Cyril Labbe
email: cyril.labbe@imag.fr

Dominique Labbe
email: dominique.labbe@iep.upmf-grenoble.fr

Inter-textual distance and autorship attribution Corneille and Molière

The calculation proposed in this paper, measures neighborhood between several texts. It leads to a normalized metric and a distance scale which can be used for authorship attribution. An experiment is presented on one of the famous cases in French literature : Corneille and Molière. The calculation clearly makes the difference between the two works but it also demonstrates that Corneille contributed to many of Moliere's masterpieces.

«Molière aurait confié à Nicolas Despréaux : Je dois beaucoup au Menteur. Lorsqu'il parut j'avais bien envie d'écrire, mais j'étais incertain de ce que j'écrirais ; mes idées étaient confuses : cet ouvrage vint les fixer» . André Le Gall, Corneille, Paris, Flammarion, 1997, p 469. The authorship research of an unknown or doubtful text is one of the oldest statistical problems applied to literature. The unknown text is to be compared with other texts where we are sure that we know the author or we are sure that he wrote at least a part of it. Usually, the study concerns the most frequent words or a selection of them, often the «function words ».

On this topic, see [START_REF] Holmes David | The Federalist revisited : new directions in autorship attribution[END_REF], (Baayen and al, 1996) and [START_REF] Binongo José | The Application of Principal Component Analysis to Stylometry[END_REF]. In this paper, we propose a calculation which considers the entire text and which gives a standardized measure of the actual distance between it and another text. This is known as «lexical connection» defined as «the intersection of two texts vocabularies» [START_REF] Muller | Principes et méthodes de statistique lexicale[END_REF]. Therefore, connection is the complement of distance, a colloquial term in statistics ; for this reason, we have chosen it.

To understand our calculation, one may consider the difference between «token» and «type». The token is the smallest measurable element in a text, and the «type» forms the vocabulary's basic element. For instance, the longest novel in French, Les misérables is made up of half a million tokens : its length or extent (noted N), while its vocabulary (noted V) is made up of less than 10 000 normalized and tagged types.

Usually, the «connection» measure is done on the vocabulary regardless of the type frequency (see [START_REF] Brunet | Une mesure de la distance intertextuelle : la connexion lexicale[END_REF]. Here , we suggest to consider the frequency of each type, that is to say, the entire texts (we use the adjective «textual» in order to show that the calculation is on N and not only on V or on a part of V).

Our metric measures whether two or several texts are relatively far from one another. It has been applied to a lot of corpora and used to set up a useful distance scale for authorship attribution. We present an application to one of the most famous cases in French literature :

Corneille and Molière. The measure makes clear the difference between their works but it also proves that Corneille probably wrote a lot of Moliere's plays.

Intertextual distance

To be allowed to say whether two texts are rather near or far from one another, if we consider their extents, we must use a «metric» with the following properties:

-non sensitive to length differences of the compared texts; -applicable to several texts and, if possible, to all texts in the same language; -varying in the same way -between 0 (the same vocabulary and similar frequency of each type in the 2 texts) and 1 (no common type) -without jump, nor threshold effect around some values.

-symmetric (given 2 texts A and B then : The absolute distance between A and B will be the union of the 2 texts less their intersection, (N a ∪ N b ) -(N a ∩ N b ), that is to say the sum of the differences between the absolute frequencies of each type in the 2 texts.

δ (A,B) = δ(B,
The relative distance can be computed in two ways: 2) is the one given by E. [START_REF] Brunet | Une mesure de la distance intertextuelle : la connexion lexicale[END_REF]. Two objections to these formulae can be found.

( ) ( , ) 1 δ a b ia ib Va ib ia Vb a b F F F F N N = - + - + ∑ ∑ ( ) ( , ) 2 1 2 δ a b ia ib Va a ib ia Vb b F F N F F N = - + -         ∑ ∑ Formula (
-( 1) and ( 2) are equivalent only when the texts lengths are equal (Na = Nb). If no type is shared, the two formulas actually give a result of 1 whatever the text length is (which is one of the conditions for our idealistic metric);

-on the other hand, the theoretical minimum can reach 0 only in the specific case of equal lengths. As a matter of fact, the greater the difference in length between the two texts, the further the minimal numerator will be from 0. For instance, in Molière's corpus, the shortest text counts 732 tokens and 274 types (it is a piece from a lost play : Pastoral Comedy). On the other hand, the longest play is the Malade imaginaire (19 920 tokens and 2 082 types). Even if the small text was completely included in the large one, the distance would not be null since there is not enough room in the small text for all the types of the long one;

-in (1) as in (2), the intersection of the 2 texts is counted twice. Therefore, more importance is given to the common types rather than to the specific vocabulary of each text.

Is it possible to overcome these two objections and allow a good approximation of the distances between several texts ?

An approximation of intertextual distance

In order to get an accurate estimate of the distance between several texts, we propose to «reshape» the largest to the size of the smallest. Define B' this reduction of B to the size of A:

N a A N b ' N a ∩ N' b N b
The mathematical expectation of every type of B with a frequency f i is:

E ia(u) = F ib * U (a,b) with U N N a b a b ( , ) =
This gives the value of N' b :

N' = E b ia(u)
Vb ∑ Consequently, we can reformulate (1) and ( 2) replacing F ib by E ia(u) and N b by N' b .

Zero, the theoretical minimum will be reached when the small text is like a model of the largest. In this case, all the types of A are present in B with a frequency F ia = E ia(u) and, consequently, the numerators of the formulae will be equal to zero. In fact, 2N a is the maximum token population that the two texts can share if they have the same size, the same vocabulary and equal frequencies for each type. Conversely, the theoretical maximum (one) means that A and B do not share any type : in this case, both numerator and denominator are equal to N a + N' b .

However, this new formulation gives no answer to the double count objection about the intersection of the two texts and does not entirely solve the «physical» problem noted above :

if the lengths of the two texts are very different, all the types of the largest cannot be used in the smallest.

To accomodate and to allow an unbiased measurement of the distance, we propose to:

-consider the intersection of the two texts only once;

-limit calculations to all the types of A and the only types of B whose frequency is high enough to expect almost one in A (E ia(u) ≥ 1). The sum of these expectations is N' b .

Consequently, calculation is done in two steps (see the figure below):

C D E (text A) (text B')
Firstly, the V a types : C and D (in the C set, E ia(u) = 0) and, secondly, the types of E : V'b(e) (in this case : F ia = 0). The absolute distance between A and B' is :

D = F E Va, b(u) ia V V ia(u) a, © b(E) ∑ -
When A and B share no type, this distance will be equal to : N a + N' b . This will be the numerator of the relative distance formula since the metric maximum is 1 and the actual result must be less than 1 when the intersection of A and B is not empty.

( )

( , ) 3 D a b = - = - + ∑ ∑ ∑ ∑ F E F E F E N N© ia V V ia(u) ia + ia(u) V V ia V V ia(u) a b a, © b(e) © b a a, © b(e)
It is worth noting that:

-the same result, rounding excepted, can be obtained by subtracting the relative frequencies of the two texts, if one considers all the vocabulary of the smallest text (A) and only the B types whose frequencies are high enough to expect at least one if B is reduced to the size of A.

-the metric accuracy is slightly reduced by rounding. In fact, the observed frequencies are always integers whereas mathematical expectations include decimals which will contribute to the distance. This drawback will increase when low frequency types are an important part of the texts, that occurs in the case of small texts. To overcome this, it is convinient not to apply the calculation to too small texts -we never applied this calculation under the limit of 1 000 tokens (so that the small excerpt of the Comédie pastorale cannot be examined) -and to avoid a too large scale of sizes (under 1/10). In the application above, the shortest text counts 3 500 tokens -it is Molière's first comedy (see the appendix) and the largest counts 20 300 (Corneille's Toison d'or)1 . For the same reasons, all results under .50 are eliminated from the numerator (|F ia -E ia(u) | < .5).

-this calculation means that, beforehand, the texts are normalized and -from our point of view -that all the tokens are tagged (in French : «lemmatisés»), i.e. attached to their dictionary entries [START_REF] Muller | Principes et méthodes de statistique lexicale[END_REF]Labbé, 1990). For example, comparing prose and poetry pieces, without reducing to lower case the verses initial upper case, automatically creates a distance of around 1/8 since a verse counts around 6-10 words. The distance calculation applied to a non-normalized corpus will place on one hand all pieces of prose and on the other hand all poetry, even if both content are not different… Other examples exist: in his letters, an author may use a lot of abbreviations (Mr for «mister», initials for names, etc.) but not in his works: is this an actual vocabulary difference ? One can see that the distance calculation implies a prior agreement on standards.

-the interpretation of the results is very easy. For example, a metric value of .50 means that we can estimate that the two texts share half of their whole extent; .25 that three quarters of the two texts are common, etc. Thus, a scale of distances can be established, which can be useful for authorship attribution.

Distance scale

This calculation has been applied to various corpora the total size of which is about 10 million of tokens all counted with the same standards : General de Gaulle's and F.

Mitterrand's speeches, Canadian and French Prime minister's adresses to parliament since 1945 (Labbé-Monière, 2000), several novels from the last 3 centuries (with E. Brunet), Trade Unions newspapers editorials (Labbé-Brugidou, 1999), economic press articles, transcription of interviews (Bergeron-Labbé, 2000). These experiments have been used to establish the following empirically distance scale. .

25

-for the same author, we always notice distances smaller than those existing between two different and contemporary authors (when they are dealing with the same topic).

-distances smaller than .20 usually do not exist between two different authors (concerning texts of the same kind with close topics). In the case of an unknown writer, authorship attribution is quite sure. If it is known that both authors are different, then one of them was «inspired» by the other.

-between 0.20 and 0.25 represents the case where the texts are very similar. In the case of only one author, a change in themes and genre is indicated. If one of the authors is unknown, attribution is possible but it will be sure only if it is proved that there are no other texts nearer and if one can provide other proofs, particularly stylistic.

-above .25, authors are probably different or genre and topics are too far to allow a comparison ;

As an example, we present an application of the calculation to Corneille and Molière's plays. These works have often been analysed by critics and even in statistic studies (especially, Muller 1967;[START_REF] Britt-Marie | Le vocabulaire de Molière dans les comédies en alexandrins[END_REF] that gives some useful references.

From the very beginning, it was rumoured that Molière was not the writer of his plays.

These rumours were intensified by a publisher's «warning» placed at the head of one play :

Psyché (1671). It was said that although Corneille wrote two thirds of the verses, it had previously been played under Molière's name (this play and the publisher's warning are published in the second volume of Corneille's complete works in La Pléiade, Gallimard).

Since then, the problem has been discussed many times; most often, Corneille is said to be the virtual author; among others, the poet P. Louys at the beginning of the XXth century, and more recently, two Belgian writers have underlined how similar the two works are (Wouters and Ville de Goyer, 1990).

Moliere's plays

Intertextual distance calculus gives some interesting information. Firstly, as an example, one can find below the distances separating the most well known Molière's plays (table II). The calculation shows an important similarity between all these plays, although their topics are very different. prose. Considering these differences, it is obvious that all these masterpieces are from the same author. Some cases seem particularly clear : Tartuffe and Dom Juan -two plays which caused scandal and were withdrawn -are written with, the first in verse and the second in prose. In spite of a lot of «patois» in the second one, which increases their distance, they remain very close (.199): this confirms that they have only one author and that they were written during the same period (the same comment can be said for l'Avare and Tartuffe).

Moliere's plays are too numerous to reproduce here their distances matrix (33 lines and 33 columns). The mean of the distances separating each play from all the others gives some information (table III). The overall mean is .249, with a small relative variation coefficient (15%). Thus Molière's works are rather homogeneous, less than Corneille's (.230), but more than Racine's (.290) although half of Molière's plays are in verse and the rest in prose and although he used a lot of «latin» words, some «patois» and imaginary language.

Table III. Overall distances between one play and all the others in Molière's works.

Corneille and Molière

The two works were merged in a single corpus (see the annex list). Besides Psyché, this corpus consists of 64 plays that is to say 917 000 tokens, the writing of which spreads over 44 years . It mingles comedies, tragedies, verses and prose plays, and tackles extremely diverse themes. However, the whole work remains more homogeneous than Racine's only dramatic work, which is much smaller (166 000 words) and all in alexandrine verses, and than all large corpora -even with a single author -we have dealt with until now.

To obtain an overall view, two classification experiments were carried out. The first one was a cluster analysis on the distances matrix. The two nearest plays are merged and the distances of this new set with respect to all other plays are calculated again for the following grouping. The classification steps are summed up in a dendrogram (Table IV): from left to right, the regrouping order and, as ordinate, distances corresponding to the different agregation stages. The origin is placed on .15 in order to enable an easy reading of the graph, but it must not be forgotten that all these plays are very near.

This first experiment brings to light that the works are different but near; the two corpora link at .28. Moreover, regrouping fits with what is expected. On the left, the most homogeneous group is made up of Corneille's mature tragedies (group A), then come his first tragedies (B) that made him famous (le Cid, Horace, Cinna…), and, finally, his comedies (C). As for Molière, the classification separates his verse comedies (D) and prose ones (E, F).

In finer detail, most basic groupings correspond to thematical proximity already remarked by critics. For instance, Molière's les Femmes savantes, l'Ecole des Maris and l'Ecole des femmes. The distance calculation combined with cluster analysis, thus provides an accurate and reliable tool. However , in this corpus, this tool detects some «anomalies» (bold lines) :

-one of Molière's play is found in the middle of Corneille's ones : Dom Garcie. Very probably, this play is from Corneille. As a matter of fact, it is very near to those he wrote in the period when Dom Garcie was created.

-the two Psyché are placed together in Corneille's work. The one ascribed to Corneille is almost at an equal distance between comedies and tragedies -this is logical -and Molière's one nearly between the two works.

-two of Corneille's comedies (the Menteur and the Suite du Menteur) are found in the middle of Molière's verse plays. This classification is very surprising because these comedies (the two last ones officially written by Corneille) dated 1642-43, while Molière's first plays were supposed to have been written at the earliest in 1656 and were played in Paris only since 1660. So, we can say that, as Corneille is the undisputed author of the two Menteurs, he probably also wrote the plays found on the dendrogram on the left of these two plays and which are all very near to one another: Tartuffe, le Misanthrope, les Femmes savantes, l'Etourdi, le Dépit amoureux, l'Ecole des maris, l'Ecole des femmes, Sganarelle, Amphytrion, la Princesse d'Elide, Mélicerte and les Fâcheux. That is to say all Molière's verse plays.

On the other hand, the relationship of the Menteur with Molière's prose plays is not so clear; although it has been noted that these Molière comedies are nearer than Psyché is to Corneille's work. Therefore, the authorship of Corneille is possible but not so clear as for the alexandrine works. These overall findings suggest a thorough examination of the existing neighbourhood particularly around the two Menteurs. Besides verse plays, Dom Juan and the Avare appear under the .25 level which make us think they were written by the same author as the Menteurs. The Psyché part written by Corneille, as well as Dom Garcie are suggesting an even higher level (.273) that is to say a probable contribution by Corneille to Amphytrion and Fourberies de Scapin and even to the Malade imaginaire (but here dog latin and italian interludes increase the distance). The two Menteurs were probably used as a model for a great number of Molière's plays, especially all the verse ones, so that the Menteurs are clustered with Molière's and not with Corneille's works, even his comedies. However, except for l'Etourdi (1660), all other distances exceed the level of .20 above which authorship becomes unsure. Time lag must also be considered : Molière's plays were written twenty years and even later the Menteurs. To check these findings, another method was used : the tree-analysis. Texts or groups of texts are no longer classified one by one, but, for each one, the best representation of its neighbourhood is compared with all the others ( Barthélémy and Guénoche 1988;[START_REF] Luong Xuan | L'analyse arborée des données textuelles : mode d'emploi[END_REF]. Each text forms an end (leaf) linked to others by branches and by trunk sections. The path to link two texts measures their proximity (Table VI). We thank M. Xuan Luong (Nice University) for this graph. Each play number is given in the annex. For a better view, we substract .15 from all the distances (as in the dendrogram above). This substraction exagerates neighborhoods between leaves and distances between main nodes).

Allmost all Corneille's plays (numbered from 1 to 33) appear closely gathered on the left, at the bottom of the graph (with two sub-sets : comedies on one hand, tragedies and tragicomedies on the other). Molière's plays (from 37 to 67) all appear at the top of the graph, clearly divided in two groups : prose writings (the top set) and verse ones (in the middle). On Above all, X. Luong's tree-analysis strengthens the main conclusion : Corneille's Menteurs (15-16) stand quite in the centre of Molière's works, though they are rather far from the rest of Corneille's plays. In other words, the Menteurs authorship is clearly the same as most of Molière's masterpieces. This finding brings out interesting questions. For instance, it can be asked which particularities may be found in Corneille's style and vocabulary that are also present in the plays he wrote for Molière. And conversely, which are their differences.

Molière's historic importance is not at all minimized by it being almost certain that Corneille contributed to most of his masterpieces. He was indeed the first French modern "theater businessman", at the same time : company manager, artistic director, theater director and actor, and, as is shown, an excellent scenario "hunter". Historians and critics will explain how and why the two men worked together and why they concealed it. This analysis does not put an end to the question of how to establish Molière's authorship.

One may go on thinking that Molière summurarised the idea of a contemporaneous life satire (Corneille seemed to have abandoned this idea after the second Menteur failure). It can be added that Molière usually directed and played Corneille's works and he could have been " immersed " in Corneille's language and ready to write in the same way as the author he prefered and of whom he knew thousands of verses. More studies will have to be carried out to answer these questions. It could be interesting to look at Molière's main contemporary authors. Some of them could have helped him to write some prose comedies that are too different to be from the pen of Corneille such as : les Précieuses ridicules, l'Impromptu or la Critique de l'école des femmes… Finally, the easiness of distance calculation programming is to be noticed. Futhermore, the results and their interpretation are within the reach of everyone, without any statistics culture.

That is why we hope other studies will strengthen of intertextual distance significance combined with cluster analysis as a tool for literary criticism especially in authorship attribution.

  A)); -as «transitive» as possible: when we «agregate» 2 texts, the distance of this «corpus» regarding other texts must reflect the prior distance in the ordering (δ (A,B) < δ(A,C) < δ(B,C) then δ (A,B) < δ{A,(B∪ C)}; -as «robust» as possible (ie: a marginal change in one of the 2 texts must be reflected by a marginal change in their distance…) Some previous studies in this field, especially Muller's and Brunet's ones, suggest the following method. Given 2 texts A and B: V a and V b : number of types in A and B F ia : frequency of the ith type in A F ib : frequency of the ith type in B N a and N b : number of tokens in A and B with N a = ∑ F ia and N b = ∑ F ib

  the right, at the bottom, five "anomalies" (represented in bold): besides a short piece of Psyché by Quinault (36) the two other Psyché by Corneille (34) and Molière (35) ; Dom Garcie (43) and (06) the Comédie des tuileries fifth act (Corneille's work commissioned by Cardinal Richelieu 37 years before Psyché).

Table 1 .

 1 Intertextual distance standardized scale 2 .

	An author		Different authors
		.65
			Minimal commun nucleus for texts in the
			same language.
	Minimal commun nucleus for texts produced by a same author.	.40
			Different genres, remote topics
	Different genres, remote topics	.30	Similar genre = remote topics
			Different genres = close topics
	Similar genre = remote topics
	Different genres = close topics		Same genre and topics
			Possible authorship attribution
		.20
	Same author, genre, topic.		Sure authorship attribution
		.10

Table II .

 II Distances between Molière's well known works.

		Ecole des	Tartuffe Dom Juan	Le	L'Avare Bourgeois	Femmes	Malade
		femmes			Misanthrope		gentilh.	savantes	imaginaire
	Ecole des femmes	0	.183	.205	0.194	0.200	.231	.198	.223
	Le Tartuffe		0	.199	.167	.199	.230	.170	.219
	Dom Juan			0	.204	.170	.207	.219	.205
	Le Misanthrope				0	.210	.239	.173	.239
	L'Avare					0	.194	.214	.187
	Bourgeois gentilh.						0	.234	.196
	Femmes savantes							0	.226
	Malade imaginaire								0

  The smallest(.167) is between Tartuffe and le Misanthrope, two plays in Alexandrines in which Molière does not use farce nor colloquial language, nor jargon. The greatest (.239) is between le Misanthrope and le Bourgeois gentilhomme or le Malade imaginaire. The first one is in verse, the two others in prose and they contain a lot of inventions in «turkish» or in «latin». More generally, distances greater than .20 separate l'Ecole des femmes, Tartuffe, le Misanthrope and les Femmes savantes -written in verseand Dom Juan, l'Avare, le Bourgeois gentilhomme and le Malade imaginaire, written in

  The bottom of the table shows that Corneille's admitted contribution appears rather far from the rest of the work but it contains a surprising fact : Molière's Psyché is further apart than Corneille's one. As a matter of fact, the only conclusion to be drawn from this last measure concerns the atypical position of Psyché in Molière's work (as well as in Corneille's one).

	Title	Year of création	Nature	Distance
	L'Avare	1668	Prose	.216
	Dom Juan	1665	Prose	.220
	L'Ecole des femmes	1662	Verse	.220
	Le Tartuffe	1664	Verse	.224
	Le Misanthrope	1666	Verse	.229
	L'Ecole des maris	1661	Verse	.230
	Femmes savantes	1672	Verse	.232
	Dépit amoureux	1658	Verse	.235
	Malade imaginaire	1673	Prose	.235
	Fourberies de Scapin	1671	Prose	.237
	L'étourdi	1656	Verse	.238
	Monsieur de Pourceaugnac	1669	Prose	.239
	Bourgeois gentilhomme	1670	Prose	.239
	Georges Dandin	1668	Prose	.240
	Princesse d'Elide	1664	Verse & prose	.241
	Le Sicilien ou l'amour peintre	1667	Prose	.243
	Amphytrion	1668	Prose	.244
	L'amour médecin	1665	Prose	.245
	Médecin malgré lui	1666	Prose	.246
	Amants magnifiques	1670	Prose	.252
	Les fâcheux	1661	Verse	.255
	Sganarelle	1660	Verse	.256
	Mélicerte	1666	Verse	.256
	Comtesse d'Escarbagnas	1671	Prose	.257
	Mariage forcé	1664	Prose	.265
	L'impromptu	1663	Prose	.266
	Précieuses ridicules	1660	Prose	.267
	Médecin volant	1659	Prose	.279
	Critique de l'Ecole	1663	Prose	.280
	Dom Garcie	1661	Verse	.284
	La jalousie	1660	Prose	.310
	Psyché Corneille		Verse	.293
	Psyché Molière		Verse	.305
	Mean Molière			.249

Comedies spread out in a very caracteristic way : the main masterpieces -l'Avare, Dom Juan, l'Ecole des femmes, l'Ecole des maris, les Femmes savantes, le Tartuffe, le Misanthrope, le Malade imaginaire -stay in the center and at small mean distances (it would be the same for the Bourgeois if the «Turkish» language was not put at the end of this play). On the other hand, some plays are apart : the first comedies Molière played before living in Paris (la Jalousie du barbouillé, le Médecin volant) or some small occasional creations like la Critique de l'Ecole des femmes et l'Impromptu de Versailles. In the same case, we find les Précieuses ridicules (first of Molière's success) and Dom Garcie, a serious verse comedy which was unsuccessful. Except these few plays, it is quite sure that all the work is from a single author.

  Table IV Cluster analysis on Corneille and Molière's plays

						.40
						.35
						.30
						.25
						. 20
						.15
	A	B	C	D	E	F
	From left to right :	Dom Garcie		D. Molière (verse)	L'avare
	A. Corneille :	B : Corneille		Le Tartuffe		Dom Juan
	Tite et Bérénice	Cinna		Le Misanthrope		Fourberies Scapin
	Pulchérie	Pompée		Femmes savantes	Médecin malgré lui
	Suréna	Le Cid		L'étourdi		M. Pourceaugnac
	Agésilas	Horace		Dépit amoureux	Malade imaginaire
	Othon	Médée		L'école des maris	Bourgeois gentil.
	Sertorius	Psyché Corneille	L'école des femmes	L'amour médecin
	Sophonisbe	C Corneille comédies	Amphytrion		Mariage forcé
	Atilla	Galerie du Palais	Sganarelle		Ctesse d'Escarb.
	Nicomède	La Suivante		Le Menteur 1		F. Molière :
	Don Sanche	Mélite		Le Menteur 2		Critique de l'école
	Polyeucte	La Veuve		Mélicerte		L'impromptu
	Théodore	La Place Royale	Les fâcheux		Précieuses ridicules
	Héraclius	L'illusion comique	Princesse d'Elide	Médecin volant
	Pertharite	Clitandre		E. Molière (prose)	La jalousie
	Andromède	Comédie des		Amants		Psyché Quinault
	Toison d'Or	Tuileries		magnifiques
	Rodogune	Psyché Molière	Le sicilien
	Oedipe			Georges Dandin

Table V Two

 V Menteurs distances compared to Molière's plays

	Plays	Le Menteur 1 Le Menteur 2
	Le Menteur 1	0,000	0,180
	Le Menteur 2	0,180	0,000
	Psyché Corneille	0,288	0,273
	Psyché Molière	0,329	0,325
	La jalousie	0,341	0,331
	Médecin volant	0,310	0,293
	L'étourdi	0,205	0,206
	Dépit amoureux	0,215	0,212
	Précieuses ridicules	0,315	0,314
	Sganarelle	0,259	0,253
	Dom Garcie	0,280	0,273
	L'école des maris	0,223	0,217
	Les fâcheux	0,248	0,248
	L'école des femmes	0,226	0,217
	Critique de l'école	0,323	0,319
	L'impromptu	0,321	0,316
	Mariage forcé	0,322	0,302
	Princesse d'Elide	0,251	0,243
	Le Tartuffe	0,242	0,228
	Dom Juan	0,259	0,248
	L'amour médecin	0,292	0,289
	Le Misanthrope	0,252	0,234
	Médecin malgré lui	0,298	0,289
	Mélicerte	0,257	0,250
	Le sicilien	0,277	0,260
	Amphytrion	0,253	0,256
	Georges Dandin	0,292	0,279
	L'Avare	0,256	0,244
	M. de Pourceaugnac	0,292	0,283
	Amants magnifiques	0,282	0,279
	Bourgeois gentilhomme	0,294	0,280
	Fourberies de Scapin	0,269	0,263
	Ctesse d'Escarbagnas	0,311	0,300
	Femmes savantes	0,260	0,248
	Malade imaginaire	0,282	0,270
	Molière's entire work mean	0,275	0,266
	Molière's verse plays mean	0,241	0,234
	Corneille's mean	0,252	0,249

Table VI .

 VI Tree-classification on Molière's and Corneille's plays

  Annex Corneille's and Molière's Plays.

	Corneille		Year of création Genre	Length
					(tokens)
	1	Mélite	1630	Comédie en vers	16 690
	2	Clitandre	1631	Comédie en vers	14 402
	3	La Veuve	1631	Comédie en vers	17 661
	4	La Galerie du Palais	1632	Comédie en vers	16 140
	5	La Suivante	1633	Comédie en vers	15 160
	6	Comédie des Tuileries	1634	Comédie en vers	3 627
	7	Médée	1635	Tragédie en vers	14 269
	8	La Place Royale	1634	Comédie en vers	13 801
	9	L'illusion comique	1636	Comédie en vers	15 428
	10	Le Cid	1636	Tragédie en vers	16 677
	11	Cinna	1641	Tragédie en vers	16 126
	12	Horace	1640	Tragédie en vers	16 482
	13	Polyeucte	1641	Tragédie en vers	16 472
	14	Pompée	1642	Tragédie en vers	16 492
	15	Le Menteur 1	1642	Comédie en vers	16 653
	16	Le Menteur 2	1643	Comédie en vers	17 675
	17	Rodogune	1644	Tragédie en vers	16 842
	18	Théodore	1645	Tragédie en vers	17 121
	19	Héraclius	1647	Tragédie en vers	17 433
	20	Andromède	1650	Tragédie en vers	15 514
	21	Don Sanche	1650	Tragédie en vers	16 947
	22	Nicomède	1651	Tragédie en vers	16 923
	23	Pertharite	1651	Tragédie en vers	17 121
	24	Oedipe	1659	Tragédie en vers	18 618
	25	Toison d'Or	1661	Tragédie en vers	20 343
	26	Sertorius	1662	Tragédie en vers	17 675
	27	Sophonisbe	1663	Tragédie en vers	16 858
	28	Othon	1664	Tragédie en vers	16 971
	29	Agésilas	1666	Tragédie en vers	18 227
	30	Atilla	1667	Tragédie en vers	16 788
	31	Tite et Bérénice	1670	Tragédie en vers	16 697
	32	Pulchérie	1672	Tragédie en vers	16 630
	33	Suréna	1674	Tragédie en vers	16 545
	Psyché
	34	Psyché Corneille	1671	Comédie en vers	10 067
	35	Psyché Molière	1671	Comédie en vers	4 816
	36	Psyché Quinault	1671	Comédie en vers	1 299
	Molière
	37	La jalousie	1660	Comédie en prose	3 501
	38	Médecin volant	1660	Comédie en prose	3 876
	39	L'étourdi	1660	Comédie en vers	18 671
	40	Dépit amoureux	1660	Comédie en vers	16 242
	41	Précieuses ridicules	1660	Comédie en prose	6 648
	42	Sganarelle	1660	Comédie en vers	6 042
	43	Dom Garcie	1661	Comédie en vers	17 049

(2007) In fact : the longest play is l'Avare (21 033 tokens).

(NB 2007) : this distance scale has been calibrated with the help of contemporaneous texts the lengths of which were over

and under 20 000 tokens.

Acknowledgements

We are grateful to Professor Pierre Hubert for suggesting us the Molière and Corneille case, to Jean-Guy Bergeron, Pierre Hubert and Denis Monière for their helpfull comments, and to Mrs Kathleen Milsted for her accurate reading of our first translation.


Hélène Canot

Emmanuel Frénod

Modeling electromagnetism in and near composite material using two-scale behavior of the time-harmonic Maxwell equations

Keywords: Harmonic Maxwell Equations, Electromagnetic Pulses, Electromagnetism, Homogenization, Asymptotic Analysis, Asymptotic Expansion, Two-scale Convergence, Effective Behavior, Frequencies, Composite Material

The main purpose of this article is to study the two-scale behavior of the electromagnetic field in 3D in and near composite material. For this, time-harmonic Maxwell equations, for a conducting two-phase composite and the air above, are considered. Technique of two-scale convergence is used to obtain the homogenized problem.

Introduction

We are interested in the time-harmonic Maxwell equations in and near a composite material with boundary conditions modeling electromagnetic field radiated by an electromagnetic pulse (EMP). An electromagnetic pulse is a short burst of electromagnetic energy. It may be generated by a natural occurrence such like a lightning strike, meteoric EMP, EMP caused by geomagnetic Storm or nuclear EMP. The paper focuses on what happens over a period of time of a millisecond during the peak of the first return stroke. We study the electromagnetic pulse caused by this lightning strike. This is the first step of a larger study which goal is to understand the behavior of the electromagnetic field and its interaction with a composite material.

EMP interference is generally damaging to electronic equipment. A lightning strike can damage physical objects such as aircraft structures, either through heating effects or disruptive effects of the very large magnetic field generated by the current. Structures and systems require some form of protection against lightning. Every commercial aircraft is struck by lightning at least once a year in average. Lightning protection for aircraft is a major concern for aircraft manufacturers. With the increasing of use of composite materials, up to 53% for the latest Airbus A350, and 50% for the Boeing B787, aircrafts offer increased vulnerability facing lightning. Earlier generation aircrafts, whose fuselages were predominantly composed of aluminum, behave like a Faraday cage and offer maximum protection for the internal equipment. Currently, in aircrafts, composite materials consisting of a resin enclosing carbon fibers have significant advantages in terms of weight gain and therefore fuel saving. Yet they are much less electrical conductors with loss of the Faraday effect: 100 to 1000 times lower than aluminum. Modern aircrafts have seen also the increasing reliance on electronic avionics systems instead of mechanical controls and electromechanical instrumentation. For these reasons, aircraft manufacturers are very sensitive to lightning protection and pay special attention to aircraft certification through testing and analysis.

There are two types of lightning strikes to aircraft: the first one is the interception by the aircraft of a lightning leader. The second one, which makes about 90% of the cases, is when the aircraft initiates the lightning discharge by emitting two leaders when it is found in the intense electric field region produced by a thundercloud, our approach applies in this case. When the aircraft flies through a cloud region where the atmospheric electric field is large enough, an ionized channel, called a positive leader, merges from the aircraft in the direction of the ambient electric field. Laroche et al [START_REF] Laroche | Experimental studies of lightning strikes to aircraft[END_REF] at an altitude of 6000m, observed an atmospheric electric field close to 50 kV/m inside the storm clouds, 100kV/m to the ground. When upward leader connects with the downward negative leader of the cloud, a return stroke is produced and a bright return stroked wave travels from aircraft to cloud. The lightning return strokes ra-diate powerful electromagnetic fields which may cause damage to aircraft electronic equipment. Our work is devoted to the study of the electromagnetic waves propagation in the air and in the composite material. In this artificial periodic material, the electromagnetic field satisfies the Maxwell equations.

We evaluate the electromagnetic field within and near a periodic structure when the period of this microstructure is small compared to the wavelength of the electromagnetic wave. Our model is composed by air above the composite fuselage and we study the behavior of the electromagnetic wave in the domain filled by the composite material, representing the skin aircraft, and the air. We build the 3D model, under simplifying assumptions, using linear time-harmonic Maxwell equations and constitutive relations for electric and magnetic fields. Composite materials consist of conducting carbon fibers, distributed as periodic inclusions in a matrix (epoxy resin). We impose a magnetic permeability µ 0 uniform and an electrical permittivity = 0 , where is the relative permittivity depending of the medium. In the future, we will enrich this model by adding complexity, we will consider non uniform magnetic permeability and electrical permittivity. Now, we account for some characteristic values. In a first place we focus on the boundary conditions as we consider them as the source. Then, we use on the upper frontier, the magnetic field induced by the peak of the current of the first return stroke

H d = I 2πr , (1)
with current intensity I = 200 kA and r the radius of the lightning strike, this is the worst aggression that can suffer an aircraft, and we deduce a characteristic electric field E = 20 kV/m. In our model we consider that we have very conductive -but not perfect conductors -carbon fibers and an epoxy resin whose conduction depends on its doping rate. The conductivity of the air is non-linear. Air is a strong insulator [START_REF] Pr | Cours : Matériaux diélectriques[END_REF] with conductivity of the order of 10 -14 S.m -1 but beyond some electric solicitation, the air loses its insulating nature and locally becomes suddenly conductive. The ionization phenomenon is the only cause that can make the air conductor of electricity. The ionized channel becomes very conductive.

Our mathematical context is periodic homogenization. We consider a microscopic scale ε, which represents the ratio between the diameter of the fiber and thickness of the composite material. So, we are trying to understand how the microscopic structure affects the macroscopic electromagnetic field behavior. Homogenization of Maxwell equations with periodically oscillating coefficients was studied in many papers. N. Wellander homogenized linear and non-linear Maxwell equations with perfect conducting boundary conditions using two-scale convergence in [START_REF] Wellander | Homogenization of the Maxwell equations: Case I. Linear theory[END_REF] and [START_REF] Wellander | Homogenization of the Maxwell equations: Case II. Nonlinear conductivity[END_REF]. N. Wellander and B. Kristensson homogenized the full time-harmonic Maxwell equation with penetrable boundary conditions and at fixed frequency in [START_REF] Wellander | Homogenization of the Maxwell equations at fixed frequency[END_REF]. The homogenized time-harmonic Maxwell equation for the scattering problem was done in F. Guenneau, S. Zolla and A. Nicolet [START_REF] Guenneau | Homogenization of 3d finite photonic crystals with heterogeneous permittivity and permeability[END_REF]. Y. Amirat and V. Shelukhin perform twoscale homogenization time-harmonic Maxwell equations for a periodical structure in [START_REF] Amirat | Homogenization of time-harmonic maxwell equations and the frequency dispersion effect[END_REF]. They calculate the effective dielectric ε and effective electric conductivity σ. They proved that homogenized Maxwell equations are different in low and high frequencies. The result obtained by two-scale convergence approach takes into account the characteristic sizes of skin thickness and wavelength around the material.

On of the parameter we account for in our model: δ = 1 √ ω σµ 0 , where σ is the characteristic conductivity and ω the order of the magnitude of the pulsation shares much with the definition of theoretical thickness skin δ = 2 ωσµ 0 . The thickness skin is the depth at which the surface current moves to a factor of e -1 . Indeed, at high frequency, the skin effect phenomenon appears because the current tends to concentrate at the periphery of the conductor. On the other side, at low frequencies the penetration depth is much greater than the thickness of the plate which means that a part of the electric field penetrates the composite plate. We use the theory of two-scale convergence introduced by G. Nguetseng [START_REF] Nguetseng | A general convergence result for a functional related to the theory of homogenization[END_REF] and developed by G. Allaire [START_REF] Allaire | Homogenization and Two-scale Convergence[END_REF].

The paper is organized as follows : in Section 2 we specify the geometry of the model and the dimensionless equations converting the problem into an equivalent one with which we work in the following sections. In Section 3 we perform the mathematical analysis of the model. In particular we introduce the weak formulation of the problem for the electric field and we regularize it using divergence term. We establish the existence and uniqueness result for the regularized Maxwell equations thanks to Lax-Milgram Theorem. We conclude this section by estimate of the electric field. The last Section is devoted to the homogenization of the problems for electric field using the two-scale convergence concept.

Modeling

This section is dedicated to the complete mathematical model we will study in this paper. In a first place, we consider a problem that seems relevant with the perspective of propagation of the electromagnetic field in the air and in the skin of aircraft fuselage made of composite material. Secondly, we make a scaling of this model and finally we operate simplifications. If desired, the reader can go directly to the mathematical analysis knowing that the problem to be studied is given by (65), (70) equipped with boundary conditions (68), (69).

Notations and setting of the problem

We consider set Ω = {( x, y, z) ∈ R 3 , y ∈ (-L, d)} for L and d two positive constants, with two open subsets Ω a and P (see Figure 1). The air fills Ω a and we consider that the composite material, with two materials periodically distributed, stands in domain P .

We assume that the thickness L of the composite material is much smaller than its horizontal size. We denote by e the lateral size of the basic cell Y e of the periodic microstructure of the material. The cell is composed of a carbon fiber in the resin. We define now more precisely the material, introducing:

P = {( x, y, z) ∈ R 3 / -L < y < 0}, (2)
which is the domain containing the material. Now we describe precisely the basic cell. For this we first introduce the following cylinder with square base:

Z e = [- e 2 , e 2 ] × [-e, 0] × R, (3)
We consider α such that 0 < α < 1, and R e = α e 2 . We set

D e = {( x, y) ∈ R 2 /( x 2 + ( y + e 2 ) 2 ) < ( R e ) 2 }. (4)
We define the cylinder containing the fiber as (see fig 1 ):

C e = D e × R. (5)
Then the part of the basic cell containing the matrix is

Y e R = Z e \ C e , (6)
and by definition, the basic cell Y e is the couple

( Y e R , C e ). ( 7
)
The composite material results from a periodic extension of the basic cell. More precisely the part of the material that contains the carbon fibers is

Ω c = P ∩ {(ie, je, 0) + C e , i ∈ Z, j ∈ Z -}, (8)
where the intersection with P limits the periodic extension to the area where stands the material. Set {(ie, je, 0) + C e , i ∈ Z, j ∈ Z -} is a short notation for

{( x, y, z) ∈ R 3 , ∃i ∈ Z, ∃j ∈ Z -, ∃(x b , y b , z b ) ∈ C e ; x = x b + ie, y = y b + je, z = z b }. (9)
In the same way the part of the material that contains the resin is

Ω r = P ∩ {(ie, je, 0) + Y e R }, (10)
or equivalently

Ω r = P ∩ {(ie, je, 0) + Z e \ C e } = (R × (-L, 0) × R)\ Ω c . (11)
So the geometrical model of our composite material is couple ( Ω c , Ω r ). Now, it remains to set the domain that contains the air:

Ω a = {( x, y, z)/0 ≤ y < d}. ( 12
)
We consider that d is of the same order as L and we introduce the upper frontier Γ d = {( x, y, z)/ y = d} of domain Ω. On this frontier we will consider that the electric field and magnetic field are given. We also introduce the lower frontier Γ L = {( x, y, z)/ y = -L} with those definitions we have

Ω a ∩ P = ∅, Ω c ∩ Ω r = ∅, P = Ω r ∪Ω c , Ω = Ω a ∪ P = Ω a ∪ Ω r ∪ Ω c
, and for any ( x, y, z) ∈ ∂ Ω = Γ d ∪ Γ L and, we write n, the unit vector, orthogonal to ∂ Ω and pointing outside Ω. We have :

n = e 2 on Γ d n = -e 2 on Γ L . (13)
In the following we need to describe what happens at the interfaces between resin and carbon fibers, and resin and air. So we define Γ ra = {( x, y, z) / y = 0} and Γ cr the boundary of the set defined by (9).

Maxwell equations

In Ω, we now write a PDE model that has to do with electromagnetic waves radiated from return stroke. We are well aware that the model we write is a simplified one. Nonetheless, it seems to be well dimensioned for our problem which consists in making homogenization. It is well known (see Maxwell [START_REF] Maxwell | A dynamical theory of the electromagnetic field[END_REF]) the propagation of the electromagnetic field is described by the Maxwell equations which write:

- ∂ D ∂t + ∇× H = J , (14)
∂ B ∂t + ∇× E = 0, (15)
∇• D = ρ , (16)
∇• B = 0, (17)
in R × Ω. In ( 14)-( 17), ∇× and ∇• are the curl and divergence operators. E (t, x, y, z) is the electric field, H (t, x, y, z) the magnetic field, D (t, x, y, z) the electric induction, B (t, x, y, z) the magnetic induction and ρ (t, x, y, z) is the charges density (see T. Abboud and I. Terrasse [START_REF] Abboud | Modélisation des phénomènes de propagation d'ondes[END_REF]).

System of Maxwell equations (( 14) -( 17)) is completed by the constitutive laws which are given in R × Ω by :

D = 0 E , (18)
B = µ 0 H . ( 19
)
where µ 0 and 0 are the permeability and permittivity of free space. is the relative permittivity of the domains defined by

| Ωa = 1, | Ωr = r , | Ωc = c , (20)
where r and c are positives constants. In order to account for energy transfer between the electromagnetic compartment and the propagation of the electric charges, we take for granted the Ohmic law, in R × Ω

J = σ E , (21)
where σ is the electric conductivity. Its value depends on the location:

σ | Ωa = σ a , σ | Ωr = σ r , σ | Ωc = σ c , (22)
where Ω a , Ω r and Ω c were defined in [START_REF] Guenneau | Homogenization of 3d finite photonic crystals with heterogeneous permittivity and permeability[END_REF], [START_REF] Crouseilles | Two-Scale Macro-Micro decomposition of the Vlasov equation with a strong magnetic field[END_REF] and (8).

Boundary conditions

For mathematical as well as physical reasons we have to set boundary conditions on Γ d and Γ L . On Γ d we will write conditions that translate that E and H are given by the source located in y = d. The way we chose consists in setting:

E × n = E d × n; H × n = H d × n on R × Γ d , (23)
where E d , H d are functions defined on Γ d for any t ∈ R. On Γ L , we chose something simple, i.e :

∇× E × n = 0 on R × Γ L , (24)
that translate that E does not vary in the y-direction near Γ L . Problem ( 14)- [START_REF] Nguetseng | A general convergence result for a functional related to the theory of homogenization[END_REF] supplemented with [START_REF] Nicaise | Limit behaviors of some boundary value problems with high and/or low valued parameters[END_REF] and [START_REF] Ouchetto | Effective constitutive parameters of periodic composites[END_REF], is considered as containing all physics we want to account for. In the following we will consider simplifications of it.

Time-harmonic Maxwell equations

The first simplification we make, consists in considering the harmonic version of the Maxwell equations ( 14)- [START_REF] Nguetseng | Asymptotic analysis for a stiff variational problem arising in mechanics[END_REF]. This simplification is used in many references studying electromagnetic phenomena and especially for lightning applications [START_REF] Hoole | Aircraft-lightning electrodynamics using the transmission line model part i: review of the transmission line model[END_REF], in spite of the fact that it considers implicitly that every fields and currents are waves of the form, for all ω ∈ R : a( x, y, z) cos(-ωt + φ( x, y, z)) = e[a( x, y, z) exp i ωt exp iφ( x, y, z) ], [START_REF] Pak | Geometric two-scale convergence on forms and its applications to maxwell's equations[END_REF] where ω is the pulsation, φ( x, y, z) is the phase shift of the wave and a( x, y, z) is its amplitude. In particular, it supposes E d , H d in ( 23) are of the form, for all w ∈ R:

E d (t, x, z) = e( E d ( x, z) exp i ωt ), (26)
H d (t, x, z) = e( H d ( x, z) exp i ωt ), (27)
where E d and H d take into account the amplitude and the phase shift of their corresponding fields. Taking [START_REF] Nguetseng | A general convergence result for a functional related to the theory of homogenization[END_REF] into account, the time-harmonic Maxwell equations, which describe the electromagnetic radiation, are written:

∇× H -i ω 0 E = σ E, Maxwell -Ampere equation ( 28)

∇× E + i ωµ 0 H = 0, Maxwell -Faraday equation (29) ∇•( 0 E) = ρ, (30)
∇•(µ 0 H) = 0, (31)
where E (t, x, y, z) = e( E( x, y, z) exp i ωt ) and H (t, x, y, z) = e( H( x, y, z) exp i ωt ), ∀t ∈ R+, ( x, y, z) ∈ Ω. The magnetic field H can be directly computed from the electric field

E H = - 1 iωµ 0 ∇× E. (32)
Now, for the electric approach, taking the curl of equation ( 32) yields an expression of ∇× H in term of ∇ × ∇× E. Inserting ∇× H in [START_REF] Wellander | Homogenization of the Maxwell equations at fixed frequency[END_REF] we get the following equation for the electric field:

∇×∇× E + (-ω 2 µ 0 0 + i ωµ 0 σ) E = 0 in R × Ω. ( 33
)
Taking the divergence of the equation ( 28) yields the natural gauge condition:

∇•[(i ω 0 + σ) E] = 0 in R × Ω. (34)
Notice that i ω 0 + σ is equal to i ω 0 + σ a in Ω a , to i ω 0 r + σ r in Ω r and to i ω 0 c + σ c in Ω c , those quantities being all nonzero. Then (34) is equivalent to:

∇• E |Ωa = 0 in Ω a , ∇• E |Ωr = 0 in Ω r , ∇• E |Ωc = 0 in Ω c . ( 35
)
with the transmission conditions

(i ω 0 + σ a ) E | Ωa . n = (i ω 0 r + σ r ) E | Ωr . n on Γ ra , (i ω 0 r + σ r ) E | Ωr . n = (i ω 0 c + σ c ) E | Ωc . n on Γ cr . (36)
Summarizing, we finally obtain the PDE model:

∇×∇× E + (-ω 2 µ 0 0 + i ωµ 0 σ) E = 0 in R × Ω. ( 37
)
According to the tangential trace of the Maxwell-Faraday equation [START_REF] Pr | Cours : Matériaux diélectriques[END_REF] we obviously obtain that using boundary condition [START_REF] Nicaise | Limit behaviors of some boundary value problems with high and/or low valued parameters[END_REF], is equivalent to using:

∇× E × e 2 = -i ωµ 0 H d ( x, z) × e 2 on R × Γ d (38)
where H d is defined in ( 27) and where we used [START_REF] Hoole | waves along an unmagnetized lightning plasma channel[END_REF]. And on Γ L we have the following boundary condition:

∇× E × e 2 = 0 on R × Γ L . (39)

Scaling

In this subsection we propose a rescaling of system ((37)-( 39)), we will consider a set of characteristic sizes related to our problem. Physical factors are then rewritten using those values leading to a new set of dimensionless and unitless variables and fields in which the system is rewritten. The considered characteristic sizes are : ω the characteristic pulsation, σ the characteristic electric conductivity, E the characteristic electric magnitude, H the characteristic magnetic magnitude. We also use the already introduced thickness L of the plate P . We then introduce the dimensionless variables:

x = (x, y, z) with x = x L , y = y L , z = z L and fields E, H and σ that are such that              E(ω, x) = 1 E E(ωω, Lx, Ly, Lz), H(ω, x) = 1 H H(ωω, Lx, Ly, Lz), σ(x) = 1 σ σ(Lx, Ly, Lz), (40)
Taking ( 22) into account, σ also reads:

σ(x) = σ a σ if 0 ≤ Ly ≤ d, (41) σ
(x) = σ r σ if (Lx, Ly, Lz) ∈ Ω r , (42)
σ(x) = σ c σ if (Lx, Ly, Lz) ∈ Ω c . ( 43
)
Doing this gives the status of units to the characteristic sizes. Since, for instance:

∂E ∂x (ω, x) = L E ∂ E ∂ x (ωω, Lx, Ly, Lz), (44)
using those dimensionless variables and fields and taking (41)-(43) into account, equation (37) gives:

E ∇×∇×E(ω, x) - L 2 ω 2 c 2 ω 2 + iσ ω ωL 2 µ 0 σ(x, ω) EE(ω, x, y, z) = 0, (45)
for any (ω, x) such that (ωω, Lx, Ly, Lz) ∈ R × Ω. Now we exhibit

λ = 2πc ω , (46)
which is the characteristic wave length and

δ = 1 √ ω σµ 0 , (47)
which is the characteristic skin thickness. Using those quantities equation (45) reads, for any (ω, x) ∈ R × Ω:

∇×∇×E(ω, x) + (- 4π 2 L 2 λ 2 ω 2 + i L 2 δ 2 σ a σ ω)E(ω, x) = 0 when 0 ≤ Ly ≤ d, ∇×∇×E(ω, x) + (- 4π 2 L 2 λ 2 r ω 2 + i L 2 δ 2 σ r σ ω)E(ω, x) = 0 when (Lx, Ly, Lz) ∈ Ω r , ∇×∇×E(ω, x) + (- 4π 2 L 2 λ 2 c ω 2 + i L 2 δ 2 σ c σ ω)E(ω, x) = 0 when (Lx, Ly, Lz) ∈ Ω c . (48)
In the following expressions, L λ and L δ appearing in the equations above will be rewritten in terms of a small parameter ε.

The boundary conditions are written

∇×E(ω, x) × e 2 = -iωωµ 0 L E H d (Lx, Lz) × e 2 when (Lx, Ly, Lz) ∈ R × Γ d , ∇×E(ω, x) × e 2 = 0 when (Lx, Ly, Lz) ∈ R × Γ L . (49)
The characteristic thickness of the plate L is about 10 -3 m and the size of the basic cell e is about 10 -5 m. Since e is much smaller than the thickness of the plate L, it is pertinent to assume the ratio e L equals a small parameter ε: e L ∼ 10 -2 = ε.

Then, in what concerns the characteristic pulsation ω, in the tables below we consider several values. The lightning is seen as a low frequency phenomenon. Indeed, energy associated with radiation tracers and return stroke are mainly burn by low and very low frequencies (from 1kHz to 300kHz). Components of the frequency spectrum are however observed beyond 1GHz (see [START_REF] Leboulch | Analyse spectrale vhf, uhf du rayonnement des éclairs[END_REF]). So, in the case when we want to catch low frequency ie we consider ω = 100 rad/s, (in our study we will consider ω = 10 6 rad/s), for medium frequency we set ω = 10 10 rad/s and for high frequency phenomena ω = 10 12 rad/s. Then, concerning the characteristic electric conductivity it seems to be reasonable to take for σ the value of the effective electric conductivity of the composite material. Yet this choice implies to compute a coarse estimate of this effective conductivity at this level.

For this we take into account that the composite material is composed of carbon fibers and epoxy resin. The resin can be doped, which increases strongly its conductivity, or not. The tables below summarize the cases when the resin is doped and also when the resin is not doped. We also account for the fact there is not only one effective electric conductivity but a first one in the fiber direction : the effective longitudinal electric conductivity (in cases 1, 2, 5 and 6 of the tables below), and a second effective electric conductivity, in the direction transverse to the fibers (considered in cases 3, 4, 7 and 8). In this context, we consider the basic model which is based on the electrical analogy and the law of mixtures. It corresponds to the Wiener limits: the harmonic average and the arithmetic average. The effective values are the extreme limits of the conductivity of the composite introduced by Wiener in 1912 see S. Berthier p 76 [START_REF] Berthier | Optique des milieux composites[END_REF].

The effective longitudinal electric conductivity corresponding of the upper Wiener limit is expressed by the equation:

σ = σ long = f c σ c + (1 -f c ) σ r , (51)
where

f c = π α 2 4
is the volume fraction of the carbon fiber. The effective transverse electric conductivity corresponding of the lower Wiener limit is expressed by

σ = σ trans = 1 fc σc + (1-fc) σr . (52)
For the computation, we take values close to reality. We consider composite materials with similar proportions of carbon and resin, this means that α is close to 1 2 . When the resin is not doped σ r ∼ 10 -10 S.m -1 is much smaller than σ c ∼ 40000S.m -1 . Then, σ = σ long is close to π α 2 4 σ c ∼ σ c and σ = σ trans is close to σr

(1-π α 2 4 )
∼ σ r . Now, we express the electric conductivity of the air in terms of σ, we consider two possibilities. The first one is relevant for a situation with a ionized channel. The second one of situation with a strong atmospheric electric field but without a ionized channel. In this situation air is not ionized and has a low conductivity. All possible situations are gathered in the tables below. Cases 5 to 8 are associated with the first situation with air conductivity σ a being σ lightning = 4242S.m -1 for an ionized lightning channel see [START_REF] Hoole | waves along an unmagnetized lightning plasma channel[END_REF]. Cases 1 to 4, to the second one, with σ a = 10 -14 S.m -1 .

All calculations of the different cases of the tables are detailed in Annex A. In our study we consider the case 6 for ω = 10 6 rad.s -1 , which corresponds to the air ionized, a resin doped and the effective longitudinal electric conductivity of the carbon fibers.

As it is well known the tables confirm that at high frequencies the thickness of the plate is much greater than the skin depth. This one depends on σ and ω and decreases strongly for high conductivity or high frequencies. For ω = 10 10 rad.s -1 and σ = 4 * 10 4 S.m -1 , the effective conductivity in the direction of the carbon fibers, which the skin effect phenomenon appears. Indeed, for high frequencies ω = 10 12 rad.s -1 and when σ is the effective conductivity is in direction of the carbon fibers i.e. in high conductivity, δ = 10 -4 m. In low frequencies and low conductivity δ is large so the electromagnetic wave can penetrate the composite material.

σ c (S.m -1 ) σ σ σ ε 7 σ ε 4 σ σ σ ε 7 σ ε 4 σ r (S.m -1 ) ε 7 σ ε 4 σ σ σ ε 7 σ ε 4 σ σ σ σ a (S.m -1 ) ε 9 σ ε 9 σ ε 2 σ ε 6 σ εσ εσ σ ε 5 σ ε 2 4πL 2 λ 2 ε 9 ε 9 ε 9 ε 9 ε 9 ε 9 ε 9 ε 9 L 2 δ 2 ε 2 ε 2 ε 10 ε 7 ε 2 ε 2 ε 10 ε 7 Table 1: for ω = 100rad.s -1 .
case 1 case 2 case 3 case 4 case 5 case 6 case 7 case 8 L(m) -3 10 -3 10 -3 10 -3 10 -3 10 -3 10 -3 10 -3 e(m) -5 10 -5 10 -5 10 -5 10 -5 10 -5 10 -5 10 -5 λ(m) 10 3 10 3 10 3 10 3 10 3 10 3 10 3 10 3 σ(S.m -1 ) 40000 40000 10 -10 10 -3 40000 40000 10 -10 10 -3 δ(m) -3 10 -3 10 5 10 10 -3 10 -3 -3 10 -3 10 -3 10 -3 10 -3 10 -3 10 -3 10 -3 e(m) -5 10 -5 10 -5 10 -5 10 -5 10 -5 10 -5 10 -5 λ(m) -1 10 -1 10 -1 10 -1 10 -1 10 -1 10 -1 10 -1 σ(S.m -1 ) 40000 40000 10 -10 10 -3 40000 40000 10 -10 10 -3 δ(m) -5 10 -5 10 3 10 -1/2 10 -5 10 -5

10 5 10 σ c (S.m -1 ) σ σ σ ε 7 σ ε 5 σ σ σ ε 7 σ ε 5 σ r (S.m -1 ) ε 7 σ ε 4 σ σ σ ε 7 σ ε 4 σ σ σ σ a (S.m -1 ) ε 9 σ ε 9 σ ε 2 σ ε 6 σ εσ εσ σ ε 5 σ ε 2 4πL 2 λ 2 ε 5 ε 5 ε 5 ε 5 ε 5 ε 5 ε 5 ε 5 L 2 δ 2 1 1 ε 8 ε 5 1 1 ε 8 ε 5
10 3 10 -1/2 σ c (S.m -1 ) σ σ σ ε 7 σ ε 4 σ σ σ ε 7 σ ε 4 σ r (S.m -1 ) ε 7 σ ε 4 σ σ σ ε 7 σ ε 4 σ σ σ σ a (S.m -1 ) ε 9 σ ε 9 σ ε 2 σ ε 6 σ εσ εσ σ ε 5 σ ε 2 4πL 2 λ 2 ε ε ε ε ε ε ε ε L 2 δ 2 1 ε 2 1 ε 2 ε 6 ε 3 1 ε 2 1 ε 2 6 3
Table 3: for ω = 10 10 rad.s -1 .

case 1 case 2 case 3 case 4 case 5 case 6 case 7 case 8 L(m) -3 10 -3 10 -3 10 -3 10 -3 10 -3 10 -3 10 -3 e(m) -5 10 -5 10 -5 10 -5 10 -5 10 -5 10 -5 10 -5 λ(m) -3 10 -3 10 -3 10 -3 10 -3 10 -3 10 -3 10 -3 σ(S.m -1 ) 40000 40000 10 -10 10 -3 40000 40000 10 -10

10 -3 δ(m) -6 10 -6 1 10 -3/2 10 -6 10 -6 1 10 -3/2 σ c (S.m -1 ) σ σ σ ε 7 σ ε 4 σ σ σ ε 7 σ ε 4 σ r (S.m -1 ) ε 7 σ ε 4 σ σ σ ε 7 σ ε 4 σ σ σ σ a (S.m -1 ) ε 9 σ ε 9 σ ε 2 σ ε 6 σ εσ εσ σ ε 5 σ ε 2 4πL 2 λ 2 1 1 1 1 1 1 1 1 L 2 δ 2 1 ε 3 1 ε 3 ε 3 ε 1 ε 3 1 ε 3 3 ε Table 4: for ω = 10 12 rad.s -1 .
limits the penetration of the electromagnetic wave to a boundary layer whose depth is about δ. Now, we will discuss on the values of E and ρ. It seems that the density of electrons in a ionized channel is about 10 10 part.m -3 . Hence we take ρ = 10 10 . When the air is not ionized, the charge density is much smaller, and we choose: ρ = 1.

For the boundary conditions, in the context of the case 6 and ω = 10 6 rad/s, we consider the peak of the current of the first return stroke. Then the magnetic field magnitude H is H d given by [START_REF] Abboud | Modélisation des phénomènes de propagation d'ondes[END_REF].

Then the dimensionless boundary conditions (38) writes:

∇×E(x, ω) × e 2 = -iωωµ 0 L E H d H d (x, z) × e 2 , (53)
where

H d H d (x, z) = H d (Lx,
D = {(ξ, ν) ∈ R 2 /ξ 2 + (ν + 1 2 ) 2 < R 2 }, (54)
and R = α 2 . The set Ω c is then defined as:

Ω c = {(i, j, 0) + C, i ∈ Z, j ∈ Z -}. ( 55
)
We denote Y r as Y r = Z\C and then the set

Ω r = {(i, j, 0) + Y r , i ∈ Z, j ∈ Z -}. (56)
Then unit cell Y is defined as Y = (Y r , C). Finally, we define the domain Ω a :

Ω a = {y = (ξ, ν, ζ) / ν > 0}. ( 57
)
Using this, we will give a new expression of the sets in which the variables range in equations (48). We see the following:

(Lx, Ly, Lz) ∈ Ω r ⇔ (Lx, Ly, Lz) ∈ P , ( L e x, L e y, L e z) ∈ Ω r , (58)
i.e.

(Lx, Ly, Lz) ∈ Ω r ⇔ (Lx, Ly, Lz) ∈ P ,

( x ε , y ε , z ε ) ∈ Ω r . (59)
In the same way:

(Lx, Ly, Lz) ∈ Ω c ⇔ (Lx, Ly, Lz) ∈ P , ( x ε , y ε , z ε ) ∈ Ω c , (60)
and:

0 ≤ Ly ≤ d ⇔ y ∈ R 2 Ly ≤ d, (61)
or

(Lx, Ly, Lz) ∈ Ω a ⇔ Ly ≤ d ( x ε , y ε , z ε ) ∈ Ω a . (62)
We define:

Σ ε (y) = Σ ε (ξ, ν, ζ) =    Σ ε a in Ω a , Σ ε r in Ω r , Σ ε c in Ω c , (63)
where

Σ ε a = σa σ L 2 δ 2 , Σ ε r = σr σ L 2 δ 2 and Σ ε c = σc σ L 2 δ
2 have their expressions in term of ε given from Tables above depending on the case we are interested in. The detail of this expressions are in appendix B. The model that we present is the case ω = 10 6 rad.s

-1 , η = 5, Σ ε a = ε, Σ ε r = ε 4 and Σ ε c = 1.
Defining also mapping

ψ ε : R 3 → R 3 (x, y, z) → ( x ε , y ε , z ε ), ( 64
) we can set Ω ε a as ψ -1 ε (Ω a ) ∩ (R × [0, d L ] × R), Ω ε r as ψ -1 ε (Ω r ) ∩ P and Ω ε c as ψ -1 ε (Ω c ) ∩ P . We also define the boundaries Γ d = {x ∈ R 3 , y = d L } and Γ L = {x ∈ R 3 , y = -L} and interfaces Γ ra = {x ∈ R 3 , y = 0} and Γ ε cr = ∂Ω c .
Hence equation (48) reads:

∇×∇×E ε + (-ω 2 ε η + i ω σ ε (x, y, z))E ε = 0 in Ω, (65)
where

Ω = Ω ε a ∪ Ω ε r ∪ Ω ε c = {x ∈ R 3 , -1 < y < d L } does not depend on ε. Only its partition in Ω ε a , Ω ε r and Ω ε c is ε-dependent where σ ε (x, y, z) = Σ ε ( x ε , y ε , z ε ), (66)
with Σ ε given by (63) and

ε η = 4π 2 L 2 λ 2 , ( 67
)
with the value of η ≥ 0 extracted from Tables, and where we replace E by E ε , to clearly state that it depends on ε. Equation ( 65) is provided with the following boundary conditions:

∇×E ε × e 2 = -iωH d (x, z) × e 2 on R × Γ d , (68)
coming from (53). And, coming from (49),

∇×E ε × e 2 = 0 on R × Γ L . (69)
From (65) we can deduce the condition on the divergence of E ε which can be written in two ways. As previously in (34), ( 35) and (36) we obtain:

∇•[(-ω 2 ε η + iωσ ε )E ε ] = 0 in Ω, (70)
which will be preferentially used with (65) and its second one is

∇•E ε |Ω ε a = 0 in Ω ε a , ∇•E ε |Ω ε r = 0 in Ω ε r , ∇•E ε |Ω ε c = 0 in Ω ε c , (71)
with the transmission conditions on the interfaces Γ ra and Γ ε cr

(-ω 2 ε η + iωΣ ε a ) E ε |Ω ε a • n |Ω ε a = (-ω 2 ε η r + iωΣ ε r ) E ε |Ω ε r • n |Ω ε r on Γ ra , (-ω 2 ε η r + iωΣ ε r ) E ε |Ω ε r • n |Ω ε r = (-ω 2 ε η c + iωΣ ε c ) E ε |Ω ε c • n |Ω ε c on Γ ε cr . (72)
Before treating mathematically the question we are interested in, we make a last simplification. Since it seems clear that physical relevant phenomena occur in the upper part of the plate. The boundary condition on the lower boundary of the plate has very little influence on the physics of what happens in the upper part, we consider that the lower boundary of Ω is located in y = -∞ in place of y = -1, making the second boundary condition useless. Besides, as L and d are of the same order it seems reasonable to set Γ d = {x ∈ R 3 , y = 1} and consequently

       Ω = {x ∈ R 3 , y < 1} = Ω ε a ∪ Ω ε r ∪ Ω ε c , with, Ω ε a = ψ -1 ε (Ω a ), Ω ε r = ψ -1 ε (Ω r ), Ω ε c = ψ -1 ε (Ω c ), (73)
with ψ ε defined in (64). We have that the border of Ω is Γ d . In the following section we will establish existence and uniqueness results.

3 Mathematical analysis of the models

Preliminaries

We are going to make precise the variational formulation. First of all, we need to introduce the following functional spaces. We have the standard function spaces

L 2 (Ω ε ) = [L 2 (Ω ε )] 3 H(curl, Ω) = {u ∈ L 2 (Ω) : ∇×u ∈ L 2 (Ω)}, H(div, Ω) = {u ∈ L 2 (Ω) : ∇•u ∈ L 2 (Ω)}, (74)
with the usual norms:

u 2 H(curl,Ω) = u 2 L 2 (Ω) + ∇×u 2 L 2 (Ω) , u 2 H(div,Ω) = u 2 L 2 (Ω) + ∇•u 2 L 2 (Ω) . (75)
They are well known Hilbert spaces. Above, the curl and the divergence are defined in the weak sense as:

(∇×u, V ) = (u, ∇×V ), ∀V ∈ C 1 (Ω), (∇•u, V ) = -(u, ∇V ), ∀V ∈ C 1 (Ω). (76)
We use in this paper, the trace spaces H -12 (curl, Γ d ) and H -1 2 (div, Γ d ) defined by

H -1 2 (curl, Γ d ) = {u ∈ H -1 2 (Γ d , R 3 ), (n • u) |Γ d = 0, curl Γ d u ∈ H -1 2 (Γ d , R 3 )}, ( 77
)
H -1 2 (div, Γ d ) = {u ∈ H -1 2 (Γ d , R 3 ), (n • u) |Γ d = 0, div Γ d u ∈ H -1 2 (Γ d , R 3 )} (78)
where the surface divergence div Γ d u and the surface rotation curl Γ d u are defined by

(div Γ d u, V ) L 2 (Γ d ) = -(u, ∇ Γ d V ) L 2 (Γ d ,R 3 ) , ∀ V ∈ C 1 (Γ d ) curl Γ d u = n • (∇ × u |Γ d ) (79)
and the surface gradient ∇ Γ d V is defined by the orthogonal projection of ∇ on Γ d , i.e n × (∇ Γ d × n), n denotes the outward unit vector normal to Γ d .

Finally we recall the trace theorems, see J.C Nédélec [START_REF] Nédélec | Acoustic and electromagnetic equations; integral representations for harmonic problems[END_REF] for the demonstration, stating that the traces mappings

γ T : H(curl, Ω) -→ H -1 2 (curl, Γ d )
, that assigns any u ∈ H(curl, Ω) its tangential components n × (u × n) is continuous and surjective, that is:

γ T (u) H -1 γ t : H(curl, Ω) -→ H -1 2 (div, Γ d )
, that assigns any u ∈ H(curl, Ω) its tangential components u × n, is continuous and surjective:

γ t (u) H -1 2 (div,Γ d ) ≤ C γt u H(curl,Ω) , ∀u ∈ H(curl, Ω).
Moreover, H -1 2 (div, Γ d ) is the dual of H -1 2 (curl, Γ d ) and one has the Green's formula:

Ω (∇×u • V -u • ∇×V )dx = u × n, V T Γ d ∀(u, V ) ∈ H(curl, Ω). ( 80
)
We define the next space:

X(Ω) = {u ∈ H(curl, Ω) | ∇•u |Ω ε a ∈ L 2 (Ω ε a ), ∇•u |Ω ε r ∈ L 2 (Ω ε r ), ∇•u |Ω ε c ∈ L 2 (Ω ε c )}. ( 81
)
Our variational space is:

X ε (Ω) = {u ∈ X(Ω) | (-ω 2 ε η + iωσ ε |Ω ε a )u |Ω ε a • e 2 = (-ω 2 ε η ε r + iωσ ε |Ω ε r )u |Ω ε r • e 2 , (-ω 2 ε η r + iωσ ε |Ω ε r )u |Ω ε r • n ε |Ω ε r = (-ω 2 ε η ε c + iωσ ε |Ω ε c )u |Ω ε c • n ε |Ω ε c . (82)
Finally

X ε (Ω) = {u ∈ X(Ω) | (-ω 2 ε η + iωΣ ε a )u |Ω ε a • e 2 = (-ω 2 ε η ε r + iωΣ ε r )u |Ω ε r • e 2 , (-ω 2 ε η r + iωΣ ε r )u |Ω ε r • n ε |Ω ε r = (-ω 2 ε η ε c + iωΣ ε c )u |Ω ε c • n ε |Ω ε c }. ( 83
)
This space is equipped with the norm

u 2 X ε (Ω) = u 2 L 2 (Ω) + ∇•u |Ω ε a 2 L 2 (Ω ε a ) + ∇•u |Ω ε r 2 L 2 (Ω ε r ) + ∇•u |Ω ε c 2 L 2 (Ω ε c ) + ∇×u 2 L 2 (Ω) .

Weak formulation

Now, we introduce the variational formulation of our problem (65), ( 68) and (69) for the electric field. Integrating (65) over Ω and using the Green's formula and (68) we obtain

Ω ∇×E ε • ∇×V dx + Ω ε a (-ω 2 ε η + iωΣ ε a )E ε • V dx + Ω ε c (-ω 2 ε η c + iωΣ ε c )E ε • V dx + Ω ε r (-ω 2 ε η r + iωΣ ε r )E ε • V dx = Γ d (∇×E ε × e 2 ) • V T dσ = Γ d -iωH d × e 2 • V T dσ ( 84
)
where V is the complex conjugate of V and V T = (e 2 × V ) × e 2 . We introduce the sesquilinear form depending on parameters η and ε:

     For E ε , V ∈ X ε (Ω), a ε,η (E ε , V ) = Ω ∇×E ε • ∇×V dx + i=a,r,c Ω ε i (-ω 2 ε η i + iωΣ ε i ) E ε • V dx. (85)
Hence, the weak formulation of (65), ( 68) and ( 69) that we will use is the following, is:

   Find E ε ∈ X ε (Ω) such as ∀ V ∈ X ε (Ω)
we have :

a ε,η (E ε , V ) = -iω Γ d H d × e 2 • V T dσ. ( 86
)
Integrating by parts in the variational formulation (84), we find the following transmission problem:

                     ∇×∇×E ε + (-ω 2 ε η + i ω Σ ε a )E ε = 0 in Ω ε a , ∇×∇×E ε + (-ω 2 ε η r + i ω Σ ε r )E ε = 0 in Ω ε r , ∇×∇×E ε + (-ω 2 ε η c + i ω Σ ε c )E ε = 0 in Ω ε c . E ε |Ω ε a × e 2 = E ε |Ω ε r × n |Ω ε r on Γ ra , E ε |Ω ε r × n |Ω ε r = E ε |Ω ε c × n |Ω ε c on Γ ε cr , ∇×E ε |Ω ε a × e 2 = ∇×E ε |Ω ε r × n |Ω ε r on Γ ra , ∇×E ε |Ω ε r × n |Ω ε r = ∇×E ε |Ω ε c × n |Ω ε c on Γ ε cr , (87)
where e 2 is the unit outward normal to Ω ε a , n |Ω ε r is the unit outward normal to Ω ε r and n |Ω ε c is the unit outward normal to Ω ε c . We refer to Annex C for the proof that transmission problem (87) is equivalent to ((65), (68), (69), (71)).

Regularized Maxwell equations for the electric field

The sesquilinear form a ε,η is not coercive on X ε (Ω), so we regularize it adding terms involving the divergence of E ε in Ω ε a , Ω ε r and Ω ε c . Thanks to the additional terms, existence and uniqueness of the regularized variational formulation solution will be established by the Lax-Milgram theory. Let s be an arbitrary positive number, we define the regularized formulation of problem (86):

                       Find E ε ∈ X ε (Ω) such that for any V ∈ X ε (Ω) a ε,η R (E ε , V ) = a ε,η (E ε , V ) + s Ω ε a ∇•E ε ∇•V dx + s Ω ε r ∇•E ε ∇•V dx + s Ω ε c ∇•E ε ∇•V dx = -iω Γ d H d × e 2 • V T dσ. (88)
For any ε > 0 and any η ≥ 0, sesquilinear form a ε,η R (., .) is continuous over X ε (Ω) thanks to the continuity conditions. We will show that it is also coercive. The following proposition was inspired by article [START_REF] Costabel | Corner singularities of maxwell interface and eddy current problems[END_REF] Lemma 1.1. Proposition 3.1. For any ε > 0, for any η ≥ 0 and for any s > 0, there exists a positive constant ω 0 which does not depend on ε and such that for all ω ∈ (0, ω 0 ), there exists a positive constant C 0 depending on ε r , ε c , s, ω but not on ε such that:

∀ E ε ∈ X ε (Ω), [exp(-i π 4 ) a ε,η R (E ε , E ε )] ≥ C 0 E ε X ε (Ω) (89)
Proof. We have:

[exp(-i π 4 ) a ε,η R (E ε , E ε )] = a ε R (E ε , E ε ) - Ω ε a ω 2 ε η |E ε | 2 dx - Ω ε r ω 2 ε η r |E ε | 2 dx - Ω ε c ω 2 ε η c |E ε | 2 dx. (90)
with

a ε R (E ε , E ε ) = Ω |∇×E ε | 2 dx + s Ω ε a |∇•E ε | 2 dx + s Ω ε r |∇•E ε | 2 dx + s Ω ε c |∇•E ε | 2 dx + Ω ε a ωΣ ε a |E ε | 2 dx + Ω ε r ωΣ ε r |E ε | 2 dx + Ω ε c ωΣ ε c |E ε | 2 dx. (91)
We have the following estimate:

|a ε R (E ε , E ε )| ≥ min{1, ω, s}( ∇×E ε 2 L 2 (Ω) + ∇•E ε 2 L 2 (Ω ε a ) + ∇•E ε 2 L 2 (Ω ε r ) + ∇•E ε 2 L 2 (Ω ε c ) + E ε 2 L 2 (Ω) ). ( 92
)
Then we have:

| a ε R (E ε , E ε )| ≥ min{1, ω, s} E ε 2 X ε (Ω) . (93)
Returning to formulation (89), for η ≥ 0, since max(Σ ε a , Σ ε r , Σ ε c ) > ε η , inequality (89) is valid with C 0 = min{1, ω, s} as soon as ω 2 min{1, ε r , ε c } < min{1, ω, s} or ω < ω 0 = min{1,ω,s} min{1,εr,εc} . This ends the proof of Proposition 3.1.

Thanks to Proposition 3.1 we can state the existence and uniqueness of the solution to regularized problem (88). Theorem 3.2. Under the assumptions of Proposition 3.1, there exists a unique solution E ε to regularized problem (88).

Proof. Since the sesquilinear form a ε,η R is continuous, bounded, coercive thanks to Proposition 3.1 and the right hand side is continuous on X ε (Ω), problem (88) has a unique solution in X ε (Ω) thanks to the Lax-Milgram Theorem.

Existence, uniqueness and estimate

Theorem 3.3. For any ε > 0, for any η ≥ 0, there exists a positive constant ω 0 which does not depend on ε and such that for all ω ∈ (0, ω 0 ), there exists a unique solution of (87) or ((65), ( 68), ( 69), ( 71)).

Proof. It is obvious that any solution of (87) or of (( 65), ( 68), ( 69),( 71)) is also solution to (88). Indeed, since from (87) or from ((65), ( 68), ( 69),( 71)) we have

∇•E ε |Ω ε a = 0, ∇•E ε |Ω ε r = 0, ∇•E ε |Ω ε c = 0, the additional terms s Ω ε a ∇•E ε ∇•V dx + s Ω ε r ∇•E ε ∇•V dx + s Ω ε c ∇•E ε ∇•V dx cancel in (88).
Uniqueness follows from that if E ε 1 and E ε 2 are two solutions to (65) with the boundary condition (69) their difference e ε = E ε 2 -E ε 1 satisfies the problem (65) with (69). Then it comes

Ω |∇×e ε | 2 dx + Ω ε a (-ω 2 ε η + iωΣ ε a )|e ε | 2 dx + Ω ε c (-ω 2 ε η c + iωΣ ε c )|e ε | 2 dx + Ω ε r (-ω 2 ε η r + iωΣ ε r )|e ε | 2 dx = 0. ( 94
)
Taking the imaginary part of the expression we get

Ω ε a ωΣ ε a |e ε | 2 dx+ Ω ε c ωΣ ε c |e ε | 2 dx+ Ω ε r ωΣ ε r |e ε | 2 dx = 0
and then e ε = 0. Let us consider the reciprocal assertion, according to the same proof of S. Hassani, S. Nicaise, A. Maghnouji in [START_REF] Nicaise | Limit behaviors of some boundary value problems with high and/or low valued parameters[END_REF], we define

H 1 0 (Ω ε c , ∆) the subspace of ψ ∈ H 1 0 (Ω ε c ) such that ∆(ψ) ∈ L 2 (Ω ε c )
. Let E ε be the solution of the regularized formulation (88). In (88) we take a test function V = ∇ψ where ψ ∈ H 1 0 (Ω ε c , ∆), extended by zero outside Ω ε c . We get:

Ω ε c s∇•E ε ∇•(∇ψ) dx + Ω ε c (-ω 2 ε η c + iωΣ ε c )E ε • ∇ψ dx = 0. ( 95
)
By Green's formula, ∀ψ ∈ H 1 0 (Ω ε c , ∆), we obtain:

Ω ε c ∇•E ε (∆ψ + ω 2 ε η c -iωΣ ε c s ψ) dx = 0. ( 96
)
Thus, if we choose s such that ω 2 ε η c-iωΣ ε c s is not an eigenvalue of (∆ dir , Ω ε c ): the Laplacian operator in Ω ε c with Dirichlet condition on its boundary, then for all ϕ ∈ L(Ω ε c ) 2 there exists

ψ ∈ H 1 0 (Ω ε c , ∆) solution of ∆ψ + ω 2 ε η c -iωΣ ε c s ψ = ϕ. ( 97
)
Then, we conclude that

∇ • E ε |Ω ε c = 0. ( 98
) A similar argument in Ω ε a yields ∇•E ε |Ω ε a = 0 for s such that ω 2 ε η -iωΣ ε a s
is not an eigenvalue of (∆ dir , Ω ε a ). In the same way, we obtain in

Ω ε r , ∇•E ε |Ω ε r = 0 with s such that ω 2 ε η r -iωΣ ε r s
is not an eigenvalue of (∆ dir , Ω ε r ).

So, (88) becomes (84). Applying Green's formula, we find (65).

Theorem 3.4. Under the assumptions of Theorem 3.2, E ε ∈ X ε (Ω) solution of ( 88) satisfies

E ε X ε (Ω) ≤ C (99) with C = Cγ t Cγ T C 0 H d H(curl,Ω)
Proof. The sesquilinear form a ε,η R (E ε , V ) is coercive, weak formulation (88) becomes:

C 0 E ε 2 X ε (Ω) ≤ (exp(-i π 4 )a ε,η R (E ε , E ε )) ≤ | exp(-i π 4 ) • a ε,η R (E ε , E ε )| = |a ε,η R (E ε , E ε )| ≤ | Γ d -iωH d × e 2 • E ε T dσ| ≤ H d × e 2 H -1 2 (div,Γ d ) E ε T H -1 2 (curl,Γ d ) ≤ C γt C γ T H d × e 2 H(curl,Ω) E ε H(curl,Ω) (100)
where

E ε T = e 2 × (E ε × e 2 )
and the continuous dependence of the trace norm with

C = Cγ t Cγ T C 0 H d H(curl,Ω) gives: E ε 2 X ε (Ω) ≤ C E ε H(curl,Ω) ≤ C E ε X ε (Ω) . ( 101
)
4 Homogenization

With the aim to obtain a convergence result for the problem (65), ( 68) and ( 69), we propose an approach based on two-scale convergence. This concept was introduced by G. Nguetseng [START_REF] Nguetseng | A general convergence result for a functional related to the theory of homogenization[END_REF], [START_REF] Nguetseng | Asymptotic analysis for a stiff variational problem arising in mechanics[END_REF] and specified by G. Allaire [START_REF] Allaire | Homogenization and Two-scale Convergence[END_REF], [START_REF] Allaire | Multiscale convergence and reiterated homogenization[END_REF] which studied properties of the two-scale convergence. M. Neuss-Radu in [START_REF] Neuss-Radu | Some extensions of two-scale convergence[END_REF] presented an extension of twoscale convergence method to the periodic surfaces. Many authors applied two-scale convergence approach D. Cionarescu and P. Donato [START_REF] Cionarescu | An introduction to homogenization[END_REF], N. Crouseilles, E. Frénod, S. Hirstoaga and A. Mouton [START_REF] Crouseilles | Two-Scale Macro-Micro decomposition of the Vlasov equation with a strong magnetic field[END_REF], Y. Amirat, K. Hamdache and A. Ziani [START_REF] Amirat | Homogenisation of degenerate wave equations with periodic coefficients[END_REF] and also A. Back, E. Frénod [START_REF] Back | Geometric two-scale convergence on manifold and applications to the Vlasov equation. Discrete and Continuous Dynamical Systems -Serie S[END_REF]. This mathematical concept were applied to homogenize the time-harmonic Maxwell equations S. Ouchetto, O. Zouhdi and A. Bossavit [START_REF] Ouchetto | Effective constitutive parameters of periodic composites[END_REF], H.E. Pak [START_REF] Pak | Geometric two-scale convergence on forms and its applications to maxwell's equations[END_REF].

In our model, the parallel carbon cylinders are periodically distributed in direction x and z, as the material is homogenous in the y direction, we can consider that the material is periodic with a three directional cell of periodicity. In other words,

introducing Z = [-1 2 , 1 2 ] × [-1, 0] 2 , function Σ ε given by (63) is naturally periodic with respect to (ξ, ζ) with period [-1 2 , 1 2 ] × [-1
, 0] but it is also periodic with respect to y with period Z. Now, we review some basis definitions and results about two-scale convergence.

Two-scale convergence

We first define the function spaces

H # (curl, Z) = {u ∈ H(curl, R 3 ) : u is Z-periodic} H # (div, Z) = {u ∈ H(div, R 3 ) : u is Z-periodic} (102)
and where H(curl, R 3 ) and H(div, R 3 ) are defined by (74) with Ω ε replaced by R 3 . We introduce

L 2 # (Z) = {u ∈ L 2 (R 3 ), u is Z-periodic}, (103)
and

H 1 # (Z) = {u ∈ H 1 (R 3 ), u is Z-periodic}, (104)
where H 1 (R 3 ) is the usual Sobolev space on R 3 . First, denoting by C 0 # (Z) the space of functions in C 0 (R 3 ) and Z-periodic, C 0 0 (R 3 ) the space of continuous functions over R 3 with compact support, we have the following definitions:

Definition 4.1. A sequence u ε (x) in L 2 (Ω) two-scale converges to u 0 (x, y) ∈ L 2 (Ω, L 2 # (Z)) if for every V (x, y) ∈ C 0 0 (Ω, C 0 # (Z)) lim ε→0 Ω u ε (x) • V (x, x/ε) dx = Ω Z u 0 (x, y) • V (x, y) dxdy. (105)
If (105) happens, u ε (x) two-scale converges to u 0 (x, y):

u ε (x) u 0 (x, y). Proposition 4.2. If u ε (x) two-scale converges to u 0 (x, y) ∈ L 2 (Ω, L 2 # (Z)), we have for all v(x) ∈ C 0 (Ω) and all w(y) ∈ L 2 # (Z) lim ε→0 Ω u ε (x) • v(x)w( x ε ) dx = Ω Z u 0 (x, y) • v(x)w(y) dxdy. (106)
Theorem 4.3. (Nguetseng). Let u ε (x) ∈ L 2 (Ω). Suppose there exists a constant c > 0 such that for all ε

u ε L 2 (Ω) ≤ c.
Then there exists a subsequence of ε (still denoted ε) and u 0 (x, y) ∈ L 2 (Ω, L 2 # (Z)) such that:

u ε (x) u 0 (x, y). (107)
Proposition 4.4. Let u ε (x) be a sequence of functions in L 2 (Ω), which two-scale converges to a limit u 0 (x, y)

∈ L 2 (Ω, L 2 # (Z)). Then u ε (x) converges also to u(x) = Z u 0 (x, y)dy in L 2 (Ω) weakly. Furthermore, we have lim ε→0 u ε L 2 (Ω) ≥ u 0 L 2 (Ω×Y ) ≥ u L 2 (Ω) . (108)
Remark 4.5. : -For any smooth function u(x, y), being Z-periodic in y, the associated sequence u ε (x) = u(x, x ε ) two-scale converges to u(x, y). -Any sequence u ε that converges strongly in L 2 (Ω) to a limit u(x), two-scale converges to the same limit u(x).

-If u ε admits an asymptotic expansion of the type u ε (x) = u 0 (x, x/ε)+εu 1 (x, x/ε)+ ε 2 u 2 (x, x/ε) + ... , where the functions u i (x, y) are smooth and Z-periodic in y, twoscale convergence allows to identify the first term of the expansion u 0 (x, y) with the two-scale limit of u ε and the two-scale limit of (Frénod,Raviart and Sonnendrucker [11]). Proposition 4.6. Let u ε (x) be bounded in L 2 (Ω). Up to a subsequence, u ε (x) twoscale converges to u 0 (x, y) ∈ L 2 (Ω, L 2 # (Z)) such that:

u ε (x)-u 0 (x, x ε ) ε with u 1 (x, y) see
u 0 (x, y) = u(x) + u 0 (x, y), (109)
where

u 0 (x, y) ∈ L 2 (Ω, L 2 # (Z)) satisfies Z u 0 (x, y) dy = 0, (110)
and u(x) = Z u 0 (x, y) dy is a weak limit in L 2 (Ω).

Proof. Due to the a priori estimates (99), u ε (x) is bounded in L 2 (Ω), then by application of Theorem 4.3, u ε we get the first part of the proposition. Furthermore by defining u 0 as

u 0 (x, y) = u 0 (x, y) - Z u 0 (x, y)dy, (111)
we obtain the decomposition 109 of u 0 .

Defining

∇ x = ( ∂ ∂x ; ∂ ∂y ; ∂ ∂z ), ∇ y = ( ∂ ∂ξ ; ∂ ∂ν ; ∂ ∂ζ ),
we have Proposition 4.7. Let u ε (x) be bounded in H(curl, Ω). Then, up to a subsequence, there exists a function

u 1 ∈ L 2 (Ω, H # (curl, Z)) such that ∇×u ε (x) ∇ x × u 0 (x, y) + ∇ y × u 1 (x, y), (112)
where u 0 is given by Proposition 4.6.

Proof. From Theorem 4.3, since u ε and ∇×u ε are bounded in L 2 (Ω) then up to a subsequence they two-scale converge to

u 0 (x, y) ∈ L 2 (Ω, L 2 # (Z)) and η 0 (x, y) ∈ L 2 (Ω, L 2 # (Z)). So we have for all V (x, y) ∈ C 0 0 (Ω; C 0 # (Z)): lim ε→0 Ω u ε (x) • V (x, x/ε) dx = Ω Z u 0 (x, y) • V (x, y)dxdy, (113)
lim ε→0 Ω ∇×u ε (x) • V (x, x/ε) dx = Ω Z η 0 (x, y) • V (x, y)dxdy. (114)
Next, by integration by parts, we have:

Ω ∇×u ε (x) • V (x, x/ε) dx = Ω u ε (x) • (∇ x × V (x, x/ε) + 1 ε ∇ y × V (x, x/ε)) dx. (115)
If we choose a test function V ∈ C 0 0 (Ω, C 0 # (Z)) such that ∇ y × V = 0, passing to the limit in the left-hand side (114) we get

Ω ∇ x × u ε (x) • V (x, x/ε) dx → Ω Z u 0 (x, y) • ∇ x × (V (x, y)) dxdy = Ω Z ∇ x × u 0 (x, y) • V (x, y) dxdy. (116)
This means that with the difference between (114) and (116):

Ω Z [η 0 (x, y) -∇ x × u 0 (x, y)] • V (x, y) dxdy = 0, (117)
for all functions V ∈ C 1 0 (Ω) with ∇ y × V = 0. It follows that function η 0 (x, y) -∇ x × u 0 (x, y) is orthogonal to functions with zero rotational in L 2 (Ω, H # (curl, Z). This implies that there exists a function

u 1 ∈ L 2 (Ω, H # (curl, Z)) such that ∇ y × u 1 (x, y) = η 0 (x, y) -∇ x u 0 (x, y). (118)
Thus

∇×u ε (x) = ∇ x × u 0 (x, y) + ∇ y × u 1 (x, y). (119)
Proposition 4.8. Let u ε (x) be bounded in H(curl, Ω).

∇×u ε (x) ∇ x × u(x) weakly in L 2 (Ω). ( 120
)
where u(x) is given by Proposition 4.6.

Proof. Choosing a function test V (x) ∈ C 1 (Ω), an integration by parts yields

Ω ∇×u ε (x) • V (x) dx = Ω u ε (x) • ∇×V (x) dx. (121)
Taking the two-scale limit we have

lim ε→0 Ω u ε (x) • ∇×V (x) dx = Ω Z u 0 (x, y) • ∇×V (x) dxdy = Ω ∇×u(x) • V (x) dx. (122)
Proposition 4.9. Let any two-scale limit u 0 (x, y), given by Proposition (4.6), can be decomposed as

u 0 (x, y) = u(x) + ∇ y Φ(x, y). (123)
where

Φ ∈ L 2 (Ω, H 1 # (Z)
) is a scalar-valued function and where u ∈ L 2 (Ω). Proof of (123), integrating by parts, for any

V (x, y) ∈ C 1 0 (Ω, C 1 # (Z)), we have ε Ω ∇×u ε (x) • V (x, x ε ) dx = ε Ω u ε (x) • ∇×V (x, x ε ) dx = Ω u ε (x){ε∇ x × V (x, x ε ) + ∇ y × V (x, x ε )} dx. (124)
Taking the two-scale limit as ε → 0 we obtain

0 = Ω Z u 0 (x, y) • ∇ y × V (x, y) dxdy, (125)
which implies that ∇ y × u 0 (x, y) = 0. To end the proof of the proposition we use the following result:

Proposition 4.10. If u 0 ∈ L 2 (Ω) satisfies ∇ y × u 0 (x, y) = 0, (126)
then there exists u ∈ L 2 (Ω) and Φ ∈ L 2 (Ω, H 1 # (Z)) such that u 0 (x, y) = u(x) + ∇ y Φ(x, y).

Applying this proposition we obtain equality (123) ending the proof of Proposition (4.8).

These results are important properties of the two-scales convergence. We note that the usual concepts of convergence do not preserve information concerning the micro-scale of the function. However, the two-scale convergence preserves information on the micro-scale.

Homogenized problem

We will explore in this section the behavior of electromagnetic field E ε using the twoscale convergence to determine the homogenized problem. We place in the context of the case 6 with δ > L and ω = 10 6 rad.s -1 , then we have η = 5 and Σ ε a = ε, Σ ε r = ε 4 , Σ ε c = 1 which gives the following equation:

∇×∇×E ε -ω 2 ε 5 k( )E ε + iω[(1 ε C ( x ε ) + ε 4 1 ε R ( x ε ))1 {y<0} + ε1 {y>0} ]E ε = 0, ( 127
)
where for a given set A, 1 A stands for the characteristic function of A and where

1 ε A (x) = 1 A ( x ε )
, hence 1 ε C and 1 ε R are the characteristic functions of the sets filled by carbon fibers and by resin. And where k( ) = ( c 1 ε C (x) + r 1 ε R (x))1 {y<0} + 1 {y>0} . On this purpose, we have the following Theorem: Theorem 4.11. Under assumptions of Theorem 3.4, sequence E ε is solution of (88) or (87) or ((65), (68), (69), (71)). E ε two-scale converges to E(x) + ∇ y Φ 0 (x, y), where E ∈ L 2 (Ω) and Φ 0 ∈ L 2 (Ω, H 1 # (Z)), the unique solution of the homogenized problem:

                   ∇ x × ∇ x × E(x) + iωθE(x) = 0 in Ω, -∇ y • {∇ x × ∇ x × ∇ y Φ 0 (x, y)} -iω∇ y • {1 C (y)∇ y Φ 0 (x, y)} = iωE(x)∇ y • {1 C (y)} in Ω × Z, ∇ x × E(x) × e 2 = -iωH d × e 2 on Γ d , ∇ x × E(x) × e 2 = 0 on Γ L , (128)
where θ = Z 1 C (y) dy is the volume fraction of carbon fiber.

Proof.

Step 1: Two-scale convergence. Due to the estimate (99), E ε is bounded in L 2 (Ω). Hence, up to a subsequence, E ε two-scale converges to E 0 (x, y) belonging to L 2 (Ω, L 2 # (Z)). That means for any V (x, y) ∈ C 1 0 (Ω, C 1 # (Z)), we have:

lim ε→0 Ω E ε (x) • V (x, x ε ) dx = Ω Z E 0 (x, y) • V (x, y) dydx. ( 129
)
Step 2: Deduction of the constraint equation. We multiply Equation (127) by oscillating test function

V ε (x) = V (x, x ε ) where V (x, y) ∈ C 1 0 (Ω, C 1 # (Z)): Ω ∇×E ε (x) • (∇ x × V ε (x, x ε )) + 1 ε ∇ y × V ε (x, x ε + [-ω 2 ε 5 k( ) + iω 1 ε C ( x ε ) + ε 4 1 ε R ( x ε ) 1 {y<0} + ε1 {y>0} ]E ε • V ε (x, x ε ) dx = -iω Γ d H d × e 2 • (e 2 × V (x, 1, z, ξ, 1 ε , ζ)) × e 2 dσ. (130)
Integrating by parts, we get:

Ω E ε (x) • (∇ x × ∇ x × V ε (x, x ε ) + 1 ε ∇ y × ∇ x × V ε (x, x ε ) + 1 ε ∇ x × ∇ y × V ε (x, x ε ) + 1 ε 2 ∇ y × ∇ y × V ε (x, x ε )) + [-ω 2 ε 5 k( ) + iω 1 ε C ( x ε ) + ε 4 1 ε R ( x ε ) 1 {y<0} + ε1 {y>0} ]E ε (x) • V ε (x, x ε ) dx = -iω Γ d H d × e 2 • (e 2 × V (x, 1, z, ξ, 1 ε , ζ)) × e 2 dσ. (131)
Now we multiply (131) by ε 2 and we pass to the two-scale limit, applying Theorem 4.3, using (129) we obtain:

Ω Z E 0 (x, y) ∇ y × ∇ y × V (x, y) dydx = 0. ( 132
)
We deduce the constraint equation for the profile E 0 :

∇ y × ∇ y × E 0 (x, y) = 0. ( 133
)
Step 3. Looking for the solutions to the constraint equation. Multiplying Equation (133) by the conjugate of E 0 and integrating by parts over Z leads to

Z ∇ y × ∇ y × E 0 (x, y)E 0 (x, y) dy = Z |∇ y × E 0 (x, y)| 2 dy = 0. ( 134
)
We deduce that equation ( 134) is equivalent to

∇ y × E 0 (x, y) = 0, (135)
Moreover since a solution of ( 135) is also solution of ( 133), ( 133) and ( 135) are equivalent.

Hence, from Proposition 4.10 we conclude that E 0 (x, y) can be decomposed as

E 0 (x, y) = E(x) + ∇ y Φ 0 (x, y). ( 136
)
Step 4. Equations for E(x) and Φ 0 (x, y). Now, we seek what E satisfies. For this, we build oscillating test functions satisfying constraint (136) and use them in weak formulation (131). We define test function V (x, y) = α(x) + ∇ y β(x, y),

V (x, y) ∈ C 1 0 (Ω) × C 1 0 (Ω, C 1 # (Z)
) and we inject in (131) test function V ε = V (x, x ε ), which gives:

Ω E ε (x) • ∇ x × ∇ x × V (x, x ε ) + 2 ε ∇ x × ∇ y × V (x, x ε ) + 1 ε 2 ∇ y × ∇ y × V (x, x ε ) + [-ω 2 ε 5 k( ) + iω 1 ε C ( x ε ) + ε 4 1 ε R ( x ε ) 1 {y<0} + ε1 {y>0} ]E ε (x) • V (x, x ε ) dx = -iω Γ d H d × e 2 • (e 2 × V ‡ (x, 1, z, ξ, ζ)) × e 2 dσ, (137)
with V (x, 1, z, ξ, ν, ζ) = V ‡ (x, 1, z, ξ, ζ) the restriction on V which does not depend on ν. The term containing the constraint, the third one, disappears. Passing to the limit ε → 0 and replacing the expression of V by the term α(x) + ∇ y β(x, y), we have

∇ x × ∇ y × V (x, y) = ∇ x × ∇ y × [α(x) + ∇ y β(x, y)] = ∇ x × ∇ y × (α(x)) + ∇ x × ∇ y × (∇ y β(x, y)) = ∇ x × ∇ y × (∇ y β(x, y)). (138)
Since ∇ y × (∇ y ) = 0, the term 2 ε ∇ x × ∇ y × V (x, y) vanishes. Therefore, (137) becomes:

Ω Z E 0 (x, y) • ∇ x × ∇ x × (α(x) + ∇ y β(x, y) + iω 1 C (y)E 0 (x, y) • (α(x) + ∇ y β(x, y) dydx = -iω Γ d H d × e 2 • (e 2 × (α(x, 1, z) + ∇ y β(x, 1, z, ξ, ζ))) × e 2 dσ. (139)
Now in (139) we replace Expression E 0 giving by (136). We obtain

Ω Z (E(x) + ∇ y Φ 0 (x, y)) • ∇ x × ∇ x × (α(x) + ∇ y β(x, y)) + iω(1 C (y)(E(x) + ∇ y Φ 0 (x, y)) • (α(x) + ∇ y β(x, y)) dydx = -iω Γ d H d × e 2 • (e 2 × (α(x, 1, z) + ∇ y β(x, 1, z, ξ, ζ))) × e 2 dσ.
(140) Now using the fact that Ω Z ∇ y Φ 0 (x, y) • ∇ x × ∇ x × α(x) dydx = 0 we have:

Ω Z E(x) • ∇ x × ∇ x × α(x) + E(x) • ∇ x × ∇ x × ∇ y β(x, y) + ∇ y Φ 0 (x, y) • ∇ x × ∇ x × ∇ y β(x, y) + iω 1 C (y)(E(x)α(x) + ∇ y Φ 0 (x, y)α(x) + E(x)∇ y β(x, y) + ∇ y Φ 0 (x, y)∇ y β(x, y)) dydx = -iω Γ d H d × e 2 • (e 2 × α(x, 1, z)) × e 2 dσ. (141)
Then taking β(x, y) = 0 in (141) and integrating by parts, we get

Ω ∇ x × ∇ x × E(x) • α(x) + iω Z 1 C (y) dy E(x)α(x) dx = -iω Γ d H d × e 2 • (e 2 × α(x, 1, z)) × e 2 dσ.
(142) which gives the following well posed problem for E(x)

         ∇ x × ∇ x × E(x) + iω( Z 1 C (y) dy)E(x) = 0 in Ω, ∇ x × E(x) × e 2 = iωH d × e 2 on Γ d , ∇ x × E(x) × e 2 = 0 on Γ L . (143)
Now taking α(x) = 0 in (141), we obtain

Ω Z ∇ x × ∇ x × ∇ y Φ 0 (x, y)∇ y β(x, y) + iω1 C (y)(E(x) • ∇ y β(x, y) + ∇ y Φ 0 (x, y)α(x)∇ y β(x, y)dydx = 0. ( 144
)
Integrating by parts

Ω Z -∇ y • {∇ x ×∇ x × ∇ y Φ 0 (x, y)}β(x, y) -iω∇ y • {1 C (y)∇ y Φ 0 (x, y)}β(x, y) dydx = iω Ω Z ∇ y • {1 C (y)}E(x)β(x, y) dydx.
(145) which gives the microscopic problem for Φ 0 (x, y)

-∇ y • {∇ x × ∇ x ×∇ y Φ 0 (x, y)} -iω∇ y • {1 C (y)∇ y Φ 0 (x, y)} = iωE(x)∇ y • {1 C (y)}. ( 146
)
This concludes the proof of Theorem 4.11.

Conclusion

We presented in this paper the homogenization of time harmonic Maxwell equation by the method of two-scale convergence. We started by studying the time harmonic Maxwell equations with coefficients depending of ε. We remind that λ is the wave length, δ is the skin length, L is thickness of the medium and e the size of the basic cell and then ε = e L is the small parameter. We find for low frequencies the macroscopic homogenized Maxwell equations depending on the volume fraction of the carbon fibers and we find also the microscopic equation.

Annexes

A Presentation of all cases of tables 1, 2, 3 and 4 -The case 1 corresponds to the air not ionized, a resin not doped and σ is the effective electric conductivity in the direction of the carbon fibers. We have for the effective electric conductivity σ = σ c ∼ 40000S.m -1 , the resin conductivity is about σ r ∼ 10 -10 S.m -1 and the conductivity in the air is about 10 -14 S.m -1 . So when we want to calculate the ratio in (41)-(43) depending on ε we get: σr σ ∼ ε 7 and σa σ ∼ ε 9 . -In case 2, the air is not ionized, the resin is doped and σ is the effective conductivity is in direction of carbon fibers. We have like the case 1 σ = σ c ∼ 40000S.m -1 . The resin conductivity is about σ r ∼ 10 -3 S.m -1 and the conductivity in the air is about 10 -14 S.m -1 . So σr σ ∼ ε 4 and σa σ ∼ ε 9 . -In case 3, the air is not ionized, the resin is not doped and σ is the effective conductivity is orthogonal to the fibers. σ = σ r ∼ 10 -10 S.m -1 . The carbon fiber conductivity is about σ c ∼ 10 4 S.m -1 and the conductivity in the air is about 10 -14 S.m -1 . σc σ ∼ 1 ε 7 and σa σ ∼ ε 2 . -Case 4 corresponds to the air non ionized, the resin doped and σ is the effective conductivity orthogonal to the fibers. The effective electric conductivity is σ = σ r ∼ 10 -3 S.m -1 . The carbon fiber conductivity is about σ c ∼ 40000S.m -1 and the conductivity in the air is about 10 -14 S.m -1 . σc σ ∼ 1 ε 4 and σa σ ∼ ε 6 . -In case 5, the air is ionized, the resin is not doped and σ is the effective conductivity is in the direction of the carbon fibers. This one is equal σ = σ c ∼ 40000S.m -1 , the resin conductivity is about σ r ∼ 10 -10 S.m -1 and the conductivity in the air is now about 4242S.m -1 . σr σ ∼ ε 7 and σa σ ∼ ε. -Case 6 corresponds to the air ionized, the resin doped and σ is the effective conductivity in direction of the carbon fibers. This one is equal σ = σ c ∼ 40000S.m -1 , the resin conductivity is about σ r ∼ 10 3 S.m -1 and the conductivity in the air is now about 4242S.m -1 . σr σ ∼ ε 4 and σa σ ∼ ε. -Case 7 corresponds to the air ionized, the resin not doped and σ is the effective conductivity orthogonal to the fibers. The effective conductivity is σ = σ r ∼ 10 -10 S.m -1 , the carbon fibers conductivity is about σ c ∼ 40000S.m -1 and the conductivity in the air is now about 4242S.m -1 . σc σ ∼ 1 ε 7 and σa σ ∼ 1 ε 6 . -Case 8 corresponds to the air ionized, the resin doped and σ is the effective conductivity orthogonal to the fibers. The effective conductivity is σ = σ r ∼ 10 -3 S.m -1 , the carbon fibers conductivity is about σ c ∼ 40000S.m -1 and the conductivity in the air is now about 4242S.m -1 . σc σ ∼ 1 ε 4 and σa σ ∼ 1 ε 2 .

B Structure of the equations depending of ε

For ω = 100rad.s -1 , we have Case 1

η = 9 and Σ ε a = ε 11 , Σ ε r = ε 9 , Σ ε c = ε 2 . ( 147
) Case 2 η = 9 and Σ ε a = ε 11 , Σ ε r = ε 6 , Σ ε c = ε 2 . ( 148
) Case 3 η = 9 and Σ ε a = ε 12 , Σ ε r = ε 10 , Σ ε c = ε 3 . ( 149
) Case 4 η = 9 and Σ ε a = ε 13 , Σ ε r = ε 7 , Σ ε c = ε 3 . ( 150
) Case 5 η = 9 and Σ ε a = ε 3 , Σ ε r = ε 9 , Σ ε c = ε 2 . ( 151
) Case 6 η = 9 and Σ ε a = ε 3 , Σ ε r = ε 6 , Σ ε c = ε 2 . ( 152

C The transmission Maxwell problem

Taking a test function V ∈ C 1 (Ω) with compact support in Ω ε c , in weak formulation (86) associated with the problem ((65), (68), (69)). Since

Ω ∇×E ε |Ω ε c • ∇×V dx = ∇×∇×E ε |Ω ε c , V Ω ε c , (179)
we deduce the third equation in (87). Similarly, taking V ∈ C 1 (Ω) with compact support respectively in Ω ε r and Ω ε a , we obtain the first and the second equation in (87). Now, since E |Ω ε a ∈ H(curl, Ω ε a ) and E |Ω ε r ∈ H(curl, Ω ε r ), let V ∈ C 1 (Ω ε a Ω ε r ) integrating by parts we get

Ω ε a Ω ε r E • ∇×V dx = Ω ε a E |Ω ε a • ∇×V dx + Ω ε r E |Ω ε r • ∇×V dx = Ω ε a ∇×E |Ω ε a • V dx + Ω ε r ∇×E |Ω ε r • V dx + Γra (E |Ω ε a × e 2 -E |Ω ε r × n |Ω ε r ) • V ds. (180)
Since on every point of Γ ra e 2 = -n |Ω ε r the assumed continuity require

E |Ω ε a × e 2 = E |Ω ε r × n |Ω ε r , (181)
we obtain the fourth relation in (87). With the same argument on Γ ε cr , we obtain the last relation in (87). This shows that (86) implies (87). And, if E ε is solution to (87) following that for any regular set Ω in Ω the Stokes's formula gives, for more details see p 57, 58 of P. Monk's book [START_REF] Monk | Finite Element Methods for Maxwell's Equations. Numerical Mathematics and scientific computation[END_REF]:

∀ E, V ∈ H(curl, Ω) Ω ∇×E • V -E • ∇×V dx = E × n Ω , V T ∂ Ω (182)
H(curl, Ω) has the same definition as H(curl, Ω) with Ω replaced by Ω and where V T = (n × V ) × n, and n Ω is the unit outward normal of ∂ Ω. For all V ∈ H(curl, Ω),

V |Ω ε r ∈ H(curl, Ω ε r ), V |Ω
Ω ε r ∇×E ε |Ω ε r • ∇×V dx = Ω ε r ∇×∇×E ε |Ω ε r • V dx + ∇×E ε |Ω ε r × n |Ω ε r , V T Γra + ∇×E ε |Ω ε c × n |Ω ε c , V T Γ ε cr . (183)
Doing the same for Ω ε c , we have

Ω ε c ∇×E ε |Ω ε c • ∇×V dx = Ω ε c ∇×∇×E ε |Ω ε c • V dx + ∇×E ε |Ω ε c × n ε |Ω ε c , V T Γ ε cr . ( 184
)
Finally for Ω ε a , we have

Ω ε a ∇×E ε |Ω ε a • ∇×V dx = Ω ε a ∇×∇×E ε |Ω ε a • V dx + ∇×E ε |Ω ε a × e 2 , V T Γ d -∇×E ε |Ω ε a × e 2 , V T Γ ε ra . (185)
Summing the relations above since in every point of Γ ra n |Ω ε r = -e 2 and in every point of Γ ε cr n |Ω ε c = -n |Ω ε r , it comes

Ω ∇×E ε • ∇×V dx = Ω ∇×∇×E ε • V dx+ < [∇×E ε × n], V T > Γra + [∇×E ε × n], V T Γcr -iω Γ d H d × n ε • V T dσ. (186)
According to (86) and the first, second and third equations in (87) we have

∇×E ε |Ω ε c × n |Ω ε c , V T Γ ε cr -∇×E ε |Ω ε r × n |Ω ε r , V T Γ ε cr + ∇×E ε |Ω ε r × n |Ω ε r , V T Γra + ∇×E ε |Ω ε a × e 2 , V T Γra = 0, (187)
for all V ∈ H(curl, Ω) which causes the last two equalities in (87) and concludes the first part of the proof. Reciprocally, integrating by parts (87) we have:

∀ V ∈ X ε (Ω), Ω ∇×E ε • ∇×V dx + Ω ε a (-ω 2 ε η + iωΣ ε a )E ε • V dx = -iω Γ d H d × n ε • V T dσ, (188)
and

∀ V ∈ X ε (Ω), Ω ∇×E ε • ∇×V dx + Ω ε r (-ω 2 ε η r + i ωΣ ε r )E ε • V dx = 0, (189)
and

∀ V ∈ X ε (Ω), Ω ∇×E ε • ∇×V dx + Ω ε c (-ω 2 ε η c + i ωΣ ε c )E ε • V dx = 0. ( 190
)
By adding these three integrals, we get the variational formulation (86) associated with the problem ((65), (68), (69)).

Taking the divergence of the first three equations of (87) we get (71).

Figure 1 :

 1 Figure 1: The global domain

Figure 2 :

 2 Figure 2: Left: The global microstructure in 2D. Right: Z-cell of the periodic structure.

  Lz) and where ωµ 0 L E H d being of order 1, with the characteristic electric field E = 20 kV/m. From the physical spatial coordinates ( x, y, z) ∈ Ω we define y = (ξ, ν, ζ) with ξ = x e , ν = y e , ζ = z e or equivalently ξ = x ε , ν = y ε , ζ = z ε . And we now introduce Y , the basic cell. It is built from: Z = [-1 2 , 1 2 ] × [-1, 0] × R and the set C = D × R with the disc D defined by:

) Case 7 η 8 η 1 1 η 2 η 3 η 5 η 6 ηFor ω = 10 10 rad.s - 1 1 η

 7811235611 = 9 and Σ ε a = ε 5 , Σ ε r = ε 10 , Σ ε c = ε 3 . (153) Case = 9 and Σ ε a= ε 5 , Σ ε r = ε 7 , Σ ε c = ε 3 . (154)For ω = 10 6 rad.s -Case = 5 and Σ ε a = ε 9 , Σ ε r = ε 7 , Σ ε c = 5 and Σ ε a = ε 9 , Σ ε r = ε 4 , Σ ε c = 1. (156) Case = 5 and Σ ε a = ε 10 , Σ ε r = ε 8 , Σ ε c = ε. = 5 and Σ ε a = ε, Σ ε r = ε 7 , Σ ε c = 5 and Σ ε a = ε, Σ ε r = ε 4 , Σ ε c Case = 1 and Σ ε a = ε 7 , Σ ε r = ε 5 , Σ ε c = 1 ε 2 . (163) Case η = 1 and Σ ε a = ε 7 , Σ ε r = ε 2 , Σ ε c and Σ ε a = 1 ε , Σ ε r = ε 2 , Σ ε c = 1 ε 2 . (168) Case η = 1 and Σ ε a = 1 ε , Σ ε r = ε 6 , Σ ε c = 1 ε . (169) Case η = 1 and Σ ε a = ε, Σ ε r = ε 3 , Σ ε c = 1 ε . (170)For ω = 10 12 rad.s -1Caseη = 0 and Σ ε a = ε 6 , Σ ε r = ε 4 , Σ ε c = 1 ε 3 .(171)Caseη = 0 and Σ ε a = ε 6 , Σ ε r = ε, Σ ε c = 1 ε 3 . (172) Case η = 0 and Σ ε a = ε 5 , Σ ε r = ε 3 , Σ ε c and Σ ε a = 1 ε 2 , Σ ε r = ε 3 , Σ ε c = 1 ε 4 .(177)Caseη = 0 and Σ ε a = 1 ε , Σ ε r = ε, Σ ε c = 1 ε 3 .(178)

  ε a ∈ H(curl, Ω ε a ) and V |Ω ε c ∈ H(curl, Ω ε c). Hence, fixing any E ∈ H(curl, Ω) according to the second equation in (87), we have ∇×E ε|Ω ε r ∈ H(curl, Ω ε r ) then applying (182) in Ω ε r with E = ∇×E ε |Ω εr and V we get

Table 2 :

 2

	case 1 case 2	case 3	case 4	case 5	case 6	case 7	case 8
	L(m)

for ω = 10 6 rad.s -1 .

(curl,Γ d ) ≤ C γ T u H(curl,Ω) , ∀u ∈ H(curl, Ω)