F Baccar
email: fedia.baccar@ims-bordeaux.fr

F Le Henaff

L Théolier

S Azzopardi

E Woirgard

Electrical Characteristics Evolution of the Deep Trench Termination Diode Based on a Finite Elements Simulation Approach

The main contribution of this work consists in showing the possibility to use the Cyclotene 4026-46 BCB (BenzoCycloButen) resin in thick layer to realize Deep Trench Termination (DT 2 ). The development of the DT 2 in power devices strongly depends on its reliability. 2D finite elements simulations were used to determinate the electrical characteristics after mechanical stresses created in the structure. It appears that void created inside the structure does not affect the structure's characteristic; however it changes when a quantity of charge was added at interface Silicon/BCB.

Introduction

Recently, deep trench structure has become one of the design options for semiconductor devices due to the advancement in the Micro Electro Mechanical Systems (MEMS) process technology [START_REF] Kamibaba | Design of trench termination for High Voltage Device[END_REF]. The DT 2 was presented in 2009 [START_REF] Théolier | A new junction technique: the Deep Trench Termination (DT 2 )[END_REF] and then, several studies [START_REF] Kamibaba | Design of trench termination for High Voltage Device[END_REF][START_REF] Seto | Universal trench edge termination design[END_REF] have proposed electrical improvements. Many studies demonstrate the possibility to use BCB passivation layers in Wafer Level Chip Scale Package (WLCSP) process [START_REF] Lee | Thermo-Mechanical Reliability of the Benzocyclebuten (BCB) film in a WLCSP process[END_REF] or use it in order to create moisture sensor [START_REF] Tetelin | Water solubility and diffusivity in BCB resins used in microelectronics packaging and sensor applications[END_REF], but no reliability study has been done on power devices.

The first aim of this work is to study the reliability of a 1200 Volts Deep Trench Termination diode. DT 2 structure is one of the candidates for future high voltage power devices edge termination design for reducing the chip area and improves the reverse voltage. Then, in a second step, it will be interesting to avoid long sample preparation for optical analyses and to propose simulations in order to explain ageing phenomena. Indeed, it takes a lot of time to prepare samples for optical observations, due to the care needed to go through coating and polishing. Moreover, the water used for polishing may inflate the BCB volume and defects may appear which will not be the result of ageing. We propose to use finite elements simulation, to justify the electrical variations that can be observed after passive thermal ageing. Several degradations have been implemented in the simulated structure such as creating voids and adding traps to the interface between BCB and silicon.

Presentation of the Deep trench Termination structure

The top view schematic of the 1200V structure is presented in Figure 1. Figure 2 shows the 2D cross section structure of the DT 2 diode. The diode is composed by a 10 14 cm -3 N - concentrations substrate with a P + implantation in top layer and a N + implantation in the bottom layer. A deep trench (105µm x 75µm) is filled by BCB in order to realize the termination. A field plate is needed to draw out the electrostatic potential in the trench.

Presentation of the limits of the optical observations

Figure 3 describes electrical reverse characteristics of two DT 2 diodes before and after 50 hours passive of thermal ageing (-40°C, +125°C). This figure exhibits the breakdown voltage variations after ageing. An increase or decrease of 100 volts can be observed. In order to explain these variations, samples are prepared for optical observations. Figure 4 presents optical observation of the real component, after 50 hours passive thermal ageing (-40°C, +125°C). Optical observation does not highlight that peeling of the resin and silicon can explain these variations, and hence, more analyses are requested. Therefore, a simulation approach is proposed.

Thermal variations induce mechanical stress; so it is possible that a separation of materials along the Si/BCB interface and fixed charges appear.

ANSYS is used to localize the maximum stress in the structure after an increase in temperature and TCAD SENTAURUS is used to visualize the electrical characteristics after degradations created in the structure.

ANSYS simulations

First of all, using ANSYS Modeler software, a 3D structure of our chip composed of silicon, BCB and gold layers (see table 1 for material properties) was created. Due to the presence of two symmetries, the final structure is 1/4 of the initial structure, as shown in Figure 5. This simplification was used to reduce the simulation time. The major part of the structure was meshed into rectangle elements except for the area near the curved lines where regular trapezia elements were used. The model loading was an isothermal temperature applied to the entire structure in order to obtain the mechanical stress distribution in the structure. FEM analysis has been useful to understand the behavior of the chip under thermal environment (125°C and isotherm cases) and areas under important mechanical stress have been located on the interface BCB/silicon).

TCAD SENTAURUS simulations

The simulated structure is composed by two finite element cells, a base cell and the termination (Figure 8). In order to consider the real die, it is necessary to take into account the different surfaces for each cell. It is necessary to define the Area Factor for the base cell regardless of the termination one. Sentaurus Device is a single-device simulator, and a mixed-mode device and circuit simulator. A singledevice command file is defined through the mesh, contacts, physical models, and solve command specifications [START_REF][END_REF]. For a multi-device simulation, the command file must include specifications of the mesh (File section), contacts (Electrode section), and physical models (Physics section) for each device. A circuit netlist must be defined to connect the devices ( Figure 9), and solve commands.

Figure 9: multi device simulation and its circuit netlist.

Sentaurus Device also provides a number of compact models for use in mixed-mode; in this work Spice has been implemented for defined the system of the circuit (Figure 10): D1, DT present respectively the central and elementary cell.

A. Void created in the structure

A.1. Breakdown voltage variations

At the interface Si/BCB, more precisely in internal interface of the termination (Area 1) and external interface of the termination (Area 2), a void with different width (Wv) has been created to observe its impact on the structure.

Figure 11 presents the reverse characteristics after created the void in different parts of the device. A same observation can also be obtained in Area 1 and Area 2 (Figure 8). These figures exhibit that a separation at the interface does not affect the breakdown voltage. This is explained by the fact that the electric field is concentrated at the end of the field plate and does not make change in this region.

The insertion of a void in the structure at the interface Si/BCB modifies the distribution of potential lines due to the low permittivity of the void (Figure 12). This change creates a local increase of the electric field, but remains very low and has no impact in the breakdown voltage of the structure.

Figure 13 presents the impact of the void created in the structure on electrical field along a horizontal cutline. It is interesting to observe that electrical field increases in the region near the void; however it has no effect in other regions, that confirms a separation at the interface does not affect the breakdown voltage.

A.2. Parasitic capacitance variation

When a reverse voltage is applied to a PN junction, the holes in the P-region are attracted to the anode terminal and electrons in the N-region are attracted to the cathode terminal. The resulting region contains almost no carrier, and is called the depletion region. The depletion region similarly acts to the dielectric of a capacitor, hence the creation of the capacity of the junction.

To check also the proper function of the structure, its capacitance is simulated. Figure 14 presents the capacitance of structure for different value of void. Figure 14 illustrates that the void created at the interface has no effect on the parasitic capacity. All components have parasitic capacitances as illustrated Figure 15. where C j1 and C j2 are the junction capacitances, C BCB is the termination capacitance and the C Void is the void capacitance.

The analytical expression of capacitance is given by equation

(1) (1)
Where is the vacuum permittivity ( 8.854 10 , is the relative permittivity of material, e is the distance between the capacitor's plates, and A is the area of the capacitor's plate (Figure 17). The different values of capacitance at -1000V have been calculated in order to decrease the junction capacitances:

BCB Capacitance: • =3.9 • =3 10 5 µm 2 C BCB =98nF • =105µm Void capacitance !"# $%&' ( ( • !"# =1 • =4 10 2 µm 2 C Void =33pF • =105µm Junction Capacitance D1 ) *& + + • ) =11.7 • ) =10 6 µm 2 C j1 =1030nF • ) =100µm Junction capacitance DT ) *& +( +( • ) =11.7 • ) =12 10 4 µm 2 C j2 =124nF • ) =100µm
It can be noticed that:

C Void << C BCB<< C j2<< C j1.
This result confirms that a vacuum created in the structure has no effect on the electrical characteristic of diode.

B. Charge interface insertion

B.1. Breakdown voltage variations

Another type of degradation of the simulated structure is the insertion of charge in the interface Si/BCB. Figure 18 illustrates the I-V reverse characteristics with Si/BCB interface charge in Area 1 and Area 2. The breakdown voltage is modified by the insertion of charge interface in Area 1, but it is not modified in Area 2. Holes insertion creates a virtual P + vertical doping area and increases the breakdown voltage; while electrons insertion creates a virtual N + vertical doping area decreases the breakdown voltage. Holes and electrons in Area 2 have not effect in electrical characterization, indeed there is no depletion in this area but charges could have an influence on dynamic behavior. Figure 19: Effect of the traps on the parasitic capacitance (Area1) for the low value of reverse voltage.

Indeed holes insertion creates a virtual P + vertical doping area, so a new P -N horizontal junction as expressed in Figure 20, and in this case the capacitance increases very fast, until the depletion region of the new junction is of maximum width. If it can be imagined that the diode capacitance can be likened to a parallel plate capacitor, then as the plate spacing (i.e. the depletion region width) increases, the capacitance should decrease. Thus, the junction has a variable capacitance that depends on the evolution of the voltage which is applied on it. It is shown that the holes insertion increases the breakdown voltage therefore increases the depletion region (Figure 22) which causes the decrease of its capacitance (Figure 21); while electrons insertion decreases the breakdown voltage, therefore decreases the depletion region which causes the increase of its capacitance.

Figure 21: Effect of the traps on the parasitic capacity (Area1) for the higher value of BV.

It is observed that for negative charges (e-), the depletion region is smaller than for initial structure. This explains the capacitance rise for a same voltage. Furthermore, it seems the traps can be observed by a capacitance variation.

(a) (b) Figure 22: distribution of potential lines in the structure with e-traps (a) and without traps (b).

Conclusion

This paper presents the possibility to use the BCB resin in thick layer to realize deep trench termination.

It can be observed that reverse characteristics changing after 50 hours of power cycling. Optical microscopy could not permit to observed structural modification in order to explain the electrical variations. Thus, TCAD software is used to correlate structural modification with electrical characteristics in order to confirm the presence of the structural modification by combined characteristics.

Delamination on the Si/BCB interface has no impact on the proper function of the interface can not be observed by electrical variation. Moreover, the increasing or the decreasing of the breakdown voltage can be explained by the appearance of a quantity of charge at this interface.

Figure 1 :

 1 Figure 1: Top view of the schematic structure.

Figure 2 :

 2 Figure 2: Cross section of the schematic structure.

Figure 3 :

 3 Figure 3: Measured I-V reverse characteristics after 50 hours of thermal ageing.

Figure 4 :

 4 Optical observation (a) of the entire device and (b) a focus on the trench.

Figure 5 :

 5 Figure 5: base model of the chip for ANSYS FEM analysis (green = silicon, grey = BCB and blue = gold).

Figure 6 and

 6 Figure6and Figure7show the mechanical stress distribution in the BCB and silicon after the thermal load. A logarithmic scale was used to obtain a visual distribution of those mechanical stresses. As expected, the most stressed areas for both materials are located near the BCB/silicon interface on the right angle edge. Indeed, from a geometrical point of view, this geometrical singularity, shown in Figure7, is the most complex part of the chip structure. Furthermore, the most stressed area of the gold layer is on top of this singularity and the interface BCB/silicon around the trench present high mechanical stress.

Figure 6 :

 6 Figure 6: Mechanical stress distribution in the BCB (left) and the silicon (right).

Figure 7 :

 7 Figure 7: Zoom of the BCB and the silicon parts where the most stressed areas are located.

Figure 8 :

 8 Figure 8: Base cell and termination.

Figure 10 :

 10 Figure 10: Circuit for SENTAURUS mixed mode.

Figure 11 :

 11 Figure 11: void width effect on the reverse characteristic in Area 1 (a) and Area 2 (b).

  Figure 12: The distribution of potential lines in the structure and its zoom with (a) and without (b) void.

  Electrical filed along a horizontal cut-line (a) and its zoom (b).

Figure 14 :

 14 Figure 14: Effect of the void on the parasitic capacity (Area1).

Figure 15 :Figure 16 :

 1516 Figure 15: Parasitic capacitances of the structure.Therefore, there are three capacitances: the junction capacitance associated to a depletion region, capacitance associated to the void, and to the BCB. The total capacitance is presented in Figure16:

Figure 17 :

 17 Figure 17: Capacity model.

  Traps Effect on the reverse characteristic in Area 1 (a) and Area 2 (b).

2 .

 2 Parasitic capacitance variationFigure19points out the traps effect in the parasitic anode-cathode capacitance for the low value of reverse voltage.

  structure capacitance (a) and its equivalent circuit (b).

Table 1 :

 1 Material properties for FEM analysis.

		Density (Kg.cm -3 )	CTE (10 -6 /°C)	Young's modulus (GPa)	Poisson's ratio
	Gold	1930	14.2	79	0.44
	BCB	1000	42	2.9	0.34
	Silicone	2300	2.8	130	0.28

Acknowledgments

This work was supported by the French National Research Agency (ANR) through SUPERSWITCH program (ANR 2011 BS09 033).


Nathan Souviraà-Labastie

Anaik Olivero

Emmanuel Vincent

Frédéric Bimbot

AUDIO SOURCE SEPARATION USING MULTIPLE DEFORMED REFERENCES

Keywords:

This paper deals with audio source separation guided by multiple audio references. We present a general framework where additional audio references for one or more sources of a given mixture are available. Each audio reference is another mixture which is supposed to contain at least one source similar to one of the target sources. Deformations between the sources of interest and their references are modeled in a general manner. A nonnegative matrix co-factorization algorithm is used which allows sharing of information between the considered mixtures. We run our algorithm on music plus voice mixtures with music and/or voice references. Applied on movies and TV series data, our algorithm improves the signal-todistortion ratio (SDR) of the sources with the lowest intensity by 9 to 12 decibels with respect to original mixture.

INTRODUCTION

Source separation is a cross-cutting field of research dealing with various types of problems and data. This field is rapidly growing taking advantage of its inherent diversity and progress made in each of its components. In the case of audio source separation, achieving the natural human ability of hearing and describing auditory scenes still remains a far end goal. Many approaches have been investigated such as Nonnegative Matrix Factorization (NMF) [START_REF] Févotte | Nonnegative matrix factorization with the itakura-saito divergence. with application to music analysis[END_REF], sparse representations [START_REF] Plumbley | Sparse representations in audio and music: from coding to source separation[END_REF] and others.

Blind source separation is an ill-posed problem, and a key point is to embed a maximum amount of a priori information about the sources to guide the separation process [START_REF] Liutkus | An overview of informed audio source separation[END_REF][START_REF] Vincent | From blind to guided audio source separation[END_REF]. For instance, the general framework presented in [START_REF] Ozerov | A general flexible framework for the handling of prior information in audio source separation[END_REF] proposes to take into account spatial and spectral information about the sources. More recently, a number of approaches have been proposed to exploit information about the recording conditions, the musical score [START_REF] Simsekli | Score guided audio restoration via generalised coupled tensor factorisation[END_REF], the fundamental frequency 𝑓 0 [START_REF] Durrieu | Musical audio source separation based on user-selected F0 track[END_REF], the language model [START_REF] Mysore | A non-negative approach to language informed speech separation[END_REF], the text pronounced by a speaker [START_REF] Magoarou | Textinformed audio source separation using nonnegative matrix partial co-factorization[END_REF], or a similar audio signal [START_REF] Simsekli | Score guided audio restoration via generalised coupled tensor factorisation[END_REF][START_REF] Magoarou | Textinformed audio source separation using nonnegative matrix partial co-factorization[END_REF][START_REF] Gerber | Professionally-produced music separation guided by covers[END_REF][START_REF] Smaragdis | Separation by humming : User-guided sound extraction from monophonic mixtures[END_REF]. We focus on the latter category of approach where additional information comes from an extra signal called reference.

In [START_REF] Simsekli | Score guided audio restoration via generalised coupled tensor factorisation[END_REF], the authors propose a model for piano spectrogram restoration, based on generalized coupled tensor factorization where additional information comes from an approximate musical score and spectra of isolated piano sounds. The framework described in [START_REF] Magoarou | Textinformed audio source separation using nonnegative matrix partial co-factorization[END_REF] proposed a separation model between voice and background guided by another speech sample corresponding to the pronunciation of the same sentence. The speech reference is either recorded by a human speaker or is created with a voice synthesizer based on the available text pronounced by the speaker of the mixture to be separated. A nonnegative matrix co-factorization (NMcF) model is designed so that some of the factorized matrices are shared by the mixture and the speech reference. The authors of [START_REF] Durrieu | Musical audio source separation based on user-selected F0 track[END_REF] incorporate knowledge of the fundamental frequency 𝑓 0 in a NMF model, by fixing the source part of a source-filter model to be a harmonic spectral comb following the known 𝑓 0 value of the target source over time. In the context of audio separation of movie soundtracks, the separation can be guided by other available international versions of the same movie [START_REF] Liutkus | Separation of music+effects sound track from several international versions of the same movie[END_REF]. A cover-informed source separation principle is introduced in [START_REF] Gerber | Professionally-produced music separation guided by covers[END_REF] where the authors assume that cover multitrack signals are available and are used as initialization of the NMF algorithm. The original mixture and the cover signals are timealigned in a pre-processing step.

In this paper we propose a general framework for referencebased source separation which enables joint use of multiple deformed references. For most of previously cited approaches, the framework can either express it as a special case or model the kind of information used in a common formalism : reference signals can be directly available and symbolic additional information can either be synthesized or used to initialize the model. For instance, text-informed separation [START_REF] Magoarou | Textinformed audio source separation using nonnegative matrix partial co-factorization[END_REF], score informed [START_REF] Simsekli | Score guided audio restoration via generalised coupled tensor factorisation[END_REF], separation by humming [START_REF] Smaragdis | Separation by humming : User-guided sound extraction from monophonic mixtures[END_REF], cover guided separation [START_REF] Gerber | Professionally-produced music separation guided by covers[END_REF] are some of them. In our case, we assume that the musical references have been automatically discovered using an algorithm such as [START_REF] Catanese | MODIS: an audio motif discovery software[END_REF], in the context of the separation of voice and music in long audio sequence of movies or TV series. This paper is organized as follows. We first describe the NMcF model with audio references in Section 2 and we discuss how it generalizes some of the existing approaches. We also provide an example use case for the separation of a music plus voice mixture guided by music or/and voice references, on single channel audio data coming from TV series and movies. Section 3 provides an algorithmic implementation of the general framework, and Section 4 reports experimental results. We conclude in Section 5.

GENERAL FRAMEWORK

Input representation

The observations are 𝑀 single-channel audio mixtures x 𝑚 (𝑡) indexed by 𝑚. We assume that x 1 (𝑡) is the mixture to be separated, and x 𝑚 (𝑡) for 𝑚 > 1 are the references used to guide the separation process. Each mixture x 𝑚 (𝑡) is assumed to be the sum of sources y 𝑗 (𝑡) indexed by 𝑗 ∈ 𝐽 𝑚 :

x 𝑚 (𝑡) = ∑ 𝑗∈𝐽𝑚 y 𝑗 (𝑡) with x 𝑚 (𝑡), y 𝑗 (𝑡) ∈ ℝ. (1)
In the time-frequency domain, equation ( 1) can be written as :

x 𝑚 𝑓𝑛 = ∑ 𝑗∈𝐽𝑚 y 𝑗,𝑓 𝑛 with x 𝑚 𝑓𝑛 , y 𝑗,𝑓 𝑛 ∈ ℂ. (2)
The power spectrogram of each source 𝑗 of the mixture 𝑚 is denoted as 𝑉 𝑗 ∈ ℝ 𝐹 ×𝑁 + , and the mixture spectrum as 𝑉 𝑚 = ∑ 𝑗∈𝐽𝑚 𝑉 𝑗 . Following the general framework in [START_REF] Ozerov | A general flexible framework for the handling of prior information in audio source separation[END_REF], each 𝑉 𝑗 is split as the product of an excitation spectral power 𝑉 𝑒 𝑗 and a filter spectral power 𝑉 𝜙 𝑗 . The excitation part (resp. the filter part) is decomposed by an NMF separating the spectral content 𝑊 𝑒 𝑗 ∈ ℝ ). 𝐷 𝑒 and 𝐷 𝜙 denote the size of the NMF decomposition of the excitation and the filter. The following decomposition holds :

𝑉 𝑗 = 𝑉 𝑒 𝑗 ⊙ 𝑉 𝜙 𝑗 = 𝑊 𝑒 𝑗 𝐻 𝑒 𝑗 ⊙ 𝑊 𝜙 𝑗 𝐻 𝜙 𝑗 ( 3
)
where ⊙ denotes the point wise multiplication, and :

• 𝑊 𝑒 𝑗 aims to capture the pitch of the source (e.g., frequency range of an instrument or a speaker, and harmonicity) • 𝐻 𝑒 𝑗 the corresponding temporal activations (e.g., piano roll or 𝑓 0 track [START_REF] Durrieu | Musical audio source separation based on user-selected F0 track[END_REF][START_REF] Smaragdis | Separation by humming : User-guided sound extraction from monophonic mixtures[END_REF]) • 𝑊 𝜙 𝑗 will capture the spectral envelope (e.g., phoneme dictionary in the case of speech sources [START_REF] Magoarou | Textinformed audio source separation using nonnegative matrix partial co-factorization[END_REF] or spectral information about an instrument such as isolated notes [START_REF] Simsekli | Score guided audio restoration via generalised coupled tensor factorisation[END_REF]) • 𝐻 𝜙 𝑗 the corresponding temporal activations (e.g., phoneme alignment for speech or instrument timber changes) As in [START_REF] Ozerov | A general flexible framework for the handling of prior information in audio source separation[END_REF], the matrices 𝑊 and 𝐻 can be either fixed (i.e., unchanged during the estimation) or free (i.e., adapted to the mixture).

Modeling relationships between the sources of different mixtures

As the different mixtures are composed of similar sources, the matrices 𝑊 and 𝐻 can in addition be shared (i.e., jointly estimated) between two sources 𝑗 (from mixture 𝑚 = 1) and 𝑗 ′ (from a reference mixture 𝑚 ′ ). We model the deformations between those sources by adding transformation matrices 𝑇 in the corresponding NMF decomposition (e.g., in the case of one shared matrix

𝑊 𝑒 𝑗 ′ = 𝑇 𝑓𝑒 𝑗𝑗 ′ 𝑊 𝑒 𝑗 𝑇 𝑑𝑒 𝑗𝑗 ′ ).
Matrices 𝑇 can be either fixed or free as 𝑊 and 𝐻. When all the matrices 𝑊 and 𝐻 are shared, the relation becomes :

𝑉 𝑗 ′ = (𝑇 𝑓𝑒 𝑗𝑗 ′ 𝑊 𝑒 𝑗 𝑇 𝑑𝑒 𝑗𝑗 ′ 𝐻 𝑒 𝑗 𝑇 𝑡𝑒 𝑗𝑗 ′ ) ⊙ (𝑇 𝑓𝜙 𝑗𝑗 ′ 𝑊 𝜙 𝑗 𝑇 𝑑𝜙 𝑗𝑗 ′ 𝐻 𝜙 𝑗 𝑇 𝑡𝜙 𝑗𝑗 ′ ) (4)
where

𝑉 𝑗 ′ ∈ ℝ 𝐹 ′ ×𝑁 ′ +
, and :

• 𝑇 𝑓𝑒 𝑗𝑗 ′ (resp. 𝑇 𝑓𝜙 𝑗𝑗 ′ ) ∈ ℝ 𝐹 ′ ×𝐹
+ models the frequency deformations of the excitation (resp. filter) such as equalization or frequency shift (resp. changes in vocal tract length [START_REF] Magoarou | Textinformed audio source separation using nonnegative matrix partial co-factorization[END_REF]). Note that when 𝐹 ′ and 𝐹 are different, this also enables use of different time-frequency representations.

• 𝑇 𝑑𝑒 𝑗𝑗 ′ (resp. 𝑇 𝑑𝜙 𝑗𝑗 ′ ) ∈ ℝ 𝐷 𝑒 ×𝐷 𝑒 + (resp. ∈ ℝ 𝐷 𝜙 ×𝐷 𝜙 +
) is a dictionary of deformations of the excitation (resp. filter), and can model pitch shifting, (resp. timber correspondence or different dialects).

• 𝑇 𝑡𝑒 𝑗𝑗 ′ (resp. 𝑇 𝑡𝜙 𝑗𝑗 ′ ) ∈ ℝ 𝐹 ′ ×𝐹 +
is the temporal deformation of the excitation (resp. filter), and it is used to time-align the signals. Dynamic time warping can be used to initialize such matrices [START_REF] Magoarou | Textinformed audio source separation using nonnegative matrix partial co-factorization[END_REF], given that 𝑁 ′ and 𝑁 are usually different. It should also be noticed that using matrices 𝑇 𝑡 will only align the power spectrum of the mixture. Using phase aligned signals is one of our axis of improvements.

Separation guided by speech and/or music references

In the following, we describe with more details one use case of the previously described general framework, that suits the problem of separating speech and music from old recorded single-channel movies and TV series. To guide and enhance the separation, we consider speech or/and music references. The music references discovered using [START_REF] Catanese | MODIS: an audio motif discovery software[END_REF] are intrinsically deformed and contain additional sources such as sound effects. Speech references correspond to the same sentences uttered by different speakers without noise.

In the particular setup reported here, speech sources are numbered 1 and 2, music sources 3 and 4 and noise sources ). The fixed matrices 𝑇 set to identity are removed from the notations. The mixture to be separated is modeled as :

𝑉 1 = 𝑉 1 + 𝑉 3 + 𝑉 5 (5) = 𝑊 𝑒 1 𝐻 𝑒 1 ⊙ 𝑊 𝜙 1 𝐻 𝜙 1 + 𝑊 𝑒 3 𝐻 𝑒 3 ⊙ 𝑊 𝜙 3 𝐻 𝜙 3 + 𝑊 5 𝐻 5

A voice reference mixture

The second mixture is composed of the speech reference alone :

𝑉 2 = 𝑉 2 = 𝑊 𝑒 2 𝐻 𝑒 2 ⊙ 𝑇 𝑓𝜙 12 𝑊 𝜙 1 𝐻 𝜙 1 𝑇 𝑡𝜙 12 (6)
𝐻 𝑒 1 and 𝐻 𝑒 2 are estimated separately to model the different intonations between the speakers, whereas the filter matrices 𝑊 𝜙 1 and 𝐻 𝜙 1 are estimated jointly to model similar phonetic content. 𝑇 𝑡𝜙 12 models the time realignment between the two pronounced sentences. 𝑇 𝑓𝜙 12 is constrained to be diagonal and it models both the equalization and the speaker's difference. This model is equivalent to the one used in [START_REF] Magoarou | Textinformed audio source separation using nonnegative matrix partial co-factorization[END_REF].

A music reference mixture

The third mixture is composed of the music reference 𝑉 4 supposed to be similar to 𝑉 3 , and some noise 𝑉 6 :

𝑉 3 = 𝑉 4 + 𝑉 6 = 𝑊 𝑒 4 𝐻 𝑒 3 𝑇 𝑡𝑒 34 ⊙ 𝑊 𝜙 3 𝐻 𝜙 3 𝑇 𝑡𝜙 34 + 𝑊 6 𝐻 6 (7)
𝑇 𝑡𝑒 34 and 𝑇 𝑡𝜙 34 models the time realignment between the two music examples. Surprisingly it appears that keeping the matrix 𝑇 𝑡𝜙 34 fixed yields better results.

Combining references of different kinds

These two reference models can easily be combined in order to jointly use the three mixtures x 1 , x 2 and x 3 during the separation process. The common notations enable us to optimize parameters for each reference.

Extensions of our approach

The proposed framework generalizes the state-of-the-art approaches in [START_REF] Simsekli | Score guided audio restoration via generalised coupled tensor factorisation[END_REF][START_REF] Magoarou | Textinformed audio source separation using nonnegative matrix partial co-factorization[END_REF]. In [START_REF] Simsekli | Score guided audio restoration via generalised coupled tensor factorisation[END_REF], the source reference is composed of isolated notes and can be modeled with a shared 𝑊 𝑒 𝑗 , a free 𝐻 𝑒 𝑗 and by setting the product of 𝑊 𝜙 𝑗 and 𝐻 𝜙 𝑗 to a matrix of ones (no excitation-filter model). The approach described in [START_REF] Magoarou | Textinformed audio source separation using nonnegative matrix partial co-factorization[END_REF] is exactly expressed by ( 5) and [START_REF] Simsekli | Score guided audio restoration via generalised coupled tensor factorisation[END_REF].

Our framework can also model the same kind of information used in [START_REF] Simsekli | Score guided audio restoration via generalised coupled tensor factorisation[END_REF][START_REF] Durrieu | Musical audio source separation based on user-selected F0 track[END_REF][START_REF] Magoarou | Textinformed audio source separation using nonnegative matrix partial co-factorization[END_REF][START_REF] Gerber | Professionally-produced music separation guided by covers[END_REF][START_REF] Smaragdis | Separation by humming : User-guided sound extraction from monophonic mixtures[END_REF]. The separation by humming approach in [START_REF] Smaragdis | Separation by humming : User-guided sound extraction from monophonic mixtures[END_REF] can be implemented by sharing the excitation part (i.e., 𝑊 𝑒 𝑗 and 𝐻 𝑒 𝑗 ) between the target source and the reference. Symbolic music or speech information can be used after being synthesized as in [START_REF] Magoarou | Textinformed audio source separation using nonnegative matrix partial co-factorization[END_REF], or directly in the model as in [START_REF] Simsekli | Score guided audio restoration via generalised coupled tensor factorisation[END_REF][START_REF] Durrieu | Musical audio source separation based on user-selected F0 track[END_REF] by constraining 𝐻 𝑒 𝑗 . Cover-guided separation as in [START_REF] Gerber | Professionally-produced music separation guided by covers[END_REF] is also possible by aligning the cover and the mixture to be separated using matrices 𝑇 𝑡 . In [START_REF] Gerber | Professionally-produced music separation guided by covers[END_REF] the cover is used to initialize the sources but not used during the source estimation. We explain below how our framework can model the deformations between the cover and the source of interest and hence enable the use of the cover during the source estimation.

In addition, this framework can model other kind of information and thus lead to new scenarios of use. We here briefly describe some of them that have not be investigated yet to our knowledge :

• using multiple references for a specific source, i.e., several 𝑗 ′ for a single 𝑗. This will leads to more robust separation, especially in the case of references with additive sources (like when references are automatically obtained with an algorithm such as [START_REF] Catanese | MODIS: an audio motif discovery software[END_REF]). For instance in the case of multi-speaker source separation, the speech sources can be guided by several references, ideally containing the same words uttered by the same speaker. • music source separation for a verse guided by an other verse. In that case the speech sources will have a shared excitation (𝐻 𝑒 𝑗 and 𝑊 𝑒 𝑗 ) but a different filter (𝐻 𝜙 𝑗 ) over time. The approach is similar to [START_REF] Rafii | Repeating pattern extraction technique (repet): A simple method for music/voice separation[END_REF] but we consider the voice as a repeated deformed pattern instead of modeling the background music only.

• cover guided music separation with explicit models for the deformations. The change of an instrument or a singer can be modeled by setting matrix 𝑊 𝜙 𝑗 to not-shared or adapting matrix 𝑇 𝑓𝜙 . Covers played in minor/major or in another tone can also be considered by using 𝑇 𝑑𝑒 to model note changes or frequency transposition.

ALGORITHMIC ASPECTS

In this section, we describe a general algorithm based on multiplicative updates (MU) as well as the initialization used for the matrices from the example of subsection 2.3.

Mutiplicative updates

Following [START_REF] Févotte | Nonnegative matrix factorization with the itakura-saito divergence. with application to music analysis[END_REF], the Itakura-Saito NMF (IS-NMF) model is well-adapted to audio data and provides the following cost function :

𝒞(Θ) = 𝑀 ∑ 𝑚=1 𝐹,𝑁 ∑ 𝑓,𝑛=1 𝑑 𝐼𝑆 (𝑋 𝑚 𝑓𝑛 |𝑉 𝑚 𝑓𝑛 ) ( 8
)
where Θ is the set of parameters to be estimated, i.e., matrices 𝑊 , 𝐻 and 𝑇 that are not fixed.

𝑋 𝑚 = [|x 𝑚 𝑓𝑛 | 2 ]
𝑓𝑛 and 𝑉 𝑚 are respectively the observation and the estimated spectrum and 𝑑 𝐼𝑆 (𝑎|𝑏) = 𝑎/𝑏 -𝑙𝑜𝑔(𝑎/𝑏) -1 is the Itakura-Saito divergence. Following a standard NMF algorithm [START_REF] Févotte | Nonnegative matrix factorization with the itakura-saito divergence. with application to music analysis[END_REF], multiplicative updates (MU) are easily derived from (3), ( 4) and [START_REF] Mysore | A non-negative approach to language informed speech separation[END_REF]. Due to a lack of space, we will just derive the update for two representative examples : a non-shared free variable 𝑊 𝑒 𝑗 [START_REF] Magoarou | Textinformed audio source separation using nonnegative matrix partial co-factorization[END_REF] and a shared free variable 𝑊 𝑒 𝑗 [START_REF] Gerber | Professionally-produced music separation guided by covers[END_REF] between source 𝑗 of mixture 𝑚 = 1 and 𝐽 ′ sources 𝑗 ′ of mixtures 𝑚 ′ ∕ = 1. We can notice that if for a given 𝑗 the set of 𝑗 ′ is empty (4) becomes [START_REF] Liutkus | An overview of informed audio source separation[END_REF]. The final source estimate are obtained using an adaptive wiener filter.

𝑊 𝑒 𝑗 ← 𝑊 𝑒 𝑗 ⊙ [𝑉 𝜙 𝑗 ⊙ 𝑉 𝑚 .[-2] ⊙ 𝑋 𝑚 ][𝐻 𝑒 𝑗 ] 𝑇 [𝑉 𝜙 𝑗 ⊙ 𝑉 𝑚 .[-1] ][𝐻 𝑒 𝑗 ] 𝑇 (9) 𝑊 𝑒 𝑗 ← 𝑊 𝑒 𝑗 ⊙ [𝑉 𝜙 𝑗 ⊙ 𝑉 𝑚 .[-2] ⊙ 𝑋 𝑚 ][𝐻 𝑒 𝑗 ] 𝑇 + ∑ 𝑗 ′ [𝑇 𝑓𝑒 𝑗𝑗 ′ ] 𝑇 [𝑉 𝜙 𝑗 ′ ⊙ 𝑉 𝑚 ′ .[-2] ⊙ 𝑋 𝑚 ′ ][𝑇 𝑑𝑒 𝑗𝑗 ′ 𝐻 𝑒 𝑗 𝑇 𝑡𝑒 𝑗𝑗 ′ ] 𝑇 [𝑉 𝜙 𝑗 ⊙ 𝑉 𝑚 .[-1] ][𝐻 𝑒 𝑗 ] 𝑇 + ∑ 𝑗 ′ [𝑇 𝑓𝑒 𝑗𝑗 ′ ] 𝑇 [𝑉 𝜙 𝑗 ′ ⊙ 𝑉 𝑚 ′ .[-1] ][𝑇 𝑑𝑒 𝑗𝑗 ′ 𝐻 𝑒 𝑗 𝑇 𝑡𝑒 𝑗𝑗 ′ ] 𝑇 (10)

Initialization

NMF is known to be sensitive to the initialization of the matrices. We also give here some details on our initialization choices for the use case described in subsection 2.3. Let us also note that, as we work with MU, zeros in the parameters remain unchanged over the iterations.

The fixed excitation spectral patterns 𝑊 𝑒 𝑗 for 𝑗 = 1, 2, 3, 4 are a set of harmonic components computed as in [START_REF] Ozerov | A general flexible framework for the handling of prior information in audio source separation[END_REF]. We initialize the synchronization matrices 𝑇 𝑡𝜙 12 , 𝑇 𝑡𝑒 34 , and 𝑇 𝑡𝜙 34 with Dynamic Time Warping (DTW) [START_REF] Ellis | Dynamic time warping in matlab[END_REF] matrices computed on MFCC vectors [START_REF] Ellis | PLP and RASTA (and MFCC, and inversion) in Matlab[END_REF] for speech sources and on chroma vectors for music sources. Following [START_REF] Magoarou | Textinformed audio source separation using nonnegative matrix partial co-factorization[END_REF], we allow the temporal path to vary within an enlarged region around the estimated DTW path. As long as we work with deformed and noisy data (especially for music), we weight this enlarged path by coefficients of the similarity matrix, in order to avoid obvious initialization errors. We invite the reader to refer to [START_REF] Magoarou | Textinformed audio source separation using nonnegative matrix partial co-factorization[END_REF] and following works for details on this strategy and a discussion on its influence on the results. The spectral transformation matrix 𝑇 𝑓𝜙 12 is initialized by the identity matrix. Choosing this matrix to be diagonal leads to time-invariant spectral deformations. The others matrices ( 𝐻 𝑒 1 , 𝐻 𝑒 2 , 𝑊 5 , 𝐻 5 , 𝑊 6 , 𝐻 6 , 𝑊 𝜙 1 , 𝐻 𝜙 1 , 𝐻 𝑒 3 , 𝑊 𝜙 3 , 𝐻 𝜙 3 ) are initialized with random values. In addition, we perform 10 iterations of the classical IS-NMF with MU on the reference signals ( 6) and [START_REF] Durrieu | Musical audio source separation based on user-selected F0 track[END_REF] alone, where the shared matrices (𝑊 𝜙 1 , 𝐻 𝜙 1 , 𝐻 𝑒 3 , 𝑊 𝜙 3 , 𝐻 𝜙 3 ) and the noise parameters (𝑊 6 , 𝐻 6 ) are updated whereas matrices 𝑇 are not. For both references, this guided initialization leads to better separation results. After those initializations, 𝑊 6 and 𝐻 6 are then set once again to random values, we then perform 10 updates of the main NMcF.

EXPERIMENTS

Data

As our underlying goal is to separate old audio-visual recordings, we generate the mixture and the references signals to depict such situations. The musical samples and the corresponding references are obtained using the algorithm in [START_REF] Catanese | MODIS: an audio motif discovery software[END_REF] that allows the discovery of non-exact repetitions in long audio stream, here movies or TV series. The discovered samples are characterized by distortion of the source of interest (rhythm changes, fade in ) and additional sources (mainly sound effects). Speech examples are taken from the database in [START_REF] Benezeth | BL-Database: A french audiovisual database for speech driven lip animation systems[END_REF] in which 16 different speakers uttered the same 238 sentences. We keep 4 musical examples and 4 sentences (two female and two male speakers) to generate the mixtures.

We consider two voice-to-music ratio levels : -6 dB (music as foreground and voice as background, and 12 dB (the inverse case). These levels are close to those effectively observed in movies and TV series. We synthesized such examples in order to obtain objective measures for the evaluation and compare our estimated sources with the original ones. Combining those parameters leads to 32 original mixtures 𝑋1 . The original mixtures and the references are about eight seconds long and they are sampled at 16 kHz. Some examples are available online 1 .

Results

We here analyze the performance obtained for the use case described in subsection 2.3 with 10 iterations for the NMcF decomposition and initialized according to subsection 3.2. Table 1 shows the signal-to-distortion ratio (SDR), the signal-tointerference ratio (SIR) and the signal-to-artifact ratio (SAR) [START_REF] Vincent | Performance measurement in blind audio source separation[END_REF]. Even when the music ground truth is corrupted, the result is still relevant as it gives a lower bound to the actual non-measurable performance. We consider the three cases described in subsection 2.3 respectively corresponding to the combination of the mixtures ( 5)-( 6), ( 5)- [START_REF] Durrieu | Musical audio source separation based on user-selected F0 track[END_REF], and ( 5)-( 6)- [START_REF] Durrieu | Musical audio source separation based on user-selected F0 track[END_REF].

Bold values indicate the best SDR. As expected, the best results are most often obtained when all available references are used. The improvements can be deduced from the values in Table 1 after subtraction of the original source-mixture ratio, and a quality improvement is observed in almost all cases. For each voice-to-music ratio levels, the best improvement are achieved for the sources with the lowest intensity, i.e., 9 dB for voice and 12 dB for music.

CONCLUSION AND FUTURE WORKS

In this paper, we presented a general way to use audio information to separate a given mixture. This model is general enough to take different kinds of audio references into account which are possibly deformed in the frequency domain and in the temporal domain. We described in details a voice and music separation example guided by speech and/or music references using this general framework. Music separation from international versions [START_REF] Liutkus | Separation of music+effects sound track from several international versions of the same movie[END_REF] take advantage of multichannel that is not handled by this framework yet. In the future, we plan to extend our algorithm to the multichannel case following the multiplicative rules described in [START_REF] Ozerov | Multichannel nonnegative tensor factorization with structured constraints for user-guided audio source separation[END_REF] or [START_REF] Duong | Underdetermined reverberant audio source separation using a full-rank spatial covariance model[END_REF]. A first perspective of this work is to use an EM-like algorithm.

Voice-to-music ratio levels

A more general perspective will be the design of some automatic processes to choose the initializations of the parameters we have to estimate. Our model can also be improved by adding well-chosen constraints on the parameters. For instance, smoothness constraints on the spectral transformation matrices 𝑇 𝑓𝜙 𝑗 ′ 𝑗 can help to derive a more relevant spectral deformation between the target sources and the references.

  5, and 6. Fixed variables are in black (𝑊 𝑒 1 , 𝑊 𝑒 2 , 𝑊 𝑒 3 , 𝑊 𝑒 4 , 𝑇 𝑡𝜙 34 ). Free variables are in green ( 𝐻 𝑒 1 , 𝐻 𝑒 2 , 𝑇 𝑓𝜙 12 , 𝑇 𝑡𝜙 12 , 𝑇 𝑡𝑒 34 , 𝑊 5 , 𝐻 5 , 𝑊 6 , 𝐻 6 ). And variables that are both free and shared are in red or violet ( 𝑊 𝜙 1 , 𝐻 𝜙 1 , 𝐻 𝑒 3 , 𝑊 𝜙 3 , 𝐻 𝜙 3

Table 1 .

 1 28|-2.69|9.98 2.61|22.03| 5.04 8.57 |15.15|14.24 -2.64|2.23|5.04 Music reference 1.99 | 7.14 |3.00 9.64|17.21|11.00 6.54 |25.54|10.30 -0.30|3.66|3.00 Speech and music references 3.86 | 8.84 |5.38 7.76|17.45| 8.69 11.93|26.04|13.16 0.34 |4.05|2.93 Comparison of separation guided by speech or/and music references in terms of average SDR|SIR|SAR (dB).

		-6 dB		12 dB
		voice	music	voice	music
	Speech reference	-3.

http://maia.gforge.inria.fr/demo/eusipco2014.html

* Work supported by Maia Studio and Bretagne Region scholarship


Patricia Ern

Nicolas Brosse

Interaction of two axisymmetric bodies falling side by side at moderate Reynolds numbers

Keywords: multiphase flow, wakes, vortex interactions

We consider the interaction of two identical disks freely falling side by side in a fluid at rest for Reynolds numbers ranging from 100 to 300, corresponding to rectilinear and oscillatory paths. For the three aspect ratios of the disks investigated, we observed that the bodies always repel one another when the horizontal distance between their centres of gravity is less than 4.5 diameters. They never come closer for distances spanning between 4.5 and 6 diameters. Beyond the latter distance, the disks appear indifferent to each other. For both rectilinear and periodic paths, the repulsion effect is weak, leading to an overall horizontal drift lower than 3 % of the vertical displacement. We propose a model for the repulsion coefficient C r , which decreases with the separation distance between the bodies and is inversely proportional to the aspect ratio of the bodies, C r thus being stronger for the thicker ones. Furthermore, in the case of the oscillatory paths, we show that the effect of the interaction reduces to the repulsion effect, since the characteristics of the oscillatory motion of each disk appear unaffected by the presence of the companion disk and no synchronization is observed between the paths, nor between the wakes, of the two disks.

Introduction

The problem of the hydrodynamical interaction of two bodies, either free to move in a fluid or fixed in an incident flow, is a question common to biomechanics, multiphase flow and fluid-structure interaction. Primary issues lie in the strength and nature (attractive or repulsive) of the force due to the hydrodynamical interaction between the two objects, and in their relationship with the motion of the bodies, when these are free to move. In this work, we consider the problem of two freely falling bodies in the paradigm situation of bodies placed side by side. In the case of two objects placed side by side in an incoming flow or moving side by side at moderate Reynolds numbers, two mechanisms compete to drive their interaction. The first is provided by potential flow analysis and results in an attractive effect since the pressure is lower between the two bodies due to the increase in flow velocity. The second is related to the vorticity produced at the surface of a body. Vorticity diffuses into the boundary layer which is formed on the body and is advected in the wake. The asymmetry induced in the vorticity field by the presence of the other body results in a repulsive force. The relative strength of the two effects depends on the Reynolds number and on the distance separating the two bodies, as discussed by [START_REF] Legendre | Hydrodynamic interactions between two spherical bubbles rising side by side in a viscous liquid[END_REF] in the case of two spherical bubbles, but also on the boundary condition on the body (no-slip for a solid body or shear-free for a bubble), as discussed by [START_REF] Takemura | The transverse force on clean and contaminated bubbles rising near a vertical wall at moderate Reynolds number[END_REF] in the twin problem of a clean or contaminated bubble interacting with a wall.

For fixed three-dimensional bodies displaying a stationary wake, substantial progress concerning the nature of the interaction between two solid spheres [START_REF] Kim | Three-dimensional flow over two spheres placed side by side[END_REF] and two spherical bubbles [START_REF] Legendre | Hydrodynamic interactions between two spherical bubbles rising side by side in a viscous liquid[END_REF]) placed side by side has been achieved thanks to numerical simulations. For solid spheres and Reynolds numbers spanning from 50 to 150, [START_REF] Kim | Three-dimensional flow over two spheres placed side by side[END_REF] showed the existence of a critical distance ∆ h1 below which the spheres repel and above which they weakly attract one another up to a distance at which they have no effect on each other of about ∆ h ≃ 21 (∆ h denotes the centre-to-centre distance between the two bodies made dimensionless using their diameter d). Moreover, they showed that ∆ h1 decreases with the Reynolds number (∆ h1 ≃ 8 for Re = 50, ∆ h1 ≃ 4 for Re = 100 and ∆ h1 ≃ 3.5 for Re = 150); the range of separation distances for which the attractive potential effect is dominant over the repulsive vortical effect is extended when the Reynolds number increases. The hydrodynamical interaction between two fixed bubbles placed side by side in an incoming flow was investigated by [START_REF] Legendre | Hydrodynamic interactions between two spherical bubbles rising side by side in a viscous liquid[END_REF]. Below a critical Reynolds number close to 30, the bubbles repel one another. For larger Reynolds numbers, two critical distances were identified, ∆ h1 and ∆ h2 with ∆ h1 < ∆ h2 , that depend on Re (∆ h1 ≃ 1.5 for Re = 30 and decreases when Re increases; ∆ h2 ≃ 2 for Re = 30 and increases to ∆ h2 ≃ 6 for Re = 100). In the interval between these two relative positions, the bubbles attract each other, outside it they repel. Note that the equilibrium distance ∆ h1 is smaller for bubbles than for solid spheres, in line with a production of vorticity greater in the case of a no-slip boundary condition leading to a repulsive force stronger for solid spheres than for spherical bubbles at the same Re. A model for the coefficient C L associated with the transverse force acting between two fixed bubbles was developed by [START_REF] Legendre | Hydrodynamic interactions between two spherical bubbles rising side by side in a viscous liquid[END_REF]. For Re 30 and ∆ r 3,

C L = -6∆ -4 r 1 - 40 Re + O(Re -3/2 ) + ∆ -3 r + O(Re -1 ∆ -2 r , ∆ -5 r ) (1.1)
where ∆ r = 2 ∆ h is the distance separating the centres of the bubbles normalized with their radius d/2. The first term, in ∆ -4 r , corresponds to the attractive potential effect whereas the term in -1/Re corresponds to the repulsive effect of vorticity.

The case of fixed bodies placed side by side has received equal attention when the wakes of the bodies are unsteady. For a Reynolds number of 300 corresponding to unsteady wakes for solid spheres, [START_REF] Yoon | Flow-induced forces on two nearby spheres[END_REF] found numerically that spheres placed side by side repel one another for all the relative distances investigated, namely ∆ h 5, indicating that repulsion between two side-by-side fixed spheres occurs for separation distances that are larger for unsteady wakes than for steady ones. Results related to the interaction between the unsteady wakes of the bodies were also provided. Different modes of wake instability and of wake synchronization were observed depending on the distance between the bodies. In particular, [START_REF] Williamson | Evolution of a single wake behind a pair of bluff bodies[END_REF] investigated experimentally the wakes of two cylinders placed side by side in an incident flow with Reynolds numbers ranging from 100 to 200. For ∆ h > 7, no visible interaction was observed between the wakes, whereas synchronization of vortex shedding occurred for 3 < ∆ h < 7. For ∆ h < 3, vortices are shed at different frequencies from the two cylinders, while in the far wake they merged into a single street. [START_REF] Peschard | Coupled wakes of cylinders[END_REF] considered the same problem theoretically and experimentally. They also reported vortex shedding either in phase or phase opposition but observed two additional regimes: first, an asymmetrical regime in which vortices are released at the same frequency but with an amplitude larger for one cylinder than for the other; and second, a quasi-periodic regime identical to the previous one but with phase and amplitude differences between the vortex shedding of the two bodies that slowly evolve in time. [START_REF] Schouveiler | Interactions of the wakes of two spheres placed side by side[END_REF] studied numerically and experimentally the interaction of two spheres placed side by side in an incident flow at a Reynolds number of 300. They observed for ∆ h < 1.05 a single wake, for 1.05 < ∆ h < 1.3 two asymmetrical wakes, and for 1.3 < ∆ h < 3 two symmetrical coupled wakes. For larger distances, the wakes appear uncorrelated in space and time.

In the case of freely moving bodies, the translational and rotational degrees of freedom of the bodies may affect the interaction and modify the hydrodynamical loads observed in the case of fixed bodies. A question also arises concerning the existence of specific modes of interaction between the wake of the bodies and their consequences on the bodies motion. Only a few studies have dealt with the interaction of freely moving solid bodies placed side by side. [START_REF] Wu | Dynamics of dual-particles settling under gravity[END_REF] studied experimentally the interaction between two side-by-side spherical particles for Reynolds numbers between 0.02 and 2000 for a fixed density ratio between the bodies and the fluid (ρ s /ρ f = 1.38). They observed that at Re = 100 the final distance between the particles is between 3 and 7 diameters and that their fall velocity is the same as that of an isolated particle. Synchronization of oscillatory paths was observed numerically for two-dimensional ellipsoids and for Reynolds numbers smaller than 6 by [START_REF] Aidun | Direct analysis of particulate suspensions with inertia using the discrete Boltzmann equation[END_REF] and for three-dimensional ellipsoids in a narrow channel at Re = 100 by [START_REF] Pan | Simulating the dynamics of fluid-ellipsoid interactions[END_REF]. In both studies, they found that both the inclination and the path of the two ellipsoids oscillate in phase opposition.

In this paper, we investigate the behaviour of two identical solid bodies freely falling in the side-by-side configuration, and more specifically of two axisymmetric bodies that can be viewed as finite-thickness disks. The goal of the present paper is twofold: to characterize the kinematics of the interaction of these bodies and propose an empirical model for its driving force; and to investigate if specific modes of body motion arise due to wake interaction, in particular when periodic vortex shedding occurs. The kinematics and the dynamics of finite-thickness disks were investigated in detail in the isolated body case by [START_REF] Fernandes | Oscillatory motion and wake instability of freely rising axisymmetric bodies[END_REF] and [START_REF] Fernandes | Dynamics of axisymmetric bodies rising along a zigzag path[END_REF]. These studies showed notably the strong impact of the aspect ratio of the body (the diameter-to-thickness ratio of the disk) on the characteristics of its path and on the loads governing its motion, in particular on those related to the vorticity produced at the body surface. The body geometry is therefore expected to also have an important effect on the hydrodynamical interaction between the two bodies.

The layout of the paper is the following. Section 2 presents the experimental setup used for the investigation. Section 3 focuses on the kinematics associated with the interaction of the two disks in the case of rectilinear paths. A model for the horizontal repulsion between the bodies is then proposed in § 4. Finally, the interaction of disks displaying oscillatory paths is considered in § 5. The main results of the paper are outlined in § 6.

Experimental setup

The bodies are released in a large glass tank (1.70 m high with a square cross-section of 0.4 m width) containing salted water of density ρ f = 1010 kg m -3 and kinematic viscosity ν = 1.020 mm 2 s -1 (see figure 1 of [START_REF] Fernandes | Oscillatory motion and wake instability of freely rising axisymmetric bodies[END_REF]). The bodies are finite-thickness disks (or short-length cylinders) of density ρ s ≃ 1020 kg m -3 (the density ratio between the bodies and the fluid is thus close to unity). Their diameters d (resp. heights h) range from 5 to 20 mm (resp. 1 to 5 mm). The aspect ratio χ = d/h is chosen at the values of 3, 6 and 10, determined with an accuracy of ±1 %. The motion of the body depends on the Archimedes number Ar defined by Ar = (( ρ/ρ f ) g r eq ) 1/2 r eq /ν, where ρ = |ρ f -ρ s |, r eq is the radius of the sphere having a volume equal to that of the body, and g is the gravitational acceleration. Note that Ar corresponds to a Reynolds number based on a gravitational velocity. Three characteristic values of Ar are considered here, about 35, 45 and 105, allowing us to investigate both rectilinear and periodic motions [START_REF] Fernandes | Oscillatory motion and wake instability of freely rising axisymmetric bodies[END_REF]. When the bodies are falling alone in the tank, these values of Ar correspond to the Reynolds numbers Re = U m d/ν ≃ 110, 140 and 280, U m being the mean vertical velocity of the body.

In this paper, we focus on the hydrodynamical interaction between two disks that can be considered within experimental accuracy as identical. Special care was taken to select the pairs. In fact, a difference of 1 % between the mean vertical velocities of the disks (due to tiny differences in geometry or density) leads to a vertical separation of 1.5 cm at the bottom of the tank, larger than a disk diameter. Therefore only the pairs of disks for which the vertical separation distance remained smaller than a body diameter all over their paths were considered for the experiments. The two disks were released simultaneously in one or two vertical parallel tubes using a sliding plate on which the bodies rested at the time of release. A single tube or two separate tubes were used according to the size of the bodies and to the initial separation distance investigated. The release tubes enabled the bodies to accelerate while being restrained in their transverse motion and rotation. At the exit of the tubes, the bodies were falling broadside to the vertical and side by side, at a velocity close to their terminal velocity, and were free to interact. The motion of the bodies was then recorded by means of two perpendicular travelling cameras. The image-and signal-processing techniques used to determine the time evolution of the coordinates of the centre of the bodies and the angles defining the inclinations of their symmetry axes are described in detail in [START_REF] Fernandes | Oscillatory motion and wake instability of freely rising axisymmetric bodies[END_REF]. The use in this case of two PCO 2000 cameras of spatial resolution 2048 × 2048 pixels provided an accuracy of ±0.06 mm for the position and of ±0.75 • for the inclination. Each camera recorded the positions of the centre of gravity of the bodies in a vertical plane, denoted (X, Z) and (Y, Z), where Z is the vertical coordinate and X and Y are horizontal perpendicular coordinates. In what follows, the separation distance between the bodies is characterized by the vertical distance ∆ * z = Z 1 -Z 2 and the horizontal distance

∆ * h = ((X 1 -X 2 ) 2 + (Y 1 -Y 2 ) 2 ) 1/2
, where indices 1 and 2 each stand for a body.

Kinematics of repulsion of disks having rectilinear paths

We first consider the case of disks displaying a rectilinear path when falling alone in the tank. Along this path, the axis of symmetry of the body is aligned with the vertical. The effect of the interaction is illustrated here for disks with χ = 10 but the same behaviour is observed for the aspect ratios χ = 3 and 6. Figure 1 In all these, the disks are initially very close, ∆ h (t = 0) < 1.5 (and > 1 to avoid contact). The curves were repositioned at the origin to account for the differences in initial separation distances. In all cases, horizontal repulsion between the bodies occurs, which is stronger when the bodies are closer. There is however some scatter between the tests although the dimensionless parameters that control the problem are identical within experimental accuracy and the initial conditions very close. For instance, the final separation distance (at t = 130) spans between 4 and 4.5 diameters. This difference might be due to the fact that the bodies are not perfectly horizontally aligned during their fall. The vertical distance separating the centres of gravity of the disks normalized with their diameter d, ∆ z = ∆ * z /d, is always less than ∆ z < 0.5 but varies in time as shown in figure 1(b). However, no clear relationship could be established between the horizontal and vertical evolutions. Note also that the sideways motion of the disks due to the repulsion is very slight. The horizontal drift of the disks in the side-by-side situation represents about 2.5 % of their overall vertical displacement (typically 3 cm compared to 1.2 m). [START_REF] Fernandes | Oscillatory motion and wake instability of freely rising axisymmetric bodies[END_REF] observed for isolated disks a random drift related to experimental imperfections, such as residual fluid movements or small defects in the geometry or the homogeneity of the disks, of the same order of magnitude. The clear and systematic trend observed for ∆ h seems to indicate that specific disturbances in the path of each disk do not significantly affect the repulsion effect but this may explain the scatter in the evolution of ∆ h and the irregularities observed for ∆ z . We measured that, as soon as the bodies leave the release tubes, they reach their terminal vertical velocities in a few seconds, so that the paths are performed at constant vertical velocity. As could be expected from the weakness of the repulsion effect, the mean fall velocity U m of the disks during the interaction is within measurement accuracy identical to those measured when the disks are falling alone. We also explored the behaviour of the disks when the initial separation distance is larger. Typical results are shown in figure 2(a), for χ = 10 and Re = 115. For a horizontal separation distance ∆ h > 6, the behaviours of the disks appear as decoupled. For these distances, the interaction effect between the disks is of the same order of magnitude as the disturbances present in the trajectory of each disk, so that the three behaviours, attraction, repulsion and indifference, can be observed, either consecutively along the same path or for different trials with the same bodies. We carried out a large number of trials for separations close to this distance, approximately 15 per aspect ratio, in order to determine the limit below which a significant interaction effect can be observed, i.e. a reproducible effect that is stronger than the background experimental noise. We obtained that for all χ , when ∆ h < 4.5, the disks always repel each other; for ∆ h < 6, they never come closer and for ∆ h > 6 no significant interaction seems to occur between the disks.

We now focus our attention on the effect of the aspect ratio on the repulsion effect observed for disks falling sufficiently close together. Figure 3(a) presents the evolution of the horizontal distance ∆ h as a function of time for the aspect ratios χ = 3, 6 and 10 and for Re = 115 ± 10, and figure 3(b) the corresponding evolution of ∆ z . Seven cases are shown revealing, rather surprisingly, the same evolution for thin and thick bodies. The scatter of the results between the different aspect ratios is of the same order of magnitude as the scatter of the results for one aspect ratio so that the kinematics of repulsion can be considered in a first approximation as independent of the aspect ratio (and therefore of the body thickness). For all χ and ∆ h 1.5, the mean evolution of the set of curves displayed in figure 3(a) can be fitted by the power law shown with a dashed red line, which is

∆ h ∼ a t b with a = 1.1 and b = 2 7 .
(3.1)

The upper and lower limit of the set of curves can then be obtained by varying a by ±2.5 % or b by ±1.5 %.

A model for the repulsion between two disks falling side by side

The aim of this section is to model the repulsive force on each disk that leads to the kinematics determined in the previous section. Following the approach of [START_REF] Takemura | The transverse force on clean and contaminated bubbles rising near a vertical wall at moderate Reynolds number[END_REF] for a bubble interacting with a fixed wall, we consider that within a quasi-steady hypothesis a transverse drag force balances the repulsion force related to the modification of the flow about the body induced by the other body. For rectilinear paths of the bodies, the transverse drag force can be estimated from the results for a fixed body placed with a given incidence angle in a uniform flow. [START_REF] Fernandes | Etude expérimentale de la dynamique de corps mobiles en ascension dans un fluide peu visqueux[END_REF] determined by numerical simulations the force acting on a fixed disk of aspect ratio χ placed with different angles of attack in a uniform flow. He showed that the transverse drag force (i.e. the component of the force along the transverse direction y of the body) can be written as a classical drag force, i.e. as the product between the norm U * of the velocity of the disk, the transverse velocity v * and a dimensionless coefficient C y that depends on the aspect ratio of the body. For 2 χ 10, the numerical values obtained for C y can be conveniently modelled by C y = 2/χ. Denoting as F r the repulsion force induced by the interaction and as C r the dimensionless repulsion force coefficient such that F r = C r ρ f S ref U * 2 /2, the quasi-steady force balance in the transverse direction of a disk is then

1 2 C r ρ f S ref U * 2 = 1 2 C y ρ f S ref U * v * (4.1)
where S ref = πd 2 /4. This equation becomes, in non-dimensional form, C r = 2v/χ, where v = v * /U * is the dimensionless transverse velocity. Note that at leading order U * is close to the mean fall velocity U m of the body, so that U m will be applied in the following. We now use (3.1), characterizing the experimental kinematics of repulsion of the two bodies, to obtain the power-law dependence of the transverse velocity v and of C r on the horizontal distance ∆ h . For each disk, we have

v = 1 2 d∆ h dt = a 2 bt (b-1) = 1 2 a 1/b b∆ h (b-1)/b (4.2)
so that we get for the repulsion coefficient

C r = 1 χ a 1/b b∆ h (b-1)/b . (4.3)
Using the experimental values a = 1.1 and b = 2/7 leads to

C r ≃ 0.4 1 χ ∆ h -5/2
for 1.5 < ∆ h < 5. (4.4) Equation ( 4.4) indicates that the repulsive force acting between two identical disks falling side by side is inversely proportional to their aspect ratio (it varies as 1/χ) and decreases with their separation distance as ∆ h -5/2 . In the case of spherical bubbles, the repulsive force coefficient decreases more rapidly as ∆ h -4 . This difference is in line with a production of vorticity that is stronger for solid disks than for bubbles. Finally, note that the χ -1 dependence of the repulsion coefficient C r thus being stronger for thicker bodies, leads to the kinematics of repulsion that is independent of the aspect ratio observed in figure 3(a).

Interaction of two bodies having periodic paths

Above a critical Reynolds number that depends on the aspect ratio, the bodies display a quasi-planar zigzag motion, their velocity and orientation oscillate and periodic vortex shedding occurs in the wake of the bodies [START_REF] Fernandes | Oscillatory motion and wake instability of freely rising axisymmetric bodies[END_REF]. For these periodic paths, we observed that the predominant effect of the interaction for two bodies falling side by side is repulsion, as also happens in the case of rectilinear paths. Figure 4(a) presents the evolution of the horizontal distance ∆ h between the centres of gravity of the bodies as a function of the dimensionless time t for Re ≈ 280 ± 5 and figure 4(b) the corresponding vertical separation distance. For readability, only two cases are plotted. Since the disks do not oscillate in phase, both horizontal and vertical separation distances oscillate. The scatter of the results for ∆ h is larger than in the case of rectilinear paths. For instance the relative distance ∆ h varies between 3 and 3.8 at the dimensionless time t = 70. The envelope containing the mean evolution of ∆ h for all tests is plotted with dashed lines in figure 4(a). The two limit curves evolve as in the case of the rectilinear paths as ∆ h = at b with b = 2/7 for ∆ h 1.5. The factor a is equal to 0.9 for the lower limit and 1.1 for the upper limit. This upper limit corresponds to the case where the repulsion velocity is the most important and is identical to the expression obtained for disks with rectilinear trajectories (3.1). Oscillating disks therefore repel each other slightly slower than disks having rectilinear motion, and expression (4.3) remains a good approximation for the repulsion coefficient driving the sideways motion on a timescale much longer than the period of oscillation of the bodies.

We observed that the characteristics of the oscillatory motion of the disks are unaffected by the side-by-side repulsion: within experimental accuracy, the mean fall velocity U m , the Strouhal number and the amplitudes of oscillation in position and inclination are unchanged compared to those measured for isolated disks [START_REF] Fernandes | Oscillatory motion and wake instability of freely rising axisymmetric bodies[END_REF]. The eventuality of synchronization of the motion of the bodies due to synchronization of the vortex shedding in their wakes was also carefully investigated. As mentioned in the introduction, vortex shedding for two fixed bodies may synchronize. For objects in free fall investigated numerically in a two-dimensional geometry by [START_REF] Aidun | Direct analysis of particulate suspensions with inertia using the discrete Boltzmann equation[END_REF] and in a confined geometry by (2005) synchronization was also observed leading to oscillations of the trajectory and of the inclination of the bodies in phase or in phase opposition. In our case, despite extensive investigation, phase synchronization of the oscillations of the two disks was not observed. This may be due to the fact that in the case of freely falling bodies in a three-dimensional configuration, additional degrees of freedom influence the interaction between the wakes of the bodies and cause the absence of phase-locking between the paths of the two disks. In particular, in the three-dimensional configuration, the vertical planes containing the paths of the two bodies are generally distinct, they depend on initial conditions and may rotate slightly, as happens in the case of isolated bodies [START_REF] Fernandes | Oscillatory motion and wake instability of freely rising axisymmetric bodies[END_REF]).

Conclusion

This paper is devoted to the hydrodynamical interaction of two identical freely falling disks placed side by side for Reynolds numbers ranging from 100 to 300, corresponding to rectilinear and oscillatory paths. Careful experiments allowed us to bring to light a weak, though clear and systematic, repulsion effect between the disks for the three aspect ratios investigated. For bodies initially nearly in contact, this effect typically leads to a sideways motion of the body of about 2 diameters for a vertical displacement of 120 diameters. We observed that the disks always repel one another when the horizontal distance between their centres of gravity is less than 4.5 diameters. They never come closer for distances between 4.5 and 6 diameters. For distances greater than 6 diameters, the disks appear to have no effect on each other. For disks displaying rectilinear paths, we present a model for the repulsion coefficient C r , which decreases with the separation distance between the bodies and is inversely proportional to their aspect ratio, varying like 1/χ, so that the coefficient is stronger for thicker bodies. The proposed model satisfactorily reproduces the kinematics of interaction observed experimentally. Though we did not determine the dependence of C r on the Reynolds number, it appears that for the oscillatory paths the strength of the repulsion effect decreases slightly, being at most 20 % lower at Re = 300 than at Re = 100, whatever χ . However, for the oscillatory paths, the most notable result is that no synchronization is observed between the paths and between the wakes of the two disks, and that the only apparent effect of the interaction is the weak horizontal repulsion occurring until the bodies reach the distance at which they have no detectable effect on each other of about 6 diameters. Our three-dimensional experiments indicate that the degrees of freedom of the bodies are able to inhibit the synchronization phenomenon observed for fixed bodies or in constrained geometries for freely moving bodies. It is also noteworthy that the characteristics of the oscillatory motion of each disk appear unaffected by the interaction, indicating that in the range of parameters investigated the dynamics of the oscillatory motion of a disk are robust with respect to the flow generated by the presence of an identical disk at its side.

FIGURE 1 .

 1 FIGURE 1. Examples of the evolution of the dimensionless (a) horizontal distance ∆ h and (b) vertical distance ∆ z between the centres of gravity of the bodies as a function of the dimensionless time t = t * U m /d for five cases with χ = 10, Re = 115, d = 9 mm, U m = 13.2 mm s -1 .

  (a) presents the evolution of the horizontal distance separating the centres of gravity of the two disks normalized with their diameter d, ∆ h = ∆ * h /d, as a function of the dimensionless time t = t * U m /d, for χ = 10 and Re = 115. Several cases are presented.

FIGURE 2 .

 2 FIGURE 2. Examples of the evolution of the dimensionless (a) horizontal distance ∆ h and (b) vertical distance ∆ z between the centres of gravity of the bodies as a function of the dimensionless time t = t * U m /d for eight cases with χ = 10, Re = 115, d = 9 mm, U m = 13.2 mm s -1 .

FIGURE 3 .

 3 FIGURE 3. Dimensionless (a) horizontal distance ∆ h and (b) vertical distance ∆ z between the centres of gravity of the bodies as a function of the dimensionless time t: two cases for χ = 3, Re = 105, d = 5.7 mm, U m = 18.8 mm s -1 ; two cases for χ = 6, Re = 125, d = 7.2 mm, U m = 14.9 mm s -1 ; three cases for χ = 10, Re = 115, d = 9 mm, U m = 13.2 mm s -1 . The dashed red line corresponds to (3.1).

FIGURE 4 .

 4 FIGURE 4. Dimensionless (a) horizontal distance ∆ h and (b) vertical distance ∆ z between the centres of gravity as a function of the dimensionless time for oscillating disks: χ = 3, Re = 285, d = 12 mm, U m = 23.9 mm s -1 ; χ = 10, Re = 275, d = 18 mm, U m = 15.4 mm s -1 . Dashed lines in (a): (3.1) with a = 0.9 (lower line) and a = 1.1 (upper line), b = 2/7.

† Email address for correspondence: ern@imft.fr

Acknowledgements

The authors are grateful to Hervé Ayroles, Sébastien Cazin and Gregory Ehses for the technical support provided for the experiments.