Éric Gaudard

Philippe Druault
email: philippe.druault@upmc.fr

Régis Marchiano

Franc ¸ois Van Herpe

POD and Fourier analyses of a fluid-structure-acoustic interaction problem related to interior car noise

Keywords: Aeroacoustic analysis, Proper Orthogonal Decomposition, Fourier analysis, interior car noise. Nomenclature

In order to approach a flow configuration revealing the aerodynamic noise contribution in the interior of road vehicles due to the A-pillar vortex, a numerical simulation of a Forward Facing Step (FFS) coupled with a vibrating structure is performed. This numerical study is based on a weak coupling of three solvers to compute (i) the flow field in interaction with the FFS, (ii) the vibration of the structure and (iii) the acoustic radiation in the open cavity. The purpose of this work is then to evaluate the ability of two different post-processing methods: Proper Orthogonal Decomposition and Fourier Decomposition to identify the origin of the noise radiated into a cavity surrounded by an unsteady flow. Fourier and POD decompositions are then successively performed to extract the part of the aeroacoustic wall pressure field impacting the upper part of an upward step mainly related to the radiated acoustic pressure in the cavity. It is observed that the acoustic part, extracted from the wavenumber frequency decomposition (Fourier analysis) of the wall pressure field generates a non-negligible part of the interior cavity noise. However, this contribution is of several orders smaller than the one related to the aerodynamic part of the pressure field. Moreover, it is shown that the most energetic part of the pressure field (POD analysis) is due to the shear flapping motion and mainly contributes to the low-frequency noise in the cavity. Such post-processing results are of particular interest for future analyzes related to the noise radiated inside a car.

Turbulent convection wavenumber k s

Structural wavenumber k x wavenumber along the x direction (L x , L y ) physical dimension of the FFS flow domain (L x , L y ) physical dimension of the cavity domain n t number of instants (n x , n y ) number of grid points of the FFS flow configuration (n x , n y ) number of grid points of the cavity domain p(x, y, t) pressure signal at position (x, y) and time t U ∞ uniform streamwise flow velocity s entropy, X = (x, y) two-dimensional space variables x vib

x-coordinate of the one dimensional vibrating structure, Y transverse displacement of the structure p(k x , f ) spatio-temporal FT of the pressure signal ∆t time step, (∆x, ∆y) mesh-grid discretization along (x, y) directions ∆ f frequency resolution

Φ (n) (x)
n th spatial POD eigenfunction ρ s density of the structure ρ density of the fluid τ viscous stress tensor

Introduction

The development of quieter car engines and of hybrid road vehicles has led to consider aerodynamic noise as an important source of interior noise especially at mid and high driving speeds. The aerodynamic noise comes primarily from the vortex shedding at the A-Pillar location which interacts with the Turbulent Boundary Layer (TBL) developing on panel windows. This vortex shedding on the window creates wall pressure excitations that cause noise inside the vehicle, leading to some discomfort for the driver and passengers. In order to reduce the interior noise limiting the driver's fatigue, the level of noise induced by structural vibrations in the cavity has to be predicted. Then the development of post-processing tools aimed at characterizing the flow-structural interaction is essential to better understand and predict the interior acoustic field. Such analysis performed in a realistic flow configuration remains today quite difficult. We then propose to test the ability of two postprocessing tools applied to a simplified flow-structure-acoustic interaction problem which approaches the A-pillar vortex flow configuration. A numerical study of the acoustic radiation in an open cavity induced by a vibrating structure forced by an unsteady flow is then considered in this paper. It is assumed that there is no source of sound in the open cavity and that the only mechanism of sound production is given by the coupling between the cavity and the vibrating structure. The purpose of this work is to study the aeroacoustic part of the wall pressure field which is responsible for the acoustic noise in the cavity. In this sense, we do not want to generate additional acoustic modes associated with the cavity properties that justify the choice of an open cavity. The vibrations of the structure are then only due to the unsteady pressure acting on its surface: the wall pressure. The wall pressure can be viewed as the superposition of two components: (i) the pressure due to acoustics waves propagating in the flow, (ii) and the unsteady aerodynamic pressure flow. A simple configuration to observe this situation is to consider the flow around a forward facing step (FFS) and to consider that the elastic structure is located behind the FFS (see Figure 1 for a schematic representation). In practice, this configuration occurs in presence of detached flows impinging an elastic structure and when the sound field behind the structure matters. For instance, this configuration can be viewed as a very simplified reproduction of the aerodynamic noise contribution in the interior of a road vehicle due to the A-pillar, even though the simulation is here two-dimensional. Nevertheless, this study does not claim to reproduce the real 3D configuration but is an attempt to link aerodynamic events to acoustic radiation in a cavity for a two-dimensional configuration.

The study of aerodynamic noise generated from flow-excited structures involves the investigation of the wall pressure fluctuations on the structure, the structural vibrations excited by the surface pressure field, and finally the sound radiated from the vibration of the structure. The complete investigation of such flow-structure coupling and of the resulting sound radiation is quite complex and time consuming. Only few authors have already performed such an analysis on a simplified flow configuration (1; 2). A full study needs to take into account the complex exterior flow field (interaction of A-pillar vortex, TBL flow, door mirror) that generates unsteady wall pressure fields impacting the vibrating panel window and leading to interior noise. Apart from the difficulties in numerically investigating such a complex flow, simplified flow configurations are generally retained allowing the investigation of particular flow events responsible for interior noise in the cabin. For instance, a lot of previous studies have proposed to model the structural loading analytically based on deterministic or random excitations (3; 4; 5; 6). In this case, the structural loading usually includes random, distributed excitations, such as diffused acoustic fields and aerodynamic excitations. Then, the structural vibrations and acoustic radiations are computed from the analytical model representing wall pressure excitation [START_REF] Hekmati | Numerical synthesis of aeroacoustic wall pressure fields over a flat plate: Generation, transmission and radiation analyses[END_REF].

The difficulty of such modeling concerns the restitution of the complex aeroacoustic wall pressure field excitation and its associated physical mechanisms. Indeed, this excitation is composed of a turbulent component and an acoustic one (3; 7; 8) and both components contribute to the interior noise. Another solution may consist in performing a numerical simulation of similar flow configurations. Thus, to reproduce a flow configuration approaching the A-pillar vortex, a Forward Facing Step (FFS) configuration is considered. This simplified geometry allows the generation of an aeroacoustic wall pressure field impacting the upper part of the step corresponding to a vibrating structure.

Even though forward facing step has been intensively studied in the past in incompressible subsonic flows (see for instance (9; 10; 11) and references therein), the analysis of compressible FFS flow has been much less investigated. Based on previous experimental and numerical database, it has been shown that the main region containing acoustic sources is located in the recirculating flow close to the step corner. The far-field acoustic of FFS flow configuration has been experimentally investigated by several authors (12; 13; 14; 15). Recent acoustic measurements exhibit the dependence of the step height on the acoustic spectra [START_REF] Catlett | Flow induced noise from turbulent flow over steps and gaps[END_REF]. Moreover, the acoustic source seems to be of dipolar nature (12; 13; 14). Numerical simulations have also been performed for aeroacoustic analysis of FFS flow configuration (16; 17; 18; 19). For instance, based on incompressible Large Eddy Simulations (LES) of the FFS flow coupled with Lighthill's analogy formulation to access the far-field acoustic, Ji and Wang [START_REF] Ji | Sound generation by turbulent boundary-layer flow over small steps[END_REF] showed that the front step acts primarily as a dipole source aligned in the streamwise direction. Previous studies have assumed that, for low-Mach number (M ≈ 0.1) FFS configurations, the dominant acoustic contribution comes from the step front flow impact. However, very recent experimental and theoretical analyses (20; 21) demonstrated that the noise due to wall pressure fluctuations may dominate the other contributions for various Mach numbers FFS flow. Awasthi et al. [START_REF] Awasthi | The disturbance of a high Reynolds number turbulent boundary layer by small forward steps[END_REF] have also demonstrated that the wall pressure spectrum levels are related to the step height of the forward-facing step. Thus, the characterization of the wall pressure signal including not only acoustic but also hydrodynamic fluctuations, remains a great challenge to elucidate the flow mechanisms responsible for the structure vibration and then for the radiated noise in a cavity.

As the present study is a preliminary one focusing on the flow-structure-acoustic interaction, it needs to be simplified to better investigate the origin of the radiated sound in the cavity. It is then proposed to perform a direct noise computation of the two-dimensional (2D) FFS configuration. Note that the numerical simulation of similar fluid-structure-acoustic interaction in a 3D context leads to a very high computational cost. But, even if a 2D computation is not able to take into account the 3D flow structures developing in the recirculation area (23; 24), it allows the description of the main flow characteristics. Moreover, Wilhelm et al. The objective for the present study is to identify the wall pressure contribution which is mainly related to the radiated acoustic pressure in a cavity. More precisely, we attempt to identify unsteady flow characteristics (shear layer flapping motion, eddy structure interacting with the wall, . . . ) that could be mainly related to the noise radiated in an open cavity located under the step. For such an investigation, two common post-processing methods are tested: Proper Orthogonal Decomposition (POD) and Fourier decomposition. These modal analyses are ones of the most popular flow decomposition techniques used for the analysis of the experimental and/or numerical space-time database. POD and Fourier analysis both allow a space-time description of the flow dynamics as a set of statistical quantities referred as eigenfunctions. For the POD technique, eigenfunctions are determined based on the energy content [START_REF] Holmes | Turbulence, coherent structures, dynamical systems and symmetry[END_REF]. Conversely, for Fourier decomposition, the modes are related to the wavenumber and/or frequency content of the available database. These flow decompositions have demonstrated in the past their efficiency in identifying the flow structures present in turbulent flow. More precisely, POD technique allows the detection and the characterization of the large scale energetic coherent structures [START_REF] Holmes | Turbulence, coherent structures, dynamical systems and symmetry[END_REF] while Fourier analysis extracts periodic flow structures. Note also that the Fourier technique is generally more devoted to noise and vibration analysis thanks to spectral analysis. In a previous paper, Gaudard et al. [START_REF] Gaudard | About the pod application for separating acoustic and turbulent fluctuations from wall pressure synthesised field[END_REF] have already analyzed the effectiveness of these two methods but in a different context. Indeed, in this last paper, a synthetic wall pressure field was used to test the effectiveness of both mathematical tools (POD and FFT) to discriminate the acoustic and aerodynamic parts of the synthesized aeroacoustic wall pressure field. In the present paper, we propose to test the ability of these two methods in a fluid-structure-acoustic interaction context related to a simplified flow configuration allowing to state about which aeroacoustic events that can be associated with the main noise contribution in the cavity.

In section 2, the selected configuration and numerical solvers are presented. Section 3 is devoted to the analysis of the forward-facing step flow. The results of the decomposition of the wall pressure using two different processing tools are discussed in section 4. The analysis of the radiation of the elastic structure is given in section 5.

Numerical solvers and fluid-structure configuration

A schematic illustration of the configuration and the computational domains is given in figure 1. As already mentioned, the goal of this study is to find links between a pressure recorder in the open cavity and the wall pressure generated by the flow in the exterior domain on the vibrating structure. From a theoretical point of view, each physical domain can be associated with a particular model. In the exterior domain, as explained in the introduction, we would like to simulate the effect of the unsteady flow over the structure as well as the acoustic waves generated by the flow and radiated. Therefore, one has to deal with the compressible Navier-Stokes equations. The vibrating structure is a beam. Assuming small amplitudes of deformation it is reasonable to consider a linear and isotropic Euler/Bernoulli model. Finally, the open cavity is supposed to be at rest with no flow. Moreover, the propagation can be considered linear, hence the use of the classical wave equation to model that domain.

To predict the acoustic pressure field in the cavity, three successive computations are done: 1) a two-dimensional Navier-Stokes solver is used in the exterior domain especially to compute the wall pressure, 2) a one-dimensional beam solver simulates the structural vibrations due to the unsteady wall pressure, 3) a two-dimensional acoustic solver allows to access to the radiated acoustic field in the open cavity.

In the following parts these three solvers are briefly described and the associated numerical parameters used in simulations are provided.

Navier-Stokes solver and numerical flow parameters

The wall pressure is computed in the exterior domain from the Navier-Stokes equations:

∂ρ ∂t + ∇.(ρv) = 0, (1)
ρ ∂v ∂t + (v.∇)v + ∇p = ∇.τ, (2)
∂s ∂t + (v.∇)s = 0. ( 3
)
with p(X, t) and ρ(X, t) the pressure and density field depending on the space and time variables. v(X, t) is the velocity vector and s(X, t) the entropy field. τ is the viscous stress tensor. The fluid is supposed to be newtonian. These equations are solved with a 2D numerical solver called CAAMELEON and already described in the following references (27; 28; 29). CAAMELEON solver allows the Direct Numerical Simulation of the compressible 2D Navier-Stokes equations. However, in the present test case, the grid resolution is not sufficient to ensure that all time-space flow scales of the flow are resolved. Thus, here, the grid discretization acts as a filtering procedure for the Navier-Stokes equation resolution. This code solves the pseudo-characteristic formulation of the Navier-Stokes equations that provides a decomposition of the pressure, velocity and entropy fluxes (28; 30). Equations are written in generalized coordinates and the code is able to solve any configurations on a curvilinear grid [START_REF] Sesterhenn | A characteristic-type formulation of the Navier-Stokes equations for high order upwind schemes[END_REF]. To enforce both numerical stability and accuracy, a fourth-order accurate upwind DRP scheme is used and this scheme is modified near the computational domain boundaries [START_REF] Lu | Pseudo-characteristic formulation and dynamic boundary conditions for computational aeroacoustics[END_REF]. Time integration is performed using a third-order TVD Runge-Kutta scheme. The code is parallelized with a domain decomposition.

To reproduce previously published FFS flow configurations (9; 16), the step height is fixed to h = 0.0127m and a uniform streamwise velocity field U ∞ = 25m.s -1 is imposed in the far field. The Reynolds number Re h = U ∞ h/ν is around 20000. The physical computation domain extends from -14h to 14h (L x = 28h) in the streamwise direction and from -h to 22h (L y = 23h) in the transverse direction (see figure 1). The computational meshgrid is not uniform in both directions. As mentioned before the code can deal with curvilinear grids. The transverse grid discretization is uniform dy/h = 1/300 for y/h ∈ [-1 : 0.5] and then the transverse discretization is regularly stretched along that direction until reaching a maximum mesh size of dy/h = 1/10. In a similar manner, the streamwise discretization is uniform dx/h = 1/100 for x/h ∈ [-1 : 1] and it is regularly stretched in both directions x/h < -1 and x/h > 1 with a maximum mesh step of dx/h = 1/10. The total number of points for the physical domain under investigation is (n x , n y ) = (510, 720) excluding the sponge regions. The CFL number is chosen equal to 0.5 based on the smallest mesh size. Considering this CFL number and the meshgrid, the time step of the simulation is equal to 5.947.10 -8 s. The coordinate system origin is located at the upper corner of the step. To avoid any spurious reflexions, sponge regions are added along the streamwise direction (left and right-boundary) and also at the upper y-boundary. Sponge layers are made by applying a stretching on the grid in the direction of interest. The stretching is progressive to avoid any spurious reflection at the boundary between the physical domain and the sponge layer (the stretching coefficient is 0.028). The size of the sponge layers is chosen to be large enough to ensure the damping of the waves and of the aerodynamic structures before the computational boundary (168 points in the x-direction and 137 points in the y-direction corresponding approximately to 12.5% of the computational domain for each of them). Note that, in the following figures, flow variables will only be represented in the physical domain.

Such a flow configuration is very sensitive to initial conditions [START_REF] Wilhelm | Computational analysis of the two-dimensional-threedimensional transition in forward-facing step flow[END_REF]. To initialize it, the initial state (set of pressure, density and velocities in the whole domain) is computed by solving the incompressible RANS equations with the open source CFD software OpenFoam (31; 32). It provides a statistically well converged mean flow field related to the FFS flow configuration. A uniform and cartesian mesh of (1000 × 750) points is used to discretize the (x, y) domain of the FFS configuration. A laminar boundary layer is imposed in inflow. The model for turbulence is the classical k-model. This method allows us to save an important time comparatively with a naive initialization of the code with, for instance, a uniform state in the whole domain. Nevertheless, this procedure requires to interpolate the steady solution obtained by OpenFoam to the grid described above.

The initial conditions are now available and correspond to a uniform mean streamwise flow velocity of U ∞ = 25m.s -1 outside the mean Blasius velocity field obtained near the wall. Based on the pseudo-characteristics formulation of the Caameleon solver, it is not necessary to generate at each time step this streamwise velocity field at the left boundary domain [START_REF] Lu | Pseudo-characteristic formulation and dynamic boundary conditions for computational aeroacoustics[END_REF]. Indeed, in such a formulation, only the flow derivatives have to be imposed at the left streamwise boundary of the computational domain. Thus, imposing a null derivative of the velocity derivative fluxes at this boundary is similar to generate a uniform velocity field. Regarding the grid resolution associated with the Boundary Layer (BL) flow, the classical normalized values of x + and y + are respectively equal to 13.2 and to 4.4 based on the determination of the friction velocity when the TBL flow is developed. These values are obtained at x/h ≈ -3.5. Such a grid resolution is sufficient to accurately predict the aeroacoustic flow variables. More than 3.6 10 6 time steps have been computed (about 108 hours on 64 CPUs). Once the initial condition is entirely evacuated, the flow variables are stored every 500 time steps, therefore the sampling frequency is 33630 Hz. We consider that the data are significant after 2000 time steps. Indeed, beyond that number of time steps, the different statistical quantities studied in the next paragraphs are converged. Hence, the database containing the flow variables is made of 5000 samples for each point of the mesh grid. That leads to an analyzable time duration of 0.1488s which corresponds approximately to ten times the time travel of a fluid particle advected through the complete flow domain at the uniform speed of the flow. Note that the time duration of the simulation may place some constraints on the following stochastic data analysis. However, when performing Fourier, the frequency domain covers [0 : 16815]Hz with a frequency step of 6.7Hz. These values seem quite sufficient for the present investigation. Indeed, the flow structure passage is largely inferior to 10000Hz and the frequency resolution provides a sufficient accuracy for the characterization of the present aerodynamic events.

The wall pressure is extracted from this database. The spatial coordinate of the vibrating structure will be referred to as to x vib indicating that x is varying from 0 to 14h and y = 0. Thus, the wall pressure denoted p(x vib , t) is then used as a source term for the elastic solver.

Euler-Bernoulli beam model solver and numerical parameters

The model used here is based on the hypothesis that one-way interactions occur: first, the fluid outside the cavity engenders vibrations of the structure, then, the vibrations of the structure radiate acoustical waves in the cavity. We do not consider a full-coupling approach. Some preliminary tests based on a full-coupling approach in a very simplified configuration were performed during the Ph.D. Thesis of Gaudard [START_REF] Gaudard | Développement d'outils pour l'analyse de champs aéroacoustiques à l'origine du bruit dans un habitacle automobile[END_REF]. In this context, the error done for a non-full coupling approach was less than 1% compared to the full-coupling approach. In this study, it is expected that the full-coupling approach would not modify the present results. Indeed, the aeroacoustic wall pressure field is of very high energy content comparatively to the energy's level of the acoustic radiation in the cavity and then of the internal acoustic vibrations. Then, we assume that the one way coupling leads to modify only less than a few percents the solution comparatively to the solution of a full-coupling approach. Moreover, the interaction between the flow and the structure is assumed to be small enough to consider only small deformations. It is also supposed that the structure is unaffected by the acoustic response inside the cavity. Only the transverse displacements (in the y-direction) of the structure Y(x vib , t) are considered. According to these assumptions, the vibrations are modeled with the one-dimensional Euler-Bernoulli equation with supported ends. The fluid load excitation denoted ∆p(x vib , t) corresponds to the difference between the wall pressure field described above (p(x vib , t)) and the uniform pressure field initially present in the open cavity:

m ∂ 2 Y(x vib , t) ∂t 2 + EI ∂ 4 Y(x vib , t) ∂x 4 = ∆p(x vib , t), (4)
where E, I and m are respectively the Young modulus, the second moment of inertia and the linear mass of the structure. This structure is supposed to be a glass with the following properties: thickness e = 0.001m, Young modulus E = 70 × 10 9 Pa, density ρ s = 2500kg.m -3 . The transverse displacement is computed based on the finite element method. The structure is meshed with 139 equally spaced elements [START_REF] Gaudard | Développement d'outils pour l'analyse de champs aéroacoustiques à l'origine du bruit dans un habitacle automobile[END_REF] and the time integration is done by using a Newmark's scheme [START_REF] Newmark | A method of computation for structural dynamics[END_REF] with a time step of 3 10 -5 s.

Acoustic wave propagation solver

Knowing the transverse displacement of the structure, it is possible to compute the normal velocity of the structure and to use it as an imposed boundary condition to compute the acoustic field in the open cavity. To compute the acoustic field in the cavity, the previously described two-dimensional Navier-Stokes solver is used. The dimensions of the cavity located under the vibrating structure are (L x , L y ) = (14h, 15h) that corresponds to a uniform mesh in both directions of (n x × n y ) = (140 × 150) points with a constant space discretization of dx /h = dy /h = 1/10. The time step of this simulation is 2.478 10 -6 s. The initial condition corresponds to a medium at rest with a uniform pressure field, p 0 = 101325Pa. Instantaneous fluctuating velocity field computed based on the beam model solver is used as inflow condition (upper transverse boundary domain) at each time step of this numerical simulation. As the time step of the beam solver response is smaller than the time step of the Navier-Stokes solver, a linear interpolation of the transverse velocity field in time is done before its generation as inflow condition. In the other boundaries, a sponge region is imposed to avoid any spurious reflexions. More than 60000 time steps are simulated. Once the acoustic field is established in the cavity, 12000 instantaneous acoustic pressure fields are stored with a sampling frequency of 403560Hz. That corresponds to a time duration of 0.03s. We have in mind that any limited time duration places some constraints on data analysis, especially in the frequency resolution which is then only of 33.6Hz. However, for the present acoustic data analysis, this discretization seems to be sufficient to properly describe the nature of the acoustic signals in the cavity and also to perform a comparative analysis of the efficiency of flow decomposition methods.

3. Aeroacoustic analysis of forward-facing step flow

Mean flow properties

The streamwise mean velocity profiles extracted at selected x-locations are presented in figure 2. The thickness of the turbulent boundary layer developing before the flow separation is of an order of 2.4h. Two separation regions are retrieved in front of and on the step (see streamlines of the mean velocity field on the figure 3). The recirculation region downstream of the step extends to 2h which is in reasonable agreement with previous investigations (11; 35). Such recirculation area corresponds to the flapping motion of the shear layer. As expected the turbulent kinetic energy is higher in the upper recirculation region than in other regions due to the increase in the shear of the mean velocity field (33).

Wall pressure analysis

The pressure fluctuations obtained at the wall (before and after the step) denoted p wall are now investigated. Figure 4 displays two representative wall pressure coefficients defined as follows: 10 y/h

where an overbar indicates the time average and a prime is related to the fluctuating pressure field. The pressure coefficient C p wall increases as the flow velocity decreases (see figure 4). The maximum value is obtained at the corner of the step. This coefficient drastically decreases in the separation region located above the step. By comparing with previous 3D results (16; 20), a similar tendency is observed but the amplitude levels are quite different. This is directly related to the 2D character of present simulation. Moreover, the streamwise evolution of C RMS p wall coefficient follows a classical behavior (16; 20). The wall pressure fluctuation begins to increase from x/h = -3 and has its maximum value (x/h ≈ 1) just downstream the corner of the step and before the reattachment point. Then wall pressure fluctuations decrease due to the flow relaxation process following reattachment region. As it was previously observed [START_REF] Farabee | Measurements of fluctuating wall pressure for separated/reattached boundary layer flows[END_REF], present results confirm that the flow reattachment zone is of higher amplitude than he one of other zones. It may be a main source for the structure excitation.

A spectral analysis of the wall pressure is conducted to elucidate the spectral behavior of the wall pressure field. Figure 5 shows the power spectral density for six different pressure signals computed by the Welch periodogram method. The signals are extracted at selected x-locations related to different areas: i) far upstream the step -TBL flow; ii) near the corner of the step; iii) in the shear layer flapping motion area; iv) far downstream the recirculation region. First, it is shown that the spectrum magnitudes of the wall pressure in the recirculation area are higher than the other wall pressure spectrum ones. Second, each frequency spectrum exhibits a low frequency high magnitude region followed by two successive power law decay regions with similar slopes of -1 and -2.5. This is in agreement with previous works dealing with similar flow configurations (16; 20).

Far acoustic pressure field analysis

In the far-field acoustic (y/h = 20), the frequency spectra are broadband for frequencies lower than f 0 = 1000Hz. For higher frequencies ( f > f 0 ), a power law decay region is observed with a slope of -2.5 [START_REF] Gaudard | Développement d'outils pour l'analyse de champs aéroacoustiques à l'origine du bruit dans un habitacle automobile[END_REF]. Such a frequency corresponds to a Strouhal number S t = f 0 h/U ∞ of 0.5. To determine the nature of the acoustic sources, the directivity patterns at r = 10h (circle of radius r centered at the corner of the step) of the Root Mean Square (RMS) of the acoustic pressure field are presented in figure 6. This figure presents: i) the directivity of the original acoustic pressure field, ii) the directivity of the low-frequencies filtered pressure field ( f ∈ [0 : 1000]Hz), iii) the directivity of the middle-frequencies filtered pressure field ( f ∈ [1000 : 5000]Hz). More than 3000 time steps are used to compute the RMS of acoustic pressure field. The directivity pattern of the RMS of the acoustic pressure indicates a pronounced effect for angle in [140 • : 180 • ], especially for the low pass filtered pressure field. For the middle-frequency filtered pressure field, two main pronounced effects are recovered for an angle around 80 • and for an angle in [140 • : 180 • ]. This last result may be related to a dipolar nature of the source as previously noted in a similar 3D flow configuration [START_REF] Ali | Aeroacoustic study of a forward facing step using linearized Euler equations[END_REF]. Globally, even though some pronounced peaks may be observed in the directivity pattern which are quite similar to those previously obtained (18; 36) it is not possible to properly state about the nature of acoustic sources in such a flow configuration.

Conclusive remark

Even though the two-dimensional approach of the problem is a limited modeling, the main features of the physics of a FFS are recovered. This is then sufficient for the purpose of the present study that is to test the effectiveness of POD and Fourier techniques in such a fluid-structure-acoustic interaction problem. x/h =4

and acoustic computations

Table 1: Notations used in the following for the five test cases based on different inflow wall pressure fields. Each p method i wall pressure field, with i = 1, 2 and method = FT or POD, will be defined in the following.

Flow decomposition analysis of the forward-facing step flow

In this section, advanced post-processing mathematical tools are implemented to analyze and to decompose the wall pressure field, p(x vib , t). The classical Discrete Fourier Transform (referred to as FT in the following) and the Proper Orthogonal Decomposition (POD) are successively considered for the investigation.

In the following, the reference wall pressure field extracted on the step will be linearly decomposed into two parts either by FT method or POD one. Consequently, FT and POD applications lead to obtain 4 partial wall pressure fields: 2 related to a POD decomposition and 2 related to a Fourier decomposition of the reference wall pressure signal. These four pressure fields will be denoted: p method i with i = 1, 2 and method = FT or POD. Then, five test cases will be successively done to examine the fluid-structure-acoustic interaction problem as a function of the inlet wall pressure field. The notations given in table 1 will be used in the following.

Brief description of Proper Orthogonal Decomposition (POD)

POD is an efficient statistical technique providing an optimal decomposition in representing the flow variable energy (here the energy is related to the RMS of the fluctuating pressure field). This maximization leads to a Fredholm integral eigenvalue problem [START_REF] Holmes | Turbulence, coherent structures, dynamical systems and symmetry[END_REF] which consists in determining the spatial discrete orthogonal eigenfunctions denoted Φ (n) of the two point spatial correlation tensor R of the random pressure field:

D R(X, X')Φ (n) (X') = λ (n) Φ (n) (X) (6)
λ is the eigenvalue and D is the spatial domain under investigation. Using such POD basis, each instantaneous fluctuating pressure field can be expressed as a linear combination of POD eigenfunctions:

p(X, t) = N modes n=1 a (n) (t)Φ (n) (X), (7)
where a (n) (t) are the random time coefficients of projection satisfying a (n) (t)a (m) (t) = δ nm where δ is the Kronecker symbol. N modes is the total number of POD modes. The first POD modes capture most of the fluctuating pressure energy of the flow with a minimum of terms, compared to any flow decomposition. That explains the great development of POD application to turbulent flows due to its efficiency in extracting the large scale coherent structures present in turbulent flows (38; 39).

Analysis of the wall pressure decomposition

The wall pressure field stored data consists of 5000 instantaneous snapshots sampled at f e = 33630Hz with a sampling resolution of ∆ f = 6.7Hz. For postprocessings, the wall pressure signal has to be available on a regular x-mesh grid. In this sense, to avoid any effect of the interpolation onto post-processing tools, a wall pressure signal discretized on the largest mesh step discretization is retained: dx/h = 1/10. Then N s = 140 points are regularly extracted from the step wall.

Figure 7 (graphs (a), (b) and (c)displays the wall pressure field in the physical space p(x vib , t) and in the spectral space p(k x , f ) where k x is the wavenumber along the streamwise direction. The physical representation is quite similar to the one given in [START_REF] Moon | A hybrid prediction method for low-subsonic turbulent flow noise[END_REF]. For x/h < 1, the great unsteady character of the flow is observed revealing the high pressure fluctuation footprints in the shear layer flapping motion area. When approaching the reattachment point x/h ≈ 2, the highest pressure fluctuations are observed. Far upstream (x/h > 3), the space-time convection of the large scale flow structures are clearly indicated. The spectral representation exhibits the signature of these large scale flow structures that convect at 25m.s -1 . This aerodynamic contribution is mainly observable at low frequencies ( f < 1000Hz) and has a higher amplitude than the acoustic contribution related to low wavenumbers. The main characteristics of the present wall pressure field can then be regarded as similar to the ones of the wall pressure field impacting a side car window (2; 8).

Mathematical tools are now implemented to decompose the wall pressure field as follows:

p(x vib , t) = p method 1 (x vib , t) + p method 2 (x vib , t) (8)
where the method can be either FT or POD. According to mathematical tool properties, p 1 and p 2 pressure fields will differ in

• their wavenumber-frequency content (FT application)

• their energy content (POD application)

Fourier analysis of the wall pressure field

The wall pressure field is of aeroacoustic nature. It contains an acoustic component and an aerodynamic one. The acoustic part propagates at the sound velocity while the aerodynamic component convects at the flow velocity, about 25m.s -1 . As phase velocity c is linked to frequency f and wavenumber k by the relation c = 2π f k , the acoustic contribution is associated, for a selected frequency, to the low wavenumbers. A spatiotemporal FT can then be used to distinguish both contributions of the wall pressure field as it was previously done (3). Practically, one considers that the component convecting at a flow velocity superior to 200m.s -1 cannot be associated with an aerodynamic component. It is then assumed to correspond to the denoted acoustic component. More precisely, the choice of the threshold convection velocity value is directly linked to the wavenumber and frequency resolutions. Here, the wavenumber resolution is 35.3m -1 and the frequency resolution is 6.7Hz. Thus, for the first wavenumber k = 35.3m -1 , two flow events convected at 300m/s or 200m/s correspond to a frequency of 1686Hz or 1124Hz respectively. Present high wavenumber discretization induces then a condition on the determination convection velocity for the FFT filtering procedure. Indeed, to keep the low-wavenumber part of the acoustic field, one prefers retaining a convection velocity of 200m.s -1 to discriminate both acoustic and aerodynamic contributions.

In this sense, to extract this acoustic component, a triangle area is isolated in the Fourier space (see figure 7-(e)). The three points delimiting the triangle area are (k x , f ) = (0, 0), (k x , f ) = (500, 16000) and (k x , f ) = (-500, 16000). Then for the filtering of the acoustic component, one first extracts the spectral content only contained in this area, and then one performs an inverse FFT to retrieve the acoustic component of the signal in the physical domain. To access the aerodynamic part, the inverse FT is computed over the remaining area. By applying such a filter, p(k x , f ) is then decomposed based on the summation of pFT 1 (k x , f ) and pFT 2 (k x , f ) pressure fields. In this case, pressure spectrum pFT 1 is mainly related to the acoustic component of p(k x , f ) while pFT 2 is mainly related to the aerodynamic one. An illustration of the filtering process is given in figure 7. By realizing an inverse FT , the (x vib , t) space representation of p FT 1 (x vib , t) and of p FT 2 (x vib , t) is obtained. These last two pressure fields are also displayed in figure 7 (graphs (e) and (f)). The amplitude of pFT 1 (k x , f ) (acoustic component) is quite smaller than the one related to p FT 2 . The energy of the pressure signal is mainly contained in the energetic aerodynamic component of the reference signal. As observed in figure 7 (graphs (b) and (c)) in the physical space, the amplitudes of p FT 1 (x vib , t) and p FT 2 (x vib , t) differ by a factor of 10. These two filtered pressure fields will be separately used in section 5 allowing the investigation of their associated radiated acoustic fields in the open cavity.

POD analysis of the wall pressure field

To perform a direct comparison with previous FFT analysis, the POD is implemented to the same wall pressure data. This simplified 1D stored data is also retained due to the fact that in the recirculation area, some most energetic flow events are observed compared to those observed downstream of this area. Based on such data, it seems easier to differentiate both flow events using the POD energetic flow partitioning. As the number of available grid points is small compared to the number of snapshots, the classic POD formulation introduced by Lumley (40) is used. That leads to obtain N modes = N s (see equation 7). An analysis of the POD mode convergence similar to the one detailed in [START_REF] Hekmati | About the convergence of POD and EPOD modes computed from CFD simulation[END_REF] has been conducted [START_REF] Gaudard | Développement d'outils pour l'analyse de champs aéroacoustiques à l'origine du bruit dans un habitacle automobile[END_REF]. This analysis shows that the first 10 POD modes at least are statistically converged. Figure 8 presents the POD mode convergence. The first 4 cumulated POD modes contain 36% of the total fluctuating pressure energy respectively. Figure 9 displays selected POD eigenfunctions Φ (n) (x vib ). Figure 10 represents the Fourier spectra of the first five temporal POD coefficients and also of the 20 th one. Then, these first POD modes clearly extract the main energetic contribution of the flow, which is the one associated with the shear layer flapping instability (x/h < 2.5). Conversely when regarding high order POD modes, a high frequency component is extracted in the recirculation area and a lower frequency component is retrieved far away (x/h > 5) with a similar amplitude [START_REF] Gaudard | Développement d'outils pour l'analyse de champs aéroacoustiques à l'origine du bruit dans un habitacle automobile[END_REF]. These high order POD t Φ (1) modes are then mainly related to the eddy structure interacting with the wall far downstream the shear layer flapping motion. To select in an objective way, a POD cut-off mode number allowing the separation of organized motions and background turbulent part remains often questionable. It does not exist a universal criterion for such an investigation. However, it is generally expected that the first POD modes contain the main part of the energetic organized flow pattern of the flow. In this work, the objective is to isolate the energetic vortex in the recirculation area and after several trials, it seems that the combination of the first four POD modes provides a nice extraction of these vortices. Using POD basis, the instantaneous wall pressure field is then decomposed following equation 8:

(s) (a) t (s) (b) t (s) (c) f (Hz) (d) f (Hz) (e) f (Hz) (f)
(x vib ) Φ (2) (x vib ) Φ (3) (x vib ) Φ (4) (x vib ) Φ (5) (x vib ) Φ (20) (x vib )
p POD 1 (x vib , t) = 4 n=1 a (n) (t)Φ (n) (x vib ) and p POD 2 (x vib , t) = N modes n=5 a (n) (t)Φ (n) (x vib ) (9)
In this context, the p POD 1 pressure field corresponds to the wall pressure field projected onto the first four POD modes and it is associated with the main energetic flow pattern of the wall pressure field. The p POD 2 field (called the residual part of the wall pressure field) corresponds to the wall pressure field projected onto the POD mode remainder and it is associated with the low energetic flow pattern of the wall pressure field. Note that the purpose of the current POD application is not to discriminate acoustic and aerodynamic parts of the pressure field as previously tested (8; 26). Indeed, POD application is optimal in an energetic context and then a specific POD mode may also contain information about several distinct flow events (energetic or not). In this context, POD cannot be viewed in the present study as a method to interpret the acoustic source terms. In this study, contrary to FT filtering decomposition, this method offers a new way for decomposing the wall pressure field based on an energetic criterion. Figure 11 illustrates the resulting POD flow decomposition by plotting p POD ) exhibits a low energetic vortex signature in the downstream area (x/h < 3). This last contribution is then mainly related to the convected flow structures interacting with the wall after the recirculation area. Spectral representations show that the acoustic component is shared by p POD 1 and p POD 2 .

Analysis of the structure vibro-acoustic response and of the acoustic radiation in the cavity

The vibroacoustic response of the wall pressure loading is now investigated. The properties of the vibrating structure have already been given (see §2.2). As a preliminary result, the first radiated modes of the vibrating structure are computed theoretically. They are given in table 2 and presented in a graph on figure 12. Turbulence convection wavenumber k c and acoustic wavenumber k 0 at the structural resonance frequencies are also shown. The hydrodynamic coincidence condition is below the first beam resonance frequency and the acoustic coincidence condition is above the seventh beam resonance frequency. This knowledge of the modal frequencies is of interest in the following interpretation of the vibro-acoustic response.

Beam solver response

The reference wall pressure field p(x vib , t) is used as an inflow condition to compute the reference velocity response. Recall that the present numerical simulation is based on a simplified coupling method between the pressure distribution and the flow field and the vibrating structure. It is not based on an interacting method but based on a weak coupling. However, note that previous tests [START_REF] Gaudard | Développement d'outils pour l'analyse de champs aéroacoustiques à l'origine du bruit dans un habitacle automobile[END_REF] have been performed based only on one computation taking into account the vibro-acoustic coupling. Deduced results have demonstrated that the sound due to the vibrating structure itself has no effect on the FFS flow or on the radiated acoustic field in the cavity. An illustration of the beam solver computation is presented in figure 13 (top) displaying the vibration velocity response. Each wall pressure field p method i (x vib , t) (with i = 1, 2 and method = FT, POD) is also successively used as inflow conditions (see table 1). The resulting vibration transverse velocities are represented 20 in figure 13. As a first observation, the amplitude of transverse velocity computed from the test case 2 is much smaller than the one computed from other test cases. This is directly related to previous comments (see §4.2.1) according to the content of p FT 1 wall pressure field. Moreover, the results of the test case 3 approach the ones of the reference test case 1. That seems to argue that the aerodynamic component of the wall pressure field is mainly responsible for the structure vibration. On the other hand, even if the wall pressure p POD 1 field is less energetic than the p POD 2 one, the amplitude of resulting transverse velocity is quite similar in both cases. Nevertheless, the frequency contents of both beam solver responses differ. For the test case 4, based on p POD 1 pressure field, a low frequency component seems to dominate in the vibration transverse velocity response. This behavior could be directly due to the low frequency component of the energetic flow structures associated with the shear layer flapping motion which is well extracted based on the first POD modes (see the spectral representations of the first POD temporal coefficients in figure 10). These large scale flow structure instabilities impact the wall regularly and lead to force the low frequency excitation of the vibrating structure.

Acoustic radiation in the cavity

The transverse velocity computed with the beam solver is now used as an inflow condition for the numerical simulation of the acoustic propagation in the cavity. As previously, five test cases are performed (see energy values computed from the test case 2 differs from more than 20dB, to the one related to the test case 3, demonstrating that the main acoustic energy in the cavity is related to the aerodynamic part of the wall pressure field. Indeed, this part allows the recovering of the whole acoustic energy. As for POD test cases 4 and 5, a quasi similar energy level is retrieved slightly inferior to the one of the reference test case 1.

During each simulation when the flow is established, instantaneous acoustic pressure field is stored at the fixed point located exactly at the center of the cavity. Figure 14 displays the time evolution of the acoustic pressure signal and its associated spectral representation. To enhance the comparative analysis, the acoustic spectrum deduced from the reference test case is always superimposed onto the acoustic spectra deduced from other test cases. Moreover, figure 16 represents similar representations but for the acoustic signals stored at a point close to the middle of the beam. These two figures show quasi similar results detailed below. Several peaks appear in the acoustic spectrum computed from the reference test case. These peaks are well separated and they correspond to the structural modes of the vibrating wall (see table 2). Note that to avoid any confusion about the dependence of the time duration for the post-processing, we performed similar postprocessing analysis based on twice the current simulation time duration. Deduced results show that no real difference is noticeable compared to the present ones. Apart from structural mode number 2, all the structural modes contribute to the acoustic response in the cavity. Structural modes 1 and 3 (that correspond to frequencies: 164Hz and 904Hz respectively) present a large magnitude and dominate the others. Note that it may not be possible to retrieve exactly the frequencies of the structural mode due to the dependence of frequency resolution imposed by the time duration of the simulation.

In each test case, the acoustic spectrum also shows peaks around the structural mode of the wall. However, in each case the amplitude of the frequency peak differs. According to results of test cases 2 and 3, the main noise contribution radiated in the cavity is due to the filtered aerodynamic pressure field, p FT 2 . Indeed, test case 3 exhibits a similar time evolution and spectral representation to the ones obtained with the reference test case. Even if the acoustic signal computed from test case 2 is of very low amplitude, the associated acoustic radiation in the cavity is non-negligible, especially for high frequencies (see figure 15), confirming previous statements [START_REF] Arguillat | Measured wavenumber-frequency spectrum associated with acoustic and aerodynamic wall pressure fluctuations[END_REF]. It is noticeable to observe that the test case 2 shows that the beam mode of 2287Hz is entirely recovered. It also contains each frequency observed in the reference test case. Moreover, as the FFT filtering procedure is quite dependent on the wavenumber-frequency resolution [START_REF] Gaudard | About the pod application for separating acoustic and turbulent fluctuations from wall pressure synthesised field[END_REF], it may be possible that the acoustic contribution is not entirely extracted based on the present filtering procedure especially at very low frequencies. Then such an acoustic wall pressure field could also have a higher contribution in the radiated acoustic pressure field in the cavity. Concerning the test cases 4 and 5, the acoustic pressure signal computed from the most energetic p POD duced from the test case 5 (p POD 2 pressure field). The acoustic pressure fields from test cases 4 and 5 contain the same frequency peaks as the reference test case. The beam modes 4 and 5 (1517Hz and 2287Hz) are better restituted thanks to the test case 5 (p POD 2 wall pressure field). Conversely, test case 4 exhibits a first frequency peak ( f ≈ 164Hz corresponding to the first beam mode) of several orders superior to the one related to the test cases 1 and 5. It seems then that the structural vibration response leading to contribute to the first structural acoustic mode is quite related to the shear layer flapping motion that corresponds to the most energetic large scale flow structure. This result emphasizes the fact that the low frequency acoustic component obtained in the cavity is mainly associated with the recirculation area. Then, to reduce the low frequency acoustic noise in a similar flow configuration, one has to implement future control strategies allowing to reduce and/or limit the separated flow. For the investigation, POD could be used to control the flow separation (42).

Conclusion

The noise radiated by a vibrating structure into a cavity is of particular interest in modern automobile and aerospace industry. The numerical investigation of realistic flow configuration related to plane or vehicle applications remains quite complex today. In this sense, we propose in this work to consider a simplified flow configuration allowing the generation of similar fluid-structure-acoustic interactions to those observed in realistic industrial application. A FFS flow configuration is retained and the upper part of the step is assumed to be a vibrating plate permitting the examination of radiated noise in an open cavity located under the plate. Three successive computations are performed. First, the numerical FFS flow characteristics are observed to be similar to the ones previously given in the literature even if a 2D flow configuration is considered. Second, based on each wall pressure field defined from FT or POD applications, the radiated acoustic pressure emitted in the cavity is determined. It is then observed that the main contribution of the interior noise is due to aerodynamic component extracted based on a frequency-wavenumber filtering procedure using FT . However, it is also confirmed that the acoustic pressure structural loading may have a non-negligible contribution in the radiated acoustic pressure. POD applications lead to the following results: i) the most energetic wall pressure field corresponds to the shear layer flapping motion; ii) this energetic flow field mainly contributes to the low frequency noise in the cavity; iii) the POD flow partitioning allows the discrimination of two components which contribute in a similar manner to the noise radiated in the cavity. This study permits to test the ability of two post-processing mathematical tools to elucidate the aeroacoustic contribution of an unsteady flow which impacts a vibrating structure and associated with the main noise radiated in a cavity located under the structure. These post-processing tools offer then some great prospects for the future analysis of a more realistic fluid-structure-acoustic interaction problem occurring in automobile industry. Based on these preliminary results obtained in a simplified 2D flow configuration, the potential of both mathematical tools has been emphasized. Thus, a prospect of this study is to perform a more realistic 3D forward step flow configuration allowing to confirm these preliminary results. In this case, both FFT and POD methods will be implemented to decompose the 2D wall pressure field. The FFT and POD algorithms remain the same and have already been applied to similar 2D wall pressure field but without the same aim (5; 26). However, 3D applications need some great additional computational costs. In each spectral representation, the acoustic spectrum computed from the reference test case is also plotted (with dotted-line) for comparison.

Figure 1 :

 1 Figure 1: Schematic representation of the three flow domains: exterior forward facing step, vibrating structure and open cavity (not to scale). The sponge regions of the open cavity domain and also at the upper transverse boundary domain are not represented for clarity.

Figure 2 :

 2 Figure 2: Near wall profiles of mean streamwise velocity extracted at selected x-locations.

Figure 3 : 2 ∞

 32 Figure 3: Streamlines of the mean velocity field.

Figure 4 :

 4 Figure 4: Streamwise evolutions of the wall pressure coefficient (C p wall , left hand side) and of the fluctuating wall pressure coefficient (C RMS p wall , right hand side).

Figure 5 :Figure 6 :

 56 Figure 5: Power Spectra Densities (dB scale, ref 4.10 -10 Pa 2 /Hz) computed at selected x locations.

Figure 7 :NFigure 8 :

 78 Figure 7: Representation of the wall pressure signal: Top, in the physical space (t, x/h) expressed in Pa. Bottom, in the wavenumber-frequency domain ( f, k x ), expressed in dB scale. Left: Reference wall pressure field. Center: low wavenumber filtered field (acoustic component) and Right: high wavenumber filtered field (hydrodynamic component).

Figure 9 :Figure 10 :

 910 Figure 9: Spatial evolution of the first 5 spatial POD eigenfunctions and also the 20 th one.

1 (

 1 x vib , t) and p POD 2 (x vib , t) pressure fields. The first POD modes clearly extract favorably the energetic wall pressure field in the recirculation area. The residual part (pressure field p POD 2

Figure 11 :

 11 Figure 11: Representation of the wall pressure signal: Top, in the physical space (t, x/h) expressed in Pa. Bottom, in the wavenumber-frequency domain ( f, k x ), expressed in dB scale. Left: p POD 1 (x vib , t) and Right: p POD

2 (

 2 x vib , t).

Figure 12 :

 12 Figure 12: Theoretical structural modes of the vibrating glass structure. Test cases 1 (Reference) 2 3 4 5 Acoustic energy 106.47 106.65 92.40 104.71 102.03

Figure 14 :

 14 Figure 14: Left: Time evolution of the radiated acoustic pressure stored at the center of the cavity. Right: Corresponding Power Spectra Densities (dB scale, ref 4.10 -10 Pa 2 /Hz). In each spectral representation, the acoustic spectrum computed from the reference test case is also plotted (with dotted-line) for comparison.

Figure 15 :

 15 Figure 15: Power Spectra Densities (dB scale, ref 4.10 -10 Pa 2 /Hz) of the radiated acoustic due to the aerodynamic component (line) and due to the aeroacoustic component (dotted line).

Figure 16 :

 16 Figure 16: Left: Time evolution of the radiated acoustic pressure stored at the point close to the middle of the beam. Right: Corresponding Power Spectra Densities (dB scale, ref 4.10 -10 Pa 2 /Hz).In each spectral representation, the acoustic spectrum computed from the reference test case is also plotted (with dotted-line) for comparison.

Table 2 :

 2 Theoretical structural modes of the vibrating glass structure.

Table 3 :

 3 Global radiated energy (expressed in dB) in the open cavity for each test case.

table 1 )

 1 . As a first investigation, table3gives the global radiated energy (expressed in dB) in the open cavity. The acoustic 21

		Reference test case 1
	t (s)
	Test case 2	Test case 3
	t (s)	t (s)
	Test case 4	Test case 5
	t (s)	t (s)

Figure 13: Beam solver computation. Vibration transverse velocity response of the structure for different test cases (see table 1

).

pressure field is of slightly higher amplitude than the one de-


JOB SEEKER'S SPATIAL EQUILIBRIUM IN A FUZZY MATHEMATICAL FRAMEWORK

Keywords:

This paper incorporates complex behavior in a job seeking strategy model.

Since the state of information is imperfect, a job seeker faces an imprecise labor market. His behavior is fuzzy and variable. It is shown how this complex behavior can be described by fuzzy subsets theory and bifurcation analysis. The latter examines the discontinuous phenomena.

Then it is proved that a job seeker' s fuzzy equilibrium with a spatial constraint exists.

The empirical literature on job seek theory is often focused on the studies of the linkage between two elements of a job seek strategy : migration and job seek efficiency [d' Avirsenet (1979), Goss E. P. and Schoening N. V. (1984), Greenwood M. J. , Hunt G.L. and Me Dowell J.M.

(1986), Lesourne J. (1986), Renault E. and Lesourne J. (1986)] ; wage requirement and job seek efficiency [START_REF] Barron | Search Effort in The Labor-Market[END_REF], [START_REF] Kahn | The Returns to Job Search : a Test of Two Models[END_REF]] ; imperfect information and job seeking strategy [Laffond G. and Lesourne J. (1986), Rea S.A. Jr (1981)] and job seeker' s qualifications and job choices [Lösch A. (1954)].

The aim of this paper is to combine these different elements 1.e. to introduce the complex behavior of a job seeker in a job seek strategy. More precisely, we consider that the labor market is an Imprecise space where job seekers have a fuzzy and spatial behavior. The job seeker' s fuzzy behavior results from many influences of the different characteristics of the jobs : wage, qualifications, job location and of the job seeker' s ones : qualifications, socio-demographic characteristics, residential location and wished wage.

Since informations on vacancies, wages and job locations are imperfect, job seekers cannot have precise and constant behaviors. So they are not able to say whether a job belongs to their sets of preferences or not (without any other choice) but they are able to say that a job imperfectly belongs to them. That is designated by "fuzzy behavior".

The job seeker' s spatial behavior is analyzed as a choice between migration and geographical immobility. First the analysis is job oriented (given a residence look for jobs). Then, if the job seeker does not choose a job in his residential labor market, he decides to migrate.

THE ANALYTICAL FRAMEWORK

A particular economic framework is given.

The job seek strategy depends on four principal factors : wage, information, space and qualification -which are interdependant-. For example, the level of qualification determines the level of wage, or a man who does not want to migrate will accept a lower wage and if the Information is costly the job seeker will accept more easily a lower wage.

Finally, we can set down that the wage absorbs most of the influences of the other factors. A job seek strategy will be mainly articulated around the wages wished by Job seekers and offered by employers.

The job seeker fuzzy behavior will consist in obtaining the maximum wage which also depends on qualifications, space and information.

We set down that the wage utility measures the job utility because of the following comments : first the wage levels take into account some characteristics linked to the practise of a job : hard job, tiresome job, unhealthy job. Second, other characteristics are not systematically taken into account by the wage because their "monetary" integrations depend on the work contract négociation between a job seeker and an employer. For example, a job seeker who lives far from his future job location will try to obtain an higher wage to compensate the distance costs. Finally, wage utility is a good approximation of job utility if most of the influence of these factors : space, information and qualification is taken into account.

However, we suppose that these last factors constrain the job seek strategy. More precisely, we set down that a job seeker faces only with the spatial constraint. In fact, the state of information determines the set of Jobs known by a job seeker and his qualification determines the jobs he can can obtain. So these two factors influence the job seek strategy before it begins. On the contrary, the influence of the factor "space" appears during its progress.

The spatial behavior of a job seeker can be resumed as follows :

a job seeker tries to make his residential location and his job location coincide. So, either he does not want to migrate and he prefers to move daily from home to job location or he accepts to migrate when these daily moves are too costly and tiresome.

This spatial behavior is a passage from a refusal to migrate to an acceptance to migrate, i.e. is a spatial behavior breaking off. It can be analyzed by the bifurcation theory.

THE HODEL

A job seeker follows a strategy of a fuzzy wage utility maximization under a fuzzy spatial constraint. We study this behavior with the theory of fuzzy subsets and with the bifurcation theory. We will obey the following plan : first we will define the jobs sets and second we will describe the objective and constraint.

We draw the attention on the following point : only ordinary (non fuzzy) mathematical concepts are underlined. For example A c X is a fuzzy subset of the X referential.

Jobs sets

A job seeker prospects on a specific labor market which is the set of a priori possible jobs that we note £. 8 is characterized by the state of the labor market and by its capacity to deliver information.

The state of the labor market depends on economics circumstances and we will suppose that the unemployment rate is rather high. Jobs are perfectly localized and types of jobs are known through available Information. So for each Job man asks some qualifications and man offers a certain wage.

We define 6 as the set of jobs offered in a given location for a certain level of qualification and for a given wage.

We set down that :

• { ' ' -3 1 )
where 1 represents the job locations 1 = 0. ..m q represents the qualifications asked for the jobs q = l...n and s represents the offered wages s = l...r. 1 3 For example, the job e^ is a job offered in Lille (a town whose code is 1) for a professional whose qualification level is 3 and for a wage of x FF (code 4 in the wages scale).

However, the space of offered jobs is different from the space of accessible jobs. The latter depends on the job seeker' s qualifications.

We will note E the set of technical possible jobs.

Then E c § and E = -| [ ^e^ ] j-where q ' = 1. . . t ; t s n. q ' are the job seeker' s qualifications. The Job seeker' s decision belongs to E.

The objective

A job seeker tries to maximize a fuzzy wage utility which is imprecise because it depends on the different effects of the different characteristics of the jobs : the wage offered by the employer, the job location, the qualifications required for the job, and it depends on the job seeker' s requirements : the wished minimum wage (the reservation wage)

and the wished job locations.

Remarks 1)

The maximization wage utility objective is compatible with the reservation wage concept. So, for constant wage requirements, the objective is to obtain a wage equal to or higher than the reservation wage. If wage requirements can be reviewed falling, the objective is to obtain the nearest wage from the reservation wage. If wage requirements can be reviewed rising, the objective is to obtain the highest possible wage.

2.)

The state of information is imperfect, so the reservation wage is not exactly known. This also justifies the use of a fuzzy wage utility.

Finally, a job seeker will prefer more or less a job following its offered wage actually observed for this type of jobs on the labor market.

Let F be the fuzzy objective of a job seeker and be the r membership function which designates the fuzzy wage utility.

We define F and (ip as follows :

F c E F = j e s , (ip ; V e g e E ; Mpie«,) e [0, l]j where eg = [ ] and : (ip

: E i -----» [0, 1] es ' -----> * V (es) € [°' 1 L
* Let s be the reservation wage and s be the average wage really 1 h ^ observed for the job eg which is payed for a wage s.

We will note e = s -s the estimation error of the reservation wage.

* ® * * -If s is overestimated, c is positive and max (s , s ) = s . m * * -If s is underestimated, e is negative and max (s , s ) = s . m m F Mf(es ) -0 • V ' S ' ' 1 /iF (es ) € ]0, It if s < min (s , s ) m * if s i max (s , s ) m * * if min (s , s ) s s < max (s , s ) m m
If the reservation wage is exactly estimated, we find again the 1 h * traditional case, i.e. a job seeker will accept the job e if s £ s = * ® s , then u_(e ) = 1 and he will refuse it if s < s = s , then LU,(e ) = 0. m r s m F s

The constraint

The job seeker' s spatial behavior depends on the labor market area he prospects. If we define this area as a space centred on the job seeker' s residential location, we define a linkage between the job seeker' s mobility and the labor market area he prospects. The latter is

Limited by the maximum distance between the residential location and the * job location he will accept. We note R this maximum distance.

Since the distances are costly, R depends on the job seeker' s earnings. Since job seeker wage requirements are imprecise and since a job seeker can have other earnings over wage (financial yields, spouse wage, game incomes, work on the side), the whole job seeker' s earnings is * imprecise, so R is not perfectly determined and the spatial constraint is fuzzy.

We can define the spatial fuzzy constraint C as follows :

C = {e1, n c , V e 1 € E, M ^e 1) € [0, 1]>
where e* = [^g] and is the membership function of a job to the C fuzzy subset.

We set down :

E i ------» [0, 1] = e 1 i ------» e *°' ^
Let 1q be the residential location and 1 be the Job places with

1 = 0. . .m. / M ^e 1 ) = 0 if d(lQ, 1) * R* -Mc (e1) -1 if d(l0, 1) = 0 -^( e 1) € ]0, 1[ if 0 < d(l0 , 1) < R*
where d(lQ , 1) is the distance between 1Q et 1.

We show that decreases when d(lg, 1) increases

V [* eg»]j e E x E where 1' £ 1 ; h ' * h ; s' <* s if d d 0> i) £ d(iQ , 1 ^ then s Now
we can define the job seeker' s fuzzy and spatial equilibrium.

JOB SEEKER' S FUZZY AND SPATIAL EQUILIBRIUM

A job seeker must find a compromise between the wages he can obtain and the location of the jobs. In order to solve this problem, we must define the spatial behavior of a job seeker before we shall determine the job seeker' s fuzzy and spatial equilibrium.

The spatial behavior

A job seeker prospects on a more or less large labor market area because of his mobility constraint. Since daily moves are costly and tiresome, he will measure the profitability of the offered jobs. For a given wage, the labor market area is a disk whose center is lnand whose « « radius is R . We can set down the following assumptions : HI : The residential utility at 1^ is noted and it measures the profitability of the job offered at 1. For every wage s, the residential utility is maximum when 1 = 1^.

H2

For a given wage s, the residential utility decreases when the distance <1(1^, 1) increases.

H3 : For a given wage s, the residential utility equals 0 when d(l0 , 1) = R*. * H4 : If cU I q , 1) = R , a Job seeker will prefer to migrate and go to live in 1, where the new residential utility will become maximum.

Under these different assumptions the spatial behavior of a Job seeker can be interpreted as follows : for a given wage, the residential utility equals 0 when it is more interesting to migrate than to move daily from home to job location. So, a job seeker will change his behavior, i.e.

he will accept to migrate after he has refused to do it.

Finally, the problem of migration is to determine a maximum * distance R = d(lg, 1) for which we observe a spatial behavior break off.

This problem can be solved by the bifurcation theory since a bifurcation point is a point for which the type of the set of solutions * changes. Here, for a distance R , the residential utility U^ equals 0 but it becomes maximum in 1. We actually have a bifurcation point. We set down Sq = a s (consumption spent is a constant part of the wage s).

Then UQ = (J -twx -n R ^) ((1 -a) s -nC1 R ) Remark

*» 1)

The residential utility can be minimum for a distance R , solution of the equation :

ô R i.e. R ., V ^s U -a)) + CjCT -tWj) 2n V1 Cj * ' 11 2 R = I (R j ♦ R j)
Then if Ug is minimum, its value is negative and we cannot accept it in economics.

* 1 2 2) We choose R = min (R^, R^) in order to respect the traditional behavior of leisure preference.

* 3) R is a maximum radius of a labor-market area centred on the residential location of a job seeker who refuses to migrate. * All the points of the frontier of the disk (1^, R ) are bifurcation points, i.e. UQ equals 0 but it becomes maximum when a job seeker migrates from 1q to 1. So, if a job seeker changes his behavior, we have a discontinuous phenomena that we can represent as follows : * R Figure 1 Remarks The new residential utility value U depends on the cost of life at 1 so it can be lower, higher or equal to Un (figure 1).

* u

The value of R mainly depends on offered wages. So for each wage level, a specific set of solutions is symbolized by a circle (figure 2) and the higher the wage is, the more slowly the residential utility decreases (figure 1).

Bifurcation phenomenas are symbolized by the dotted lines (figure 1) which represent the discontinuity of the residential utility function. A numerical application using French data illustrates our purpose. With the property of decreasing of the sequence we have :

V (oij, a2 ) € [0, 1] x [0, 1] if <f s g then C S C
and Sup Mp(eg ) s Sup Mp(eg ) =* We know that a fuzzy subset is strictly convex if and only if its membership function is strictly quasi-concave. We are going to prove that jip is strictly quasi-concave.

Up is strictly quasi-concave if and only if the following condition is true.

V e € Pe*1] g F V e , € le*1] e F with e * e , S

[ sj S L sj s s

3

) if M p (es ) £ Mp(© ) * V A € ]0, 1] |ip(X eg+ (1 -A)eg , ) £ Mp(es, )
Then fip is strictly quasi concave and F is strictly convex. So we have to prove proposition (3).

If e is different from e , , these two jobs are different either by the s s location or by the wage or by the qualification or by any combination of two or three of these elements. We have to examine seven different cases.

First case s * s' ; h = h ' and 1 = 1*

The two jobs are only different by the wage.

So in terms of wage (As + (1 -A) s' ) > min (s, s' )

Then the job (Ae + (1 -X) e ,) is preferred to the job e , s s s and pu(\e) + (s -A) e , ) > fi (e . ). * e^j where 1" = A1 + (1 -A)l' we will have

Uptes"* > then * V ^es + ^ ^ es ' * > iV * es' ^'

Fourth case s * s' ; h * h ' and 1 = 1' If e is preferred to e , because of the combination (e, h) then, s s according to the individual division of labor theory (Lösch [13]), s is higher than s' . We find again the first case and MF (Aes + (1 _ X) es' ) > Fifth case s * s' ; h = h ' and 1 * 1 ' V es " " If e is preferred to e , then the residential utility for the job e is s s s higher than the residential utility for the job e ,. s Then the value of the residential utility for the job (Xe + (1 -X)e ,) s s will be higher than the residential utility for the Job e s Therefore jip(Xeg + (1 -X) eg, ) > Mp(es> )• Sixth case s = s' ; h * h ' and 1 * 1 ' If e is preferred to e , because of an optimum combination of 1 and h ; s s then all the compromise will be still preferred to eg , which is the worth wished situation. Then (ip(Xeg + (1 -X) eg, ) > (ip(eg, ).

Seventh case s * s' ; h * h ' and 1 * 1 '

The jobs e and e , are completely different. We can find again every l h " e € E Now we must examine if this solution is unique, i.e. if the two following conditions are confirmed, first the function ¡1-, must be strictly 1 h quasi concave (that is true) and second V e e (C -C.) we must have S Ot 1

Uppe^j * 1 and we cannot confirm this condition because we can find a job eg so that jippegj = 1.

Finally, a solution of the problem of the job seeker' s fuzzy and spatial equilibrium exists but it is not unique.

This conclusion does not prevent our model to be relevant to describe a job seek strategy when the labor market is considered to be an imprecise space. And a numerical example will illustrate the model while it will be explaining how the solution can be found.

NUMERICAL APPLICATION *

The solution e belongs to C e E, i.e. it belongs to the open disk (1q, R) and it requires the qualifications h of the job seeker.

*h

The job seeker has for each of his qualifications a reservation wage s , *h a fuzzy utility F(s ) and a seek labor market area determined by Uq (s ).

A job seeker follows an optimum strategy if he begins to seek the jobs whose combinations wage qualification are the best.

If he does not find a job, he will have to change his spatial behavior and move to go to live at the job location. Before we realize a numerical application, we are going to define numerically the membership function fZp and n^.. #ip is defined as follows :

E i-----» [0, 1] *V ' eg i ----» e f°> ^
where Mp(es ) ■ = 0 If we choose the first mathematical programming we determine A.

" f ' V = 1 if s < min (s , s ) m if s s max (s , s ) m minis, s ) -s " m u"(e ) = -------------- if min (s ,
The optimal Job e belongs to A and it has the highest value fi-te ). A job seeker will choose the job eg which is located 17 km for from his residential location and whose wage is 6 000 FF per month. This wage is

CONCLUSION

The following conclusions can be drawn.

1. The theory of fuzzy subsets allows to describe less strictly the job seek strategy than with the reservation wage theory since a job seeker wishes to maximize the fuzzy wage utility.

2. The bifurcation theory allows to analyse the job seeker' s spatial behavior as a variable behavior that refuses the traditional constant behavior assumption.

3. Finally, the problem of the job seeker' s fuzzy and spatial equilibrium admits solutions. The solution is not often unique but this result does not penalize the model because a job seeker can yet choose among the different optimal solutions. 4. The theory of fuzzy subsets allows the complex behavior analysis and their applications to performant models for decisions making.

Finally

0 !

 0 Now we are going to define it. The residential utility is a function of the time man can pass there and of the money man can spend there. Therefore, we set down that U^ = x if, where U^ = residential utility in 1^, = total leisure time passed in ln , 0 Pq = wage left over. This definition allows us to count all type of leisure either by their costs or their time if they are free. Remark The introduction of time supposes that we actualize the different costs compared with the job seeker' s age. But we set down that the reference period is smaller or equal to one year. This condition avoids the actualization and is compatible with information on Jobs which are quite precise for a year. More precisely we have : UQ = (7 -twx -n R : V x) (twxS -n ^ R 1 -SQ AQ ) where T = reference period (week, month, year...), tWj -work time on T for the job 1, n = number of moves from 1^ to 1 and from 1 to 1^ during T, Vj = move time from 1^ to 1, Rj = d(l0 , 1) (km), S = wage rate per work time, We note tw^ S = s = weekly wage or monthly wage or yearly wage, = move cost per km for d(lg, 1), Sq = cost of living at 1^, Aq = Consumption at 1^, We note A^ = s^ : consumption spent during T.

1)

  When d(l0 , 1) = 0 then UQ = U max = U* U = iff -tw^) (1 -a) Sq 2) Migration costs (psychological and monetary costs) are not counted because they are negligible compared with n x C^ x R^and n x x R^ because they represent only one move from 1^ to 1

For

  we suppose that 70 % of earnings is spent for the consumption of food, clothes, health and home). We obtain : Rj = 1032 km or R£ = 23,4 km So : R* = min (1032, 23,4) = 23,4 km Then a man who earns 5 000 FF for 4 weeks will have to migrate to live at his job location if it is 23,4 km or more far from his residential location. The value R^ = 1032 km is explained because total leisure time includes total sleeping time and this result proves the importance of the time over earning one. 4.2. Job seeker' s fuzzy and spatial equilibrium * The optimum job e is the best for the fuzzy objective and for the fuzzy constraint. This type of job is a fuzzy optimal decision which does not allow improving the objective without deterioring the constraint and vice-versa. * Therefore e e D with D = F r> C where D c E is defined as follows : D -f es • • * V V les e £• ^( les) -Fles> " " c '* 1'} where is the membership function of a job to the D fuzzy subset.To solve a problem of maximization of non fuzzy objective under non fuzzy constraint we need a convexity assumption for the C subset. In case of fuzzy subsets, convexity can be required for F or C. We choose to put convexity on F.The solution e is determined in the following manner MD (e') = Sup MnpeÜ) must find a so that Sup fi^Ce ) = a = Sup Mn (e ) if and only if the function #>(a) has a fixed point, i.e. if and only if <p(a) is continuous and decreasing over the interval [0, 1] due to the following theorem.TheoremIf the function Sup fip.pegj is continuous and decreasing over that ^p(a) is decreasing.

  will prove that <p(a) is continuous over [0, 1] with the theorem due to Tanaka-Okuda-Asal [21]. Theorem If the fuzzy subset F is strictly convex, then the function Sup ti"(e ) is continuous.

  ' ; h * h ' and 1 = 1' The two jobs are only different by the qualification. If eg is preferred to eg, because of the qualification, the job seeker will prefer again a job where he can use this qualification h to a job eg, where he cannot use it. Then MpCX eg + (1 -A) eg, eg ) > Mp(es, ) then d(lQ , 1) < d(lQ , 1' ).

  s s previous case by determining what is or what are the different elements that make eg preferred to eg . 2 Therefore, V (eg ; eg, ) € F if (ip(eg ) > Mp(eg. ) then V X e ]0, 1] Mr (Xe + (1 -X) e , ) > |i"(e , ) job seeker can be indifferent for eg and eg , or he cannot compare the two jobs. If the job seeker is Indifferent for eg and for eg,, the preference for a combination of this two jobs is evident. So we obtain (4) *V^es^ S *V^es^ * ^V ^s + ^ ~ es ' > ^F ^s ' If the Job seeker cannot compare the two jobs we have the axiome of the fuzzy subsets : Mp(0 ) = 0. So jjp(eg ) = Mp(©s») = 0 and the proposition (4) is confirmed. This proposition (4) proves that (ip is strongly quasi concave, then (ip is strictly quasi-concave, V (e , e ,) e F^. s s Then F is strictly convex and <p(a) is continuous over [0, 11. * Moreover, <p(a) is decreasing over [0, 1], Then ^p(a) has a fixed point a so that a = Sup ¿ip(es ) = Sup ^( ' e 9

  leg «£/«(. (leJ) * Mp = {* •; € E / "c (le s) * "} So we can calculate a from A or from a.

  the second mathematical programming we determine for each value of a e [0, 1] the corresponding fuzzy subset C . * a The optimum Job e belongs to C * and it has the highest value jip(e ).If we find some jobs equally placed, the job seeker will choose among them the one which satisfies other wishes. Now we state a numerical example.The job seeker residential location is 1^. He has three qualifications h = {1, 2, 3} and he can choose among ten jobs.


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