Pedro Campinho

Martin Behrndt

Jonas Ranft

Thomas Risler

Nicolas Minc

Carl-Philipp Heisenberg
email: heisenberg@ist.ac.at

Tension-oriented cell divisions limit anisotropic tissue tension in epithelial spreading during zebrafish epiboly

Epithelial spreading is a common and fundamental aspect of various developmental and disease-related processes such as epithelial closure and wound healing. A key challenge for epithelial tissues undergoing spreading is to increase their surface area without disrupting epithelial integrity. Here we show that orienting cell divisions by tension constitutes an efficient mechanism by which the Enveloping Cell Layer (EVL) releases anisotropic tension while undergoing spreading during zebrafish epiboly. The control of EVL cell-division orientation by tension involves cell elongation and requires myosin II activity to align the mitotic spindle with the main tension axis. We also found that in the absence of tension-oriented cell divisions and in the presence of increased tissue tension, EVL cells undergo ectopic fusions, suggesting that the reduction of tension anisotropy by oriented cell divisions is required to prevent EVL cells from fusing. We conclude that cell-division orientation by tension constitutes a key mechanism for limiting tension anisotropy and thus promoting tissue spreading during EVL epiboly.

Introduction

In zebrafish gastrulation, the EVL is formed as a squamous epithelial cell layer at the animal pole of the embryo and spreads over the entire spherical yolk cell during the course of epiboly (4 -10 hours post fertilization/hpf), thereby rapidly increasing its surface area [START_REF] Lepage | Zebrafish epiboly: mechanics and mechanisms[END_REF] . Actomyosin contraction within the yolk syncytial layer (YSL), to which the EVL is connected at its margin, is thought to drive EVL epiboly movements by pulling on the EVL margin in direction of the vegetal pole [START_REF] Behrndt | Forces driving epithelial spreading in zebrafish gastrulation[END_REF][START_REF] Cheng | Organization and function of microfilaments during late epiboly in zebrafish embryos[END_REF][START_REF] Köppen | Coordinated cell-shape changes control epithelial movement in zebrafish and Drosophila[END_REF] . However, the mechanisms by which the epibolizing EVL rapidly increases its surface area and, at the same time, maintains its epithelial integrity during zebrafish epiboly remain unclear.

Studies in another teleost fish, Fundulus heteroclitus, have suggested that the EVL increases its surface area during epiboly by both passive cell spreading in response to pulling forces from the YSL, and active cellular rearrangement adjusting the shape of the EVL to the spherical geometry of the yolk cell on which it spreads [START_REF] Trinkaus | Mechanism of Fundulus epiboly-a current view[END_REF] .

Consistent with the idea of EVL cells passively spreading as a result of the YSL pulling on the EVL margin are observations in zebrafish that marginal EVL cells become increasingly elongated along the axis of tissue spreading and in Fundulus of tension building up within the plane of the EVL during the course of epiboly [START_REF] Köppen | Coordinated cell-shape changes control epithelial movement in zebrafish and Drosophila[END_REF][START_REF] Trinkaus | Mechanism of Fundulus epiboly-a current view[END_REF] .

Cellular rearrangements have been noted to particularly occur at the margin of the EVL in both zebrafish and Fundulus with individual cells constricting at their leading edge and eventually being displaced from the EVL margin [START_REF] Köppen | Coordinated cell-shape changes control epithelial movement in zebrafish and Drosophila[END_REF][START_REF] Keller | Rearrangement of enveloping layer cells without disruption of the epithelial permeability barrier as a factor in Fundulus epiboly[END_REF] . However, whether EVL cell spreading and rearrangement constitute the sole mechanisms mediating EVL surface expansion during fish epiboly, or whether other mechanisms might also be involved, is still unclear.

Cell divisions have profound effects on epithelial tissue morphogenesis [START_REF] Guillot | Mechanics of epithelial tissue homeostasis and morphogenesis[END_REF] , and it is thus conceivable that they also play a critical role in EVL epiboly movements. In zebrafish, EVL cells have been shown to undergo divisions within the plane of the tissue during epiboly [START_REF] Kimmel | Origin and organization of the zebrafish fate map[END_REF] . Moreover, planar tension within the EVL has been speculated to function in maintaining EVL cell lineage by keeping the mitotic spindle of dividing cells oriented in the plane of the cell sheet [START_REF] Kimmel | Origin and organization of the zebrafish fate map[END_REF][START_REF] Kane | Life at the edge: epiboly and involution in the zebrafish[END_REF] . Here, we investigate how cell divisions contribute to EVL epiboly movements, and how tension within the EVL relates to the EVL cell-division rate and/or orientation.

Results

Anisotropic tissue tension controls cell division orientation within the EVL

To obtain insight into the cellular processes underlying EVL tissue spreading during zebrafish epiboly, we first asked how the increase of total surface area of the EVL during epiboly correlates with changes in surface area, height and volume of individual EVL cells and with their division pattern (Fig. 1; Supplementary Video S1).

We found that EVL tissue spreading is accompanied by pronounced EVL cell flattening, as recognized by an increase in apical surface area and concomitant decrease in cell height along the apical-basal axis of individual EVL cells (Fig. 1b; Supplementary Fig. S1a,b). In contrast, EVL cell volume remained largely constant during flattening and was cut in half once these cells underwent division (Fig. 1c; Supplementary Fig. S1c). EVL cells exclusively divided within the plane of the epithelium with higher frequency at early stages of EVL epiboly and at central/animal compared to marginal regions of the EVL (Fig. 1d). Moreover, our observation that the total number of EVL cells more than doubles during the early stages of epiboly (Fig. 1e), suggests that every EVL cell on average divides at least once within this period. Taken together, these observations suggest that the EVL tissue spreads by cell flattening without significant increase in tissue volume.

Oriented cell divisions have been hypothesized to facilitate tissue spreading [START_REF] Ranft | Fluidization of tissues by cell division and apoptosis[END_REF] . To investigate their role in EVL spreading, we first analyzed division orientation of individual EVL cells during the course of epiboly. We found that EVL cells preferentially divided along the axis of EVL tissue spreading (animal-vegetal/AV axis of the embryo) during all stages of epiboly (Fig. 1f; Supplementary Video S1).

This stereotypical division orientation (SDO) of EVL cells is consistent with a potential role of oriented cell divisions in EVL epiboly progression.

We next asked how SDO of EVL cells is controlled during EVL epiboly. Mechanical tension has been suggested to represent a powerful mechanism for orienting the divisions of isolated cells in culture [START_REF] Fink | External forces control mitotic spindle positioning[END_REF] . Moreover, studies in Fundulus heteroclitus have suggested that the epibolizing EVL is under tension both around the circumference and along the AV axis of the embryo [START_REF] Trinkaus | Mechanism of Fundulus epiboly-a current view[END_REF][START_REF] Trinkaus | A study of the mechanism of epiboly in the egg of Fundulus heteroclitus[END_REF] . We therefore hypothesized that anisotropic tension distribution within the EVL may control SDO of EVL cells.

To test this hypothesis, we used a laser-cutting device to map tension within the EVL during the course of epiboly [START_REF] Behrndt | Forces driving epithelial spreading in zebrafish gastrulation[END_REF][START_REF] Mayer | Anisotropies in cortical tension reveal the physical basis of polarizing cortical flows[END_REF] . To detect tension along the AV axis of the EVL, we cut the apical actomyosin cortex of the EVL along a 100-µm-long line parallel to the EVL margin and determined the recoil velocity of the cortex as a readout for AV tension (Fig. 2a). Similarly, to detect EVL tissue tension perpendicular to the AV axis, we determined the recoil velocity of the actomyosin cortex in cuts oriented perpendicular to the EVL margin (Fig. 2a; Supplementary Video S2). We found that tension along the AV axis of the EVL is higher than along the circumference of the embryo, and that this anisotropic tension distribution is particularly pronounced at mid-to late-gastrulation stages (Fig. 2b). This global correlation between EVL tissue tension anisotropy and SDO of EVL cells is consistent with a function of tissue tension in orienting EVL cell divisions.

To determine whether there is a causal link between EVL tissue tension and celldivision orientation, we locally induced anisotropic tissue tension within the EVL and analyzed resultant changes in cell-division orientation. We locally induced anisotropic tension within the EVL by simultaneously ablating two small groups of EVL cells (2-3 cells each) positioned close to each other (≈ 120 µm), which led to the extrusion of the ablated cells and, consequently, considerable stretching of the EVL tissue between the two ablation sites (Fig. 2c; Supplementary Video S3). We then analyzed changes in spindle orientation of cells that were located between the two ablation-sites and had their metaphase spindle axis oriented perpendicular to the axis of induced tension before tension was applied (Fig. 2c). We found that in these cells the mitotic spindle preferentially re-oriented along the main axis of tension before cytokinesis (Fig. 2c), demonstrating that tissue tension anisotropy influences the cell-division orientation of EVL cells.

The orientation of EVL cell divisions by tension involves cell elongation and requires myosin II activity

We next asked how anisotropic tissue tension controls EVL cell-division orientation.

Many cell types have been shown to preferentially orient their mitotic spindle along their longest axis [START_REF] Minc | Influence of cell geometry on division-plane positioning[END_REF][START_REF] Gibson | Control of the mitotic cleavage plane by local epithelial topology[END_REF][START_REF] Minc | Predicting division plane position and orientation[END_REF][START_REF] Théry | Experimental and theoretical study of mitotic spindle orientation[END_REF] . Notably, EVL cells display flat two-dimensional defined geometries and do not round up at their apical side during mitosis (Supplementary Video S4), and thus cell-shape anisotropies could influence metaphase spindle orientation through putative length-dependent astral microtubule forces [START_REF] Minc | Influence of cell geometry on division-plane positioning[END_REF] .

Considering that EVL cells appear preferentially elongated along the main axis of tension shortly before undergoing cytokinesis (AV axis; Fig. 3a), we therefore hypothesized that anisotropic EVL tissue tension may control SDO of EVL cells by elongating cells along the main axis of tension. To assess the potential role of EVL cell elongation on division orientation, we made use of a computational model that predicts the preferred division orientation based on cell shape [START_REF] Minc | Influence of cell geometry on division-plane positioning[END_REF] . We found that EVL cell elongation is a reliable predictor for division orientation (Figs. 3b,c), supporting our hypothesis that EVL tissue tension controls cell-division orientation by cell elongation.

In addition to cell shape, external forces have also been proposed to control spindle orientation in a myosin II-dependent manner [START_REF] Fink | External forces control mitotic spindle positioning[END_REF] . To determine whether myosin II is required for tension-controlled EVL cell-division orientation, we analyzed division orientation in embryos with normal and reduced myosin II activity. To reduce myosin II activity, we exposed embryos to the myosin II inhibitor Blebbistatin. We then compared the effect of cell elongation on division orientation between control and Blebbistatin-treated cells and found that EVL cell shape less accurately predicted spindle orientation in cells with reduced myosin II activity (Fig. 3b,c).

Moreover, the failure of the mitotic spindle to align with the longest axis of the cell and thus the main axis of tension in Blebbistatin-treated embryos was often accompanied by increased spindle fluctuations (Fig. 3d; Supplementary Video S5).

The effect of myosin II inhibition on spindle orientation could in principle be due to defects in cell elongation and/or spindle alignment with the longest cell axis. To distinguish between these two possibilities, we compared EVL cell elongation between control and Blebbistatin-treated embryos and analyzed how accurately our computational model can predict experimental spindle orientation on the basis of these measured cell shapes. Consistent with a critical function of actomyosin contraction in cell shape changes [START_REF] Salbreux | Actin cortex mechanics and cellular morphogenesis[END_REF] , we found that EVL cell elongation was reduced in Blebbistatin-treated embryos (Fig. 3e), suggesting that myosin II affects spindle orientation by cell elongation. However, even when taking these differences in cell shape into account, myosin II inhibition still increased the deviation of experimental spindle orientation in EVL cells from model predictions made on the basis of cell shape only (Fig. 3f). This comparison of model predictions with experimental observations suggests that myosin II, in addition to controlling EVL cell elongation, might be required to align the mitotic spindle with the longest cell axis.

Tension-oriented EVL cell divisions can reduce tissue tension anisotropy and facilitate tissue spreading

Having shown that anisotropic tissue tension can reorient cell divisions along the main axis of tension, questions arise as to the function of these tension-oriented cell divisions for EVL epiboly movements. It has been suggested that oriented cell divisions facilitate tissue spreading by decreasing tissue tension along the axis of division [START_REF] Ranft | Fluidization of tissues by cell division and apoptosis[END_REF] . We therefore first asked whether individual cell divisions affect EVL tissue tension. On the characteristic timescale of a single cell-division event, stresses due to individual cell deformations may have relaxed but deformations of the cellular junctional network imply persistent elastic stresses. When modeling the forces exerted by a dividing cell as a point force dipole in a continuous elastic medium, tensions and elastic deformations show that a cell division decreases elastic stresses in the surrounding tissue along the axis of division (Fig. 5a; Supplementary Theory). To test this prediction for the EVL, we turned to our minimal tension assay where we locally induce anisotropic tension within the EVL by ablating two small groups of EVL cells in close distance from each other (Fig. 4a, left). We first compared the degree of EVL tension anisotropy in cases where a cell division oriented along the axis of induced tension occurred between the ablation sites, to cases where no such cell division was observed. We found that tension anisotropy was significantly diminished in the presence of tension-oriented cell divisions (Fig. 4a right; Supplementary Video S6), indicating that EVL cell divisions reduce tissue tension along their axis of division. We then determined whether the reduction of tissue tension along the axis of cell division also facilitates EVL tissue spreading by comparing the degree of tissue spreading in cases where a tensionoriented cell division occurred between the ablation sites, to cases without such division. Theoretically, the displacement field following a cell division shows maximum displacement along the division axis (Supplementary Theory). To monitor tissue spreading in our ectopic tension assay, we analyzed the degree by which the EVL extends along the ectopic tension axis between the two ablation-sites (Fig. 4b, left). We found that EVL spreading was significantly increased in the presence of tension-oriented cell divisions (Fig. 4b, right; Supplementary Video S7), indicating that the reduction of tissue tension along the axis of cell division facilitates tissue spreading within the EVL.

Tension-oriented EVL cell divisions prevent ectopic cell fusions

To investigate the consequences of these findings made at the level of individual or small groups of cells for overall EVL spreading, we developed a model of the EVL that is based on our observations described so far, namely that (i) external tensions can bias the orientation of the EVL cell-division axis, and that (ii) cell divisions release tissue tension and facilitate tissue spreading along the axis of division. We describe the EVL tissue as a continuous material on length scales large compared to that of individual cells, with a rheology that incorporates the effect of tensionoriented cell divisions in an effective shear viscosity (Fig. 5a; Supplementary Theory). With these ingredients, our model predicts an anisotropic tension profile along the AV axis of the EVL and, as a result of this, a global pattern of cell-division orientation within the EVL (Supplementary Fig. S2). This prediction fits with the experimental observations reported above (Figs. 2b and1f, respectively). Furthermore, our model predicts a decrease in tissue flow and an increase in tissue tension anisotropy when the mechanism of tension-oriented cell divisions is defective (Fig. 5b; Supplementary Theory and Fig. S2). To address these predictions, we analyzed EVL epiboly progression in embryos, in which we either reduced cell divisions during gastrulation by exposing them to the cell-division inhibitors Aphidicolin and Hydroxyurea [START_REF] Lyons | erbb3 and erbb2 are essential for Schwann cell migration and myelination in zebrafish[END_REF] , or interfered with cell-division orientation by injecting them with an α-Dynein antibody [START_REF] Quesada-Hernández | Stereotypical cell division orientation controls neural rod midline formation in zebrafish[END_REF] (Supplementary Fig. S3a). Consistent with the model predictions, embryos with strongly reduced cell divisions showed slightly reduced EVL epiboly movements at late stages of epiboly (Fig. 5c). However, EVL tissue tension and flow appeared largely normal in embryos with no preferred cell-division orientation (Fig. 5d; Supplementary Fig. S3b), contrary to our model assumptions that orienting cell divisions by tension constitutes the main adaptive mechanism for limiting tension anisotropy within the spreading EVL. Instead, we found EVL cells fusing within the plane of the epithelium when cell divisions were reduced or misoriented (Fig. 6a,b; Supplementary Video S8). These fusions occurred with no preferred orientation, and fused cells remained integrated within the EVL and expanded their apical area at similar rates as their non-fused counterparts (Fig. 6c,d). Moreover, cell fusions were not the result of incomplete cell divisions, as fusions usually did not occur between sister cells (Supplementary Fig. S4).

Importantly, EVL cell fusions were accompanied by a rapid extension of the collapsing junction directly prior to the fusion (Fig. 6e), indicative of tension release along this junction. These observations led us to hypothesize that EVL cell fusions might be caused by augmented tissue tension in embryos with reduced or misoriented cell divisions, and that they might lead to a reduction of tissue tension anisotropy, partially compensating for the lack of tension-oriented cell divisions in this process. To experimentally address this hypothesis, we sought to subject the EVL to augmented tissue tension by deforming the embryos into a cylindrical shape [START_REF] Behrndt | Forces driving epithelial spreading in zebrafish gastrulation[END_REF] and thereby expanding their surface area, assuming that the embryo volume is conserved (Fig. 7a). Consistent with our hypothesis of augmented tissue tension inducing cell fusions, we found EVL cell fusions to be strongly increased upon surface expansion (Fig. 7b,c; Supplementary Video S9). Moreover, our previous observation that EVL epiboly movements appear largely unaffected in cylindrical embryos [START_REF] Behrndt | Forces driving epithelial spreading in zebrafish gastrulation[END_REF] is compatible with the assumption that cell fusions, similar to tensionoriented cell divisions, promote EVL epiboly movements by releasing ectopic tissue tension.

Yet, questions remain as to the mechanisms by which tissue tension induces EVL cell fusions and, vice versa, EVL cell fusions release tissue tension. While difficulties in predicting and/or inducing fusions of EVL cells did not allow us to directly address the reciprocal relationship between tension and cell fusion using our ectopic tension assay, our findings so far support the notion that globally elevated tissue tension can induce cell fusions, and that cell fusions can locally release tissue tension. Whether cell fusions can also reduce global tissue tension anisotropy is not yet entirely clear.

However, considering that collapsing borders oriented along the main axis of tissue tension would be expected to release more tension than collapsing borders with other orientations, the combined effect of all EVL cell fusions without preferred orientation will likely reduce overall EVL tissue tension anisotropy.

Discussion

Our study identifies the orientation of cell division by external tension as an efficient mechanism to release anisotropic tissue tension during EVL spreading. The degree of anisotropic tissue tension thus depends on the ability of oriented cell divisions to reduce tension anisotropy and, vice versa, the stereotypical orientation of cell divisions depends on the degree of anisotropic tension within the tissue. We expect that these reciprocal dependencies balance each other so that the EVL tissue generally displays a low degree of tension anisotropy, facilitating its spreading.

Consistent with this, we observe low tension anisotropy at early stages of EVL spreading when the rate of cell divisions is high, and high tension anisotropy at later stages of epiboly when the rate of cell divisions is comparatively low (Figs. 1d and2b).

Tension has been proposed to control cell-division orientation through different mechanisms. One prime effector mechanism is cell elongation, as cell-shape anisotropies are thought to influence metaphase spindle orientation through putative length-dependent astral microtubule forces [START_REF] Minc | Influence of cell geometry on division-plane positioning[END_REF] . Our finding that within the epibolizing EVL, spindle orientation, cell elongation and the main axis of tissue tension are all aligned to each other, suggests that tension controls EVL cell division orientation by cell elongation. However, tension has also been shown to orient the mitotic spindle in a myosin II-dependent manner independently of its effect on cell shape [START_REF] Fink | External forces control mitotic spindle positioning[END_REF] . While we have no direct evidence for a tension mediated polarization of myosin II to orient the division plane within the EVL, our observation that myosin II might be required for spindle orientation in EVL cells in addition to its function in cell elongation, points to similar mechanisms for cell division orientation within the EVL. Yet, to prove such mechanism, further experiments will be needed to show how tension modulates myosin II activity and whether myosin II indeed mediates the function of tension in orienting the mitotic spindle in EVL cells.

Whereas there is increasing evidence for tissue tension controlling cell division orientation [START_REF] Minc | Predicting division plane position and orientation[END_REF][START_REF] Mao | Differential proliferation rates generate patterns of mechanical tension that orient tissue growth[END_REF][START_REF] Legoff | A global pattern of mechanical stress polarizes cell divisions and cell shape in the growing Drosophila wing disc[END_REF] , considerably less is known about the effect of tension-oriented cell divisions on tissue tension anisotropy. Importantly, the effect of oriented cell divisions on tension anisotropy we observed is not a consequence of tissue growth along the division axis, as EVL cells do not increase their volume between divisions (Fig. 1c). Rather, tension-oriented cell divisions lead to cellular rearrangements releasing tension along this axis of division. Similarly, cell fusions are expected to release tissue tension, and thereby partially compensate for the lack of oriented cell divisions, by fusion-mediated cell rearrangements. However, as the degrees of cell rearrangements associated with division and fusion seem similar (Supplementary Videos S4 and S8) but the number of cell fusions compared to divisions is small (21 fusions vs. 576 divisions per embryo from sphere to 55 % epiboly stage; for details on how the total number of fusions and divisions per embryo was estimated see Supplementary Methods), cell fusions are unlikely to be sufficient to fully compensate for the function of oriented cell divisions in reducing tissue tension anisotropy. Observations in different teleost fish species of EVL cells undergoing junctional remodeling during epiboly [START_REF] Köppen | Coordinated cell-shape changes control epithelial movement in zebrafish and Drosophila[END_REF][START_REF] Keller | Rearrangement of enveloping layer cells without disruption of the epithelial permeability barrier as a factor in Fundulus epiboly[END_REF][START_REF] Kageyama | Cellular basis of epiboly of the enveloping layer in the embryo of the medaka, Oryzias latipes. II. Evidence for cell rearrangement[END_REF] , and in flies of anisotropic tension distribution within the spreading epidermis during dorsal closure being associated with cell elongation along the axis of main tension [START_REF] Kiehart | Multiple forces contribute to cell sheet morphogenesis for dorsal closure in Drosophila[END_REF] , point at junctional remodeling and cell elongation as additional mechanisms for tension release within the EVL. In agreement, we observe preferential junction disassembly within the EVL along the circumference of the embryo, which appears more pronounced when tensionoriented cell divisions are impaired (Supplementary Fig. S5). Thus, anisotropic tension release during EVL tissue spreading in epiboly relies on distinct cellular mechanisms that act in concert to limit tissue tension anisotropy and facilitate epiboly movements. S1). (c) Volume of individual EVL cells both before and after cell division as a function of time after sphere stage (4 hpf). Vertical lines (dashed) indicate cell divisions; plotted values, mean ± s.e.m. normalized to the average value at t = 120 min (volume120min = 6536 ± 600 µm 3 ); n, number of cells/time-point (for details see Supplementary Table S1). (d) Average percentage of EVL cells undergoing divisions as a function of their position along the animalvegetal (AV) axis for sequential stages of early epiboly; note that EVL cell divisions become very rare after 55 % epiboly stage [START_REF] Kane | Mitotic domains in the early embryo of the zebrafish[END_REF] (for 60 % epiboly stage < 0.26 % of all EVL cells, n = 10 embryos), and have therefore not been spatiotemporally analyzed.

For sphere and 30 % epiboly stage embryos (4 and 4.66 hpf, respectively), the EVL was subdivided along its AV axis into three bins (I, II and III), whereas for 55 % epiboly stage embryos (5.66 hpf), the EVL was subdivided into five bins (I, II, III, IV (f) Energy penalty values (arbitrary units) for individual EVL cells computed using the shape model in myosin II-inhibitor (Blebbistatin)-treated (blue squares, n = 32 divisions, 16 embryos) and control embryos (red circles, n = 36 divisions, 17 embryos) plotted over the angles between predicted and observed spindle axis. The energy penalty quantifies the deviation between the observed angle of the spindle axis from that predicted by the shape model taking differences in shape anisotropy between cells into account. For instance, the same angular deviation between observed and predicted spindle orientation would result in a lower energy penalty for cells with a small degree of shape anisotropy compared to cells with a high degree of anisotropy. Thus, if the deviation between the observed and predicted spindle orientations in Blebbistatin-treated embryos was due only to a lower degree of shape anisotropy in those cells (e), we would expect the energy penalty to level off with similar maximal values in control and Blebbistatin-treated conditions.

Instead, the energy penalties in these two conditions are significantly different (p = 0.0396), suggesting that the effects of Blebbistatin treatment on spindle orientation are not solely due to changes in cell-shape anisotropy. A.U., arbitrary units. Number

Figure- 1 :

 1 Figure-1: EVL cell-shape changes and divisions. (a) Exemplary images of the EVL in an embryo expressing GPI-RFP to outline EVL cells (upper panel) and respective segmented cell boundaries (lower panel) with cell divisions marked in yellow; t = 0 min corresponds to sphere stage (4 hpf). (b) Apical cell area and apical-basal cell height of individual EVL cells as a function of time after sphere stage (4 hpf); plotted values, mean ± s.e.m. normalized to the average values at t = 120 min (area120min = 838 ± 9 µm 2 , height120min = 9 ± 0.29 µm); n, number of cells/time-point (for details see Supplementary TableS1). (c) Volume of individual EVL cells both before and

  and V); plotted values, mean ± s.e.m. normalized to the number of EVL cells per bin; see e for number of analyzed cells and embryos; (e) The total number of EVL cells per stage was on average 496, 660 and 1072 for sphere (n = 12 embryos), 30 % (n = 10 embryos), and 55 % (n = 10 embryos) epiboly stage, respectively; plotted values, mean ± s.e.m. (calculated by using embryo numbers). (f) Rose diagram of the celldivision axes at cytokinesis (yellow; n = 524 divisions, 6 embryos) for EVL cells dividing during the course of gastrulation; p (division orientation) = 0.0067 (calculated by using division numbers). Number of independent experiments = 6 (area), 5 (height) (b), 5 (c), 1 (d,e), 6 (f).

Figure- 2 :

 2 Figure-2: EVL tissue tension and cell-division orientation. (a) Exemplary images of embryos at 65 % epiboly stage for UV laser cuts of the apical actomyosin cortex at the animal pole (blue) or perpendicular (red) and parallel (green) to the EVL margin in Tg(actb2:myl12.1-eGFP) embryos. Cuts were 100 µm long and placed 3 to 6 cell rows away from the EVL margin (for parallel and perpendicular cuts); scale bar, 20 µm. (b) Average initial recoil velocities for UV laser cuts throughout the course of epiboly; error bars, s.e.m.; n, number of cuts; note that only one cut/embryo was performed; p for perpendicular vs. parallel recoil velocities (40 -50 %) = 0.97; p (50 -60 %) = 0.11; p (60 -70 %) < 0.0001; p (70 -80 %) = 0.17 (c) Alignment of the cell-division axis with the axis of induced tension in Tg(actb2:myl12.1-mCherry) embryo at 40 % epiboly stage injected with tau-GFP mRNA to mark spindle microtubules. Tension was induced orthogonally to the initial axis of the spindle (yellow) by creating two constricting wounds in the EVL (red crosses). The resulting spindle alignment ('spindle axis') was determined by measuring the angle between the final spindle axis directly prior to cytokinesis and the induced tension axis. For controls, no wounds were induced and the endogenous rotation of the spindle from its initial axis was quantified in the same manner, i.e. with the initial spindle axis orthogonal to the control axis (Supplementary Methods). Histograms show the frequency distributions of the observed angles for both control and induced tension cases; p < 0.0001; n, number of divisions; note that only one division/embryo was analyzed; cell contour, white; scale bar, 20 µm. Number of independent experiments = 29 (b), 14 (c).

Figure- 3 :

 3 Figure-3: Effects of cell shape and myosin II on spindle orientation and positioning. (a) Rose diagram of the orientations of the longest cell axis of EVL cells undergoing division during the course of EVL epiboly (n = 514 cells, 6 embryos) 5 min prior to the onset of cytokinesis; p (longest axis) = 0.0369 (calculated by using cell numbers). (b) Alignment of the observed axis of cell-division orientation determined by the spindle axis (yellow) and the predicted axis of cell division (blue)

  of independent experiments = 6 (a), 13 (b-f).

Figure- 4 :

 4 Figure-4: Oriented cell divisions and EVL tissue tension. (a) Exemplary images and average initial recoil velocities for UV laser cuts of the apical actomyosin cortex perpendicular (blue) or parallel (orange) to the division axis in Tg(actb2:myl12.1-eGFP) embryos at 30 to 40 % epiboly stage after induction of tension either in the presence (bottom left; bar plot -right) or absence (top left; bar plot -left) of an EVL cell division (white cell contour) oriented along the axis of tension. Cuts (blue or orange lines) were 50 µm long; error bars, s.e.m.; n, number of cuts; note that only one cut/embryo was performed; p perpendicular recoil velocities (no division) vs. (division) = 0.0112; p perpendicular vs. parallel recoil velocities = 0.0028 (no division) and = 0.7789 (division); red crosses mark the ablation sites where wounds were induced; scale bar, 20 µm. (b) Exemplary images of the spreading displacement of an EVL cell (white cell contour) and average spreading displacement in Tg(actb2:myl12.1-eGFP) embryos at 30 -40 % epiboly stage after induction of tension either in the presence (bottom left; scatter plot -squares) or absence (top left; scatter plot -circles) of a cell division (white cell contour) oriented along the axis of tension. The spreading displacement corresponds to the change in distance between the edges of the ablated cells (green lines) before ablation compared to 252 sec after ablation; error bars, s.e.m.; n, number of embryos; note that only one measurement/embryo was performed; p = 0.0165; cell/wound contours, blue; red crosses mark the ablation sites where wounds were induced; scale bar, 20 µm. Number of independent experiments = 16 (a), 8 (b).

Figure- 5 : 5 (

 55 Figure-5: Oriented cell divisions and EVL epiboly progression. (a) Illustration of the main ingredients of the theoretical model proposed to describe the role of oriented cell divisions for EVL tension and spreading during epiboly (Supplementary Theory). A single cell division redistributes the stresses in the tissue on the timescale of the division (left) and is itself oriented by local stresses

Figure- 6 :

 6 Figure-6: EVL cell fusions in embryos with reduced or misoriented cell divisions. (a) EVL cell fusions binned from sphere -30 % epiboly and from 30 -55 % epiboly stage, plotted as mean ± s.e.m. in cell division inhibitor-treated (n = 6 embryos) or α-Dynein antibody-injected embryos (n = 4 embryos); control embryos for the inhibitor and Dynein antibody experiments were incubated in 1 % DMSO (n = 5 embryos) or injected with the antibody supernatant/ascites (n = 4 embryos), respectively; error bars, s.e.m. (b) Time course of an exemplary EVL cell fusion event (arrowheads) in a cell division inhibitor-treated embryo from sphere stage (t = 0 min) onwards. Cell membrane and spindle microtubules were outlined by GPI-RFP and Tau-GFP, respectively; scale bar, 20 µm. (c) Histogram showing the frequency distribution of the orientation of fusing EVL cell-cell junctions; n = 21

Figure- 7 :

 7 Figure-7: EVL cell fusions in cylindrically deformed embryos. (a) Low magnification of an exemplary Tg(actb2:GFP-utrCH) cylindrical embryo at sphere (t = 0 min) and 70 % epiboly stage (t = 211 min). Arrowheads point at multinucleated EVL cells resulting from cell fusions; scale bar, 100 µm. (b) EVL cell fusions binned from sphere -30 % and from 30 -55 % epiboly stage, plotted as mean ± s.e.m. in normal (n = 6 embryos) and cylindrically deformed embryos (n = 4 embryos). (c) Time course of an exemplary EVL cell fusion event (arrowheads) in a cylindrical embryo from sphere stage (t = 0 min) onwards. Arrows point at cell divisions, of which one gave rise to a daughter cell subsequently undergoing fusion. Cell membrane and nuclei were marked by GPI-RFP and H2A-Cherry, respectively; scale bar, 20 µm. Number of independent experiments = 10

Acknowledgements

We are grateful to Benoit Aiguy, Suzanne Eaton, Stephan Grill, Robert Hauschild and Michael Sixt for advice, and the imaging and zebrafish facilities of the IST Austria and MPI-CBG for continuous help. We are particularly thankful to Jean-Francois Joanny for discussions regarding the theory part of this work, and Buzz Baum for sharing data prior to publication. This work was supported by the IST Austria, MPI-CBG and a grant from the Fonds zur Förderung der wissenschaftlichen Forschung (FWF) (I930-B20) to C.-P.H.


N Eguémann

L Giger

M Roux

C Dransfeld

F Thiébaud

D Perreux

COMPRESSION MOULDING OF COMPLEX PARTS FOR THE AEROSPACE WITH DISCONTINUOUS NOVEL AND RECYCLED THERMOPLASTIC COMPOSITE MATERIALS

Keywords: discontinuous, cradle to cradle, thermoplastic, compression moulding, complex

General Introduction

Major structures of modern aircrafts such as the long range types from Boeing: 787 [START_REF] Marsh | Boeing's 787: trials, tribulations, and restoring the dream[END_REF] and from Airbus: the A350XWB [START_REF] Marsh | Airbus A350 XWB update[END_REF], are mostly manufactured from carbon-fibre-reinforced polymers (CFRP). In these CFRP primary structures, constituted of skins and reinforcing profiles, hundreds of complex metallic load introduction elements such as brackets, hinges or fittings are attached. Those numerous components are actually remaining metallic, but complex metallic load introduction elements have several drawbacks in aerospace applications. Aluminium parts integrated within aircraft or rotorcraft CFRP airframes have to be mounted with several cost-intensive protections against galvanic corrosion, as shown in Figure 1.

The use of steel for this kind of application adds an increasing weight penalty, while the use of titanium results in high production costs [START_REF] Eguémann | Manufacturing and Recycling of Complex Thermoplastic Parts for Aerospace Applications[END_REF] and the need for a secured supply-chain [START_REF] Seiner | The Need for a Secure Titanium Supply-Chain[END_REF].

To best comply with the latest EU regulations, such as the Waste Framework Directive [START_REF]European Bioplastics, Fact Sheet, Mechanical Recycling, Waste framework Directive[END_REF] and the Directive on End Of Life Vehicles [START_REF] Eu | End of Life Vehicle Directive[END_REF], thermoplastic CFRP (TP-CFRP) will be preferred to thermosets CFRP (TS-CFRP). TS-CFRP, due to their cross-linked matrix, cannot be moulded again in comparison to TP-CFRP that can be easily remelted [START_REF] Pickering | Recycling technologies for thermoset composite materials -current status[END_REF]. A full life cycle assessment performed on a complete helicopter door also showed the global superiority of TP-CFRP over TS-CFRP [START_REF] Parlevliet | EcoDesigned Thermoplastic Composite Helicopter Airframe Structures[END_REF].

In order to use TP-CFRP for this specific type of complex application, Toray Industries, Inc. conducted early research that, in 1990, led to a patent [START_REF] Kimoto | Thermoplastic composite plate material and products molded from the same[END_REF] clearly describing how strip pieces (chips) made of chopped thermoplastic pre-impregnated unidirectional fibres (UD tapes) are distributed in a plane parallel to the surface cavity of a tool, heated and consolidated under pressure. The length of the chips is defined to be long enough to obtain mechanical properties similar to those of a quasi-isotropic laminate while being short enough to maintain the advantage of fitting well. Meanwhile, very few studies have been published reporting the mechanical properties of this specific kind of material, which is the goal of the present work.

The application case delivered by Eurocopter Germany GmbH is a CNC-machined steel hinge from the rotorcraft model EC 135 (Figure 2), weighing 135 grams and mounted on a CFRP door structure.

The overall part dimensions are 112mm x 42mm x 22.5mm (Figure 3). The steel hinge has to withstand a static maximal load case of 2.172kN for a perpendicular bearing force.

An original steel hinge was tested using a specially designed jig (Figure 4) at a displacement rate of 1mm/min. The resulting curve (Figure 5) shows typical elasto-plastic behaviour with an elastic zone up to 3kN and plastic zone up to the ultimate failure at 8kN. For security reasons and maintenance convenience, the new TP-CFRP hinges, which retain similar geometrical interfaces and functions, will have to withstand the above-mentioned critical load and show the same progressive failure behaviour with large plastic deformation as the original steel hinge.

Hinge Production

Several types of TP-CFRP materials were selected. They are all constituted of AS4 carbon fibres embedded in a PEEK matrix. Only the fibre length and fibre content may vary according to Table 1 and Figure 6.

While it is easily understandable what kind of materials UD tapes, chopped tapes and injection moulding granules are, we will pay more attention to explaining how "recycled fragments" are obtained. Prior trials to recycle TP-CFRP parts using warm sulphuric and nitric acids or mechanical shredding [START_REF] Eguémann | Manufacturing and Recycling of Complex Thermoplastic Parts for Aerospace Applications[END_REF] delivered poor results, pushing us to test another proven technology coming from the mining industry called high-voltage fragmentation. The material to be recycled (in our case, door hinges) is placed in water between an anode and a cathode (Figure 7).

Pulsed electric discharges are applied to the material over a short period, creating pressure waves that lead to the cracking of weak regions (mainly resin-rich areas) in the composite material, generating recycled fragments as shown in Figure 8.

In this specific case, the fragments were obtained by recycling TP-CFRP door hinges produced from 20mm chopped tapes and directly pressed into the mould after drying. So it was possible to demonstrate a real cradle-to cradle recycling process. An extensive testing campaign [START_REF] Eguémann | Industrial Manufacturing of Parts, Brackets and Fittings in Carbon PEEK Compounds[END_REF] on coupons helped to determine the properties of the different materials.

The shear web shape was optimised through re-design in order to obtain an I-profile (Figure 9), resulting in a new part weight of 22g (without bearing).

A tooling solution using two sliders and a stamp was developed to obtain the four fixation holes, the bearing sit and the I-profiled shear web moulded into a net shape (Fig. 10). For the hinges produced from chopped tapes, recycled fragments and injection moulding granules, the required amount of material was gravimetrically pre-dosed and placed into the tool.

Then the tool and material were heated above the melting temperature of PEEK (345°C). Once the melting temperature was reached, a 100-bar pressure was applied to the material. Finally, the tool was cooled in a homogeneous way under the glass transition temperature of PEEK (140°C). Then the consolidated part was ejected and ready to be tested. Door hinges were also produced using a more classical laminate approach. Customized preforms were cut from UD tapes and put at the right place in the tool, as shown in Figure 11, with the following ply lay-up:

 Zone 1: UD 0°, spar caps  Zone 2: +/-45°, shear web  Zone 3, quasi-isotropic, basis plate Although the production of laminate-based door hinges is time intensive, it is of general interest to generate a challenging benchmark study.

Results

Figure 12 shows representative load-deformation curves for each type of material. We can observe that the hinges produced from injection moulding granules and recycled fragments show brittle failure behaviour. Furthermore, the failure occurs directly in the bearing sit for both material types. Hinges produced with chopped tapes and laminated UD tapes show a tough failure with a large plastic deformation.

The failure occurs exclusively in the shear web. In Figure 13, we can observe the mean ultimate load and corresponding deviation measured for each type of material. NOVEL AND RECYCLED THERMOPLASTIC COMPOSITE MATERIALS Laminate-based hinges show the best behaviour but, as mentioned before, their production is time consuming and expensive. The hinges produced from 20mm chopped tapes are almost as good as the laminate-based ones but their deviation measured over 5 samples is too high and could be problematic when determining the statistical minimum values, also called A and B values [START_REF]Handbook Metallic Materials and Elements for Aerospace Vehicle Structure[END_REF].

This type of material is known to be sensitive to statistical size effects [START_REF] Harper | Representative volume elements for discontinuous carbon fibre composites -Part 2: Determining the critical size[END_REF] and, for this reason, the use of the shortest chips could improve the global behaviour by reducing the deviation. This is exactly what is observed with hinges produced from 10mm chopped tapes, although the measured mean value is a little bit lower. Recycled door hinges only show 15% less strength than the original hinges they were produced from. Although their failure behaviour is not adequate, the cradle-to cradle recycling can be considered as successful with a really low downcycling effect. Recycled hinges still show a higher ultimate load than hinges produced from injection moulding granules.

Conclusion

Hinges produced with 10mm chopped tapes seem to be the best candidates to replace metallic hinges: their production time is short, they are less sensitive to size effects, their failure behaviour is appropriate and a weight gain of 80% could be reached compared to CNC-machined steel hinges.

When computing the allowables based on the values measured on the eight tested samples according to the procedure13 shown in Figure 14, the values in Figure 15 are obtained. We can observe that even the computed minimal statistic A and B values are over the above-mentioned threshold load of 2.172kN. This makes this lightweight alternative also appropriate for critical applications such as aerospace parts.

Figure 1 Figure 3 Figure 4 Figure 5

 1345 Figure 1 EASA recommended protection against galvanic corrosion

Figure 6 Figure 7 8 Figure 9 Figure 10 Figure 11 Figure 12 10 Figure 13

 67891011121013 Figure 6 From left to right, UD tape, 20mm chopped tapes, 10mm chopped tapes, recycled fragments, injection moulding granules

Table 1

 1 Raw Material Types

	material type	volume fibre content Fiber Length
	UD tape	58%	continuous
	20mm chopped tapes	58%	20mm
	10mm chopped tapes	58%	10mm
	recycled fragments	>58%	<10mm
	injection molding granules	24%	<1mm

Acknowledgments

The authors would like to acknowledge Eurocopter Germany GmbH, the Swiss Commission for Technology and Innovation and the School of Engineering of FHNW and its Excellence Programme for supporting this experimental study. The study was also supported by SUPREM SA through the donation of material.


M Collet

M O U I S S E A

M Ichchou

R Ohayon

Computation of 2D vibroacoustic wave's dispersion for optimizing acoustic power flow in interaction with adaptive metacomposites

Keywords: Distributed control, 2D Waves Dispersion, Bloch Theorem, Shunted piezoelectric, Mid-Frequency Optimization

This paper presents an integrated methodology for optimizing vibroacoustic energy flow in interaction between an adaptive metacomposite made of periodically distributed shunted piezoelectric material glued onto passive plate and open acoustic domain. The computation of interacting Floquet-Block propagators is also used to optimize vibroacoustic. The main purpose of this work is first to propose the numerical methodology to compute the fluid-structure multi-modal wave dispersions. In a second step optimization of electric circuit is used to control the acoustic power flow. 3D standard computation is use to confirm the efficiency of the designed metacomposite in term of acoustic emissivity and absorption.

INTRODUCTION

Tailoring the dynamical behavior of wave-guide structures can provide an efficient and physically elegant approach for optimizing mechanical components with regards to vibration and acoustic criteria, among others. However, achieving this objective may lead to different outcomes depending on the context of the optimization. In the preliminary stages of a product's development, one mainly needs optimization tools capable of rapidly providing global design directions. Such optimization will also depend on the frequency range of interest. Thus, piezoelectric materials and other adaptive and smart systems are employed to improve the vibroacoustic quality of structural components, especially in the Low Frequency range [START_REF] Preumont | Vibration control of structures : An introduction[END_REF][START_REF] Nelson | Active Control of Sound[END_REF][START_REF] B A N K Sa N Dy | Smart material structures Modeling Estimation and Control[END_REF] even if ditributed transducers are used. [START_REF] Tzou | A study of segmentation of distributed piezoelectric sensors and actuators, part i: Theorical analysis[END_REF] Recently, much effort has been spent on developing new multi-functional structures integrating electro-mechanical systems in order to optimize their vibroacoustic behavior over a larger frequency band of interest. [START_REF] Collet | Wave Motion Optimization in Periodically Distributed Shunted Piezocomposite Beam Structures[END_REF][START_REF] Thorp | Attenuation and localization of wave propagation in rods with periodic shunted piezoelectric patches[END_REF][START_REF] Beck | Broadband vibration suppression assessment of negative impedance shunts[END_REF][START_REF] Beck | Experimental analysis of a cantilever beam with a shunted piezoelectric periodic array[END_REF][START_REF] Bartoli | Modeling wave propagation in damped waveguides of arbitrary cross-section[END_REF][START_REF] Foda | Vibration suppression of composite laminated beams using distributed piezoelectric patches[END_REF][START_REF] Dawood | Three-dimensional static shape control analysis of composite plates using distributed piezoelectric actuators[END_REF][START_REF] Jiang | Decentralized robust vibration control of smart structures with parameter uncertainties[END_REF][START_REF] Ren | Finite difference based vibration simulation analysis of a segmented distributed piezoelectric structronic plate system[END_REF] For Medium Frequency band some specific approaches have been developped know as metacomposite concept firstly introduced in. [START_REF] Collet | Smart metacomposites for semi-active control of sound radiation[END_REF][START_REF] Collet | Structural energy flow optimization through adaptive shunted piezoelectric metacomposites[END_REF] This concept couples two different aspects in vibration control. The first concept is connected to periodic structures theories usually connected to metamaterial developments. In this case, it is well known that the dynamic behavior is fully connected to periodicity ratios and existing pass bands and blocked bands can be of real use in vibration control. The second concept is associated to vibration control through piezoelectric and smart materials. Specifically, shunted piezoelectric smart materials are employed for the metacomposite achievement by integrating into the metamaterial electronics and numerical components allowing implementation of adaptive and controlled behavior. We also tend to extend the notion of programmable matter within the meaning of work presented in [START_REF] Toffoli | Programmable matter: Concepts and realization[END_REF] to vibroacoustic programming. First application of this concepts has been already done for controlling mechanical energy power flow in [START_REF] Collet | Structural energy flow optimization through adaptive shunted piezoelectric metacomposites[END_REF] or by processing acoustic radiation effect in. [START_REF] Collet | Semi-active optimization of 2D wave dispersion into shunted piezo-composite systems for controlling acoustic interaction[END_REF] In this paper, we present an integrated methodology for optimizing vibroacoustic energy flow in interaction between an adaptive metacomposite made of periodically distributed shunted piezoelectric material glued onto passive plate and open acoustic domain. Extension of shifted cell operator methodology to fluid-structure interaction is presented. The computation of interacting Floquet-Block propagators is also used to optimize vibroacoustic indicators as noise absorption and emission. The main purpose of this work is first to propose the numerical methodology to compute the fluid-structure multi-modal wave dispersions. In a second step optimization of electric circuit is used to control the acoustic power flow.

After recalling the Floquet-Bloch theorems, we introduce a new numerical formulation for computing the multi-modal damped wave numbers dispersion in the whole first Brillouin domain of a periodical smart structure made of periodically distributed shunted piezoelectric patches. Based on this wave modeling, optimization of the electrical impedance of the shunted circuit is made in order to decrease group velocity of flexural waves. The obtained optimal impedance is also tested in controlling the HF response of a semi-distributed system.

APPLICATION OF THE FLOQUET-BLOCH THEOREM TO MODEL ADAPTIVE FLUID STRUCTURE INTERACTIONS

In this section the application of teh Floquet-Bloch theorem is presented for piezo-elastodynamic and acoustic interaction problem. Based on the well known results obtained by Floquet 18 in one-dimensional and later rediscovered by Bloch [START_REF] Bloch | Über die Quantenmechanik der Electron in Kristallgittern[END_REF] in multidimensional problems, we propose an original application to bi-dimensional piezo-elastodynamical and acoustic problem leading to very general numerical implementation for computing waves dispersion for periodically smart distributed mechanical systems incorporating electronic components and damping effects. [START_REF] Collet | Smart metacomposites for semi-active control of sound radiation[END_REF][START_REF] Collet | Structural energy flow optimization through adaptive shunted piezoelectric metacomposites[END_REF][START_REF] Collet | Numerical tools for semi-active optimization of 2d waves dispersion into mechanical system[END_REF] 2.1 The shift cell orperator for fluid-structure interaction problem

Application to the Floquet-Bloch theroem for computing complex waves dispersion in piezo-elastodynamical problem via the resolution of the shift cell operator eigensolutions is described in. [START_REF] Collet | Smart metacomposites for semi-active control of sound radiation[END_REF][START_REF] Collet | Structural energy flow optimization through adaptive shunted piezoelectric metacomposites[END_REF] In this paper we concider a fluid-structure periodic system made of unitary cells as depicted in figure . It is made of a supporting aluminum plate on which a circular piezoelectric patch is glued. On the other plate face, we consider an acoustic parallelepiped domain bounded by a rigid wall far from the elastic plate. All dimensions are presented on figure . The fluid structrure interaction is written by imposing continuity of the normal velocity and acoustic pressure is applied on the structural domain.

The shift cell operator, its weak formulation for computing waves dispersion into periodical piezocomposite lattice is preciselly described in. [START_REF] Collet | Structural energy flow optimization through adaptive shunted piezoelectric metacomposites[END_REF] We also just introduce here the acoustic application of the previoulsy described methodology. The Helmoltz equation gouverning the acoustic equilibrium and its corresponding boundary conditions are given by :

ω 2 C 2 o p(x)+∇ 2 p(x)=0 ∀x ∈ Ω f (1) p(x) Γ x -= p(x) Γ x + (2) p(x) Γ y -= p(x) Γ y + (3) n∇p(x)=0 ∀x ∈ Γ ∞ (4) n∇p(x)=-ω 2 ρ o n.w(x) ∀x ∈ Γ § ( 5
)
where ω is the angular frequency, C o the speed of sound in the medium, ρ o the fluid mass density, n the unitary normal vector, w the plate displacement. The first boundary condition 4 represent the hard wall condition, the last one, in equation 5, is the fluid structure interaction condition corresponding to the continuity of the normal displacements. We note that the adjoint condition is applied by imposing acoustic pressure acting on the mechanical structure. The continuity conditions on lateral faces Γ x -,Γ x + ,Γ y -and Γ y + of the primitive cell of the periodic problem are direcly linked to the use of the Floquet-Bloch theorem for computating of the Bloch eigenmodes and the associated dispersion curves. The main idea is to search the generalized eigenvalue problem given by equations ( 1),( 4), ( 5) when the pressure p(x) (and also displacement and voltage) can be written as :

p(x)=p n,k (x)e ik.x (6)
( ω 2 C 2 o -k 2 )p n,k (x)+2ik∇p n,k (x)+∇ 2 p n,k (x)=0 ( 7 )
with the associated bounadry conditions :

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ p n,k (x) Γ x -= p n,k (x) Γ x + p n,k (x) Γ y -= p n,k (x) Γ y + n(∇p n,k (x)+ikp n,k (x)) = 0 ∀x ∈ Γ ∞ n(∇p n,k (x)+ikp n,k (x)) = -ω 2 ρ o n.w n,k (x) ∀x ∈ Γ § (8)
where k = k ⎡ ⎣ cos(φ) sin(φ) 0 ⎤ ⎦ = kΦ where φ represent the direction angles into the reciprocal 2D lattice domain.

These acoustic equations are associated with the piezoelatodynamical one presented in. [START_REF] Collet | Smart metacomposites for semi-active control of sound radiation[END_REF] The proposed formulation is also based on the computation of the Floquet vectors (equation ( 7)), instead of computing the Floquet propagators commonly used for elastodynamic applications. Our approach allows to obtain the full 2D waves dispersions functions and to clearly introduce damping and electrical impedance into the acousto-piezo-elastodynamic 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 Bloch's waves

The numerical implementation is obtained by using a standard finite elements method to discretize the adapted weak formulation as specified in. [START_REF] Collet | Smart metacomposites for semi-active control of sound radiation[END_REF] The assembled matrix equation is given by:

(K(Z(iω n (λ, φ)) + λL(φ, Z(iω n (λ, φ))) -λ 2 H(φ, Z(iω n (λ, φ))) -ω 2 n (λ, φ)M )u n,k (φ)=0, (9)
where u n,k (φ) represents the extended degrees of freedom vector made of displacements w, voltage V and pressure p, λ = ik, M and K(Z(iω n (λ, φ))) are respectively the standard symmetric semi-definite mass and stiffness matrices (the mass matrix is semi definite because elastostatic equation are condensed into the equation), L(φ, Z(iω n (λ, φ))) is a skew-symmetric matrix and H(φ, Z(iω n (λ, φ))) is a symmetric semi-definite positive matrix.

When k and φ are fixed and Z does not depend on ω the system ( 9) is a linear eigen value problem allowing us to compute the dispersion functions ω 2 n (k,φ) and the associated Bloch eigenvector u n,k (φ).

This approach has been widely used for developing homogenization techniques and spectral asymptotic analysis like in the work of. [START_REF] Allaire | Bloch waves homogenization and spectral asymptotic analysis[END_REF] It can also be applied for computing wave's dispersion even if Floquet propagators is preferred for 1D or quasi 1D computation, as indicated in. [START_REF] Ichchou | Guided waves group and energy velocities via finite elements[END_REF][START_REF] Houillon | Wave motion in thin-walled structures[END_REF][START_REF] Mencik | Multi-mode propagation and diffusion in structures through finite elements[END_REF] Nevertheless these approaches have been only developed for undamped mechanical systems that is to say represented by a set of real matrices. In this case, most of the previously published works present techniques based on the mesh of a real k-space (i.e k or λ and φ) inside the first Brillouin zone for obtaining the corresponding frequency dispersion and the associated Floquet vectors. For undamped system only propagative or evanescent waves exist corresponding to a family of eigen solutions purely real or imaginary. Discrimination between each class of waves is easy. If a damped system is considered (K, L, H are complex frequency dependent) or frequency dependence of the electrical shunt impedance is considered, the obtained eigenvalue problem is not quadratic and a complex specific numerical methodology has to be implemented. Furthermore, evanescent part of propagating waves appear as the imaginary part of ω 2 n (λ, Φ). It then becomes very difficult to distinguish the propagative and evanescent waves but also to compute the corresponding physical wave's movements by applying spatial deconvolution.

Another much more suitable possibility for computing damped system, dedicated for time/space deconvolution and for computation of diffusion properties as defined by, [START_REF] Collet | Wave Motion Optimization in Periodically Distributed Shunted Piezocomposite Beam Structures[END_REF][START_REF] Mencik | Multi-mode propagation and diffusion in structures through finite elements[END_REF] is to consider the following generalized eigen value problem:

(K(Z(ω) -ω 2 M + λ n (ω, φ)L(φ, Z(ω)) -λ 2 n (ω, φ)H(φ, Z(ω)))u n (ω, φ)=0. (10)
In this problem, the pulsation ω is a real parameter corresponding to the harmonic frequency. Wave's numbers and Floquet vectors are then computed. An inverse Fourier transformation in the k-space domain can lead us to evaluate the physical wave's displacements and energy diffusion operator when the periodic distribution is connected to another system as in. [START_REF] Collet | Wave Motion Optimization in Periodically Distributed Shunted Piezocomposite Beam Structures[END_REF] Another temporal inverse Fourier transformation can furnish a way to access spatio-temporal response for non-homogeneous initial conditions. As L is skew-symmetric, the obtained eigen values are quadruple (λ, λ, -λ, -λ) collapsing into real or imaginary pairs (or a single zero) when all matrices are real (i.e. for an undamped system). In this case a real pair of eigen values correspond to evanescent modes oriented in two opposite directions on the k-space and imaginary values to two traveling waves propagating in opposite direction.

As previously mentioned, the real part of k = kΦ vector is restricted to stand inside the first Brillouin zone. In the quadratic eigen value problem (10) nothing restricts computation to only find eigen values satisfying this condition. For direction vector Φ orthogonal to the lattice facelets (i.e. for Φ p1 =[ 1 , 0] T and Φ p2 =[ 0 , 1] T in bi-dimensional rectangular cell), the periodical conditions expressed for one dimensional wave guide are still valid: if λ j (ω, Φ p ) is an eigen value associated to w j (ω, Φ p )then∀m ∈ Z 3 , λ + i.Φ T p (G.m) is also an eigen value associated to w j (ω, Φ p )e -i.Φ T p (G.m)x . Thus, for undamped systems, all obtained eigenvalues are periodically distributed in the k-space along its principal directions.

Computation of the group velocity and evanescence criterion

The main aim of this paper is to provide a numerical methodology for optimizing the piezoelectric shunt impedance Z(ω) for controlling energy flow into the periodically distributed acousto-piezo-composite structure. For doing this, we need to define a suitable criterion. The waves group velocities indicate how energy is transported into the considered system and allow to distinguish the 'propagative' and 'evanescent' waves. If one Bloch eigen solution (i.e u n (ω, φ),k n (ω)) is considered, the associated group velocity vector [START_REF] Maysenhölder | Körperschall-energie Grundlagen zur Berechnung von Energiedichten und Intensitäten[END_REF] is given by :

C g n (ω, φ)=∇ k ω = S e tot = I E tot ( 11
)
where : is the spatial and time average respectivelly on one cell and one period, S is the density of energy flux defined in, [START_REF] Maysenhölder | Körperschall-energie Grundlagen zur Berechnung von Energiedichten und Intensitäten[END_REF] I the mean intensity and e tot , E tot the total piezomechanical energy and its time average on a period (see [START_REF] Maysenhölder | Körperschall-energie Grundlagen zur Berechnung von Energiedichten und Intensitäten[END_REF] for details). In this problem, we only consider mechanical energy transportation as the electrostatic coupling is decentralized and can be condensed as a mechanical interface as proved in [START_REF] Hagood | Damping of structural vibrations with piezoelectric materials and passive electrical networks[END_REF] and generally computed in. [START_REF] Collet | Modal Synthesis and Dynamical Condensation Methods for Accurate Piezoelectric Systems Impedance Computation[END_REF] So we also compute the intensity vector I by :

I n = - ω 2 Re( Ω C(ε n (x,ω,φ)+ikΞ n (x,ω,φ)).(w * n (x,ω,φ)) + (∇ + ikΦ)p n (x,ω,φ)p * n (x,ω,φ) (ρ o iω) dΩ/V ol (12)
where . * is the complex conjugate and V ol the domain volume. As the spatio-temporal average of the system Lagragian is null (see [START_REF] Maysenhölder | Körperschall-energie Grundlagen zur Berechnung von Energiedichten und Intensitäten[END_REF] ), the total energy average is approximated by only computing the kinetic energy average into the solid part and acoustic energy into the fluid :

E tot = 1 2 ( Ω ρω 2 w n (x,ω,φ).w * n (x,ω,φ)+(ρ o C 2 o ) * p n (x,ω,φ).p * n (x,ω,φ)dΩ/V ol (13)
The group velocity vectors C g n (ω, φ) is 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, φ)= 1 2 ( Ωx ρω 2 wz n (x,ω,φ)wz * n (x,ω,φ)dΩ/V ol E tot (14)
with wz n (x,ω,φ)be i n gt h e( Oz) component of vector w n (x,ω,φ).

OPTIMIZATION OF THE FLEXURAL MECHANICAL ENERGY FLOW AND ACOUSTIC EFFECTS

The considered piezo-composite 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 is given in table 1. The fluid medium is made of air : C o = 343 m.s -1 and ρ o =1.25 kg.m -3 .

The used methodology for optimizing the shunt impedance Z(iω) is based on the minimization of the maximal group velocity collinear to the wave number vector (11) for waves having a ratio of transported flexural kinetic energy (14) greater than 0.8. The used criterion can also be written as:

Crit(Z(iω),φ)=max n/Ind(n,ω,φ)>0.8 (C g n (ω, φ).Φ) (15)
The used numerical optimization of the criteria is based on a multidimensional unconstrained nonlinear minimization (Nelder-Mead).

We consider optimization by considering frequency dependent complexe impedance. We present in figure 3 the obtained real parts of the propagative wave number kx n (iω) along (Ox) axis. The red circles mark the dispersions curves for Z = 0 and the blue crossed the optimal dispersion. The corresponding group velocities along (Ox) is presented in figure 3 while the real and imaginary parts of the optimal impedance are plotted in figure 4. We immediately observe that the optimization of the shunt impedance leads to greatly modify the group velocity of the A o mode and to create an evanescent mode. We notice an increase of the imaginary parts of the wave number of A 1 mode indicating an increase of the spatial decay rates even if this mode remains propagative. The acoustic radiation is greatly modified by the cancellation of A o mode as shown in figure ??.c. The imaginary part of kz n stay unchanged at very low values for radiations induced by A 1 . The optimal impedance values are almost real, and correspond to those obtained if a constant negative capacitance is used. The corresponding average value is -233.66 pC.V -1 . Some imaginary parts of the optimal impedance are negative, 1.5319 Ω which indicate that the optimization leads to provide energy to the system for controlling mechanical damping effect introduced with hysteretic damping ratios into the model, and, also, obtain a fully conservative system. Acoustic wave numbers plotted on figure 3 Lz where L z =0.5 m (see figure 1). The coincidence frequency for which the in plane wave number is real clearly appears on figures 3, 3 on which we observe a familly of acoustic waves denoted F i . Control effects on acoustic waves (is quiet neglectible in the low frequency band even if some decreases on the energy velocity can be observed in high frequency on figure 3 when frequency tends to the free acoustic coincidence (here 12.8 kHz 14 )

Piezoelectric Material

CONCLUSIONS

This paper presents a numerical procedure able to compute the wave's dispersion functions in the whole first Brillouin domain of multi dimensionnal acousto-piezo-elastodynamical wave guides. The method was applied for determining the optimal impedance allowing to minimize the group velocities of the flexural waves. The control caracterisation has been done from mechanical but also acoustic point of view. We also underline a strong influence of the designed shunt circuits in the dynamical response of the mechanical system and none on the fluidic part in the tested low frequency band. We also demonstrated that our developed numerical procedures can be used for optimizing the energy diffusion operator of such adaptive mechanical interface. To do so, additional work has to be done for optimizing the complete interface scattering and for controlling the acoustic waves interacting with the mechanical system. The proposed methodology can also be used for studying particular dissipation phenomenon such as those induced by complex shunted piezoelectric patches as proposed by [START_REF] Beck | Broadband vibration suppression assessment of negative impedance shunts[END_REF] and, [START_REF] Casadei | Vibration control of plates featuring periodic arrays of hybrid shunted piezoelectric patches[END_REF] or even foams or complex polymers behaviors. The proposed method furnishes an efficient tool for future optimization of distributed smart cells as proposed in the case of 1D-2D wave guide by [START_REF] Collet | Wave Motion Optimization in Periodically Distributed Shunted Piezocomposite Beam Structures[END_REF][START_REF] Collet | Smart metacomposites for semi-active control of sound radiation[END_REF][START_REF] Collet | Structural energy flow optimization through adaptive shunted piezoelectric metacomposites[END_REF] by extending the approach toward complex multiphysic systems.

Figure 1 .

 1 Figure 1. Generic 3D piezocomposite periodic cell

Figure 2 .Figure 3 .

 23 Figure 2. Real parts of the propagative wave number kxn(iω)a l o n g( Ox), the red circles mark values obtained for Z =0 and the blue crossed the optimal ones

Figure 4 .

 4 Figure 4. Real and imaginary parts of the optimal impedance

  correspond to classical dispersion on acoustic 2D tube in which the normal wave numbers k z are ruled by the Neumann boundary condition applyed on the external side of the fluidic domain. If fluid structure coupling is neglectible (i.e in laow frequency band) the normal wave numbers are k zi = i * π

Table 1 .

 1 Piezoelectric patch characteristics

	Symbol	Value	Property
	s E 11 = s E 22 = s33 E 11.6 × 10 -12 Pa -1 s E 12 = s E 13 = s E -3.33 × 10 -12 Pa -1 12, 13 and 23 compliance matrix coefficient 11, 22 and 33 compliance matrix coefficients 23 s E 44 = s E 55 = s E 66 45.0 × 10 -11 Pa -1 44, 55 and 66 compliance matrix coefficients
	η	0.1%	Hysteretic Damping ratio
	d 31 = d 32	-6 × 10 -11 C/N	31 and 32 piezoelectric matrix coefficients
	d 33	15.2 × 10 -11 C/N	33 piezoelectric matrix coefficient
	d 24 = d 15	730 × 10 -12 C/N	24 and 15 piezoelectric matrix coefficients
	ρ	7600kg/m 3	Density
	ε T 11 = ε T 22 ε T 33	504.1 ε o C/V/m 270 ε o C/V/m	Dielectric Permittivity Dielectric Permittivity

ACKNOWLEDGMENTS

This work was carried out with a grant of French agency ANR number NT09-617542. We gratefully acknowledge the French ANR and CNRS for supporting this program.