion of linear elasticity (see e.g. [8, §. 5.1]),

σ m, + ρω 2 u m = 0, (1)
as well as Hooke's law (see e.g. [8, Eq. (A.5.13)]),

σ m = ρc 2 T (u ,m + u m, ) + ρc 2 L (1 -2γ 2 )δ m u n,n ], (2)
with δ m -the Kronecker delta; subscript , k-partial derivative with respect to x k ; ρmaterial density; c L , c T -speeds of longitudinal and transverse motions, respectively; γ = c T /c L < 1; and the summation convention implied. Also let the boundary be subjected to an integrable incident traction (see e.g. [8, Eq. (5.1.39)])

t in = σ in • e 3 , (3)
and let the stress-free boundary condition be (see e.g. [8, Eq. (6.1.13)])

σ sc 3 (x)| x 3 =0 + = -t in (x)| x 3 =0 + .
On top of that, let us assume the usual radiation condition (see e.g. [8, §1.1.7]) that when k T is complexified with a small complex part all outgoing waves decay at infinity. The above boundary value problem can be formulated via the Green's Theorem (see e.g. [13, Eq. (5.57)]) to represent the scattered field u sc (x) as convolution of the free-space Green's stress tensor of the third rank σ G (x) and (an unknown) surface displacement u tot (x ) = u in (x ) + u sc (x ),

u sc (x) = - ∞ -∞ ∞ -∞ σ G 3k (x -x ) u tot k (x ) dx 1 dx 2 , (4)
with x = (x 1 , x 2 , 0)-an arbitrary surface point; and stress tensor σ G associated with the second rank free-space Green's tensor G k -the displacement due to δ k δ(xx ), the delta function type point source at x . The standard ka utilises this relationship and assumption that on the shadow side of the boundary, u tot (x ) is zero and on the irradiated side, it is the sum of incident and reflected field (see e.g. [13, §6.6]).

Let us now introduce the Lamb's Green's tensor of the second rank G Lamb (x) and stress tensor of the third rank σ Lamb (x), solutions of Equation (1) supplied with the same radiation condition as above and boundary condition

σ Lamb 3k (x)| x 3 =0 + = -δ k δ(x)| x 3 =0 + . (5)
It is easy to check by substitution that convolution

u sc (x) = ∞ -∞ ∞ -∞ G Lamb k (x -x ) t in k (x ) dx 1 dx 2 (6)
is an alternative representation to (4). In physical terms, (6) represents the Huygens principle: the displacement is the superposition of the displacements generated by the elementary point loads on the scatterer surface. Below we derive expressions for G Lamb and construct the aka by substituting into (6) the far-field asymptotics of this tensor.

Incident tractions

Let us assume that the incident field in ( 6) is due to a CPS.

G k = 1 ρc 2 T k -2 T (I T -I L ) , k + I T δ k , I α = e ik α s in 4πs in , (7)
with

s in = (x 1 -a 1 ) 2 + (x 2 -a 2 ) 2 + (x 3 -a 3 ) 2 ; and α = L or T .
Let us complexify the first coordinate of a. This operation turns the spherical field into a beam, which propagates along the axis singled out by the complexification process (see e.g. [START_REF] Deschamps | Gaussian Beam as a Bundle of Complex Rays[END_REF]). Introducing the beam Cartesian coordinate systems {ē 1 , ē2 , ē3 } shown in Figure 2, this beam axis lies in the (ē 1 , ē3 ) plane; the beam coordinates of the source a are (-ā 1 , 0, 0); and the beam coordinates of its complexified version A are (-ā 1 + ia R , 0, 0). Figure 2 introduces ϑ in , the supplementary angle between e 1 and beam axis, and below we sometimes make use of its counterpart θin = -ϑ in . For a point load fδ(x -A) the resulting displacement is

U CPS = U 0 G(x, A)f. (8)
Introducing the distance along the beam axis 2 , the complexified distance between an arbitrary point x and the complexified source A can be approximated in the following manner:

∆ x1 = x1 + ā1 , in the paraxial approximation x2 2 + x2 3 (∆ x1 )
(∆ x1 -ia R ) 2 + x2 2 + x2 3 ≈ ∆ x1 -ia R + 1 2 x2 2 + x2 3 ∆ x1 -ia R .
Therefore on the surface of the scatterer, inside the beam, ∆ x1 can be treated as constant and we can write Īα ≈ -i 4π

w 0 w( x1 ) e k α       a R - x2 2 + x2 3 2w 2       e ik α       ∆ x1 + x2 2 + x2 3 2       -iζ , (9)
demonstrating that in the paraxial approximation the CPS field behaves as a threedimensional Gaussian beam [START_REF] Pott | Scattering of an acoustic Gaussian beam from a fluidsolid interface[END_REF], with 2w 2 (∆ x1 )/k α -beam spot (the radius of the beam at distance ∆ x1 from the waist); 2w 2 (0)/k α -beam spot at the waist; a R -the Rayleigh length (the distance along the propagation direction of the beam from its waist to where the beam spot increases by

= ∆ x1        1 + a R ∆ x1 2        , w 2 = a R        1 + ∆ x1 a R 2        , ζ = 1 2 tan -1 ∆ x1 a R . Above, w = w(∆ x1 ). It is reasonable to select the constant U 0 = exp(-k α a R ).

Surface tractions due to Complex Point Source

Let us now find the incident surface tractions t in (x 1 , x 2 , 0) generated by a CPS. Applying (2) to the complexified version of ( 7), the beam coordinates of the corresponding stress tensor become

σG mk = ρc 2 T [ Ḡ k,m + Ḡmk, + γ -2 (1 -2γ 2 ) Ḡnk,n δ m ]. ( 10
)
Using beam coordinates ( x1 , x2 , x3 ) and their complexified version

X1 = ∆ x1 -ia R , X2 = x2 , X3 = x3 ,
the derivatives of I T -terms and I L -terms in [START_REF] Huet | Modeling of corner echo ultrasonic inspection with bulk and creeping waves, Ultrasonic Wave Propagation in Non Homogeneous Media[END_REF] become

Īα, = I α S 2 in (ik α S in -1) X , Īα, m = I α S 4 in 3 -3ik α S in -k 2 α S 2 in ) X Xm + S 2 in (ik α S in -1)δ m , Īα, mk = I α S 6 in (-15 + 15ik α S in + 6k 2 α S 2 in -ik 3 α S 3 in ) X Xm Xk + S 2 in (3 -3ik α S in -k 2 α S 2 in )( X δ mk + Xm δ k + Xk δ m ) .
Using (3), ( 8) and the beam coordinates of both the normal n = e 3 and point load f, at the surface point x , the components of the incident surface traction are tin = σG mk nm fk .

The Lamb's Green's tensor

The components of the Lamb's Green tensor have no analytical expressions but their Fourier transforms do. These are derived below.

4.1. The Lamb's Green's tensor in the Fourier space We introduce the relevant Fourier Transforms first. In an isotropic and homogeneous space, there is no difference between directions, but the plane boundary singles out the direction of its normal. For this reason, we utilise the double spatial transform defined for any integrable function

f (x 1 , x 2 ) as f (ξ 1 , ξ 2 ) = ∞ -∞ ∞ -∞ f (x 1 , x 2 ) e -ik T (ξ 1 x 1 +ξ 2 x 2 ) dx 1 dx 2 . ( 11
)
Its inverse is the usual spectrum of the outgoing waves

f (x 1 , x 2 ) = k 2 T 4π 2 ∞ -∞ ∞ -∞ f (ξ 1 , ξ 2 ) e ik T (ξ 1 x 1 +ξ 2 x 2 ) dξ 1 dξ 2 . ( 12
)
Above, ξ = (ξ 1 , ξ 2 , ξ 3 ) are dimensionless wave vectors; introducing ξ ⊥ = ξ 2 1 + ξ 2 2 and γ α = c T /c α , their magnitudes are ξ α = γ α ; and their third components are given by

ξ α 3 = γ 2 α -ξ 2 ⊥ , (13)
with the root chosen to be principal, so that the outgoing waves u exp i(-ωt + k T ξ • x) decay at infinity and thus satisfy the radiation condition. Applying [START_REF] Miller | The Field and Radiation Impedance of Mechanical Radiators on the Free Surface of a Semi-Infinite Isotropic Solid[END_REF] to (1) the double Fourier Transforms of each column k of the Lamb's Green's tensor can be represented as a sum of T V (transverse vertical), T H (transverse horizontal) and L (longitudinal) components,

G Lamb k (ξ 1 , ξ 2 , x 3 ) = U T V k e ik T ξ T 3 x 3 + U T H k e ik T ξ T 3 x 3 + U L k e ik T ξ L 3 x 3 . (14)
The amplitudes

U α = U α (ξ 1 , ξ 2 ) = U α (ξ 1 , ξ 2 ) d α (ξ 1 , ξ 2
) can be found by applying [START_REF] Miller | The Field and Radiation Impedance of Mechanical Radiators on the Free Surface of a Semi-Infinite Isotropic Solid[END_REF] to [START_REF] Deschamps | Gaussian Beam as a Bundle of Complex Rays[END_REF], see Appendix A.

4.2. The Lamb's Green's tensor in the physical space Using Appendix A and [START_REF] Pott | Scattering of an acoustic Gaussian beam from a fluidsolid interface[END_REF], in the physical space, the columns of Lamb's Green's tensor can be represented as double integrals

G Lamb k (x -x ) = k 2 T 4π 2 U T k e ik T ξ T •(x-x ) + U L k e ik T ξ L •(x-x ) dξ 1 dξ 2 , (15)
where

U T k = U T V k + U T H k . It is convenient to introduce cylindrical coordinates (ξ ⊥ , ϕ), such that ξ 1 = ξ ⊥ cos ϕ, ξ 2 = ξ ⊥ sin ϕ, x 1 -x 1 = r cos φ , x 2 -x 2 =
r sin φ and express the TV components of the Lamb's Green's tensor as

G Lamb(T V) k (x -x ) = ik T 2πρc 2 T ∞ 0 A k g (2) k ξ 2 ⊥ (1 -2ξ 2 ⊥ ) R ξ 2 ⊥ e ik T ξ T 3 x 3 dξ ⊥ . ( 16
) [8] Above, g (2) 1 = ξ T 3 , g (2) 2 = ξ T 3 , g (2)
3 = -ξ ⊥ ; and, using Euler's formula, the standard definition of the Bessel function J p (z) [1, Eq.(9.1.21)] as well as identity J -p (z) = -J p (z), with p = 0 -treated as p = 0 + , the components of the symmetric matrix A are

A 11 = 1 2 [J 0 (k T ξ ⊥ r ) -i sin (2φ )J 2 (k T ξ ⊥ r )], A 22 = 1 2 [J 0 (k T ξ ⊥ r ) + i sin (2φ )J 2 (k T ξ ⊥ r )], A 12 = -i cos (2φ )J 2 (k T ξ ⊥ r ), A 13 = i cos(φ )J 1 (k T ξ ⊥ r ), A 23 = i sin(φ )J 1 (k T ξ ⊥ r ), A 33 = J 0 (k T ξ ⊥ r ).
Integrals ( 16) can be evaluated with a two-dimensional variable step version of the Simpson rule [START_REF] Fradkin | A semi-numerical model for near-critical angle scattering[END_REF]. There is no need to derive similar results for L and T H components, because their stationary phase asymptotics give satisfactory results-see below.

Numerical evaluation of integral (6)

It is easy to check that at the Rayleigh distance a R from the waist, the boundary value of the intensity of the Gaussian beam falls from its axial value by the factor of exp(-2). Therefore choosing a 1 = a R , a reasonable approximation to (15) can be achieved by reducing the integration domain to the foorprint of the beam on the scattering surface (see Figure 2). This footprint is an ellipse with the axes a min = 2 √ a R /k α and a max = 2 √ a R /k α / cos ϑ in . Hence its boundary can be described by the parametric equations x 1 = a max cos ϕ , x 2 = a min sin ϕ = a max cos θin sin ϕ .

Let us introduce the corresponding circular coordinates (a , ϕ ),

x 1 = a cos ϕ , x 2 = a cos θin sin ϕ .

The Jacobian of transformation from (x 1 , x 2 ) coordinates to (a , ϕ ) coordinates is a cos θin . It follows that ( 6) can be approximated by using

u sc(α) k (x) cos θin × (17) 2π 0 a max 0 G Lamb(α) k (x 1 -a cos ϕ , x 2 -a cos ϑ in sin ϕ , x 3 )t in (a , φ )a da dϕ .
In order to evaluate this integral numerically we employ a 2D Simpson's scheme, with the intervals [0, a max ] and [0, 2π] divided, respectively, into N f and M f subintervals and also specify the grid of evenly distributed observation points. Below we plot results in the x 1 x 3 -plane only, and N o and M o are the numbers of nodes in x 1 -and x 3 -directions, respectively. When dealing with the exact expression for the Lamb's Green's tensor considerable reduction in runtime is achieved by specifying the footprint grid first and then specifying a convenient grid of observation points.

The Alternative Kirchhoff Approximation and its Validation

If the Lamb's Green's tensor in (17) is evaluated numerically the resulting codes have long run times. Fortuitously, the ultrasonic inspections are normally conducted in the far field |k L x 3 | 1, with x 3 -the vertical distance from the observation point to scatterer. In this approximation the double integrals (15) contain a rapidly oscillating exponent and slowly varying amplitude and therefore can be evaluated using the uniform stationary phase method [START_REF] Borovikov | Uniform Stationary Phase Method[END_REF]. The method describes asymptotic contributions of such critical points as stationary points of the phase and singularities of the amplitude, both in the geometrical regions in the scattered field, where the critical points are isolated and transition regions, where some of them coalesce.

It is easy to check that in (15) the phases k α ξ T • (xx ) possess stationary points, and the amplitudes have poles as well as branch points ξ ⊥ = ±γ, where ξ L 3 = 0. These amplitude singularities give rise to the Rayleigh wave and head waves, respectively. Calculation of the Rayleigh wave is easy and lies outside the scope of this paper. The simplest situation giving rise to a head wave is discussed in [START_REF] Fradkin | The Two-Component Representation of Time-Harmonic Elastic Body Waves in the high-and intermediate-frequency regimes[END_REF] and presented in Figure 3, where the fronts of longitudinal, transverse and head waves are designated by L, T and H, respectively. The T and L fronts are semi-spherical and the conical H front exists only for θ > θ H .

To evaluate the contributions of the isolated stationary phase points, for each surface point x let us choose the Cartesian coordinate system with the origin at this point. The associated cylindrical coordinates are (r , s , θ ), where ∆x n = x n -

x n , n = 1, 2; r = ∆x 2 1 + ∆x 2 2 ; s = r 2 + x 2 3 ; and sin θ = r /s (see Figure 1). In these coordinates the phase functions in (15) are

f α (ξ 1 , ξ 2 ) = ξ 1 ∆x 1 + ξ 2 ∆x 2 + γ 2 α -ξ 2 ⊥ x 3 .
It is easy to check that each has one stationary point (ξ α(GE)

1 , ξ α(GE) 2 ) = (γ α ∆x 1 /s , γ α ∆x 2 /s ); f α (ξ α(GE) 1 , ξ α(GE) 2 ) = s ; ξ α(GE) ⊥ = γ α sin θ ; and the eigenvalues of each Hessian ( f α ) (ξ α(GE) 1 , ξ α(GE)
2 ) are positive. The above considerations allow us to produce asympotic description of T V, T H and L waves generated by each Huygens source, see Appendix B. The superscript GE emphasises that the same description can be obtained using the Geometrical Elastodynamics.

The asymptotic contributions of branch points ξ ⊥ = ±γ, where ξ L 3 = 0 (see Equation ( 13)) are also well known [START_REF] Cěrvenỳ | Seismic Ray Theory[END_REF]. However, in industrial steels these asymptotics do not work well [START_REF] Gridin | A fast method for simulating the propagation of pulses radiated by a rectangular normal transducer into an elastic half-space[END_REF]. For this reason we do not use them and advocate the aka insteadthat is, substitute into Equation ( 6) the amplitudes G Lamb(β,GE) k (β = T V, T H or L) of [START_REF] Huet | Modeling of corner echo ultrasonic inspection with bulk and creeping waves, Ultrasonic Wave Propagation in Non Homogeneous Media[END_REF] the semi-spherical waves from Appendix B. Note that the components G Lamb(T V,GE) k possess branch points at sin θ = ±γ and therefore retain information about the branch points in the Fourier space.

The aka has been tested by simulating scatter of Gaussian beams oscillating at f = ω/2π = 5MHz, a typical frequency of industrial transducers, from the surface of a block of stainless steel with c L = 5890 m/s, c T = 3210 m/s and ρ = 8050 kg/m 3 . The corresponding critical incident angle is ϑ in = cos -1 γ ≈ -57 o (θ in ≈ 33 o ).

Figures 4(c) and 4(d) simulate reflection of the subcritical T Gaussian beam. For comparison, reflection of the L beam is shown in Figures 4(a) and 4(b). As expected, the L beams have a wavelength twice as large and display larger footprints. In both L -L or T -T cases the reflection takes place according to Snell's law for elastic waves, see e.g. [START_REF] Graft | Wave Motion In Elastic Solids[END_REF]Eq. (6.1.33)]. By contrast, Figure 4(e) shows the full scatter of the T beam into a weaker upper beam (reflected according to the Snell's law) and stronger lower beam (propagating at the critical angle). In principle, beam splitting could be due to variation in the incidence angle over the footprint, but inspection of the incident beams in Figure 4 shows no comparable variation. We conclude that the lower beam is formed by the head wave. This conclusion is supported further by the fact that the beam appears or disappears depending on whether, when imaginary, the radical γ 2sin 2 θ is supplied with the + or -sign: such behaviour is typical to branch point contributions [START_REF] Fradkin | A semi-numerical model for near-critical angle scattering[END_REF]. Figures 4(f) and 4(g) simulate scatter of the critical Gaussian T beam. They demonstrate a small Goos-Hänchen shift to the right: the reflected T wave and head wave both propagate at the same angle, and their interference breaks the incident/reflected beam symmetry.

Let us now compare compare ka and aka. Figures 5(a) and 5(b) demonstrate that the two approximations produce similar scattered L beams-the fact that the ka beam is somewhat stronger might be due to the fact that ka does not allow for the head wave and the head wave energy must be distributed between L and T modes. For the same reason, when the incidence is subcritical, the ka T beam is stronger than aka T beam (see Figures 5(c) and 5(d); c f. Figure 5(e)). At the critical incidence, ka produces a much stronger T beam than aka and no Goos-Hänchen shift.

Finally, let us compare both aka and ka to the exact numerical solution based on Equation (17). Figure 6 shows that when the incident Gaussian T beam is critical, unlike ka, aka produces reasonable results: it reproduces the footprint, angle of scatter and even the scattered intensity. Note that using the Intel (R) Core (TM) i3-3227U processor CPU @1.90 GHz, Figures 6(a) and 6(b) have been produced in 11 hours and four minutes, respectively.

Conclusions

We have shown that while the ka is incapable of simulating head waves, the aka simulates the effect similarly to the exact numerical solution, reproducing a small lateral Goos-Hänchen shift and-more importantly-the correct amplitude of the scattered critical Gaussian beam. The result is not obvious, because the asymptotic (a) ( expressions of the Lamb's Green's tensor utilised in the aka describe only the semispherical L and T waves generated by each Huygens source and not the associated head waves. On the other hand, the amplitudes of the T waves contain the square roots, retaining information about the square roots in their Fourier Transforms, and it is these roots (more precisely, the fact that the integrands used in the aka are doublevalued functions) that gives rise to the head waves. The findings are of interest in ndt applications.

A. The Lamb's Green's tensor in Fourier space

Each component G Lamb k = G Lamb k (x 1 , x 2 , x 3 ) of the Green's Lamb's tensor represents the th component of the displacement G Lamb k = G Lamb k (x 1 , x 2 , x 3 )
, which is due to a unit traction acting in the direction e k . Therefore the original problem can be decomposed into three, two-with the tangential unit traction acting in the direction e 1 and e 2 and one-with the traction acting normally to the surface. In order to find the Green's Lamb's tensor we first rewrite Equation [START_REF] Abramovitz | Handbook of Mathematics Functions with Formulas, Graphs and Mathematical Tables[END_REF] as

L m G Lamb k + ρω 2 G Lamb mk = 0, ( 18
)
where L m = L m (∂ 1 , ∂ 2 , ∂ 3 ) are components of the operator matrix

L = ρc 2 L           (1 -γ 2 )∂ 2 1 + γ 2 ∇ 2 (1 -γ 2 )∂ 1 ∂ 2 (1 -γ 2 )∂ 1 ∂ 3 (1 -γ 2 )∂ 1 ∂ 2 (1 -γ 2 )∂ 2 2 + γ 2 ∇ 2 (1 -γ 2 )∂ 2 ∂ 3 (1 -γ 2 )∂ 1 ∂ 3 (1 -γ 2 )∂ 2 ∂ 3 (1 -γ 2 )∂ 2 3 + γ 2 ∇ 2          
, with ∂ -the partial derivative with respect to x and

∇ 2 = ∂ 2 1 + ∂ 2 2 + ∂ 2 3 .
Applying to (18) the double Fourier Transform (11) we obtain the system of ordinary differential equations

L m (ξ 1 , ξ 2 , ∂ 3 ) G Lamb k (ξ 1 , ξ 2 , x 3 ) + ρω 2 G Lamb mk (ξ 1 , ξ 2 , x 3 ) = 0. ( 19
)
Let us seek the outgoing solutions of this system in the form

G β (Lamb) k (ξ 1 , ξ 2 , x 3 ) = U β k (ξ 1 , ξ 2 )e ik T ξ 3 x 3 = U β k (ξ 1 , ξ 2 ) d β (ξ 1 , ξ 2 )e ik T ξ 3 x 3 . (20)
Substituting ( 20) into (19) we obtain

L(ξ) U β k (ξ 1 , ξ 2 ) = -ρω 2 U β k (ξ 1 , ξ 2 ), (21)
where the operator L = L(ξ) has components

L m = -ρω 2 [(γ -2 -1)ξ m ξ + ξ 2 δ m ]. (22)
Equation ( 21) can be embedded into the so-called Kelvin-Cristoffel equation

L d β = -ρω 2 ν β d β , (23)
which possesses a non-zero solution only if we have

| L + ρω 2 νI| = -ρω 2 (ξ 2 -ν) 2 (γ -2 ξ 2 -ν) = 0,
where I is the unit matrix. It follows that the operator L has a double eigenvalue ν TV = ν TH = ξ 2 and simple eigenvalue ν L = γ -2 ξ 2 , with the corresponding eigenvectors

d β = d β (ξ 1 , ξ 2 )
given by

d T V =                    1 ξ T ξ ⊥             ξ 1 ξ T 3 ξ 2 ξ T 3 -ξ 2 ⊥             , e 1 , d T H =                    1 ξ ⊥             ξ 2 -ξ 1 0             if ξ ⊥ 0, e 2 if ξ ⊥ = 0, d L = 1 ξ L           ξ 1 ξ 2 ξ L 3           . ( 24
)
As usual, there is arbitrariness in choosing eigenvectors corresponding to a double eigenvalue. The choice made in Equation (24) ensures that lim ξ ⊥ →0 (ξ 1 /ξ ⊥ ) = 1 and lim ξ ⊥ →0 (ξ 2 /ξ ⊥ ) = 0. The eigenvectors (24) become solutions of Equation ( 22) when

ν β = 1, that is, when |k T ξ β | = k β .
The unknown amplitudes U β k = U β k (ξ 1 , ξ 2 ) can be found from the boundary condition (5) re-written as

B mk G Lamb k | x 3 =0 + = -δ mk δ(x)| x 3 =0 + , (25)
with B mk = B mk (∂ 1 , ∂ 2 , ∂ 3 )-elements of the operator matrix

B = ρc 2 L           γ 2 ∂ 3 0 γ 2 ∂ 1 0 γ 2 ∂ 3 γ 2 ∂ 2 (1 -2γ 2 )∂ 1 (1 -2γ 2 )∂ 2 ∂ 3           . The double Fourier Transform of (25) is B G Lamb k | x 3 =0 + = -e k , (26)
with the operator matrix B = B(ξ 1 , ξ 2 , ∂ 3 ) given by

B(ξ 1 , ξ 2 , ∂ 3 ) = ρc 2 L           γ 2 ∂ 3 0 γ 2 ik T ξ 1 0 γ 2 ∂ 3 γ 2 ik T ξ 2 (1 -2γ 2 )ik T ξ 1 (1 -2γ 2 )ik T ξ 2 ∂ 3          
.

Combining (20), ( 24) and (26) produces the linear algebraic system

A U k = c k , (27)
where we use notation

A =                                                                                         ξ 1 (1 -2ξ 2 ⊥ ) ξ 2 ξ T 3 2γ -1 ξ ⊥ ξ 1 ξ L 3 ξ 2 (1 -2ξ 2 ⊥ ) -ξ 1 ξ T 3 2γ -1 ξ ⊥ ξ 2 ξ L 3 -2ξ T 3 ξ 2 e k .
For ξ ⊥ 0, |A| = -γ -1 ξ 3 ⊥ ξ T 3 R ξ 2 ⊥ , with the Rayleigh function

R ξ 2 ⊥ = 4ξ 2 ⊥ ξ T 3 ξ L 3 + [1 -2ξ 2 ⊥ ] 2 . ( 28
)
and R T (sin 2 θ ) = 4 sin 2 θ cos θ γ 2sin 2 θ + 1 -2 sin 2 θ 2 , R L (sin 2 θ ) = 4γ 3 sin 2 θ cos θ 1 -γ 2 sin 2 θ + 1 -2γ 2 sin 2 θ 2 ,

where φ and θ are specific to the source. (Similarly to Appendix A, the formal rule applied above is lim (x 1 ,x 2 )→(x 1 ,x 2 ) (x 1x 1 )/r = 1 and lim (x 1 ,x 2 )→(x 1 ,x 2 ) (x 2x 2 )/r = 0.) The stationary phase contributions to the components of

G Lamb(GE) k = G Lamb(GE) k (x - x ) become G Lamb(T V,GE) 1 =                        - cos φ 2πρc 2 T cos θ (1 -2 sin 2 θ ) R T (sin 2 θ ) e ik T s s d T V , if r 0, - 1 2πρc 2 T e ik T s s δ 1 , if r = 0, G Lamb(T H,GE) 1 =                  - sin φ 2πρc 2

Figure 1 .

 1 Figure 1. The scatterer Cartesian coordinate system and associated spherical polar coordinate system. Thick line-boundary of a solid half-space.

Figure 2 .

 2 Figure 2. Scatterer / beam coordinate systems {e 1 , e 2 , e 3 } / {ē 1 , ē2 , ē3 }.

3. 1 .

 1 Complex Point Source To introduce a CPS let us first consider an ordinary point source U 0 δ(xa) at point a. It radiates the outgoing spherical field U 0 G, where the reduced free-space Green's tensor has components [3, Eq. (2.5.54)]

√ 2 )

 2 ; = (∆ x1 )-radius of curvature of the beam wavefront; and ζ = ζ(∆ x1 )-the Gouy phase shift, where

Figure 3 .

 3 Figure 3. Wave fronts generated by a normal point load.

Figure 4 .

 4 Figure 4. Simulation of scatter of Gaussian beams using aka. The scalar displacement amplitude in the beam u is traditionally defined as |Re u 2 1 + u 2 2 + u 2 3 |. The color bar represents displacement amplitudes in pm. ϑ in L = -75 • : (a) incident L beam, (b) reflected L beam; ϑ in T = -75 • : (c) incident T beam, (d) reflected T beam; (e) reflected T beam and head wave beam; ϑ in T = -ϑ H : (f) incident T beam, (g) scattered T beam. N o = 120, M o = 120, N f = 24, M f = 40.

Figure 5 .Figure 6 .

 56 Figure 5. Comparison of ka and aka of scattered Gaussian beams. ϑ in L = -75 • , reflected L beam simulated using (a) ka, (b) aka; ϑ in T = -75 • , reflected T beam simulated using (c) ka, (d) aka ; reflected T beam and head wave beam simulated using (e) aka; ϑ in T = -ϑ H , scattered T beam simulated using (f) ka, (g) aka. N f = 24, M f = 40, N o = 80, M o = 80. The color bar as above.

Acknowledgements This work has been carried out within the framework of the CIVAMONT 2020 collaborative project. We thank Mr Flavien Deniaux for his contribution.

Solving (27), components of vectors

Each column of the Lamb's Green's tensor can be represented as a sum [START_REF] Zernov | Wedge diffraction of a critically incident Gaussian beam[END_REF]. Similar results have been obtained earlier for a linear source [START_REF] Miller | The Field and Radiation Impedance of Mechanical Radiators on the Free Surface of a Semi-Infinite Isotropic Solid[END_REF].

B. Stationary phase asymptotics of the Lamb's Green's tensor

Let us associate with each Huygens source x a Cartesian coordinate system {e 1 , e 2 , e 3 } with the origin at this source, and let all such systems be oriented the same way as the scatterer coordinate system {e 1 , e 2 , e 3 }. Then all vectors have the same components in all of them. Evaluating the stationary phase asymptotics of (15), for each Huygens source the unit displacement vectors (24) and the Rayleigh functions (28) become, respectively,

sin φ e 1cos φ e 2 , if r 0, e 2 , if r = 0, d L (xx ) = sin θ (cos φ e 1 + sin φ e 2 ) + cos θ e 3 ,


Léa Pacini
email: lea.pacini@lecnam.net

Jérôme Dupire

Isabelle Barbet

Olivier Pons

Camille Guinaudeau

Vincent Mousseau

Céline Hudelot

Caroline Huron

Textbook's accessibility for children with dyspraxia and visual disability

Keywords: Accessibility, Special Education, Single Case Design, Disabilities Context

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

In France, the 11th February 2005 law, "for equality of rights and opportunities, participation and civic rights" supports inclusion of children with disabilities in schools. However, the access to scholarship and to the compensation means, the materials and infrastructures remains low. Our work is part of the ANR project "MALIN" (MAnuels scoLaires INclusifs, ie. Inclusive Textbooks). Its purpose is to make digital textbooks more accessible. To this day, only few textbooks are adapted for children with disabilities and these process are still time-and moneyconsuming. The MALIN project seeks, on one hand, the automation of the process and, on the other hand, the conception of a platform allowing to choose for the most fitting adaptation depending on the user. The aim of the project is to allow for a better access of textbooks for the children with disabilities but the overall goal is to provide a way to adapt the textbooks to try to compensate the difficulties children can face at school during their learning process. The core of the project focuses first on two specific populations (dyspraxic people and visually disabled people), but once expanded with other type of targeted user profiles, the created basis will make all of the students able to use the same pedagogical material. First, our goal is to better comprehend the execution of the textbooks transcription process, including the issues and needs the experts face. In order to do that, we chose an expert-centered method. The activity of transcription will be analyzed through shadowing observations. We will work with the two following experts. The first one is the "Institut National des Jeunes Aveugles" (ie. Young Blind People National Institute, in Paris, France). This institute aims to develop the autonomy of children and provide adapted individual project with constant followup care during the whole education time. Hence, it also deals with transcription of textbooks and learning materials for people with visual disabilities to make access to knowledge easier. The second one is the association "Le Cartable Fantastique" (ie. The Fantastic Schoolbag), which provides resources for teachers to facilitate the education of children with dyspraxia. Dyspraxia is a neuro-developmental disorder of the coordination which means that people struggle to automate gestures. One of the resource provided is an online platform allowing to recreate the exercises of a French language textbook on a web page. The exercises are then displayed on the computer and the dyspraxic children are able to do the same thing as their peers in class, without having to hand write. The expertise of both the association and the institute will allow us to better understand the process of creation of textbooks adapted for people with dyspraxia and with visual disabilities. We will then propose a modeling of the process, which will be integrated in the platform in the form of adaptive user profile. Then, we aim to understand how these experts take into consideration the drivers, constraints and needs of the population they work for and with. This expertise and knowledge will help to think about the conception and implementation of adaption and the various features of the user profile. This will be tested by end-users using the Single-Case Design (SCD) methodology.

Methodology

The main issue about disabilities is the important heterogeneity between profiles, hence the consequences on the reliability of the statistical evaluations of software, hardware and other solutions implemented. For instance, the tiredness and fine motor skills difficulties will vary a lot among people with dyspraxia. Added to this, the high comorbidity in these specific population is another problem researchers may face in randomized trials. For these reasons, it can be difficult to obtain significant results when using classical statistical analyses, as the large inter-individual variation is responsible for the excess of within variance inside the experimental group. To address this issue, there is a methodology called Single Case Design (SCD), as mentioned above. These types of design are not in opposition to classical randomized design, but rather complementary. To sum it up, each participant are at the same time the control "group" and the experimental "group". [START_REF] Horner | The Use of Single-Subject Research to Identify Evidence-Based Practice in Special Education[END_REF] define SCD as "experimental, rather than correlational or descriptive" and whose goal is to show "causal (. . . ) relationships between independent and dependent variables". They are designs often used in special education, due to the suitability of the procedure to assess the effectiveness of the independent variable, in a participant centered manner. That is why dependent variables in SCD are mostly behaviors, for instance self-injuring behaviors. The independent variables, (i.e. what is evaluated), are called interventions and need to be actively manipulated in the experiment. To evaluate if an intervention is effective, the experimenter has to follow a precise procedure depending on the type of SCD, and meet standards set by experts of SCD [START_REF] Horner | The Use of Single-Subject Research to Identify Evidence-Based Practice in Special Education[END_REF][START_REF] Kratochwill | Single-Case Designs Technical Documentation[END_REF][START_REF] Kratochwill | Single-Case Intervention Research Design Standards[END_REF][START_REF] Plavnick | Single-Case Experimental Designs in Educational Research: A Methodology for Causal Analyses in Teaching and Learning[END_REF]. Even if some features may differ depending on the type of design, some other are to apply in every case. The basic procedure of a SCD consists in first a baseline phase (Phase A), where the experimenter measure the performance of each participant without the intervention, then apply the intervention and measure again the performance (Phase B). The phases are measured over time. This basic design is called the AB Design. Every other type of SCD requires a baseline phase and an intervention phase. The analysis of SCD is done in two steps: [START_REF] Horner | The Use of Single-Subject Research to Identify Evidence-Based Practice in Special Education[END_REF] with visual analysis and (2) with statistical analysis. According to [START_REF] Ledford | Systematic Use of Visual Analysis for Assessing Outcomes in Single Case Design Studies[END_REF], visual analysis can not be replace by any other type of analysis to represent changes in dependent variable in a comprehensive way, but "quantitative metrics can be helpful for summarizing data characteristics". The process of visual analysis have to be very well understood by SCD researchers and requires to follow correctly the guidelines and standards [START_REF] Lane | Visual analysis in single case experimental design studies: Brief review and guidelines[END_REF][START_REF] Manolov | How Can Single-Case Data Be Analyzed? Software Resources, Tutorial, and Reflections on Analysis[END_REF], and not every data analysis techniques are appropriate depending on the data [START_REF] Manolov | Recommendations for Choosing Single-Case Data Analytical Techniques[END_REF]. We plan to use a specific design of SCD, called Alternating Treatment Design. This design allows to compare two or more interventions in a rapidly alternated manner. We want to compare the different adaptations using SCD to see if we can find any trends in the results. In our case, we want to know which are more facilitating for dyspraxic children in terms of right answers, speed of response, but also to study their strategies when answering. We also want to assess their preferences about the different adaptations. The goal is to understand what are the abilities and the limitations of these children in order to better adapt all the various types of exercises of textbooks. In the long term, this work aims to provide resources and a "user" template that could be re-used in other accessibility contexts, in particular in school contexts, and that could be overall applied not only to dyspraxic and visually deficient children, but to any that face difficulties at some time. At the time of the AAATE Conference, the two models from the analyze of the textbooks transcription activity as well as initial results from SCD about assessment of one of our exercise adaptations for dyspraxic people will be presented.

  Copyright: © AAATE and L.Pacini et al. 2023. This is an openaccess extended abstract of a contribution (Oral contribution) given at the AAATE 2023 Conference in Paris. It is distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0), which permits copy and reproduction of the material in any medium or format, provided the original author and source are credited.


Vocal analysis of emotions during the recall of Self-Defining Memories (SDM) among antisocial population

INTRODUCTION

SDM are autobiographical memories with the particularity of establishing a sense of identity ("Self") and to persist over time due to their emotional charge. They refer to specific moments in the individual's life which have generated a "life lesson" that will change its perception of Self and life. SDM can be classified based on four features: a) specificity (specific or generic), b) valence (positive, negative, mixed, or neutral), c) integration (integrated or non-integrated) and d) theme (threatening life events, pleasant life events, interpersonal relationships, goal achieving or substance abuse). Antisocial Personality Disorder (ASPD) implies a deficit in the emotional sphere such as high impulsivity and aggressivity and a disregard for others' psychological/physical integrity. While most studies on ASPD forensic inpatients have addressed impulsivity, empathy or emotion decoding domains, few have addressed emergence of emotions despite the key role of Self and negative affect regulation on antisocial behavior. To our knowledge, no study has analysed acoustic parameters of SDM. In contrast to less ecological experimental design, these memories allow the emergence of emotions in a natural context.

Objective: Analyse acoustic parameters of vocal cues from naturalistic expressed emotions during the recall of SDM among antisocial forensic inpatients.

METHOD

Participants 17 Belgian male inpatients from the High-Security Forensic Hospital (Tournai) with ASPD diagnosis (SCID-II) participated on a voluntary basis. The mean age was 46.18 (SD = 13.77) and the mean length of hospitalization was 10.55 years (SD = 7.93). The mean I.Q. score (WAIS-IV) was 75.87 (SD = 14.47) and the mean Social Desirability score (MC-SDS) was 16.94 (SD = 5.39).

Instruments and Procedure

Data analysis First, we presented the descriptive statistics about SDM (N = 83) recalled by ASPD inpatients. In absence of normality of distribution (Kolmogorov-Smirnov test), we performed non-parametric comparison group analyzes (Wilcoxon W) on vocal cues between emotional and unemotional utterances. Finally, we performed non-parametric correlation (Spearman ) between acoustic parameters and IRI/CERQ scores.

RESULTS

DISCUSSION

•

Prevalence results (Table 1) concerning the highest recalled SDM valence (neutral) suggest an emotional detachment as previously found in literature. However, prevalence and acoustic parameters results (Table 2) support the hypothesis that ASPD are less emotionally impaired as expected. Results about non-integrated SDM support Baumeister's theory that impairment creates and maintains a coherence of Self in individuals with ASPD. Unexpectedly, they recalled specific SDM, exhibiting the ability to plan and remain focus on the recalling of a memory. However, ASPD forensic inpatients showed difficulty in retrieving very specific single events and rather recalled memories comprising multiples events. Finally, the two main SDM themes recalled were partially expected considering the antisocial population. Indeed, literature supports that antisocial offenders experienced more and greater childhood adversity experiences than general population.

• Literature supports the assumption that emotional utterances are more activating at the prosodic level. Results highlight a significant variability of the speech, but not of the amplitude, between EMO and nonEMO utterances. Previous research suggested that ASPD forensic inpatients mainly expressed anger at the level of facial expression during the SDM recall task, whatever the valence SDM. This overexpression of anger may be associated to the high prevalence of threatening life-events recalled, and potentially to childhood adversity experiences.

• Correlational results (Table 3) suggest that the use of adaptative strategies contributes to the emotional vocal expression and this whether the lexical content is emotional or nonemotional. This result highlights the interest of the multi-level analysis of emotions. With regard to empathy, there was a negative correlation between perspective-taking (PT) abilities and level of activation of emotion. This result may be explained by the nature of the task (cognitive empathy) asking to shift from one's own point of view to other's point of view. This result is congruent with the definition of the cognitive/affective dimension of the mentalization process. Perspective taking does not imply connection and sharing with emotions. However, the integration of emotional and cognitive aspects is necessary to understand mental states. This result may illustrate the callous empathy dimension of individuals with psychopathic traits.

Future perspectives

• Undertake an in-depth analysis of the SDMs themes (ex: life-threatening event-others, life-threatening event-accident/illness, life-threatening event-physical, life-threatening event-unclassifiable)

•
Assess the Psychopathic Personality Disorder (PPD) in order to describe and compare both the classification features of recalled SDM and the acoustic parameters between PPD and ASPD • Conduct a correlational analysis between emotional regulation strategies (adapted and non adapted) and acoustic parameters both among ASPD and PPD populations • Implementing a bottom-up approach, starting from the acoustic parameters distribution in order to identify emotionally charged utterances among each SDM •

Implementing a multi-level analysis (facial and vocal expressions of emotions, physiological activity, empathic abilities, self-regulatory strategies, mentalization) of SDM among antisocial populations Tiberi L.A. 1 , Saloppé, X.

  Note: EMO = Emotional utterances; nonEMO = Nonemotional utterances; AS = Adaptative strategies; NAS = Non adaptive strategies; PT = Perspective-taking ; EC = Empathic concern; PD = Personal distress; F = Fantasy; *p < .05; **p < .001

2,3,4 , Lavallée, A. 3 , Degouis, F. 1,3 , Sens, A. 5 , Nandrino, J-L. 3 , Pham, T.H. 1,2,6 , & Delvaux, V. 7 1 Forensic Psychology Department, University of Mons (UMONS), Belgium 2 Center of Research in Social Defense (CRDS), Belgium 3 Univ. Lille, CNRS, UMR 9193 -SCALab -Sciences Cognitives et Sciences Affectives, F-59000 Lille, France 4 Service de Psychiatrie, Hôpital de Saint-Amand-les-Eaux, France 5 Unité Baudelaire, Centre Hospitalier Psychiatrique (CHP) Le Chêne aux Haies, Belgium 6 Philippe Pinel Institute, Canada 7 Language Sciences and Metrology Unit, University of Mons (UMONS), Belgium

	IRI Empathy	CERQ Emotion Regulation	SDM Task Recall 5 SDM (recorded speech and video)	EMOTAIX Text Analysis 3 valences + 2 unspecified emotions 6 meta-categories Emotional & Nonemotional utterances	Praat Acoustic parameters (MeanF0, MedianF0, SDF0, MinF0, MaxF0, SpeechRate)
		Prevalence (%)
	SDM Classification
	features
	Valence
	Positive	19.28
	Negative	25.30
	Mixed	22.89
	Neutral	32.53
	Specificity
	Generic	44.58
	Specific	55.42
	Integration
	Integrated	10.84
	Non-integrated	89.16
	Theme
	Moral choice	1.20
	Susbtance abuse	2.41
	Pleasant life event	4.82
	Goal reaching	7.23
	Other	25.30
	Interpersonal relationships	28.92
	Threatening life event	28.92

Table 2 -

 2 Acoustic parameters descriptive statistics (Hz) of SDM

	EMO (n = 78)	nonEMO (n = 83)	W (r)

Note: EMO = Emotional utterances; nonEMO = Nonemotional utterances; W = Wilcoxon signed-rank; r = effect size; *p < .05; **p < .001

Table 3 -

 3 Correlation () between acoustic parameters and Self-Questionnaires

	CERQ

Table 1 -

 1 SDM descriptive statistics (%)

Contact : luca.tiberi@umons.ac.be

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