Mohamed Ez-Zaouia
email: ezzaouia.mohamed@gmail.com

Elise Lavoué
email: elise.lavoue@univ-lyon3.fr

EMODA: a Tutor Oriented Multimodal and Contextual Emotional Dashboard

Keywords: CCS Concepts, Human-centered computing ➝ Visualization Visualization application domains → Visual analytics, Applied computing➝ Education ➝ Interactive learning environments, Emotions, Interactive visualizations, Learner monitoring, Tutor dashboard, Language training, Multimodal data

Learners' emotional state has proven to be a key factor for successful learning. Visualizing learners' emotions during synchronous on-line learning activities can help tutors in creating and maintaining socio-affective relationships with their learners. However, few dashboards offer emotional information on the learning activity. The current study focuses on synchronous interactions via a videoconferencing tool dedicated to foreign language training. We collected data on learners' emotions in real conditions during ten sessions (five sessions for two learners). We propose to adopt and combine different models of emotions (discrete and dimensional) and to use heterogeneous APIs for measuring learners' emotions from different data sources (audio, video, self-reporting and interaction traces). Based on a thorough data analysis, we propose an approach to combine different cues to infer information on learners' emotional states. We finally present the EMODA dashboard, an affective multimodal and contextual visual analytics dashboard, which allows the tutor to monitor learners' emotions and better understand their evolution during the synchronous learning activity.

INTRODUCTION

Learning analytics (LA) is a fast-growing field with a strong potential of making on-line learning environments competitive, attractive and, more importantly, efficient and fit for purpose [START_REF] Siemens | Penetrating the Fog: Analytics in Learning and Education[END_REF]. Several studies position the learner at the center of interest in measuring performance in the learning environment ecosystem [START_REF] Wu | A study of student satisfaction in a blended e-learning system environment[END_REF][START_REF] Sun | What drives a successful e-Learning? An empirical investigation of the critical factors influencing learner satisfaction[END_REF]. At the same time, teachers lack visual insights that can help them monitor learners during on-line learning activities, especially with indicators on their emotions.

Several recent studies demonstrate that emotions are an essential factor in the learning process [START_REF] Pekrun | The Control-Value Theory of Achievement Emotions: Assumptions, Corollaries, and Implications for Educational Research and Practice[END_REF][START_REF] Boekaerts | The crucial role of motivation and emotions in classroom learning[END_REF] and, more precisely, that positive emotions play a crucial role in fostering successful learning [START_REF] Pekrun | Measuring emotions in students' learning and performance: The Achievement Emotions Questionnaire (AEQ)[END_REF][START_REF] D'mello | Towards an Affect-Sensitive AutoTutor[END_REF]. Therefore, considering learners' emotions can have a huge impact on learning and achievement. Research into the emotional dimension of learning environments has thus started to attract more interest in the community (e.g. [START_REF] Mega | What makes a good student? How emotions, self-regulated learning, and motivation contribute to academic achievement[END_REF][START_REF] Ruiz | Supporting learning by considering emotions: Tracking and Visualization[END_REF][START_REF] Ghasemaghaei | A Dashboard for Affective E-learning: Data Visualization for Monitoring Online Learner Emotions[END_REF][START_REF] Happy | Automated Alertness and Emotion Detection for Empathic Feedback during e-Learning[END_REF]). Consequently, to optimize the learning experience, it is important for tutors to be aware of and understand the emotional characteristics experienced by the learners.

The majority of the proposed dashboards considering learners' emotions are powered only by learners' interaction traces stored in the learning environment [START_REF] Mazza | CourseVis: A Graphical Student Monitoring Tool for Supporting Instructors in Web-Based Distance Courses[END_REF][START_REF] May | TrAVis to Enhance Self-Monitoring in Online Learning Supported by Computer-Mediated Communication Tools[END_REF]. Other dashboards are based on a single channel [START_REF] Ghasemaghaei | A Dashboard for Affective E-learning: Data Visualization for Monitoring Online Learner Emotions[END_REF][START_REF] Happy | Automated Alertness and Emotion Detection for Empathic Feedback during e-Learning[END_REF] or on self-reported data only [START_REF] Ruiz | Supporting learning by considering emotions: Tracking and Visualization[END_REF]. However, several studies concerning the affective computing domain, particularly emotion recognition, show that user behavior has to be considered in a global and multimodal fashion (voice, image, interaction, self-report, etc.). In this way, affective multimodal systems are more accurate than unimodal approaches [START_REF] Caridakis | Multimodal emotion recognition from expressive faces, body gestures and speech[END_REF][START_REF] Pantic | Affective Multimodal Human-Computer Interaction[END_REF]. By considering multimodal data, the visual insights proposed on dashboards can go beyond simple reporting visual displays to attain a multimodal visual analytics tool combining different cues with their associated contextual information. Such rich visualizations can facilitate actionable feedback and decisionmaking for tutors in the online learning context. The aim of our paper is to propose a multimodal and contextual emotional dashboard called EMODA. The design of this dashboard is based on a thorough analysis of learners' emotional data collected in real conditions. We more precisely answer the following research questions: 1) How to collect data on learners' emotions during online learning sessions? 2) How to exploit and combine different cues for inferring learners' emotions? 3) How can learners' inferred emotions be visualized by tutors to facilitate actionable feedback?

We conducted an experimental study using a real-time web videoconferencing learning platform called SpeakPlus. This solution aims at training learners in foreign language speaking, based on online training sessions between a tutor and a learner. We applied both dimensional and discrete models for measuring learners' emotions. We used heterogeneous APIs for emotion classification based on different cues and in non-intrusive settings. We provided an exploratory analysis comparing learners' inferred emotions and an approach for unifying different models of emotions. Based on the results, we designed an affective multimodal dashboard that presents the measured emotions with their associated contextual information. These visualizations can help tutors better monitor and understand the emotional characteristics of the learners and, consequently, improve the socio-affective relationship between them and their learners.

The paper is organized as follows. Section 2 presents the theoretical and technological background. Section 3 details the context and the experimental settings of our study. Section 4 focuses on emotional data collection and analysis. Section 5 presents the EMODA dashboard, a multimodal and contextual emotional dashboard. The last section presents our conclusions and future directions.

THEORETICAL AND TECHNOLOGICAL BACKGROUND

This section gives an overview of research in learning analytics and emotions. In the first part we present some recent affective learning systems. The second part is dedicated to emotion theories, emotion recognition and measurement methods.

Emotion and Learning

Over the last decade, researchers in the educational field have showed huge interest in emotions. Recent studies underscore their key role in learning, regulation processes and strong coupling to motivation and achievement [START_REF] Boekaerts | The crucial role of motivation and emotions in classroom learning[END_REF][START_REF] Pekrun | Measuring emotions in students' learning and performance: The Achievement Emotions Questionnaire (AEQ)[END_REF][START_REF] Mega | What makes a good student? How emotions, self-regulated learning, and motivation contribute to academic achievement[END_REF][START_REF] Lavoué | Reflection-in-Action Markers for Reflection-on-Action in Computer-Supported Collaborative Learning Settings[END_REF]. Learning is more likely to be successful if tutors, as well as learning platforms, help minimize negative emotions (e.g. fear, frustration, stress) and facilitate positive ones (e.g. happiness, enjoyment). This has led to the increased importance of emotional data in online learning and teaching environments. Designing learning environments that are enjoyable, motivating and inspiring for learners is a key issue for the LA community [START_REF] Bouvier | A trace-based approach to identifying users' engagement and qualifying their engagedbehaviours in interactive systems: Application to a social game[END_REF].

In a traditional learning environment, tutors can, more or less, easily monitor and evaluate the emotions experienced by their learners. However, in computer-mediated learning settings, this might not be an easy task for tutors. Therefore, emotional analytics can provide tutors with the opportunity to monitor and better understand their learners' emotions (e.g. [START_REF] Ruiz | Supporting learning by considering emotions: Tracking and Visualization[END_REF]). The development of such affective monitoring systems depends on the measurement of learners' emotional state. Methods for measuring emotions can be either Objective/Subjective, Snapshot (before and after learning)/Continuous and Qualitative/Quantitative [START_REF] Afzal | Modelling Affect in Learning Environments-Motivation and Methods[END_REF].

Recent research has emerged on this topic by tracking learners' emotions to promote awareness, decision-making and information visualization. Ruiz et al. [START_REF] Ruiz | Supporting learning by considering emotions: Tracking and Visualization[END_REF] propose an emotional dashboard to support students' self-reflection. The authors opted particularly for subjective methods to measure students' emotions using a questionnaire addressed to learners (before and after the learning activity -snapshot type). Their results show that the students' emotions are correlated to their performance in class. GhasemAghaei et al. [START_REF] Ghasemaghaei | A Dashboard for Affective E-learning: Data Visualization for Monitoring Online Learner Emotions[END_REF] propose a dashboard to help and support tutors in reflecting on students' emotions experienced during learning. The authors used objective and continuous data, mainly images extracted from students' webcams during the learning activity, to infer students' emotions using a dedicated software. Happy et al. [START_REF] Happy | Automated Alertness and Emotion Detection for Empathic Feedback during e-Learning[END_REF] propose a system that can estimate both alertness and attention levels from ocular parameters and the emotional state using objective data, mainly visual cues (video).

Most of the proposed dashboards in the educational domain are rather unimodal-based (e.g. [START_REF] Ruiz | Supporting learning by considering emotions: Tracking and Visualization[END_REF][START_REF] Ghasemaghaei | A Dashboard for Affective E-learning: Data Visualization for Monitoring Online Learner Emotions[END_REF][START_REF] Michel | Supporting Awareness and Self-Regulation In Project-Based Learning through Personalized Dashboards[END_REF]). Some tools collect subjective data from questionnaires that are easy to handle but cannot detect emotional reactions at given times during the learning session. Furthermore, unimodal data and subjective questionnaires are not as accurate as global multimodal approaches as explained in the next section. The latter introduces emotion theories and the main tools for automatic emotion recognition.

Emotion Recognition Theories

Emotions represent a complex psychological state of the person. Researchers examining the development of our internal emotional aspects have come to diverse opinions. This diversity leads to different perspectives for emotions.

Discrete and dimensional are the two most important and widely adopted emotion theories. Discrete emotions are a small set of distinct emotions (e.g. anger, disgust, fear, happiness, sadness, surprise), called universal or basic emotions, which constitute the core of all humans' emotions regardless of the socio-cultural factors of the person [START_REF] Ekman | The Facial Action Coding System: A Technique for the Measurement of Facial Movement[END_REF]. Dimensional emotions are rather structured in a dimensional space varying along certain dimensions, such as the degree of the valence representing the positiveness of the emotion felt (varying between positive and negative) and the arousal that represents the physical response and the intensity of that emotional manifestation (varying between low and high) [START_REF] Barrett | Independence and Bipolarity in the Structure of Current Affect[END_REF]. It should be noted that, while other dimensions have been proposed (e.g. 'control' [START_REF] Scherer | CoreGRID and MiniGRID: Development and validation of two short versions of the GRID instrument[END_REF]), valence and arousal are still the most widely adopted dimensions. At the same time, different studies have been derived, mapping one theory into the other (e.g. [START_REF] Scherer | CoreGRID and MiniGRID: Development and validation of two short versions of the GRID instrument[END_REF]).

The interest in emotions has rapidly increased in human-computer interaction applications, more commonly known as affective computing [START_REF] Picard | Affective Computing[END_REF]. Research in psychology, neuroscience and computer science, especially Artificial Intelligence (AI), has been combined to "teach" computers how to infer humans' emotions. Researchers rely on the intensity "emitted" (internally or externally) by the emotion felt in order to infer the person's emotions. Subjective emotions of a person can be collected through self-reporting. Two other approaches are used to collect objective measurements: perception-based measurements include all the manifestations expressed by the person (facial, vocal, gesture, textual, etc.) while physiological measurements include all human body responses (heart-beat, blood pressure, brain activity, etc.) [START_REF] Cernea | A survey of technologies on the rise for emotion-enhanced interaction[END_REF].

The automatic emotion recognition (AER) process can be divided into three main stages: feature extraction aims to filter, extract and normalize the suitable features, feature reduction aims to reduce the high dimensionality of the extracted features, and classification aims to classify emotions using machine learning techniques [START_REF] Konar | Emotion Recognition: A Pattern Analysis Approach[END_REF]. Our research follows these three AER steps to feed the visualizations offered in our dashboard.

Three main features are used for automatic emotion recognition in non-intrusive settings. Facial expression recognition can be based on two techniques: a geometric model-based technique that relies on distinctive facial features such as the position of nose, eyes, mouth, and an appearance model-based technique that considers the face as an array of intensity values (pixels). Thus, facial expressions can be mapped to emotions through theories such as those proposed by [START_REF] Ekman | The Facial Action Coding System: A Technique for the Measurement of Facial Movement[END_REF]. In terms of tools we can cite for example FaceReader 1 and Microsoft Emotion Service API2 that we adopted in our study. Other automatic emotion recognition features are based on voice (sound) and are attracting increasing interest with researchers. The most commonly used features are the prosodic (pitch, intensity and first formant frequency) [START_REF] Lee | Toward detecting emotions in spoken dialogs[END_REF] and the spectral features, (Mel-frequency spectral coefficients (MFCC) [START_REF] Lee | Toward detecting emotions in spoken dialogs[END_REF]. In terms of tools, we can cite for example OpenEAR [START_REF] Eyben | OpenEAR -Introducing the Munich Open-Source Emotion and Affect Recognition Toolkit[END_REF] and Beyond Verbal Emotion Service API 3 that we chose for our study. Textual cues are also considered. However, we do not consider them since the learning activity we observe is not textual-oriented.

The most important current trend in emotion recognition is the combination of different cues (visual, vocal, textual, etc.) for inferring emotions. In fact, many studies (e.g. [START_REF] Caridakis | Multimodal emotion recognition from expressive faces, body gestures and speech[END_REF][START_REF] Pantic | Affective Multimodal Human-Computer Interaction[END_REF]) show that the multimodal approach for measuring people's emotions is more accurate than the unimodal approach. Different strategies exist for data fusion. The first one, called feature level fusion, relies on feature vectors for the different cues that are stacked together and used as input to the classifier. The second strategy is called decision level fusion; each cue is classified separately with its classifier, and the result of each classifier is combined and used as input for a final classifier. We opted for this second strategy.

In our study, the means of communication between the tutor and the learner is a videoconferencing: audio-visual cues are thus the main data channel. We propose to apply both dimensional and discrete emotion theories using several APIs for emotion recognition. We combine these data with self-reported data and interaction traces. We thus aim to acquire rich data on learners' emotions to produce a multimodal and contextual emotional dashboard for tutors.

CONTEXT AND METHODOLOGY

This section is dedicated to the description of the context and the procedure of our study, mainly the learning environment and the experimental settings.

SpeakPlus: a Web-based Learning Environment for Foreign Language Training

SpeakPlus is a commercial Software as a Service (SaaS), a webbased environment dedicated to improving oral communication skills in a foreign language. English, French and Spanish are supported by the platform. The platform connects qualified tutors and learners around the world in a way that the learning activity focuses mainly on the goals set by the learner him/herself (e.g. improving oral presentation skills, preparing for travel abroad). The platform is designed in a modular fashion to be used as a research tool by SpeakPlus researchers' partners, mainly for testing and evaluating new approaches and didactic models in the educational domain.

The learning activity in SpeakPlus can be divided into two main steps. The first step takes place before the learning session: at this level, the tutor can create or customize learning materials (activities) for each learning session by creating new materials or reusing old ones from his/her library. By combining the materials, the tutor can create a learning plan for each session. The learning activity has a duration and can be associated with documents, instructions for the learner or personal notes for the tutor. After preparing the learning materials, the tutor can conduct a synchronous learning session with a learner (see Figure 1). The tutor and the learner communicate in real-time by means of a dedicated videoconferencing technology optimized for online learning. The tutor has the possibility of sharing the prepared materials with the learner (e.g. pdf, word, image, audio, and video) or of communicating via the chat (mainly to point out some expressions, words, etc.).

Experiment Procedure and Participants

Our study is based on authentic and non-intrusive settings. In fact, 3 http://www.beyondverbal.com/api the data were collected using sensors already used in the platform for the learning activity, namely a webcam and a microphone for video and audio recording, respectively. The platform has been slightly augmented to collect and process data (see section 4).

Two graduate female students (in their twenties) agreed to participate in our pilot study. They were contacted after their enrollment in the platform to train in French language speaking for professional goals, more precisely for preparing a job interview. One learner is from Latin-America and the other from Asia. Each learner trained for five learning sessions with the same tutor. The learners had one learning session every week. Each session lasted 45 minutes. Data collection lasted around eight [START_REF] Mega | What makes a good student? How emotions, self-regulated learning, and motivation contribute to academic achievement[END_REF] weeks as the students did not start their learning sessions at the same time. We collected a total of 7 hours and 30 minutes of audio/video data.

EMOTIONAL DATA ACQUISITION AND ANALYSIS

This section presents the first three stages of our research: learners' emotions data acquisition, data exploration, and data analysis, that we applied to produce visualizations on the proposed emotional dashboard for tutors.

Emotional Data Acquisition

Throughout this section we aim to respond to research questions 1) and 2) identified in the introduction. Therefore, different trackable cues have been chosen to investigate how they can be used to track learners' emotions in the SpeakPlus learning environment. We adopted a multimodal approach and identified four data sources: audio, video, self-report, and interaction traces. Audio and video are the primary communication channels in the platform as the learning activity aims at improving oral skills via videoconferencing technology. Self-reported data inform on learners' subjective emotions before and after the learning session. Finally, interaction traces have been used to contextualize the measured emotions and investigate into whether or not there are any particular triggers behind learners' emotions.

According to our multimodal approach, we used heterogeneous cloud APIs for measuring emotions. These APIs were chosen to apply both dimensional and discrete emotions models. The Microsoft emotion recognition Service API (MS API) classifies emotions based on facial expressions (video) according to a discrete emotion model. The Beyond Verbal service API (BV API) classifies emotions based on voice (audio), according to a dimensional emotion model (arousal and valence).

Figure 2 shows the global architecture of the system we have built to collect emotional data during learning sessions. The learning sessions between the tutor and the learner are recorded (audiovisual). Records are stored along with the session in the cloud (AWS S3). At the end of the session, audiovisual records are downloaded from S3 and processed via the BV API and the MS API. The resulting classified emotions are then stored in a database, together with the interaction traces and self-reported data, via a developed API for data collection. The stored data are then processed and aggregated to produce visualizations. The following subsections describe for each data source how the data are collected and stored and how emotions are classified.

Self-Reported Emotion Collection

We developed an interactive interface that enables learners to express their emotion(s) before and after the learning session, according to both dimensional and discrete emotions. The dimensional interface is inspired from the MoodMap tool proposed in [START_REF] Mora | Supporting Mood Awareness in Collaborative Settings[END_REF]. Basically, it proposes a two-dimension (valence and arousal) squared area (Figure 3-a). The learner can click on it to express his/her valence and arousal levels. Four colors were combined in a conic gradient form to represent the four levels: positive and negative valence by green and red colors respectively, and positive and negative arousal by blue and yellow colors respectively. A mouse hover event has been added to track and display the position of the mouse. This information is reflected properly in a mirror bar chart at the bottom of the squared area to guide the learner when reporting his/her twodimensional emotion. Figure 3-b presents the interface that enables learners to report their emotions in a discrete manner. A slider for each emotion (neutral, happiness, surprise, sadness, disgust, contempt, anger) is used to report its intensity on a continuous scale from 1 to 100. An emoticon and a tooltip displaying the emotion name (when the mouse is hovering the slider) are used to distinguish between each emotion. The value of the slider is tracked and displayed in a marker associated with the slider and inside a bubble next to it.

Figure 3. Interface for self-reporting of emotions

In our study, learners were asked to express their emotions both at the start and at the end of the learning session. This emotional information is complementary to the data automatically collected during the learning session. The collected self-reported data are stored in a database as illustrated in Figure 2.

Facial Expression Acquisition and Analysis

The learning sessions are recorded in the SpeakPlus platform. Each session may have one or more archive folder(s) depending on whether or not there were any interruptions during the session. Each archive folder contains two files: one for the learner recorded stream and the other for the tutor recorded stream. An algorithm based on the FFmpeg library (ffmpeg.org) was used to concatenate the video streams from the archive folders when needed. Once the video sequence has been constructed, emotions are classified and stored using a simple algorithm that first uses FFmpeg to extract frames (one frame per second) from the video stream and then calls the MS API to classify the emotions identified for each frame, before storing the result in a database. More precisely, the MS API returns a face entry and its associated emotion scores. An empty response indicates that no face was detected. An emotion entry contains the fields listed in Table 1.

Table 1. Description of MS API response (& JSON example)

Field Description faceRectangle

Rectangle location of the face in the image.

scores Emotion scores for each face in the image; (neutral, happiness, surprise, sadness, disgust, contempt, fear, anger). application/JSON [ { "faceRectangle": { "left": 68, "top": 97, "width": 64, "height": 97 }, "scores": { "anger": 0.00300731952, "contempt": 5.14648448E-08, "disgust": 9.180124E-06, "fear": 0.0001912825, "happiness": 0.9875571, "neutral": 0.0009861537, "sadness": 1.889955E-05, "surprise": 0.008229999 } } ]

Audio Data Collection and Analysis

The same algorithm as for video was used to construct the sound sequence from archive folders (only the learner channel). A call of BV API then returns an array of time interval entries and their associated emotion scores. An empty response indicates that no emotions were detected in the audio file. An emotion entry contains the fields listed in Table 2. The classified emotions are then stored in a database as illustrated in Figure 2. -Value: between 0 and 100. -Group: low, neutral or high application/JSON "analysisSegments": [ { "analysis": { "Arousal": { "Group": "low", "Value": "4.35" }, "Valence": { "Group": "positive", "Value": "82.28" } "duration": 37410, "offset": 4576 } ]

Interaction Trace Collection

We decided to track all actions performed by the tutor and the learner on the SpeakPlus environment during the learning session. This information can be used to contextualize the collected emotional data. We list in Table 3 the events tracked (in real-time) and stored in database using WebSocket.

Emotional Data Exploration and Analysis

This part presents the data exploration and analysis stage of our study. It addresses mainly research question 2), i.e. investigating whether or not audio and video data are correlated. We do not include self-reported information in the analysis because it is not collected at the same time (before and after the session). Throughout this section we answer more specifically the following sub-questions: a) Is there a correlation between audio and video data? and b) What valuable emotional information should be visualized?

Emotional Data Comparison

Use of two different models for emotions (dimensional based on audio and discrete based on video) was the most challenging part of this comparison. Thus, to be able to compare these cues both models have to be unified. Several studies have been conducted in this context to convert discrete emotions into a dimensional model (valence and arousal). We used the valence coordinates of the discrete emotions (neutral, happiness, surprise, sadness, disgust, contempt, fear, anger) as proposed in [START_REF] Scherer | CoreGRID and MiniGRID: Development and validation of two short versions of the GRID instrument[END_REF]. The coordinates provided are on a scale from -100 to 100, from negative valence to positive valence (neutral emotion has a valence equal to zero).

As described in section 3.1.3, audio valence is on a scale from 0 to 100 (as returned by the BV API). For instance, neutral valence is equal to 50. Therefore, scale unification was needed between the coordinates from [START_REF] Picard | Affective Computing[END_REF] and the returned result from the VB API.

To this end, we used the formula (𝑓1) that maps a domain [a, b] interval to a range interval [c, d] to scale VB interval from [0, 100] to [-100, 100].

𝑓 𝑥 = 𝑑 -𝑐 (𝑥 -𝑎) 𝑏 -𝑎 + 𝑐 (𝑓1)
As mentioned in 3.1.2, a successful call to the MS API returns the emotion scores of the frames (images). By combining the coordinates from [START_REF] Picard | Affective Computing[END_REF] and the returned scores, we were able to compute for each frame the corresponding value of the valence. The weighted mean was used for this purpose, as illustrated in the formula 𝑓2 :

𝑣𝑎𝑙𝑒𝑛𝑐𝑒 !"# !"#$% ! = 𝑠𝑐𝑜𝑟𝑒 !"#$% ! ! * 𝑣𝑎𝑙𝑒𝑛𝑐𝑒 ! ! ∈ {!"#$%&',!"##$%&'', !"#$#%!&,!"#$%!!, !"#$%#&,!"#$%&'$, !"#$,!"#$%} (𝑓2) - score !"#$% ! !
: is the score of the emotion i ∈ { neutral, happiness, surprise, sadness, disgust, contempt, fear, anger } in the result returned by the MS API for the frame n (frame ! ) -valence ! : the valence corresponding to the emotion i. -valence !"# !"#$% ! : the weighted valence average (weighted mean) of the frame n.

As described in 3.1.2, a call to the BV API returns an array of time segment entries with their associated emotion scores (valence and arousal). Thus, a simple algorithm has been implemented to map the images extracted from video, every second, to its corresponding audio time segments. Once the images have been grouped, the valence is computed first for each image (with 𝑓2 ).

The average over images belonging to each time segment is then computed.

Emotional Data Correlation

This section responds to the sub-question a). Once the two data models were unified, the aim was to investigate whether there is any correlation between both variables: audio and video valence.

There are several coefficients for measuring dependencies between variables. Spearman's and Pearson's coefficients are those most used in the literature. However, they can detect only linear or non-linear monotonic correlations as stated in [START_REF] Posnett | Check: A Correlation Tactic for ESE Data[END_REF].

Conversely, the Maximal Information Coefficient (MIC) may be more powerful as it can detect several associations between variables, such as linearity, nonlinearity, asymmetry and even non-functionality [START_REF] Posnett | Check: A Correlation Tactic for ESE Data[END_REF]. Table 4 presents the correlation coefficients (MIC, Spearman, and Pearson coefficients) of audio and video valence. Regarding the MIC coefficient, the variables (audio and video valence) are correlated (contrary to Pearson and Spearman). We deduce that there is no clear linear correlation between the video and audio variables, but there must be a correlation between the peaks (either positive or negative) of both variables, which is a non-functional correlation that the MIC can detect. At this stage, an exploratory analysis using audio-video records was necessary to further investigate and understand this non-functional dependency between both cues.

Emotional Data Exploration

To explore audio and video valences, we draw a line chart for the ten learning sessions. The result reveals an explicit dependency between both valences. Figure 4 and Figure 5 are examples of one learning session valence timeline. The x-axis represents the timesegment indices, while the y-axis represents the corresponding valence score between -100 and 100. As annotated with blue color rectangles, many apparent similarities and dissimilarities exist between the peaks in audio and video valences.

Figure 4 presents some examples of dissimilarities comparing raw audio and video data for one learning session. Table 5 presents the associated interpretations and comments for the first four segments. The learner was speaking loudly and confidently. Thus valence increased.

Figure 4. Audio and video valence -examples of dissimilarities

The majority of images have been classified as 'neutral'. Thus valence decreased.

Neutral was the dominant emotion for the video while the audio indicated a high level of valence which is not reflected on the face.

Similarly, Figure 5 presents some similarity examples comparing raw audio and video data for the same learning session as above. Table 6 presents the associated interpretations and comments for the first two segments in the figure. The remainder of the segments is more or less similar to the commented segments.

Figure 5. Audio and video valence -examples of similarities

The comparative analysis conducted in this section reveals some interesting points. First, by adopting different models of emotions and using heterogeneous APIs for emotion recognition, we were able to identify many similarities and dissimilarities between the measured emotions from both audio and video data. This result underscores a dependency between these cues. Second, as described in Table 5 and Table 6, audio data can reveal some particular levels of valence that video data fail to detect and vice versa. We deduce that both cues should be considered for inferring learners' emotions. The learner was obliged to interrupt the session in order to talk to a member of his/her family.

'Neutral' emotion

This proves that the result might be more accurate when both cues are combined.

31

There was a joke in this segment that leads to positive valence, as well as a peak in the audio valence.

Only one image has been classified as 'happiness'.

This also proves that the result might be more accurate when both cues are combined.

Emotional Data Integration

This section responds to sub-question b) and tends to aggregate data to create information for the visualizations offered in the dashboard. The amount of data collected for each session is huge, and only a subset of these data is likely to constitute an effective emotional state. Hence, both dimensional and discrete emotions need to be filtered and synthetized to keep only valuable emotional reactions.

Concerning dimensional emotions (based on audio), we use arousal (intensity of the emotion) to filter the valence of audio data. A list of arousal thresholds [START_REF] Ghasemaghaei | A Dashboard for Affective E-learning: Data Visualization for Monitoring Online Learner Emotions[END_REF][START_REF] Ekman | The Facial Action Coding System: A Technique for the Measurement of Facial Movement[END_REF][START_REF] Posnett | Check: A Correlation Tactic for ESE Data[END_REF]40,50) was defined. When a threshold is selected, only the time-segment entries with the highest corresponding arousal are considered of interest during a learning session.

Concerning discrete emotions (based on video), time series analysis was used to synthetize the collected emotions. First, the set of emotions (neutral, happiness, surprise, sadness, disgust, contempt, fear, anger) is divided into three categories: neutral, positive (happiness, surprise) and negative (sadness, disgust, contempt, fear, anger). Second, the scores of these emotions are computed as the sum of the corresponding emotions for each category and so, for each image (frame), (see

(𝑓3) (𝑓4) (𝑓5)).
Third, once the scores have been computed for each image, we identify the dominant type of emotion (neutral, positive, and negative) as the most frequent over all the frames of the segment.

𝑠𝑐𝑜𝑟𝑒 !"#$%&' !"#$% ! = 𝑠𝑐𝑜𝑟𝑒 !"#$% ! !"#$%&' (𝑓3) 𝑠𝑐𝑜𝑟𝑒 !"#$%$&' !"#$% ! = 𝑠𝑐𝑜𝑟𝑒 !"#$% ! ! ! ∈ {!"##$%&'', !"#$#%!&} 𝑓4 𝑠𝑐𝑜𝑟𝑒 !"#$%&'" !"#$% ! = 𝑠𝑐𝑜𝑟𝑒 !"#$% ! ! ! ∈ {!"#$%!!, !"#$%#&,!"#$%&'$,!"#$,!"#$%} 𝑓5 - score !"#$% ! !
: is the score of the emotion i ∈ { neutral, happiness, surprise, sadness, disgust, contempt, fear, anger } in the result returned by the MS API for the frame n.

We then identified two patterns: the positive moments and the negative moments (neutral is dropped as it may not be interesting for learner monitoring). Thus, two measurements have been defined, 'True positive' and 'True negative', to track these patterns. 'True positive' detects the moments where the average of positive points is highest or equal to a fixed threshold. 'True negative' is defined to detect moments where negative emotions are defined as significant, i.e. the gap between the average of negative points and the maximum score of the remaining emotions is less than or equal to a fixed threshold. The detected positive/negative moments for both dimensional and discrete emotions are considered as the most important time-segments that are more likely to represent effective emotional reactions during the learning session, hence, they will be used to power our visualizations described in the next section.

EMOTIONAL DATA VISUALIZATION

This section addresses our research question 3). We defined four design principles (DPs) to build easily readable and understandable visualizations: dp1) Unified design, dp2) Selfexplanatory, dp3) Easy to use and dp4) Interactive design. By adopting these principles, we aim to make the dashboard interface intuitive and thus increase the learning curve.

Dashboard Overview

The emotional dashboard we propose is illustrated in Figure 9. It is a tutor oriented web-based application. The tutor connects to the dashboard by selecting a learner from his/her learners' list. The dashboard is then powered by the data for the selected learner. The dashboard is designed so as to present information from general-to-specific at different levels of abstraction: 1) Global information about the learning session reflects in a summarized manner the emotional state of the learner during the learning session, combining both subjective and objective emotions; 2) Timeline information integrates contextual information to illustrate the learning session with the emotions experienced by the learner along the learning activity; 3) Timesegment information is designed to provide more details about each positive or negative emotion identified in the timelines.

The dashboard consists of a toolbar at the top and a (see Figure 9). The toolbar provides quick access to the filtering functions. The tutor can select in the drop-down menu a learning session corresponding to the selected learner. The tutor can also select an arousal threshold in the drop-down menu. The toolbar is also used to quickly show insights about learners' emotions during the session by displaying the number of positive and negative emotions in the form of markers. The body of the dashboard groups all the emotional visualizations related to the selected session. When the session changes, all the visualizations in the body are updated accordingly.

The top level of the body is designed to convey emotional insights about the whole learning session. It is divided into two parts: the left part presents the emotions expressed by the learner (selfreported) before and after the learning session, while the right part presents the average of the emotions measured over the session. The middle part of the interface provides details on a selected time-segment. It is divided into two parts: the left part presents contextual information related to the time-segment, in particular learners' facial expressions recorded during the learning session and the audio record, while the right part displays quantified information on associated emotions. The bottom part is composed of three timelines, where the degree of detail increases from top to bottom. On the top, bubbles represent the positive and negative moments. In the middle, we placed contextual and emotional information in a multiline-based chart. At the bottom, a segmentbased chart presents the top three emotions for each time-segment.

Color is an important abstraction widely adopted in visual design and particularly in emotional visualizations (e.g. [START_REF] Ruiz | Supporting learning by considering emotions: Tracking and Visualization[END_REF][START_REF] Ghasemaghaei | A Dashboard for Affective E-learning: Data Visualization for Monitoring Online Learner Emotions[END_REF]). Hence, we decided to define a color scheme for our visualizations. Figure 6 presents three sets of colors: the first for discrete emotions (neutral, happiness, surprise, sadness, disgust, contempt, fear, anger), the second for what we call markers: positive/negative emotions and events (chatting and sharing documents between tutor and learner) and the third for the audio/video valence.

Global Information Level

The left part of the global level concerns the subjective information expressed by the learner. As explained in section 3.1.1., learners can report their emotions immediately before the beginning of the live learning session and just after its end. Figure 7 presents the implemented visualization of both dimensional (valence and arousal) and discrete emotions at both times. This representation facilitates the comparison between the two times (before and after). Two different charts are used to visualize selfreported dimensional emotions and discrete emotions. A pie donut chart is adopted to visualize dimensional emotions, where the corresponding name and value (on a scale from 0 to 100) are placed inside the pie chart. Chart color is green or red, depending on whether the value of the emotion is positive or negative, respectively. As the learner can choose several discrete emotions, a light bar-based chart is adopted to represent them. Each bar represents an emotion, and the width corresponds to the value of the emotion as expressed by the learner on a scale from 0 to 100. The emotion name and value are displayed on the top of each bar. The right part presents the average information of the inferred emotions (using APIs). Three visualizations are proposed (see Figure 8), ranging from the most general (on the left) to the most detailed (on the right). The visualization in Figure 8-(a) is based on a multi-arc pie donut chart. It presents the ratio of each type of discrete emotion, corresponding to the abstraction of positive, negative and neutral emotions as described in section 4.2 (emotional data integration). Each type is represented by a color (red, green and blue); the percentage is also displayed with

Timeline Level of the Learning Session

Timeline visualizations offer more explanatory (dp1) representations of emotions as they incorporate the time notion of the learning session. Our main timeline visualization is based on the analysis described in part 4.2. As illustrated in Figure 10, a specific timeline chart is implemented to combine both discrete and dimensional emotions. We represent positive/negative markers for positive and negative time-segments, respectively. We choose a bubble shape to represent discrete emotion markers and a star shape to represent dimensional emotions. The size of the bubbles and stars depends on the computed score for each emotion which is also represented with the height of the marker.

Figure 10. Positive/Negative markers visualization

The duration of each marker (time-segment) is added as a small horizontal bar over the x-axis, which represents learning session time in minutes. As mentioned above, a green color is chosen for positive markers and a red color for negative ones. When the session or arousal threshold changes (using drop-down menus in the toolbar), the visualization is updated accordingly. Integrating both markers (dimensional and discrete) in the same timeline facilitates the comparison between the emotions from both audio and video data sources.

Figure 11 presents a second timeline displaying only the top three discrete emotions associated with each marker. The x-axis represents the time in minutes, and each emotion is represented by a horizontal bar with the associated color. The width of the bar corresponds to the duration of the time-segment, while the position of the bar in the y-axis corresponds to the emotion score on a scale from 1 to 100. This visualization simplifies the comparison between the top three emotions.

Figure 11. Top 3 discrete emotion visualization

The third timeline visualization (see Figure 12) allows the tutor to monitor the discrete (top three) and the dimensional (audio/video valence) emotions over time with the associated contextual information (events). We collected two main events: chatting and document sharing (pdf, audio, video or image). Emotions are represented by lines with different colors. Events are added to the top of the chart using bubbles as markers, with different icons to distinguish events by their types. The duration of the event is shown by a horizontal bar and two small icons (eye, eye-slash) placed at both ends of the bar (showing the start and end of document sharing). This setting not only allows better monitoring of learners' emotions but also correlation between emotions and the events and activities that may have triggered these emotions. For instance, we can observe on this chart that the learner was greatly surprised at ~ 14 minutes while sharing an image.

By providing tutors with different levels of abstraction of learners' emotions and contextualized information on these emotions, we aim at helping them easily identify specific emotional moments and reactions in the learning sessions (both positive and negative) and explore the reasons (context) of such moments.

Time Segment Detail Visualization

We provide tutors with details on emotional information corresponding to each particular time-segment (marker). These visualizations offer a detailed view of learners' emotions at a given moment selected in the timeline described above. The dashboard is interactive (dp4): when the mouse hovers a marker in the timeline. The detailed visualizations are updated accordingly with the data related to the hovered marker.

To compare time-segment information and global information, the same visualization presenting the average of the objective emotional information was used and stacked vertically (see Figure 9). We also help tutors better understand quantified information on learners' emotions by showing either facial expressions and the sound corresponding to each time-segment (see Figure 9). Images are updated accordingly when the mouse hovers the markers (see Figure 10). Likewise, the tutor can listen to the sound of a selected time-segment by clicking on its associated marker. To know whether or not the learner is talking, we implemented a bar-based chart to display audio frequency with green or red colors (depending on whether the selected marker is positive or negative). Audio-visual contextual information is placed at the same level as time-segment details visualizations.

Pilot Study on Tutors' Perception

We conducted an initial study in order to evaluate the EMODA dashboard and obtain feedback on how tutors perceive its usability and utility as recommended in [START_REF] Lam | Empirical Studies in Information Visualization: Seven Scenarios[END_REF]. This pilot evaluation was conducted with two tutors using the SpeakPlus platform. For this purpose, the evaluation was conducted in three steps. The first step consisted of a case study with 8 associated questions (e.g. 'At what time did the learner feel a peak of fear during the learning session?'). The second step consisted of 6 questions based on a 7-Likert Scale regarding global perception on the dashboard (e.g. 'How do you find the display and organization of data on the screen?', 'To what extent do you think the dashboard will help you as a tutor to improve the learning experience?). The third step consisted of 7 grid questions with 7-Likert scales for each visualization (e.g. 'How do you find the learner image carousel displayed in part (II-a)?'). We added to each question a text area requesting more explanations.

The answers to the questionnaire associated with the case study confirmed that the dashboard is easy to use, as the tutors were able to answer all questions correctly without any help. Regarding global perception, it was clear that our dashboard has a problem in displaying timeline visualizations. Tutors state that 'reading is a little bit difficult' and 'Conversely, timeline visualizations were not rapidly comprehensible... for me it would be better to keep only one simplified visualization'. This suggests that one simplified visualization of the three timeline visualizations would be easier to understand, for instance by adding a tooltip with the mouse hover events to the marker timeline (Figure 10). This apart, the perception of the dashboard was rather positive. From the last part of the questionnaire, we note that the idea of combining emotions and events was very interesting for both tutors. In fact, they suggested further "promoting" this visualization: "should be better valued", "the idea of combining events and emotions is interesting, but the chart is not big… it's a good idea but should be simplified". Therefore, more focus should be given to the contextualization part as it provides essential understanding of the emotions experienced by learners.

CONCLUSION AND FUTURE WORK

Our initial investigations prove the feasibility of using heterogeneous APIs for emotion recognition in online learning environments. By applying both dimensional and discrete emotion models, we showed that we can detect different learners' emotions during such settings. This highlights the need for a multimodal approach to emotion collection and recognition. We believe that providing tutors with an awareness tool to visualize and track their learners' emotions will help them better maintain a sustainable socio-affective relationship as well as adjust learning activity materials accordingly. Furthermore, our study has a key side benefit for the designers of the SpeakPlus platform. The feedback on learners' emotions can help them for instance monitor the performance of the learning sessions and learners' satisfaction.

In this study we opted for a post-processing (off-line) approach for emotion tracking, analysis and visualization to build a first prototype of our dashboard based on real data. Through an iterative design process, the pilot study will help us to make the dashboard evolve according to tutors' feedbacks and needs. In fact, the current architecture should be altered to offer a real-time feedback to tutors in order to optimize the learning session and make it more enjoyable.

Our study is learner-centric. As such, we also think it is important to investigate whether there are correlations between the emotions felt by the tutor and by the learner. Furthermore, our study focuses on universal discrete emotions and dimensional emotions (arousal and valence). Several recent studies (e.g. [START_REF] Pekrun | Measuring emotions in students' learning and performance: The Achievement Emotions Questionnaire (AEQ)[END_REF]) show that the learning context is more concerned with domain-specific emotions (e.g. confused, bored, frustrated, pride). Thus, investigating such emotions or combining them with universal ones might be a promising research issue for future work.

Figure 1 .

 1 Figure 1. Live learning session on the SpeakPlus platform

Figure 2 .

 2 Figure 2. EMODA global architecture

Figure 6 .

 6 Figure 6. Color Scheme

Figure 7 .Figure 8 .

 78 Figure 7. Self-reported emotion visualization

Figure 9 .

 9 Figure 9. An overview of the EMODA dashboard. A positive marker is selected in the first timeline

Figure 12 .

 12 Figure 12. Contextual event and emotion visualization

Table 2 . Description of BV API response (& JSON example)

 2

	Field		Description
	analysisSegments	The array containing analysis segments.
	Offset	The offset of the segment from the beginning of
		the sample being analyzed (in ms).
	duration	Duration of the analysis segment in the sample
		being analyzed.
	analysis	Analysis object. Contains analysis values for the
		segment. The content of the object is provided as
		an example.
	valence	Valence object score (has value and group):
		-	Value: a value between 0 and 100.
		-	Group: positive, neutral or negative.
	arousal	Arousal object score (has value and group).

Table 3 . List and description of events collected in interaction traces with the SpeakPlus environment

 3

	Field		Description
	action_name	The name of the action (event) triggered
		between the actors, it can be:
		-	SHARING_DOC: the tutor starts sharing a
			document
		-	STOP_SHARING_DOC: the tutor stops
			sharing a document
		-	ACTIVITY_TASK: an action sent to the
			learner by the tutor
		-	FREE_TEXT: message chatting
		-	SHOW_DOC: the learner enables display
			of the shared document
		-	HIDE_DOC: the learner disables display
			of the shared document
	action_content_type	The type of event:
		-	PDF: Pdf document sharing
		-	AUDIO: Audio file sharing
		-	VIDEO: Video file sharing
		-	IMAGE: Image file sharing
		-	TEXT: Text action sent to the learner by
			the tutor
		-	SHOW_HIDE_DOC: for the
			SHOW_DOC or HIDE_DOC action
	activity	Activity Id
	item	Item Id in the activity (an activity might have
		one or more items)
	document	Document Id (pdf, audio, video, etc.)
	chat_message	Message-Id

Table 4 . Correlation coefficients between Audio / Video valence -MIC vs. Pearson vs. Spearman

 4

			Coefficients
		MIC	PEARSON-R	SPEARMAN-R
	Valence Audio Vs Video	0.787	0.052	0.172

Table 5 . Interpretation of valence dissimilarities (audio/video)

 5

	Index	Audio	Video	Comments
	8	The correction of	Almost half of	Even if the learner's
		a mistake by the	the images have	voice showed regret,
		tutor triggers a	been classified	there was no sign of
		regret reaction in	as 'happiness'.	this on his/her face.
		the learner. Thus	Thus valence
		valence	increased.
		decreased.
	13	A kind of 'aaaah'	Neutral emotion	Audio might be a
		expressed by the		better indicator for
		learner. Thus		some specific
		valence		emotional vocal
		decreased.		expressions.
	55	The learner's	Many images	The learner was
		voice was calm	have been	smiling, which was a
		(neutral).	classified as	sign of 'happiness',
			'happiness'.	but a calm voice was
			Thus valence	a sign of 'neutral' for
			increased.	audio.
	91

Table 6 . Interpretation of valence similarities (audio/video)

 6

	Index	Audio	Video	Comments
	25

http://www.noldus.com

https://www.microsoft.com/cognitive-services/en-us/emotion-api

Acknowledgements

This research is conducted within the EmoViz project funded by the Région Auvergne-Rhône-Alpes. We thank the SpeakPlus (https://speakplus.fr) company for funding this research work. We also thank the learners and tutors who participated to the study.


Keywords:

In numerical simulations, the transfer of fields between the different meshes is a key step. Interpolation is probably the most popular method for transferring data between meshes. Actually, it must ensure the consistency, the continuity and the accuracy of the solutions among the meshes. We present in this research report several conservative interpolation methods from a donor mesh to a target mesh. We start by introducing the basics of the transfer of fields, then we present the common mapping methods used nowadays and finally we give a detailed comparative study of the conservative interpolation methods found in the literature.

Introduction

Thermal Treatment describes the multifaceted operations in heating furnaces and quenching tanks, performed on a material in the solid state, for the purpose of altering its microstructure and properties. The output of this step is the input of all the following manufacturing steps such as forging, rolling processes and even the prediction of microstructure evolution. Therefore, any lack of control in this upstream operation will affect the global manufacturing chain, and the consequences are then immediate such as prohibiting better quality, higher availability and adaptability of products. In particular, during quenching process, the boiling phenomena taking place is a concentrate of various physical phenomena (see Fig 1) that cannot be modeled as a simple global heat exchange coefficient. The process realism implies an advanced fluid-solid coupling. This subtle step needs to be efficiently reproduced to reach a predictive and exploitable numerical result.

Figure 1: Different steps of the quenching process Thus, the main purpose of this project is to develop numerical tools in order to simulate the evolution of both the thermal and mechanic properties of the immersed solid during the quenching at an industrial scale. This project will be done in collaboration with Montupet, an industry leader in the manufacture of complex cast aluminium components for the automotive industry worldwide. Recall also that the developed framework will be validated using experimental results given by Montupet.

The initial approach is to consider two domains, see Fig 2. First of all, the fluid-solid domain in which the solid is totally immersed in the fluid. For that, we will refer to the use of our immersed volume framework [START_REF] Hachem | Modeling of heat transfer and turbulent flows inside industrial furnaces[END_REF]. Indeed, a full Eulerian framework that simulates the quenching process has been established. It takes into account three ingredients that can be resumed to : (i) geometric : flexibility for multidomain simulation,(ii) fluid mechanics: accounting for turbulent boiling, (iii) physics : phase change and phase transformation, [START_REF] Hachem | Stabilized finite element solution to handle complex heat and fluid flows in industrial furnaces using the immersed volume method[END_REF]. As for the solid part, we recall that the quenching leads to important deformations such as cracks and defects inside the solid, so we want to study the effect of the residual stresses. Montupet is using Zset, a software dedicated to the solid that is able to simulate the heat treatment of mechanical pieces with phase change and analyze the residual stresses. The fluid mechanical part and the turbulent boiling will be handled during my PhD and implemented in the library CimLib-CFD, that handles different fields of applications : finite element solvers for heat transfer simulations ( [START_REF] Brogniez | A high-order discontinuous Galerkin method with Lagrange multipliers for advection diffusion problems[END_REF], [START_REF] Hachem | A three-field stabilized finite element method for fluid-structure interaction: elastic solid and rigid body limit[END_REF]), anisotropic mesh adaptation ( [START_REF] Mesri | Automatic coarsening of three dimensional anisotropic unstructured meshes for multigrid applications[END_REF], [START_REF] Hachem | On the stabilized finite element method for steady convection-dominated problems with anisotropic mesh adaptation[END_REF], [START_REF] Mesri | Gestion et contrôle des maillages anisotropes non structurés : Applications a l'aérodynamique[END_REF], [START_REF] Mesri | On optimal simplicial meshes for minimizing the Hessian-based errors[END_REF], [START_REF] Jannoun | Anisotropic meshing with timestepping control for unsteady convection-dominated problems[END_REF]) and high parallel computing ( [START_REF] Mesri | Hierarchical adaptive multi-mesh partitioning algorithm on heterogeneous systems[END_REF], [START_REF] Mesri | Parallel Adaptive Mesh Refinement for Capturing Front Displacements : Application to Thermal EOR Processes[END_REF]). Thus, a coupling between the two codes and therefore interpolation of fields and fluxes of temperature must be done in an accurate and conservative manner. The first step of this project consists in transferring the fields of the first fluid-solid domain to the solid domain. Indeed, the two domains are completely different and interpolation is required. Moreover, during adaptive numerical simulations, we use the interpolation of the fields not only when we wish to transfer from a donor mesh to a target mesh, but also when we adapt the mesh following the evolution of the physical properties. However, one serious drawback of many mesh adaptivity algorithms on unstructured meshes is the necessity of interpolating solution fields from the initial mesh to the newly adapted mesh. Such interpolation destroys conservation of important physical quantities and leads to errors in the solution fields. Thus, this bibliographic report will investigate several conservative interpolation methods from a donor mesh to a target mesh. We will start by introducing the basics of the transfer of fields, we will present the common mapping methods used nowadays and then give a detailed comparative study of the conservative interpolation methods found in the literature.

Common mapping methods

In numerical simulations, the transfer of fields between the different meshes is a key step. Interpolation (and sometimes extrapolation when a point falls outside the range of the source mesh) is probably the most popular method for transferring data between meshes. Actually, it ensures the consistency, the continuity and the accuracy of the solutions among the meshes. While resolving a mechanical problem, we distinguish for instance the nodal fields of type P1 such as velocity, pressure and temperature, and P0 fields defined at the Gauss points (stress and strain tensor), see Fig 3. While resolving the constitutive equations, we need to transfer the solutions which are defined on various discretisations using the transfer of fields.

Linear interpolation

The most common interpolation functions are polynomial because they are easier to integrate and to derive, unlike the trigonometric functions that lead to additional computational time [START_REF] Ern | Elements finis : théorie, applications, mise en oeuvre[END_REF]. The choice of the interpolation polynomial is based on the type and the degree of the finite element defined in the initial mesh. Most of the time, the interpolation polynomials are defined throughout the shape functions of the elements but are not necessary of the same degree of the element at issue. Nowadays, the standard interpolation method is the linear interpolation. It is a two-step procedure defined as follows : first of all, one needs to determine the position of the new node in the old mesh; for that, several localization algorithms can be used and will be detailed in the annex of this report (section 5). For each node p T ∈ T T , a containing element K D is identified in the donor mesh T D using the localization algorithm, and then the solution u D is evaluated at the physical location of the target node p T . The linear interpolation uses the source functions, it assigns the value at the node p T of the target mesh to be :

Π 1 u(p T ) = ∑ i Φ i (p T )u(p i ) (1)
with Π 1 the P1-interpolation operator, Φ i the shape functions associated with the source mesh and u(p i ) the solution defined in the nodes of the element that contains the new vertex.

Moving least-squares approximation

The moving least square method developed by [START_REF] Breitkopf | Consistency approach and diffusive derivation in element free methods based on moving least-squares approximation[END_REF] is a method for reconstructing continuous functions from a set of unorganized point samples via the calculation of a weighted least squares measure around the point at which the reconstructed value is requested. The approximation of the exact solution at a point is defined based on its nodal values at a limited number of neighbor points : only the closest points to the current optimum are taken into account. It is computed as follows:

u app (x) = p T (x)a(x) (2)
with p the basis functions and a the adjusting coefficients that are computed by minimizing a norm of the weighted difference between the estimated values at nodes and the nodal values u i (see [START_REF] Breitkopf | Moving least squares response surface approximation : Formulation and metal forming applications[END_REF] for details):

J(a) = ∑ i w i ( x i -x )(p T (x i -x)a -u i ) 2 (3)
The basic principle of this method is that the influence of a node is governed by a decreasing weighting function w i , that is equal to zero outside the domain of influence of the node. The weight function plays an important role, it gives a local character to the approximation by influencing the way that the coefficients a i depend on the location of the designed point x.

Patch recovery methods

In finite element method, the use of numerical integration, approximation and interpolation leads to an accumulation of the errors. We recall the concept of superconvergent patch recovery methods (SPR) that was first introduced by Zienkiewicz and Zhu [START_REF] Zienkiewicz | The superconvergent patch recovery (SPR) and adaptive finite element refinement[END_REF], [START_REF] Zienkiewicz | Superconvergence and the superconvergent patch recovery[END_REF] in order to estimate the errors made when using a finite element method. The use of a unique interpolation polynomial coupled with a patch recovering procedure was suggested. The basic principle of this approach consists in recovering the value of the nodal fields by least square fit method and then interpolating the nodal values using standard shape functions. For that, an element patch is defined at each node (see Fig 5), it contains all the elements to which the node belongs to. For each node, an improved solution is computed by determining a polynomial expansion over the patch. However, for a given patch, only the values on the integration points are conserved; the derivatives values over the boundaries are not taken into account. For that, when an interpolation point is located in the intersection between two patches, a weighted average procedure of the different values must be done. Moreover, an error estimate is defined in order to minimize the error between the finite element solution gradient and the new improved solution obtained using a least square projection of the gradient. A study developed by Zhang [START_REF] Zhang | Analysis of the superconvergent patch recovery technique and a posteriori error estimator in the finite element method (I)[END_REF] shows that the patch recovery technique of Zienkwicz and Zhu gives ultra convergent results when finite element spaces have the same order and local uniform meshes are used. Over the years, several techniques inspired from the SPR method were proposed. For instance, the Recovery by Equilibrium in patches (REP) proposed by Boroomand and Zienkiewicz [START_REF] Boroomand | Recovery by equilibrium in patches (REP)[END_REF], which avoids the use of superconvergent points and uses a weighted form of equilibrium equation to produce recovered solution. Gu and al. [START_REF] Gu | A modified superconvergent patch recovery method and its application to large deformation problem[END_REF] increased the robustness and the accuracy of the SPR approach for non linear problems using integration points as sampling points and introducing additional nodes. The stress recovery method was later used to recover nodal fields from integration points in order to transfer data between a donor and a target mesh [START_REF] Khoei | The superconvergence patch recovery technique and data transfer operators in 3D plasticity problems[END_REF]. Kumar and Forment [START_REF] Kumar | Remapping method for transferring data between two meshes using a modified iterative SPR approach for parallel resolution[END_REF] proposed a consistent technique easier to implement in a parallel environment that deals with the boundary points with the same order of accuracy as the interior points.

Conclusions and remarks

In fluid-solid interaction applications, when we wish to interpolate a field, the data transfer must be numerically accurate and physically conservative. However, the mapping methods listed above suffer from many drawbacks. First of all, conservation : the integral of the interpolant on the target mesh is not the same as the integral of the field on the donor mesh :

T D u = T T π 1 u (4)
Due to the accumulation of numerical errors during interpolation, the conservation of mass and energy is not necessarily respected which is crucial for industrial applications.

Secondly, the maximum and the minimum of the solution will be lost during interpolation :

min q∈T D u(q) ≤ Π 1 u(p T ) ≤ max q∈T D u(q) (5)
Therefore, we will focus in the following section on the various methods found in the literature that answer the problem of conservation and maximum principle.

Conservative interpolation methods

Several conservative interpolation operators that preserve the global integrals of the solution fields can be found in the literature. Some of them require building an auxiliary mesh, either as the intersection between the elements of the different meshes or as the union of the donor and target meshes in order to facilitate the use of projection operators. Some techniques depend upon the use of a weighted averaging procedure and others on the resolution of an optimization problem.

Conservative interpolation via intermediate mesh building

Many conservative interpolation methods based on the use of an auxiliary mesh can be found in the literature, in the following section we decide to detail the most recent methods : the P1-conservative algorithm and the one based on the construction of a supermesh.

P 1 -conservative technique by mesh intersection

In [START_REF] Alauzet | P1-conservative solution interpolation on unstructured triangular meshes[END_REF], the author proposes a P1-conservative interpolation operator that satisfies the maximum principle. This operator is based on local mesh intersections. The approach begins with the localization procedure based on the barycentric coordinates presented in the previous section, then uses the mesh intersection algorithm. It is a local procedure that consists in computing the intersection between an element of the target mesh and all the elements of the donor mesh that it overlaps. This algorithm gives us a precise intersection list that contains for each element of the new mesh, all the triangles of the background mesh that it overlays. For each couple of triangles K T and K j D , a new mesh of the intersection with the following definition is introduced :

T j = K T ∩ K j D (6)
The solution is piecewise linear by element. For each triangle of the donor mesh, we know the values of:

• The mass m K D = K D u
• The constant gradient of the solution ∇u K D Hence, we use a quadrature formula in order to compute the exact quantity of mass and the value of the gradient of the element K T on the target mesh :

m K T = K T π c 1 u = ∑ j T j u (7) (∇π c 1 u) |K T = ∑ j T j ∇u | K T | ( 8
)
This scheme is P 1 -exact, respects the mass conservation and fulfills the maximum principle by reconstructing the mass field and its gradient with the elemental intersections between both the donor and target mesh, see [START_REF] Alauzet | A parallel matrix-free conservative solution interpolation on unstructured tetrahedral meshes[END_REF] for more details.

The obtained numerical results show the influence of the conservative interpolation method, see Fig 6. The properties of this algorithm have been verified numerically on analytical examples and adaptive simulations. While preserving the mass, the results with the P1-conservative interpolation scheme are more accurate and ensure better conservation of the mass than the ones obtained with the classical linear interpolation. However, the difficulty of this algorithm lies in the construction of the list of intersections : many degenerated topological cases can be encountered.

Interpolation via common-refinement or supermesh construction

Another recent technique is the construction of a common-refinement or supermesh. Grandy [START_REF] Grandy | Conservative remapping and region overlays by intersecting arbitrary polyhedra[END_REF] proceeds by mapping from donor to target mesh by calculating the intersection volume of overlapping polyhedra between the donor and the target mesh, as opposed to Bailey [START_REF] Bailey | An algorithm for the conservative interpolation of data between two-dimensional structured or unstructured triangular meshes[END_REF] that approximates the solution over the area of intersection using a Galerkin projection. One of the main fields of application of conservative interpolation is the transmission of loads between interfaces, for instance in coupled problems such as fluid structure. Cebral and Lohner [START_REF] Cebral | Conservative load projection and tracking for fluidstructure problems[END_REF], proposed the conservation of the load along the interface using a nodeprojection scheme, whereas Jiao and Heath [START_REF] Jiao | Common-refinement-based data transfer between non-matching meshes in multiphysics simulations[END_REF] proposed a method also based on the use of an auxiliary mesh that is highly recommended by Jamain et al [START_REF] Jaiman | Assessment of conservative load transfer for fluid-solid interface with non matching meshes[END_REF]. The idea is to define a common-refinement which is composed of the intersection of the fluid and solid meshes along the interface, the algorithms used for surface meshes are described in [START_REF] Jiao | Overlaying surface meshes (part 1): algorithms[END_REF]. It is an accurate, conservative data transfer algorithm based on the use of a common-refinement of two non matching surface meshes in order to allow an efficient Galerkin projection of the results, which minimizes the L 2 -norm of the interpolation error, see [START_REF] Jiao | Common-refinement-based data transfer between non-matching meshes in multiphysics simulations[END_REF]. Several methods are compared for transferring data once the common-refinement has been constructed, and it has been shown that the common-refinement based scheme with L 2 minimization on linear basis functions leads to errors which grow more slowly with iteration number compared to other interpolation methods. On the other hand, Farell [START_REF] Farrell | Conservative interpolation between unstructured meshes via supermesh construction[END_REF] was the first to present a minimally diffusive bounded interpolation algorithm between unrelated unstructured meshes that ensures the maintenance of the maximum principle and mass conservation. It is based on building a supermesh as the union of the original and the target mesh. The purpose of this union is to facilitate the use of projection operators as mesh to mesh interpolators. An example of the construction of a supermesh is shown in Fig 7 . The supermesh must respect the following conditions :

N S ⊇ N D ∪ N T ( 9
)
with N S the nodes of the supermesh T S , and N D , N T the nodes belonging to the parent meshes, respectively T D and T T . By definition, if a node belongs to a parent mesh it has to be in the supermesh; and for each element K S ∈ T S , the intersection of K S with any element of the parent meshes, has to be either empty or the element itself. The utility of this supermesh is that it provides a decomposition of elements in T D and T T as elements in T S . This is done using mapping functions χ SD and χ T S that allow to determine the parent element of K S ∈ T S , in other words, whether it belongs to the donor or the target mesh, and respectively the children, see [START_REF] Farrell | Conservative interpolation between volume meshes by local galerkin projection[END_REF] (section 2) for more details. Let us recall that u is the function whose integral need to be conserved :

T D u = T T π c u (10)
Let us use a discrete form of eq. ( 11) over the meshes T D and T T :

∑ K D ∈T D K D u = ∑ K T ∈T T K T π c u (11)
We express K T using the mapping functions, so the problem reduces to interpolating from T D to T S in a conservative manner such that :

∑ K D ∈T D K D u = ∑ K S ∈T S K S π c u (12)
We have the elemental integrals of the solution, the last step is to compute its nodal values by means of a local Galerkin projection, [START_REF] Farrell | Conservative interpolation between volume meshes by local galerkin projection[END_REF]. We obtain the coming equation :

M T u T = M T D u D (13)
with u T , u D and M T respectively the discrete solutions on the target and donor meshes, and the mass matrix composed of the basis functions of the target mesh. The difficulty arises in computing the right hand-side of the equation ( 14). Indeed, the mixed matrix M T D is composed of multiplications of basis functions on the donor and target meshes. The issue that arises in the evaluation of this integral is that the donor mesh basis functions are only guaranteed to be continuous over each element of the donor mesh. If the integral of the basis functions is evaluated at Gauss quadrature points on the target mesh, it will lead to a loss of conservation and accuracy. The use of the supermesh ensures the continuity of the basis functions of the donor mesh. The mixed matrix is assembled by decomposing the elements of T T into its children (elements belonging to the supermesh T S ) using the mapping functions presented earlier, hence by construction this scheme is conservative. An example of the use of the supermesh construction and the Galerkin projection for conservative interpolation with control-volume meshes is given by Adam and al. [START_REF] Adam | Higher-order conservative interpolation between control-volume meshes: Application to advection and multiphase flow problems with dynamic mesh adaptivity[END_REF]. It is achieved by first mapping the control-volume field into a finite element representation on the donor mesh by Galerkin projection, then interpolating onto the target mesh by constructing a finite element supermesh from the intersection of the donor and target meshes and finally projecting back to a control-volume representation on the target mesh.

The algorithm based on the construction of a supermesh makes possible the use of Galerkin projection between unrelated unstructured meshes. It was proved to be efficient and accurate in three dimensions and also for adaptive meshing simulations.

Conservative redistribution via local remapping

The techniques for achieving the conservation of certain quantities during the interpolation consist in either interpolating the results to a new random grid after computing the intersections between the target and donor mesh, as presented in the previous section, or by locally modifying the original one, it is called local remapping. These methods grew out of the development of arbitrary Lagrangian-Eulerian (ALE) methods, [START_REF] Donea | Arbitrary Lagrangian-Eulerian Methods[END_REF]. In ALE settings, there is a Langragian phase in which the nodes and the elements are advected with the flow, and then a rezoning phase in which the nodes of the mesh are moved to a more optimal configuration, then a remapping phase in which the Lagrangian solution is interpolated onto the rezoned grid. Indeed, conservative remapping is a simple form of data transfer that consists in computing the value of an element of the target mesh by calculating the weighted average of the values of the source elements in contact with it. Dukowicz [START_REF] Dukowicz | Conservative rezoning (remapping) for general quadrilateral meshes[END_REF] first introduced this remapping method that computes the intersection between meshes with a surface integral and then builds the interpolated field for quadrilateral meshes : the problem of computing the volume intersection of old and new cells into a surface integral is simplified by using the divergence theorem. The approach proposed by Grandy [START_REF] Grandy | Conservative remapping and region overlays by intersecting arbitrary polyhedra[END_REF] differs from that of Dukowicz [START_REF] Dukowicz | Conservative rezoning (remapping) for general quadrilateral meshes[END_REF] in the manner in which the volume of intersection is computed. J. D. Ramshaw [START_REF] Ramshaw | Conservative rezoning algorithm for generalized twodimensional meshes[END_REF] extended this approach for all kind of 2D domains. In 1985, this method was modified with not only considering the case with constant density cell, but also a more general case with a linear density distribution, which improved the accuracy of the scheme (less diffusion), [START_REF] Dukowicz | Accurate conservative remapping (rezoning) for arbitrary lagrangien-eulerian computations[END_REF].

A conservative local interpolation step in ALE algorithms is given by Margolin and Shashkov [START_REF] Margolin | Second-order sign-preserving conservative interpolation (remapping) on general grids[END_REF], it is based on estimating the mass exchanged between the neighboring cells at their common interfaces. It is an accurate scheme that divides the new cells by taking into account the intersection with the old ones and guarantees the mass conservation along the interfaces; the mass is exchanged between the neighboring cells. It was extended to 3D domains by Garimella et al. [START_REF] Garimella | An efficient linearity and bound preserving conservative interpolation (remapping) on polyhedral meshes[END_REF]. The authors defined an underlying function that represents the density throughout the domain; the only given information about this density function is its mean value in each cell of the old mesh :

ḡ(K D ) = K D g(r)dV V (K D ) (14)
with r = (x, y, z), K D an element of the donor mesh T D , g the underlying function, ḡ the mean value of the function g, and V (K D ) the volume of the cell K D . This function will be used to define an efficient linearity and bound preserving remapping method. Indeed, the first step of the algorithm is the reconstruction of the underlying function. The gradient of the function in all the cells of the old mesh is computed in order to obtain a reconstructed piecewise linear function. Then, a quadrature formula is used to compute the approximate integration of the reconstructed function onto the new mesh, in order to obtain the mean values in the new cells (see section 4.2.1 of [START_REF] Garimella | An efficient linearity and bound preserving conservative interpolation (remapping) on polyhedral meshes[END_REF] for details). The last step is the conservative redistribution of the mass in each cell in order to conserve the local bounds. Kucharik and Shashkov [START_REF] Kucharik | Extension of efficient, swept-integration-based conservative remapping method for meshes with changing connectivity[END_REF] recommended the use of local remappers that are simpler and computationally more efficient than global remappers. Indeed, all the possible coincidences between donor ans target nodes, edges , and facets have been handled without altering the position of nodes, thus allowing to conserve mass to nearly machine precision. Moreover, the remapping methods can also be applied to unstructured meshes [START_REF] Lin | A local rezoning and remapping method for unstructured mesh[END_REF], however, they are restricted to the use of an ALE framework.

Matrix based conservative interpolation with restrictions

A conservative formulation based on the projection of the physical properties has been proposed by Chippada and al, [START_REF] Chipada | A projection method for constructing a mass conservative velocity field[END_REF]. It is not based on grid operations or algorithms but on the physics of the problem. In this article, the authors present the example of the transport of the velocity field. The idea is to compute an approximation of the solution field defined in the initial mesh; this approximated solution has to satisfy the mass conservation law. It will not only depend on the initial solution but also on a corrector term that ensures mass conservation without modifying the vorticity of the velocity field. Brancherie and al. [START_REF] Brancherie | Field transfer in nonlinear structural mechanics based on diffuse approximation[END_REF] presented an adaptive remeshing technique based on the diffusive approximation and preserves local equilibrium and ensures conservation of dissipated and strain energy, whereas, Chesshire and Henshaw [START_REF] Cheshire | A scheme for conservative interpolation on overlapping grids[END_REF] discussed conservative interpolation of fluxes between overlapping strucutured meshes; the interpolation coefficients are assumed to be free parameters, then constraints are applied to the integration weights and the interpolation coefficients.

Pont, Codina and Baiges, [START_REF] Pont | Interpolation with restrictions between finite element meshes for flow problems in ALE setting[END_REF] proposed an approach so that one will not be restrained to mass and momentum equations. Thus, the following formulation is proposed. First of all, the dicretised solution field on the target mesh is computed using a classic interpolation. For that, two choices are possible : either the use of a projection in the L 2 norm, or of a standard Lagrangian interpolation. Let us assume we computed the interpolated solution u h,T on the target mesh. This solution has the drawback of being non-conservative. So, the idea proposed by the authors consists in obtaining ũh,T by resolving an optimization problem with Lagrange multipliers. A series of constraints, such as the conservation of the mass or energy are applied to the interpolated field trough the Lagrange multipliers. The interpolated field must satisfy the two following restrictions :

• ũh,T must be the nearest solution to u h,T in the L 2 norm

• ũh,T must respect some physical properties of u h,D , solution on the donor mesh

In order to verify the second condition, restriction operators must be defined :

R n,i : V h,n → R (15)
where n refers either to donor mesh or target mesh, i = 1, ..., m represents the restriction counter. V h,n is the subspace of discretised functions, V h,n ⊂ V , the space of functions where the continuous solution lives. It is proposed to minimize this functional :

L : V h,T × R m → R L(v h,T , µ) = 1 2 ∑ n T N n T T (V n T T -U n T T ) 2 L 2 (Ω) - m ∑ i=1 λ i ∑ n T R n T T,i V n T T -∑ n D R n D D,i U n D D ( 16
)
where n D and n T are the nodes on the donor and target mesh and λ i the Lagrange multipliers.

N n T T represents the shapes functions on the target mesh; U n D D ,U n T T and V n T T are respectively the nodal values of u h,D , u h,T and v h,T . After differentiating the functional L with respect to the unknowns , we obtain an algebraic system that is different whether we consider a linear or non linear problem, more details in [START_REF] Pont | Interpolation with restrictions between finite element meshes for flow problems in ALE setting[END_REF] (section 2.2 & 2.3). Solving the obtained system of equations will allow determining the new solution ŨT that satisfies the conservation of mass and energy. Figure 8, shows how the imposition of momentum and L 2 norm conservation reduces most of the dissipation. These results are very encouraging. Indeed, the method is easy to implement and general for different resolutions. However, its extension to mesh adaptation as well as to anisotropic meshing remains a challenge.

Conclusion and perspectives

The interpolation of numerical solutions between computational meshes is a well known procedure used in many applications : mesh adaptivity to reduce the computational cost, coupling between different physical domains or moving meshes which are used in order to follow rigid bodies in their movement [Phd Thesis Wafa DALDOUL]. The accuracy of the data transfer is the main concern since the accumulation of generalized diffusion can lead to numerical errors and create convergence problems. Thus, this bibliographic reports gives a concise review of the data transfer methods in the literature, both conservative and non conservative.

Indeed, many conservative interpolation methods were introduced over the years. Among them, the remapping methods that are appealing, but we are using an Eulerian framework and these methods where only tested in ALE settings. Some approaches depend upon the use of an auxiliary mesh, either as the intersection of the elements of the initial and final mesh, or as a union of the elements. These methods have the advantage of being local, and therefore easily parallelized. Moreover, they provide satisfying results when tested in a context of adaptive meshing. Finally, another type of methods that are very attractive : the matrix based with restrictions. Indeed, they are based on applying restrictions to interpolated solutions, which can easily be implemented. Moreover, the interpolation with restriction was only tested with simple cases that do not require mesh adaptation. Therefore, it could be interesting to extend it to adaptive anisotropic meshing, [START_REF] Dervieux | Continuous Mesh Adaptation Models for CFD[END_REF].

In the context of this research project, the method that we will be implementing needs to be operational in a parallel environment, robust and optimal in computational time. On the other side, the conservative property is mandatory for industrial applications. Today, the linear interpolation scheme is used in CEMEF, it works in parallel and deals with triangular and tetrahedral meshes. First of all, we will focus on modifying this method to increase the accuracy and ensure a conservative transfer in terms of mass and energy. Different approaches can be investigated. Indeed, the supermesh construction proposed by Farell ( section 3.1.2) and the matrix based (section 3.3) will be analyzed, implemented and tested for different applications. And finally, we will apply conservative interpolation to the industrial problem, in order to transfer accurately the fields from a fluid-solid mesh to a solid mesh.

Annex : Localization algorithms

The localization algorithm consists in identifying the element of a mesh that contains a given point. The localization of a point in a mesh is necessary when we shall transfer data from one mesh to another. Hence, the algorithm consists in finding the elements of the old mesh that contains the new nodes of the target mesh. For that, different approaches can be used, a detailed comparison of the efficiency of the different methods can be found in [START_REF] Lohner | Robust, vectorized search algorithms for interpolation on unstructured grids[END_REF]. We recall here two major contributions.

Topological search method

Alauzet, [START_REF] Alauzet | P1-conservative solution interpolation on unstructured triangular meshes[END_REF] proposes a topological method based on the barycentric coordinates. Here, for the sake of simplicity, we will consider a two dimensional space. Let us consider the following situation: we have a node p T that belongs to the new mesh T T and we want to find the element K D ∈ T D in which it is located. The element K D is defined by its three vertices [P 0 , P 1 , P 2 ], and the barycentric coordinates are computed as follows :

β i = A K i A K ( 17
)
with i = [0, 1, 2], A K the surface of the triangle K D and A K i , the surface of the triangle whose vertex p i is replaced with the vertex of the new mesh p T .

The idea is to study the sign of β i (Fig 9 ), three cases are possible : • if only one β i is negative, then we have to move to the neighbor K i that shares the edge e i with K D

• if two β i are negative, then we have to try the two neighbors.

We reiterate this process till all the coordinates are positive which means we found the triangle in which p T is located, an example of the localization algorithm in a three dimensional space is given in Fig 10. Starting from K 0 , triangles K 1 , K 2 , K 3 , K 4 and K 5 are visited, bringing us back to K 0 , [START_REF] Alauzet | P1-conservative solution interpolation on unstructured triangular meshes[END_REF] Tree search algorithms

An alternative to the topological approach is the use of tree search algorithms. For instance, the quadtree algorithm for a two dimensional case that consists in partitioning the two dimensional space into four quad (Fig 11) and the octree algorithm by subdividing the 3D space into 8 regions.. We start with one rectangle, and then for each node of the new mesh we check :

• if that new node fits completely inside the rectangle

• if yes, we subdivide the rectangle into four children, then recursively do the first step

• if not, then continue to the next rectangle until there are no nodes left

The vertex of the new mesh is located in the smallest rectangle that can contain it. Here in CEMEF, we use the octree algorithm but also the R-tree [START_REF] Guttman | R-Trees: A dynamic index structure for spatial searching[END_REF]. Indeed, the structure is designed so that a spatial search requires visiting only a small number of nodes. In this case, the rectangles are not regular, we group nearby elements and represent them with their minimum bounding rectangle.

FluidFigure 2 :

 2 Figure 2: Division into two domains : Fluid-Solid and Solid

Figure 3 :

 3 Figure 3: Left: nodal fields -Right: fields defined at Gauss points

Figure 4 :

 4 Figure 4: Interpolation method applied to a transfer of field

Figure 5 :

 5 Figure 5: Top, Example of an element patch in a regular mesh. Bottom : Sampling points (triangles) on a patch, Right P1, Left P2

Figure 6 :

 6 Figure 6: Gaussian analytical function. Top left, 3D representation of the function. Top right, the mass variation for the transfer T D → T T . Bottom left, error for the transfer T D → T T . Bottom right, error for the transfer T D → T T → T D ,[START_REF] Alauzet | P1-conservative solution interpolation on unstructured triangular meshes[END_REF]

Figure 7 :

 7 Figure 7: (a) and (b) Quadrilateral meshes, (c) A triangular supermesh of (a) and (b) coloured to show the elements of (a) -(d) is the same supermesh coloured to show the elements of (b),[START_REF] Farrell | Conservative interpolation between volume meshes by local galerkin projection[END_REF]

Figure 8 :

 8 Figure 8: Top left, original function. Top right, final profile of the function after interpolating without restrictions. Bottom left, final profile of the function after interpolating with the restrictions. Bottom right, superposition of all the profiles at y = 0.5,[START_REF] Pont | Interpolation with restrictions between finite element meshes for flow problems in ALE setting[END_REF]

Figure 9 :

 9 Figure 9: Sign of the barycentric coordinates

Figure 10 :

 10 Figure 10: Left, a possible path to locate the vertex P of the new mesh starting from the triangle K 0 of the donor mesh. Right, cyclic path leading to an already checked element.Starting from K 0 , triangles K 1 , K 2 , K 3 , K 4 and K 5 are visited, bringing us back to K 0 ,[START_REF] Alauzet | P1-conservative solution interpolation on unstructured triangular meshes[END_REF]

Figure 11 :

 11 Figure 11: Quadtree algorithm

Chahrazade BAHBAH