Corentin Trégouët

Thomas Salez

Nadège Pantoustier

Patrick Perrin

Mathilde Reyssat

Cécile Monteux
email: cecile.monteux@espci.fr

C Tregouet

Probing the adsorption/desorption of amphiphilic polymers at the air-water interface during large interfacial deformations

Probing the adsorption/desorption of amphiphilic polymers at the air-water interface during large interfacial deformations

Introduction

Polymers are widely used as interfacial stabilizers, emulsifiers or suspension dispersants. As they adsorb at liquid/liquid or solid/liquid interfaces, some monomers adsorb into trains while some monomers form loops or tails extending into the liquid, which provide a steric protection for droplets or particles against coalescence or flocculation. Although the desorption energy of one monomer is of the order of k B T, the overall desorption energy of one polymer chain with N>>1 adsorbed monomers is of the order of Nk B T, hence the adsorption of a polymer chain is irreversible. However, the low anchoring energy per monomer enables quick exchanges of monomers between adsorbed trains and loops or tails. Amphiphilic polymers, composed of hydrophilic and hydrophobic parts, such as random and block copolymers or polymers grafted with hydrophobic anchors adsorb more strongly to interfaces with aqueous solutions than homopolymers thanks to the high anchoring energy of the hydrophobic anchors. The structure of amphiphilic polymer layers adsorbed at solid and liquid interfaces has been the object of theoretical and experimental research in the 90s [START_REF] Fleer | Polymers at Interfaces[END_REF]. For block copolymers, it was predicted that the hydrophobic parts form 2D coils while the hydrophilic parts form 3D coils swollen in the bulk solution [START_REF] Aguie-Beghin | [END_REF]3]. At high surface polymer concentration, the hydrophilic parts stretch perpendicularly to the interface thus forming a brush [START_REF] Aguie-Beghin | [END_REF][3][4][5][6][7][8][9]. Brushes are also expected for other amphiphilic polymers such as telechelic [10], end-functionalized [11] or hydrophobically modified polymers [12][13][14][15][16][17], for which hydrophobic grafts adsorb to the interface and hydrophilic parts stretch into the bulk solution.

The dynamical properties of amphiphilic polymer layers under an interfacial compression [18][19][20][21][22] has attracted much less attention than their structure, even though it is relevant in foaming or emulsification processes as well as for foam and emulsion stability [23,24]. For example, during bubble rearrangements in a foam, which sometimes leads to bubble coalescence, bubble interfaces and thin liquid films can be strongly stretched or compressed. A fine understanding of how the polymer architecture influences the dynamical properties of the interfacial layers would help rationalizing the design of polymer molecules for foams and emulsions with improved stability. The pendant drop method [7,[20][21][22] or Langmuir trough experiments [6,10] are useful methods to investigate the behavior of adsorbed layers during a reduction or increase of the surface area. In the case of amphiphilic diblock or triblock copolymers such as poly(ethylene oxide)poly(styrene), PEO-PS, or poly(propylene oxide)poly(ethylene oxide) -poly propylene oxide) copolymers, PPO-PEO-PPO or telechelic polymers, a mushroom to brush transition was observed upon large compression of the layers [4,6,7,10], as initially adsorbed PEO monomers, desorb from the interface to form stretched loops while the more hydrophobic blocks, PPO or PS remain adsorbed at the interface.

Hydrophobically modified polymer molecules, such as PAAH--C n composed of a poly(acrylic acid), PAAH, backbone, covalently and randomly grafted with hydrophobic alkyl anchors, C n , with n the number of carbon atoms on the alkyl chain and  the mole fraction of grafted monomers, are known to be very efficient foam and emulsion stabilizers [19,[START_REF] Cardenas-Valera | Colloids Surfaces A Physicochem[END_REF][26][27]. Moreover their chemical architecture -molar mass, number of grafts, degree of ionization and length of grafted alkyl chain-can be easily modified, therefore they are good candidates to investigate how chemical parameters influence the interfacial compression and dilatation of amphiphilic polymers. In a previous article, we have investigated the adsorption dynamics of a series of PAAH--C n , [17]. We found that at short times, the surface tension is lower for an increasing degree of grafting, while at longer times, as the adsorption dynamics becomes logarithmic, the evolution of the surface tension becomes slower as the degree of grafting increases. Our results were consistent with the formation of a polymer brush at the interface, with PAAH loops extending in solution between adsorbed C n hydrophobic anchors. We suggested a coarse-grained model where the adsorption dynamics at long times is limited by a free energy-barrier, which results from both the deformation of the incoming polymer coils and the deformation of the adsorbed brush. As the grafting degree increases, the number of monomers between C n anchors decreases, and hence the length of the loops decreases. This leads to a thinner brush and less stretching of the adsorbing chains and thus to a faster adsorption.

In the present article, we study behavior of adsorbed layers of PAAH--C n upon large interfacial compressions and dilatations. By comparing the measured interfacial tension to the one that we expect in the case of a constant number of adsorbed monomers, we deduce that the polymer chains with the longest hydrophobic grafts desorb for larger compressions than the polymers with the shortest grafts, because of the larger desorption energy. Furthermore, counter intuitively we observe more desorption upon compression for higher grafting densities. We attribute this counter intuitive result to the fact that for high grafting densities, the length of the loops is shorter. Hence upon compression, the elastic penalty is larger for these layers, leading to a faster desorption. The dilatation experiments reveal that the number of adsorbed anchors remains constant in the case of chains with low grafting density while chains with the highest degree of grafting seem to show some degree of adsorption during the dilatation. Therefore for these highly grafted chains there may be unadsorbed grafts remaining in the vicinity of the interface, which may adsorb quickly to the interface upon dilatation.

Materials

We use a series of hydrophobically modified poly(acrylic acid), consisting in a PAA backbone (poly(acrylic acid)) of molar mass of M W = 100 000 g/mol, covalently grafted with a grafting density  of hydrophobic alkyl grafts of length n. The PAA backbone is provided by Polysciences Inc and the grafting reaction was done according to Ref. [28]. In our work,  ranges between 0.5/100 and 5/100 and n = 8 or 12.

To prepare the polymer solutions, the polymer chains are dissolved at a concentration of 1% w/w for 60 hours. The pH is adjusted to pH=3 with a few milliliters of molar hydrochloric acid solution to obtain conditions in which the PAAH chains do not bear any negative charges. The solutions are then filtered through 0.45 µm PVDF, poly(vinylidene fluoride) membranes. The solutions are stored at a temperature between 4 and 6 °C. The solutions are kept at ambient temperature for 12 hours prior to each experiment. The solutions are used within the seven days after the polymer dissolution.

Measurements.

To probe the adsorption dynamics of the polymer layers, we use the rising bubble method (Tracker from Teclis, France) which enables the measurement of the surface tension as a function of time by fitting the profile of a bubble to the Laplace equation. Details concerning the principle of this technique can be found in Ref. [29]. Briefly, a fresh millimetric air bubble is formed in the polymer solution and the polymer molecules are left to adsorb at the surface for 3600 seconds. The temperature is maintained constant at 20°C both in the vessel and in the syringe. Then the bubble surface is either reduced or increased by either sucking out air from the bubble or injecting air into the bubble. The interfacial tension and area of the bubble are recorded as a function of time during the compression or dilation of the bubble. The compression or the dilation rates, defined as the relative area variations per unit time, are either 0.007 s -1 or 0.1 s -1 .

Results

Large compressions of the adsorbed layers

To probe the behavior of the polymer layers under compression, we deflate the air bubble and we record the area and surface tension for PAAH-C n with varying grafting degrees and graft lengths. Figure 1 shows the variation of the surface tension as a function of the deformation , defined as the relative change in surface area,

, where is the initial area, and the area at the considered time. We observe that the surface tension decreases during a compression which means that the surface density of adsorbed polymer chains increases.

If the number of chains remains constant, ie the chains do not desorb upon compression, we can easily predict the evolution of the polymer surface excess, , as a function of the initial surface excess, , and the deformation : According to Aguié-Béghin et al. [START_REF] Aguie-Beghin | [END_REF] the surface pressure  of a copolymer layer in a regime where the hydrophobic parts are adsorbed to the air/water interface and the hydrophilic parts form stretched loops in solution reads ,

, ( 2
where and are the interfacial tensions of pure water and of the solution, N is the number of monomers per polymer chain and N L is the number of monomers per loop. In our case we assume that a polymer chain composed of N monomers typically contains N grafts, and N loops, which contain 1/ monomers each. Therefore eq. ( 3) becomes

    
If the number of adsorbed entities is constant during the deformation, we can predict the evolution of the surface tension,  no desorption , according to the following equation ,

with  i the initial surface tension before compression. can also be written ,

which is the surface pressure obtained if no desorption of the chains occurs. In Figure 2 we represent the measured surface tension, , as a function of . The red dashed lines are the predictions from Eq. 6. A compression with desorption would lead to higher surface tensions, i.e. to variation curves located above that prediction.

The compression curves shown in Figures 2a and 2b present two regimes. At low compression

and thus low  no desorption , the surface tension superimposes well with the prediction of eq. 6, which indicates that there is no desorption in this regime. At larger compressions and thus high  no desorption , the surface tension saturates to a plateau indicating that some desorption or partial desorption of molecules occurs, and that the surface excess reaches a maximal value due to possible steric/free-enthalpic constraints at the surface. We suggest that the desorption process concerns the hydrophobic anchors, as previous investigations [13][14][15]17] showed that the hydrophilic AAH blocks form loops extending in solution while the C n grafts adsorb at the interface. For a faster compression of the PAAH-0.7-C 12 layers the desorption occurs at a larger deformation and the transition between the two regimes is sharper (Figure 2a). Moreover, when comparing C 8 and C 12 anchors, we find that the surface tension remains higher for C 8 grafts than for C 12 grafts during compression in agreement with the lower anchoring ability of the smaller grafts. The driving mechanism for the desorption is the increase of chemical potential of the chains in the brush. In References 10 and 30, it was shown that the rate of desorption decreases with the length of the alkyl tail hence for a given compression speed, we therefore expect that C 8 grafts will desorb faster than C 12 grafts, consistently with the observation that for C 8 grafts the surface tension leaves the "no-desorption" curve at a lower deformation than for C 12 grafts.

Moreover for a given alkyl anchor length, if the compression is faster than the rate of desorption, we expect to follow the "no-desorption" curve while an infinitely low compression speed the desorption of the grafts should favor a constant interfacial tension, consistently with our results which show that the surface tension remains closer to the "no-desorption" curve for a higher compression speed.

In Figure 3a we study the influence of the grafting degree on the compression behavior for the PAAH--C 8 chains. We observe that the more grafted the chain is, the lower the initial surface tension is, as a result of a better affinity of the chains with the interface. Remarkably the surface tension at large compression is further above the "no-desorption" curve for the chains with a higher grafting degree than for the less grafted chains. of the compression parameter (see eq. 9) for the different grafted monolayers.

To rationalize this counter-intuitive finding, we provide in Figure 3b another way of plotting the results. We define the following dimensionless numbers: the relative change in surface pressure , and the compression parameter defined as follows:

,

The compression parameter increases during compression starting from zero.

Using Equations (7,8,9), the relative change in the surface pressure in the case of no desorption can be written as:

.

In Figure 3b, we plot the relative change in surface pressure versus the compression parameter for slow compression of polymer PAAH--C 8 , with different grafting densities. The red dashed line represents the theoretical prediction of Equation ( 10) corresponding to the case of no desorption. We systematically find that . The more grafted the chains, the stronger the deviation from Equation (10), which indicates a stronger desorption.

Taking into account the fact that in these pseudo-brushes the hydrophobic grafts adsorb in the interface while the hydrophilic blocks form loops away from the interface, one can reasonably argue that the number of hydrophobic anchors directly determines the length of the loops.

Assuming in first approximation that all grafts adsorb at the interface, a loop between two adsorbed grafts has a length 1/ in number of monomers, then the ideal entropic spring constant which resists the compression scales as  per loop and thus N. 2 per chain. Thus when compressing the layer, higher alpha leads to higher elastic penalty and thus faster desorption/diffusion rate.

Large dilatations of the layers

We now probe the effect of large dilatations. We plot in Figure 4 the measured surface tension as a function of  no desorption (see eq 7) during the dilatation. The surface tensions obtained for the chains with the lowest degrees of grafting ( ) follow the prediction of Equation ( 6) which corresponds to a constant number of adsorbed species, ie the absence of any adsorption or desorption of the chains. This indicates that as the interface is dilated (from right to left on the xaxis, due to the minus sign in ), the surface coverage decreases but the total number of adsorbed entities remains constant. The dilatation occurs too quickly to allow for significant adsorption during the deformation. These observations are consistent with the fact that for the chains with low degrees of grafting, all the hydrophobic grafts of the adsorbed chains are already adsorbed before dilatation, and that diffusion of new chains toward the interface is to slow to decrease the surface tension. Note that the AAH monomers comprised in the loops of the brush may in principle also adsorb upon dilatation, but we know from previous studies that the adsorption of AAH monomers does not lead to significant decrease of the surface tension [17,31]. For chains with larger degrees of grafting, , the interfacial tension is slightly lower than the dashed "no-adsorption" line. This indicates that new molecules or entities have time to adsorb while the interface is expanded (from right to left on x-axis). We suggest that some unadsorbed grafts may remain in the vicinity of the interface hence can adsorb quickly during interfacial dilatation.

Conclusion

In this article, we investigate the behaviour of PAAH--C n polymer layers at air-water interfaces when submitted to large interfacial compressions and dilatations. By comparing the interfacial tension with a prediction obtained for a constant number of adsorbed entities, the compression experiments enable to probe the forced desorption dynamics of these layers. We show that longer alkyl grafts desorb more slowly resulting in lower transient interfacial tensions than for shorter grafts. Moreover faster compressions lead to lower transient interfacial tensions because the grafts have less time to desorb. Furthermore, we observe more desorption for higher grafting densities, which is a counter intuitive result. We attribute this result to the fact that for high grafting densities, the number of monomers per loop is lower, hence the elastic penalty caused by the compression is higher. The dilatation experiments show that the number of adsorbed anchors remains constant in the case of chains with low grafting densities suggesting that all the grafts of the adsorbed chains are initially adsorbed. However, chains with the highest degrees of grafting seem to show some adsorption during the dilatation. Hence, for these chains there may be unadsorbed grafts remaining in the vicinity of the interface, which may adsorb quickly to the interface upon dilatation.

Figure 1 :

 1 Figure 1: Evolution of the surface tension of PAAH--Cn layers during compression at a rate of (a) 0.007 s -1 and (b) and 0.1 s -1 .

Figure 2 :

 2 Figure 2: Surface tension as a function of the reference surface pressure (see eq. 7) if the number of adsorbed molecules stays constant during the compression of the interface. (a) Compression of PAAH-0.7-C 12 at compression rates of 0.007 s -1 and 0.1 s -1 . (b) Compression of PAAH-0.8-C 8 and PAAH-0.7-C 12 at a rate of 0.1 s -1 .

Figure 3 :

 3 Figure 3: (a) Surface tension as a function of the surface pressure predicted for a constant number of adsorbed monomers (see eq. 7) during the compression of the interface covered by monolayers of different grafting densities.; (b) Relative change in surface pressure as a function

Figure 4 :

 4 Figure 4: Surface tension as a function of reference surface pressure (see Eq. 7) during the dilatation of the interface covered by monolayers of chains with different grafting densities and graft lengths, as indicated. (a) Dilatation rate is 0.007 Hz; (b) dilatation rate is 0.1 Hz.

Acknowledgements

The authors acknowledge ANR JCJC INTERPOL for financing this work.


Claire Guille-Biel Winder

Frédérick Tempier
email: frederick.tempier@u-cergy.fr

A theoretical framework for analyzing training situations in mathematics teacher education

Keywords: Teacher education, professional development, primary education, knowledge for teaching, analysis framework

The thoughts on the primary schoolteacher training have led to the production of many resources for primary schoolteachers. Faces of the abundance of such documents, teacher educators need some tools to identify the knowledge potentially at stake in training situations and to allow them to implement such situations according to their own objectives and context. We present a five-level analysis framework that characterizes the training tasks, according to the expected posture of the prospective teacher, to the type of the knowledge at stake and to possible degrees of decontextualization. We illustrate this analysis framework by presenting an example of a training scenario based on the principle of role-play.

Introduction

The research about primary schoolteacher education in mathematics and professional development has led to the production of many resources for educators. In France, the COPIRELEM1 group produced many of them. These resources provide training situations based on various training strategies [START_REF] Houdement | Autour des stratégies utilisées pour former les maîtres du premier degré en mathématiques[END_REF], and are generally accompanied with information about their implementation (phases, steps, instructions, elements of institutionalization) with regard to the stakes of the training. But their quality does not guarantee an accurate appropriation by teacher educators. Our questioning is: how is it possible to help teacher educators to exploit training situations in a relevant way, according to their objectives?

The research literature usually provides studies about knowledge for teaching, teacher conceptions, and their evolution [START_REF] Shulman | Those who understand: Knowledge growth in teaching[END_REF][START_REF] Houdement | Autour des stratégies utilisées pour former les maîtres du premier degré en mathématiques[END_REF][START_REF] Ball | Content Knowledge for teaching, what makes it special?[END_REF]. Other studies present one training situation, and generally focus on its effect on the prospective teachers. For example [START_REF] Horoks | Training primary school teachers through research in mathematics' didactics[END_REF] "analyse the contents and methods of an initiation course in research in mathematics education, and […] how it can influence the beginner teachers' practises" (p.2811). To our knowledge, no study focuses on the characteristics of training situations nor provides specific framework in order to analyse any training situation. This led us to develop an analysis framework for training situations. The paper presents this COPIRELEM's work in progress.

Presentation of the analysis framework

Relying on the Theory of Didactical Situations (TSD) developed by [START_REF] Brousseau | Theory of Didactical Situations in Mathematics: Didactique des Mathématiques 1970-1990[END_REF] we define a training situation as a situation2 that involves prospective teachers (students, pre-service or inservice teachers) and educators within an institution of teacher education. It is composed of a set of tasks based on a so-called "initiating task" and conducted by a teacher educator. We take into account all the tasks proposed by the teacher educator. Each task corresponds to a type of prospective teachers' activity: we name "study level of a training situation" each type of activity. We distinguish in a training situation five study levels: mathematical activity, (mathematical) analysis of a mathematical activity, didactical and pedagogical analysis of a (mathematical) activity, analysis of a didactical and pedagogical activity, problematization of professional issues.

For each type of activity we take into account three dimensions: the type of knowledge at stake; the degree of decontextualization of this knowledge; the posture of the prospective teachers expected by the teacher educator. These dimensions are specified in next sections.

Three types of knowledge

We rely on the three types of knowledge for teaching mathematics identified by [START_REF] Houdement | Autour des stratégies utilisées pour former les maîtres du premier degré en mathématiques[END_REF]: mathematical knowledge, pedagogical knowledge and didactical knowledge.

"Mathematical knowledge corresponds to mathematics that a teacher needs to know in order to prepare, regulate and evaluate his lesson and his students" (Houdement, 2013, p.12). It "includes and specifies the content knowledge" identified by [START_REF] Shulman | Those who understand: Knowledge growth in teaching[END_REF]. Moreover, the specific didactical nature of mathematical knowledge can be identified to the Specialized Content Knowledge (SCK) developed later by [START_REF] Ball | Content Knowledge for teaching, what makes it special?[END_REF].

According to [START_REF] Houdement | Au milieu du gué: entre formation des enseignants et recherche en didactique des mathématiques[END_REF], didactical knowledge is linked to the mathematical content and fed by research in the field of mathematics didactics. It corresponds to analysis of teaching and learning phenomenon and to propositions of engineering. Therefore it can be associated with at least two categories [START_REF] Ball | Content Knowledge for teaching, what makes it special?[END_REF]: Knowledge of Content and Students (KCS) and Knowledge of Content and Teaching (KCT).

Pedagogical knowledge3 is characterised as "knowledge of experience" [START_REF] Portugais | Didactique des mathématiques et formation des enseignants[END_REF]. It is related to teaching and learning conceptions and to the organisation and management of the class. It is less dependent of the mathematical content than other types of knowledge. It is important to take this knowledge into account because schoolteachers deal with various school subjects.

Three degrees of decontextualization

In TSD, [START_REF] Brousseau | Theory of Didactical Situations in Mathematics: Didactique des Mathématiques 1970-1990[END_REF] and [START_REF] Douady | The interplay between different settings. Tool-Object dialectic in the extension of ability[END_REF] identify three degrees of decontextualization of a mathematical knowledge: implicitly mobilized; explicitly mobilized in context or decontextualized (to become available in other contexts). We extend this notion to didactical and pedagogical knowledge.

A mathematical knowledge is (implicitly) mobilized in context (in act) if it is used as tool [START_REF] Douady | The interplay between different settings. Tool-Object dialectic in the extension of ability[END_REF] in a mathematical task. This task can be carried out: what is asked is effectively achieved (manipulation, elaboration and writing a solution for example). But the task can only be evocated: it mentally achieved. A mathematical is explicit in context its use (as tool) is identified formulated. At least, a mathematical knowledge is decontextualized if a status of object is given (by the educator) to the concept used previously as tool, usually during an institutionalization phase 1997).

The didactical/pedagogical knowledge is mobilized in context when the didactical/ pedagogical choices are made for the considered mathematical task. It is explicit in context during the analysis about the consequences of these choices. At least, it is decontextualized when the underlying didactical/pedagogical concepts are highlighted.

Four postures of the prospective teachers

In conjunction with the teacher trainer's relationship to the prospective teachers identified by [START_REF] Sayac | Study of the Practices of educators in Mathematics, on Pre-service Teachers Training[END_REF], we define four specific postures of prospective teachers, which are expected by the educator during a training situation Prospective teachers are in a posture of student relatively to mathematical knowledge when they have to perform mathematical activity or when they are concerned with the mathematical knowledge of this activity. They are in a student/teacher posture when they investigate mathematical tasks for students or students' works, or when they analyse the conditions of implementation of a task in the classroom. They are in a teacher posture when entering in a broader questioning on classroom practices and issues of mathematical learning. Finally, they are in a practitioner/researcher posture when they problematize a professional issue related to mathematical learning or teaching.

The five study levels

In order to analyse a training situation, we define five study levels. To each level corresponds a type of activity, that induces (implicitly or explicitly) a posture of the prospective teacher (expected by the educator), and that involves different types of knowledge in a certain degree of décontextualisation.

Level 0. A task induces a mathematical activity. This activity can be performed or evocated (mentally performed). The mathematical knowledge is mobilized (implicitly or explicitly) in context. The prospective teachers are in a posture of student (relatively to mathematical knowledge).

Level 1. A task induces a (mathematical) analysis. It highlights decontextualized mathematical knowledge. In this task, didactical and/or pedagogical knowledge can be implicitly mobilized in context. The prospective teachers are in a posture of learning mathematics student.

Level 2. A task induces a didactical and pedagogical analysis of the mathematical activity (analysis implementation conditions -actual or anticipated only). Didactical and pedagogical knowledge is explicit in context. The prospective teachers are in a student/teacher posture.

Level 3. A task induces an analysis of the didactical and pedagogical activity. It can be a questioning on classroom practice (specific learning tasks, professional actions...) or on issues of mathematical learning for one or several contents (curriculum, progressions...), or even a highlighting of didactical analysis concepts (didactic situation phases, types of tasks...). This analysis leads to the decontextualization of didactical and/or pedagogical knowledge. The prospective teachers are in a posture of teacher.

Level 4. This level corresponds to the problematization of professional issues related to classroom practices, learning issues and/or didactical analysis tools. The prospective teachers are in a posture of practitioner/researcher, especially when it comes to developing an analysis methodology of this issue and to infer results.

The following table summarizes the characteristics of the five study levels.

An example of use of the analysis framework

Definition of role-play

As [START_REF] Lajoie | Role-play by pre-service elementary teachers as a means to develop professional competencies in teaching mathematics[END_REF] state "role-play involves staging a problematic situation with characters taking roles". It is used over many years in mathematics education course in UQAM of Québec in Montréal) and is organized as follows:

First, the 'theme' on which students will need to role-play is introduced (introduction time). Second, students then have about 30 minutes to prepare in small groups (preparation time). Third comes the play itself (play time), where students chosen by the educator come in front of the classroom and improvise a teacher-student(s) interaction (sometimes, like in the case reported here, involving the whole class). Finally, we have a whole classroom discussion on the play (discussion time). (Lajoie, accepted) We designed a role-play on the teaching of proportions based on a problem from a textbook. We use the analysis framework to illustrate an example of analysis aimed at highlight the potential of this situation.

An example of role-play

The role-play presented below is intended for pre-service schoolteacher education. We describe the different phases.

Introduction time. The educator distributes to prospective teachers an excerpt from a fifth grade (10-11 year old pupils) handbook presenting a problem of proportions (Fig. 1), and various productions of pupils. The teaching issue announced by the educator is the following: to manage a class discussion about the pupils' strategies and about their ideas and solutions, in order to share them in the class community and to determine their validity and efficiency.

Preparation time. The prospective teachers have to prepare the discussion class about the pupils' strategies.

Play time. At the end of the preparation time, the educator chooses prospective teachers to play the game: some of them play pupils, one of them plays the teacher, while the others are watching the discussion class and taking notes.

Analysis of this role-play

The initiating task is a professional situation and corresponds to level 2 of the analysis framework: the prospective teachers are initially in a student/teacher posture. But they will need « to go down » to as student posture and « to go up » to a teacher posture during the phases of the scenario. We describe this more precisely below.

The preparation time of the discussion class leads the prospective teachers "to go back and forth" to the study levels 0, 1 and 2. The problem solving corresponds to level 0 and the mathematical analysis of the problem solving to level 1. Moreover there are various strategies to solve this proportion problem. Preparing the discussion class of the pupils' strategies (level 2) hence needs to analyse and rank them (from the least elaborate to the most elaborate). This analysis corresponds to level 2.

The prospective teachers don't have the same activity during the playtime. The study level is different according to the role to play: mostly levels 0 and 1 for the students' roles and level 2 for the teacher's role.

The discussion time corresponds to level 2 when the prospective teachers analyse how the discussion class has been managed. But it can also correspond to lower levels, when they discuss about pupils' strategies, difficulties, mistakes and their exploitation during the discussion class.

Various institutionalizations can be considered, according to the knowledge that was developed at different study levels. The institutionalized elements will be more or less developed according to the teacher educator's objectives and progression, the prospective teachers' knowledge, etc. Here are some propositions organized in ascending order of study levels. The teacher educator can institutionalize some mathematical knowledge at stake (level 1) and related to the proportionality field: various methods to solve a proportion problem, the mathematical justifications and the mathematical theories they are relied on. He can situate the proportion problems in the more general category of multiplicative problems, or he can explicit some didactical variables usually at stake in proportion problems (level 3). He also can identify some difficulties or mistakes revealed by the pupils' productions as « usual » and highlight mistaken conceptions: identification of quantities, choice of an adapted strategy, persistence of an « additive model », etc. At least, in regard of the announced objective of the role-play, the teacher educator also can institutionalize some didactical knowledge, relatively to the organization of a discussion class (level 3): formulation and validation in mathematics; teacher's tasks before, during and after the discussion class…

Conclusion

The example of role-play situation shows how the analysis framework can be a tool for an a priori analysis. Moreover this example shows that the organization of the study levels is not chronological but a hierarchical one: the initiating task can be on level 0, 1, 2, 3 or 4. But the transition to lower levels is often necessary. The conceptual maps of the knowledge for teaching developed by [START_REF] Houdement | Autour des stratégies utilisées pour former les maîtres du premier degré en mathématiques[END_REF] or by [START_REF] Ball | Content Knowledge for teaching, what makes it special?[END_REF] have a descriptive, predictive and prescriptive dimension (Ball & al., 2008, p.405). But beyond their interest, (Houdement, 2013, p. 21) stressed the importance of the knowledge reconfiguration in connection with the mathematical content. The analysis framework reports how, during a training situation, the types of knowledge for mathematics' teaching are hinged to one another in connection with the mathematical content.

The analysis framework allows teacher educators to identify the potentialities of a full range of training situations. We intend to extend its use to study other types of training situations (for example e-learning situations). By clarifying the stakes of the various phases of the implementation, the analysis framework reveals various possible strategies for the teacher educator. Thereafter it could be a useful tool for elaborating different training scenarios. Hence, the teacher educator should be able to implement situations in a specific context according to his objectives and constraints (time and period of training, place in a progression which take into account the mathematical, didactical and pedagogical knowledge ever studied…). Besides it is possible to consider a sequence of successive scenarios. The analysis framework can also highlight various possible "training paths", which should reveal the educator's training strategy at a more global scale. A perspective is now to study how teacher educators appropriate this framework and how it supports their teaching practises.

  Discussion time. The debate intends to highlight and to analyse the choices of the « teacher » during the play game: what worked well during the implementation of the discussion class? What was difficult? What seemed to be important? What alternative implementations could be realized? Institutionalization time 7 . The educator institutionalizes the knowledge at stake: he generalizes some elements about how to manage a discussion class or about proportion problems solving.

Figure 1 :

 1 Figure 1: A proportion problem

Table 1 : Characteristics of the five study levels

 1 Each study level is based on the study of the activity of previous levels and involves mathematical, didactical and/or pedagogical knowledge. The change from study level n to study level n + 1 is linked either to a change of the prospective teachers' posture or to a change of degree of decontextualization for at least one type of knowledge (from implicitly mobilized in context to explicit in context, from explicit in context to decontextualized). But the different activities induced by a training situation don't usually appear in a chronological order (from level 0 to level 4). For examples, see the analysis of various training situations developed in French context by the COPIRELEM group (Guille-Biel[START_REF] Winder | Proposition d'un cadre d'analyse de situations de formation des professeurs des écoles[END_REF] Bueno-Ravel and al., accepted). But we think that the analysis could be extended to based on different training strategies. That is why we present here the analysis of a training scenario 6 based on the principle of role-play developed in an international context[START_REF] Lajoie | Role-play by pre-service elementary teachers as a means to develop professional competencies in teaching mathematics[END_REF][START_REF] Lajoie | Learning to act in-the-moment: Prospective Elementary Teachers' roleplaying on numbers[END_REF]

	Study levels	Type of activity	Posture of the PT	Mathematical	Didactical Knowledge	Pedagogical
	0	Mathematical performed effectively or mentally) activity (action	Student	Mobilized implicitly or explicitly in context
	1	Analysis of the mathematical activity	Student Student/teacher		Mobilized Implicitly in context	Mobilized Implicitly in context
	2	Didactical and pedagogical analysis of the mathematical activity	Student/teacher		Explicit in context	Explicit in context
	3	Analysis of the didactical and pedagogical activity	Teacher	Decontextualized
		Problematization of professional			Decontextua-	Decontextua-
	4	issues practices, learning issues and/or related to classroom	Practitioner/re-searcher		lized	lized
		didactical analysis tools

The COPIRELEM is a commission dedicated to the education to the primary school. It is stemming from the network of IREM (French institute of research on mathematical education).

"A situation is characterised within an institution by a set of relations and mutual roles of one or several subjects (student, teacher, etc.) with a milieu, which aims at the transformation of this milieu according to a project. The milieu is composed by all what interacts (physical, cultural or social objects, humans) with the subject in a situation."(Brousseau, 2010, p.2) translated.

According to[START_REF] Houdement | Au milieu du gué: entre formation des enseignants et recherche en didactique des mathématiques[END_REF]), Ball's, Phelps' and Thames' typology doesn't seem to take into account this type of knowledge.

In institutionalization phase[START_REF] Brousseau | Theory of Didactical Situations in Mathematics: Didactique des Mathématiques 1970-1990[END_REF], the teacher gives a cultural (mathematical) status to some knowledge emerging from students' actions during the situation.

We notice that the prospective teachers are not always aware of these postures.

We voluntarily distinguish situations from "scenarios" because we intend to underline the dynamic aspect: a scenario is a set of chronologically organized tasks chosen among all the tasks that constitute the training situation.

We add this new time to the four ones proposed by[START_REF] Lajoie | Role-play by pre-service elementary teachers as a means to develop professional competencies in teaching mathematics[END_REF].


Q Akkaoui

E Capiez-Lernout

C Soize

R Ohayon

Revisiting the experiment of a free-surface resonance of a liquid in a vibration tank using a nonlinear fluid-structure computational model

Keywords: Fluid-structure interactions, Geometric structural nonlinearities, Vibrations, Reduced-order model, Sloshing, Capillarity

. The proposed simulation model allows the experimental setup to be reproduced. The structure undergoes large displacements and large deformations (geometrical nonlinear effects of the structure) that play an important role on the liquid vibrations. The experimental setup is simulated using a large-scale numerical model of the elastic cylindrical tank partially filled with water that is considered as a compressible fluid and that takes into account surface tension and sloshing effects. The results show that for a frequency external excitation in the frequency band [500, 2,500] Hz of the fluid-structure system, unexpected high-amplitude sloshing vibrations are observed in the frequency band [0, 150] Hz. The observed phenomenon, which cannot be reproduced with a linear fluid-structure model, is explained by the transfer of the vibrational energy from the frequency band of excitation into a low-frequency band (and then exciting the first sloshing modes) by a non-direct coupling mechanism between the structural modes and the sloshing modes.

Introduction

This paper deals with a computational study of a fluid-structure system that proposes an explanation to an unusual sloshing resonance observed in an experimental investigation of a vibrating elastic tank partially filled with water, exhibited and investigated in [START_REF] Lindholm | Breathing vibrations of a circular cylindrical shell with an internal liquid[END_REF][START_REF] Abramson | Some experimental studies of the dynamic stability of thin shells containing liquid[END_REF]. The experiments performed in [START_REF] Lindholm | Breathing vibrations of a circular cylindrical shell with an internal liquid[END_REF] were first designed to quantify the effects of an internal liquid on the breathing frequencies of a circular cylindrical thin-walled shell. During these experiments, an unexpected behavior was observed when the structure were subjected to a high-frequency excitation of small amplitude. The unusual results obtained in this first experiment were then investigated in further details by [START_REF] Abramson | Some experimental studies of the dynamic stability of thin shells containing liquid[END_REF]. Indeed, it was observed that a totally unexpected coupling between low-frequency liquid free-surface oscillation and high-frequency shell-wall vibration could occur for a wide range of parameter (cited from [START_REF] Abramson | Some experimental studies of the dynamic stability of thin shells containing liquid[END_REF]). This result is all the more unexpected as the frequency separation between the first eigenfrequencies of the sloshing modes and the first structural modes is of order 100. A discussion on these results is proposed in these papers, which suggests that this unexpected behavior would be based on several unusual characteristics that are not explained readily on the basis of existing theory (cited from [START_REF] Abramson | Some experimental studies of the dynamic stability of thin shells containing liquid[END_REF]). These authors concluded that the source of such coupling could possibly come from a nonlinear coupling between the liquid and the structure for which the source of nonlinearities would not be precisely known. A first attempt for explaining this unexpected phenomenon has been carried out by [START_REF] Chu | A theory for nonlinear transverse vibrations of a partially filled elastic tank[END_REF] using an analytical approach with a linear theory for the structure and a nonlinear one for the free surface of the liquid. These authors conclude that exact quantitative comparisons of theory and experiments may required a more complicated system. In this paper, an alternative is proposed for explaining the observed phenomenon using a nonlinear structure (geometrical nonlinearities) and a linear compressible liquid (acoustic fluid) with sloshing and capillarity effects using a large-scale computational model. The observed experimental results are simulated and the mechanisms of this unexpected free-surface response is detailed, which constitutes the novelty of this work.

Theory and applications of liquid sloshing in elastic structures has been investigated in literature (see for example, [START_REF] Paidoussis | Fluid-structure interactions: slender structures and axial flow[END_REF]Dodge, 2000;[START_REF] Ibrahim | Liquid Sloshing Dynamics: Theory and Applications[END_REF][START_REF] Faltinsen | Sloshing[END_REF][START_REF] Moiseyev | Dynamic stability of bodies containing fluid[END_REF]). The effects of surface tension (capillarity effects) in a rigid structure has also been analyzed (see for instance, [START_REF] Finn | On the equations of capillarity[END_REF][START_REF] Finn | The contact angle in capillarity[END_REF][START_REF] Finn | On the capillary problem for compressible fluids[END_REF]). On the computational level, variational formulations, computational models, and reduced-order models concerning vibroacoustics and fluid-structure interactions can be found, for instance, in [START_REF] Morand | Fluid Structure Interaction[END_REF][START_REF] Ohayon | Structural acoustics and vibration: Mechanical models, variational formulations and discretization[END_REF], 2014[START_REF] Ohayon | Vibration of structures containing compressible liquids with surface tension and sloshing effects. Reduced-order model[END_REF][START_REF] Belytschko | Finite element computation of sloshing modes in containers with elastic baffle plates[END_REF]Schotté andOhayon, 2005, 2013;[START_REF] Farhat | Modeling of fuel sloshing and its physical effects on flutter[END_REF][START_REF] Hambric | Engineering Vibroacoustic Analysis: Methods and Applications[END_REF].

The effects of the geometrical nonlinearities of structures containing liquids have been studied in order to quantify the influence of a fluid on the large amplitude motion of the structure [START_REF] Dowell | Comments on the nonlinear vibrations of cylindrical shells[END_REF]Amabili and Paıdoussis, 2003;[START_REF] Amabili | Nonlinear vibrations and stability of shells and plates[END_REF][START_REF] Strand | Linear sloshing in a 2d rectangular tank with a flexible sidewall[END_REF][START_REF] Alijani | Non-linear vibrations of shells: A literature review from 2003 to 2013[END_REF]. For example, many experimental studies are available for large amplitude vibrations of thin shells partially filled with liquids such as the ones proposed by [START_REF] Olson | Some experimental observations on the nonlinear vibration of cylindrical shells[END_REF][START_REF] Leissa | Vibration of shells[END_REF][START_REF] Chen | Nonlinear vibration of cylindrical shells[END_REF][START_REF] Chiba | Nonlinear hydroelastic vibration of a cylindrical tank with an elastic bottom, containing liquid. part i: Experiment[END_REF][START_REF] Koval'chuk | Experimental study of induced oscillations with large deflections of fiberglass shells of revolution[END_REF] and also by [START_REF] Carra | Experimental study of large amplitude vibrations of a thin plate in contact with sloshing liquids[END_REF] in which the experimental study that is presented could be used as a test case for future nonlinear fluid-structure interaction problem. However, to the best knowledge of the authors, only the Chu-Kanna work [START_REF] Chu | A theory for nonlinear transverse vibrations of a partially filled elastic tank[END_REF] proposed an explanation of this unusual phenomenon put in evidence in [START_REF] Lindholm | Breathing vibrations of a circular cylindrical shell with an internal liquid[END_REF][START_REF] Abramson | Some experimental studies of the dynamic stability of thin shells containing liquid[END_REF][START_REF] Abramson | Some experimental studies of the dynamic stability of thin shells containing liquid[END_REF], by using a linear structure coupled with a nonlinear free-surface effects of the liquid. The analysis of these four last references led us to consider the nonlinear effect of the structure, which was suggested but not investigated by these authors, coupled with a linear fluid. In order to quantify these effects, we have used a formulation that allows for taking into account the nonlinear geometric effects of the structure, while considering a linear formulation for the internal liquid in presence of sloshing and capillarity effects. This paper presents a numerical simulation of the experimental setup, using a large-scale computational model based on the formulation [START_REF] Ohayon | Reduced models for fluid-structure interaction problems[END_REF]Soize, 2015, 2016) and gives an alternative more accurate explanation of the phenomenon observed during the experiments [START_REF] Lindholm | Breathing vibrations of a circular cylindrical shell with an internal liquid[END_REF][START_REF] Abramson | Some experimental studies of the dynamic stability of thin shells containing liquid[END_REF].

This experimental setup is a tank that is partially filled with a compressible liquid (acoustic fluid) that has a free surface. The whole system is submitted to earth gravity field inducing sloshing of the free surface. The computational model of the coupled system is constructed using the finite element method [START_REF] Zienkiewicz | The finite element method[END_REF]. This computational model allows for taking into account the geometrical nonlinearities of the structure (Amabili and Paıdoussis, 2003;[START_REF] Wriggers | Nonlinear finite element methods[END_REF][START_REF] Crisfield | Nonlinear finite element analysis of solids and structures[END_REF][START_REF] Belytschko | Finite element computation of sloshing modes in containers with elastic baffle plates[END_REF]. A nonlinear reduced-order model (ROM) allowing for considerably reducing the computational cost is built. Such computational ROMs adapted to fluid-structure interactions have been extensively studied these last two decades. The method of proper orthogonal decomposition (POD) has been widely used to construct a ROM, for example by (Amabili et al., 2003;[START_REF] Amabili | Reduced-order models for nonlinear vibrations of fluid-filled circular cylindrical shells: comparison of pod and asymptotic nonlinear normal modes methods[END_REF][START_REF] Ballarin | POD-Galerkin monolithic reduced order models for parametrized fluid-structure interaction problems[END_REF]. In this paper, we use the ROM proposed by [START_REF] Ohayon | Reduced models for fluid-structure interaction problems[END_REF]Soize, 2015, 2016), which consists in projecting the equations of the model onto a basis constituted of the modal bases issued from each physical subset of the fluid-structure system, for the following reasons. (i) A better understanding of the couplings between each part of the fluid-structure system can be done. (ii) It allows for filtering the eigenfrequencies of the different physical subsets of the system; it is well known that the frequency gap between the (usually low-frequency) sloshing resonances and the (usually high-frequency) elastic and acoustic resonances is huge; this means that if we were to build a global reduced-order basis, we would have to compute an untractable number of (weakly energetics) sloshing eigenmodes before reaching any acoustic or elastic eigenmodes. (iii) The computation of a global reduced-order projection basis would induce some difficulties related to the non-symmetric global formulation of the coupled problem. (iv) Such ROM allows for being predictive with a reasonable computational time [START_REF] Ohayon | Reduced models for fluid-structure interaction problems[END_REF][START_REF] Ohayon | Advanced Computational Vibroacoustics: Reduced-Order Models and Uncertainty Quantification[END_REF].

As shown in [START_REF] Ohayon | Nonlinear model reduction for computational vibration analysis of structures with weak geometrical nonlinearity coupled with linear acoustic liquids in the presence of linear sloshing and capillarity[END_REF], the modal bases are computed by solving three different generalized eigenvalue problems: the first one being related to the elastic structure in presence of the liquid, considered as incompressible; the second one being related to the compressible liquid in a rigid container; the third one being related to the free surface in presence gravity field and with surface tension effects. In addition, the computation of the reducedorder bases exhibits major difficulties in terms of computational cost when studying a large-scale fluid-structure model due (i) to the high flexural motion of the thin-walled cylinder and (ii) to the free surface. These difficulties are circumvented by using the algorithms presented in [START_REF] Akkaoui | Solving generalized eigenvalue problems for large scale fluid-structure computational models with mid-power computers[END_REF]. The outline of the paper is as follows. Section 2 deals with a summary of the computational model. Section 3 is devoted to the numerical simulation of the experimental setup. The dimensions of the physical system is described, as well as the strategy used to computationally reproduce the experimental boundary conditions. A modal characterization is presented to highlight the frequency separation between the sloshing resonances and the elastic resonances. A partial validation of the computational model is performed by comparing the eigenfrequencies of the tank with the experimental ones. We then end up by showing the forced responses of the coupled system highlighting the geometrical nonlinearities effects on the dynamical behavior of the system and in particular on the amplitude of the free surface motion.

Summarizing the computational model used for the numerical simulations

In this section, we summarize the construction of the computational model detailed in [START_REF] Ohayon | Nonlinear model reduction for computational vibration analysis of structures with weak geometrical nonlinearity coupled with linear acoustic liquids in the presence of linear sloshing and capillarity[END_REF]), which will be used for simulating the experimental setup.

Boundary value problem for the nonlinear fluid-structure problem

The fluid-structure system is considered to be taken in its reference configuration defined in Figure 1. The structure occupies a bounded volume Ω U and is taken in its natural state without prestresses. The structure contains a liquid occupying a bounded domain Ω P . The liquid is modeled by a linear dissipative compressible fluid (dissipative acoustic fluid, see [START_REF] Ohayon | Advanced Computational Vibroacoustics: Reduced-Order Models and Uncertainty Quantification[END_REF]), which means that the vibrational motion of the liquid is assumed to be small enough. Gravitational and surface tension effects are taken into account but internal gravity waves are neglected. It should be noted that the geometry of Ω P is obtained by a first computation allowing the static equilibrium position of the free surface of the liquid to be found, due to the presence of capillarity effects.

The boundary of Ω P , assumed to be smooth enough, is denoted as

∂Ω P = Γ L ∪ γ ∪ Γ (with Γ L ∩ γ = ∅, Γ ∩ γ = ∅, and Γ L ∩ Γ = ∅)
, where Γ L is the fluid-structure interface, Γ is the free surface of the liquid and γ is the contact line between the structure and the liquid. The boundary of Ω U , assumed to be smooth enough, is denoted as

∂Ω U = Γ 0 ∪ Γ E ∪ Γ L ∪ γ ∪ Γ G
where Γ 0 is a part of the boundary in which there is the Dirichlet condition u = 0, where Γ E is the external surface of the structure and Γ G is the structure internal wall without contact with the liquid. The structure is submitted to a given body force field b in Ω U and to a given surface force field f on Γ E . The external unitary normals to ∂Ω U and ∂Ω P are written n u and n. Let ν and ν L be the external unit normals to γ belonging respectively to the tangent plane to Γ and to the tangent plane to Γ L . Let x = (x 1 , x 2 , x 3 ) be the generic point in a Cartesian reference system (O, e 1 , e 2 , e 3 ). The gravity vector is g = -g e 3 with g the gravitational constant. The boundary value problem is expressed in terms of the structural displacement field u(x, t), of the internal pressure field p(x, t), and of the normal displacement field of the free surface η(x, t). From here on, in order to simplify the notations, the convention for summation over repeated Greek and Latin indices is considered and parameters x and t are removed if there is no possible confusion. Let a(x, t) be a given function, the following notations are used : a , j = ∂a/∂x j , ȧ = ∂a/∂t, and ä = ∂ 2 a/∂t 2 .

The nonlinear boundary value problem in (p, η, u) is written as,

1 ρ 0 c 2 0 p - τ ρ 0 ∇ 2 ṗ - 1 ρ 0 ∇ 2 p = 0 in Ω P , (1) 1 + τ ∂ ∂t ∂p ∂n = -ρ 0 ü • n on Γ L , (2) 1 + τ ∂ ∂t ∂p ∂n = -ρ 0 η on Γ , (3)
p = ρ 0 η g (e z • n) -σ Γ 1 R 1 + 1 R 2 η + ∇ 2 Γ η on Γ , ( 4
)
∂η ∂ν = c η η + J u on γ , (5)
ρ S ü -div (F • S) = b in Ω U , (6)
u = 0 on Γ 0 , (7)
115

(F • S) • n u = f on Γ E , (8)
(F • S) • n u dΓ L = p n u dΓ L -σ Γ (J η) dµ γ on Γ L , (9)
in which the tensor F is the deformation gradient tensor defined in Ω U by

F i j = δ i j + u i, j , (10)
where δ i j denotes the Kronecker symbol such that δ i j = 1 if i = j and δ i j = 0 otherwise. In Eq. ( 10), the tensor S is the second Piola-Kirchhoff symmetric tensor in the reference configuration written as

S i j = a i jk E k + b i jk d E k dt , (11)
where the tensors a i jk and b i jk are fourth-order real tensors, verifying the symmetry and positiveness properties and

120
where the Green-Lagrange strain tensor is written as

E k = 1 2 u k, + u ,k + u m,k u m, . (12)
In Eqs.

(1) to (3), ρ 0 is the constant mass density of the homogeneous liquid, c 0 is the constant speed of sound, and τ is the constant coefficient that characterizes the dissipation in the internal liquid. In Eqs. ( 4) and ( 5), σ Γ is the surface tension coefficient, R 1 and R 2 are the main curvature radii of the free-surface, and c η is the contact angle coefficient.

In Eq. ( 6), ρ S is the constant mass density of the structure. Equations ( 4) and ( 9) correspond to the contact angle between the elastic structure and the free surface of the liquid in which the operator J is such that

J u = E u • n u - ∂(u • n u ) ∂ν L , (13)
in which E a real coefficient. In Eq. ( 9), dµ γ is a real measure on Γ L such that Γ L f (x) dµ γ (x) = γ f (x) dγ(x) (this means that the support of measure dµ γ is γ), and the term (J η) dµ γ is defined on Γ L by algebraic duality of the term J u defined on γ.

Nonlinear computational model

Let n p , n h , and n u be the number of degrees of freedom related to the fluid, to the free surface, and to the structure, issued from the finite element discretization of the system. We then denote by P, H, and U the R n p -vector, R n hvector, and R n u -vector corresponding to the finite element discretization of the fields p(x, t), η(x, t), and u(x, t). The computational finite element model of the fluid-structure system is then written as

[M] P + [D] Ṗ + [K] P -[C pη ] T Ḧ -[C pu ] T Ü = 0 , ( 14
) [C pη ] P + [K gc ] H + [C ηu ] U = 0 , ( 15
) [C pu ] P + [C ηu ] T H + [M u ] Ü + [D u ] U + [K u ] U + F NL (U) = F u . (16)
In Eq. ( 14), the (n p × n p ) symmetric positive-definite matrix [M] is the fluid "mass" matrix. The (n p × n p ) positivesemidefinite matrices [K] and [D] = τ [K] are the "stiffness" and "damping" matrices of the liquid. In Eq. ( 15), the (n h × n h ) symmetric positive-definite matrix [K gc ] is the matrix related to the sloshing/capillarity effects on the free surface. In Eq. ( 16 16) the R n u -vector F u corresponds to the discretization of the external force field applied on the structure. The R n u -vector F NL (U) is the discretization of the nonlinear conservative internal forces induced by the geometrical nonlinearities of the structure. The numerical solution of the nonlinear coupled differential equations (Eq. ( 14) to ( 16)) appears to be expensive or even impossible for large finite element models. It is thus essential to introduce an efficient adapted reduced-order model as we have explained in Section 1.

Construction of the nonlinear reduced-order computational model

The admissible space C P,H,U is decomposed in the following direct sum,

C P,H,U = C P ⊕ C H ⊕ C U , (17)
where the admissible spaces C P , C H and C U are defined as follows.

Vector basis of C P . The admissible space C P is related to the conservative part of the discretized problem in P (see Eq. ( 14)) for which a zero Neumann boundary condition for the pressure is applied on Γ L and a zero Dirichlet boundary condition for the pressure is applied on Γ ∪ γ. Therefore, a vector basis of C P can be constructed as the acoustic modes, which are the eigenvectors of the generalized eigenvalue problem (in which the boundary conditions have to be added),

[K] [Φ P ] = [M] [Φ P ] [Λ P ] . (18)
In Eq. ( 18), [Φ P ] = [ϕ 1 p , . . . , ϕ N P p ] is the (n p × N P ) real matrix whose N P columns are constituted of the eigenvectors associated with the N P first smallest positive eigenvalues sorted by increasing order such that λ 1 p ≤ . . . ≤ λ N P p contained in the diagonal matrix [Λ P ].

Vector basis of C H . The admissible space C H is related to the conservative part of the discretized problem in H (see Eq. ( 15)) for which the liquid is considered as an inviscid incompressible liquid, with sloshing and capillarity, and for which a zero Neumann boundary condition for the pressure is applied on Γ L . Therefore, a vector basis of C H can be constructed as the sloshing modes of the liquid in presence of capillarity, which are the eigenvectors of the generalized eigenvalue problem (in which the boundary condition have to be added),

[K] [Φ PH ] + [C pη ] T [Φ H ] [Λ H ] = 0 , ( 19
) [C pη ] [Φ PH ] + [K gc ] [Φ H ] = 0 . ( 20
)
In Eqs.( 19) and ( 20) Vector basis of C U . The admissible space C U is related to the conservative and homogeneous linear part of the discretized problem in U (see Eq. ( 16)) related to the structure coupled with a liquid that is inviscid and incompressible, with a zero Dirichlet boundary condition for the pressure on Γ ∪ γ (in addition to the Dirichlet condition on Γ 0 ). Therefore, a vector basis of C U can be constructed as the elastic modes of the underlying linear structure with the fluid added mass, which are the eigenvectors of the generalized eigenvalue problem

, [Φ H ] = [ϕ 1 h , . . . , ϕ N H h ] and [Φ PH ] = [ϕ 1 ph , . . . ,
[K u ] [Φ U ] = ( [M u ] + [M A ] ) [Φ U ] [Λ U ] . (21)
In Eq. ( 21), the (n u × n u ) symmetric positive-definite added-mass matrix [M A ] is defined by

[M A ] = [C pu ] [K * ] -1 [C pu ] T , (22)
where Computational aspects related to the calculation of the bases. For large-scale 3D computational models, solving the three generalized eigenvalue problems with standard algorithms induced major difficulties for mid-power computers. This is the reason why non-standard algorithms have been proposed in [START_REF] Akkaoui | Solving generalized eigenvalue problems for large scale fluid-structure computational models with mid-power computers[END_REF] for solving these generalized eigenvalue problems. The limitations of the computer resources are principally due to the RAM limitations and a prohibitive CPU time. A double projection method and a subspace iteration method have been proposed and allow for solving such a problem that cannot be treated with standard algorithms.

Reduced-order computational model. The nonlinear reduced-order computational model is constructed by projecting the equations of the fluid-structure system (defined by Eqs. ( 14) to ( 16)) on the reduced-order basis [Ψ] constituted of the vector bases constructed above. The solutions P, H, and U are then written as

          P H U           =           [Φ P ] [Φ PH ] 0 0 [Φ H ] 0 0 0 [Φ U ]                     q p q h q u           = [Ψ] Q , ( 23
)
in which Q = [q p q h q u ] T is the R N phu -vector of the generalized coordinates, with N phu = N p + N h + N u , solution of the reduced nonlinear differential equation

[M FS I ] Q + [D FS I ] Q + [K FS I ] Q + F NL (Q) = F , (24)
in which the (N phu × N phu ) matrices [M FS I ], [D FS I ], [K FS I ], and the R N phu vector F are defined as

[M FS I ] = [Ψ] T           [M] -[C pη ] T -[C pu ] T 0 0 0 0 0 [M u ]           [Ψ] , (25)
[

D FS I ] = [Ψ] T           [D] 0 0 0 0 0 0 0 [D u ]           [Ψ] , (26)
[

K FS I ] = [Ψ] T           [K] 0 0 [C pη ] [K gc ] [C ηu ] [C pu ] [C ηu ] T [K u ]           [Ψ] , (27)
F = [Ψ] T           0 0 F u           . ( 28
)
In Eq. ( 24), F NL (Q) denotes the nonlinear conservative contribution of the geometrical nonlinearities of the structure. The generalized conservative internal nonlinear forces are written [START_REF] Mignolet | Stochastic reduced order models for uncertain geometrically nonlinear dynamical systems[END_REF] as

F NL (Q) = K (2) αβγ q u β q u γ + K (3) αβγδ q u β q u γ q u δ , (29)
where the quadratic and cubic stiffness K (2) αβγ and K (3) αβγδ are such that

K (2) αβγ = 1 2 K (2) αβγ + K (2) βγα + K (2) γαβ , (30)
K (2) αβγ = Ω S a i jk ϕ α i, j ϕ β m,k ϕ γ m, dx , (31)
K (3) αβγδ = 1 2 Ω S a i jk ϕ α s,i ϕ β s, j ϕ γ m,k ϕ δ m, dx , (32)
The numerical aspects of the generalized nonlinear forces are detailed in [START_REF] Capiez-Lernout | Computational stochastic statics of an uncertain curved structure with geometrical nonlinearity in three-dimensional elasticity[END_REF].

3. Simulating the experimental setup with the nonlinear reduced-order computational model

Experimental setup, computational model, and updating the structural model with experimental eigenfrequencies of the structure in vacuo

The experimental setup is the one described in [START_REF] Lindholm | Breathing vibrations of a circular cylindrical shell with an internal liquid[END_REF][START_REF] Abramson | Some experimental studies of the dynamic stability of thin shells containing liquid[END_REF]. The structure is a steel tank constituted of a thin circular cylinder closed at both ends by circular plates. Its isotropic material properties are given by E = 2.05 × 10 11 N.m -2 , ν = 0.29, ρ S = 7 800 Kg.m -3 , and damping coefficient τ S = 10 -6 . This tank is filled with 30% water with sound velocity c 0 = 1 480 m.s -1 , mass density ρ 0 = 1 014 kg.m -3 , and damping coefficient τ = 10 -5 .

Since capillarity effects are taken into account in the computational model, a water-steel contact angle of θ = 83 • has been considered to compute the equilibrium position of the free surface of the liquid using the software Surface Evolver [START_REF] Brakke | The surface evolver[END_REF]. Let R e = 3.7833 × 10 -2 m and R i = 2.76047 × 10 -2 m be the external and the internal radii of the steel cylinder. The thickness of the end plates and the thickness of the cylindrical shell are h d = 2.54 × 10 -2 m and e = 2.286 × 10 -4 m. The total height of the tank and the fluid depth are h = 0.23876 m and h f = 0.3 h. The geometry of the computational model is described in Figure 2 (left). The origin O of the Cartesian coordinates system (O, e 1 , e 2 , e 3 ) is located at the center of the bottom of the cylindrical tank. Axis e 3 coincides with the revolution axis of the system. A particular attention is paid to the modeling of the boundary conditions of the experimental setup. Thus, the structural node located at the bottom center of the tank is locked along its directions e 1 , e 2 , and e 3 . The node located at the top center of the tank is locked along its directions e 1 and e 2 . The rotation of the bottom plate is locked with one dof on the edge of the plate. The boundary conditions described hereinabove (represented by • symbol) are displayed in Figure 2 (right figure). Moreover, the experimental pin-ended and cantilever boundary conditions are set in order to study the breathing vibrations of the cylinder. In the present numerical simulation, the structural model in vacuo has been calibrated by optimizing the thickness of the bottom and top plates such that the numerical eigenfrequencies of the structure coincides with the experimental ones. The optimization must take into account the fact that the plates have to stay rigid compared to the cylinder wall. This is why the optimal thickness of these plates has been set as h d = 2.54 × 10 -3 m. The finite element model of the fluid-structure system is constructed using 3D-solid tetrahedral finite elements with 10 nodes for the structure and for the acoustic fluid (see Figure 3). The free surface of the liquid is meshed using 2D finite elements with 6 nodes and the triple line is meshed using 1D finite elements with 3 nodes. Table 1 sums up the characteristics of the finite element mesh. This high-order finite element modeling is chosen in order to accurately simulate the highly flexible motion of the thin shell and also of the liquid free surface. This choice has been motivated by the fact that there are few difficulties for obtaining correct structural eigenfrequencies and representative sloshing eigenmodes when using linear interpolation functions for finite elements with order 1. Consequently, the finite element mesh is constituted of finite elements with order 2, which allows for accurately representing the dynamical behavior of the system. It is important to note that only the eigenfrequencies are available from the experiments.

Observation points. The dynamical response of the fluid-structure system is analyzed for different observation points of the finite element mesh. These observations points are chosen to be of best interest for the dynamical study of the system. One of them is common to both the structure, the fluid, and the free surface. It allows for seeing the correspondance between the pressure in the acoustic fluid, the displacement of the structure, and the elevation of the free surface. This common point is denoted as x p 1 in the fluid, x h 1 on the free surface, and x u on the structure. Moreover, another observation point x p 2 is the point located at the bottom of the fluid is also chosen. Finally, another observation point x h 2 is also chosen as the point located at the center of the free surface. For the sake of clarity, the coordinates of these observation points are summarized in Table 2. The quantities P i and H i denote the pressure and the elevation of the free surface at observation points x p i and x h i (for i = 1, 2), and U j is the component j (for j = 1, 2, 3), of the structural displacement at observation point x u . sloshing eigenmodes in the liquid are displayed in Figure 5 (lower figure), for which we can see that the pressure exponentially decreases with respect to the depth of the liquid, as it was expected.

Location

Structural modes with added mass effects. The geometry of the structure has preliminarily been updated in order to match, for the best, the experimentally measured eigenfrequencies for the structure in vacuo. Figure 6 quantifies this updating and displays the graph of the structural eigenfrequencies of the structure in vacuo with respect to the longitudinal and circumferential wave numbers m and n. It should be noted that all the calculations have been carried with the 3D computational model and that the longitudinal and circumferential wave numbers of the structural modes have been identified from the 3D calculations (note that the identification of the values of n and m has been done by examining the 3D plots of the modes). It must also be noted that, in the reduced-order model, all the structural modes have been taken into account in the frequency band: the structural modes n = 0 and all the modes n ≥ 1 for which their eigenfrequencies have a multiplicity greater that one (due to the symmetries). The updated computed structural eigenfrequencies are displayed in blue solid line and the experimental ones [START_REF] Lindholm | Breathing vibrations of a circular cylindrical shell with an internal liquid[END_REF] in red triangles. One can see that the updated computational model correctly fits the experimental eigenfrequencies. Concerning the structural modes for which no experiments are available, we have only the computed structural modes. Figure 7 displays two structural modes (rank 6 and 12) in vacuo (two left figures) and displays the corresponding two structural modes (rank 6 and 10) with the added mass effect involved by the liquid for the structure filled with 30% water (two right figures). It is shown that the presence of the liquid locally modifies the in vacuo structural modes in the spatial area where the liquid is in contact with the structure. A decrease of its structural eigenfrequencies is observed (as expected). It should be noted that the structural modes could be constructed for the structure in vacuo. Nevertheless, for this elastic tank partially filled with water, the use of the structural modes computed with the effects of the added mass, yields a much faster speed of convergence of the reduced-order model with respect to the truncation order. Concerning the structural modes in vacuo identified as n = 0, the rank of the first one is 89 and its eigenfrequency is 4 509 Hz. The rank of the corresponding structural mode with the added mass effects is 32 and its eigenfrequency is 1 852 Hz (there is an important shift due to the added mass). It can be seen that this first n = 0 structural mode with added mass effects has an eigenfrequency that belongs to the frequency band of excitation. There is no other n = 0 structural mode with added mass effects whose rank is less than or equal to 100.

3.4. Eigenfrequency characterization of the system in the frequency band of analysis B a Physically there are sloshing modes in the frequency band of excitation B e but their contributions in this band B e are negligible. Consequently, only the sloshing modes whose eigenfrequencies are below 500 Hz are kept for constructing the ROM. Note that the sloshing eigenfrequencies increase at a very slow rate since ν h 500 = 57.78 Hz. The first 3.5. Solver for computing the nonlinear response in the time domain

The nonlinear coupled differential equation Eq. ( 24) is solved by using an implicit iterative Newmark scheme for which, at each time step, the nonlinear algebraic equation is solved using the fixed point method or the Newton-Raphson algorithm when the fixed point method is not convergent. Once the time response has been computed, a Fourier transform is performed to analyze the frequency response of the system in the frequency band of analysis B a = [0, 6 000] Hz.

3.6. Convergence of the reduced-order model with respect to the modal truncation

The nonlinear reduced-order model presented in Section 2.3 is constructed using the modes calculated in Section 3.3. A convergence analysis of the nonlinear response is performed with respect to the order of the nonlinear reduced-order model. Since the reference solution cannot be computed for such large-scale nonlinear computational model, it is assumed that the convergence is reached when the response is no longer sensitive to parameters N p , N h or N u . We choose to normalize the convergence function with respect to the solution constructed with N p = 200, N h = 2 250, and N u = 100. The convergence functions are then defined by,

Conv X (N p , N h , N u ) = 1 B a | X(N p , N h , N u , 2πν) | 2 dν 1/2 B a | X(N p , N h , N u , 2πν) | 2 dν 1/2 , (36)
for the Fourier transform X of a solution X at a given observation point. Three convergence analyses are then performed with respect to N p , with respect to N h , and with respect to N u . Figure 8 displays the graphs of the following convergence functions,

N p → Conv X (N p , N h , N u ) , N h → Conv X (N p , N h , N u ) , N u → Conv X (N p , N h , N u ) . (37)
It is shown that the convergence analyses yields an optimal size of the ROM of N p = 40, N h = 1 500, and N u = 60. It should be noted that the large number of sloshing eigenmodes required for the convergence of the solution is due to the high modal density of the sloshing modes.

Concerning the nonlinear geometrical effect that induces a coupling between displacements of type n = 0 and of type n ≥ 1, the reduced-order model has carefully been constructed and validated by the convergence analysis presented in Section 3.6 for Nu up to 100 (rank of structural modes with added mass effects). The first structural mode with added mass effects, of type n=0, is inside the frequency band of excitation and is kept in the reduced-order model. As the boundary condition of the cylinder at its top end is free in the vertical direction, the structural modes with added mass effects, of type n=0, whose eigenfrequencies are greater than the frequency band of analysis, should a priori be less important.

3.7. Linear and nonlinear dynamical responses of the fluid-structure system As previously, the notation A(2πν) denotes the Fourier transform of quantity A(t). The geometrically nonlinear effects induced by the structure are analyzed by comparing the linear and the nonlinear dynamical responses of the system for a given external excitation. For notational convenience, we denote by superscript L the quantities calculated with the linear system and by superscript NL the quantities calculated with the nonlinear system. For i = {1, 2}, Figure 9 displays the graph of ν → | P i (2πν)| for the acoustic pressure computed with the linear ROM and with the nonlinear one. First, one can see that numerous resonances arise outside the excitation frequency band B e due to the structural nonlinearities. The contributions of most of the elastic modes located above 2 500 Hz appear in both nonlinear responses. On the other hand, below 500 Hz and for these two observation points, the sloshing modes are not detected due to the exponential decreasing of the sloshing-induced pressure with the depth. In addition, for the nonlinear responses, the resonance peaks located inside B e are largely spread out with respect to the ones obtained by the linear responses. This is the case, for example, for the resonance located at 2 089 Hz. For i = {1, 2, 3}, Figure 10 ROM and with the nonlinear one. In addition, one can see that some sloshing modes below 500 Hz contribute to the 355 structural displacements. The structural nonlinearities strongly modify the frequency responses in excitation band B e ; the resonance peaks are not only spread out, as for the acoustic pressures, but they are also less isolated and less acute; this is due to the fact that the energy of the responses in the frequency band B e is partially transferred outside this band yielding an apparent damping by this transfer of mechanical energy. The dynamical behavior of the structure is thus significantly damped as it can be seen that the resonances levels inside B e are strongly damped by a factor up to 100. Note that the frequency spreading effect is again observed for the resonance located at 872 Hz.

For i = {1, 2}, Figure 11 displays the graph of ν → | H i (2πν)| for the free-surface elevation computed with the linear ROM and with the nonlinear one. It can be seen an unexpected response in frequency band [0 , 150] Hz (below the excitation band B e ); the first sloshing modes are excited by the energy transferred in this band, [0 , 150] Hz, due to the structural nonlinearities. Let us note that, for the same reasons as the ones given for the structural displacements, an apparent damping by a factor up to 100 occurs in excitation band B e . In Figure 12, when zooming in the sub-frequency band [0 , 150] Hz, one can see that numerous isolated sloshing resonances of high amplitude appear. It is essential to underline that there is a high-amplitude low-frequency free-surface motion while no external excitation exists in this very low-frequency band. The experimental results presented in [START_REF] Lindholm | Breathing vibrations of a circular cylindrical shell with an internal liquid[END_REF][START_REF] Abramson | Some experimental studies of the dynamic stability of thin shells containing liquid[END_REF], concerning the unexpected free-surface elevation of liquid in a vibrating cylindrical shell, have been reproduced in the present work that clearly exhibits the same unexpected phenomena. The understanding of such complex mechanisms requires to quantify the influence of the different coupling operators involved in the dynamical behavior of the free surface. The investigations that have been done for understanding this phenomenon show that the dynamical response of the free surface is weakly sensitive to the coupling operator C ηu , meaning that the direct energy transfer between the structure and the free surface is almost negligible. The complete analysis of this phenomenon is detailed in the next section.

Discussion and conclusion

In this paper, we have simulated, using with a large-scale nonlinear computational model, an experimental setup made up of a thin elastic cylinder partially filled with water, for which the available experimental results are limited to the eigenfrequencies of the structure in vacuo but for which unexpected high-amplitude of sloshing phenomenon has been experimentally observed at low-frequency for a high-frequency excitation, which cannot be explained by a linear model. The updated computational model that has been used takes into account the geometrical nonlinearities of the tank, the compressibility of the liquid, the sloshing and the capillarity effects with an adapted contact angle condition. The geometrical nonlinearities of the tank allow for highlighting the phenomenon observed in the experiments.

Considering the numerical results obtained, it appears that the high-amplitude motion of the free surface in the very-low frequency band outside the frequency band of excitation, can be explained by the the couplings illustrated in Figure 13 and explained hereinafter. The geometrical nonlinearities of the structure induce a transfer of the vibrational energy from the high-frequency band of excitation in the very low-and low-frequency band (outside the frequency band of excitation). As the first acoustic modes of the liquid is greater that the upper bound of the high-frequency band of excitation, the acoustic fluid has a quasistatic behavior in the low-frequency band. Therefore, the energy transferred by the structure in the very low-frequency band is transmitted through the acoustic liquid to the first very low-frequency sloshing modes. This means that the observed phenomenon appears to be an indirect transfer of energy from the structure to the free surface through the acoustic liquid and is due to the presence of the geometrical nonlinearity of the elastic tank.

Figure 13: Scheme of the energy transfers between the nonlinear structure, the linear acoustic liquid, and the linear free surface, which explain the excitation of the first sloshing modes in the the very-low frequency band that is outside the frequency band of the external excitations.

Figure 1 :

 1 Figure 1: Reference configuration of the coupled fluid-structure system

  ), only the linear part of the damping forces are kept and are represented by the term [D u ] U. Consequently, the nonlinear forces F NL are independent of U and therefore, depend only on U. The (n u ×n u ) symmetric positive-definite matrix [M u ] is the structural mass matrix. The (n u × n u ) symmetric positive-definite matrices [K u ] and [D u ] = τ S [K u ] are the stiffness and damping matrices of the structure. The rectangular (n u × n p ), (n h × n p ), and (n h × n u ) matrices [C pu ], [C pη ], and [C ηu ] are the coupling matrices between the fluid and the structure, between the fluid and the free surface, and between the structure and the free surface. In Eq.(

  [K * ] -1 denotes the solution in [P] of the linear matrix equation [K] [P] = [I] (in which [I] is the identity matrix) for which, each column P j of [P] satisfies the constraint P j = 0 for the dofs related to Γ ∪ γ. In Eq. (21), [Φ U ] = [ϕ 1 u , . . . , ϕ N U u ] is the (n u × N U ) real matrix whose N U columns are constituted of the eigenvectors associated with the N U first smallest positive eigenvalues sorted by increasing order such that λ 1 u ≤ . . . ≤ λ N U u contained in the diagonal matrix [Λ U ].

Figure 2 :

 2 Figure 2: Dimensions of the fluid-structure system (left) and representation of the numerical boundary conditions applied on the system (right).

Figure 3 :

 3 Figure 3: Finite elements meshes of the structure with 1 981 155 dofs (left figure), of the liquid with 431 354 dofs (right figure), and of the free surface with 11 566 dofs (top of the right figure).

Figure 5 :

 5 Figure 5: Example of 4 sloshing eigenmodes of the free surface (top) and the corresponding pressure in the acoustic liquid (bottom).

  displays the graph of ν → | U i (2πν)| for the structural displacement computed with the linear

Figure 8 :

 8 Figure 8: Graph of the convergence functions for P, H, and U at the observation points with respect to N p (top left), to N h (top right), and to N u (down central).

Figure 9 :

 9 Figure 9: Linear and nonlinear responses of the acoustic pressure. Graphs of ν → | P 1 (2πν)| (left) and ν → | P 2 (2πν)| (right): pressure computed with the linear ROM (blue line or smooth down grey line for black and white printing) and pressure computed with the nonlinear ROM (red line or irregular top grey line for black and white printing).

Figure 10 :

 10 Figure 10: Linear and nonlinear responses for the structural displacement. Graphs of ν → | U 1 (2πν)| (top left), ν → | U 2 (2πν)| (top right), and ν → | U 3 (2πν)| (down central): structural displacements computed with the linear ROM (blue line or relatively smooth down grey line for black and white printing) and structural displacements computed with the nonlinear ROM (red line or irregular top grey line for black and white printing).

Figure 11 :

 11 Figure 11: Linear and nonlinear responses for the free-surface elevation. Graphs of ν → | H 1 (2πν)| (left) and ν → | H 2 (2πν)| (right): free-surface elevation computed with the linear ROM (blue line or relatively smooth down grey line for black and white printing) and free-surface elevation computed with the nonlinear ROM (red line or irregular top grey line for black and white printing).

Figure 12 :

 12 Figure 12: Zoom of the linear and nonlinear responses for the free-surface elevation. Graphs of ν → | H 1 (2πν)| (left) and ν → | H 2 (2πν)| (right) displayed over the frequency band [0 , 150] Hz: free-surface elevation computed with the linear ROM (blue line or down grey line for black and white printing) and free-surface elevation computed with the nonlinear ROM (red line or top grey line for black and white printing).

  ϕ N H ph ] are the (n h × N H ) and the (n p × N H ) real matrices whose N H columns are constituted of the eigenvectors associated with the N H first smallest positive eigenvalues sorted by increasing order such that λ 1 h ≤ . . . ≤ λ N H h contained in the diagonal matrix [Λ H ].

Table 1 :

 1 Table of the finite element mesh properties.

		Nodes	Dof	Elements
	Fluid	431 354	431 354	296 459
	Free surface 11 566	11 566	5 684
	Structure	660 385 1 981 155	334 784

Table 2 :

 2 Coordinates of the observation points for the fluid, the structure and the free surface

		Name x coordinate y coordinate z coordinate
	Fluid Free surface Structure	x p 1 x p 2 x h 1 x h 2 x u	0.0187 0.0187 -0.0144 0 -0.0144	0 0 -0.0347 0 -0.0347	0.0076 0.0708 0.1248 0.1245 0.1248

ν h 1 = 3.44 Hz ν h 20 = 8.28 Hz ν h 25 = 8.71 Hz ν h 107 = 19.67 Hz

Acknowledgements

This work has been partially funded by DGA in France through a PhD grant to the first author.

Dynamical excitation of the system

The dynamical excitation is a time-dependent force for which its Fourier transform is a constant in the frequency band of excitation B e = [ν min , ν max ] Hz, with ν min = 500 Hz and ν max = 2 500 Hz. The nonlinear responses is computed in the time domain. The Fourier transform of the time response allows for obtaining the response in the frequency domain. The external load vector, which was denoted by F u , is written as

in which α is the intensity coefficient taken as α = 4, g(t) is the time function of the dynamical excitation, and F is the normalized vector representing the spatial distribution of the external time load. The excitation is located on a small rectangular patch, which is radially oriented (see the red patch in Figure 2 left). In Eq. ( 33), the signal g(t) is written as

in which ∆ν = ν max -ν min , s = ν max + ν min 2∆ν , and sinc π (x) = sin(π x) π x . Therefore, the Fourier transform of g(t) is written as

which is 1 if ν ∈ B e and 0 otherwise. The main computational difficulty is to well capture the very low-frequency sloshing modes of the free surface. Consequently, the time duration of the response computation has to be sufficiently high for obtaining a good frequency resolution and to be such that the response at the final time corresponds to the static equilibrium. The computation is carried out on a truncated time domain [t ini , t ini + T ] with a long time duration T for which t ini = -1.28 s and T = 21 s. The sampling frequency and the number of time steps are chosen as ν e = 25 000 Hz and N t = 524 288.

Computation of the reduced-order bases

The computation of the reduced-order bases is performed by solving the generalized eigenvalue problems presented in Section 2.3. The computed elastic, acoustic, and sloshing-capillarity modes are indexed by their eigenfrequencies denoted by ν a i = (λ a i ) 1/2 /2π, for i = 1, . . . , N a with a ∈ {p, h, u}. Acoustic modes.