Philippe Dessus
email: philippe.dessus@upmf-grenoble.fr

Stefan Trausan-Matu
email: stefan.trausan@cs.pub.ro

Implementing Bakhtin's Dialogism Theory with NLP Techniques in Distance Learning Environments

Keywords: Dialogism, CSCL, Natural Language Processing, Latent Semantic Analysis 1

The purpose of this paper is to present and discuss a Natural Language Processing-based operationalization of Bakhtin's dialogism ideas, and to insert it in the context of distance learning. Three core concepts from Bakhin are introduced and their application in the domain of distance learning discussed. Then, two computer-based systems based on these ideas, PolyCAFe and Pensum are presented.

Introduction

Contemporary research on education often considers education-based events as processes developed through dialogue, embedded in a community of practice [START_REF] Wenger | Communities of practice: Learning, meaning and identity[END_REF]. This view has been taken in many research fields, like teacher education [START_REF] Burbules | Theory and research on teaching as dialogue[END_REF], pedagogy [START_REF] Egan | The Educated Mind, how cognitive tools shape our understanding[END_REF] and Computer-Supported Collaborative Learning (CSCL) [START_REF] Koschmann | Toward a dialogic theory of learning: Bakhtin's contribution to understanding learning in settings of collaboration[END_REF]. The dialogic approach considers learning as taking a perspective of another in a dialogue [START_REF] Wegerif | A dialogic understanding of the relationship between CSCL and teaching thinking skills[END_REF].

The above vision is rooted in the ideas of Mikhail Bakhtin [START_REF] Bakhtin | The dialogic imagination: Four essays[END_REF][START_REF] Bakhtin | Speech genres and other late essays[END_REF][START_REF] Bakhtin | Problems of Dostoevsky's poetics[END_REF], who extended the scope of dialogs emphasizing the role of considering the otherness, considering that any linguistic act, either spoken or written is a dialog. Moreover, the universality of dialog gives birth to dialogism, which "has been characterized as a metatheory for the human mind" [9, p. 30]. Koschmann [START_REF] Koschmann | Toward a dialogic theory of learning: Bakhtin's contribution to understanding learning in settings of collaboration[END_REF] was one of the first scholars to present a dialog-based model of CSCL, starting from Bakhtin's theories, which emphasizes the role of the language and especially of utterances, either spoken or not, language being a social mediator of knowledge building [START_REF] Hicks | Learning as a prosaic act[END_REF]. In many domains learning may be seen as mastering a speech genre, as becoming able to enter into a dialog or even to participate to a polyphony of voices [START_REF] Trausan-Matu | Polyphonic inter-animation of voices in VMT[END_REF]. These voices should be understood in a larger extend than the acoustic sense, as it will be discussed later. They may be actually present, like in a voiced conversation, but they may also be implicitly present in any text. From this perspective, Bakthin's dialogism suits not only collaborative knowledge building but also individual knowledge building -see also Stahl's two cycles model [START_REF] Stahl | Group cognition. Computer support for building collaborative knowledge[END_REF]-which implies explicit or implicit inner dialogs among learner's voice and the other voices (e.g., author's, professor's, commenters' voices).

The purpose of this paper is to present and discuss an attempt to operationalize Bakhtin's ideas on dialogism with Natural Language Processing (NLP) techniques, and to insert them in the context of distance learning. We have extended this dialogical view by using computer-based natural language technologies as a means to "expand the space of learning" by emphasizing the presence of the 'other voices' through generating automatic feedback. This feedback is given to different types of text written by students: essays, instant messenger (chats) and forums.

The remainder of this paper is as follows. We first detail the core concepts in Bakhtin's dialogism that could be at hand in every distance learning situation, namely utterances, voices and echoes. Then we present a comprehensive framework for instructional situations at a distance that take into account Bakhtin's ideas. Eventually, we present two computer-based systems (PolyCAFe and Pensum), also based on dialogical concepts. We assume that these systems can foster student's knowledge building, either individually or collaboratively.

Core Concepts in Bakhtin's Dialogism

We consider in this paper three core concepts from Bakhtin's work: utterances, voices and echoes. This section introduces each of them and explains how they can be used to describe usual instructional interactions. We further try to illustrate the application of Bakhtin's theory in the context of distance-learning environments. We claim here that this theory is particularly well suited to instructional computer-mediated situations. Let us consider students and a teacher/tutor engaged in a distance learning situation. They can either read texts, write out notes, essays, or discuss such or such topic. All the stakeholders performing these activities write texts or say something in natural language, producing "utterances" that can become "voices" populating the distance learning platform, responding to each other. The way the students can, upon a given question, gather information from multiple text sources in order to compose their own piece of text (mainly, summaries or syntheses) might be viewed as "contexts" in which they try to handle the polyphony of voices. Let us define these three core concepts and contextualize them in distance learning.

Utterances

Natural language traditional units of analysis are words and sentences. Signs or words are not the mere representation of something, but a mediation between people [START_REF] Lorino | Concevoir l'activité collective conjointe : l'enquête dialogique. Étude de cas sur la sécurité dans l'industrie du bâtiment[END_REF]. As Wegerif [5, p. 144] puts it: "Any sign taken to be a mediation between self and other, a word or a facial expression, must pre-suppose the prior opening of a space of dialogue (an opening of a difference between voices) within which such a sign can be taken to mean something." However, even if Bakhtin considers words as very complex because "they are filled with overtones of previous voices and contexts" [6, p. 293], he states that utterances should be the central unit of analysis [START_REF] Bakhtin | Speech genres and other late essays[END_REF]. Each utterance may be considered as a potential "voice" which enters in relations with other ones, either of the same emitter of the voice or of other persons, therefore appearing a multitude of voices which would be a desideratum to become a polyphony [START_REF] Trausan-Matu | Polyphonic inter-animation of voices in VMT[END_REF]. The notion of utterance differs from that of sentence, whose boundary is more precisely defined. The boundary of an utterance is the change of subject at stake in a discussion. Wegerif [5, p. 144] expressed that "the meaning of an utterance is given by its location within a dialogue", and adds further that its meaning is neither reducible to the speaker's intentions, nor the response, but emerges from these two (Bakhtin's notion of inter-animation).

An utterance is a piece of text whose boundary is the change of speech subject [7, p. 91], and is not a purely individual act. Additionally, one can answer to an utterance: "When hearing or reading, we clearly sense the end of the utterance, as if we hear the speaker's concluding dixi." [7, p. 76]. Thus the boundary between utterances is not necessarily the turn-taking, as Dong [14, p. 524] expresses it: "An utterance boundary is defined by the turntaking between speakers, that is, when one speaker stops talking and the next speaker begins. This orderly mapping can be revealed by looking at the coherence between any two utterances (communicative acts) as a function of the 'distance' between the utterance boundaries."

According to Bakhtin ([7, p. 91], see also [START_REF] Shotter | Conversational realities. Constructing life through language[END_REF]), an utterance is "[...] a link in the chain of speech communication of a particular sphere. […] Utterances are not indifferent to one another, and are not selfsufficient; they are aware of and mutually reflect one another… Every utterance must be regarded as primarily a response to preceding utterances of the given sphere (we understand the word 'response' here in the broadest sense). Each utterance refutes, affirms, supplements, and relies upon the others, presupposes them to be known, and somehow takes them into account… Therefore, each kind of utterance is filled with various kinds of responsive reactions to other utterances of the given sphere of speech communication". A major consequence is that having an utterance-based vision is forcing us to respond, to have an active, creative attitude, to interanimate.

In our vision, an utterance may be a simple word or interjection or it may be even a a whole novel, as also Bakhtin mentioned [START_REF] Bakhtin | Speech genres and other late essays[END_REF]. An utterance may contain other utterances, at a different degree of complexity, the process of delimiting an utterance being a difficult task.

Since many instructional activities are dialog-based, utterances formed by questions or answers can be analyzed from a Bakhtinian perspective: they are both acts of communication and pieces of discourse [START_REF] Linell | Rethinking language, mind, and world dialogically[END_REF] and represent the way teachers and students individually negotiate the construction of knowledge. The following notion of voice shows how utterances are embodied, having effects on the subsequent development of the conversations.

Voices and ventriloquism

An utterance may become a voice and may contain echoes and overtones of other voices [START_REF] Bakhtin | Speech genres and other late essays[END_REF]. Even if it is a central concept in Bakhtin's work, it is very hard to define exactly what a voice is and this difficulty is augmented by the phenomenon of ventriloquism [START_REF] Bakhtin | Problems of Dostoevsky's poetics[END_REF], which is the (re)emitting of a voice by another one. As defined elsewhere [START_REF] Trausan-Matu | Polyphonic inter-animation of voices in VMT[END_REF] in the context of linguistic communication, we consider a voice "not the acoustical, physical, vocal expression of a given participant in a dialog but, rather, a distinct position, an utterance, an event or a recurrent series of events of emitting utterances that are heard, remembered, discussed and have influence on the utterances emitted by the other voices. In music, for example, a voice is not fixed to an instrument; the same instrument may play several voices, and different instruments may take the position of a given voice, simultaneously or sequentially. A voice may be seen as a distinctive position in a group, a person or a group of people who have uttered something, with effects on the subsequent utterances […] Moreover, a voice has some particularities; it may have a personality, goals, beliefs, desires and emotions. Consequently, a dialog among several voices is not a dialog among impersonal entities. From another point of view, a voice may become a theme or may contribute to a theme of the discussion. " [START_REF] Trausan-Matu | Polyphonic inter-animation of voices in VMT[END_REF] Cazden [START_REF] Cazden | Vygotsky, Hymes, and Bakhtin: From word to utterance and voice[END_REF] restricts the notion of voice as follows, but to a larger extent, a voice can also be from another speaker: "Voice is Bakhtin's term for the 'speaking consciousness': the person acting-that is speaking or writing in a particular time and place to known or unknown others. Voice and its utterances always express a point of view, always enact particular values. They are also social in still a third meaning: taking account of the voices being addressed, whether in speech or writing. This dialogic quality of utterances Bakhtin calls 'responsivity' or 'addressivity'" [16, p. 198]. Linell [START_REF] Linell | Rethinking language, mind, and world dialogically[END_REF] remarks that there is an extension of the concept of voice in which it is viewed as perspectives or topics. There are generalized voices, "tied to a group of sense-makers, rather than a single individual", reflecting the "perspective on a topical domain" [9, p. 116]. Moreover, "the same person may appropriate, internalize or express several different voices, whether these voices are taken from other individuals or they are generalized voices" (ibid.).

Voices express ideas and opinions and "polyvocality" (co-presence of multiple voices) is sought in instructional distant situations. Teachers and students can hold ideas or opinions from multiple perspectives, which are encouraged in a collaborative knowledge building. The next notion of echo describes how different voices are replicated among contexts.

Contexts and echoes

From Bakhtin's perspective [7, p. 147] : "A context is potentially unfinalized; a code must be finalized. A code is only a technical means of transmitting information; it does not have cognitive, creative significance. A code is a deliberately established, killed context." From our point of view, a context may be considered as a slice of a discussion thread whose utterances have a strong cohesion. The participants of a context are either all the persons who utter at least once, or the authors of the texts under discussion. The voice may be seen as a selfcoherent line of contexts, uttered by one or several participants. The echo of a voice is a replication of it with enough strength to influence other voices in one or more contexts. Two kinds of echoes are taken into consideration: individual echoes, i.e., the way a given student internalize a voice, and collective echoes, i.e., the way several students react to a voice and enrich it.

A support is a channel dedicated to the conveying of voices (not necessarily those of the participants). Supports we consider are essays, chat discussions, texts read. The timeline is the time-to-time track of all the voices uttered during a learning period. It can be represented and visualized in a textual or a graphical form. Some information can be added to this timeline.

An example

For understanding the ideas of utterance, voice and echoes, let us consider the following problem, which was solved only collaboratively in a chat session by a group of students after they had the opportunity to solve it also individually but they didn't (see also [START_REF] Trausan-Matu | Polyphonic inter-animation of voices in VMT[END_REF]):

"Three years ago, men made up two out of every three Internet users in America. Today the ratio of male to female users is about 1 to 1. In that time the number of American females using the Internet has grown by 30,000,000, while the number of males who use the Internet has grown by 100%. By how much has the total internet-user population increased in America in the past three years? (A) 50,000,000 (B) 60,000,000 (C) 80,000,000 (D) 100,000,000 (E) 200,000,000"

In Figure 1 is a chat excerpt that includes the main utterances that contributed to the finding of the solution. The lines 360 to 364 may be seen as five utterances but they may be also considered (especially from the perspective of the discussion in section 2.1 about the boundaries of utterances) as a single utterance of Mic, which is becoming a voice which, even if it is a joke, a buffoonery, has echoes on the following voices: on Hal's utterance in line 371, which also becomes a voice, which in her turn has echo on Cosi's utterances 387, 392-396, 399-400 and 403-405, which inter-animate with Mic's and Hal's previous voices eventually becoming the voice that solves the problem. The text stating the problem and the chat excerpt constitute a context.

Utterances, Voices and Echoes All Together in Distance Learning Contexts

Detecting utterances during instructional interactions, voices expressing a given content and their echoes upon contexts can be of importance in distance learning, at least because a lot of these interactions are not multimodal (mostly text-based) and not synchronous (i.e., through chat or forums). Because of these features, the whole instructional process is opaque and the intentions of the stakeholders difficult to guess: teachers encounter difficulties to see what content is actually understood by students and the latter have difficulties to get just-intime feedback on their productions.

We present now a framework that both serves to represent the different levels of an instructional event and has to be compatible with Bakhtin's theory. Every instructional event is embedded within lesson episodes [START_REF] Leinhardt | The cognitive skill of teaching[END_REF], which have a single pedagogical intent. Each sequence can be split in segments. Figure 2 depicts the framework. Each instructional sequence can be viewed at three levels: a structural one (above the arrow), which comprises the main structural features of the episode (its goals, its taught knowledge, the planned pedagogical activities that have been designed by the teacher); a behavioral one (below the arrow), which encompasses the behaviors (either) of the stakeholders and a social one (all below), which pertains to the social scripts engaged in the instructional situation [START_REF] Weinberger | Epistemic and social scripts in computer-supported collaborative learning[END_REF]. The classical learning approaches focus on the structural level, the dialogistic perspective of Bakhtin giving light on the importance of social and behavioral levels. We can rephrase the activity of learning and understanding a new topic with Bakhtin's own words: a word is learnt when "it becomes one's own", when "the speaker populates it with his own intention, its own accent, when he appropriates the word, adapting it to his own semantic and expressive intention" [6, pp. 293-394]. First of all, from our perspective, that means that the learner uses the needed words in the chat and in summaries. Moreover, she should use them in inter-animation patterns [START_REF] Trausan-Matu | Supporting polyphonic collaborative learning[END_REF] for example, in repetitions, difference making, etc. In every instructional situation there are several interacting voices, more or less explicit, uttered by e.g., the author, the teacher, the learner that reads texts, the echoes of the voices present in each word. Moreover, from this multivocal perspective, texts become meaning generation mechanisms, facilitating understanding and creative thought [START_REF] Wertsch | Voices of the mind: A sociocultural approach to mediated action[END_REF][START_REF] Dysthe | The multivoiced classroom: Interactions of writing and classroom discourse[END_REF]. A consequence is that in education, "the interaction of oral and written discourse increased dialogicality and multivoicedness and therefore provided more chances for students to learn than did talking or writing alone" [21, p. 385]; the dialogic and multivoicedness features of any utterance, even written, may be unifying factors for the integration of several learning environments. This claim has two main consequences:

• One could use the "dialogicity" (or "inter-animation") of the interactions in an environment to assess the degree of quality of learning/teaching. The more dialogic the situation is, the more the students can learn from it. • One can use tools for analyzing the way these voices interact to each other.

Natural Language Processing techniques can be used to automatically analyze some features of the texts at hand during an instructional process at distance and to uncover some of their underlying utterances, voices and echoes. The next section is devoted to introduce two NLP-based systems (PolyCAFe and Pensum) pursuing this aim.

Dialogical NLP-based Systems

PolyCAFe and Pensum are respectively positioned on the collective and individual level of utterances. They use several NLP techniques that have been detailed elsewhere [START_REF] Trausan-Matu | Deliverable 5.2 -Learning support and feedback[END_REF][START_REF] Trausan-Matu | Deliverable D5.1 -Support and feedback design[END_REF].

PolyCAFe [START_REF] Trausan-Matu | A polyphonic model and system for inter-animation analysis in chat conversations with multiple participants[END_REF] analyses inter-animation and collaboration between participants to a distance learning situation. An utterance in the current version is considered an individual line input by the chat users, completed when the user enters a carriage-return. The echoes are determined starting from implicit or explicit links. From these links a graph is constructed connecting utterances and, in some cases, words. In this graph, contexts may be identified as groups of threads of implicit or explicit links. Simple examples of threads are repetitions of words or lexical chains. The same utterance may, of course, be included in several threads. Multivocality is detected by the presence of interwoven threads.

For example, in Figure 3, a high degree of collaboration (showed in the graph from the middle of the image) between utterances 150 to 200 (see the numbered scale) is generated due probably to the high degree of interanimation (debate) among the usage of the words 'chat' and 'forum' (whose usage is shown in the top part of the figure). The debates on these words will echo what the learners will further read and the summaries they will write.

To that end, a second system, Pensum [START_REF] Villiot-Leclercq | Helping students understand courses through written syntheses. An LSA-based online advisor[END_REF] (see Figure 4), has been devised to analyze individual threads of voices created when students read course materials and write out syntheses or summaries. In Pensum utterances are short pieces of syntheses, and not dialog-based turn-taking as in PolyCAFe. Pensum focuses on semantic relations between the authors of courses and students. The voice of the teacher should be present in the readingsummarizing loop if the learner attended and understood the lectures. Similarly, the voices of the authors of teaching documents should be present in the texts written by the students, if they read and understood the important concepts. The voice of the learner should also be present in the creative writing, in which the learned concepts are deeply internalized. All these voices are embedded in cohesive contexts formed by students' syntheses, which represent the current state of understanding of a given course content. Pensum uses Latent Semantic Analysis (LSA) for detecting the presence of teacher's and teaching document authors' voices in students' summaries.

Conclusions and Further Developments

This study is the first step of our modeling of Bakhtin's views with NLP techniques. Despite some difficulties due to the vagueness and the multiple meanings carried by these three notions, we plan to improve their operationalization with NLP techniques in trying to improve the facilities offered by the implemented tools and their effectiveness in instructional settings. A very important point is to recognize whether or not a discussion is monologic or dialogic, polyphonic. Multivocality or polyphony means that there are permanently several voices entering in competition. Each utterance is filled with "overtones" of other utterances. A first problem is to detect these overtones. The first experiments with PolyCAFe are encouraging in this sense.

Pensum allows students and teachers to instantly recognize (and be aware of) the sources ("voices") of what they are learning and teaching. This awareness is very important, first to prevent from plagiarism and also for students' self-regulated learning.

Philippe Quéau : Pendant presque vingt ans, à Imagina, ma responsabilité a été de trouver, chaque année, les présentations les plus étonnantes, les plus révélatrices, les plus stimulantes. Mais vient un moment où l'on se pose des questions d'une autre nature, où l'on ne peut plus faire l'économie de la question du sens, où l'on devient sensible à des contradictions de plus en plus flagrantes, par exemple à l'élargissement du fossé dit "numérique" (digital divide). Internet va-t-il accroître les inégalités, entre le Nord et le Sud ? Certains le pensent, d'autres croient à une progression rapide des utilisation d'Internet au Sud. Les Cassandre expliquent qu'Internet ne s'implantera pas durablement dans les pays en développement et construisent une attitude protectionniste qui ne résistera cependant pas "au vent de l'histoire ni au désir de la jeunesse de prendre sa place dans le concert mondial" .Au delà des discours volontaristes, nous distinguons A further implementation would allow us to use these two systems in an integrated way. Through the framework depicted in Figure 2, we plan to model with NLP techniques a large range of the text-based learning activities such as searching documents, reading, writing summaries or forum posts, chatting, and even "moving" in a distance learning platform.

Since chat utterances (and even short course notes) can be very short pieces of texts, their analysis with LSA is difficult. According to Bakhtin's view of polyphony, our aim is to detect the "contexts" of a chat discussion. The method may be as follows:

1. Split the discussion into cohesive pieces of text. Each of them is called a context; 2. Enrich the context. In each of the contexts, the texts each student who participated to the context read or wrote are added, with a specific weight.

Several methods have to be tested for step 2. There are many ways to give an adequate weight to the texts. For instance, the texts read are added once while those written out are added twice in the database. We alternately plan some more elaborated ways of weighting, e.g., add the enrichment proportionally to the semantic relation of the utterance to the texts read/written. For instance, when an utterance has a weak relation to the texts previously read/written, its weight will be also weak (added once); it will be added many times as its semantic relation with texts read/written increases.

350Fig. 1 .

 1 Fig. 1. An excerpt illustrating a collaborative solution construction

Fig. 2 .

 2 Fig. 2. Graphical representation of an instructional situation.

Fig. 3 .

 3 Fig. 3. Inter-animation visualization in PolyCAFe.

  4 types d'analyses autour de la diffusion d'Internet en Afrique ?celles-ci prennent souvent le domaine éducatif comme référence. Le but de cet article est de repérer les types et les fonctions des aide mémoire AM que l on trouve dans les classes d enseignants d école primaire et d en proposer une première typologie. Nous décrivons tout d abord l enseignement comme la supervision et le contrôle d un environnement dynamique, ce qui nous amènera ensuite à montrer en quoi les AM peuvent faciliter ces activités.Dans une troisième partie, nous détaillons une observation sur la présence et l usage d AM auprès de six enseignants expérimentés d école élémentaire cycle 3 , suivie d un entretien visant à les faire s exprimer sur la manière dont ils produisent et utilisent des aide mémoire dans leur pratique quotidienne.

Fig. 4 .

 4 Fig. 4. Pensum and its two main windows. The course window (top) and the synthesis window (bottom). The underlining warns the student if the corresponding sentences are not in the synthesis (top) or are not in the course (bottom).

Acknowledgements

The research presented in this paper was partially performed under the FP 7 EU STREP project LTfLL (Language Technologies for Lifelong Learning). The authors wish to thank Benoît Lemaire, Sonia Mandin, Emmanuelle Villiot-Leclercq and Virginie Zampa who collaborated on the implementation of Pensum; as well as Traian Rebedea, Mihai Dascalu, Alexandru Gartner, Costin Chiru, Dan Banica and Dan Mihaila who collaborated on the implementation of PolyCAFe. Thanks to the MSH-Alpes which provided material help.


Rémi Bonnefoi

Christophe Moy

Jacques Palicot

Framework for Hierarchical and Distributed Smart Grid Management

We use the Hierarchical and Distributed Cognitive Radio Architecture Management (HDCRAM) inherited from cognitive radio as a generic framework for smart grid management. This architecture is an efficient tool which can be used for the design of decision making algorithms in complex systems. We apply this architecture on the smart grid and we identify both intelligent entities and communication networks which compose the management part of the smart grid. Furthermore, we consider which radio standards can be used for each link.

Introduction

In a smart grid, information and communication technologies are used to improve reliability, security and efficiency of the electrical network [START_REF] Moslehi | A reliability perspective of the smart grid[END_REF]. On the one hand, decision making algorithms are used to manage the grid. On the other hand, a communication system is necessary to transfer the information between the entities which compose the smart grid, e.g. sensors and decision making (intelligent) entites. Furthermore, in a smart grid, the management entities are heterogeneous (in term of processing capabilities, energy consumption, etc.), distributed and used for various applications. In such scenario, efficient management strategies have to be built on a framework. That is why several hierarchical architecture have been proposed as frameworks for smart grid management. In [START_REF] Dou | Multi-agent based hierarchical hybrid control for smart microgrid. Smart Grid[END_REF], the authors propose a hierarchical architecture for microgrid management. In [START_REF] Lloret | An integrated iot architecture for smart metering[END_REF], a three level architecture is proposed for the management of the Advanced Metering Infrastrucutre (AMI). However, these architectures are designed for only one application or one part of the smart grid. The Smart Grid Coordination Group (SG-CG) has proposed the Smart Grid Architecture Model (SGAM) as a general framework for smart grid management [START_REF]Overview of SG-CG Methodologies[END_REF][START_REF]Smart Grid Reference Architecture[END_REF]. The SGAM architecture focuses on the interoperability between the different part of the smart grid. In [START_REF] Palicot | Application of hierarchical and distributed cognitive architecture management for the smart grid[END_REF], the authors of this paper proposed the Hierarchical and Distributed Cognitive Radio Architecture Management (HDCRAM) for the management of the smart grid. However, in our previous work, we did not consider that the different parts of the grid are managed by different mechanisms and can be managed independently. In this paper we show that the HDCRAM architecture can be used for the management of the different parts of the grid and we also identify the communication networks that should be used appropriately at each level of management.

The aim of this paper is to describe the HDCRAM architecture as a generic framework for smart grid management. We show that HDCRAM can be used either for the management of the electricity production and consumption or for transmission and distribution grids automation. We identify the management entities and the communication networks used between these entities.

The rest of this paper is organized as follows. The communication networks used for the smart grid are described in section 2. HDCRAM architecture is described in section 3. In section 4, we use HDCRAM as a framework for smart grids and we identify the role of the intelligent entities and of the communication networks in the architecture. Finally, section 5 concludes this paper.

Communication Networks for the Smart Grid

Several communication networks must be used for the management of the smart power grid. In [START_REF]Overview of SG-CG Methodologies[END_REF], the Smart Grid Coordination Group describes fourteen communication networks used for smart grid management. In table 1, we have sorted the networks which could be used for smart grid management by range: short range (e.g. Home Area Network (HAN), Neighbourhood Area Network (NAN) and Wide Area Network (WAN)) and depending on the application field. We consider three application fields; consumer side, production side and grid monitoring. The main networks used for the management of the power consumption are used for home automation and for the advanced Metering Infrastructure (AMI). The production field includes all the networks used for the management of power plants (nuclear plants, solar plants, etc.). The last field includes all the communication networks used for the distribution and transmission grids automation which is mainly done in substations. Smart grid communications are done through many networks which will use several communication technologies. In such situation with multi-standard communications, it is interesting to use a cognitive radio approach based on software defined radio (SDR) [START_REF] Palicot | Radio Engineering: From Software Radio to Cognitive Radio[END_REF].

All the networks described in table 1 have their own constraints in term of throughput, latency, reliability and security [START_REF]Communications Requirements of Smart Grid Technologies[END_REF]. As a consequence, the communication technolo- gies have to be adapted to the network requirement. We have listed in table 1 some of the wireless standards which can be used for each network. A more complete list can be found in [START_REF]Smart Grid Reference Architecture[END_REF].

Hierarchical and Distributed Cognitive Radio Architecture Management

The Hierarchical and Distributed Cognitive Radio Architecture Management (HDCRAM) is a logical architecture which was first proposed for the management of cognitive radio systems [START_REF] Godard | An executable metamodel of a hierarchical and distributed architecture management for cognitive radio equipments[END_REF]. This three levels architecture is shown in Figure 1.

This architecture comprises two managing sub-parts. The management of the production and consumption in a smart grid can be viewed at different levels. In figure 2, HDCRAM architecture is used for the local management of the grid. The proposed three level architecture shows the management from smart appliances to the local substation. Level 1 manager is in the substation and can make the link between the local grid (which can be a microgrid) and the rest of the distribution network.

In figure 3, the management architecture is used at a higher level. Smart homes, buildings, power plants and other energy producers and consumers are connected to the level 3 entities and the level 1 entity is a global manager of the production and consumption. This entity manages the production and consumption over a large area (e.g. a country).

Both proposed HDCRAM architectures of figure 2 and3 show the different communication networks which are used between the management entities. Most of the networks used for the management of the production and consumption part of the grid do not have tight constraints in reliability (packet delivery ratio) and latency. For each of these networks, a large number of communication technologies can be used as given in table 1. As an example, Wifi, Bluetooth or Zigbee can be used for local networks and cellular or Low Power Wide Area Networks can be used for long range communications.

HDCRAM for Transmission Grid Management

The management of the electricity production and consumption is not the only mechanism used for the management of the grid. Indeed, sensors and actuators are also used for distribution and transmission grids automation. The management of the transmission part of the grid is done by Phasor Measurement Units (PMUs) and Intelligent Electronic Devices (IEDs) [START_REF] Matsumoto | Wide-area situational awareness (wasa) system based upon international standards[END_REF]. The information collected by PMUs is transmitted to the Phasor Data Concentrator (PDC) which can transmit it to the transmission coordinator. HDCRAM architecture used for the management of the transmission grid is shown in figure 4. Only one communication network is used for the transmission grid management. The inter-substation network is a wide area network with stringent constraints in latency and reliability [START_REF]Smart Grid Reference Architecture[END_REF]. In [START_REF] Phadke | Communication needs for wide area measurement applications[END_REF], it is specified that the communications latency for some PMUs applications (e.g. islanding) should not exceed 50ms. Consequently, only a few technologies can be used. Most of these technologies are wired technologies but the LTE standard can meet these communication requirements as given in table 1.

HDCRAM for Distribution Grid Management

Compared to transmission grid, the distribution grid is currently being considerably improved to handle renewable energy resources. Mechanisms similar to those used for the transmission grid will be used for the distribution grid. The management of the smart distribution grid is done through smart sensors and actuators. The framework used for distribution grid management is shown in figure 5.

In [START_REF]Smart Grid Reference Architecture[END_REF], the inter-substation network is used for both the transmission and distribution part of the grid. However, with the integration of renewable energy resources and the development of microgrids, the distribution grid will require more stringent constraints than the transmission grid.

In [START_REF] Budka | Communication Networks for Smart Grids: Making Smart Grid Real[END_REF], a 4ms latency is required for distribution grid monitoring. This stringent requirements can be met with wired or wireless protocols specifically designed for ultra lowlatency communications.

Conclusion

In this paper, the HDCRAM architecture is presented as a general framework for smart grid management. We show that many networks have to be used in a smart grid. Each network has specific constraints which depends on the application and on the range. We have mapped HDCRAM architecture on the different parts of the grid and we have identified the different management entities and the networks which compose the future grid. These architectures can represent the same part of the grid with a different granularity. This can ease the interconnection between the different visions.

Acknowledgements

Part of this work is supported by the project SOGREEN (Smart pOwer GRid for Energy Efficient small cell Networks), which is funded by the French national research agency, under the grant agreement coded: N ANR-14-CE28-0025-02 and by Région Bretagne, France.

Figure 1 .

 1 Figure 1. HDCRAM has 3 management levels and is composed by two independant paths. A cognitive path from bottom to top for retrieving sensing information to cognitive units and a configuration path from top to bottom.

4

  Communication Networks in HDCRAM for Smart Grid 4.1 HDCRAM for Grid Production and Consumption ManagementOne of the main tasks of smart grid management is to balance the electricity production and consumption to avoid the deterioration of the grid. Consequently, the proposed management architecture must handle both production and consumption part of the smart grid.

Figure 2 .

 2 Figure 2. The HDCRAM architecture used as a management architecture for the local management of the electricity consumption.

Figure 3 .

 3 Figure 3. The HDCRAM architecture used as a management architecture for the global management of the electricity production and consumption

Figure 4 .

 4 Figure 4. HDCRAM architecture used for the management of the transmission grid.

Figure 5 .

 5 Figure 5. HDCRAM architecture used for the management of the distribution grid.

Table 1 .

 1 Communication networks for different levels of smart grid management

	Field	Range	networks	Wireless Technologies and standards
		Short range	Subscriber access network, Home automa-	Zigbee, Bluetooth, Wifi
	Consumer field	(HAN)	tion
		NAN	Neighborhood access network, Backbone	Zigbee, Wifi, WirelessHART
			network
		WAN	AMI backhaul	GSM, GPRS, UMTS, LTE, WiMAX, Lo-
				RaWAN, NB-IoT, Sigfox
	Grid monitoring	Short range Intra-substation Low-end intra-substation	Most of the protocols envisioned for this network are wired, e.g. IEC 61850
		WAN	Inter-substation	LTE, LTE-M, NB-IoT, 5G
	Production field	Short range Industrial fieldbus WAN Balancing network	Zigbee, Bluetooth, Wifi GSM, GPRS, UMTS, LTE, WiMAX, Lo-
				RaWAN, NB-IoT, Sigfox


D Duhamel
email: duhamel@lami.enpc.fr

Paper submitted to Engineering analysis with boundary elements FINITE ELEMENT COMPUTATION OF GREEN'S FUNCTIONS

Keywords: Green's function, periodic structure, wave, waveguide, finite element, dynamic

Green's functions are important mathematical tools in mechanics and in other parts of physics. For instance, the boundary element method needs to know the Green's function of the problem to compute its numerical solution. However, Green's functions are only known in a limited number of cases, often under the form of complex analytical expressions. In this article, a new method is proposed to calculate Green's functions for any linear homogeneous medium from a simple finite element model. The method relies on the theory of wave propagation in periodic media and requires the knowledge of the finite element dynamic stiffness matrix of only one period. Several examples are given to check the accuracy and the efficiency of the proposed numerical Green's function.

Introduction

This article presents an algorithm for the general numerical computation of Green's functions for linear homogeneous media of arbitrary types from simple finite element models. Green's functions give the solution of a problem described by a partial differential equation when it is submitted to a unit excitation. These are fundamental tools since if the Green's function for a linear differential equation is known, then it is possible to calculate, by simple convolution type integrals, the solutions of the same problem for an arbitrary loading. They are also the basic tools to formulate a boundary integral equation to solve the partial differential equation describing the physical problem. So the search for Green's functions for the various physical problems in acoustics, solid mechanics, heat conduction and other parts of mechanics and physics played an important role in the past for solving these problems.

For the simplest cases the Green's functions are known for a long time and are described in classical books like Morse and Feshbach [START_REF] Morse | Methods of Theoretical Physics[END_REF], Courant and Hilbert [START_REF] Courant | Methods of Mathematical Physics[END_REF], Carslaw and Jaeger [START_REF] Carslaw | Conduction of Heat in Solids[END_REF], Roach [START_REF] Roach | Green's functions, 2 nd Edition[END_REF] and Bonnet [START_REF] Bonnet | Boundary integral equation method for solids and fluids[END_REF]. One has, for instance, solutions for two-and three-dimensional acoustics, for two-and threedimensional elastostatics and elastodynamics or for diffusion equations. Solutions can be in the time or frequency domains according to the equation describing the physical phenomenon. These solutions are given as analytical formulas, often involving special functions.

For example, Yang and Pan [START_REF] Yang | Three-dimensional Green's functions in anisotropic trimaterials[END_REF] found the static Green's function for anisotropic trimaterials while Bai et al. [START_REF] Bai | Three-dimensional steady state Green function for a layered isotropic plate[END_REF] considered a layered isotropic plate and found the elastodynamic Green's function by using a modal summation. The Green's functions for half spaces were also studied by many authors like Pan [START_REF] Pan | Static Green's functions in multilayered half spaces[END_REF] who found the Green's functions for static multilayered transversely isotropic half spaces and Spies [START_REF] Spies | Green's tensor function for Lamb's problem: The general anisotropic case[END_REF] who considered the general static anisotropic Green's function for a half-space. He and Lim [START_REF] He | Surface Green function for a soft elastic half-space: Influence of surface stress[END_REF] studied the Green's function for a soft half-space with surface stress. More complicated cases with several behaviors were also considered by Ding et al. [START_REF] Ding | Green's functions and boundary element method for transversely isotropic piezoelectric materials[END_REF] where the authors found the Green's function for transversally isotropic piezoelectric materials. In all these cases except the simplest ones, the Green's functions are found by complex analytical expressions, often involving series expansions. For media with complex properties involving several materials or coupled mechanical models, the Green's functions are generally unknown.

Here, we present a numerical method to find such Green's functions. It is supposed that the medium is two-dimensional, linear, infinite, homogeneous or periodic but can otherwise be arbitrary. The approach is based on the theory of periodic media and one supposes that a finite element model of a period is given from standard finite element software. Similar ideas were used in Mace et al. [START_REF] Mace | Finite element prediction of wave motion in structural waveguides[END_REF] and Duhamel et al. [START_REF] Duhamel | Finite element analysis of the vibrations of waveguides and periodic structures[END_REF] in the one-dimensional case. The twodimensional case was considered by Langley [START_REF] Langley | The response of two-dimensional periodic structures to point harmonic forcing[END_REF][START_REF] Langley | The response of two-dimensional periodic structures to impulsive point loading[END_REF][START_REF] Langley | The response of twodimensional periodic structures to harmonic point loading: a theoretical and experimental study of a beam grillage[END_REF] who found expressions based on Bloch's waves for the response to point loading for harmonic and impulsive loads. But the analysis was mainly theoretical with simple examples and no connection with the finite element method was made. We consider here only two-dimensional problems to simplify. From the dynamic stiffness matrix, waves in different directions are calculated and the Green's function is found as an integral of wave functions.

The paper is divided in two parts. In the first part, the numerical method is presented and the formula giving the Green's function is obtained. In the second part, different examples are given to check the method against known analytical Green's functions and a more complex example is also presented to calculate the Green's function of a two materials periodic medium for which no simple solutions are known.

Green's functions calculation

We consider an infinite periodic or homogeneous medium. The Green's function G(x 1 , x 2 ) is the response at a point x 1 in the medium for a unitary excitation at an other point x 2 as in figure 1. As an homogeneous medium is only a special case of a periodic medium, in the following, only periodic media will be considered. The analysis is also limited to the two-dimensional case and the elementary period is defined by the domain 0 ≤ x 1 ≤ L 1 and 0 ≤ x 2 ≤ L 2 . The excitation is in a cell and the point where the response is calculated can be in another cell, see figure 2.

Pseudo periodic functions

It is proved in appendix A that a function U(x 1 , x 2 ) defined on the twodimensional periodic medium can be decomposed as an integral of pseudo periodic functions

U(x 1 , x 2 ) = π L 2 -π L 2 e ikx 2 Û(x 1 , k, x 2 )dk (1)
where Û(x 1 , k, x 2 ) is a periodic function in x 2 with period L 2 . Along direction 1, we use a decomposition in Bloch waves as it is usual in periodic media.

Finally, the general solution can be obtained from functions u(x 1 , k, x 2 ) = e ikx 2 Û (x 1 , k, x 2 ) with the following pseudo periodic properties

u(x 1 , k, x 2 + m 2 L 2 ) = e ikm 2 L 2 u(x 1 , k, x 2 ) u(x 1 + m 1 L 1 , k, x 2 ) = e im 1 µ 1 u(x 1 , k, x 2 ) (2)
where m 1 and m 2 are integers and the number k is real and belongs to

[-π L 2 , π L 2 ].
Along dimension 1 we will suppose, according to the Floquet's theorem, that µ 1 is a complex number. In the following, we are looking for numerical approximations of functions which satisfy relations (2).

Behavior of a cell

A cell is defined as a two-dimensional domain from which the whole plane can be obtained by tessellation. It is the smallest period of the medium and was defined previously as the domain 0 ≤ x 1 ≤ L 1 and 0 ≤ x 2 ≤ L 2 . A cell can be meshed with an arbitrary number of elements using the full possibilities of usual finite element software. The discrete dynamic behavior of a cell obtained from a finite element model at a frequency ω is given by

(K -iωC -ω 2 M)q = f (3)
where K, M and C are the stiffness, mass and damping matrices respectively, f is the loading vector and q the vector of the degrees of freedom. The stiffness, mass and damping matrices could be obtained from any commercial finite element software and so this allows the consideration of cells with complex structures. The size of these matrices depends on the number of elements used to mesh the cell and can be arbitrarily increased for a better precision of the results. Introducing the dynamic stiffness matrix D = K -iωCω 2 M, decomposing into boundary (B) and interior (I) degrees of freedom, and assuming that there are no external forces on the interior nodes, results in the following relation

          DBB DBI DIB DII                     q B q I           =           f B 0           (4)
The assumption that there are no forces on the interior degrees of freedom is satisfied for free waves inside the structure for which the forces on a cell are only produced by boundary forces from the adjacent cells. The interior degrees of freedom can thus be eliminated to get

f B = ( DBB -DBI D-1 II DIB )q B ( 5
)
Dropping the B index, this relation is written in the sequel as

f = Dq (6)
and only boundary degrees of freedom will be considered now.

Elimination of transverse dofs

Consider the situation described in figure 3. The boundary degrees of freedom are divided into the longitudinal dofs, denoted q l which are the dofs such that the first coordinate equals an integer number of the first period L 1 and the transverse dofs which are the other boundary dofs. These transverse dofs can be divided into two subgroups denoted q B for the bottom dofs and q T for the top dofs. The top dofs are obtained from the bottom dofs by the pseudo periodic boundary conditions [START_REF] Courant | Methods of Mathematical Physics[END_REF].

The cells are supposed to be meshed with an equal number of nodes on their opposite edges. The boundary can be decomposed into left (L), right (R), bottom (B) and top (T) parts and the associated corners (LB), (RB), (LT) and (RT) as shown in figure 4 and in Abdel-Rahman [START_REF] Abdel-Rahman | Matrix analysis of wave propagation in periodic systems[END_REF]. The behavior in direction 2 is supposed to be defined by the propagation constant e ikL 2 where

k is a wavenumber in [-π L 2 , π L 2 ].
In this case, one has

q l =                                        q L q R q LB q RB q RT q LT                                        (7)
The relations between the transverse dofs are therefore

q T = e ikL 2 q B f B + e -ikL 2 f T = 0 (8)
The behavior of a cell, given by relation [START_REF] Diaz-Contreras | Green's function applied to solution of Mindlin plates[END_REF], can also be written as

                 D ll D lB D lT D Bl D BB D BT D T l D T B D T T                                   q l q B q T                  =                  f l f B f T                  (9)
One gets from ( 8) and rows 2 and 3 of relations ( 9)

(D Bl + e -ikL 2 D T l )q l + (D BB + D T T + e -ikL 2 D T B + e ikL 2 D BT )q B = 0(10) so q B = -(D BB + D T T + e -ikL 2 D T B + e ikL 2 D BT ) -1 (D Bl + e -ikL 2 D T l )q l (11)
and finally the condensed behavior of the cell with only the longitudinal dofs is given by

f l = [D ll -(D lB + e ikL 2 D lT )(D BB + D T T + e -ikL 2 D T B + e ikL 2 D BT ) -1 ×(D Bl + e -ikL 2 D T l )]q l ( 12
)
whose notation will be simplified to

f l = D l q l ( 13
)
One can notice that we have

T D l (k) = D l (-k) (14)

Elementary degrees of freedom

We go on by a further partitioning of the left and right boundaries. The longitudinal displacement and load vectors can be partitioned as in relation ( 7)

q l =                                        q L q R q LB q RB q RT q LT                                        f l =                                        f L f R f LB f RB f RT f LT                                        (15)
So, the pseudo periodic conditions (2) lead to the following relations between the displacement degrees of freedom q R = e iµ 1 q L q RB = e iµ 1 q LB q RT = e i(µ 1 +kL 2 ) q LB q LT = e ikL 2 q LB [START_REF] Ding | Green's functions and boundary element method for transversely isotropic piezoelectric materials[END_REF] We see that the displacement components depend on an elementary set of degrees of freedom which is denoted q T r = [q L q LB ] T and is presented in figure 4. The vector q l can then be expressed as

q l = (W 0 + e iµ 1 W 1 )q r ( 17
)
where the matrices W 0 and W 1 depend on the wave number k and are given by the following relations

W 0 =                                        I O O O O I O O O O O e ikL 2 I                                        W 1 =                                        O O I O O O O I O e ikL 2 I O O                                        (18)

Spectral problem

The equilibrium conditions between adjacent cells written on the reduced set of degrees of freedom lead to the following relations between the components of the force vector

e iµ 1 f L + f R = 0 e iµ 1 f LB + f RB + e i(µ 1 -kL 2 ) f LT + e -ikL 2 f RT = 0 (19)
that can also be written as

          e iµ 1 I I O O O O O O e iµ 1 I I e -ikL 2 I e iµ 1 -ikL 2 I           f l = 0 (20)
or

(e iµ 1 W * 0 + W * 1 )f l = 0 (21)
Relations ( 13), ( 17) and ( 21) finally give

(e iµ 1 W * 0 + W * 1 )D l (W 0 + e iµ 1 W 1 )q r = 0 (22)
that can also be written as

e iµ 1 (W * 0 D l W 0 + W * 1 D l W 1 ) + e 2iµ 1 W * 0 D l W 1 + W * 1 D l W 0 q r = 0 (23) or A 0 + e iµ 1 A 1 + e 2iµ 1 A 2 q r = 0 ( 24
)
where

A 0 = W * 1 D l W 0 A 1 = W * 0 D l W 0 + W * 1 D l W 1 A 2 = W * 0 D l W 1 (25)
Relation (24) give the spectral problem to be solved to find the propagation constants e iµ 1 and the vectors q r . The propagation constant e iµ 1 , the matrices A 0 , A 1 , A 2 and the eigenvector q r depend on the transverse wavenumber k but to simplify the notation this dependence is not explicitly written. The eigenvector of (24) associated with the eigenvalue e iµ i is denoted by q i . By taking the determinant of (23) and of its transpose for -k and noting that k), using relation ( 14), it can be shown that if e iµ i is an eigenvalue for the wavenumber k then e -iµ i is also an eigenvalue for the wavenumber -k. These represent a pair of positive and negative-going waves.

W * i = t W i = t W i (-
This is true for any shape or property of the cell.

The 2n eigenvalues of equation ( 24) can be split into two sets of n + and n -eigenvalues and eigenvectors with 2n = n + + n -. They are denoted by (e iµ + i , q + i ) and (e iµ - i , q - i ), with the first set such that e iµ + i ≤ 1. From an eigenvector q i , one gets the longitudinal vector by relation [START_REF] Mace | Finite element prediction of wave motion in structural waveguides[END_REF] and the longitudinal component of the force vector applied on a cell is given by

f i = D l (W 0 + e iµ i W 1 )q i (26)
The forces of left cells on right cells are then given by

f r i = W * 0 f i = W * 0 D l (W 0 + e iµ i W 1 )q i ( 27
)
In the case |e iµ i | = 1, the set of positive-going waves must contain the waves which are such that Re iωq H i f r i > 0 where f r i is the reduced set of force degrees of freedom on the boundary obtained by relation (27). The eigenvalue e iµ - i , in the second set, is associated with waves such that Re iωq H i f r i < 0.

Green's function on a periodic array

We can now build the complete Green's function. The Green's function gives the solution of the problem defined on the periodic medium for a point force excitation. Suppose that the force is located at point (0, 0), we look for a Green's function which is such that in the domain m 1 ≥ 0 the solution is

given by

q r (m 1 L 1 , m 2 L 2 ) = π L 2 -π L 2 i=n + i=1 a + i q + i e im 1 µ + i e ikm 2 L 2 dk (28)
because the waves must propagate from the source to infinity, so the negative waves do not appear in the sum. In the domain m 1 ≤ 0, the solution must be

q r (m 1 L 1 , m 2 L 2 ) = π L 2 -π L 2 i=n - i=1 a - i q - i e im 1 µ - i e ikm 2 L 2 dk (29)
At m 1 = 0 there are continuity of the displacements and the equilibrium condition on the forces can be written by analogy with formula (21) by

i=n + i=1 W * 0 f + i + i=n - i=1 e -iµ - i W * 1 f - i = -f r ext ( 30
)
where f - i and f + i are, the respective contributions of the negative and positive waves on the forces. The expression of f ext for a point force excitation of amplitude F is given in appendix A. The relations on the displacements and forces lead for each k to

i=n + i=1 a + i q + i - i=n - i=1 a - i q - i = 0 i=n + i=1 a + i W * 0 D l (W 0 + e iµ + i W 1 )q + i + i=n - i=1 a - i W * 1 D l (e -iµ - i W 0 + W 1 )q - i = - L 2 2π F (31)
We get 2n relations for 2n unknowns. So these relations allow to get the amplitudes a + i and a - i and then the values of the Green's function from relations (28) and (29). So the Green's function has been found as an integral of solutions to one-dimensional periodic problems.

Numerical calculation of integrals

The Green's function is obtained from relations (28) and (29). These relations are of the following form

q r (m 1 L 1 , m 2 L 2 ) = π L 2 -π L 2 Q(k)e ikm 2 L
k i a Q(k)e ikm 2 L 2 dk = √ k 2 i -a 2 0 u k 2 i -u 2 Q( k 2 i -u 2 )e i √ k 2 i -u 2 m 2 L 2 dk (33)
which allows to remove singularities in 1/ k 2 i -k 2 in Q. Similar expressions are used for the singularities at lower interval bounds and for negative values of k.

At the present state no further optimization was made to improve the numerical evaluation of these integrals. For the following examples the computing time can be evaluated to a few seconds per couple of source and receiver points for each frequency. The program is written with MATLAB r and some improvement could be expected by considering the same computation in C or Fortran. One should also see from formulas (28) and (29) that the computation for one or several points in space are quite similar. Most of the computing time is devoted to finding the wavefunctions and the eigenvalues which need to be done only once. After it is easy to compute the Green's functions for different points sources and different receivers with little additional cost. One should also notice that formula (28) requires the computation of oscillatory integrals and for points far enough from the sources asymptotic formulas as the stationary phase method could be used to dramatically reduce the computing time. All these issues should be discussed in more details if the present method is used in a BEM problem.

Generalization to non rectangular cells

In the precedent discussion the elementary cell was supposed to be rectangular. Consider now a quadrilateral cell as in figure 5 which has a diamond shape. Along direction 2 the pseudo periodic condition does not change. Along direction 1 one has to substitute e i(µ 1 +kd) to e iµ 1 in the precedent relations.

For example relation ( 24) should be substituted by

A 0 + e iµ 1 (e ikd A 1 ) + e 2iµ 1 (e 2ikd A 2 ) q r = 0 (34)
One has also to pay attention that m 1 steps along direction 1 and m 2 steps along direction 2 from the origin lead to the point (m 1 L1, m 1 d + m 2 L2).

Examples

Acoustics

The Green's function of the two-dimensional acoustics is the radiating solution

of ∆p + K 2 p = -δ(r) (35)
where K = ω/c is the wavenumber and c the sound velocity. It is well known that the solution of this equation for the time dependence e -iωt is given by

G(r) = i 4 H 0 (Kr) ( 36
)
where H 0 is the Hankel function of order zero and first type. We first propose to check the numerical procedure developed in this article with this Green's function. Consider the rectangular four nodes acoustic element of size a × b presented in figure 6, the stiffness and mass matrices are given by

K =                               1 3 b 2 + a 2 ab 1 6 -2b 2 + a 2 ab - 1 6 b 2 + a 2 ab - 1 6 -b 2 + 2a 2 ab 1 6 -2b 2 + a 2 ab 1 3 b 2 + a 2 ab - 1 6 -b 2 + 2a 2 ab - 1 6 b 2 + a 2 ab - 1 6 b 2 + a 2 ab - 1 6 -b 2 + 2a 2 ab 1 3 b 2 + a 2 ab 1 6 -2b 2 + a 2 ab - 1 6 -b 2 + 2a 2 ab - 1 6 b 2 + a 2 ab 1 6 -2b 2 + a 2 ab 1 3 b 2 + a 2 ab                               M = ab 4c 2                         
                         (37)
and then the dynamic stiffness matrix is D = K -ω 2 M. As in this case there is only the q LB degree of freedom, the matrices W 0 and W 1 are given by

W 0 =                          1 0 0 e ikb                          W 1 =                          0 1 e ikb 0                          (38)
Using relations (24) the matrices of the spectral problem, which in this case reduce to simple scalars, are given by

A 0 (k) = W * 1 (k)D l (k)W 0 (k) = - 1 18 12 b a -6 a b + 2k 2 ab + (6 b a + 6 a b + k 2 ab) cos(kb) A 1 (k) = W * 0 (k)D l (k)W 0 (k) + W * 1 (k)D l (k)W 1 (k) = - 2 9 -6 b a -6 a b + 2k 2 ab + (-3 b a + 6 a b + k 2 ab) cos(kb) A 2 (k) = W * 0 (k)D l (k)W 1 (k) = - 1 18 12 b a -6 a b + 2k 2 ab + (6 b a + 6 a b + k 2 ab) cos(kb) (39)
and the solutions to the spectral problem ( 24) are

e iµ 1 = (-A 1 ± A 2 1 -4A 0 A 2 )/(2A 2 ) e -iµ 1 = (-A 1 ∓ A 2 1 -4A 0 A 2 )/(2A 2 ) (40)
The signs are selected according to the rules defined in section 2.4. Choosing the components of the displacement to one, the force components are given by relations (30) and (31)

F + (k) = W * 0 (k)f + (k) = W * 0 (k)D l (k)(W 0 (k) + e iµ 1 W 1 (k)) F -(k) = e iµ 1 W * 1 (k)f -(k) = W * 1 (k)D l (k)(e iµ 1 W 0 (k) + W 1 (k)) (41)
and the system (31) to solve is

          1 -1 F + (k) F -(k)                     a + a -           =           0 b/(2π)           (42)
which yields

a + = a -= b 2π(F + (k) + F -(k)) (43)
and so in this case the Green's function is given by using formulas (28) and

(29) which leads to

G(m 1 a, m 2 b) =                        b 2π π/b -π/b e i(m 1 µ 1 +m 2 kb) F + (k) + F -(k) dk if m 1 ≥ 0 b 2π π/b -π/b e i(-m 1 µ 1 +m 2 kb) F + (k) + F -(k) dk if m 1 ≤ 0 (44)
A more detailed computation shows that

F + (k)+ F -(k) = - 1 9b 3((6 + (Kb) 2 ) cos(kb) -6 + 2(Kb) 2 ) (45) × (-12b 2 + 6a 2 + (Kab) 2 ) cos(kb) -6a 2 -24b 2 + 2(Kab) 2 (46)
For the cases Kb << 1, Ka << 1, kb << 1, one has the approximation

F + (k) + F -(k) ≈ -2 (kb) 2 -(Kb) 2 (47)
can create instabilities for low frequencies as can be seen in the curve for the size 0.0005m. It seems reasonable to have cells with size such that Ka > 0.001.

For large frequencies the mesh must satisfy the usual requirement of having a sufficient number of nodes (5 to 10) per wavelength otherwise the displacement could not be accurately represented on a cell.

Other computations were also done for diamond shape cells as in figure 5.

Taking into account the modifications described in section 2.8 and computing the same example as for figure 7 lead to the same curves which are not reproduced here. So the proposed method is not strictly limited to rectangular cells and can be generalized to more complex shapes with appropriate changes in the equations.

Two-dimensional elastodynamics

An example similar to the precedent is calculated for the two-dimensional elastodynamic case for plane strain problems. The periodic cell is first taken as the square of size 0.01m × 0.01m. The material is steel with E = 2.10 11 P a, ν = 0.3 and ρ = 7800kg/m 3 .

In figure 10 the analytical solution is compared with the present numerical solution for the point (0.5m, 0.2m) and a unit force at origin in the x direction.

The curves show the real and imaginary parts of the x component of the displacement. A very good agreement between the two curves is observed. In figure 11 the error between the analytical and numerical solutions are plotted versus the frequency for a periodic cell made of different number of elements.

The global size of the periodic cell is still 0.01m × 0.01m but it can be divided in 1 × 1, 2 × 2 or 4 × 4 elements of degree one or in 1 × 1 element of degree two. It can be clearly seen that a denser mesh leads to lower errors and that using an element of degree two considerably improves the result.

In figure 12 the error is plotted versus the size of the periodic cell. Only one element of degree one per cell is used for these examples. It can be seen that the error decreases as the size of the cell is smaller. So it seems interesting to take cells with rather small sizes. These results are quite similar to the acoustic case.

Finally, figure 13 presents the numerical and analytical solutions for the frequency 1000Hz along the direction x for a distance varying between 0.01m (the size of a cell) and 1m. It can be observed that the agreement is very good for all distances. So the numerical computation is still correct near the force and the singularity of the solution is obtained by the proposed method which is not limited to the computation of points far from the source.

Periodic case

The last example is a truly periodic material made of a network of two materials as seen in figure 14 and plane strain problems are still considered. The A good agreement between the two solutions is found. Some differences can be seen at low frequencies for which the finite element solution is no more accurate since the size of the absorbing layer becomes insufficient for these frequencies.

The full finite element problem contains 640000 elements and 641601 nodes and a typical cell is meshed with 100 elements and 121 nodes. In the periodic computation a cell is also meshed with 100 elements and 121 nodes, so with the same density as for the full problem. For the numerical Green's function presented in this article, it is necessary to do different computations for different values of the transverse wavenumber k. However it is much more efficient to do a large number of computations on a structure with 121 nodes than a single computation on a structure with 641601 nodes. With the parameter used for the two materials the shear velocities are c 1 = 1601m/s and c 2 = 620m/s.

For the maximum frequency the wavelengths are 0.53m and 0.21m which are much larger than the size of a cell. The mesh density is sufficient at these frequencies.

Conclusion

We have presented a method to calculate numerically the Green's function for homogeneous or periodic media in the two-dimensional case. Good agreements were found in the cases where an analytical solution is known. Very general media could be considered as this method needs only to post process small size finite element models to get the mass and stiffness matrices of a period in the medium.

This process could be extended firstly to increase the numerical efficiency of the Green's function computation mainly if this has to be used in a boundary element formulation. This can be done easily if different points are to be calculated at the same time as the propagation constants and the wave shapes have to be calculated only once for different points. Asymptotic methods like the stationary phase method could also be considered. Finally, the three-dimensional case has to be considered in a future work.

It is known that the sum can be written as a periodic sum of delta functions (consider the Fourier series of the delta function). This leads to the periodic sum

f(x 1 , k, x 2 ) = L 2 2π Fe -ikx s 2 δ(x 1 -x s 1 ) +∞ m 2 =-∞ δ(x 2 -x s 2 -m 2 L 2 ) (A-8)
Taking the restriction of this function to a period gives

f(x 1 , k, x 2 ) period = L 2 2π Fe -ikx s 2 δ(x 1 -x s 1 )δ(x 2 -x p 2 ) (A-9)
where x p 2 is the number in the sequel x s 2 + m 2 L 2 which is in the considered cell.

. .

x 1 x x 1 x 2 1 G(x ,x ) 2 2 1 Fig. 1. Green's function. L one period L 1 2 x 1 x 2 . x 1 G(x ,x ) 1 2
.

1 x 2
Fig. 2. Periodic medium.

x 1

x 2

Longitudinal dofs

Transverse top dofs

Transverse bottom dofs Fig. 3. Transverse and longitudinal degrees of freedom.

two materials have the properties E 1 = 2 .

 12 10 10 P a, ν 1 = 0.3, ρ 1 = 7800kg/m 3 and E 2 = 2.10 9 P a, ν 2 = 0.3 and ρ 2 = 2000kg/m 3 . Each material occupies a square of size l 1 = 0.025m × l 2 = 0.025m, so the periodic cell has the size L 1 = 0.05m × L 2 = 0.05m. The figure15presents the comparison between the numerical Green's function and a finite element calculation with a structure made of 40 × 40 periods plus 20 layers with increasing dissipation as in the perfectly matched layer method to avoid wave reflections at boundaries.

Fig. 4 .Fig. 6 .Fig. 8 .Fig. 9 .Fig. 10 .Fig. 11 .Fig. 12 .Fig. 13 .

 468910111213 Fig. 4. Reduced set of degrees of freedom.

Fig. 14 .

 14 Fig. 14. Network of two materials.

which shows that the singularity in the integral is of square root type as expected.

An example of result is presented in figure 7. The acoustic element has the dimensions a = 0.01m and b = 0.01m. The point force excitation is supposed to be at the origin, which is also a corner of a cell, and the analysis is done for a point at x = 0.5m and y = 1m. The sound velocity is c = 340m/s and the calculation is done over the frequency band [0, 1000Hz]. One can see that the numerical Green's function is in full agreement with the analytical solution.

To have a better estimate of the relative error in the computation, this error is plotted versus the number of elements used to mesh the periodic cell in 8 that increasing the number of elements leads to a decrease in the computational error. Another possibility is to reduce the size of the periodic cell, which can be done here since the medium is homogeneous. This is presented in figure 9 where the error is plotted versus the size of the cell. In every case only one element with a = b is used for the mesh. The results show clearly that reducing the size of the cell leads to much better results. However reducing too much

A Decomposition into pseudo periodic solutions

Consider the structure shown schematically in figure 2 which has periodic mechanical properties and is defined on R 2 . The first step is to decompose functions defined on this structure into a sum of pseudo periodic functions.

A point in R 2 is denoted (x 1 , x 2 ). A square integrable function in R 2 can be decomposed by a partial Fourier transform as

The integration on k can be further decomposed into a sum of integrals by

where L 2 is the spatial period along dimension 2. This can also be written under the form

where u(x 1 , k, x 2 ) is a periodic function in x 2 given by

It can be proved, by decomposing the periodic function u into Fourier series in formula (A-3), that the function u is uniquely determined by U.

As an example of decomposition consider a point force at (x s 1 , x s 2 ) whose expression is

where F is a vector giving the amplitude and the direction of the force. The partial Fourier transform of this solution is

So in formula (A-4), the periodic function of x is given by

List of Figures