Matthieu Quignard
email: matthieu.quignard@ens-lyon.fr

Serge Heiden

Frédéric Landragin
email: frederic.landragin@ens.fr

Matthieu Decorde
email: matthieu.decorde@ens-lyon.fr

Textometric Exploitation of Coreference-annotated Corpora with TXM: Methodological Choices and First Outcomes

Keywords: coreference chain, corpus annotation, annotation tool, visualisation tool, exploration tool, statistical analysis of textual data

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Introduction

The manual annotation of a textual corpus with referring expressions [START_REF] Charolles | La référence et les expressions référentielles en français[END_REF] and coreference chains [START_REF] Schnedecker | Nom propre et chaîne de référence[END_REF], Landragin & Schnedecker, 2014) requires adapted tools. A coreference chain can cover the whole text; it is therefore a linguistic object for which the existing means of visualization and exploration are few and often perfectible. The MMAX2 tool [START_REF] Müller | Multi-level annotation of linguistic data with MMAX2[END_REF] allows for visualizing the links between referring expressions using arrows which link markables. The GLOZZ tool (Mathet & Wildlöcher, 2009) offers several means of visualization: with arrows like MMAX2, or with a specific marking in the margin or the middle of the text. The ANALEC tool [START_REF] Landragin | ANALEC: a New Tool for the Dynamic Annotation of Textual Data[END_REF] and its specific extension for coreference chains [START_REF] Landragin | Conception d'un outil de visualisation et d'exploration de chaînes de coréférences[END_REF] proposes a graphic metaphor based on the succession of coloured dots. This allows the analyst to configure visual parameters, for instance the colour which can be linked to any of the annotated properties. This type of visualization makes it possible to see at a glance the structural differences between the different reference chains of a text. That must be useful to the analyst, in addition to manual explorations and finer linguistic analyses.

Linguistic objects and methodology

In the continuity of previous works [START_REF] Heiden | The TXM Platform: Building Open-Source Textual Analysis Software Compatible with the TEI Encoding Scheme[END_REF][START_REF] Landragin | Conception d'un outil de visualisation et d'exploration de chaînes de coréférences[END_REF], we present here a set of measures -some of which can lead to specific visualisations -with the objective to enrich the possibilities of exploration and exploitation of annotated data. We focus in particular on annotations which concern discursive phenomena like coreference, i.e., annotations which are necessarily described within two levels: 1. markable, group of contiguous words to which is assigned some labels, using for instance a feature structure; 2. set of markables, or links between markables, as is it the case for any chain of annotations: anaphoric chains, textual organizers chains, textual structure elements chains, etc. A feature structure can also be assigned at level 2, i.e., to the set or to the links. As a first visualization mode, we reuse the very classic concordancer to display the elements which constitute a coreference chain. The use of such a visualization tool, which is well established in the community of corpus exploration [START_REF] Poudat | Explorer un corpus textuel : méthodes, pratiques, outils. Champs Linguistiques[END_REF], seemed natural for visualizing chains of annotations. The last version of TXM [START_REF] Heiden | The TXM Platform: Building Open-Source Textual Analysis Software Compatible with the TEI Encoding Scheme[END_REF] thus includes a concordancer which makes it possible to display in a column all the elements (e.g. referring expressions) of a chain (e.g. coreference chain), with left and right contexts for each elements. Compared to MMAX2 [START_REF] Müller | Multi-level annotation of linguistic data with MMAX2[END_REF] and GLOZZ (Mathet & Wildlöcher, 2009) visualisation choices, i.e. arrows linking marquables which are displayed directly on the text, this concordancer has the advantage of regrouping all the relevant information in a small graphic space. Fig. 1 shows the list of all referring expression to the character 'Caillette'. Sorted in the textual order, the concordancer shows the alternation of the use of proper nouns, pronouns, possessives, etc. This concordancer may also be sorted along a given property of the marquable, e.g. its POS label. This representation may then be exploited to see whether the POS annotation is consistent or not. A second mode of visualization, also very traditional, is the histogram (bar plot). The user can select one or several properties -the determination of the referring expressions, for instance, or the type of referent -and launch calculations on their occurrences: crosscounts, correlation computation and so on. TXM now includes a histogram generator, which allows for example to display the distribution of coreference chains throughout the text, as well as the distribution of chains according to the number of referring expressions they include. These calculations and their associated visualizations provide TXM with integrated functionalities which required in other state-of-the-art tools the development of scripts, in order to export the relevant data and exploit them in an external tool like a spreadsheet.

A concordancer adapted to annotations chains

Histograms for visualising distributions of annotations chains

Figure 2 compares the distribution of grammatical categories of referring expressions in three texts. Although all texts are all encyclopedical ones, the Discourse from Bossuet shows a particular profile, with a high number of proper nouns (GN.NAM).

Progression charts for annotations chains

A third (new) mode of visualization consists to graphically show the progress of each chain throughout the text. The principle is simple, but the possibilities of exploration and exploitation of the generated graph are numerous. In a two-dimensional chart the abscissa of which represents the linearity of the text, chains are displayed point by point (cf. Fig. 3): each occurrence of a referring expression increases by one notch the ordinate of the corresponding point. The resulting broken lines are all ascending but can considerably vary in their areas of progression and flat areas.

When they are visualized simultaneously, it is possible to detect the parts of the text where several referents are competitors, or on the contrary those where several referents appear alternately. Zooming (in and out) as well as focussing features allows for visualizing the characteristics of each point, thus enriching the exploration possibilities of these progression chart and the underlying coreference chains.

Discussion

The common points of these new visualization modes is not only to propose visual representations which are easy to understand (and possibly interactive, when it is possible to modify on the fly one of the properties), to allow the visualization of these representations directly in TXM, with no need to export annotated data and to use external tools, but also to facilitate the detection by the analyst of intruders, outliers and deviant examples. For instance potential annotation errors: it can be the case for a referring expression which has nothing to do in the currently visualised chain. It may be a peak or a suspect flat in one of the generated histograms. It may be a zone with a very high slope (or a very long flat) in a progression diagram. In all three cases, the analyst can directly access the suspicious annotation, in order to verify it and of course to modify it. The integration of the measurements and their visualizations in TXM allows this immediate return to the corpus annotation phase. This is particularly effective when the corpus is being annotated manually.

Conclusion and future works

One can say that it is by annotating that we can see the mistakes we make, but we still need appropriate tools to detect these errors. With the new possibilities of interaction that we propose here, we hope that we are taking a significant step in this direction. The first tests which we have carried out demonstrated the relevance of our approach.

Fig 1 :

 1 Fig 1: Concordancer with the elements of a coreference chain, dedicated to a character named "Caillette".

Fig 2 :

 2 Fig 2: Comparative barplots of grammatical categories usage by reference units in three texts: Bossuet, "Discours sur l'histoire universelle" (1681), Diderot, "Essais sur la peinture" (1759-1766), Montesquieu, "Esprit des lois" (1755).

Fig 3 :

 3 Fig 3: Progression graph of the main coreference chains at the beginning of "Essais sur la peinture" from Denis Diderot. The dots highlighted with symbols correspond to referring expressions with low accessibility.


published or not. The documents may come

Introduction

This document is inspirated of the work of David Clausen (University of Puget Sound, USA) with simplification in the proofs. The reader has to have basic knowledge on group theory including:

• Abelian and cyclic groups.

• Lagrange theorem.

• Normal subgroup and quotient group.

• Direct and semi-direct products.

• Cauchy theorem.

• Operation of groups on a set.

Preliminaries

Definition 1 (Center of a group). Let G be a group, the center of G is defined as: Z(G) = {h : ∀g, hg = gh} Definition 2 (Commutator subgroup). Let G be a group, the commutator of G is the smallest subgroup containing the commutators [g 1 , g 2 ] = g 1 g 2 g -1 1 g -1 2

Proposition 1. If the quotient group G/Z(G) is cyclic then G is Abelian and consequently G/Z(G) ∼ = {e}.

Proof. Suppose G/Z(G) = {Z(G), gZ(G), g 2 Z(G), . . . , g n-1 Z(G)}. Let g i x i and g j x j , with x i , x j ∈ Z(G) be two elements of G. Then:

g i x i g j x j = g i g j x i x j = x j g i+j x i = x j g j g i x i = g j x j g i x i

Proposition 2. If |G| = p n , where p is a prime number, then |Z(G)| = p k for k ≥ 1.

Proof. G operates on itself by conjugaison g.h = ghg -1 . For any h, the stabilizer of h is the centralizer S h = {g : gh = hg} and its orbital is the set O(h) = {ghg -1 : g ∈ G}. We recall that O(h) is in bijection with the quotient set (G/S h ) l = {gS h : g ∈ G} and consequently has the same cardinal. It is easy to show that the cardinal |O(h)| = 1 if and only if h ∈ Z(G), consequently:

|G| = |Z(G)| + h / ∈Z(G) |G| |S h | ,
where the sum is indexed for the h which are not in the same orbital.

The term h / ∈Z(G)

|G| |S h | is a multiple of p, consequently |Z(G)| also. Using the Lagrange's theorem, we deduce the result.

Theorem 1 (Burnside's theorem). If the order of a group G is equal to p 2 , with p a prime number; then G is abelian.

Proof. Easy: use the two previous propositions.

Proposition 3. The commutator subgroup, denoted D(G), is the smallest normal subgroup such that G/D(G) is Abelian.

Proof. First step: D(G) is a normal subgroup:

h[g 1 , g 2 ]h -1 = hg 1 g 2 g -1 1 g -1 2 h -1 = hg 1 g 2 g -1 1 (hg 2 ) -1 = hg 1 h -1 (hg 2 )g -1 1 (hg 2 ) -1 = hg 1 h -1 g -1 1 g 1 (hg 2 )g -1 1 (hg 2 ) -1 = [h, g 1 ][g 1 , hg 2 ] ∈ D(G)
Second step: G/D(G) is Abelian: It is trivial by construction of D(G). Indeed, g -1 1 g -1 2 g 1 g 2 ∈ D(G), consequently, g 1 g 2 D(G) = g 2 g 1 D(G). Third step: If H is a normal subgroup such that G/H is Abelian, then H contains the commutator. The conclusion is trivial.

Theorem 2 (Correspondance theorem). Let H be a normal subgroup of a group G, then there is a bijection between the set of the subgroups S of G containing H and the set of the subgroups S/H of G/H. Proposition 4. Let H and K be two subgroups of a group G. We define the set HK = {hk : h ∈ H, k ∈ K}, which is not necessarily a group. Then:

|HK| = |H| × |K| |H ∩ K|
3 Groups of order 8

We now classify all groups of order 8. The neutral element will be denoted e.

Abelian groups of order 8

Name Character presentation

Z 8 a 8 = e Z 4 × Z 2 a 4 = b 2 = e and D(G) = {e} (Z 2 ) 3 a 2 = b 2 = c 2 = e and D(G) = {e}.

No Abelian groups of order 8

Proposition 5. We have necessarily:

Z(G) ∼ = Z 2 and: G/Z(G) ∼ = Z 2 × Z 2
Proof. As G/Z(G) cannot be cyclic and |Z(G)| is a non trivial power of 2 different of 8, we deduce the result.

From the previous proposition, as G is not Abelian and G/Z(G) is Abelian; we deduce that D(G) = Z(G). Moreover, we remark that: 0,0),(1,0),(0,1),(1,1)} , has three subgroups of order 2. Consequently, by the correspondance theorem, G has three subgroups of order 4, denoted G 1 , G 2 , G 3 , containing the center.

Z 2 × Z 2 = {(
Proposition 6. If g 1 ∈ G 1 \Z(G) and g 2 ∈ G 2 \Z(G), then g 1 g 2 ∈ G 3 \Z(G).
Proof. Suppose, by the correspondance theorem, that G 1 (resp. G 2 ) corresponds to the subgroup {(0, 0), (1, 0)} (resp. {(0, 0), (0, 1)}). Then, g 1 corresponds to (1, 0) and g 2 to (0, 1). As (1, 0) + (0, 1) = (1, 1) which corresponds to g 1 g 2 ; we deduce the conclusion.

Proposition 7. If g 1 ∈ G 1 \Z(G) and g 2 ∈ G 2 \Z(G), then g 1 g 2 = g 2 g 1 .
Proof. Suppose the converse. Then g 1 commutes with the elements of Z(G), of g 1 Z(G) and of g 2 Z(G). Consequently, the centralizer of g 1 has at least 6 elements. And, as the centralizer is a subgroup of G then it has exactly 8 elements. We deduce that the centralizer of g 1 is G, so g 1 ∈ Z(G); which is contradictory.

Proposition 8. Let G i , G j be two distinct subgroups amongst G 1 , G 2 , G 3 , then: G = G i G j G i ∩ G j = Z(G)
Proof. Use the formula:

|G i G j | = |G i | × |G j | |G i ∩ G j | = 16 |G i ∩ G j | ,
and the fact that G i G j cannot have more elements than G.

As the sub-groups G i has order equal to 4, they are either isomorphic to Z 4 or Z 2 × Z 2 .

In the following, let Z(G) = {e, α} with α 2 = e.

G

1 , G 2 , G 3 ∼ = Z 2 × Z 2
In this case, there are three elements a, b, c in G\Z(G) such that:

• G 1 = {e, a, α, aα}.

• G 2 = {e, b, α, bα}.

• G 3 = {e, c, α, cα}.

From the Proposition 6., ba ∈ G 3 and consequently has order 2; so baba = e. But, as a and b have order 2, then baab = e. Consequently, ba = ab, which contradicts Proposition 7.

3.2.2 G 1 ∼ = Z 4 , G 2 , G 3 ∼ = Z 2 × Z 2
There exists three elements a, b, c in G\Z(G) such that:

• G 1 = {e, a, a 2 = α, a 3 }.

• G 2 = {e, b, α, bα}.

• G 3 = {e, c, α, cα}.

3.2.3 G 1 , G 2 ∼ = Z 4 , G 3 ∼ = Z 2 × Z 2
There exists three elements a, b, c in G\Z(G) such that:

• G 1 = {e, a, a 2 = α, a 3 }. • G 2 = {e, b, b 2 = α, b 3 }. • G 3 = {e, c, α, cα}.
We have also baba = e and baa 3 b 3 = e. Consequently ba = a 3 b 3 = a 2 abb 2 = αabα = α 2 ab = ab because α ∈ Z(G) and has order 2. ba = ab leads to a contradiction.

G

1 , G 2 , G 3 ∼ = Z 4
There exists three elements a, b, c in G\Z(G) such that:

• G 1 = {e, a, a 2 = α, a 3 }. • G 2 = {e, b, b 2 = α, b 3 }. • G 3 = {e, c, c 2 = α, c 3 }.
We have ba(ba) 3 = e and baa 3 b 3 = e. Consequently (ba) 3 = a 3 b 3 = ab; so (ba) 2 ba = ab; so αba = ab; so ba = αab = a 3 b. It is the group of quaternions denoted H. It is not a semi-direct product and its character representation is a 4 = b 4 = e, a 2 = b 2 and ba = a 3 b. It is frequent that a and b are denoted i and j, ab is denoted k and a 2 is denoted -1. With these notation

H = {1, -1, i, -i, j, -j, k, -k} with i 2 = j 2 = -1, (-1) 2 = 1, ji = -ij, ij = k.
We have classified all groups of order 8. The following table gives the non-Abelian groups of order 8.

Name

Character presentation

D 4 ∼ = Z 4 Z 2 a 4 = b 2 = e,ba = a 3 b H a 4 = b 4 = e, a 2 = b 2 , ba = a 3 b
4 Groups of order 16

4.1 Abelian groups of order 16

Name Character presentation

Z 16 a 16 = e Z 8 × Z 2 a 8 = b 2 = e, D(G) = {e} Z 4 × Z 4 a 4 = b 4 = e, D(G) = {e} Z 4 × (Z 2 ) 2 a 4 = b 2 = c 2 = e, D(G) = {e} (Z 2 ) 4 a 2 = b 2 = c 2 = d 2 = e, D(G) = {e}

Non-Abelian groups of order 16

As G/Z(G) cannot be cyclic, we show easily that |Z(G)| ∈ {2, 4}.

First case: |Z(G)| = 4

In this case, as

G/Z(G) is not cyclic, G/Z(G) ∼ = Z 2 × Z 2 . As G/Z(G) is Abelian, then D(G) Z(G).
As it has been done for the groups of order 8, we show that G has three sub-groups G 1 , G 2 , G 3 of order 8 which contain the center.

Proposition 9. The subgroups G 1 , G 2 , G 3 are Abelian.

Proof. Indeed, as Z(G) < G i , we deduce easily that Z(G) < Z(G i ). Consequently, Z(G i ) has at least 4 elements. And as the center of a group of order 8 has either 2 elements or 8 elements, then |Z(G i )| = 8 and we deduce that G i is Abelian.

The following propositions can be proved in a similar way as for the groups of order 8.

Proposition 10. If g 1 ∈ G 1 \Z(G) and g 2 ∈ G 2 \Z(G), then g 1 g 2 ∈ G 3 \Z(G). Proposition 11. If g 1 ∈ G 1 \Z(G) and g 2 ∈ G 2 \Z(G), then g 1 g 2 = g 2 g 1 . Proposition 12. Let G i , G j be two distinct subgroups amongst G 1 , G 2 , G 3 , then: G = G i G j G i ∩ G j = Z(G) A) First sub-case: Z(G) ∼ = Z 2 × Z 2 Let Z(G) = {e, α, β, γ} with α 2 = β 2 = γ 2 = e.
The Abelian groups of order 8 which have Z 2 × Z 2 for subgroup are: 1,0),(2,0),(3,0),(0,1),(1,1),(2,1),(3,1)} with: -{(0, 0), (2, 0), (0, 1), (2, 1) 0,0),(0,0,1),(0,1,0),(0,1,1),(1,0,0),(1,0,1),(1,1,0),(1,1,1)} with: -{(0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1)

• Z 4 × Z 2 = {(0, 0), (
} ∼ = Z 2 × Z 2 . • (Z 2 ) 3 = {(0,
} ∼ = Z 2 × Z 2 . -{(0, 0, 0), (0, 0, 1), (1, 0, 0), (1, 0, 1)} ∼ = Z 2 × Z 2 . -{(0, 0, 0), (0, 0, 1), (1, 1, 0), (1, 1, 1)} ∼ = Z 2 × Z 2 . -{(0, 0, 0), (0, 1, 0), (1, 0, 0), (1, 1, 0)} ∼ = Z 2 × Z 2 . -{(0, 0, 0), (0, 1, 0), (1, 0, 1), (1, 1, 1)} ∼ = Z 2 × Z 2 . -{(0, 0, 0), (0, 1, 1), (1, 0, 0), (1, 1, 1)} ∼ = Z 2 × Z 2 . -{(0, 0, 0), (0, 1, 1), (1, 0, 1), (1, 1, 0)} ∼ = Z 2 × Z 2 . A1) G 1 , G 2 , G 3 ∼ = (Z 2 ) 3 :
There exists three elements a, b, c which are not in the center Z(G) such that:

• G 1 = {e, α, β, γ, a, aα, aβ, aγ}.

• G 2 = {e, α, β, γ, b, bα, bβ, bγ}.

• G 3 = {e, α, β, γ, c, cα, cβ, cγ}.
ba has order 2, so baba = e. But, a and b have order 2, so baab = e. Consequently ba = ab, which leads to a contradiction.

A2) G 1 ∼ = Z 4 × Z 2 , G 2 , G 3 ∼ = (Z 2 ) 3 :
There exists three elements a, b, c which are not in the center Z(G) such that:

• G 1 = {e, a, a 2 = α, a 3 , β, aβ, a 2 β = γ, a 3 β}. • G 2 = {e, α, β, γ, b, bα, bβ, bγ}. • G 3 = {e, α, β, γ, c, cα, cβ, cγ}.
We show easily that ba = a 3 b (ba has order 2, a has order 4 and b has order 2).

As G = G 1 G 2 , G is generated by a, b, β. We have | < a > | × | < b > | = 8 and < a > ∩ < b >= {e}. Consequently < a, b > ∼ = D 4 . Moreover, βa = aβ and βb = bβ, as β ∈ Z(G). | < a, b > | × | < β > | = 16 = |G| and < a, b > ∩ < β >= {e}.
Consequently, the group G is:

G ∼ = D 4 × Z 2
Its character presentation is x 4 = y 2 = z 2 , yx = x 3 y, zx = xz, zy = yz. (In order to avoid confusion, we will prefer character presentation with letter x, y, z, w, ... rather than a, b, c, ...)

A3) G 1 , G 2 ∼ = Z 4 × Z 2 , G 3 ∼ = (Z 2 ) 3 :
We consider two cases: First case: There exists three elements a, b, c which are not in the center Z(G) such that:

• G 1 = {e, a, a 2 = α, a 3 , β, aβ, a 2 β = γ, a 3 β}. • G 2 = {e, b, b 2 = α, b 3 , β, bβ, b 2 β = γ, b 3 β}. • G 3 = {e, α, β, γ, c, cα, cβ, cγ}.
Then we show by considering the order of elements that ba = a 3 b 3 . Consequently, ba = a 3 b 3 = a 2 abb 2 = αabα = α 2 ab = ab, which leads to a contradiction. Second case: There exists three elements a, b, c which are not in the center Z(G) such that:

• G 1 = {e, a, a 2 = α, a 3 , β, aβ, a 2 β = γ, a 3 β}. • G 2 = {e, b, b 2 = β, b 3 , α, bα, b 2 α = γ, b 3 α} = {e, b, b 2 = β, b 3 , γ, bγ, b 2 γ = α, b 3 γ = bα}. • G 3 = {e, α, β, γ, c, cα, cβ, cγ}.
We show that ba = a 3 b 3 = αabβ = γab. Without loss of generality, one can suppose that ab = c, so ba = γc. We have bγ = γb and bc

= bab = γcb, < γ, c > ∼ = Z 2 × Z 2 , < b > ∼ = Z 4 , | < γ, c > | × | < b > | = 4 × 4 = 16 = |G| and < γ, c > ∩ < b >= {e}. Consequently, G is the semi-direct product: G ∼ = (Z 2 × Z 2 ) ϕ Z 4 , with: ϕ : Z 4 → Aut (Z 2 × Z 2 ) 1 → (1, 0) → (1, 0) (0, 1) → (1, 1)
Its minimal character presentation is x 4 = y 4 = e, yx = x 3 y 3 , x 2 , y 2 ∈ Z(G).

(another possible presentation can be x 2 = y 2 = z 4 = e, yx = xy, zx = xyz, zy = yz but needs more generators).

A4) G 1 , G 2 , G 3 ∼ = Z 4 × Z 2 :
We consider two cases: First case: There exists three elements a, b, c which are not in the center Z(G) such that:

• G 1 = {e, a, a 2 = α, a 3 , β, aβ, a 2 β = γ, a 3 β}. • G 2 = {e, b, b 2 = β, b 3 , α, bα, b 2 α = γ, b 3 α}. • G 3 = {e, c, c 2 , c 3 , . . .} (the structure of G 3 will be precised latter).
As ba is not in the center but in G 3 then ba has order 4. In all case, (ba) 2 = c 2 . Indeed, in Z 4 × Z 2 , the square of all elements of order 4 is (2, 0). We show that (ba) 2 ba = a 3 b 3 = a 2 abb 2 = αabβ = γab. Consequently, (ba) 2 = γ. Without loss of generality, one can suppose that c 2 = (ba) 2 = β; which gives the entire structure of

G 3 . Consequently, ba = βγab = αab = a 2 ab = a 3 b. We have < a > ∼ = Z 4 (idem for < b >), | < a > | × | < b > | = 16 = |G| and < a > ∩ < b >= {e}. Consequently, G is the semi-direct product: G ∼ = Z 4 Z 4
Its character presentation is x 4 = y 4 = e, yx = x 3 y.

Second case:

From the first case, one can see that amongst the square a 2 , b 2 , c 2 , at least two are equal. The second case is when a 2 = b 2 = c 2 . Without loss of generality:

• G 1 = {e, a, a 2 = α, a 3 , β, aβ, a 2 β = γ, a 3 β}. • G 2 = {e, b, b 2 = α, b 3 , β, bβ, b 2 β = γ, b 3 β}. • G 3 = {e, c, c 2 = α, c 3 , β, cβ, c 2 β = γ, c 3 β}
We show that (ba

) 3 = a 3 b 3 = a 2 abb 2 = ab, so (ba) 2 ba = ab, so ba = a 3 b. As the character presentation of < a, b > is a 4 = b 4 = e, a 2 = b 2 , ba = a 3 b, we deduce that < a, b > ∼ = H. Moreover, G = G 1 G 2 =< a, b, β >. As β ∈ Z(G), | < β > | × | < a, b > | = 2 × 8 = 16 and < β > ∩ < a, b >= {e}, we deduce that G is the direct product: H × Z 2 Its character presentation is x 4 = y 4 = z 2 = e, x 2 = y 2 , yx = x 3 y, zx = xz, zy = yz.
We have classified all groups of order 16 such that Z(G)

∼ = Z 2 × Z 2 : Name Character presentation D 4 × Z 2 x 4 = y 2 = z 2 = e,yx = x 3 y,zx = xz,zy = yz (Z 2 × Z 2 ) Z 4 x 4 = y 4 = e, yx = x 3 y 3 , x 2 , y 2 ∈ Z(G) Z 4 Z 4 x 4 = y 4 = e, yx = x 3 y H × Z 2 x 4 = y 4 = z 2 = e, x 2 = y 2 , yx = x 3 y, zx = xz, zy = yz Groups of order 16 with Z(G) ∼ = Z 2 × Z 2 B) Second sub-case: Z(G) ∼ = Z 4 . Let Z(G) = {e, α, α 2 , α 3 }.
The Abelian groups of order 8 which have Z 4 for subgroups are:

• Z 8 = {0, 1, 2, 3, 4, 5, 6, 7} with: -{0, 2, 4, 6} ∼ = Z 4 • Z 4 × Z 2 = {(0, 0), (1, 0), ( 2
, 0), (3, 0), (0, 1), (1, 1), (2, 1), (3, 1)} with:

-{(0, 0), (1, 0), (2, 0), (3, 0)} ∼ = Z 4 -{(0, 0), (1, 1), (2, 0), (3, 1)} ∼ = Z 4

We remark that the square of any element of Z 8 is in the subgroup of order 4 and the square of any element of Z 4 × Z 2 is equal to (0, 0) or (2, 0) and is in its two subgroups of order 4. We deduce that the square of any element of G i is in the center. From this fact, we deduce the following proposition.

Proposition 13. For i = j, the commutator [g i , g j ] = g i g j g -1 i g -1 j between two elements respectively from G i \Z(G) and G j \Z(G) has an order equal to 2.

Proof. As g 2 i ∈ Z(G) and using D(G) < Z(G), we deduce g j g 2 i = g 2 i g j = g i g i g j = g i [g i , g j ]g j g i = [g i , g j ]g i g j g i = [g i , g j ] 2 g j g 2
i and we deduce the conclusion.

B1) G 1 , G 2 , G 3 ∼ = Z 4 × Z 2 : By considering how Z 4 × Z 2
is built from one of its subgroup of order 4 and an element of order 2 which is not in the considered subgroup, we deduce that there exists a, b, c not in the center and of order 2 such that:

• G 1 = {e, α, α 2 , α 3 , a, aα, aα 2 , aα 3 } with |aα| = 4, |aα 2 | = 2 and |aα 3 | = 4. • G 2 = {e, α, α 2 , α 3 , b, bα, bα 2 , bα 3 } with |bα| = 4, |bα 2 | = 2 and |bα 3 | = 4. • G 3 = {e, α, α 2 , α 3 , c, cα, cα 2 , cα 3 } with |cα| = 4, |cα 2 | = 2 and |cα 3 | = 4.
The order of ba is either 2 or 4. Suppose that |ba| = 2. Then baba = e and as a and b have order 2, then baab = e. We deduce ba = ab, which leads to a contradiction. Suppose that |ba| = 4. Then ba(ba) 3 = e and baab = e. Consequently, (ba) 3 = ab. From the structure of Z 4 × Z 2 , we show easily that (ba

) 2 = α 2 , so ba = α 2 ab. As < α, a >= G 1 ∼ = Z 4 × Z 2 , < b > ∼ = Z 2 , | < α, a > | × | < b > | = 8 × 2 = 16 and < α, a > ∩ < b >= {e}; G is the semi-direct product: G ∼ = (Z 4 × Z 2 ) ϕ Z 2 with: ϕ : Z 2 → Aut (Z 4 × Z 2 ) 1 → (1, 0) → (1, 0) (0, 1) → (2, 1) Its character presentation is x 4 = y 2 = z 2 = e, xy = yx, zx = xz, zy = x 2 yz. B2) G 1 ∼ = Z 8 , G 2 , G 3 ∼ = Z 4 × Z 2 :
There exists an element a not in the center and of order 8 and two elements b, c not in the center of order 2 such that:

• G 1 = {e, a, a 2 = α, a 3 , a 4 = α 2 , . . .}. • G 2 = {e, α, α 2 , α 3 , b, bα, bα 2 , bα 3 } with |bα| = 4, |bα 2 | = 2 and |bα 3 | = 4. • G 3 = {e, α, α 2 , α 3 , c, cα, cα 2 , cα 3 } with |cα| = 4, |cα 2 | = 2 and |cα 3 | = 4.
Let us consider that |ba| = 2, then baba = e and baa 7 b = e. Consequently, ba = a 7 b = a 6 ab = α 3 ab. But α 3 , which is the commutator [b, a] is of order 4; which is contradictory. Consider |ba| = 4, then we show that (ba) 2 ba = a 7 b = α 3 ab. As (ba) 2 = α 2 ; then ba = αab; which is also contradictory.

B3) G 1 , G 2 ∼ = Z 8 , G 3 ∼ = Z 4 × Z 2 :
There exists two elements a, b which are not in the center and of order 8 and an element c not in the center of order 2 such that:

• G 1 = {e, a, a 2 = α, a 3 , a 4 = α 2 , . . .}. • G 2 = {e, b, b 2 = α, b 3 , b 4 = α 2 , . . .}. • G 3 = {e, α, α 2 , α 3 , c, cα, cα 2 , cα 3 } with |cα| = 4, |cα 2 | = 2 and |cα 3 | = 4.
If |ba| = 2, then ba = a 7 b 7 = a 6 abb 6 = α 3 abα 3 = α 2 ab. We have also |ab| = 2. Without loss of generality, one can suppose ab = c. We deduce ca = aba = aα

2 ab = aα 2 c = a 5 c. < a > ∼ = Z 8 , < c > ∼ = Z 2 , | < a > | × | < c > | = 8 × 2 = 16 and < a > ∩ < c >= {e}. Consequently, G is the semi-direct product: G ∼ = Z 8 ϕ 1 Z 2 ,
where:

ϕ 1 : Z 2 → Aut (Z 8 ) 1 → ( 1 → 5 ) Its character presentation is x 8 = y 2 = e, yx = x 5 y. If |ba| = 4, then (ba) 3 = α 2 ab. As (ba) 2 = α 2 , then ba = ab, which is contradictory. B4) G 1 , G 2 , G 3 ∼ = Z 8 :
Then, there exists a, b not in the center of order 8 and such that ba is also of order 8. We deduce (ba) 7 = a 7 b 7 = α 3 abα 3 = α 2 ab. As (ba) 6 = α 2 then ba = ab; which leads to a contradiction.

We have classified all groups of order 16 such that Z(G) ∼ = Z 4 :

Name Character presentation (Z 4 × Z 2 ) Z 2 x 4 = y 2 = z 2 = e,xy = yx,zx = xz,zy = x 2 yz Z 8 ϕ 1 Z 2 x 8 = y 2 = e,yx = x 5 y Groups of order 16 with Z(G) ∼ = Z 4 4.2.2 Second case: |Z(G)| = 2
This case is more difficult as G/Z(G) is not necessarily Abelian. In the following, we will denote Z(G) = {e, z}.

We have to study the different cases for G/Z(G). G/Z(G) is not cyclic. The different cases are presented below.

A) G/Z(G) ∼ = Z 4 × Z 2 :
The maximal subgroups of Z 4 ×Z 2 = {(0, 0), (1, 0), (2, 0), (3, 0), (0, 1), (1, 1), (2, 1), (3, 1)} are:

• {(0, 0), (1, 0), (2, 0), (3, 0)} ∼ = Z 4 and has only one subgroup of order 2.

• {(0, 0), (1, 1), (2, 0), (3, 1)} ∼ = Z 4 and has only one subgroup of order 2.

• {(0, 0), (0, 1), (2, 0), (2, 1)} ∼ = Z 2 × Z 2 and has three subgroups of order 2.

Consequently, G has 3 subgroups G 1 , G 2 , G 3 of order 8 containing the center such that G 1 , G 2 has only one subgroup of order 4 containing the center and G 3 has three subgroups of order 4 containing the center. We show that G 1 (resp. G 2 ) is Abelian. Indeed, Z(G) < Z(G 1 ). If G 1 were not Abelian, then it has three subgroups of order 4 containing Z(G 1 ) and consequently containing Z(G); which is contradictory. The following proposition conducts to a contradiction: Proposition 14. If G has at least two Abelian subgroups of order 8 then |Z(G)| ≥ 4.

Proof. Using:

|G 1 G 2 | = |G 1 ||G 2 | |G 1 ∩ G 2 | = 64 |G 1 ∩ G 2 | , we deduce that G = G 1 G 2 and |G 1 ∩ G 2 | = 4. If G 1 and G 2 are Abelian, then taking g ∈ G 1 ∩ G 2 and g 1 g 2 ∈ G. gg 1 g 2 = g 1 gg 2 = g 1 g 2 g. Consequently, G 1 ∩ G 2 ⊂ Z(G).
We deduce the result.

B) G/Z(G) ∼ = (Z 2 ) 3 : In this case, for any g ∈ G, g 2 ∈ Z(G). As G is not Abelian, then there exists an element g such that g 2 = z, so |g| = 4. The group G has no element of order 8 nor 16. Let < g >= {e, g, g 2 = z, g 3 } ∼ = Z 4 . It is a subgroup of G containing Z(G). Consequently, by the correspondance theorem, < g > /Z(G) is a subgroup of order 2 of G/Z(G). Without loss of generality, one can suppose that this subgroup is identified to the subgroup {(0, 0, 0), (0, 0, 1)} of (Z 2 ) 3 . This subgroup is contained to the following three subgroups of order 4:

• {(0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1)}.

• {(0, 0, 0), (0, 0, 1), (1, 0, 0), (1, 0, 1)}.

• {(0, 0, 0), (0, 0, 1), (1, 1, 0), (1, 1, 1)}.

Consequently, G has three subgroups G 1 , G 2 , G 3 of order 8 containing < g >. One can easily show, using the correspondance theorem, that for i = j and for

g i ∈ G i \ < g >, g j ∈ G j \ < g > then g i g j ∈ G k \ < g > for k / ∈ {i, j}.
From the proposition 14, the maximum number of Abelian subgroups of order 8 is 1. Consequently, one can suppose that G 2 and G 3 are non-Abelian. Consequently, one can take

g 2 ∈ G 2 \ < g > (resp. g 3 ∈ G 3 \ < g >) which does not commute with g. As G/Z(G) is Abelian, then D(G) = Z(G).
Consequently, gg 2 = zg 2 g and gg 3 = zg 3 g. We deduce gg 2 g 3 = zg 2 gg 3 = zg 2 zg 3 g = z 2 g 2 g 3 g = g 2 g 3 g. Consequently, g commutes with all elements of G 1 \ < g >; and as a consequence, g ∈ Z(G 1 ). Z(G 1 ) contains {e, z, g}. We deduce that G 1 is Abelian. As G does not contains element of order 8 and as g ∈ G 1 is of order 4 then G 1 ∼ = Z 4 × Z 2 . Let us denote: G 1 = {e, g, g 2 = z, g 3 , g 1 , gg 1 , g 2 g 1 , g 3 g 1 }. Let g 2 ∈ G 2 \ < g > (corresponds to a symmetry in D 4 or one of the four other elements of order 4 in H). If one of the 4 elements of G 1 \ < g > commutes with one of the elements of G 2 \ < g >, then it commutes with all elements of G 2 and the center of G 2 has at least 3 elements, which is contradictory. Consequently; none element of G 1 \ < g > commutes with one element of G 2 \ < g >. Let g 1 ∈ G 1 \ < g > and g 2 ∈ G 2 \ < g >, then g 1 g 2 = zg 2 g 1 . We show that gg 1 g 2 = gzg 2 g 1 = zgg 2 g 1 = z 2 g 2 gg 1 = g 2 gg 1 . Consequently, g 2 commutes with gg 1 ; it leads to a contradiction.

C) G/Z(G) ∼ = H:

The maximal subgroups of H = {1, -1, i, -i, j, -j, k, -k} are:

• {1, i, -1, -i} ∼ = Z 4 , which has only one subgroup of order 2.

• {1, j, -1, -j} ∼ = Z 4 , which has only one subgroup of order 2.

• {1, k, -1, -k} ∼ = Z 4 , which has only one subgroup of order 2.

Consequently, G has three Abelian subgroups of order 8. Using the proposition 14, it leads to a contradiction.

D) G/Z(G) ∼ = D 4 :
The maximal subgroups of D 4 = {Id, r, r 2 , r 3 , s, rs, r 2 s, r 3 s} are:

• {Id, r, r 2 , r 3 } ∼ = Z 4 which has only one subgroup of order 2.

• {Id, s, r 2 , r 2 s} ∼ = Z 2 × Z 2 which has three subgroups of order 2.

• {Id, rs, r 2 , r 3 s} ∼ = Z 2 × Z 2 which has three subgroups of order 2.

Consequently, G has three subgroups of order 8, G 1 , G 2 , G 3 which contain the center such that G 1 has only one subgroup of order 4 which contains the center and G 2 , G 3 have three subgroups of order 4 which contain the center. G 1 is Abelian and G 2 , G 3 are necessarily non-Abelian. We show easily Z(G) = Z(G 2 ) = Z(G 3 ) as the center of a non-Abelian group of order 8 has two elements.

If G 2 ∼ = D 4 , as r 2 = rsr 3 s, then r 2 ∈ D(G). If G 2 ∼ = H, as -1 = ij(-i)(-j), then -1 ∈ D(G).
In all cases, Z(G) < D(G) and the inclusion is strict as G/Z(G) is non-Abelian. As, in D 4 , s and rs do not commute; then there exists g 2 ∈ G 2 and g 3 ∈ G 3 such that g 2 Z(G) (identified to s) and g 3 Z(G) (identified to rs) do not commute. Consequently, there exists g ∈ G (such g Z(G) is identified to r 2 ) such g 2 g 3 Z(G) = g 3 g 2 g Z(G). We deduce that there exists z 0 ∈ Z(G) such g 2 g 3 = g 3 g 2 g z 0 . Consequently, g z 0 ∈ D(G) and as z 0 ∈ D(G), we deduce that g ∈ D(G). We have {e, g , z, g z} ⊂ D(G) and forms a group. Let us show that this group is a normal subgroup of G. We have hg zh

-1 Z(G) = hg h -1 Z(G) = g Z(G) as g Z(G) ∈ Z (G/Z(G)
). Consequently hg h -1 , hg zh -1 ∈ {g , g z}; we deduce that {e, g , z, g z} is a normal subgroup of G. As G/ {e, g , z, g z} is of order 4, it is necessarily Abelian. Consequently, D(G) = {e, g , z, g z}. Moreover, D(G)/Z(G) = Z (G/Z(G)) and corresponds to {Id, r 2 }. We deduce that D(G) < G i for any i = 1, 2, 3. Using the

formula |G i G j = |G i ||G j | |G i ∩ G j | |, we deduce G = G i G j and G i ∩ G j = D(G). D1) D(G) ∼ = Z 2 × Z 2 : G 1 cannot be isomorphic to (Z 2 ) 3 ,
as any subgroup of order 2 is included in three subgroups of order 4. The only possibility is G 1 ∼ = Z 4 × Z 2 . The unique subgroup of order 4 containing the center is then identified to {(0, 0), (2, 0), (0, 1), (2, 1)}. As the square of any element of Z 4 × Z 2 is either (0, 0) or (2, 0) and (2, 0) is in three subgroups of order 4; we deduce that there exists an element a of order 4 in G\D(G) such that:

• G 1 = {e, a, a 2 = g , a 3 , z, az, a 2 z = g z, a 3 z} ∼ = Z 4 × Z 2 .
As H has no subgroup isomorphic to Z 2 × Z 2 ; we deduce G 2 , G 3 ∼ = D 4 . As Z(G) = Z(G 2 ) = Z(G 3 ), then there exists b, c in G\D(G) of order 4 such that:

• G 2 = {e, b, b 2 = z, b 3 , g , bg = g b 3 , b 2 g = zg , b 3 g = g b}. • G 3 = {e, c, c 2 = z, c 3 , g , cg = g c 3 , c 2 g = zg , c 3 g = g c}.
By correspondance theorem, we show that baZ(G) = abg Z(G) = g abZ(G). Suppose that ba = g ab. If |ba| = 2, then ba = a 3 b 3 = a 2 abb 2 = g abz = g zab. Consequently, g = g z, which is contradictory. We have ab = g ba. If we suppose that |ba| = 4, as g is a symmetry, then |ab| = 2 and it conducts to the same contradiction. Suppose that ba = g zab. If |ba| = 4, then (ba) 2 ba = a 3 b 3 = g abz = g zab. As (ba) 2 = z then ba = g ab which is contradictory. We have ab = g zba. If |ba| = 2, as ba is not in D(G), we deduce |ab| = 4 which leads to the same contradiction. Consequently, it is impossible that D

(G) ∼ = Z 2 × Z 2 . D2) D(G) ∼ = Z 4 :
In this case, D(G) = {e, g , g 2 = z, g 3 } We have seen that G 1 cannot be isomorphic to (Z 2 ) 3 . However, as Z 4 × Z 2 has two groups isomorphic to Z 4 which contains a common subgroup of order 2; G 1 cannot be isomorphic to Z 4 × Z 2 (as the center will be included in two subgroups of order 4). Consequently, G 1 ∼ = Z 8 and there exists a ∈ G\D(G) of order 8 such that:

• G 1 = {e, a, a 2 = g , a 3 , a 4 = z, . . .}. D2-1) G 2 , G 3 ∼ = D 4 :
There exists b, c not in D(G) and of order 2 such that:

• G 2 = {e, g , g 2 = z, g 3 , b, g b = bg 3 , g 2 b, g 3 b = bg }. • G 3 = {e, g , g 2 = z, g 3 , c, g c = cg 3 , g 2 c, g 3 c = cg }.
By the correspondance theorem and the structure of the maximal subgroups of D 4 , we deduce that ba ∈ G 3 \D(G). Consequently, |ba| = 2 and ba = a

7 b. < a >= G 1 ∼ = Z 8 , < b > ∼ = Z 2 , | < a > | × | < b > | = 8 × 2 = 16 and < a > ∩ < b >= {e}.
Consequently, G is the semi-direct product, also called diedral group D 8 :

G ∼ = Z 8 ϕ 2 Z 2 ,
with:

ϕ 2 : Z 2 → Aut (Z 8 ) 1 → (1 → 7)
Its character presentation is x 8 = y 2 = e, yx = x 7 y.

D2-2) G 2 ∼ = D 4 , G 3 ∼ = H: There exists b, c not in D(G) respectively of order 2 and 4 such that:

• G 2 = {e, g , g 2 = z, g 3 , b, g b = bg 3 , g 2 b, g 3 b = bg }.

• G 3 = {e, g , g 2 = z, g 3 , c, g c, g 2 c, g 3 c}.

ba ∈ G 3 \D(G), then |ba| = 4. Consequently, (ba) 2 ba = a 7 b. As (ba) 2 = g 2 = z = a 4 , then ba = a 3 b. With the same method as previously, we deduce that G is the semi-direct product (also called semi-diedral group SD 8 ):

G ∼ = Z 8 ϕ 3 Z 2 ,
with:

ϕ 3 : Z 2 → Aut (Z 8 ) 1 → (1 → 3)
Its character presentation is x 8 = y 2 = e, yx = x 3 y.

D2-3) G 2 , G 3 ∼ = H: There exists b and c not in D(G) of order 4 such that:

• G 2 = {e, g , g 2 = z, g 3 , b, g b, g 2 b, g 3 b}.

• G 3 = {e, g , g 2 = z, g 3 , c, g c, g 2 c, g 3 c}.

|ba| = 4 then (ba) 2 ba = a 7 b 3 . As (ba) 2 = z = a 4 then ba = a 3 b 3 . It is not a semi-direct product. Its character presentation is x 8 = y 4 = e, x 4 = y 2 , yx = x 3 y 3 .

The following Table gives the groups of order 16 such that Z(G) ∼ = Z 2 :

Name

Character presentation Z 8 ϕ 2 Z 2

x 8 = y 2 = e,yx = x 7 y Z 8 ϕ 3 Z 2

x 8 = y 2 = e,yx = x 3 y Generalized Quaternions x 8 = y 4 = e, x 4 = y 2 , yx = x 3 y 3

Conclusion

The groups of order 16 are given in the following table: x 4 = y 2 = z 2 = e,yx = x 3 y,zx = xz,zy = yz {e, x 2 , z, zx 2 } ∼ = Z 2 × Z 2 (Z 2 × Z 2 ) Z 4

x 4 = y 4 = e,yx = x 3 y 3 {e, x 2 , y 2 , x 2 y 2 } ∼ = Z 2 × Z 2 Z 4 Z 4

x 4 = y 4 = e,yx = x 3 y {e, x 2 , y 2 , x 2 y 2 } ∼ = Z 2 × Z 2 H × Z 2

x 4 = y 4 = z 2 = e,x 2 = y 2 ,yx = x 3 y,zx = xz,zy = yz {e, x 2 , z, x 2 z} ∼ = Z 2 × Z 2 (Z 4 × Z 2 ) Z 2

x 4 = y 2 = z 2 = e,xy = yx,zx = xz,zy = x 2 yz {e, x, x 2 , x 3 } ∼ = Z 4 Z 8 ϕ 1 Z 2

x 8 = y 2 = e,yx = x 5 y {e, x 2 , x 4 , x 6 } ∼ = Z 4 Z 8 ϕ 2 Z 2 x 8 = y 2 = e,yx = x 7 y {e, x 4 } ∼ = Z 2 Z 8 ϕ 3 Z 2

x 8 = y 2 = e,yx = x 3 y {e, x 4 } ∼ = Z 2 Gen. Quat.

x 8 = y 4 = e,x 4 = y 2 ,yx = x 3 y 3 {e, x 4 } ∼ = Z 2

ba

  has order 2, so baba = e. a has order 4 and b has order 2, so baa 3 b = e. Consequently, ba = a 3 b. As G = G 1 G 2 , G is generated by a and b. Moreover, | < a > | × | < b > | = 4 × 2 = 8 = |G| and < a > ∩ < b >= {e}; G is the semi-direct product < a > < b > ∼ = Z 4 Z 2 . It is the diedral group D 4 . Its character representation is a 4 = b 2 = e and ba = a 3 b.

  = G Z 8 × Z 2 a 8 = b 2 = e,ba = ab Z(G) = G Z 4 × Z 4 a 4 = b 4 = e,ba = ab Z(G) = G Z 4 × (Z 2 ) 2 a 4 = b 2 = c 2 = e,D(G) = {e} Z(G) = G (Z 2 ) 4 a 2 = b 2 = c 2 = d 2 = e,D(G) = {e} Z(G) = G D 4 × Z 2


Flaviano Tateo
email: flaviano.tateo@femto-st.fr

M Collet

M Ouisse

M N Ichchou

K A Cunefare

Experimental characterization of a bi-dimensional array of negative capacitance piezo-patches for vibroacoustic control

Keywords: negative capacitance, smart materials, metacomposite, Bloch theorem, experiment

A recent technological revolution in the fields of integrated MEMS has finally rendered possible the mechanical integration of active smart materials, electronics and power supply systems for the next generation of smart composite structures. Using a bi-dimensional array of electromechanical transducers, composed by piezo-patches connected to a synthetic negative capacitance, it is possible to modify the dynamics of the underlying structure. In this study, we present an application of the Floquet-Bloch theorem for vibroacoustic power flow optimization, by means of distributed shunted piezoelectric material. In the context of periodically distributed damped 2D mechanical systems, this numerical approach allows one to compute the multi-modal waves dispersion curves into the entire first Brillouin zone. This approach also permits optimization of the piezoelectric shunting electrical impedance, which controls energy diffusion into the proposed semi-active distributed set of cells. Furthermore, we present experimental evidence that proves the effectiveness of the proposed control method. The experiment requires a rectangular metallic plate equipped with seventy-five piezo-patches, controlled independently by electronic circuits. More specifically, the out-of-plane displacements and the averaged kinetic energy of the controlled plate are compared in two different cases (open-circuit and controlled circuit). The resulting data clearly show how this proposed technique is able to damp and selectively reflect the incident waves.

INTRODUCTION

The design of innovative structures having new physical functionalities represents a big challenge for both industrial and academic communities. In the vibration domain a very important issue is the capacity of controlling the dynamic behavior of a structure by acting on the mechanical power flow in terms of transmitting and absorbing properties. Among all proposed solutions, distributed control system integrating shunted piezoelectric patches represents an interesting solution 9 . Different circuit's layouts can be considered, for example a control strategy based on resonant circuit where the modification of the external circuit's parameters allows the suppression of a resonant peak. 8 A different approach based on negative capacitance circuit allows a widening of the control frequency band. In order to magnify this effect a lattice of actuators connected to the circuits is introduced. Up to now this technique has been mainly applied to mono-dimensional structures such bars and beams 1 or bi-dimensional structures such plates and shells. 3 The aforementioned works are characterized by the presence of multiple actuators placed over the entire surface of the controlled structure. In this work the previous idea is modified introducing the concept of active interface which is the part of the plate containing the actuators that allows the modification of the system's response.

THE BLOCH THEOREM APPLIED TO THE ELASTODYNAMIC

In this section, the application of the Floquet-Bloch theorem is presented for piezoelastodynamic problems. Based on the results obtained by Floquet (Floquet, 4 1883) for monodimensional problems and later rediscovered by Bloch (Bloch, 2 1928) in multidimensional problems, an original application to bidimensional visco-elasto-dynamic problem has been proposed recently (Collet et al., 6

The Bloch theorem

The Bloch Theorem stipulates that any function u (x) ∈ L 2 R 3 , C n can be expressed as

u (x) = Ω k e ikx u (x, k) dk (1)
where the Bloch amplitude u (x, k) is Ω x -periodic and has the representations

u (x, k) = n u (k + Gn) e iGnx = Ω x (2π) 3 n u (x + Rn) e ik(x+Rn) (2)
where u (k) stands for the Fourier transform of u (x). It can also be demonstrated that the mean value of the Bloch amplitude is the Fourier amplitude of u (x) for the corresponding wave vector. Based on that theorem, the expansion functions v m (x, k) can be defined. They are called the Bloch eigenmodes and can be used to represent the Bloch amplitudes of any solution of periodic problem as

u (x, k) = m u m (k) v m (x, k) (3)
and at the same time diagonalize the partial derivative operator.

Application to piezoelastodynamic

Let us consider a piezoelastodynamic problem made of infinite periodic distribution of unitary cell. The harmonic homogeneous dynamical equilibrium of system is driven by the following partial derivative equations

ρ ẅ -∇ • σ = 0 ∀x ∈ Ω x -∇ • D = 0 ∀x ∈ Ω x ( 4
)
where w is the displacement vector, σ represents the Cauchy stress tensor and D the electric displacement. The linear constitutive material behavior relationships can be written as

σ = C E ε -e T E D = eε + S E (5)
where E = -∇V is the electric field vector (V being the voltage), C E the elasticity tensor at constant electrical field, ε the Green strain tensor, e T the piezoelectric coupling tensor, and S the dielectric permittivity at constant strain. We add to this set of equilibrium equations an output expression

q app = - St D • ndS (6)
allowing the introduction of the charge measurement on the piezoelectrics top electrode and hence the dual counterpart of the imposed electrical Dirichlet boundary condition for applying the shunt impedance operator. The equations above are consistent for each kind of material to the extent that null piezoelectric and permittivity tensors can be used when passive materials are considered. All these tensors also depend on the spatial location vector x and are Ω x periodic. As the problem is 2D infinitely periodic, mechanical boundary conditions are included in the formulation, while electrostatic boundary conditions have to be considered on each cell

V = 0 ∀x ∈ S b V = V app ∀x ∈ S t D • n = 0 ∀x ∈ S l (7)
where S b is the grounded bottom electrode of the piezoelectric layer, S t is the top electrode connected to the external shunt, and S l the lateral electrode. The top electrode applied feedback voltage V depends on the shunt characteristic and on the collected charges q app and can be expressed in the Fourier space by

V app = Zq app (8)
The proposed formulation is then based on the computation of the Floquet vectors, instead of computing the Floquet propagators commonly used for elastodynamic applications. The full 2D wave dispersion functions can then be obtained, while damping and electrical impedances can clearly be introduced into the piezoelastodynamic operator. The adopted methodology allows the computation of the complete complex map of the dispersion curves incorporating computation of evanescent waves and allowing the introduction of damping and shunt operator if any.

Numerical computation of the Blochs waves

Replacing the solution expressed in term of the Bloch modes the problem 4 assume the following discretized form:

K (Z (ω)) + λ n (ω, θ) L (θ, Z (ω)) -λ 2 n (ω, θ) H (θ, Z (ω)) -ω 2 M u n (ω, θ) = 0 (9)
where the angular frequency ω is a real parameter corresponding to the harmonic frequency. Wave numbers and Floquet vectors are then computed. This approach allows one to easily take into account the frequency dependency of various system's terms (i.e. damping and electric impedance). An inverse Fourier transformation in the k-space domain can lead us to evaluate the physical wave displacements and energy diffusion operator when the periodic distribution is connected to another system as described in Collet et al. 5 (2009). Another temporal inverse Fourier transformation can furnish a way to access spatio-temporal response for nonhomogeneous initial conditions.

Computation of the evanescence and damped power flow criteria

One aim of this article is to provide a numerical methodology for optimizing the piezoelectric shunt impedance Z (ω) for controlling energy flow into the periodically distributed piezocomposite structure. The first criterion which is considered for describing the capability of the metacomposite for transmitting structural energy is based on the computation of the wave group velocities. Indeed, they indicate how energy is transported into the considered system and allow to distinguish the propagative and evanescent waves. If a Bloch eigensolution is considered, the associated group velocity vector is given by

C g n = ∇ k ω = S e tot = I E tot ( 10
)
where . is the spatial and time average, respectively, on one cell and one period of time; S is the density of energy flow; I is the mean intensity; and e tot and E tot are the total piezomechanical energy and its time average, respectively, on a period. The intensity vector I is expressed as where * is the complex conjugate, real stands for real part, and V ol the domain volume.

I n = real - ω 2 Ωx C E (ε n + ikΞ n ) • w * n dΩ V ol (11)
As the spatio-temporal average of the system Lagrangian is null (see Maysenholder, 10 1994), the total energy average is approximated by only computing the kinetic energy average

E tot = 1 2V ol Ωx ρω 2 w n • w * n dΩ (12)
The group velocity vectors C g n (ω, Φ) are computed for all wave numbers at each frequency. In order to focus our analysis on only flexural modes (S and SH ones), we introduce an indicator allowing to select them by computing the ratio of kinetic energy average on out-of-plane displacement as

Ind (n, ω, θ) = 1 2V ol Ωx ρω 2 w z n • w z * n dΩ E tot (13)
w z n (x) being the component of vector w n (x) on z direction. Optimization of the shunt impedance Z (ω) is based on the minimization of the maximal group velocity collinear to the wave number vector for waves having a ratio of transported flexural kinetic energy greater than the threshold value of 0.8. The criterion can then be written as

Crit 1 (Z (ω) , θ) = max (C g n • Φ) ∀p ∈ n/Ind (n, ω, Φ) > 0.8. ( 14
)

Optimization of the flexural energy flow inside the shunted periodic piezocomposite

The considered piezocomposite cell is presented in Figure 1. The supporting plate material is standard aluminum with 0.1% of hysteretic damping ratio, and the piezoelectric material characteristics are given in Table 1.

The optimization of the transmission capability of the designed adaptive metacomposite is considered by using Crit 1 . The objective is to avoid any energy transportation when flexural waves are excited into the periodically distributed shunted piezocomposite cells. The numerical optimization procedure is based on a multidimensional unconstrained nonlinear minimization algorithm (NelderMead). The optimization is done by considering an active/reactive electronic circuit represented by its impedance Z (ω). The analysis is initialized with an arbitrary complex value of the shunt impedance. Optimization steps are then proceeded using Crit 1 and lead us to obtain frequency-dependent complex impedance for describing the circuit behavior. The real and imaginary parts of the optimal impedance are plotted in Figure 2 for all angles θ. The optimal impedance values almost correspond to a constant negative capacitance in all directions. Equivalent resistances corresponding to the active part of the shunt impedance are negative which indicates that the optimization leads to provide energy to the system for controlling mechanical damping effects introduced by hysteretic terms in the model.

-

EXPERIMENTAL TEST

This section introduces the design and testing of the smart metacomposite.

Periodic Lattice design

An aluminum plate with the geometric properties listed in Table 1 is suspended to a rigid frame through metallic wires. The plate is equipped with 75 piezoelectric patches from P Z26 series (Ferroperm industries) arranged in a regular 15 × 5 array. The dimension of the piezoelectric ceramics are listed in Table 1 and are justified by the following assumptions: the plate's dispersion relation of the flexural mode at 5000 Hz has a wavelength of 30 cm. This value imposes a lenght constraint of the piezoceramic in the propagation direction, the best coupling is obtained when the length of the ceramics and the wavelength of the targeted mode respect the ratio of 1/4. The thickness of the actuator was finally defined considering the restraints of the electric circuit and the nature of the control technique. Different authors argued that the best controlling effect is obtained when the circuit is tuned in correspondence of the biggest ceramic's capacitance value. This property strongly depend on the material properties and the geometry. Once the material properties and the two dimension of the piezoelectric actuator are chosen the only parameter which may be amended is the thickness. Small thicknesses correspond to larger values of the intrinsic capacitance, however, this parameter can not be reduced indefinitely due to the weakening of the piezoelectric ceramic itself. For these reasons a thickness of 0.5 mm was adopted.

Shunt circuit design

The results derived from the optimization of the external circuit permit to design it in term of real and imaginary part of a general impedance. In Figure 4 is depicted the actual circuit used during the experiment. This circuit contains some passive components such as the resistances R s , R, R tot , the capacitance C and an active component, namely an operational amplifier that actually allows the circuit to reproduce the intended behavior. This control technique requires a tuning of the synthetic capacitance around the capacitance value of the piezoelectric ceramic, this value is intrinsic and depends upon the material properties and the geometry. The intrinsic capacitance of the piezoelectric ceramic can be easily estimated measuring the capacitance observed by an RLCmeter when the specimen is free of stresses. In the present case the measured value was 52nF . The actual circuit was tuned varying the position of the potentiometer R tot with small increments. The variation of this parameter determine a variation in the synthetic capacitance according to the formula:

C eq = -αC 2 ( 15
)
where α is defined as R 4 /R 3 and R pot is the resistance of the potentiometer and it represents the sum of R 3 and R 4 . The use of the potentiometer simplifies the layout of the circuit because it permits to reproduce the capacitive effect with just one component.

The regulation phase starts setting the synthetic capacitance for very high values and then slowly decreasing it until the desired value is reached. During this phase some problems can occur because of the limitation connected to the nature of the analog circuit. When the external circuit approaches the set-point the system tend to be unstable, an increase of voltage is observed. The operational amplifier is therefore saturated. In order to mitigate this effect the circuit's layout proposed by Beck 1 were modified adding a complementary circuit able to detect this specific instability. Basically a comparator was introduced in correspondence of the op-amp output in order to identify these voltage overloads. This circuit is based on a comparator that measures the differential input between the positive component of the voltage delivered by the operational amplifier in correspondence of the output gate and a reference DC signal from the power supply circuit. Whenever the differential signal results bigger than zero the gate is opened, delivering a current able to illuminate a light emitting diode (LED). This simple solution is very helpful since it allows one to detect this source of instability avoiding the use of the oscilloscope. This feature becomes very useful in this context where a huge number of actuators are involved. For example when several piezoelectric patches are excited a loss of performance can be observed due to the electromechanical coupling occurring between adjacent actuator. In this case, observing the state of the LEDs on the control panel is possible to correctly reset each external circuit.

EXPERIMENTAL RESULTS

The experimental procedure involves the measurement of the mobility

Y (ω) = ẇ F (16)
where ẇ represent the out-of-plane velocity of the controlled plate measured by means of a laser vibrometer and F the shaker's force signal.

In figure 5 the mobility FRF is provided in term of the average of the magnitude using third octave bands, in order to estimate the response reduction of the control system compared with the reference measure on the same structure without activating the electronic circuit. On frequency range of interest, it is possible to reduce the transmitted energy of at least 5 dB. However, this is an information, it is obtain, locally, even better results, for example a 15 dB reduction around 2 kHZ. The figure presents two diagrams: the first on the top is related to a measurement made at the excitation point P 1 and the other corresponds to measurements made in the opposite corner of the plate and therefore located downstream with respect the active interface, point P 2 . Comparing the two diagrams, an interesting property can be deduced: the plate impedance seen by the excitation is not modified by the control system whereas the part of the plate beyond the active interface acts as the structure would be strongly damped. This effect can be used with profit in case one wants to isolate a source of disturbance. These results show of the proposed strategy, it is clear that in the frequency band of 50 -5000 Hz the control system properly works, reducing the magnitude of the vibrations. However for low frequency values the system seems to be unable to correctly control the underlying structure: the response is amplified during the control phase, this is only due to the deterioration of the measured data, the coherence function shows a poor quality. In order to clarify the system's behavior at low frequency other measurements were performed. In Figure 6 is depicted the plate's frequency response function measured at point P 1 and P 2 of the controlled plate. The mobility function is measured over the frequency range of 0 -5000 Hz, in each row, however, is presented just a small portion of the total FRF in order to point out the details connected dynamics of the system, keeping the best possible values for the coherence function.

In the low frequency range (0 -500 Hz) the system cannot be very effective for both positions upstream and downstream, however, beyond 200 Hz the control system is activated, causing an appreciable vibration reduction. On this frequency band the response of the controlled system is constantly smaller than the reference measurement: the control system's main function is to reduce the vibrations adding a certain amount of damping. In Figure 7 a measurement of the plate's velocity distribution is provided. In the top of the image is presented the velocity field of the plate of the reference measurement (control off) against the controlled one in the frequency range where the system is not effective (low frequencies). As a consequence, the control system can not influence the plane and the response level are comparable. The slight modification of the shape of the mode observed can be justified as follows: the controlled plate in this case is experiencing a stiffness properties' modification (frequency shifting). In the active frequency band the response level are decreased in a uniform fashion over the whole region. In this case the wave confinement effect is still present demonstrated by an increase of response level in the portion of the plate behind the active interface delimited by the two horizontal lines.

CONCLUSIONS

This article presents a numerical procedure able to compute the damped wave dispersion functions in the whole first Brillouin domain of multidimensional damped piezoelastodynamical wave guides. The method was applied for determining the optimal shunt impedance allowing the minimization of the group velocities of the flexural waves and the damping capability of the distributed system. The proposed methodology can also be used for studying particular dissipation phenomenon such as those induced by complex shunted piezoelectric patches or even foams or complex polymers behaviors. From the experimental point of view, we showed how it is possible to modify the dynamics of the plate equipped by a series of piezoelectric actuators shunted to negative capacitance circuits and arranged in a periodic fashion. The aim of the project was achieved overcoming several difficulties such as the circuit instability around the set point and the electrical coupling between the different piezoelectric actuators. These preliminary results are encouraging, however further analysis will be required in order to thoroughly understand the mechanisms related to this control technique. In particular it will be necessary to better match the theoretical results with the experimental measurements.

  2011) and extended to piezo-elasto-dynamic problem (Collet et al., 7 2012).

Figure 1 .

 1 Figure 1. Actuators periodic lattice and its primitive cell Ωx.

Figure 2 .

 2 Figure 2. Optimal electric impedance using Crit1.

Figure 3 .

 3 Figure 3. Layout of the electric circuit. Schematic (left), real-life circuit (right).

Figure 4 .

 4 Figure 4. Layout of FRF measurement system.

Figure 5 .

 5 Figure 5. FRF plot comparison between the uncontrolled Yr and controlled system Yc expressed in third of octave band.

Figure 6 .

 6 Figure 6. FRF plot comparison between the uncontrolled (blue line) and controlled system (green line). P1 measurement (left column), P2 measurement (right column).

Figure 7 .

 7 Figure 7. FRF velocity field comparison. Top: inactive region for the control system. Bottom: active region for the control system.

Table 1 .

 1 Geometry and physical properties of the system.

	Plate	Value	Unit
	Length	2100	mm
	Height	1050	mm
	Thickness	3	mm
	Mass density	2700	kg/m 3
	Young's modulus	70 • 10 9	N/m 2
	Poisson's ratio	0.33	-
	Piezo	Value	Unit
	Length	50	mm
	Height	50	mm
	Thickness	0.5	mm
	Mass density	7650	kg/m 3
	Poisson's ratio	0.31	-
	Dielectic loss	< 0.05	%
	Coupling factor	0.51	-

ACKNOWLEDGMENTS

This study is a collaborative effort supported by the French Research Agency under grant number NT09-617542.


Flaviano Tateo
email: flaviano.tateo@femto-st.fr

Manuel Collet

Morvan Ouisse

Mohamed N Ichchou

DESIGN AND EXPERIMENTAL VALIDATION OF A METACOMPOSITE MADE OF AN ARRAY OF PIEZOPATCHES SHUNTED ON NEGATIVE CAPACITANCE CIRCUITS FOR VIBROACOUSTIC CONTROL

Design and experimental validation of a metacomposite made of an array of piezopatches shunted on negative capacitance circuits for vibroacoustic control

INTRODUCTION

In the last few decades, researchers have given a lot of attention to new engineered materials with the purpose of developing new technologies and devices such as mechanical filters, low frequency sound and vibration isolators, and acoustic waveguides. In the field of light propagation, several efforts were made in order to design and construct photonic crystals showing photonic band gaps that prevent light from propagating in certain directions with specified frequencies. Many other efforts were made in order to create photonic crystal able to propagate light in anomalous and useful ways (i.e. negative refraction and artificial magnetism). In the acoustic domain, similar studies were carried out with the aim of preventing the propagation of elastic waves within the medium. In both cases, the band gap is obtained by periodically modulating some electromagnetic or mechanical properties [START_REF] Yang | Ultrasound tunneling through 3D phononic crystals[END_REF]. This technique presents two main limitations: The spatial modulation must be of the same order as the wavelength in the gap, the position of the band gap cannot be easily changed since it strongly depends on the materials involved (Bragg's band gap). A possible solution for these problems is found using composites with locally resonant units. The periodicity of the crystal creates a stop band that can be shifted by modifying the properties of the resonators. Liu et al. [START_REF] Liu | Locally resonant sonic materials[END_REF] had demonstrated that a resonant sonic crystal with building blocks of rubber-coated lead balls exhibits a low-frequency sonic band gap, and the resonance can provide a maximum impedance mismatch to shield the airborne sound. The same effect can be obtained using Helmoltz resonators as showed by Fang et al. [START_REF] Ambati | Surface resonant states and superlensing in acoustic metamaterials[END_REF][START_REF] Fang | Ultrasonic metamaterials with negative modulus[END_REF] or Hu [START_REF] Hu | Two-dimensional sonic crystals with helmholtz resonators[END_REF]. A further improvement can be obtained using shunted piezoelectric materials employed simultaneously with electronic components and controllers in order to achieve new functionalities. Adaptive metacomposite can be designed to create local resonances by coupling piezoelectric patches to a resonant external circuit, as illustrated in [START_REF] Thorp | Attenuation and localization of wave propagation in rods with periodic shunted piezoelectric patches[END_REF]. This concept involves a periodic array of simple RL-shunted piezos mounted on the structure to passively control the propagation of elastic waves and the subsequent vibration field. Periodically induced impedance-mismatch zones generate broader stop bands, i.e., frequency bands where waves are attenuated. The tunable characteristics of shunted piezo-patches allow the equivalent mechanical impedance of the structure to be tuned so that stop bands are generated over desired frequency ranges. The presence of a resistance in the shunt circuit generates a damped resonance of the electrical network. The resistance also allows the energy dissipation mechanism of shunted piezos to be exploited, which dampens the amplitude of vibration also outside the stop bands. In this paper, the considered metacomposite consists of several piezoelectric patches periodically arranged over the surface of the two-dimensional waveguide and shunted to an external electric circuit with a negative capacitance. Unlike the distributed resonant circuits, which are based on the energy exchange between the controlled structure and the resonant circuit, the negative capacitance circuit allows to broaden the control frequency for controlling either energy velocity or wave absorption properties. The main contribution of this article is the capacity of modifying and controlling the dynamic of the two-dimensional waveguide, in terms of reflected or absorbed energy flows. This effect is achieved by physically separating both ends of the waveguide with a periodic lattice of piezoelectric actuators.

System's design

An aluminum plate with the geometric properties listed in Table 1 is suspended to a rigid frame through metallic wires in order to reproduce the free boundaries conditions. The plate is equipped with 75 piezoelectric patches from PZ26 series (Ferroperm Industries) arranged in a regular 15 × 5 array (Figure 1). The dimension of the piezoelectric ceramics are listed in persion relation of the flexural mode at 5000 Hz has a wavelength of 30 cm. This value imposes a length constraint of the piezoceramic in the propagation direction. According to Livet [START_REF] Livet | Structural multi-modal damping by optimizing shunted piezoelectric transducers[END_REF] the piezoelectric coupling is increased when the ratio between the length of the ceramics and the wavelength is bigger than 1/4. Under this condition, the piezoelectric actuator can experience an acceptable strain level. The thickness of the actuator was finally defined considering the restraints of the electric circuit and the nature of the control technique. Different authors have shown that the best controlling effect is obtained when the circuit is tuned in correspondence with the biggest ceramic's capacitance value [START_REF] Forward | Electronic damping of vibrations in optical structures[END_REF][START_REF] Livet | Structural multi-modal damping by optimizing shunted piezoelectric transducers[END_REF]. This property strongly depends on the material properties and the geometry. Once the material properties and the two dimensions of the piezoelectric actuator are chosen, the only parameter which may be amended is the thickness. Small thicknesses correspond to larger values of the intrinsic capacitance, however, this parameter cannot be reduced indefinitely due to the weakening of the piezoelectric ceramic itself. For these reasons, a thickness of 0.5 mm was adopted. The actual circuit's layout considered contains some passive components, such as the resistances R s , R 2 , R 3 , R 4 , the capacitance C 2 and an active component, namely an operational amplifier that actually allows the circuit to reproduce the intended behavior. This specific layout was chosen as opposed to others [START_REF] Fleming | Synthetic impedance for implementation of piezoelectric shuntdamping circuits[END_REF] because of its simplicity and its effectiveness in the frequency range of interest. The variation of these parameters determines the variation in the synthetic impedance according to the following formula:

Z = R s - R 3 R 2 R 4 (1 + iωR 2 C 2 ) ; (1)
From the practical point of view, the circuit is further modified by adding some diodes in order to protect the operational amplifier.

The resistances R 3 and R 4 were merged in a single potentiometer with the aim of simplifying the circuit layout. This control technique requires a tuning of the synthetic capacitance around the capacitance value of the piezoelectric ceramic; this value is intrinsic to the electromechanical coupling and depends upon the material properties, the geometry of the patches and the plate stiffness. The intrinsic capacitance of the piezoelectric ceramic can be easily estimated measuring the capacitance observed by an RLC-meter when specimen is free of external forces. In the present case the measured value is 52 nF. The actual circuit was tuned by varying the position of the potentiometer in small increments from high values of the synthetic capacitance to a value very close to the targeted intrinsic capacitance value. When the external circuit approaches the setpoint the system tends to be unstable, and an increase of voltage is observed [START_REF] Livet | Structural multi-modal damping by optimizing shunted piezoelectric transducers[END_REF]. The operational amplifier is therefore saturated. In order to mitigate this effect, the circuit's layout proposed by Forward [START_REF] Forward | Electronic damping of vibrations in optical structures[END_REF] was modified adding a complementary circuit able to detect this specific instability.

Experimental set-up

The analyzed structure, shown in Figure 1, consists of an aluminum plate equipped with 75 piezoelectric actuators connected to external circuits and arranged so as to form a periodic array of 5 × 15 units. The wave propagation properties of the structure are characterized by means of two-dimensional wavefield measurements, recorded with a scanning laser vibrometer (Polytec PSV-400). Measurements are conducted over a grid of 37 × 81 scan points with a spatial resolution of 20 mm in the horizontal and vertical directions. At each grid point, the laser measures the time history of the out-of-plane velocity. Elastic waves are excited by a shaker located in the lower corner of the plate, 680 mm away from the active interface as shown in figure 2. The transducer is driven by an input signal consisting of a random broad-band excitation and amplified by an LDS power amplifier.

Scattering properties computation

The active interface described in the previous section can be considered as a infite periodic lattice of piezo-actuators. In order to correctly describe such type of system, a general mathematical method has been developed [START_REF] Collet | Structural energy flow optimization through adaptive shunted piezoelectric metacomposites[END_REF]. It based on the Floquet-Bloch theorem and allows one to estimate the dispersion properties of a two-dimensional waveguide incorporating, among others, electronic components, damping effects, or any other frequencydependent characteristics.

2D Bloch theorem applied to piezo-elastodynamic

Let us consider a piezoelastodynamic problem made of infinite periodic distribution of unitary cell. The harmonic homogeneous dynamical equilibrium of system is driven by the follow- ing partial derivative equations:

ρ ẅ -∇ • σ = 0 -∇ • D = 0 ( 2
)
where w is the displacement vector, σ represents the Cauchy stress tensor, ε = ∇ s w the Green strain tensor, and D the electric displacement. The linear constitutive material behavior relationships can be written as:

σ = C E ε -e T E D = eε + ε S E (3)
where E = -∇V is the electric field vector (V being the voltage), C E the elasticity tensor at constant electrical field, e T the piezoelectric coupling tensor, and ε S the dielectric permittivity at constant strain. We add to this set of equilibrium equations an output expression

q 0 = - S t D • ndS (4)
allowing the introduction of the charge measurement on the piezoelectrics top electrode and hence the dual counterpart of the imposed electrical Dirichlet boundary condition for applying the shunt impedance operator.

The equations above are consistent for each kind of material to the extent that null piezoelectric and permittivity tensors can be used when passive materials are considered. All these tensors also depend on the spatial location vector x and are Ω x periodic. By applying a Fourier transform, the piezoelastodynamic equilibrium can also be written as

ρω 2 w -∇ • C E ∇ s w + e T ∇V = 0 -∇ • (e∇ s w -ε S ∇V ) = 0. ( 5
)
As the problem is 2D infinitely periodic, mechanical boundary conditions are included in the formulation, while electrostatic boundary conditions have to be considered on each cell

V = 0 on S b V = V 0 on S t D n = 0 on S l (6)
where S b is the grounded bottom electrode of the piezoelectric layer, S t is the top electrode connected to the external shunt, and S l the lateral electrode. The top electrode applied feedback voltage V 0 depends on the shunt characteristic and on the collected charges q 0 given in equation ( 4) and can be expressed in the Fourier space by

V 0 = -Zq 0 (7)
Considering a primitive cell of the periodic problem Ω x , the Bloch eigenmodes and the dispersion functions can be computed by searching the eigensolutions of the homogeneous problems (5) with mechanical periodic boundary conditions and electric ones given by equation [START_REF] Hu | Two-dimensional sonic crystals with helmholtz resonators[END_REF]. This solution has the form:

u = u n,k e ik•x (8)
where u n,k is composed of Ω x periodic functions. By introducing expression [START_REF] Livet | Structural multi-modal damping by optimizing shunted piezoelectric transducers[END_REF] in the piezoelastodynamic equations, is then possible to determine the functions w n,k , V n,k , and ω n , solutions of the generalized eigenvalues problem

0 = ρω 2 w n -∇ • C E ∇ s w n + e T ∇V n + ik (C E ∇ s w n ) • Φ + ∇ • C E Ξ n,k + ik e T ∇V n • Φ + ∇ • e T ∇V n • Φ + -k 2 C E Ξ n,k • Φ + e T ∇V n • Φ • Φ + 0 = -∇ • (e∇ s w -ε S ∇V ) + -k ∇ • eΞ n,k + (e∇ s w) • Φ + +k (∇ • (ε S ∇V ) • Φ + (ε S ∇V ) • Φ) + +k 2 eΞ n,k • Φ -(ε S ∇V • Φ) • Φ + (9)
with the associated boundary conditions

w out n,k = w in n,k on S R , S T V n,k = 0 on S b V n,k = -Zq n,k on S t D • n = 0 on S l (10)
In these equations k = Φk where Φ = [cos θ , sin θ , 0] T represents the direction angles into the reciprocal 2D lattice domain and

Ξ n,k = 1/2 w n,k • Φ T + Φ • w T n,k
is the symmetric dyadic tensor or the dyadic product of the displacement w and direction vector Φ. S R are the interfaces of the cells. In the electrical boundary conditions, q 0 is given by

q 0 n,k = S t (-e∇ s w + ik∇eΞ + ε S (∇V + ikV • Φ)) • ndS (11)

Numerical computation of the Blochs waves

The numerical implementation is obtained by using a standard FE method to discretize the piezoelastodynamic equations formula. The assembled matrix equation is given by

K (Z) + λ L (θ , Z) -λ 2 H (θ , Z) -ω 2 M u (θ , k) = 0 (12)
where λ = ik; M and K are the standard symmetric semidefinite mass and stiffness matrices (the mass matrix is semidefinite because elastostatic equation is condensed into the equation), L is a skew-symmetric matrix; and H is a symmetric semidefinite positive matrix. u is the generalized eigenvector defined on all degrees of freedom of the used FE model. When k and f are fixed and Z does not depend on ω, the system (12) is a linear eigenvalue problem allowing us to compute the dispersion functions ω 2 and the associated Bloch eigenvector u n . This approach has been widely used for developing homogenization techniques and spectral asymptotic analysis. Nevertheless, these approaches have been only developed for undamped or lightly damped mechanical systems. In these cases, most of the previously published works present techniques based on the mesh of a real k-space following the boundary of the first Brillouin zone for obtaining the corresponding dispersion curves and the associated Floquet vectors. For undamped system, only propagative or evanescent waves exist corresponding to a family of eigensolutions purely real or imaginary. Discrimination between each class of waves is easy. If a highly damped system (K, L, and H are complex frequency dependent) and a frequencydependent electrical shunt impedance are considered, the obtained eigenvalue problem is not quadratic and specific numerical methodology has to be implemented. Furthermore, evanescent parts of propagating waves appear as the imaginary part of ω 2 . It then becomes much more difficult to distinguish the propagative and evanescent waves as all solution appear complex. Another much more suitable possibility for computing damped system is to consider the following generalized eigenvalue problem

K (Z) + λ L (θ , Z) -λ 2 H (θ , Z) -ω 2 M u (θ , ω) = 0 (13)
In this problem, the angular frequency ω is a real parameter corresponding to the harmonic frequency. Wave numbers and Floquet vectors are then computed. As L is skew-symmetric, the obtained eigenvalues are quadruple collapsing into real or imaginary pairs when all matrices are real (i.e. for an undamped system). In this case, a real pair of eigenvalues corresponds to evanescent modes oriented in two opposite directions on the kspace and imaginary values to two traveling waves propagating in opposite direction. As previously mentioned, the real part of k = Φk vector is restricted to stand inside the first Brillouin zone. In the quadratic eigenvalue problem (13), nothing restricts computation to only find eigenvalues satisfying this condition. For direction vector Φ orthogonal to the lattice facelets, the periodical conditions expressed for 1D wave guide are still valid. Thus, for undamped systems, all obtained eigenvalues are periodically distributed in the k-space along its principal directions.

Group velocity and electric power estimation

In order to correctly decribe and qualify the properties of this distributed control system some new quantities must be in-troduced. The first one, which is considered for describing the capability of the metacomposite for transmitting structural energy, is the wave group velocities. They indicate how energy is transported into the considered system and allow to distinguish the propagative and evanescent waves. If a Bloch eigensolution is considered, the associated group velocity vector is given by:

C g n = ∇ k ω = S e tot = I E tot ( 14
)
where is the spatial and time average, respectively, on one cell and one period of time; S is the density of energy flow; I is the mean intensity; and e tot and E tot are the total piezomechanical energy and its time average on a period. In this problem, we only consider mechanical energy transportation as the electrostatic coupling is decentralized on each cell and cannot induce spatial energy transportation.

The intensity vector I is expressed as

I = - ω 2 real C E (ε + ikΞ) w * dΩ . (15)
As the spatiotemporal average of the system Lagrangian is null, the total energy average is approximated by only computing the kinetic energy average

E tot = Ω ρww * dV. (16)
The group velocity vectors C g n are computed for all wave numbers at each frequency. In order to focus our analysis on only flexural modes (S and SH ones), we introduce an indicator allowing to select them by computing the ratio of kinetic energy average on out-of-plane displacement as

Ind z = T z T (17)
T z being the time-averaged kinetic energy in the out-of-plane direction, and T the total kinetic energy. The characterization of the absorbing properties of the control system rely on the quantity of energy dissipated within the external circuit at each cycle. The damped electric power is expressed as

P el = 1 2 real V 0 iωq 0 * (18)

RESULTS AND DISCUSSION

In this section some theoretical and experimental results are given in order to show the effectiveness of the proposed distributed control technique. First, from the theoretical point of view a comparison between the dispersion curves of the waveguide will show how the dynamic of the waveguide is influenced by the external electric circuits. In order to prove the consistency of the results for the real bounded system a full-field measurement will further illustrate the main features of the control system, characterizing the reflective and absorbing properties of the considered metacomposite.

Analysis of the dispersion properties

The propagative properties of this smart system will be analyzed comparing the dispersion diagrams, the electric power and the group velocity of the metacomposite. Three different configuration will be analyzed: open circuit, negative capacitance with reflecting (case R) and absorbing (case A) capabilities. The reflecting properties are increased setting the external circuit as follows: the capacitance have a value of -62 nF and a small resistance of 40 Ω. The damping effect is obtained by setting the external capacitance at -62 nF with a bigger resistance of 400 Ω. Figure 3 shows the dispersion curves using the propagative parts of the wave numbers k x along (Ox) axis. This figure shows the ability of the procedure to catch all Blochs solutions including evanescent ones that generally do not appear in literature because of their imaginary character. In the context of structural dynamics, including damping effects is mandatory for real-life applications and all wave numbers become complex. By using a suitable wave filter based on the group velocity, it is possible to retain the only propagative solutions responsible for energy transportation. It then becomes easy to observe branches similar to standard S 0 , A 0 , A 1 , and SH waves. The corresponding group velocities along (Ox) are presented in the same figure. It can be immediately observed that the use of the shunt impedance leads to a larger decrease of the group velocity of the A 0 modes. The bending waves also propagate energy with a very slow velocity and can be considered as evanescent. Flexural energy is, also, only transported by the A 1 mode after the cutting frequency. The propagative part of the controlled wave is strongly modified, as illustrated in Figure 3 (red line versus black line): from two complex conjugated solutions, the A 0 wave changes after control to four solutions (two complex conjugates and their opposites). It can also be observed that a nondispersive symmetric wave is also affected by the control, even if it was not targeted. Another strategy for optimizing the adaptive metacomposite consists in focusing on the damped power flow inside the electric shunts. In this case, the objective is to improve the absorption capability of the smart structure. The propagative parts of the wave number k x along (Ox) axis appear as shown in Figure 3 along with the corresponding group velocities along (Ox) and the electrically damped power. The first observation is that the optimization of the shunt impedance for improving the absorption characteristics of the system induces modifications of the group velocities of the controlled waves, while the propagative part of the wave numbers remain unchanged. This can be explained by a large improvement of the ratio between the real and imaginary parts of the wave numbers, which physically corresponds to the forcing of propagating effects to increase damping effects: energy can propagate inside the periodically distributed set of active cells for allowing electrical energy conversion, the dissipated power is largely increased when the shunt is connected to the patch (blue line).

Kinetic energy distribution

The modification of the internal dynamic of the waveguide can be proved through the estimation of the kinetic energy distribution calculated integrating the measured velocity field over the total plate's area. In particular, observing Figure 2 we can identify three different region that ideally separate the two ends of the plate and the active interface. Figure 4 presents the kinetic energy field over the whole plate's surface. The top shows a comparison of the kinetic energy distribution among three different circuits' configuration at the tuning frequency of 2500 Hz. On the left, (control OFF) the kinetic energy is evenly spread over the entire plate. In the middle, the system's response is slightly modified using a small resistance R s (case R). On the contrary, as a result of increasing the value of the resistance R s (case A), the kinetic energy distribution of the plate is clearly modified; the waves are confined in the portion of the plate below the active interface. Considering higher frequency values, for instance 3000 Hz, the behavior of the system is radically modified. In case R, the vibratory energy of the plate is confined in the region Ω 1 . In case A, the reflective behavior is no longer present, and the driving control mechanism is a pure dissipative effect that makes the kinetic energy field weaker.

CONCLUSIONS

This paper describes the analysis of wave propagation in a periodic two-dimensional waveguide with shunted piezoelectric patches. The coupling between the plate and this smart interface allows the modification of its internal dynamic. The wave propagation characteristics of the piezoelectric waveguide are first predicted through the application of the Floquet-Bloch theorem. The dispersion analysis highlights the generation of a control band where the attenuation or the reflection of the waveguide is increased. In particular selecting appropriately the external circuit's parameter is possible to reduce the group velocity of the incoming waves, that is, the active interface prevent the wave to propagate. By increasing the resistance R s the waveguide expe- riences an increase of the damping over a broad frequency band. Experimental evidence of the internal controlling behavior of the waveguide is then provided through measurements of the out-ofplane velocity of the plate. The experimental results effectively confirm the numerical predictions.

FIGURE 1 .

 1 FIGURE 1. Metacomposite structure: 2D waveguide and active interface (dashed rectangle).

FIGURE 2 .

 2 FIGURE 2. Layout of the measurement system.

FIGURE 3 .

 3 FIGURE 3. Comparison of the dispersion relation (top), dissipated electric power (middle) and group velocity (bottom) of the smart waveguide. Three cases have been considered: OC (black markers), RC neg case R (red markers) and RC neg case A (blue markers).

2 FIGURE 4 .

 24 FIGURE 4. Plate's kinetic energy field: OC (left), RC neg case R (middle) and RC neg case A (right). Two different frequency values have been considered 2500 Hz (top) and 3000 Hz (bottom).

Table 1

 1 and are justified by the following assumptions: The plate's dis-

	Plate Length	2100	mm
	Height	1050	mm
	Thickness	3	mm
	Mass density	2700	kg/m 3
	Young's modulus 70 • 10 9 N/m 2
	Poisson's ratio	0.33	-
	Piezo Length	50	mm
	Height	50	mm
	Thickness	0.5	mm
	Mass density	7650	kg/m 3
	Poisson's ratio	0.31	-
	Dielectic loss	< 0.05 %
	Coupling factor	0.31	-

TABLE 1 .

 1 Geometry and physical properties of the system.

ACKNOWLEDGMENT

This study is a collaborative effort supported by the French Research Agency under grant number NT09-617542 and is supported by the Labex Action.