Marie Lefevre
email: marie.lefevre@liris.cnrs.fr

Stéphanie Jean-Daubias
email: stephanie.jean-daubias@liris.cnrs.fr

Nathalie Guin
email: nathalie.guin@liris.cnrs.fr

Ethan James

Jules Liam

Stanley Lisa

Number

An approach

Keywords: Personalization of learning, tool for teachers, unified approach. Pedagogical strategy

published or not. The documents may come

Introduction

Personalization of learning consists in modifying activities proposed to learners according to a given teaching situation. This situation includes the learner's specificities, the teacher's pedagogical goals and the learning context. Personalization of learning activities is a complex and lengthy task for two main reasons. Firstly, there is a great variety of paper activities, as well as educational software focusing on different subjects; and secondly, pedagogical situation are very diverse. To assist teachers in the task of personalization, it seems relevant to provide them with a tool for personalization of both working sessions on paper and using educational software. This tool should take into account the specificities of each learner, and the goals and pedagogical habits of teachers. Building such a tool implies then to know how to exploit the available information on learners to take into account their individualities; to be able to adapt an activity according to needs and teaching habits of teachers; and to identify the necessary knowledge to select activities to be addressed to learners.

Section 2 illustrates the difficulties encountered by teachers during their work towards personalization. Section 3 presents the strengths and weaknesses of existing approaches to help them and describes our approach for building a tool to support personalization. Section 4 presents the principles of the PERSUA2 model on which our tool relies. Our propositions are discussed in the concluding section.

Usage scenario

We present a scenario showing the challenges for teachers while performing personalization of learning activities for students. Let's consider the example of Joey, teacher of eight-year-old French pupils. At the beginning of the school year, he gets the results of national assessments in French and Mathematics performed by his students at the end of the previous year. Besides, Joey's students work in autonomy with AMBRE-add [START_REF] Nogry | An ITS to Teach Solving Arithmetic Word Problems[END_REF], an ITS for additive word problems. In Joey's school, personalized support is available to underachieving students. Each concerned student receives two hours of extra teaching. To prepare the content of these extra teaching sessions, teachers meet one hour per week. Thus, each teacher sets the objectives for his students and provides his colleagues with a list of exercises to do, extracted from exercises books. Joey would like to rely on all the information available on the knowledge of his students (data from national assessments and learners profiles created by AMBRE-add) to propose remedial sessions either on paper, or with learning software, suited to each student.

This scenario show the complexity of the teacher's work for designing sessions in the light of the information he has on his students: he must parse the information himself, and learn how to handle various tools (generators and / or interface setting tools) to get resources adapted to his students and to his educational goals.

Approaches related to personalization of teaching

In the context of learning, researches on personalization try to meet several needs. Proposed approaches can be considered according to three analysis criteria [START_REF] Lefevre | Processus unifié pour la personnalisation des activités pédagogiques : métamodèle, modèles et outils[END_REF]. Considering individualities of learners can be done through the use of stereotypes [START_REF] Delozanne | From errors to stereotypes: Different levels of cognitive models in school algebra[END_REF] or learners' profiles [START_REF] Mitrovic | A Knowledge-Based Teaching System for SQL[END_REF]. In both cases, the difficulty consist in capturing relevant data about learners, and in enabling the teacher or the system implementing personalization to easily access to these data. Considering the pedagogical needs and habits of teachers is possible when they use authoring tools [START_REF] Murray | Authoring Tools for Advanced Technology Learning Environments[END_REF], when they define learning scenarios [START_REF] Lejeune | A taxonomy for scenario-based engineering[END_REF] or when they configure educational software by themselves [START_REF] Murray | Authoring Tools for Advanced Technology Learning Environments[END_REF]. But the heterogeneity of systems is a limit for teachers. Indeed, to adapt activities issued from various sources (produced by generators, described in scenarios, embedded in educational software) to their educational goals, teachers must learn how to use the interfaces of many tools. Assigning an activity to a learner can be done automatically by the system [START_REF] Mitrovic | A Knowledge-Based Teaching System for SQL[END_REF], or manually either by the learner [START_REF] Melis | ActiveMath: A Generic and Adaptive Web-Based Learning Environment[END_REF] or by the teacher [START_REF] Jean-Daubias | AMBRE-teacher: a module helping teachers to generate problems[END_REF]. In general, the pedagogical choices involved in personalization, and thus the assignment of an activity to a learner, must be made by the teacher if one wants the personalization to reflect pedagogical goals. However, systems that fulfill these requirements are infrequent.

Our analysis of current researches did not enable us to find any approach providing a unique tool supporting personalization of educational activities offered to learners, exploiting their profiles, while respecting the specificity of the educational context. However, this study enabled us to underline the limits of current approaches for personalization of learning. Firstly, to exploit information on learners coming from different sources (paper evaluations, teacher observations, profiles extracted from educational software) a teacher must currently process information himself, for each and every learner. Secondly, to generate paper exercises or to define the parameters of educational software, teachers must learn to manipulate various tools (generators or setting interfaces). Finally, it is currently impossible to automatically customize software that does not have a dedicated module managing this customization.

PERSUA2: a model for a unified personalization of activities

Fig. 1. Principe of the PERSUA2 model and its exploitation process

We propose a unified personalization of learning, supported by a single tool used by the teacher. This tool integrates a wide variety of information on learners and is able to adapt activities while taking into account the pedagogical needs of teachers. For this purpose, this tool exploits the PERSUA2 model, formalized in [START_REF] Lefevre | Processus unifié pour la personnalisation des activités pédagogiques : métamodèle, modèles et outils[END_REF], allowing the teacher to define a pedagogical strategy and a context for using this strategy (see Fig. 1). A pedagogical strategy defines how to assign activities for learners. This assignment is described using a set of links organized into a hierarchy 3. Links are called assignment rules. An assignment rule associates constraints on learners' profiles to one or more constraints on educational activities. Constraints on profiles , formalized in the cPMDL model [START_REF] Lefevre | Processus unifié pour la personnalisation des activités pédagogiques : métamodèle, modèles et outils[END_REF], enable to select values of the learners' profiles. Constraints on activities 2, following the GEPPETO approach [START_REF] Lefevre | Generation of pencil and paper exercises to personalize learners' work sequences: typology of exercises and meta-architecture for generators[END_REF][START_REF] Lefevre | Supporting Acquisition of Knowledge to Personalize Interactive Learning Environment through a Meta-Model[END_REF] enable to select or to generate an activity corresponding to the pedagogical strategy to be implemented. An activity can be a set of exercises to print or activities requiring an external educational software associated with the environment configuration A context of utilization  defines a set of information to characterize the situation in which the learner is when he carries out the work session. Information concerns, among others, the list of learners for who personalized sessions are required, their profiles, but also more general features on the sequence of activities (number of activities, time, material available…). The context may comprise, in addition to general information for all learners, exceptions for some of them. These exceptions enable one to take into account characteristics of some learners that do not appear in their profiles. The independence of the two parts of the personalization model enables associating the same pedagogical strategy in several contexts of utilization and vice versa. In addition, teachers can define as many strategies as they want, each of which can reflect different pedagogical goals. PERSUA2 is associated with a process of exploitation: teachers define pedagogical strategies and the contexts of personalized sessions. Using this personalization model, the system filters assignment rules given the profile of each learner 5, filters activities to generate depending on the rules activated by profiles and context of utilization 6, and generates activities according to lists of activities to be generated 7. Ultimately, the system provides as many personalized sequences as learners profiles, and produces a report for the teachers 8.

Fig. 2. Examples of profiles for two students: Jules and Lisa

We illustrate this exploitation processes by the scenario presented in Section 2. To prepare these sessions, Joey creates, for each students, a profile containing information about their skills in French and Mathematics (see Fig. 2). During the support sessions, Joey wants to have the students who have not mastered the skills of conjugation work on a type A paper exercise and those who partially master this skill on two paper exercises (one type B and one type C). For this, we have to look for the value of "French -Conjugation" in the student's profile . This element is the average of its sub-elements. The scale associated with this value is the enumerated list "not mastered, partially mastered, mastered". If the value for the student is "not mastered" then he will have a type A exercise, (resp. one type B and one type C for "partially mastered", and no exercise at all for "mastered"). Joey also wants to make work his students on addition tables. He wants that students which do not master this skill work on a type D paper exercise, (resp. type E for "partially mastered", and type F for "mastered"). The value of this skill  is defined by an integer grade from 0 to 10. If the value for the student is strictly lesser than 5 then he will get a type D exercise, (resp. type E if the value is comprised between 5 and 8 excluded, and type F in other cases). Finally, Joey wants to have his students work on multiplication, using the available software in the computer room where he works. He wants students not mastering the multiplication to do a type G activity on the software and others do a type H and I activities on the software. Here also the value of the skill  is defined by an integer value from 0 to 10. Thus if the value for the student is strictly less than 5, he will get a type G activity on the multiplication software, otherwise he will get one type H and one type I. For the next support session, Joey wants his students to work mostly on the conjugation, then on the multiplication tables. In addition, he wants each student to get between three and four activities done. This personalization strategy can be formalized with the PERSUA2 model (see Fig. 3).

To exploit this personalization model, first, we compare the values contained in the profiles with the conditions of assignment rules defined by the teacher to obtain, for each student, the list of rules activated by his profile, and the list of constraints on activities involved for each rule. Let us illustrate this exploitation with the profile of Jules (see Fig. 2). To interpret the assignment rules AR 1 and AR 2 , we obtain the value of the element "French -Conjugation"  by averaging the values of its sub-elements. Jules got a "Partially master" for this item. Thus only the rule AR 2 is selected. To interpret the assignment rules AR 3 , AR 4 and AR 5 , we exploit the element "Mathematics -Addition -Know your tables" . Jules got a 10 for this item. Thus only the rule AR 5 is selected. To interpret the rule AR 6 , the element "Mathematics -Multiplication"  is used. Jules got a 8 for this item We therefore select the rule AR 6 with the second list of constraints on activities.

Fig. 3. Schema of the personalization model of Joey

From the two lists (assignment rules and activated constraints on activities), it is possible to take into account the context of utilization for defining the final list of activities to generate for each learner. Let's consider the example of Jules. The result of applying the assignment rules on his profile provides five activities to be generated (types B, C, F, H, I). However, the utilization context specifies that the maximum is four. To remove one activity, we use the priority level of the assignment rule: the activity to be deleted is that associated with the assignment rule of lower priority level, i.e. the rule RA5 (activity F). Conversely, for Lisa, the assignment rules give only two activities. We must therefore add an activity to the list of activities to be generated to meet the constraints specified in the context of utilization. This additional activity will be that associated with the assignment rule of highest priority, i.e. the rule RA 6 and a second type G activity. Now that the list of activities to be generated has been established for each student, the principles of the GEPPETO approach [START_REF] Lefevre | Generation of pencil and paper exercises to personalize learners' work sequences: typology of exercises and meta-architecture for generators[END_REF][START_REF] Lefevre | Supporting Acquisition of Knowledge to Personalize Interactive Learning Environment through a Meta-Model[END_REF] are used to generate these activities (exercises to print and sessions on the multiplication software). At the end of the process, we obtain for Jules, a printable worksheet with two exercises of type B and C, and a configuration file for the multiplication software to do two activities of type H and I; for Lisa, a printable worksheet containing an exercise of type E, and a configuration file for the software to make two activities of type G (these two activities have not the same statement, but have characteristics in common).

Discussion

According to our analysis of current researches, no attempt to unify approaches of personalization of learning has been proposed yet. Depending on the problems they intend to address, proposals deal only with a subset of the aspects of personalization of learning. Some approaches use rules to adapt teaching to the learner. For example, several authoring tools [START_REF] Murray | Authoring Tools for Advanced Technology Learning Environments[END_REF] enable teachers to create educational software to set up models of pedagogical strategies. However, in these systems, a pedagogical strategy is associated (even included) in a given ILE. A teacher cannot define a strategy leading to the creation of a work sequence on multiple media. The PERSUA2 model is, as far as we know, the first model enabling a teacher, whatever the area he teaches, to create individual activities sessions specific to each learner. These sessions can involve various media, paper and / or software. The PERSUA2 model was defined after interviews with teachers aiming at identifying their practices for personalization.

If not exhaustive, it has the advantage of being open. Indeed it has been created to be extensible and can evolve to include new practices and educational needs. This model has been implemented in the ADAPTE software [START_REF] Lefevre | A Teacher-dedicated Tool Supporting Personalization of Activities[END_REF] that provides sequences of work suited for the profile of each student and to the pedagogical goals of the teacher. This software demonstrates the technical feasibility of our proposals and has allowed us to evaluate our models thought experiments involving teachers. These experiments, available in [START_REF] Lefevre | Processus unifié pour la personnalisation des activités pédagogiques : métamodèle, modèles et outils[END_REF], showed the utility of our models and relevance of personalization provided by our software. They also showed the complexity of the task required from teachers during the formalization of their personalization rules. For this critical aspect of the personalization process, it is therefore essential to assist the teacher with adapted tools. We therefore seek to improve our system by offering assistance at two levels: a classic help to guide the user through technical issues related to software and an "intelligent" assistance guiding the user in his choices of personalization.


Blandine Ginon

Pierre-Antoine Champin

Stéphanie Jean-Daubias

Taking into account users' knowledge, abilities and preferences to personalize animated assistant agents

Keywords: User assistance, animated assistant agents, personalization

In this paper, we present our approach to personalize existing and various animated agents, taking into account the context and the users' knowledge, abilities and preferences. For this purpose, we propose a grammar to describe animated agents in a common formalism, and specify their characteristics and abilities. According to this description, it is then possible to define actions with parameters that can be used and personalized in a description of the required assistance for a given application, which is used by a generic assistant to provide each user with personalized assistance.

Introduction

Animated agents are met in many applications and in many fields, like commercial websites, institutional web-sites or recreational applications. In particular, more and more ILE (Interactive Learning Environments) implement pedagogical animated agents [START_REF] Johnson | Animated pedagogical agents: face-toface interaction in interactive learning environments[END_REF] to facilitate learning. These agents differ a lot in their appearance: it can be human-like, animal, robot or imaginary (genius, object…). These agents can assist learners by means of demonstrations or explanations; they can guide learners and interact with them through verbal and non-verbal communication [START_REF] Johnson | Animated pedagogical agents: face-toface interaction in interactive learning environments[END_REF]. Some studies had shown the interest of these pedagogic agents: they motivate learners and help them in their learning process [START_REF] Nunes | Animated Pedagogical Agent in the Intelligent Virtual Teaching Environment[END_REF]; they also produce greater reported satisfaction and enjoyment by learners [START_REF] Yan | Life-Like Animated Virtual Pedagogical Agent Enhanced Learning[END_REF]. Furthermore, personalization is an important issue. It allows proposing more relevant content, possibly in a more appropriate form, taking into account the users' knowledge, preferences and abilities [START_REF] Wenger | Artificial Intelligence and Tutoring Systems[END_REF] [START_REF] Brusilovsky | Adaptive Hypermedia[END_REF]. In this paper, we present our approach to personalize the actions of animated agents, taking into account the context and the user's knowledge, abilities and preferences. In section 2, we present the context of this work, with an overview of a framework allowing personalized user assistance by means of formally describing the assistants, the application to assist, and user's profiles and preferences. Then in section 3 we focus on the description of animated agents. This description details every characteristic of an existing animated agent and the values that these characteristics can take. The description also includes actions with parameters, complying with the description of the animated agent and showing the agent's abilities. An example of using actions with parameters is given. Finally section 4 concludes and describes the next steps of our work.

Context

The research presented in this paper is part of a research project on user assistance [START_REF] Ginon | Towards a generic model for user assistance[END_REF]. We have first proposed a typology of user assistance, which presents the user needs and compares the different techniques and approaches of assistance that can meet these needs [START_REF] Ginon | Assistance à l'utilisateur : typologie et outils épiphytes[END_REF]. Among the approaches of assistance, we find counselor systems, adaptive interfaces, recommender systems, tutorials… In this paper, we focus on one particular approach of assistance: animated agents. We have then proposed a model of generic assistant that will provide personalized assistance to each user [START_REF] Ginon | Towards a generic model for user assistance[END_REF], complying with the description of the required assistance for a given application and the user model. For this purpose, the generic assistant will use a set of epiphyte assistants that can be grafted onto an application to perform the required assistance actions.

An epiphyte assistant is an assistant that can be grafted onto a given application, without perturbing its work [START_REF] Paquette | Épitalk, un outil générique pour la construction de systèmes conseillers[END_REF]. For instance, Microsoft proposes an epiphyte assistant than can be grafted onto any application compatible with Windows [START_REF]MSAgents[END_REF]. This assistant provides a set of animated agents (like Merlin or Robby the robot), able to communicate with the user by means of gestures (like pointing at an interface component), facial expressions (like frowning), or messages, textual or read by a speech synthesis. Our generic assistant model aims at allowing the personalization of assistants already integrated in an application. For this reason, it is necessary to describe in a unified formalism an existing assistant, epiphyte or integrated in an application, in order to be able to run it, or personalize its actions with respect to respect the description of the required assistance.

In order to adapt the assistance to users, their knowledge, preferences, capacities, goals and experiences, our model required a user model (also called user profile) that can contains any information about a user that can be useful in this context. For that purpose, we chose to use the profile modeling language PMDLe [START_REF] Eyssautier-Bavay | PMDL: a modeling language to harmonize heterogeneous learners profiles[END_REF] [START_REF] Ginon | Evolutive learners profiles[END_REF] that allows the definition of reusable user profiles. PMDLe is fully implemented in an environment that makes it possible to define and use of such profiles [START_REF] Ginon | Evolutive learners profiles[END_REF]. An extension to PMDLe, cPMDLe [START_REF] Ginon | Evolutive learners profiles[END_REF], allows to define constraints on user profile respecting PMDLe formalism. These constraints are used to personalize assistance according to the user specificities. For instance, an assistance action could be performed only for users that are novice in the application, or only for users whose hearing level is between 50% and 80%.

The formalism we use to describe the assistance required for a given application is based on a model proposed by [START_REF] Dufresne | Vers un modèle conceptuel générique de système d'assistance pour le téléapprentissage[END_REF]. The required assistance is described by rules of the form "If assistance condition Then assistance action". The assistance conditions contain a start condition (like a click on a component of the application's interface) and any cPMDLe constraint on the user's profile, in order to allow the personalization according to both the context and the user specificities. What's more, the assistance action involved an assistant, epiphyte or integrated into the application, and must respect the description of this assistant.

Propositions

We propose here an approach to describe the possible actions of animated assistant agents in order to personalize their behaviour. This approach is based on a grammar for describing animated assistant agents that presents the characteristics of an animated agent, and proposes actions with parameters that can be perform by this animated agent to provide personalized assistance to a user. These actions can then be used in the description of the required assistance for a given application. Contrary to existing approaches to describe animated agents [START_REF] Mancini | Dynamic behavior qualifiers for conversational agents[END_REF], our aim is not to define new animated agents but to describe in a common formalism existing animated agents and their abilities, whatever their origin and intern formalism. A description in a common formalism of varied animated agents makes it possible to reuse them in various contexts. Let's suppose that MSAgents, WebLéa and Cantoche animated agents are describe in a common formalism, with for each a description of an action called "greet" that depends on their individual abilities. Then, the designer of an ILE can specify that an animated agent must greet learners when starting. The choice of the agent will depend on the learners' preferences for instance, provided that the chose agent provides an action called "greet".

Grammar for animated agent description

We propose an abstract grammar for animated assistant agents' description. This grammar is used to describe any animated agent, epiphyte or integrated in an application. Our description of an animated agent consists in five parts, graphically represented on Fig. 1. In this graphical representation, the symbol + means at least one, the symbol ? means 0 or 1, the symbol * means zero, one or more, and the presence of "…" after an element means that it contains others elements (described elsewhere). The five parts of an animated agent description are: information about the involved animated agent (like its name and identifier), information about the description itself (like its author and date of creation), a characteristics_list, a message_channels_list, and an action_definition. The former is used to unambiguously identify the element for programs, while the latter is used to display the element to a user. Headings can be provided in different languages, hence the possibility to have several of them. The aim of the characteristics_list of an animated agent is to list all the characteristics (or attributes) of this animated agent and the values they can take; an example will be given in Fig. 3. For instance, an animated agent can have a characteristic "avatar", which can take several values specifying the appearance of the assistant. The characteristics_list contains at least one characteristic, each associated with an Id, some heading, and at least one value. A value is associated with an Id, some heading and either a possible_value_list, an interval (associated with a max, a min, and a scaling) or a type (like color, font…). The aim of the message_channels_list is to describe all the ways that the animated agent can pass a message to the user. For instance, an animated agent can pass a given message by means of a cartoon balloon or orally. The message_channels_list contains a set of message_channel, each associated with an Id, some heading and at least one value acting as parameters for that channel (i.e. font size or voice gender). The action_definition (cf. Fig. 2 ) contains a set of postures and a set of action_with_parameters. A posture is associated with an Id, some heading and at least one constraint that must be chosen among the characteristics of the animated agent. A posture aims at being used in an action_with_parameters. In a posture, all constraints refer to a characteristic of the animated agent, but the possible values associated with this characteristic are reduced. For example, if the characteristic is associated with an interval, the constraint can be associated with a subset of this interval; if the characteristic is associated with a set of 12 possible values, the constraint can be associated with a subset of these possible values, or even with only one value among them. An example of postures and their use in an action with parameters is given in Fig. 4 and Fig. 5.

An action_with_parameters is associated with an Id, some heading, an animation and an optional message. The aim of an animation is to describe the behaviour of the animated agent during this action. An animation consists of a postures_sequence and a repetitions number or repetition duration of this sequence. A message consists of a text and at least one message_channel that describe how this text will be presented to the user. Similarly to characterstics in postures, the provided message_channel must correspond to one of the available channels of the assistant, and its values, if provided, must restrict the available values of that channel. All the values that are still under-constrained (i.e. for which there are more than one possible value left) are called the parameters of the action.

Fig. 2 Action_definition for animated agents

Let's take the example of WebLéa [START_REF] Martin | WebLéa[END_REF], an animated assistant agent than can be integrated in any web application. A graphical representation of an extract of the characteristics list from the description of WebLéa complying with our grammar is given in Fig. 3. We can see that six characterstics of this animated agent, each associated with a list of possible values. For instance, the characteristic "C1", with heading "Avatar", can take the values "Léa", "Marco", "Julien" or "Djinn". Similarly, Fig. 4 and Fig. 5 illustrate some part of the action_definition for WebLéa that will be discussed in the next section.

Use of actions with parameters

When describing the required assistance for a given application, we will use the actions with parameters in assistance rules. These actions with parameters must comply with the ones defined in the description of the assistant that will perform them. The use of parameters will enable the personalization according to the user's profile. Let's take an example of an action with parameters for the animated agent WebLéa. The left part of Fig. 4 represents a posture that describes a smiling agent with the right arm at right up and the left arm drop. We give examples of WebLéa agents taking this posture on the right part of Fig. 4, with the four possible avatars. We can see that the Id of this posture is P1, it contains a constraint on five of the six characteristics of WebLéa (cf. Fig. 33). For these five characteristics, the posture definition reduces the possible value list. For instance, the characteristic mouth (with Id C4) can only take one value in this posture: "open_smile"; the characteristic left_arm (with Id C3) can only take three values: "drop", "drop2" and "drop3". One characteristic of WebLéa, avatar, is not present in the description of the P1: its means that in this posture, the characteristic avatar can take any value from its possible value list. Fig. 5 shows the description of an action with parameters for WebLéa animated agent, with Id A1 and heading Greet. This action uses postures P1 and P2 (P2 is another posture that describes an agent with the right arm at hand up left). In action A1, the agent switches between postures P1 and P2 during 30 seconds, causing it to wave its right hand to greet the user. The agent also communicates with the user by means of a message using message_channel S1 (not described because of space limitation; we will assume it displays the text in a baloon). The parameters of action A1 are the text of the message, that is not defined in the action, and all characteristics of the agent that are not associated with a single possible value in the action, i.e. the choice of the avatar (that is free), and the choice of the value for left_arm (reduced to three possible values). The action with parameters can now be used in the rules describing the required assistance for a given application. The action parameters will allow the personalization. For instance, an assistance rule could be "If {application start and user_level="novice"}, then {WebLéa performs (action_id=A1; text='Welcome! To discover this application, please clic on the button "Start with this application"'; C1=(preferences); C4=(random)}. This action will make a WebLéa animated agent appear and greet a novice user with its right hand, while saying the text in a baloon. According to this assistance rule, the choice of the avatar (corresponding to characteristic C1) will be done according to the user's preferences (express in the user profile) among the four possibilities ("Léa", "Marco", "Julien" and "Djinn"). The choice of the left_arm value (corresponding to characteristic C4) will be done randomly among the three possibilities defined in the action description ("drop", "drop2" and "drop3").

Conclusion

Assistance to users is a very important issue, in order to help users with both the handling and the common use of an application. One popular assistance approach is the use of animated agents that interact with users. The context of this work is to define a generic approach to personalize the action of animated agents (as well as other kinds of assistants). For this purpose, we have proposed in this paper a grammar for describing animated agents, their characteristic and their abilities. The grammar also allows the definition of actions with parameters that rely on the animated agent's characteristics' description. Assistance rules using actions with parameters will make allow the personalization of the assistance according to the users' profile (that can contain information on their knowledge, capacities, preferences, goals, experiences etc.). On the one hand, we can personalize assistance rules thanks to cPMDLe constraints on the user's profile. Indeed, an assistance action could be performed only for some users, whose profiles meet some cPMDLe constraints. On the other hand, we can personalize assistance thanks to constraints on the assistance actions. Indeed, the parameters of an action can depend on the user profile.

Finally, a generic assistant will use the description of the required assistance of an application, containing assistance rules, to provide each user with personalized assistance complying with this description. It will determine which assistance action should be performed and by which assistant. As a consequence, an action should comply with the description of animated agents involved in the assistance action, and the generic assistant should be able to pilot the assistant according to its description, in order to make it perform the correct action. We are currently working on implementing such a generic assistant, based on the assistant descriptions described in this paper. We are also working on extending the definition of our grammars to describe others kinds of assistant, like e.g. counselor systems.

Fig. 1

 1 Fig. 1 Grammar for animated agents' descriptionMany elements in our grammar contain an Id and optionally one or several Headings. The former is used to unambiguously identify the element for programs, while the latter is used to display the element to a user. Headings can be provided in different languages, hence the possibility to have several of them. The aim of the characteristics_list of an animated agent is to list all the characteristics (or attributes) of this animated agent and the values they can take; an example will be given in Fig.3. For instance, an animated agent can have a characteristic "avatar", which can take several values specifying the appearance of the assistant. The characteristics_list contains at least one characteristic, each associated with an Id, some heading, and at least one value. A value is associated with an Id, some heading and either a possible_value_list, an interval (associated with a max, a min, and a scaling) or a type (like color, font…). The aim of the message_channels_list is to describe all the ways that the animated agent can pass a message to the user. For instance, an animated agent can pass a given message by means of a cartoon balloon or orally. The message_channels_list contains a set of message_channel, each associated with an Id, some heading and at least one value acting as parameters for that channel (i.e. font size or voice gender). The action_definition (cf. Fig.2) contains a set of postures and a set of action_with_parameters. A posture is associated with an Id, some heading and at least one constraint that must be chosen among the characteristics of the animated agent. A posture aims at being used in an action_with_parameters. In a posture, all constraints refer to a characteristic of the animated agent, but the possible values associated with this characteristic are reduced. For example, if the characteristic is associated with an interval, the constraint can be associated with a subset of this interval; if the characteristic is associated with a set of 12 possible values, the constraint can be associated with a subset of these possible values, or even with only one value among them. An example of postures and their use in an action with parameters is given in Fig.4and Fig.5.An action_with_parameters is associated with an Id, some heading, an animation and an optional message. The aim of an animation is to describe the behaviour of the animated agent during this action. An animation consists of a postures_sequence and a repetitions number or repetition duration of this sequence. A message consists of a text and at least one

Fig. 3

 3 Fig. 3 Characteristics list from the description of the WebLéa animated agent.

Fig. 4

 4 Fig. 4 Description of the posture P1 for WebLéa and illustration.

Fig. 5

 5 Fig. 5 Description of an action with parameters for WebLéa animated agent.


Marie Lefevre
email: marie.lefevre@liris.univ-lyon1.fr

Nathalie Guin
email: nathalie.guin@liris.univ-lyon1.fr

Stéphanie Jean-Daubias
email: stephanie.jean-daubias@liris.univ-lyon1.fr

A Generic Approach for Assisting Teachers During Personalization of Learners' Activities

Keywords: personalization, pedagogical activity, pedagogical intention of teachers, process, meta-model of knowledge, model of knowledge

Personalization of learning is a complex task to implement. This task is even more complex if one wants to adopt a generic approach taking into account the many different teaching situations, and the wide variety of pedagogical activities that may exist. In this paper we describe the GEPPETO approach designed to assist teachers during personalization of pedagogical activities. This approach includes knowledge, models, and processes to adapt pedagogical activities to learners with respect to teachers' pedagogical goals.

Introduction

Personalization of learning consists in modifying activities that are proposed to a learner in order to match a given teaching situation. Personalization is a complex task since there are many different teaching situations and many different pedagogical activities. One of the first reasons of activities heterogeneity is the medium on which the activity is provided, e.g. paper activities or exercises to be done on an ILE (Interactive Learning Environment). In addition, pedagogical systems are diverse because they have diverse environments and because these environments offer various contents. Hence a pedagogical system can take the form of an intelligent tutor, a microworld, a simulator, a hypertext, etc. Each one of these forms is associated with a use mode (free or guided), content (predefined sequence of activities, set of objects that can be manipulated...), but also various educational goals (acquisition of a method, of a set of knowledge, of a practice...).

This paper focus on how to adapt a learning activity to take into account teaching needs and habits of a teacher, still in a generic context. This adaptation concerns both paper activities and software activities. For paper activities, the issue is to generate an exercise corresponding to the needs of a teacher. For software activities, the issue is to define parameters of an educational system so that both the system's content and its environment suit the needs of the teacher. To make possible this adaptation of pedagogical activities in a generic way, we propose the GEPPETO approach. This approach relies on generic models and generic processes to adapt the activities according to pedagogical intentions of teachers and by using constraints on activities. We applied this approach to enable the adaptation of paper activities (GEPPETO P ) and the adaptation of software activities within ILEs (GEPPETO S ).

This paper is organized as follows. Section 2 presents a review of solutions offered to teachers to adapt pedagogical activities to their needs, and shows why these are not usable for providing a generic approach. Section 3 presents the GEPPETO approach to support adaptation of pedagogical activities, and its specifications to adaptation of paper activities and software activities. The paper ends with concluding remarks on research issues opened by this work.

Related Work

In order to take into account pedagogical needs and habits of teachers, several solutions have been proposed: using authoring tools, defining pedagogical scenarios or settings up educational systems. Authoring tools allow one to create resources (exercises, worksheets of exercises or pedagogical systems) suited to each teacher [START_REF] David | Hypermedia exercises prototyping and modelising[END_REF][START_REF] Van Joolingen | Authoring tools for advanced technology educational software: Toward cost-effective production of adaptive, interactive, and intelligent educational software[END_REF]. For this purpose, the teacher must fully define the educational content. Hence, the use of authoring tools is time consuming. Pedagogical scenarios allow one to define activities given to learners. This involves the specification of the context in which they are themselves, the roles of all participants, the actions to do on resources, etc. [START_REF] Lejeune | A taxonomy for scenario-based engineering[END_REF]. However tools for defining a scenario do not create resources. They allow one to combine existing resources that are either contained in the application on which the scenario will be implemented, or manually created by the teacher, or created using a specific application. Therefore the adaptation to the teacher needs focuses only on the used educational resources. Only a few educational systems have a specific part for the teacher to define the parameters of the learner environment (teaching content and/or interface) [START_REF] Murray | Authoring tools for advanced technology educational software: Toward cost-effective production of adaptive, interactive, and intelligent educational software[END_REF][START_REF] Jean-Daubias | AMBRE-teacher: a module helping teachers to generate problems[END_REF]. Moreover, these parts are different from one system to another. Thus, a teacher wishing to use multiple systems has to master various tools settings. Whatever the approach chosen, the heterogeneity of the systems is in itself a limit for the teacher. Indeed, to adapt activities from various sources (coming from generators, described in the scenarios, contained in pedagogical systems, etc.) to their educational goals, teachers must learn how to use interfaces of many tools. Moreover, there is no unified approach to assist teachers in their tasks of adaptation of activities.

The approach that we propose consists in giving teachers a unique tool allowing them, among others functionalities, to define parameters of pedagogical activities of various types (on paper or contained in ILE) to match their needs. This tool is then in charge of producing adapted activities.

GEPPETO, or How to Adapt Pedagogical Activities According to the Needs of Teachers

To adapt pedagogical activities to the needs of a teacher, it is necessary to constrain the choice or the generation of these activities. To express constraints on activities, it is necessary to have a model of activities to adapt. Obviously, it is not possible to provide a generic model for any existing activity. However, it should be possible to define a meta-model that could serve as a basis to guide the establishment of a model for any particular activity. For this purpose, we propose the GEPPETO approach (GEnerics models and processes to Personalize learners' PEdagogical activities according to Teaching Objectives). This approach provides models and processes for adapting the pedagogical activities given to learners with respect to the educational goals of each agent of the personalization. In the context of personalization of learning, adaptation of activities given to learners is done mainly to take into account pedagogical choices of the teacher. However, effective adaptation can be performed by two different actors, the teacher or the ILE. The teacher may adapt activities directly. The ILE can handle the adaptation task taking into account teacher's requirements. This is the reason why we use the term "agent" to name the actors involved in the personalization process.

In the GEPPETO approach, a meta-model of activities (see  in Fig. 1) defines the knowledge that an expert must provide in order to describe pedagogical activities. This knowledge once provided by the expert, form either the model of a type of paper activity, or the model of an ILE for adaptation. From such a model of activities (see 2 in Fig. 1), it is possible to define constraints to guide the choice of activities. These constraints on activities (see 3 in Fig. 1) are defined by each agent (teacher or ILE) according to its adaptation needs. Then, these constraints are interpreted by a system implementing GEPPETO to adapt the pedagogical activities provided to learners (see 4 in Fig. 1). The GEPPETO processes are made of two phases. The first phase, represented by a dotted frame in Fig. 1, is an initialization phase. It is only done once. In this phase, an expert uses the meta-model of activities to define a model of activities for each paper activity or for each ILE that the agent (teachers or systems) wish to personalize. This process of creation of models of activities is represented by arrows (a) in Fig. 1.

The second phase, represented by a plain frame in Fig. 1, is the normal use of GEPPETO. In this phase, an agent uses a model of activities created by the expert to constrain the personalization of the activities proposed to learners. This process of definition of constraints on activities (see (b) in Fig. 1) consists of two steps: in the first one, the system proposes an interface adapted to the model of activities that limits the possibilities when defining constraints on activities and, in the second one, the system records these constraints. Then, constraints on activities are used by the system to personalize the pedagogical activities offered to learners. This process of generation of activities (see (c) in Fig. 1) is also decomposed in two steps: use of constraints to create an activity and formatting of this activity either to be printed or to be incorporated into an ILE.

The GEPPETO approach can be used to personalize paper activities or ILEs. The models involved in the specialization of the GEPPETO approach for adaptation of paper activities and relations between these models are shown in Fig. 2. This specialization, called GEPPETO P (P stands for "Paper"), is based on a typology of paper activities containing eight categories of activities [START_REF] Jean-Daubias | Generation of exercises within the PERLEA project[END_REF]. GEPPETO P uses a pattern structure that describes both the generic structure and all the metadata common to the eight patterns of exercises. Each pattern of exercises describes the elements necessary for generating one of the eight categories of activities contained in the previous typology. Each of the patterns of exercises can be used by a generator of exercises to create structures of exercises. Each structure of exercises can then be used by the same generator to create a set of exercises. Generators of exercises defined in the GEPPETO P approach are semi-automatic generators. They all have the same generic architecture [START_REF] Jean-Daubias | Generation of exercises within the PERLEA project[END_REF]. The models involved in the specialization of the GEPPETO approach for adapting software activities and relations between these models are shown in Fig. 2. This specialization, called GEPPETO S (S stands for "Software") allows to adapt software activities. A software activity is defined as a sequence of activities on a pedagogical system. Therefore adapting software activities consists in defining parameters on the content of the sequence and on the system. These parameters ensure that the system will be well suited to the learner.

We have studied thirty educational systems having various types, origins, target publics and domains. This study allows us to say that [START_REF] Lefevre | A Meta-Model to Acquire Relevant Knowledge for Interactive Learning Environments Personalization[END_REF]: the personalization of an ILE can cover five facets: selection or creation of activities, organization of these activities to form sequences of work, functionality available for the learner, interface and feedback provided by the ILE; the parameters influencing these facets can be represented in configuration files or modified through the interface of the ILE; and each of these parameters can be described using a common formalism to be presented uniformly to a teacher. This homogenization makes easier the customization of heterogeneous ILEs. Therefore the customization of an ILE requires having a description of all the parameters related to activities, sequences of activities, functions and/or interface of the ILE. In addition, to be able to act on an ILE, we must be able to modify configuration files. Therefore it is necessary to have technical information on the ILE such as the presence or absence of an exercises generator, the location and the content of configuration files. The study of existing norms and standards [START_REF]Dublin Core Metadata Initiative[END_REF][START_REF][END_REF]10,[START_REF] Rebaï | To Store and Retrieve Software Components for Interactive Learning Environments: the ECR Repository[END_REF] led us to observe that they do not allow an expert to describe an ILE with such a granularity.

In the approach GEPPETO S , we describe the AKEPI meta-model (Acquisition of Knowledge Enabling Personalization of Interactive learning environments), a metamodel for the acquisition of knowledge for personalization of ILEs [START_REF] Lefevre | A Meta-Model to Acquire Relevant Knowledge for Interactive Learning Environments Personalization[END_REF]. We combine this meta-model with two processes. First, the AKASI process allows an expert to instantiate the AKEPI meta-model with knowledge specific to an ILE x in order to create the OKEP/x model (Operational Knowledge Enabling Personalization of the ILE x), a model of operational knowledge for customizing the ILE x. Second, the OPIKSI process allows a system to use the OKEP/x model to propose to the teacher an interface enabling him to customize the ILE x (see Fig. 2).

Concluding Remarks

In this paper, we presented the GEPPETO approach. This approach relies on generic models and generic processes to adapt activities according to pedagogical intentions of teachers, by making use of constraints on activities. We applied this approach to enable the adaptation of paper activities (GEPPETO P ) and the adaptation of software activities within ILEs (GEPPETO S ). Both specializations of the approach have been implemented in Adapte [START_REF] Lefevre | A Teacher-dedicated Tool Supporting Personalization of Activities[END_REF], a system that provides each learner with activities suited to his profile, while respecting the pedagogical choices of their teacher. This implementation allowed us to demonstrate the technical feasibility of our proposals, and to evaluate them by practical experiments involving teachers [START_REF] Lefevre | Processus unifié pour la personnalisation des activités pédagogiques : métamodèle, modèles et outils[END_REF].

The evaluations of the GEPPETO approach allowed us to validate the principle and feasibility of the implementation of the approach. However, the typology of paper activities used by the GEPPETO P specialization must still be validated by experts in Education Sciences. For software activities, we defined the application framework of the AKEPI meta-model to allow an expert to describe an ILE in the GEPPETO S specialization. This application framework limits the use of the approach to the adaptation of pedagogical system providing individual learning activities. However, the AKEPI meta-model is open and can be extended to take into account ILEs offering collective and/or collaborative learning situations.

Hence, an immediate issue of this work is to describe this new class of ILEs with our AKEPI meta-model. It will then be necessary to adapt the activities of these types of collaborative systems according to our methodology, in order to verify that the models and tools that we propose can overcome new obstacles inherent to the specificity of these systems.

Fig. 1 .

 1 Fig. 1. Graphical representation of the GEPPETO approach.

Fig. 2 .

 2 Fig. 2. Graphical representation of the GEPPETO P and GEPPETO S approach.


R Ravaud

G Lemarquand

V Lemarquand

Magnetic Pressure and Shape of Ferrofluid Seals in Cylindrical Structures

Keywords: ferrofluid seals, analytical approach, permanent magnets, magnetic pressure

This paper presents a three-dimensional analytical model for studying the shape and the pressure of ferrofluid seals submitted to intense magnetic fields produced by permanent magnet ironless structures. This three-dimensional analytical approach is based on the exact calculation of the magnetic field components created by ring permanent magnets whose polarizations are either radial or axial. We assume that the ferromagnetic particles of the ferrofluid are saturated and the static behaviour of the ferrofluid seal depends on both the magnetic field produced by the permanent magnets and the saturation magnetization of the ferrofluid particles. In our applications, ferrofluid seals are always submitted to very high magnetic fields.

Consequently, the accurate knowledge of the ferrofluid seal shape as well as the magnetic pressure inside the ferrofluid seal is very useful for the design of devices using both permanent magnets and ferrofluid seals. It is emphasized here that our structures are completely ironless and thus, there are no iron-base piston for these structures. Then, this paper makes a review of the main structures using ring permanent magnets and ferrofluid seals. For each ironless structure, the shape and the pressure of the ferrofluid seals are determined. To our knowledge, this way of characterizing the static behaviour of ferrofluid seals submitted to intense magnetic fields appears for the first time in the literature. Indeed, we think that a three-dimensional analytical model is interesting for such a study because it is simple to use and has a very low computational cost.

I. INTRODUCTION

The interest of using ferrofluid in the new technologies increases clearly with the understanding of its chemical and physical properties [START_REF] Choi | Simulation of hydrostatical equilibrium of ferrofluid subject to magneto-static field[END_REF]- [START_REF] Raikher | Orientational dynamics in magnetic fluids under strong coupling of external and internal relaxations[END_REF]. Ferrofluids are in fact a unique class of material that have been discovered in the 1960s. A ferrofluid is a stable colloidal suspension of sub-domain magnetic nano particles in a liquid carrier. These magnetic particles are coated with a stabilizing dispersing agent that prevents generally particle agglomeration. Such a property is very useful because in some applications (as the ones presented in this paper), the magnetic fields used are very intense. Besides, it is noted that we discuss the utility of using intense magnetic fields in this paper. Some industries use ferrofluids for realizing airtightness seals or in audio engineering to decrease the temperature of coils in loudspeakers.

As this choice allows us to improve greatly these devices, a care study of the shape of a ferrofluid seal must be carried out because this property is directly linked to its magnetic energy and its mechanical properties. Their properties prove to be useful and efficient in various engineering areas such as damping systems [START_REF] Bajkowski | A model for a magnetorheological damper[END_REF], heat transfers, motion control systems, sensors [START_REF] Rosensweig | Ferrohydrodynamics[END_REF] [START_REF] Doutres | On the use of a loudspeaker for measuring the viscoelastic properties of sound absorbing materials[END_REF]. They also have very promising medical applications [START_REF] Li | Cfd study on the magnetic fluid delivering in the vessel in high-gradient magnetic fields[END_REF] [START_REF] Park | New design of the magnetic fluid linear pump to reduce the discontinuities of the pumping forces[END_REF]. However, they are more commonly used in bearings and seals for rotating devices.

The pioneering work regarding the ferrofluid lubrication was done by Tarapov [START_REF] Tarapov | Movement of a magnetizable fluid in lubricating layer of a cylindrical bearing[END_REF]. Indeed, he considered a plain journal bearing lubricated by ferrofluid and submitted to a non-uniform magnetic field. Since then, numerous studies in the field of ferrofluid dynamic bearings have been carried out. Both static and dynamic characteristics of these bearings have been studied theoretically [START_REF] Walker | Ferrohydrodynamics thrust bearings[END_REF]- [START_REF] Cunha | A new boundary integral formulation to describe three-dimensional motions of interfaces between magnetic fluids[END_REF]. Moreover, recent trends in the ferrofluid lubrication applications are described and discussed [START_REF] Shah | Ferrofluid squeeze film in a long bearing[END_REF]- [START_REF] Meng | An analysis on the magnetic fluid seal capacity[END_REF]. Ferrofluids are also used in electrodynamic loudspeakers in which they fulfill several functions: they ensure the airtightness, they play a heat transfer part and they work as a radial bearing. Furthermore, a ferrofluid seal can replace the loudspeaker suspension and leads to a better linearity of the cone movement [START_REF] Rosensweig | Study of audio speakers containing ferrofluid[END_REF]- [START_REF] Ravaud | Mechanical properties of a ferrofluid seal: three-dimensional analytical study based on the coulombian model[END_REF]. This paper presents a three-dimensional analytical model for studying the shape and the static pressure of ferrofluid seals submitted to intense magnetic fields produced by ironless structures. We use the static magnetic pressure for studying the shape of the ferrofluid seals. Our three-dimensional analytical approach is based on the exact calculation of the magnetic field components created by ring permanent magnets whose polarizations are either radial or axial. Moreover, the approach taken is based on the coulombian model of permanent magnets. Consequently, it gives a great flexibility for the review of the different structures studied in this paper.

In addition, it must be emphasized here that our experimental conditions in which the ferrofluid is used differ from the other experiments proposed by the authors. Indeed, our applications (ironless loudspeakers, bearings) require very high magnetic fields. Consequently, we think that the most important parameter representing the pressure in the ferrofluid seal must be determined carefully. This term is often simplified in many studies because it is difficult to determine. However, this term is not only the most important parameter that must be determined carefully but this is also probably the only term that allows us to study the ferrofluid seal shape. The gravitational energy or the surface tension energy do not influence a ferrofluid seal shape used in these conditions. This is why we think that it is more useful to obtain an exact three-dimensional analytical expression of the magnetic pressure by only considering the preponderant term in the equation of equilibrium in the ferrofluid seal rather than using several simplifying terms whose influence is questionable in our applications.

As the ferrofluid considered in this paper (Ferrotec APG05) has a saturation magnetization of at least 32 kA/ m and the magnetic field is greater than 400kA/m, the value of this pressure is higher than 12800 N/m 2 . On the other hand, the surface tension exists. But when the values of both the surface tension coefficient, A, (A equals 0.0256kg/s 2 for the used ferrofluids) and the radius of curvature are considered, the effect of the surface tension can be omitted: this latter does not deform the free boundary surface.

II. THREE-DIMENSIONAL ANALYTICAL MODEL

First, we present the calculation tools to study the different configurations using one or several ring permanent magnets. As the goal of this paper is to determine the shape of the ferrofluid seals for different magnet configurations, the magnetic field created by the magnets must be calculated. Then, the way of obtaining the seal shape is discussed.

It is emphasized here that we do not use a two-dimensional model for calculating the magnetic field created by these arc-shaped permanent magnets because such an approach is not accurate in the enar-field [START_REF] Ravaud | Analytical calculation of the magnetic field created by permanent-magnet rings[END_REF]. This is the reason why we use only a three-dimensional approach. Fig. 2. Arc-shaped permanent magnet whose polarization is axial: the upper surface is charged with the magnetic pole surface density +σ * and the lower surface is charged with the magnetic pole surface density -σ * , the inner radius is r in , the outer one is rout

A. Magnetic field created by the ring magnets

The magnetic field created by the ring magnets is determined by using the coulombian model [START_REF] Babic | Improvement in the analytical calculation of the magnetic field produced by permanent magnet rings[END_REF] [START_REF] Ravaud | Magnetic field produced by a tile permanent magnet whose polarization is both uniform and tangential[END_REF].

It is emphasized here that the authors use analytical approaches because they are more accurate than numerical methods [START_REF] Akyel | Mutual inductance calculation for non-coaxial circular air coils with parallel axes[END_REF]. The magnets are assumed perfect (flawless). Consequently, each permanent magnet is represented by two charged surfaces [START_REF] Ravaud | Analytical expression of the magnetic field created by tile permanent magnets tangentially magnetized and radial currents in massive disks[END_REF]. In the case of a permanent magnet whose polarization is radial, the magnetic poles are located on both curved surfaces of the ring and the magnetic pole surface density is denoted σ * (Fig. 1). The rings are assumed radially thin enough to neglect the magnetic pole volume density related to the magnetization divergence. In the case of a permanent magnet whose polarization is axial, the magnetic pole surface density σ * is located on the upper and lower faces of the ring (Fig. 2). The magnetic field components have been entirely determined in previous papers [START_REF] Ravaud | Discussion about the analytical calculation of the magnetic field created by permanent magnets[END_REF]. Since the azimuthal component equals zero because of the cylindrical symmetry [START_REF] Ravaud | The three exact components of the magnetic field created by a radially magnetized tile permanent magnet[END_REF], only two components must be calculated. Therefore, the magnetic field depends on both radius r and altitude z.

B. Potential energy and magnetic pressure of the ferrofluid seal

The potential energy of the ferrofluid is very important parameter for the study of ferrofluid seals.

Its calculation requires several assumptions, generally admitted by the authors. First, the ferromagnetic particles of the ferrofluid are assumed to be small saturated spheres which can be freely oriented in all the directions of space. Thus, all the particles of the saturated ferrofluid are aligned with the permanent magnet orienting field. In addition, when submitted to these intense magnetic fields, the ferrofluid does not modify the field created by the permanent magnets and the field created by the ferrofluid itself is omitted. Furthermore, the aggregation in chains of the ferrofluid particles is omitted [START_REF] Ivanov | Ferrofluid aggregation in chains under the influence of a magnetic field[END_REF]. It is noted that, when the device is at laboratory temperature and at rest, the aggregation phenomenon is observed. In short, the magnetic pressure is expressed as follows:

p m (r, z) = µ 0 M s .H(r, z) = µ 0 M s H r (r, z) 2 + H z (r, z) 2 (1)
where both magnetic field components H r (r, z) and H z (r, z) are analytically calculated [START_REF] Ravaud | Analytical calculation of the magnetic field created by permanent-magnet rings[END_REF]. Thus, the potential energy of the ferrofluid seal, E m , can be deducted :

E m = - (Ω) H(r, z).(µ 0 M)dV = - (Ω) p m (r, z)dV (2)
where (Ω) is the ferrofluid seal volume. As a remark, the magnetic pressure is given in N/m 2 and the potential energy in J. The analytical expression of the magnetic pressure depends on the number of ring permanent magnets used in the ironless structure. For example, if we consider a structure made of two ring permanent magnets radially magnetized, with opposite magnetizations, the magnetic pressure p m (r, z)

inside the ferrofluid seal is given by [START_REF] Raikher | Orientational dynamics in magnetic fluids under strong coupling of external and internal relaxations[END_REF].

p m (r, z) = µ 0 M s H r,2 (r, z) 2 + H z,2 (r, z) 2 (3)
where M s is the intensity of magnetization of a magnetic particle. The radial component H r,2 (r, z) of the magnetic field created by the two permanent magnets is defined by ( 4).

H r,2 (r, z) = σ * 2πµ 0 (ς(u 1 ) -ς(u 2 )) (4)
with

ς(u) =    2i(1 + u) d(-1+u) c+e 1 +du (-(a 1 d + b 1 (c + e 1 ))) F * i sinh -1 [ √ -c+d-e1 √ c+e 1 +du ], c+d+e 1 c-d+e 1 d √ -c + d -e 1 e 1 d(1+u) c+e1+du √ 1 -u 2    +    2i(1 + u) d(-1+u) c+e 1 +du (b 1 c -a 1 d)Π * e 1 c-d+e 1 , i sinh -1 [ √ -c+d+e1 c+e 1 +du ], c+d+e 1 c-d+e 1 d √ -c + d -e 1 e 1 d(1+u) c+e1+du √ 1 -u 2    +    2i(1 + u) d(-1+u1) c+e 2 +du 1 (-(a 2 d + b 2 (c + e 2 ))) F * i sinh -1 [ √ -c+d-e2 √ c+e 2 +du ], c+d+e 2 c-d+e 2 d √ -c + d -e 2 e 2 d(1+u) c+e2+du √ 1 -u 2    +    2i(1 + u) d(-1+u) c+e 2 +du (b 2 c -a 2 d)Π * e 2 c-d+e 2 , i sinh -1 [ √ -c+d+e2 c+e 1 +du ], c+d+e 2 c-d+e 2 d √ -c + d -e 2 e 2 d(1+u) c+e2+du √ 1 -u 2    -    2i(1 + u) d(-1+u) e 3 +du ((a 3 d -b 3 e 3 )) F * i sinh -1 [ √ -d-e3 √ e 3 +du ], -d-e 3 d+e 3 d √ -d -e 3 (-c + e 3 ) d(1+u) e3+du √ 1 -u 2    -    2i(1 + u) d(-1+u) e 1 +du (b 3 c -a 3 d)Π * -c+e 3 d+e 3 , i sinh -1 [ √ -d+e3 e 3 +du ], -d+e 3 d+e 3 d √ -d -e 3 (-c + e 3 ) d(1+u) e3+du √ 1 -u 2    -    2i(1 + u) d(-1+u) e 4 +du (a 4 d -b 4 e 4 ) F * i sinh -1 [ √ -d-e4 √ e 4 +du ], -d+e 4 d+e 4 d √ -d -e 4 (c + e 4 ) d(1+u) e4+du √ 1 -u 2    -    2i(1 + u) d(-1+u) e 4 +du (b 4 c -a 4 d)Π * -c+e 4 d+e 4 , i sinh -1 [ √ -d-e4 e 4 +du ], -d+e 4 d+e 4 d √ -d -e 4 (-c + e 4 ) d(1+u) e4+du √ 1 -u 2    (5)
The axial component of the magnetic field created by the two ring permanent magnets is given by [START_REF] Doutres | On the use of a loudspeaker for measuring the viscoelastic properties of sound absorbing materials[END_REF].

H z,2 (r, z) = σ * πµ 0   -r in K * -4rr in (r-rin) 2 +z 2 (r -r in ) 2 + z 2 + r in K * - 4rr in (r-rin) 2 +(z-h) 2 (r -r in ) 2 + (z -h) 2   - σ * πµ 0   r in K * 4rrin (r-rin) 2 +z 2 (r -r in ) 2 + z 2 -r in K * 4rrin (r-rin) 2 +(z+h) 2 (r -r in ) 2 + (z + h) 2   (6)
where K * [m] is given in terms of the incomplete elliptic integral of the first kind by ( 7)

K * [m] = F * [ π 2 , m] (7)
F * [φ, m] is given in terms of the elliptic integral of the first kind by ( 8):

F * [φ, m] = θ=φ θ=0 1 1 -m sin(θ) 2 dθ (8) Π * [n, φ, m]
is given in terms of the incomplete elliptic integral of the third kind by ( 9)

Π * [n, φ, m] = φ 0 1 (1 -n sin(θ) 2 ) 1 -m sin(θ) 2 dθ (9)
The parameters used in [START_REF] Rosensweig | Ferrohydrodynamics[END_REF] are defined in Table II-B. Moreover, when we input (4) in Mathematica, we have to take the real part of H r (r, z) because of the noise calculus. What's more, the parameter i used in ( 5) is the imaginary number (i 2 = -1).

C. Parameters used for describing the ironless structures

The further calculations are presented for magnets with 1T remanent magnetization in order to normalize the results. In fact, the magnets used in the prototypes are Neodymium Iron Boron ones for which the remanent magnetization can reach 1.5T. Furthermore, the interesting regions of space are the ones where the ferrofluid goes. With these 1T normalized magnets and for the proposed configurations, the magnetic field intensity there is greater than 400 kA/m. Of course, all the field values are proportional to remanent magnetization value. Then, we use commercial ferrofluids, either from the company Ferrotec or Ferrolabs . Such ferrofluids have a saturation magnetization, M s , smaller than 32 kA/m and a particle concentration [START_REF] Rosensweig | Ferrohydrodynamics[END_REF] below 5,5 %. It is to be noted that for bearing or loudspeaker applications, a great bearing effect is seeked which requires high saturation magnetizations. Therefore, the magnetic field, H, created by the permanent magnets is far higher than the ferrofluid critical field [START_REF] Matthies | Numerical simulation of normal-field instability in the static and dynamic case[END_REF]. So, the ferrofluid is totally saturated and its magnetization is denoted M s .

Parameters a 1 r in rz b 1 -r 2 in z c r 2 + r 2 in d -2rr in e 1 z 2 a 2 -r in r(z -h) b 2 r 2 in (z -h) e 2 (z -h) 2 a 3 r in rz b 3 -r 2 in z e 3 r 2 + r 2 in + z 2 a 4 r in r(-z -h) b 4 -r 2 in (-z -h) e 4 r 2 + r 2 in + (z + h) 2
Color plots of the magnetic pressure will be done in the further sections in order to visualize the characteristics of the studied configurations. The warmest color (red) is used for the regions of space where the magnetic pressure is the greatest and the coldest color (blue) for the ones where the magnetic pressure is the smallest. Moreover, the same color corresponds to the same value throughout this paper: the corresponding scale is given in Table II in which the magnetic pressure and the related magnetic field values are given. can be either made out of permanent magnet -as here the middle ring-or out of non-magnetic material -like the upper and lower rings-. For each configuration, the ferrofluid seals are located in the air gap between the piston and the rings; their number, their position as well as the magnetization direction of the ring magnets will be discussed.

III. SHAPE OF THE FERROFLUID SEAL IN VARIOUS MAGNETIC FIELDS

A. Description of the generic structure

For each configuration, we have also represented the radial component of the magnetic field created by the ring permanent magnets: indeed, this parameter can also be optimized by changing the ring dimensions in electric machines.

It is assumed throughout this paper that the forming shapes of ferrofluid are considered to be proportional to the magnetic pressure. Such an assumption is possible because the magnetic field produced by the ring permanent magnets is very high.

The coordinate system (0, u r , u θ , u z ) is placed into the centre of the magnetic system in Figs 6 and7 and is axially moved off the centre in Figs 4, 5, 8, 9, 10 and 11. In addition, the radial scale of the permanent magnets has been divided by 2 in these figures.

B. Structures using one ring magnet

The first structure considered corresponds exactly to the configuration shown in Fig. 3. All the rings have the same square cross-section whose side equals 3 mm. The middle ring is a radially magnetized permanent magnet and the upper and lower rings are non-magnetic. The magnetic field created by the magnet in the air gap is calculated at a distance in length from the rings and along the Z axis which equals 0.1 mm. Moreover, its radial component H r shown in Fig. 4 is quite uniform in front of the magnet.

Two field gradients exist in front of the edges of the magnet. We can point out that the magnetic pressure calculated in the air gap between the rings and the piston is plotted on Fig. 4 as well. The ferrofluid seeks the regions of both intense field gradient and high magnetic energy. The color plot shows that, for seals thicker than 0.5 mm, the seal expands along the whole magnet height. A smaller amount of ferrofluid would give two separate seals, but they would be too thin to have interesting mechanical properties. This result is linked to the magnet square section: were the magnet higher along Z, two separate seals would appear. It can be noted that this structure is the simplest one which can be used in electrodynamic loudspeakers.

The second structure differs from the first one only in the magnetization direction which becomes axial. Figure 5 shows the structure, the calculated radial field component and the corresponding magnetic pressure. The radial component of the magnetic field is no longer uniform in front of the magnet; it shows instead a rather large gradient all over the magnet length and the non-magnetic rings. Nevertheless, the magnetic pressure deserves the same comments as in the first structure and is somewhat similar to the previous one. However, it is obvious that it differs in form from one structure to the other.

C. Structures using two ring magnets

The structures studied here derive from the preceding ones by stacking a second permanent magnet ring directly on the first one. Both ring magnets have the same dimensions and opposed magnetization directions. The radial field component and the magnetic pressure are shown in Fig. 7 when the ring magnets are radially magnetized and in Fig. 6 when the ring magnets are axially magnetized. The magnetic field in both cases results from the superposition of the single magnet fields. As a consequence, each radial magnet creates a region of uniform field in front of itself and the field directions are opposite. The field intensity in each uniform region is higher than in the single magnet structure because the leakage is decreased. In effect, three field gradients exist, and the one that appears in front of the magnets' interface is twice as high as those at the edges. From the gradient point of view, Fig. 7 can be compared with Fig. 5, and the former will prove more useful because the gradient is steeper. The axial double structure creates progressive field gradients with no peculiar interest. By contrast, the radial double structure is used to design two-coil loudspeakers. The repartition of the magnetic energy density in the structure using two ring magnets does not derive from the superposition of the ones in the structures using one ring magnet.

The main reason is due to the expression of the energy which depends on the square of the field. Both repartitions for radial and axial magnets seem alike at first sight, but the radial structure is in fact "more energetic" and the magnetic energy in it decreases slower at an increasing distance from the magnets.

Nevertheless, the maximum energy density is in front of the magnets' interface and the ferrofluid seal will be located there. We can point out that the axial length of the seal in structures using one magnet is smaller than the one in structures using two ring magnets. Furthermore, its energy density is approximately doubled for the radial magnets.

D. Structures using three ring magnets

All the structures considered now consist of three stacked ring permanent magnets whose magnetization directions undergo a 90 degrees rotation from one magnet to the neighboring one. Such configurations with a magnetization progressive rotation are related to Halbach cylinders [START_REF] Halbach | Design of permanent multiple magnets with oriented rec material[END_REF] and to their applications in the design of electrical motors [START_REF] Marinescu | Compensation of anisotropy effects in flux-confining permanent-magnet structures[END_REF] or of passive magnetic bearings [START_REF] Yonnet | Permanent magnet bearings and couplings[END_REF]. Contrary to electrical machines, the presented structures do not have a periodical magnet pattern.

Figure 8 shows the field radial component and the magnetic pressure for an assembly of a radially magnetized middle ring and two axially magnetized upper and lower rings. Moreover, the axial magnets have opposed magnetization directions. In Fig. 9, the middle ring is axially magnetized, both other rings are radially magnetized with opposed directions.

The energy density color plots show that two ferrofluid seals form in front of the magnets' interfaces and that those seals have a great magnetic energy. Consequently, they own good mechanical properties (great radial stiffness for example). The seals are rather similar in both structures. However, the magnetic field radial component shown in Fig. 8 is fairly uniform in the area in front of the middle magnet while it varies with no particularly interesting properties in Fig. 9. Therefore, the structure with the radial middle magnet is the useful one, especially for the loudspeaker design, and the field uniformity zone may be optimized.

The axial height of the middle magnet can be varied regarding the axial height of the upper and lower magnets. For example, the middle magnet is twice as high as each other magnet in Fig. 10 and half as small in Fig. 11. As a result, the length of the field uniformity area follows closely the middle magnet height. If the latter increases, the former increases too. We can say that the ratio middle of the magnet . height over the upper and lower magnet height is important as well. Indeed, if it increases, the field uniformity region grows but the field intensity decreases, and inversely. It also influences the repartition of the magnetic pressure. Indeed, when the upper and lower magnets become too small, the ferrofluid can expand over their whole axial length and the seals can become fairly voluminous, depending on the ferrofluid available quantity. The field gradients are steep and the seal magnetic energy is quite high.

Consequently, the seals are well-fixed to the structure and have high mechanical performances. Inversely, when the middle magnet becomes too small, both seals gather to form a single one which expands over the whole height of the middle magnet. The energy in the seal is still high for radial thickness smaller than the ones presented in the previous case.

E. Discussion

So far, different magnet assemblies have been presented. Each assembly creates magnetic fields and magnetic field gradients so as to trap and fix ferrofluid to make seals. Their characteristics will be discussed and compared now.

The structures considered are either very simple or more complex. The complexity can be characterized by the number of separate parts necessary to build devices. We can say that from this point of view, the single magnet structures are the simplest ones. The complexity also lies in the magnets' magnetization direction. Indeed, an axial magnetization is easy to achieve for ring magnets whereas a radial one is technically more difficult. Thereby, ring magnets whose polarization is radial are more expensive than the ones whose polarization is axial. Nevertheless, some magnet manufacturers begin to deal with radial magnetization and to sell such products.

The utility of a structure depends on the intended applications. For example, a device can be designed solely to make ferrofluid seals with good mechanical properties. It can also be designed to create a radial magnetic field or to enable a ferrofluid seal to form as it is required in loudspeakers. A high magnetic field gradient and a high magnetic pressure are required to achieve robust seals. From the preceding sections, we deduce that multi-magnet structures are more efficient than single magnet structures in creating high field gradients. Furthermore, the magnetic pressure is higher in the air gap near the magnets.

It can be emphasized here that the shape of the ferrofluid seal in the device depends on both the magnetic pressure and the ferrofluid volume. It can be noted that this latter one is related to the air gap size. The air gap exists because the inner part has to move, in rotation or in translation, inside the outer part without rubbing against it. The air gap size depends on mechanical considerations, such as the practical machining and the related possible tolerances. The color plots show clearly that the theoretical size of the air gap which allows an efficient ferrofluid seal depends on the magnetic structure dimensions.

For example, the structures shown in Fig. 11 or in Fig. 8 lead to seals thinner than the ones shown in Fig. 7 because the region of high magnetic pressure is radially thinner. Ferrofluid seals are robust when their capacity is high; this means that they can resist to quite high axial pressures because they exert themselves an opposing force. We can say that this force is related to the energy variation of the ferrofluid seal. The higher the energy varies with the axial displacement, the more important the force is.

Moreover, a great force is attained with thin seals whose energy is high and varies extremely with axial displacements. Therefore, the smaller the air gap is, the more favorable the mechanical structure to build ferrofluid seals is.

The color plots can help optimizing both the number and the shape of ferrofluid seals for given ferrofluid volumes.

For example, the plots shown in Figs. 10 and 11 point out that two seals appear in these three magnet structures for a small volume of ferrofluid. Moreover, they show that the increase in the ferrofluid volume results in the growth of the seals which eventually join and can form a single big one. The color plots can also be considered as tools to determine the adequate quantity of ferrofluid which will confer the intended mechanical properties on the seals. Furthermore, this study shows that some structures have greater performances than others for given magnet and ferrofluid volumes.

The requirements for the loudspeaker applications are different. Indeed, the magnetic structure is expected to fulfill several purposes, namely the creation of a radial uniform magnetic field as well as the creation of field gradients. Besides, a loudspeaker coil moves thanks to the magnetic field. The linearity performances are related to the field uniformity and the efficiency is proportional to the field level. Furthermore, the ferrofluid seal is used to ensure the airtightness, to transfer the heat from the moving part to the steady one, to work as a radial bearing and to replace the loudspeaker suspension.

Thereby, it contributes to the improvement of the loudspeaker linearity. The air gaps in loudspeakers are generally 0.1 to 0.5 mm wide: this explains why the presented magnetic field calculations have been carried out along a path at a distance in length which equals 0.1 mm from the ring magnets. Thinner air gaps would be more advantageous for ferrofluid seals but are not technically possible because of the machining tolerances or the recognition of the moving part thermal dilation. Therefore, the structures shown in Figs. 3, 7 and 10 are the ones that are the more useful for loudspeakers. The three magnet structures are the more complex, but they offer the possibility of a multi-criteria optimization. We can say that this optimization is enabled and simplified by the analytical formulations of both the magnetic field and the magnetic energy density. Eventually, such structures show very good performances.

IV. CONCLUSION

This paper has presented a three-dimensional analytical model for studying the shape of the static pressure of ferrofluid seals subject to intense magnetic fields produced by permanent magnet ironless structures. Our analytical approach has been applied to well-known structures composed of one, two or three stacked ring permanent magnets whose polarization is either radial or axial. Our analytical appraoch allows us to predict easily the shape and the pressure of ferrofluid seals. Then, we have discussed the interest of using simple or complex structures with ferrofluid seals. The single magnet structures are the simplest ones but the less efficient. The use of several magnets enhances both magnetic and mechanical properties: when the created magnetic field is doubled, the magnetic energy is quadrupled. Furthermore, three magnet structures enable to create one or two seals which have a high energy. The structures can also create a high uniform magnetic field over defined regions as well as efficient ferrofluid seals: this is useful for loudspeaker applications. The results obtained can be used for the design of cylindrical structures using both ring permanent magnets and ferrofluid seals.

H

Fig. 1 .

 1 Fig.1. Arc-shaped permanent magnet whose polarization is radial: the inner curved surface is charged with the magnetic pole surface density +σ * and the outer curved surface is charged with the magnetic pole surface density -σ * , the inner radius is r in , the outer one is rout

Figure 3 Fig. 3 .

 33 Figure3shows the generic structure of all the devices studied. They consist of three outer stacked rings, of an inner non-magnetic piston and of ferrofluid seals. The piston is radially centered with the rings. The rings' inner radius, r in , equals 25 mm and their outer radius, r out , equals 28 mm. The rings

Fig. 4 .

 4 Fig.4. Right: upper and lower non-magnetic rings, a middle ring permanent magnet axially magnetized. Left: magnetic pressure in front of the rings. Bottom: magnetic field radial component at a 0.1 mm distance from the rings, along the Z axis.

Fig. 5 .

 5 Fig.5. Right: upper and lower non-magnetic rings, a middle ring permanent magnet radially magnetized. Left: magnetic pressure in front of the rings. Bottom: magnetic field radial component at a 0.1 mm distance from the rings, along the Z axis.

Fig. 6 .

 6 Fig.6. Right: Two ring permanent magnets with opposed axial magnetization. Left: magnetic pressure in front of the rings. Bottom: magnetic field radial component at a 0.1 mm distance from the rings, along the Z axis.

Fig. 7 .

 7 Fig.7. Right: Two ring permanent magnets with opposed radial magnetization. Left: magnetic pressure in front of the rings. Bottom: magnetic field radial component at a 0.1 mm distance from the rings, along the Z axis.

Fig. 8 .

 8 Fig.8. Right: axially magnetized upper and lower rings, radially magnetized middle ring. Left: magnetic pressure in front of the rings. Bottom: magnetic field radial component at a 0.1 mm distance from the rings, along the Z axis.

Fig. 9 .

 9 Fig.9. Right: radially magnetized upper and lower rings, axially magnetized middle ring. Left: magnetic pressure in front of the rings. Bottom: magnetic field radial component at a 0.1 mm distance from the rings, along the Z axis.

Fig. 10 .

 10 Fig.10. Right: axially magnetized upper and lower rings, radially magnetized middle ring, each axial magnet is half as small as the radial one. Left: magnetic pressure in front of the rings. Bottom: magnetic field radial component at a 0.1 mm distance from the rings, along the Z axis.

Fig. 11 .

 11 Fig.11. Right: axially magnetized upper and lower rings, radially magnetized middle ring, each axial magnet is twice as high as the radial one. Left: magnetic pressure in front of the rings. Bottom: magnetic field radial component at a 0.1 mm distance from the rings, along the Z axis.

TABLE I DEFINITION

 I OF THE PARAMETERS USED IN

TABLE II H

 II , MODULUS OF THE MAGNETIC FIELD (kA/m); p m , MAGNETIC PRESSURE (N.m -2 ); M s = 32kA/m


F Chedevergne

M Dahan

A Burtheret

B Parratte

Walking test to validate an equipped slipper for measurement of foot-to-ground pressure distribution

Introduction

For many purposes in sport and medical areas, it is worth knowing the barefoot plantar pressure distribution during natural walk, over many steps, with a reliable, resistant and easily usable equipment.

Our laboratory has developed such an equipment. Two comfortable medical slippers are equipped with eight strain gauge sensors throughout the way of the normal roll of the foot during the stance phase: two sensors under the heel, two under the external edge of the foot, three under the metatarsal heads and one under the hallux (figure 1).

After our previous validation of the mechanical behavior of the sensor during tensile tests and during on foot-standing tests (Chedevergne et al. 2006), we present the second part of the clinical validation on healthy patient.

Materials and methods

One healthy 25 year-old men of 72.6 kg, who takes size 42 in European system (8.5 in US sizes; 8 in UK sizes) is asked to realize a 10 m return walk (figure 2).

The sensors strain is recorded (10 Hz) during all test long and then translated into the force exerted by the foot on the ground. The sum of these recorded data at each instant gives the vertical force exerted by the whole foot on the ground along the walk.

Results and discussion

To analyze more easily the spread of the vertical force over the whole foot, we observe four areas: the heel, the external foot side, the external metatarsal heads and the hallux coupled with the first metatarsal head. Each of these areas is defined by two sensors. When observing one single stance phase, we obtain curves as in figure 3.

Figure 3 shows a normal roll of the foot during the stance phase, with normal force values: 130% of the body weight at the heel strike and the propulsion, and 90% at mid-stance. But over the thirty steps recorded, this normal *Corresponding author. Email: fany.chedevergne@univ-fcomte.fr shape of the vertical force curve was not often reproduced by our healthy subject (figure 4). This lack of consistency can provide from a dysfunction of the offset.

Conclusion

Our device allows to record barefoot pressure under the normal way of the foot roll. To confirm the reliability of the recorded data, we must imporve the offset. We also have to compare these results with those of a force-plate or another validated device.

Figure 1 .

 1 Figure 1. Location of the eight sensors used in an adult shoe and assembly of each sensor.

Figure 2 .

 2 Figure 2. Ten metre return walk.

ReferenceF.

  Chedevergne, A. Faivre and M. Dahan, "Paper presented at the IEEE ICMA Meeting", Luoyang, Henan, China, 25-28 June 2006.

Figure 3 .

 3 Figure 3. Force recorded by each area (expressed in pourcentage of body weight) over one stance phase.

Figure 4 .

 4 Figure 4. Vertical force recorded under the whole foot during several steps.


Ionel Sorin Ciuperca

Ronan Perrussel

I Ciuperca

C Poignard
email: clair.poignard@inria.fr

Influence of a rough thin layer on the potential

come

Influence of a rough thin layer on the potential

I. INTRODUCTION

In the domains with a rough thin layer, numerical difficulties appear due to the complex geometry of the rough layer when computing the steady-state potentials. We present here how these difficulties may be avoided by replacing this rough layer by appropriate transmission conditions. Particularly, we show that considering only the mean effect of the roughness is not sufficient to obtain the potential with a good accuracy.

A. Statement of the problem

Ω O 0 ε O 1 O m ε Γ Γ ε Fig. 1. Geometry of the problem.
Let Ω be a smooth bounded domain of R2 with connected boundary ∂Ω. For ε > 0, we split Ω into three subdomains: O1 , O m ε and O 0 ε . O 1 is a smooth domain strictly embedded in Ω (see Fig. 1). We denote by Γ its connected boundary. The domain O m ε is a thin oscillating layer surrounding O 1 . We denote by Γ ε the oscillating boundary of O m ε :

Γ ε = ∂O m ε \ Γ. The domain O 0 ε is defined by: O 0 ε = Ω \ (O 1 ∪ O m ε ).
We also denote by O 0 = Ω\O 1 . Two piecewise-constant conductivities on the domain Ω have to be defined:

σ(z) =      σ 1 , if z ∈ O 1 , σ m , if z ∈ O m ε , σ 0 , if z ∈ O 0 ε . σ(z) = σ 1 , if z ∈ O 1 , σ 0 , if z ∈ Ω \ O 1 .
where σ 1 , σ m and σ 0 are given positive constants 1 . Let u ε and u 0 be defined by:

∇. (σ∇u ε ) = 0, in Ω, u ε | ∂Ω = g, , ∇. σ∇u 0 = 0, in Ω, u 0 | ∂Ω = g, (1)
where g is a sufficiently smooth boundary data. We present how to define the potential u 1 such that u ε is approached by

u ε = u 0 + εu 1 + o(ε 3/2 ) for ε tending to zero 2 .

II. HEURISTICS OF THE DERIVATION OF THE CONDITIONS

Suppose Γ is a smooth closed curve of R 2 of length 1 and parameterize it by the curvilinear coordinate

Γ = {Ψ(θ), θ ∈ [0, 1]} . Let n be the (outward) normal to ∂O 1 . Γ ε is described by Γ ε = {Ψ(θ) + εf (θ/ε)n(θ), θ ∈ [0, 1]},
where f is a smooth 1-periodic and positive function, which describes the roughness of the layer.

A. Boundary layer corrector in the infinite strip

The key-point of the derivation of the equivalent transmission conditions consists in taking advantage of the periodicity of the roughness. This is performed by unfolding and upscaling the rough thin layer into the infinite strip R × [0, 1].

Define the closed curves C 1 and C 0 , which are trigonometrically oriented by

C 0 = {0} × [0, 1], C 1 = {(f (y), y), ∀y ∈ [0, 1]} .
The outward normals to C 0 and C 1 equal

n C0 = 1 0 , n C1 = 1 1 + (f ′ (y)) 2 1 -f ′ (y) . (2)
According to [START_REF] Ciuperca | Approximate transmission conditions through a rough thin layer. The case of the periodic roughness[END_REF] there exists a unique couple (A 0 , a 0 ) where A 0 is a continuous vector field and a 0 is constant such that

A 0 is 1-periodic in y, ∆A 0 = 0, in R × [0, 1], (3a) σ 0 ∂ n A 0 | C + 1 -σ m ∂ n A 0 | C - 1 = (σ m -σ 0 )n C1 , (3b)
σ m ∂ n A 0 | C + 0 -σ 1 ∂ n A 0 | C - 0 = -(σ m -σ 0 )n C0 , (3c)
A 0 → x→-∞ 0, A 0 -a 0 → x→+∞ 0, (3d)
where the convergences at infinity are exponential. We emphasize that a 0 is not imposed but is a floating potential.

B. Approximate transmission conditions

Our transmission conditions are then obtained with the help of the constant vectors D 1 and D 2 defined by:

D 1 = (σ 0 -σ m ) 1 0 f (y) dy n C0 + 1 0 A 0 (f (y), y) dy + (σ m -σ 1 ) 1 0 A 0 (0, y) dy -σ 0 a 0 , D 2 = (σ m -σ 0 ) 1 0 A 0 (f (y), y)f ′ (y) dy - 1 0 f (y) dy 0 1 .
The potential u 1 is then defined by3 :

               ∆u 1 = 0, in O 0 ∪ O 1 , u 1 | ∂Ω = 0, σ∂ n u 1 Γ = -κD 1 • ∂ n u 0 | Γ+ ∂ t u 0 | Γ + + D 2 • ∂ t ∂ n u 0 | Γ+ ∂ t u 0 | Γ + , u 1 Γ = a 0 • ∂ n u 0 | Γ+ ∂ t u 0 | Γ + ,
where ∂ t and ∂ n denote the tangential and the normal derivatives along Γ and κ is the curvature of Γ.

We emphasize that our conditions are different than if we would only consider the mean effect of the roughness. In this case, denoting by f the mean of f , the conditions would be (see [START_REF] Ciuperca | Approximate transmission conditions through a rough thin layer. The case of the periodic roughness[END_REF], [START_REF] Poignard | Approximate transmission conditions through a weakly oscillating thin layer[END_REF], [START_REF] Poignard | Approximate conditions replacing thin layer[END_REF]):

σ∂ n u 1 Γ = (σ 0 -σ m )f ∂ 2 t u 0 | Γ , u 1 Γ = σ 0 -σ m σ m f ∂ n u 0 | Γ + .

III. NUMERICAL SIMULATIONS

In order to verify the convergence rate stated in Section I, we consider a problem where the geometry and the boundary conditions are ε-periodic. The computational domain Ω is delimited by the circles of radius 2 and of radius 0.2 centered in 0, while O 1 is the intersection of Ω with the concentric disk of radius 1. The rough layer is then described by f (y) = 1 + 1/2 sin(y). One period of the domain is shown Fig 2(a). The Dirichlet boundary data is identically 1 on the outer circle and 0 on the inner circle.

The mesh generator Gmsh [START_REF] Geuzaine | Gmsh mesh generator[END_REF] and the finite element library Getfem++ [START_REF] Renard | Getfem finite element library[END_REF] enables us to compute the four potentials u ε , u 0 , u 1 and u 1 .

The rough thin layer is supposed slighty insulating. The conductivities σ 0 , σ 1 and σ m respectively equal to 3, 1 and 0.1. The computed coefficients4 issued from Problem (3) are given in Table I. The numerical convergence rates for the H 1 -norm in O 1 of the three following errors u εu 0 , u εu 0εu 1 and u ε -u 0 -ε u 1 as ε goes to zero are given Figure 2. As predicted by the theory, the rates are close to 1 for the order 0 and for the order 1 with the mean effect, whereas it is close to 2 for the "real" order 1 equal to u εu 0εu 1 . Fig. 3. H 1 -Error in the cytoplasm vs ǫ for three approximate solutions.

Fig. 2 .

 2 Fig.2. Representation of one period of the domain and the corresponding errors with approximate solutions u 0 and u 0 + ǫu 1 . ǫ = 2π/30. Do not consider the error in the rough layer because a proper reconstruction of the solution in it is not currently implemented.

The same following results hold if σ 0 , σ 1 , and σm are given complex numbers with imaginary parts (and respectively real parts) with the same sign.

The notation o(ε

3/2 ) means that ' ' u ε -(u 0 + εu 1 ) ' ' goes to zero faster than ε 3/2 as ε goes to zero. We refer to Theorem 1.1 of[START_REF] Ciuperca | Approximate transmission conditions through a rough thin layer. The case of the periodic roughness[END_REF] for a precise description of the involved norms and the accuracy of the convergence.

We denote by [w] Γ the jump of a function w on Γ.

The convergences at the infinity in Problem (3) are exponential hence we just have to compute problem (3) for |x| ≤ M , with M large enough to obtain a 0 with a good accuracy.