Nicolas François

Yacine Amarouchene

Brahim Lounis

Hamid Kellay

N Franc ¸ois

Polymer conformations and hysteretic stresses in nonstationary flows of polymer solutions

Keywords: 47, 57, Ng -Polymers and polymer solutions 47, 57, Qk -Rheological aspects 47, 61, -k -Micro-and nano-scale flow phenomena

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

The conformational state of a polymer in a flow field may undergo a first-or second-order transition from a coiled to a stretched state when the deformation rate exceeds a threshold value dependent on the polymer relaxation time [1,2]. This was successfully confirmed by single molecule experiments and numerical simulations [3][4][5][6]. Recently, a remarkable slowdown of the relaxation of a polymer in extensional flows and in some random flows, in the vicinity of the coil-stretch transition has been found theoretically and numerically [7]. This has been attributed to hysteresis in polymer conformations as well as to the large variety in polymer conformations as observed in experiments [3,8]. This transition and the slowdown in relaxation have important consequences on the flow properties of polymer solutions such as the recently observed glassy-like dynamics [9] in filament stretching experiments [10]. However, few experiments have examined the link between the macroscopic phenomena and the microscopic details [11]. In this letter we aim to bridge this gap further and present both a macroscopic and a microscopic view of the transient dynamics when a polymeric solution relaxes toward a static or a dynamic stationary state by studying the interaction of a polymer-laden nonstationary flow with a cylinder embedded in it. In the stationary case, we had shown that the drag exerted on a small cylinder by a flowing polymer solution can be enhanced and is nonlinear vs. the flow velocity [12] even though the solution used has a constant shear viscosity. The nonlinearity of the stress sets in for shear rates greater than a critical shear γc given by the inverse of the relaxation time τ of the solution. This is characteristic of the coilstretch transition and is related to the formation of a microscopic shear band, exhibiting large molecular elongation and strong velocity fluctuations, in the immediate vicinity of the cylinder. The presence of the cylinder wall and its curvature turned out to be crucial for such large flow modifications. Here, we study a similar configuration but in the nonstationary case. The results suggest a connection between the dynamic slowdown of the macroscopic stress exerted on the obstacle and the significant hysteretic effects and slow relaxation of the polymer conformations.

We measure the stress variation upon a change of the flow velocity using our previously described optical fiber sensor and study the polymer conformations by seeding the solution with a minute amount of fluorescently labeled DNA probes [12]. The stress is measured through the deflection d of the bright end of the optical fiber cantilever (radius R = 30 µm), used as a cylinder immersed in the flow, with a CCD camera working at 1000 frames per second. The force resolution of this system is 0.1 µN. The flow was produced in square channels of H = 6 mm in side and a length of 20 cm. Using an electro-valve, we were able to create a variety of nonstationary flows with variations occurring with a characteristic time T v as small as 20 ms or as large as 1 s. The dynamics of the deflection d(t) is recorded along with the velocity variations U (t) in the center of the channel where the velocity profile, in the cross section of the channel, goes through a maximum. The average velocity in the square 1 channel is noted U mean (t), which is given by an average of the velocity profile in a cross section of the channel and is deduced from the maximum value of the velocity profile U (t). In the stationary case U mean (t) is independent of time and is given by Q/H 2 , where Q is the flux through the channel. The semidilute polymer solutions use linear polyethyleneoxide (PEO, M w = 4 • 10 6 amu) that was dissolved, at a concentration of 3000 parts per million by weight (ppm) in deionized water with 50% sugar by weight. The relaxation time τ of this solution was found to be 8 ms. Another linear polymer (polyacrylamide (PAam)), with M w = 18 • 10 6 amu and in the presence of small amounts of salt, 0 and 0.17 M, with τ = 1.1 and 0.06 s, respectively), and other PEO concentrations (1500 ppm with τ = 5 ms) were also used for certain runs. The relaxation times were estimated as τ (η0-ηS )Mw cρRT , where η 0 is the zero shear viscosity of the solution, η S is the viscosity of the solvent, R is the gas constant, T is the temperature, ρ is the density, and c is the polymer concentration in ppm [START_REF] Larson | The Structure and Rheology of Complex Fluids[END_REF]. This time is the longest relaxation time in the system and was found comparable to relaxation times obtained from normal stress measurements, from the relaxation of the stress, and from the frequency dependence of the viscoelastic moduli of the solution. A second setup is used to infer the polymer conformations in the immediate vicinity of the cylinder in a band of about 10 µm from the cylinder wall. We had previously shown that polymer elongation occurs only near the cylinder [12]. We produced 20 mm long, 1 mm wide, and 160 µm high PDMS (polydimethylsiloxane) microchannels with isolated 40 µm diameter and 160 µm high cylindrical obstacles embedded in it. A 3000 ppm PEO solution, in a buffer of pH 7 and 65% by weight of added sucrose, seeded with a small amount of fluorescently labeled DNA molecules (T4-DNA of contour length 68 µm and a relaxation time of 80 s) was studied in this part. The relaxation time of this solution is τ = 0.3 s. Since the PEO molecules cannot be visualized, the best that can be done at present is to use the DNA molecules as probes of the PEO conformations even though labeled DNA can be stiffer than the nonlabeled polymers. We had previously argued that visualization of the DNA probes gives information about the conformations of the PEO molecules [12] 1 . We observed the stained molecules using an inverted fluorescence microscope (×60, 1.45 NA) and a highly sensitive camera. Measurements were made at the center of depth of the channel, where the channel-induced shear rate is negligible. Inversion of the flow was performed with a system of shunting pipes and two syringe pumps. In both setups the high sucrose concentration of the solutions allows to neglect shear thinning effects.

Our optical fiber sensor, in a Newtonian fluid, is the analog of an overdamped oscillator. Its associated restoring force can be written as F = 3πR 4 E 4l 3 d, where E is the Young modulus and l is the length of the fiber. In a stationary flow, the deflection of the fiber is linear vs. U mean [12]. This remains true for the nonstationary flows used here. The deflection d(t) and the velocity U mean (t) measured during a stop flow from a constant velocity to zero follow the same dependence vs. time. When this variation is represented in terms of stress σ = F 2πRl and shear rate γ [12,[START_REF] Faxèn | Proc. R. Swed. Acad. Eng. Sci[END_REF], a linear relation is obtained as shown in the inset of fig. 1(a) giving a constant viscosity η ∼ 0.05 Pa s which compares very well with rheological measurements. This linearity confirms that fiber inertia (the fiber's characteristic time is 1 ms which is far below T v ) and transient effects inherent for a Newtonian fluid (added mass and the viscous history force) are negligible [START_REF] Gondret | [END_REF]. Thus, our cantilever is a viscometer that works both in the stationary and nonstationary cases.

= 2•Umean(t) R•ln( H 2R -1)
For the polymer solutions the variation d(t) and U mean (t) are shown in fig. 1(a) for a similar stop-flow variation as the Newtonian case discussed above. Note that, contrary to the Newtonian solution, the deflection d(t) lags behind U mean (t). What is more striking is that even when the velocity changes sign and becomes negative, the deflection remains nonzero and positive showing a clear anti-correlation characteristic of an elastic response. A long time relaxation is also observed after the flow has been turned off. The dynamic stress exerted on the fiber turns out to be different from the stationary one, as shown in the inset. This complex response, resembling that of a yield stress fluid, exhibits both drag reduction at high shear rates and drag enhancement at lower shears with respect to the stationary case. This dynamic response actually depends on the shape of the velocity variation. For a step flow-like variation a difference between the stationary and nonstationary cases is seen only above γc . If the velocity is varied in a range where the shear rate stays below γc , d(t) and U mean (t) follow each other giving a linear stress shear rate relation. However, when the velocity is in the range where the shear rate is higher than γc , d(t) and U mean (t) do not follow each other as shown in fig. 1(b). The full behavior of the stress strain curve from different step flows is shown in the inset of fig. 1(b) showing no effect below γc and strong hysteresis above it. This response is different from the one of fig. 1(a) which shows that for a stop flow, the difference with the stationary case persists even for shear rates below γc , suggesting that history or memory effects are also important. The characteristic closing time T v is much greater than τ suggesting an adiabatic change. Nonetheless, a very large difference between the dynamic stress and the stationary stress is observed. These observations are compelling evidence for the existence of an elastic history-dependent force F el (t) acting on the fiber. We propose a new elastic stress

σ el (t) = F el (t) 2πRlimm = - t 0 C • exp(-(t-τ ) λ ) ∂U ∂τ ∂σstat
∂U dτ , where σ stat is the stress measured in the stationary case, C is a constant, and the memory function is exponential with a characteristic time λ. In fig. 1 we plot the result of the total nonstationary drag force σ = σ stat + σ el (t), which gives an accurate description for both step and stop-flow force measurements despite the large differences between them. This additional elastic force accounts for the hysteretic effects, the anti-correlation as well as the strong dependence on the flow history. Note that the clear slowdown in the dynamics introduces a long relaxation time λ ∼ 60 ms for the examples of fig. 1, a value 8 times greater than the relaxation time τ . The value of λ depends on the initial Weisenberg number W i 0 = γ0 τ , ( γ0 is the initial shear rate given by the initial plateau value of the velocity) and while it is small below W i c = γc τ , it increases as W i 0 increases, goes through a maximum near W i 0 ∼ 3, and decreases to values close to the relaxation time of the solution for large W i 0 as shown in fig. 1(c) where different values of γ0 , τ , and T v have been used. Note here that we have used data from different polymer solutions and that λ depends on how the velocity changes in the sense that even for an initial W i 0 below threshold, if the transient shear exceeds γc , a long time relaxation is observed. While the variation of λ mimics the slowdown of the polymer relaxation time near the transition [7], the times measured here seem to be large for large W i. Collective behavior is probably important in our case considering the high concentrations used. Also, the presence of the cylinder wall and the fact that a shear band, with a large number of stretched molecules, nucleates in its vicinity [12] is another possible reason for such differences.

What is the link between such complex dynamics and the dynamics of polymer conformations? We probe the microscopic details of the polymer solution through visualizations of the polymers near the cylinder. We use the second setup to image the conformations of DNA molecules, considered as probes of the PEO conformations [12], within the PEO solution during a flow inversion. As in our previous work, we note U c the critical velocity to trigger the coil-stretch transition in our setup. The variation of the velocity, measured at 30 µm from the cylinder (a location where the polymers are coiled), vs. time is shown in fig. 2(A). It starts out at a constant value above U c and goes from a positive to a negative value in a time interval of about 10 s. The snapshots in fig. 2 display the conformations of a molecule at different positions around the cylinder taken at different times during the flow inversion. This molecule starts out as a small dot far from the cylinder (coil), starts to stretch abruptly near the upper stagnation point whose importance for the unwinding of the molecules had been stressed before [12] (images a and b), and gets further stretched as it travels farther downstream (image c). Image d is taken at the moment of inversion showing that the molecule stays stretched and remains so as it travels backward due to the flow inversion (image e). This remarkable feature was observed for a large number of molecules we have analyzed and is in agreement with previous observations [8]. The polymer conformation has not relaxed to a coil despite the long duration of the velocity inversion; the velocity stays below U c for roughly 2 s which is much longer than the relaxation time of the polymer solution. behavior. A variety of molecular conformations is observed during the flow inversion as shown in fig. 2 (images d+ and e+): simple dumbbell, "squeezed spring", folded dumbbell, stem and flower configuration, etc. showing the complexity of molecular shape relaxation [3]. Such a variety of conformations as well as the hysteresis observed are the key features of the slowdown discussed previously [7].

In the stationary flow, the extension of the molecules depends on the location θ around the cylinder and the mean velocity of the flow. In order to take such variation into account and separate the effects due to the nonstationary part of the flow from the effects observed in the stationary case, we use the ratio defined as:

= L(θ,t) (Lstat(θ)-2•Req)• U (t) U stat +2•Req
, where L(θ, t) is the time variation of the molecular extension, R eq ∼ 0.8 µm is the measured equilibrium radius, and L stat (θ) is the mean local extension, measured separately from experiments in the stationary case when U = U stat . This expression takes two facts into account, first that the extension depends on the angular position θ through L stat (θ) and second that this length depends on the velocity of the flow which we assume to be linear for simplicity. We plot the ratio in fig. 3(a) using data from 20 molecules reversing at different locations around the cylinder. Notice the large variation, from molecule to molecule, in the temporal dynamics of the extension. Despite this variety, a trend is obtained by computing the mean value of which is small at the high velocity end showing that the extension is similar to the stationary case far from the imposed flow variation, but increases during the flow inversion and reaches values as high as 3 or 4 showing that the molecules are much more stretched than for the stationary case. Histograms of this ratio for low velocity and high velocity values are also reported in the inset showing that a high fluctuation level exists near the flow inversion [5,6] while far from it, the molecular extensions are close to their stationary state. The ratio is asymmetric around the flow inversion point and shows a hysteretic-like behavior. To make a link with the stress measurements, we plot in fig. 3(b) the dynamic stress of fig. 1(a) to which we subtracted the stationary stress. This curve presents a clear elastic storage at γ = 0 s -1 and shows a hysteretic effect during the flow inversion. Figure 3 shows that the nonstationary part of the stress and of the molecular extension display very similar features: an asymmetry around the flow inversion point and a persistent nonzero response as the flow vanishes. Since the additional stresses related to the presence of polymers are directly related to the molecular extension, we propose that the slow dynamics of the stress relaxation is related to the slow relaxation dynamics of the molecular conformations.

In conclusion, the behavior of a polymer solution in a nonstationary flow around a cylinder is dominated by a long time relaxation both for the stress and the molecular conformations. The time scale of this relaxation is much longer than the longest relaxation time of the molecules or of the solution. While slow dynamics is expected on theoretical grounds in the vicinity of the coil stretch transition, our results show that this slowdown occurs even for high deformation rates and suggest a direct link between the macroscopic response of a flowing polymer solution and the behavior of individual molecules.

Fig. 1 :

 1 Fig. 1: (Colour on-line) Fiber deflection and velocity vs. time: (a) Polymer solution and a stop flow. (b) Polymer solution and a step flow for γ > γc. Insets: σ vs. γ (open symbols) and σstat vs. γ (solid symbols and solid lines) as well as the extent of the linear regime below γc. The Newtonian (sucrose solution) case is shown in the inset of panel (a). The solid lines in (a) and (b) are fits using our history-dependent elastic force with λ = 60 ms and C = 0.6 in (a) and 0.78 in (b). (c) λ/τ vs. W i0 for different polymers and different flow variations, the arrows indicate the variation of γ and the dashed line is a guide to the eye.

Figure 2 (Fig. 2 :

 22 Fig. 2: (Colour on-line) T4-DNA molecules near the cylinder at different moments as shown in the schematic and indicated by letters in A which shows U (t) during the flow inversion. The solid line is a guide to the eye. (B) The extension of the molecule vs. time. The photographs in d+ and e+ show other conformations seen at positions d and e.

Fig. 3 :

 3 Fig. 3: (Colour on-line) (a) Nonstationary molecular extension: thin lines are for individual molecules and thick lines are for average values. (b) Excess stress vs. shear extracted from the deflection and velocity vs. time shown in the inset. The data are taken between points A and B where the velocity changes sign as is shown in the inset. The arrows indicate the direction of the velocity change.

The PEO solutions showed drag enhancement above a critical W ic = γcτ close to 1 indicating, according to theory and experiment, that these molecules are elongated. Visualization of two different DNA molecules (λ and T4-DNA whose relaxation times are different from each other and much larger than that of the PEO solution) embedded in the PEO solutions showed that they are highly elongated only at or above W ic, determined in the drag enhancement measurements, independently of the DNA used. Since the characteristic time for the transition to highly elongated DNA molecules is that of the PEO solution, which is much smaller than that of the DNA molecules, we have deduced that they are probes of the PEO conformations and that when they are elongated so are the PEO molecules and vice versa.


M Aucejo

N Totaro

J-L Guyader

Identification of source velocities on 3D structures in non-anechoic environments: Theoretical background and Experimental Validation of the inverse Patch Transfer Functions Method

Keywords: Holography, FEM, Patch, Transfer Functions

Identification of source velocities on 3D structures in non-anechoic environments: Theoretical background and experimental validation of the inverse patch transfer functions method

Introdution

The Nearfield Acoustical Holography (NAH), firstly introduced by Williams et al. [START_REF] Williams | Sound source reconstructions using a microphone array[END_REF][START_REF] Williams | The nearfield acoustical holography (NAH) experimental method applied to vibration and radiation in light and heavy fluids[END_REF], is based on the two-dimensional Fourier Transform of the complex pressure field at a given frequency measured on a hologram near the source. Measurements of the pressure field are generally made on a plane surface and make possible the determination of the three-dimensional source velocity field. However, the NAH is mainly applicable to simple geometries (planes [START_REF] Williams | Fourier Acoustics: Sound Radiation and Nearfield Acoustical Holography, chapter 3 -The Inverse Problem: Planar Nearfield Acoustical Holography[END_REF], cylinders [START_REF] Williams | Fourier Acoustics: Sound Radiation and Nearfield Acoustical Holography, chapter 5 -The Inverse Problem: Cylindrical NAH[END_REF] and spheres [START_REF] Williams | Fourier Acoustics: Sound Radiation and Nearfield Acoustical Holography, chapter 7 -Spherical Nearfield Acoustical Holography[END_REF]). Furthermore, problems of discontinuities exist at boundaries, because of the lack of information beyond the zone of measurements nevertheless necessary to compute the two-dimensional Fourier Transform. Thus, to avoid the influence of boundary effects, measurements have to be made in the acoustical nearfield of the structure.

The iFRF method is based on the evaluation and the inversion of transfer matrices, which can be obtained either experimentally or numerically. In the last case, if the computation of the transfer functions is based on the Boundary Element Method, it is called iBEM. This method can be used to reconstruct acoustic radiation on arbitrary surfaces [START_REF] Bai | On the reconstruction of the vibro-acoustic field over the surface enclosing space using the boundary element method[END_REF]. In that way, Martinus [START_REF] Martinus | An advanced noise source identification technique using the inverse boundary-element method[END_REF] uses this method to determine the distribution of particle velocity on the open end of a rectangular duct and shows that sound pressure field measurements need not to be made in the nearfield of the source to obtain good results. However, the main drawback of iBEM is that it requires an excessive amount of measurements to determine the acoustic field on a complex structure, since it needs a fine mesh definition (six nodes per wavelength). The difficulty of the method comes also from the inversion of the transfer matrices, which are often ill-conditioned and thus requires the use of regularization methods.

Other methods are also developed to overcome the difficulties and limitations inherent to NAH or iBEM. Thus, a hybrid NAH was introduced [START_REF] Wu | Hybrid nearfield acoustical holography[END_REF][START_REF] Zhao | Reconstruction of vibro-acoustic fields using hybrid nearfield acoustic holography[END_REF]. This method is based on a modified HELS method (Helmholtz Equation Least Squares) [START_REF] Wu | Reconstructing interior acoustic pressure field via helmholtz equation least-squares method[END_REF][START_REF] Rayess | Experimental validations of the hels method for reconstructing acoustic radiation from a complex vibrating structure[END_REF] and allows a reconstruction of the pressure field on a complex shape surface very close to the source surface. As NAH applications, the hybrid NAH needs regularization, because of measuring uncertainty and incompleteness of acoustic pressure field, which leads to ill-conditioned trans-fer matrices. Furthermore, the modified HELS method can be also combined with iBEM [START_REF] Wu | Combined helmholtz equation least-squares method for reconstructing acoustic radiation from arbitrarily shaped objects[END_REF] to reconstruct acoustic quantities on a virtual sphere enclosing the source structure, which can lead to the loss of nearfield information. Finally to avoid a great number of measurements, the PSP (Principal Source Projection) method is introduced [START_REF] Leclère | An alternative acoustic imaging technique to improve capabilities of microphone array measurements[END_REF][START_REF] Leclère | Application of an innovative acoustic imaging technique to assess acoustic power maps of a gasoline engine[END_REF]. This method is based on the evaluation of a radiation operator between the source surface, represented by a distribution of elementary sources, and a grid of measurements. For traditional methods, like iBEM, the number of identified acoustic sources is limited to the number of microphones in order to guarantee the uniqueness of the solution. On the contrary, for PSP method, the number of quantified sources is higher than the number of microphones, which means that the system is under-determined. In addition, to identify the data on the source surface, the transfer functions, calculated between a point of the elementary source and a point of measurement, have to be inverted. The transfer matrix is often ill-conditioned, which implies to use regularization methods like Truncated Singular Values Decomposition (TSVD).

In this article, an alternative method is introduced to identify source velocities: the iPTF method. This method is derived from the Patch Transfer Functions (PTF) method [START_REF] Maxit | Airbone noise prediction using patch acoustic impedance[END_REF][START_REF] Ouisse | Patch transfer functions as a tool to couple linear acoustic problems[END_REF][START_REF] Totaro | Extension of the Patch Tranfer Functions method (PTF Method) to high frequency domain (sub-cavities decomposition)[END_REF]. In its direct formulation, the PTF method is a tool to predict pressure inside and outside a cavity containing acoustic sources and apertures, thanks to the use of substructuring and impedance concepts. The acoustical medium is indeed divided into subdomains and PTF are evaluated by suitable methods for each subdomain (FEM, Rayleigh approach, measurements, . . . ). Subdomains are then coupled through their common surface, divided into patches. Coupling conditions are written for each patch as pressure and velocity equations, describing the local equilibrium. A system of linear equations is finally obtained, where unknowns are coupling patch velocities. The system is solved to determine coupling velocities and finally pressures at any point of the acoustical domain.

The aim of the inverse approach (iPTF method) is to determine source velocities from measured patch pressures and velocities on a virtual cavity. Theoretically, iPTF method allows identifying source velocities in a noisy environment (i.e. in the presence of other sources) thanks to double measurements of pressure and particle velocity fields on a surface surrounding the source, what is not possible with classical methods, which only use sound pressure measurements. Measurements can thus be made, for instance, on a part of an engine in use. However, as classically observed in inverse methods, the inversion of an ill-conditioned matrix is required. But this drawback can be alleviate by using suitable regularization techniques like TSVD or Tikhonov. Finally, one of the advantages of iPTF method is the use of the Finite Element Method (FEM) as a solver. When other methods need analytical or BEM model to estimate transfer functions, iPTF only needs modal basis of a virtual cavity to compute Patch Transfer Functions. This makes the method applicable even to sources with complex geometries.

The present article deals with the theoretical background of the method and its numerical and experimental application on a simple case (two baffled pistons driven in antiphase). This test case is voluntarily simple in order to easily study the main features of the proposed method, namely the size of the virtual cavity, the number of measurement patches, the ability to localize and separate sources and finally the robustness of the method with respect to an external stationnary disturbing source. Experimental validations of the iPTF approach confirm expected advantages and demonstrate the applicability of the method.

Theoretical background of the iPTF method

Basic concept of the iPTF method: the integral formulation

Let us consider the acoustic cavity presented in Fig. [START_REF] Williams | Sound source reconstructions using a microphone array[END_REF]. The acoustical domain Ω is delimited by a rigid surface S r , an absorbing surface S a , a vibrating surface S v and a surface with Dirichlet conditions S d . Furthermore, an acoustic source S 0 is located inside the acoustic volume Ω, but outside a part Ω c of Ω. Consequently, the problem to solve is expressed by Eq. ( 1)

                     ∆p(M ) + k 2 p(M ) = S 0 (M ) ∀M ∈ Ω ∂p ∂n (Q) = 0 ∀Q ∈ S r ∂p ∂n (Q) = -jρωV (Q) ∀Q ∈ S v p(Q) = p 0 ∀Q ∈ S d p(Q) = - Z jρω ∂p ∂n (Q) ∀Q ∈ S a (1)
Where ∆ is the Laplacian operator, ∂ ∂n the normal derivative, ρ the density of air, V (Q) the normal velocity imposed on S v , p 0 the pressure imposed

Ωc ∆Φ + k 2 Φ Ψ -∆Ψ + k 2 Ψ Φ dΩ c = Sc Φ ∂Ψ ∂n -Ψ ∂Φ ∂n dS c
(2) The previous expression is very general and functions Φ and Ψ are arbitrary. In the present work, we suppose first that the function Φ is the acoustic pressure p. In the cavity Ω c , where no sources acting, the pressure satisfies the homogeneous Helmholtz equation as expressed in Eq. (3),

∆p(M ) + k 2 p(M ) = 0 ∀M ∈ Ω c (3)
The function Ψ is chosen as the Green's function G satisfaying Eq. ( 4).

   ∆G(M, M ′ ) + k 2 G(M, M ′ ) = 0 ∀(M, M ′ ) ∈ Ω c ∂G ∂n (Q, Q ′ ) = δ(Q -Q ′ ) ∀(Q, Q ′ ) ∈ S c (4)
Consequently, by introducing Eqs. (3) and (4) in Eq. ( 2), we finally obtain the integral equation given by Eq. [START_REF] Williams | Fourier Acoustics: Sound Radiation and Nearfield Acoustical Holography, chapter 7 -Spherical Nearfield Acoustical Holography[END_REF].

p(Q ′ ) = Sc G(Q, Q ′ ) ∂p ∂n (Q)dQ (5)
The key point of this formulation is that external sources located outside the cavity do not modify the integral equation. In other words, this formulation is independent of acoustic sources external to Ω c . Furthermore, another asset of integral formulation method is that the Green's function G can be arbitrarily chosen. To illustrate this point, let us consider the Green's function G ′ satisfaying Dirichlet's boundary condition as expressed in Eq. ( 6)).

∆G ′ (M, M ′ ) + k 2 G ′ (M, M ′ ) = 0 ∀(M, M ′ ) ∈ Ω c G ′ (Q, Q ′ ) = δ(Q -Q ′ ) ∀(Q, Q ′ ) ∈ S c (6)
The introduction of Eqs. ( 3) and ( 6) in Eq. ( 2) allows obtaining the integral formulation given by Eq. ( 7), which is independent of external acoustic sources too.

∂p ∂n (Q ′ ) = Sc p(Q) ∂G ′ ∂n (Q, Q ′ )dQ (7)

Basic equation of the iPTF method

Let us consider the practical situation given in Fig. [START_REF] Williams | The nearfield acoustical holography (NAH) experimental method applied to vibration and radiation in light and heavy fluids[END_REF], where a virtual cavity of surface S c = S m ∪ S v surrounds the source and separates the acoustical domain into a virtual cavity Ω c and an exterior domain Ω e .

In the iPTF formulation, the Green's function used is that of virtual cavity where boundary surface S c is supposed rigid. It is important to notice that these rigid wall boundary conditions have no physical reality. They just provide a mathematical tool to solve the virtual cavity problem. The integral formulation to solve is consequently the one given by Eq. (5). When using Euler's equation in Eq. ( 5), the acoustic pressure p(Q ′ ) at the point Q ′ is related to the normal velocity V n (Q) at each point Q of the surface S c as expressed in Eq. [START_REF] Wu | Hybrid nearfield acoustical holography[END_REF].

p(Q ′ ) = -jρω Sc G(Q, Q ′ )V n (Q)dQ (8)
Numerical solution of the previous integral equation is based on the discretization of the surface S c into N elementary areas A r called patches. The 8) on patches basis leads to Eq. ( 9), describing the relation between the space average normal velocity vr on the patch r and the space average pressure pj on the patch j. One can notice that this relation remains true provided that the size of patches is small compared to the wavelength.

pj = N p=1 Z jr vr (9)
In Eq. ( 9), the space average pressure and velocity are related by the patch impedance Z jr . Consequently, the patch impedance Z jr between the excited patch p and the receiving patch j is defined as the ratio of the space average pressure pj on a patch j and the space average normal velocity vr on a patch p as expressed in Eqs. [START_REF] Wu | Reconstructing interior acoustic pressure field via helmholtz equation least-squares method[END_REF].

Z jr = pj vr ( 10
)
Where the space average is defined as

•r = 1 Ar Ar • dA r
The basic equation of the iPTF is derived from Eq. ( 9), where the contribution of patches is separated into patches belonging to the source surface S v and patches belonging the virtual surface S m as expressed by Eq. [START_REF] Rayess | Experimental validations of the hels method for reconstructing acoustic radiation from a complex vibrating structure[END_REF].

pj = P k=1 Z jk vk + M i=1 Z ji vi (11)
Where j and i belong to S m , k belongs to S v and N = M + P Now, the iPTF formulation can be obtained from Eq. ( 11) in a straightforward manner. For sake of simplicity, Eq. ( 11) is written in matrix form (see Eq. ( 12)) and following notations are considered:

P j = pj , V i = vi and V k = vk {P j } = [Z jk ]{V k } + [Z ji ]{V i } (12)
One has to notice that in matrices [Z ji ] and [Z jk ], the number of line j represents the number of patch pressure measurements {P j }, while the number of column k corresponds to the number of identification patches of source velocity {V k } and i represents the number of patch velocity measurements {V i }.

The vector of source velocities {V k } is thus easily obtained after simple matrix manipulation and is the basic equation of the iPTF method.

{V k } = [Z jk ] -1 ({P j } -[Z ji ]{V i }) (13)
Insofar as the impedances matrices [Z jk ] and [Z ji ] are numerically computed, only pressures {P j } and velocities {V i } have to be measured on each patch of the virtual surface S m to calculate the mean patch source velocities {V k }. In the iPTF method, the patch pressure {P j } and the patch velocity {V i } are measured with a pressure-velocity (PU) probes. Such PU probes have been assessed by Jacobsen and Jaud [START_REF] Jacobsen | A note on the calibration of pressure-velocity sound intensity probes[END_REF][START_REF] Jacobsen | Statistically optimized near field acoustic holography using an array of pressure-velocity probes[END_REF]. One important point is that velocity field on the virtual surface can be due to direct field from the vibrating surface as well as reflected sound by obstacles placed outside the virtual cavity. Thus, this method is not restricted to anechoic environment and is theoretically independent of sources located outside the measurement area in virtue of the Kirchhoff's integral. Nevertheless, the inversion of the impedances matrices can be difficult, since these matrices are rectangular and ill-conditioned as it will be presented in the section 2.4.

Computation of patch impedance Z jr

As already said in section 2.2, the patch impedance matrices [Z jk ] and [Z ji ] are numerically computed. They are derived from the integral formula-tion (see Eq. ( 8)) and the expansion on virtual rigid wall cavity modes. To obtain these quantities, we have to calculate the function G(Q, Q ′ ) satisfaying Eq. ( 4). For this purpose, we expand the solution on normal modes of the rigid wall cavity. This leads to the Green's function given by Eq. ( 14). In appendix A, the detailed calculation of the Green's function is presented.

G(Q, Q ′ ) = - n c 2 φ n (Q)φ n (Q ′ ) Λ n (ω 2 n -ω 2 + jη n ω n ω) (14)
Where Λ n is the norm of the mode n, η n the modal damping, avoiding singularities when ω = ω n .

Then, the pressure p(Q ′ ) created by a constant normal velocity vr imposed on a patch r of surface S r can be calculated from Eq. [START_REF] Maxit | Airbone noise prediction using patch acoustic impedance[END_REF].

p (Q ′ ) = -jρωc 2 S r n φnr φ n (Q ′ ) Λ n (ω 2 n -ω 2 + jη n ω n ω) vr (15)
The patch impedance Z jp between the excited patch p and the receiving patch j is thus given by Eq. ( 16), where the space average pressure over receiving patches is introduced.

Z jp = pj vp = - n jωρc 2 S p Λ n (ω 2 n -ω 2 + jη n ω n ω) φnp φn j ( 16
)
Where S p is the surface of the patch p.

One can also notice that the natural frequencies ω n and the mode shapes of the virtual rigid wall cavity φ n are obtained by FE calculation, which permits to deal with complex source geometries.

Some explanations about ill-conditioned impedance matrices

In the iPTF method, impedance matrix [Z jk ] has to be inverted. However, in many inverse problems, matrices can be ill-conditioned, which is the case in the present method. Consequently, it is important to identify factors causing the ill-conditioning of [Z jk ] matrix. For this purpose, the conditioning number is a good indicator and serves as base for the following discussion. One reminds that [Z jk ] matrix relates the number of patches j, where patch pressures {P j } are measured to the number of patches k,where patch source velocities {V k } are identified.

Influence of pressure information on source surface

Let us consider the source surface presented in Fig. (3). To avoid particular cases, this source surface has no symmetries. A virtual cavity is then defined as well as measurement and source patch meshes as shown in Fig.

(3). The dimensions of the virtual cavity are about 500 × 350 × 400 mm 3 . First of all, we consider the standard way of using the method, where pressure is measured on all virtual surfaces but not on the source surface. In this case, conditioning numbers versus frequency are presented in Fig. (4). The analysis shows that impedance matrix is ill-conditioned. Conditioning number globally decreases with frequency and presents peaks on resonance frequencies of the cavity. To clarify this tendency, we consider a second situation, corresponding to the first one with addition of pressure measurement on the source surface.

The conditioning number, presented in Fig. [START_REF] Williams | Fourier Acoustics: Sound Radiation and Nearfield Acoustical Holography, chapter 5 -The Inverse Problem: Cylindrical NAH[END_REF], clearly indicates that impedance matrix is better conditioned. Pressure information on source surface is thus important to decrease the conditioning number. Unfortunately, practically speaking, pressure measurement on source surface is often not possible in real situation. However, it may be sufficient to measure only few pressure data on the source surface to really decrease the conditioning number. In this way, we assumed that pressure is measured on all the virtual surfaces and on only one patch located in the center of source surface. Result is presented in Fig. [START_REF] Williams | Fourier Acoustics: Sound Radiation and Nearfield Acoustical Holography, chapter 5 -The Inverse Problem: Cylindrical NAH[END_REF], with one additional pressure point on the source surface. Above 1200 Hz, the conditioning number is divided by two which represents a real improvement. However, at lower frequency, no significant modifications of the conditioning number appear. Pressure measured on all the virtual surfaces, (-.-) Pressure measured on all the virtual surfaces and on the source surface and (--) Pressure measured on all the virtual surfaces and on one patch located in the center of source surface

Influence of the number of pressure control points

From a practical point a view, one can wonder if all the pressure information on virtual surfaces is necessary. For this purpose, we used the same source geometry and virtual cavity as in section 2.4.1, but we only used pressure information measured on the top virtual surface (see Fig. (3)), instead of using pressure information on all the virtual surfaces. Conditioning numbers versus frequency are presented in Fig. [START_REF] Williams | Fourier Acoustics: Sound Radiation and Nearfield Acoustical Holography, chapter 7 -Spherical Nearfield Acoustical Holography[END_REF] for the previous situations. The analysis shows as expected that it is preferable to measure pressure field on all the virtual surfaces to limit the conditioning number.

The conclusion is thus simple, the more measured patch pressure, the lower the conditioning number. Practically, it is recommended to measure pressure and velocity data on all patches of the virtual surface and if possible on additional points on the source surface. In the rest of this paper, pressure information on source surface is voluntarily not measured insofar as the aim is to find a method to identify properly source velocity without measuring data on the source surface.

Influence of virtual cavity dimensions

In the iPTF method, the virtual cavity can be arbitrary defined. It is thus interesting to know, if it is preferable to perform measurements on a small or large virtual cavity. To compare results, we consider the same source geometry as in the previous subsection (see Fig. [START_REF] Bai | On the reconstruction of the vibro-acoustic field over the surface enclosing space using the boundary element method[END_REF]). The virtual cavity smaller than the previous one is almost a parallelepiped with dimensions about 500 × 350 × 120 mm 3 and the patch mesh is defined on virtual surfaces as presented in Fig. [START_REF] Bai | On the reconstruction of the vibro-acoustic field over the surface enclosing space using the boundary element method[END_REF]). Furthermore, we assumed that pressure and particle velocity are measured on all the virtual surfaces. Fig. [START_REF] Martinus | An advanced noise source identification technique using the inverse boundary-element method[END_REF] presents conditioning numbers for the large and the small virtual cavities (see Figs.

(3) and ( 6)). This figure shows that patch impedance matrix conditioning number for a small cavity is better at low frequency than that obtained for a large cavity. But when modal overlap is achieved both cavities have similar conditioning numbers. A small cavity is therefore preferable to limit the conditioning number. This point can be associated to Pressure-Velocity measured on all the virtual surfaces for (-) the large virtual cavity and (--) the small virtual cavity

Influence of virtual cavity modes

Ill-conditioning can also be due to a bad definition of the virtual volume and patch mesh. Ouisse et al. [START_REF] Ouisse | Patch transfer functions as a tool to couple linear acoustic problems[END_REF] define a numerical criterion to limit the ill-conditioning of patch impedance matrices. This criterion stipulates that the order of virtual cavity modes in one direction has to be greater than or equal to the number of patches in the same direction. It corresponds in fact to the number of degrees of freedom necessary to ensure that patches motions are independent. To demonstrate the validity of this criterion, conditioning number is evaluated for a parallelepiped virtual cavity which dimensions are 450 × 350 × 130 mm 3 . A patch mesh is then defined on measurement and source surface and natural modes have been extracted up to 2kHz, 3kHz, 4kHz and 5kHz. In each case, patch impedance matrices have been computed in the frequency range of interest [10, 2000 Hz]. As shown in Table 1, the criterion is not respected for modes extrated up to 2 kHz and as a consequence the patch impedance matrix is extremely ill-conditioned (see Fig. [START_REF] Wu | Hybrid nearfield acoustical holography[END_REF]). Furthermore, the criterion is respected above 3 kHz and conditioning numbers are very much better than that obtained for modes extracted up to 2 kHz. Nevertheless, a regularization would be compulsory in this configuration but this criterion is required to limit the ill-conditioning of impedance matrices. When virtual cavity has a complex geometry, this criterion is not directly applicable. However, one can say that the number of virtual cavity modes must be at least equal to the number of patches.

Numerical and experimental validation of the iPTF method

In this section, we seek to validate the iPTF method numerically and experimentally from a simple set-up. The retained set-up consists of two baffled pistons driven in antiphase as shown in Fig. [START_REF] Zhao | Reconstruction of vibro-acoustic fields using hybrid nearfield acoustic holography[END_REF]. The chosen test case is voluntarily simple in order to clearly establish how the method is able to localize and separate vibrating sources. The identification area (dimensions:

. Validation of the basic principle

To validate the basic principle given by Eq. ( 13), we define a parallelepiped virtual cavity measuring 450 × 350 × 20 mm 3 around the source (see Fig. (10a)). Moreover, in the PTF method, the definition of patch meshes is driven by a wavelength criterion according Ouisse et al. [START_REF] Ouisse | Patch transfer functions as a tool to couple linear acoustic problems[END_REF]. They have demonstrated that in its direct formulation, the PTF method is applicable with a λ ac /2 criterion for patch meshes. However, for iPTF method, an additional condition must be considered; the measurement mesh has to be fine enough to obtain a sufficient number of measurement data. In other words, the number of measurement patches has to be larger than the number of identification patches to avoid under-determination of the problem. Based on this observation, the measurement surface is divided into 139 patches and pressure and particle velocity fields are analytically computed at the center of each measurement patch with standard Rayleigh integral method (see appendix B for details). The process allows identifying properly the source velocity, since reference and identification maps at 240 Hz slightly differ as shown in Figs. [START_REF] Rayess | Experimental validations of the hels method for reconstructing acoustic radiation from a complex vibrating structure[END_REF] and [START_REF] Wu | Combined helmholtz equation least-squares method for reconstructing acoustic radiation from arbitrarily shaped objects[END_REF].

Robustness of the iPTF method -Influence of the measurement noise

The measurement noise is an important parameter in the results accuracy. In order to analyse the influence of measurement uncertainties, we propose to slightly modify the exact pressure and particle velocity fields analytically computed by introducing two independent Gaussian random variables ∆Q and ∆φ, simulating errors on the magnitude and the phase of measurement data [START_REF] Pezerat | Two inverse methods for localization of external sources exciting a beam[END_REF]:

P = P exact .∆Q p .e j∆φ V = V exact .∆Q v .e j∆φ (17)
Where ∆Q p = p|P exact |X + 1 and ∆φ = arctan(p)Y , with p the noise percentage and (X,Y) two independent Gaussian random variables. Figs. ( 13) and ( 14) present the influence of measurement uncertainties on identifications for p = 30%. We can thus notice that the measurement noise has a limited influence on the identification accuracy, insofar as the magnitude and spatial distribution of source velocities are only marginally affected. Consequently, the proposed method appears to be robust with respect to measurements uncertainties. Source velocity (

Robustness of the iPTF method -Influence of correlated disturbing source

When measurements are performed in situ, correlated sources can modify the pressure and the particle velocity fields measured around the source to identify. Consequently, virtual sources are likely to appear on the hologram due to the bad estimation of acoustic field in the measurement area. As underlined in section 2, the iPTF method is theoretically independent of the presence of external sources. To prove this theoretical fact, a third piston is inserted in the rigid baffle, whereof velocity is 5 times bigger than that of the first piston (see Fig. [START_REF] Maxit | Airbone noise prediction using patch acoustic impedance[END_REF]). This third disturbing piston modifies the acoustic field computed at the center of the measurement patches. As an exemple, the mean square pressure is increased by 6.5 dB at 240 Hz, while the mean square particle velocity is increased by 3.5 bB at the same frequency. 16) presents the comparison of the spatial distribution of reference source velocity with the identified one obtained in presence of a disturbing acoustic source. Maps prove that the source velocity can be properly identified in presence of a disturbing source.

Influence of the virtual cavity dimensions

The dimensions of the virtual cavity is essential in the identification process since the more we measure far from the source, the more we lose the near field information related to the evanescent waves. In that it is hard to identify precisely the source velocity because of the lack of information. To demonstrate the influence of the cavity dimensions on the identification, we define a virtual cavity which dimensions are 450 × 350 × 130 mm 3 . Furthermore, the measurement surface is divided into 259 patches. As shown in Fig. [START_REF] Jacobsen | A note on the calibration of pressure-velocity sound intensity probes[END_REF], the iPTF method succeeds in identifying source velocities. However, as classically observed in other holographic methods (NAH or iBEM), the

Experimental validation 3.3.1. Experimental set-up

The experimental validation consists in reproducing as reliably as possible the set-up used for the numerical validation, in order to evaluate the applicability and the robustness of the method in a non anechoic environment. For this purpose, the acoustic baffle consists of a wooden thick plate of 700 × 600 × 40 mm 3 , in which two loudspeakers fed by a white noise are inserted. Furthermore, the identification area is materialized on the rigid baffle as shown in Fig. [START_REF] Jacobsen | Statistically optimized near field acoustic holography using an array of pressure-velocity probes[END_REF].We can also notice that measurements were performed in a non anechoic chamber and consequently reflected sound from room boundaries was present. Finally, reference and acoustic field measurements were carried out with a PU probe in the very nearfield of the source (5 mm from the acoustic source) and a microphone was used as phase reference.

Like in previous sections, we define around the identification area the same patch meshes as those used in the numerical validation. It is relevant to remind that space averaged data of pressure and particle velocity on patches are represented by one measurement point at the center of the patch. By this way, a positioning error of the PU probe can influence the quality of the experimental results. Of course, other factors can come into account like normal vectors orientation for particle velocity measurements or uncertainties due to transducers quality. In the following, we use an indicator ǫ which allows thoroughly dertermining the identification error of the iPTF method. This indicator is define as in Eq. ( 18).

ǫ = 10log V 2 id V 2 ref ( 18
)
Where V ref is the source velocity measured in the very nearfield of the source and V id is the identified source velocity.

Validation of the basic principle

The experimental validation of the basic principle is performed by reproducing the numerical experiment used in section 3.1. When applying the iPTF method from experimental data, a good agreement between measured and identified source velocity is observed both in magnitude and spatial distribution (see Figs. [START_REF] Pezerat | Two inverse methods for localization of external sources exciting a beam[END_REF] and ( 21)). The analysis of Fig. [START_REF] Pezerat | Two inverse methods for localization of external sources exciting a beam[END_REF] shows actually that the identification error on the first excited patch ((x, y) =(0.1125 m,0.245 m)) does not exceed 1 dB on all the frequency range.

Robustness of the iPTF method -Influence of correlated disturbing source

To test the robustness of the method in presence of a correlated disturbing source, we used the same virtual cavity as in the previous case and a third loudspeaker located outside the measurement area was added (see Fig. [START_REF] Totaro | Identify velocity of a complex source with iPTF method[END_REF]). A reference measurement was performed again to take into account the acoustic field modifications due to the disturbing source.

The pressure power spectrum measured on a patch of the measurement surface with and without disturbing source highlights the influence of the disturbing source. Disturbing pressure was in general of the same order as the primary source pressure and at certain frequencies differed from more than

Influence of the virtual cavity dimensions

As already evoked in the numerical validation of the method, the more we measure the sound field far from the source, the more we lose information related the evanescent waves. The aim of this section is to know how experi-mentally the absence of near field information affects the identification accuracy. For this purpose, we use the virtual cavity of dimensions 450×350×130 mm 3 defined in section 3.2.1. In this case too, the iPTF method gives quite satisfaying results in both magnitude and spatial distribution (see Figs. ( 26) and ( 27)). However, Fig. (26) shows that the identification is less accurate in magnitude than that performed on a cavity of reduced dimensions. The discrepancy is however acceptable up to 425 Hz, where the height of the virtual cavity reaches λ 6 ( 340 6×425 ≈ 0.13 m). Nevertheless, the method succeed in localizing the two louspeakers below 425 Hz.

Conclusion

The iPTF method allows identifying source velocities on complex structures. In this paper, the theoretical background as well as the measurement methodology have been introduced. In the method, both pressure and particle velocity fields have to be measured on the virtual cavity surfaces. This is quite simple when using PU probes. To prove the ability of the method, an experimental validation has been set up on two baffled pistons driven in antiphase. As classically observed in holographic methods, one of the main drawbacks remains the loss of the near field information when measuring acoustic field far from the source. However, the combined use of integral formulation, FEM as a solver and the double measurement of pressure and particle velocity fields allows overcoming some limitations of classical methods associated to simple structure geometry. Thus, the independence of the method with respect to external stationnary sources and its applicability to complex shape structures are the main advantages of the iPTF method. Finally, further applications of the method on more complex structures than that presented in this article, namely an L-shape plate and a vehicle precatalyst, can be found in Refs. [START_REF] Aucejo | Identification of source velocities with iPTF (inverse Patch Transfer Functions method)[END_REF] and [START_REF] Totaro | Identify velocity of a complex source with iPTF method[END_REF]

Figure 1 :

 1 Figure 1: Integral problem

Figure 2 :

 2 Figure 2: Basic identification problem

Figure 3 :

 3 Figure 3: Geometry of the vibrating source and patch mesh definition on surface of the virtual cavity

Figure 4 :

 4 Figure 4: Conditioning number of patch impedance matrices -Comparison between (-)Pressure measured on all the virtual surfaces, (-.-) Pressure measured on all the virtual surfaces and on the source surface and (--) Pressure measured on all the virtual surfaces and on one patch located in the center of source surface

Figure 5 :

 5 Figure 5: Conditioning number of patch impedance matrices -Comparison between (-) Pressure measured on all the virtual surfaces, (--) Pressure measured on the top virtual surface

Figure 6 :

 6 Figure 6: Geometry of the vibrating source and patch mesh definition Geometry of the vibrating source and patch mesh definition on surface of a 500 × 350 × 120 mm 3 virtual cavity

Figure 7 :

 7 Figure 7: Conditioning number of patch impedance matrices -Comparison between Pressure-Velocity measured on all the virtual surfaces for (-) the large virtual cavity and (--) the small virtual cavity

Table 1 :

 1 Comparison between the maximal order of cavity modes and the number of patches along the directions x,y and z Modal frequency limit 2kHz 3kHz 4kHz 5kHz Maximal order of modes 5 × 4 × 1 7 × 6 × 2 10 × 8 × 3 13 × 10 × 3 Number of measurement patches 6 × 5 × 2 Number of source patches 6 × 5

Figure 8 :

 8 Figure 8: Influence of the maximal cavity modes order on the conditioning number of patch impedance matrices for a 450 × 350 × 130 mm 3 virtual cavity and modes extracted up to (-) 2 kHz, (--) 3 kHz, (..) 4 kHz and (-.-) 5 kHz

Figure 9 :

 9 Figure 9: Definition of the validation set-up -Two baffled pistons driven in antiphase

Figure 10 :

 10 Figure 10: Definition of (a) the virtual cavity and (b) the patch mesh around the source

Figure 11 :Figure 12 :

 1112 Figure 11: Numerical validation for a 450 × 350 × 20 mm 3 virtual cavity -Comparison of (a) the reference map and (b) the identified map obtained with the iPTF method at 240 Hz

Figure 13 :

 13 Figure 13: Numerical validation for a 450 × 350 × 20 mm 3 virtual cavity -Comparison between (a) the reference map and the identified maps obtained for (b) (b)p = 30% at 240 Hz

Figure 14 :

 14 Figure 14: Numerical validation for a 450 × 350 × 20 mm 3 virtual cavity -Comparison of source velocity along a line (x,0.245 m) across the prediction plane at 240 Hz, (-) Reference, (--) Identification for p = 30%

Figure 15 :

 15 Figure 15: Location of the disturbing piston outside the identification area

Fig. (

  Fig.[START_REF] Ouisse | Patch transfer functions as a tool to couple linear acoustic problems[END_REF] presents the comparison of the spatial distribution of reference source velocity with the identified one obtained in presence of a disturbing acoustic source. Maps prove that the source velocity can be properly identified in presence of a disturbing source.

Figure 16 :Figure 17 :Figure 18 :

 161718 Figure 16: Numerical validation for a 450 × 350 × 20 mm 3 virtual cavity in presence of an external disturbing source -Comparison of (a) the reference map and (b) the identified map obtained at 240 Hz

Figure 19 :

 19 Figure 19: Experimental set-up

Figure 20 : 1 )Figure 21 :

 20121 Figure 20: Experimental validation for a 450 × 350 × 20 mm 3 virtual cavity -Identification error on the first excited patch (x, y) =(0.1125 m,0.245 m)

5

  dB (see Fig. (23)).

Figure 22 :Figure 23 :Figure 24 :Figure 25 :

 22232425 Figure 22: Experimental set-up -Location of the disturbing source

Figure 26 :

 26 Figure 26: Experimental validation for a 450×350×130 mm 3 virtual cavity -Identification error on the first excited patch (x, y) =(0.1125 m,0.245 m)

Figure 27 :

 27 Figure 27: Experimental validation for a 450 × 350 × 130 mm 3 virtual cavity -Comparison between (a) the reference map and (b) the identified map obtained experimentally with the iPTF method at 240 Hz

A. Calculation of Green's function by modal approach

The Green's identity defined in Eq. ( 2) is then used, where Φ is the mode shape φ n of the virtual rigid wall cavity Ω c satisfying Eq. [START_REF] Jacobsen | Statistically optimized near field acoustic holography using an array of pressure-velocity probes[END_REF] and Ψ is the function G satisfying Eq. [START_REF] Pezerat | Two inverse methods for localization of external sources exciting a beam[END_REF].

Consequently, using Eqs. [START_REF] Williams | The nearfield acoustical holography (NAH) experimental method applied to vibration and radiation in light and heavy fluids[END_REF], where Φ = φ n and Ψ = G, (20) and ( 19), we obtain the integral equation given by Eq. [START_REF] Aucejo | Identification of source velocities with iPTF (inverse Patch Transfer Functions method)[END_REF].

The function G is now determined by expansion on the modal basis φ n (Q):

Using this decomposition in Eq. ( 21), one obtains:

Where the norm of modes is such as Ωc φ n (Q)φ q (Q)dΩ c = Λ n δ nq . One obtains therefore the following expression:

The patch impedance between a source patch k and a reception patch j is then:

Obviously in the Eqs.( 24) and (25), this modal composition presented problems at resonant frequencies. To avoid these difficulties, one introduces the modal damping of fluid η n , which is a constant in this study. Therefore, one replaces in the denominator the term ω 2 nω 2 by the term ω 2 nω 2 + jη n ω n ω.

B. Radiation of a baffled piston

The Rayleigh integral formulation of a baffled piston is used to compute its radiation as expressed in Eq. ( 26).

However, insofar as the dimension of the piston is not negligible, the integral over the piston area is discretized into n x × n y points. Consequently, Eqs. ( 26) and (27) become:

Where (x Mr , y Mr , z Mr ) are the coordinates of the center of the patch r and (x k , y k , z k ) are the coordinates of the point k belonging to the piston.

The particle velocity is then easily obtained at the center of the patch r by using the Euler's equation defined in Eq. (29):

Where n e is the external normal vector to the virtual cavity.