Mahmoud Karimi
email: mahmoud.karimi@uts.edu.au

Laurent Maxit

Valentin Meyer

Steffen Marburg

R Kirby

Non-Negative Intensity for Planar Structures Under Stochastic Excitation

Keywords: Acoustic radiation, non-negative intensity, surface contribution, stochastic excitation, turbulent boundary layer, acoustic diffuse field

Identification of regions on a vibrating structure which radiate energy to the far field is critical in many areas of engineering. Non-negative intensity is a means to visualize contributions of local surface regions to sound power from vibrating structures. Whilst the non-negative intensity has been

used for structures under deterministic excitation due to structural forces or harmonic incident acoustic pressure excitation, it has not been considered for analyzing a structure under stochastic excitation. This work analytically formulates non-negative intensity in the wavenumber domain to investigate the surface areas on a vibrating planar structure that are contributing to the radiated sound power in the far field. The non-negative intensity is derived in terms of the cross spectrum density function of the stochastic field and the sensitivity functions of either the acoustic pressure or normal fluid particle velocity. The proposed formulation can be used for both infinite planar structure and finite plate in an infinite baffle. To demonstrate the technique, a simply supported baffled panel excited by a turbulent boundary layer as well as an acoustic diffuse field is considered and those regions contributing to the radiated sound power are identified. It is demonstrated that the nonnegative intensity distribution is dependent on the stochastic excitation. It is also found that for a panel under stochastic excitation the more the nonnegative intensity distribution is concentrated within the panel surface, the

INTRODUCTION

Reconstruction techniques of sound sources such as near-field acoustic holography (NAH), inverse boundary element method (BEM) and the equivalent sources methods are widely used in industry [START_REF] Magalhães | Sound sources reconstruction techniques: A review of their evolution and new trends[END_REF]. In many engineering applications, it is important to identify the regions on a vibrating structure which radiate energy to the far field. This identification can help design engineers to gain a deeper understanding about the noise generation mechanism, and it also allows targeted mitigation strategies to be explored. For example, noise reduction can be achieved by modifying geometry and structural properties. Acoustic intensity can help with identifying hot spots on the structure. However, intensity is usually highly bipolar and has positive and negative values that correspond to energy sources and sinks on the surface of the radiating structure. Therefore, the near-field cancellation effects occur when integrating the positive and negative components of the normal acoustic intensity over the surface of the structure. Williams [2; 3] introduced the supersonic intensity (SSI) formulation in the wavenumber domain. The SSI was employed to locate the areas on the source surface which effectively contribute to the far-field pressure. The SSI eliminates the contribution to the pressure and the velocity on the source of the high wavenumber components (subsonic components), which are evanescent and do not contribute to the far field. The modified velocity and pressure obtained by considering only the wavenumber in the acoustic circle were termed supersonic velocity and supersonic pressure respectively.

The SSI was computed in the space domain using a two-dimensional convolution between the acoustic field and a spatial filter mask by Fernandez-Grande et al. [START_REF] Fernandez-Grande | Direct formulation of the supersonic acoustic intensity in space domain[END_REF]. The filter corresponds to the space domain representation of the acoustic circle. Hence, only the acoustic waves that propagate effectively to the far field were taken into account. The numerical technique was validated by an experimental study on planar radiators. Fernandez-Grande and Jacobsen [START_REF] Fernandez-Grande | Conservation of power of the supersonic acoustic intensity[END_REF] quantitatively examined the accuracy of the supersonic intensity. They quantified the error introduced by the finite measurement aperture. It was demonstrated that the error was substantial at low frequen-cies. The study showed that using an extended aperture and/or an increased cut-off frequency the error can be diminished. Valdivia et al. [START_REF] Valdivia | Equivalent sources method for supersonic intensity of arbitrarily shaped geometries[END_REF] employed supersonic acoustic intensity to locate radiating regions on a vibrating structure of arbitrarily shaped geometries. They removed the evanescent waves from the NAH measurement. A method based on a stable invertible representation of the radiated power operator was proposed. The stable invertible operator was derived using the equivalent source formulation and a complete spectral basis. The proposed method was validated using experimental data from a vibrating ship-hull structure.

Magalhães and Tenenbaum [START_REF] Magalhães | Supersonic acoustic intensity for arbitrarily shaped sources[END_REF] extended the SSI technique to consider arbitrarily shaped sources. Their work was based on the BEM and singular value decomposition. Marburg et al. [START_REF] Marburg | Surface contributions to radiated sound power[END_REF] formulated the non-negative intensity (NNI) using the BEM to identify the surface areas of a vibrating structure that contribute to the radiated sound power. The acoustic radiation modes were employed to compute the surface contributions of the structure for all boundaries of the acoustic domain. Williams [START_REF] Williams | Convolution formulations for non-negative intensity[END_REF] proposed two analytical formulae for the NNI based on the pressure and normal fluid particle velocity for planar structures under deterministic excitation. It was shown that both formulae yield almost identical results in prediction of the regions of a structure that emit sound to the far field.

Junior and Tenenbaum [START_REF] Junior | Useful intensity: A technique to identify radiating regions on arbitrarily shaped surfaces[END_REF] proposed an equivalent technique to the SSI based on the BEM called useful intensity. The technique does not require the construction of a hologram to evaluate the acoustic pressure from the known normal velocity field on the vibrating surface. Both the analytical SSI and the numerical useful intensity methods were used by Ferreira et al. [START_REF] Ferreira | Power operator dimensional reduction to obtain the useful intensity in rectangular plates with several boundary conditions[END_REF] to examine the sound radiated from rectangular baffled panels. Eight different combinations of classical boundary conditions were considered. It was shown that the results obtained using the useful intensity were not strictly the same as those obtained using the SSI. The NNI based on the BEM was also employed to identify the surface areas of a rigid sphere and a rigid cylinder that contributes to the scattered sound power [START_REF] Liu | Surface contributions to scattered sound power using non-negative intensity[END_REF]. The same technique was applied to localize the surface areas of vibrating structure to radiated sound power [13; 14]. The surface contribution from a panel to the radiated sound power for different modes was numerically investigated [START_REF] Liu | Nonnegative intensity and back-calculated non-negative intensity for analysis of directional structure-borne sound[END_REF]. The numerical results were validated by NAH measurements. Similar distributions of numerical and experimental NNI were observed at each mode. Liu et al. [START_REF] Liu | Non-negative intensity for structures with inhomogeneous damping[END_REF] used the NNI based on the BEM to investigate the effect of inhomogeneous Rayleigh damping on the surface contributions to radiated sound power. It was found that traveling waves propagate to the regions with higher damping. Wilkes et al. [START_REF] Wilkes | Non-negative intensity for coupled fluid-structure interaction problems using the fast multipole method[END_REF] applied the NNI method to a fluid-loaded steel spherical shell excited by a point/ring force. A hybrid finite element and fast multipole boundary element method (FMBEM) was used to solve the structural-acoustic problem. The boundary field was then used in the FMBEM solver to compute the NNI.

Identification of source velocities on 3D structures in non-anechoic environments using the inverse patch transfer functions (IPTF) method was first introduced by Aucejo et al. [START_REF] Aucejo | Identification of source velocities on 3D structures in non-anechoic environments: Theoretical background and experimental validation of the inverse patch transfer functions method[END_REF]. The direct patch transfer functions method can be used to predict the structural velocity or the sound pressure of a domain containing acoustic sources by calculating acoustic impedances of uncoupled sub-domains. The IPTF method can identify the unknown sources by measuring the coupling velocity at an arbitrarily defined surface surrounding the source. Vigoureux et al. [START_REF] Vigoureux | Inverse patch transfer functions method as a tool for source field identification[END_REF] investigated rigorous criteria needed to obtain accurate results using IPTF to identify sources in a non-anechoic or reverberant environment on an irregularly shaped structure.

Further, a procedure was proposed to compute intensity of the source and wall pressure without any additional measurement. A frequency band was detected for which the IPTF method was not providing accurate results. This was attributed to the presence of evanescent waves. Valdivia [19; 20] developed a method based on the spectral decomposition of the power operator that yielded an NNI expression to efficiently compute the supersonic components from acoustic pressure measurements for arbitrary geometries.

Using numerical models it was shown that the proposed NNI matched the SSI.

Stochastic excitations such as turbulent boundary layer (TBL) and acoustic diffuse field (ADF) are widely encountered in transportation systems [START_REF] Ciappi | Flinovia -Flow Induced Noise and Vibration Issues and Aspects: A Focus on Measurement, Modeling, Simulation and Reproduction of the Flow Excitation and Flow Induced Response[END_REF][START_REF] Ciappi | Flinovia-Flow Induced Noise and Vibration Issues and Aspects-II: A Focus on Measurement, Modeling, Simulation and Reproduction of the Flow Excitation and Flow Induced Response[END_REF][START_REF] Karimi | Numerical prediction of turbulent boundary layer noise from a sharp-edged flat plate[END_REF]. For example, aircraft, satellite, marine vessels, high speed trains and cars are subject to random and non-deterministic excitations throughout their operations. While surface contribution techniques such as the SSI and NNI have been developed for structures under deterministic excitation, they have not been applied for analyzing a structure under stochastic excitation.

In this work, the NNI is analytically formulated for planar structures under stochastic excitation in the wavenumber domain. The proposed formulation is valid for both infinite planar structure and finite plate in an infinite baffle.

Two formulae are developed for the NNI which are in terms of the cross spectrum density function of the stochastic field and the sensitivity functions of either the acoustic pressure or normal fluid particle velocity. The technique is implemented to identify the regions of a vibrating simply supported baffled panel contributing to the radiated sound power. Both TBL and ADF excitations are considered to illustrate the proposed technique.

Radiated Acoustic Power

The radiated acoustic power of an infinite planar structure or a finite plate in an infinite baffle under stochastic excitation can be obtained by integrating the normal active intensity I act , corresponding to the cross spectrum between the sound pressure and the normal fluid particle velocity denoted by S pv f , over the infinite boundary surface as follows [24; 25]

Π rad (ω) = ∞ I act dx = ∞ Re S pv f (x, ω) dx, (1)
where x = (x, y), and ω is the angular frequency. The cross spectrum is given by the following analytical expression [START_REF] Marchetto | Experimental characterization of the vibroacoustic response of panels under random excitations by sensitivity functions[END_REF]

S pv f (x, ω) = 1 4π 2 ∞ H p (x, k, ω)H * v (x, k, ω)φ pp (k, ω)dk, (2)
where * denotes the complex conjugate. H p (x, k, ω), H v (x, k, ω) are sensitivity functions for the radiated pressure and the normal fluid particle velocity on the surface of structure, respectively. The sensitivity functions in the spatial domain are related to the spectral sensitivity functions in the wavenumber domain k, denoted by Hp ( k, k, ω) and Hv ( k, k, ω), by inverse Fourier transform as follows

H p (x, k, ω) = 1 4π 2 ∞ Hp ( k, k, ω)e i kx d k, (3)
H * v (x, k, ω) = 1 4π 2 ∞ H * v ( k, k, ω)e -i kx d k. (4)
Using Eqs. ( 1)-( 4), the radiated acoustic power of a planar structure under stochastic excitation can be written as follows [START_REF] Karimi | Analytical and numerical prediction of acoustic radiation from a panel under turbulent boundary layer excitation[END_REF] Π

rad (ω) = Re 1 4π 2 2 ∞ ∞ Hp ( k, k, ω) H * v ( k, k, ω)φ pp (k, ω)dkd k , (5)
where φ pp (k, ω) is the cross spectrum density (CSD) function of the stochastic force. The sensitivity function of the normal fluid particle velocity on the panel surface is related to the sensitivity function of the sound pressure in the wavenumber domain as follows [START_REF] Williams | Fourier acoustics: sound radiation and nearfield acoustical holography[END_REF] Hp

( k, k, ω) = ρ a ω kz ( k) Hv ( k, k, ω), (6)
where

kz ( k) =    k 2 a -k2 x -k2 y , k 2 a ≥ k2 x + k2 y i k2 x + k2 y -k 2 a , otherwise    , (7)
and k a is the acoustic wavenumber, ρ a is the fluid density, and k = ( kx , ky ).

Substituting Eq. ( 6) in Eq. ( 5), the radiated acoustic power can be written either in terms of sound pressure or normal fluid particle velocity sensitivity functions as follows

Π rad p (ω) = Re 1 16π 4 ρ a ω ∞ ∞ k * z ( k) Hp ( k, k, ω) 2 φ pp (k, ω)d kdk . (8)
Π rad v (ω) = Re ρ a ω 16π 4 ∞ ∞ 1 kz ( k) Hv ( k, k, ω) 2 φ pp (k, ω)d kdk . (9)
The subscripts p and v correspond to the formulations based on the pressure and velocity sensitivity functions, respectively. Considering that the φ pp (k, ω) is always real, the only function which could make the integrand in Eqs. ( 8) and ( 9) complex is kz ( k). According to Eq. ( 7), kz ( k) becomes purely imaginary when the wavenumbers are outside the acoustic circle de-

fined by Ω a = k ∈ R 2 , k ≤ k a .
Therefore, only wavenumbers inside the acoustic circle contribute to the radiated acoustic power. Hence, Eqs. ( 8) and ( 9) can be rewritten as

Π rad p (ω) = 1 16π 4 ρ a ω ∞ k∈Ωa k 2 a -k2 x -k2 y Hp ( kx , ky , k, ω) 2 φ pp (k, ω)d kdk,( 10
)
Π rad v (ω) = ρ a ω 16π 4 ∞ k∈Ωa 1 k 2 a -k2 x -k2 y Hv ( kx , ky , k, ω) 2 φ pp (k, ω)d kdk.( 11
)

Non-Negative Intensity

In this section, an analytical formulation is presented for non-negative intensity (the active normal intensity) for planar structures under stochastic excitation to identify the areas of the vibrating structure that produce radiation to the far-field. The aim here is to develop a formula for I N (x, ω) which meets the two following conditions:

1. The NNI must be always non-negative. This will prevent acoustic shortcircuit in the adjacent areas on the surface of the structure.

2. When integrating the NNI over the infinite boundary surface, it must produce the total sound power.

To meet the first condition, similar to works by Marburg et al. [START_REF] Marburg | Surface contributions to radiated sound power[END_REF] and

Williams [START_REF] Williams | Convolution formulations for non-negative intensity[END_REF] the NNI can be defined as follows

I N (x, ω) = 1 4π 2 ∞ β(x, k, ω)β * (x, k, ω)dk = 1 4π 2 ∞ |β(x, k, ω)| 2 dk, (12)
where β(x, k, ω) is a complex function which is not physically meaningful.

It has been introduced in Eq. ( 12) to ensure that the NNI is always nonnegative by definition. This satisfies the necessary condition for defining the NNI. The second condition for the NNI states that the total radiated acoustic power must be obtained by integrating the NNI over the infinite boundary surface

Π rad (ω) = ∞ I N (x, ω)dx. (13)
Eq. ( 13) can be rewritten in terms of β(x, k, ω) as follows

Π rad (ω) = 1 4π 2 ∞ ∞ β(x, k, ω)β * (x, k, ω)dkdx = 1 4π 2 ∞ ∞ |β(x, k, ω)| 2 dkdx.( 14
)
To meet the second condition, we propose two new formulae for β(x, k, ω),

β p (x, k, ω) = φ pp (k, ω) 4π 2 √ ρ a ω k∈Ωa 4 k 2 a -k2 x -k2 y Hp ( kx , ky , k, ω)e i kx d k. ( 15
)
β v (x, k, ω) = ρ a ωφ pp (k, ω) 4π 2 k∈Ωa 1 4 k 2 a -k2 x -k2 y Hv ( kx , ky , k, ω)e i kx d k. ( 16
)
As can be seen from Eqs. ( 15) and ( 16), the integral domain is confined within the acoustic circle ( k ∈ Ω a ) which means that k 2 a ≥ k2

x + k2 y and kz ( k) is real. These wavenumbers are associated with supersonic waves as their trace speeds are faster than the speed of sound. Whilst for the wavenumbers outside the acoustic circle, kz ( k) is purely imaginary and the corresponding waves are called subsonic waves since they travel at phase speeds less than the speed of sound. The purpose of defining NNI is to identify local surfaces on a structure that are contributing to the far-field radiated sound. It is the farfield sound pressure that is normally of interest in engineering applications because this is the quantity to which a potential observer is typically exposed.

The NNI enables the design engineers to identify the locations of unwanted sources of sound on the structure that make the most significant contributions to the far field. Therefore, only contributions of supersonic waves are taken into account and the subsonic components, which are evanescent and do not propagate to the far-field, are excluded.

To prove that the two formulae given by Eqs. ( 15) and ( 16) result in the radiated sound power as that given by Eqs. ( 10) and [START_REF] Ferreira | Power operator dimensional reduction to obtain the useful intensity in rectangular plates with several boundary conditions[END_REF], Eq. ( 14) should be evaluated using Eqs. ( 15) and [START_REF] Wilkes | Non-negative intensity for coupled fluid-structure interaction problems using the fast multipole method[END_REF]. In what follows, the proof is given for

β v (k, ω)
and similar approach can be used to verify that β p (k, ω) also meets this condition. β * v (k, ω) can be written as follows

β * v (x, k, ω) = ρ a ωφ pp (k, ω) 4π 2 k∈Ωa 1 4 k 2 a -k2 x -k2 y H * v ( kx , ky , k, ω)e -i kx d k,( 17
)
substituting Eqs. ( 16) and ( 17) into Eq. ( 14)

Π rad v (ω) = ρ a ω 16π 4 ∞ k∈Ωa k∈Ωa
Hv ( kx , ky , k, ω)

4 k 2 a -k2 x -k2 y H * v ( kx , ky , k, ω) 4 k 2 a -k2 x -k2 y d k (18) 1 4π 2 ∞ e i( k-k)x dx d kφ pp (k, ω)dk,
using the integral in the parenthesis in Eq. ( 18) corresponds to the Dirac delta function which is given by [27]

1 4π 2 ∞ e i( k-k)x dx = δ( k -k), (19)
using this definition, Eq. ( 18) can be simplified to

Π rad v (ω) = ρ a ω 16π 4 ∞ k∈Ωa 1 k 2 a -k2 x -k2 y Hv ( kx , ky , k, ω) 2 φ pp (k, ω)d kdk.( 20
)
This equation is exactly the same as Eq. [START_REF] Ferreira | Power operator dimensional reduction to obtain the useful intensity in rectangular plates with several boundary conditions[END_REF]. The NNI formulae can be obtained by substituting Eqs. ( 15) and ( 16) into Eq. ( 12)

I N p (x, ω) = 1 (4π 2 ) 3 ρ a ω ∞ k∈Ωa 4 k 2 a -k2 x -k2 y Hp ( kx , ky , k, ω)e i kx d k 2 |φ pp (k, ω)| dk, (21)
I N v (x, ω) = ρ a ω (4π 2 ) 3 ∞ k∈Ωa 1 4 k 2 a -k2 x -k2 y Hv ( kx , ky , k, ω)e i kx d k 2 |φ pp (k, ω)| dk.( 22
)
Due to the magnitude operation, these formulae are guaranteed to yield nonnegative results.

In Eq. ( 22), the term in the denominator tends to zero for the wavenumbers on the acoustic circle. Generally, singular integrals can be numerically evaluated as described in Refs [28; 29]. However, Singularity in Eq. ( 22) can be analytically removed using the following conversion formulae kx = kr cosθ; ky = kr sinθ,

Eq. ( 22) can then be transformed to polar wavenumber coordinates as follows

I N v (x, ω) = ρ a ω (4π 2 ) 3 ∞ θ=2π θ=0 kr=ka kr=0 kr 4 k 2 a -k2 r (24)
Hv ( kr cosθ, kr sinθ, k, ω)e i kr(xcosθ+ysinθ) d kr dθ

2 |φ pp (k, ω)| dk.
Finally, the change of variable, kr = k a sinγ analytically removes the singularity from the integral. As such, Eq. ( 24) can be expressed by

I N v (x, ω) = ρ a ωk 3 a (4π 2 ) 3 ∞ θ=2π θ=0 γ= π 2 γ=0 sinγ √ cosγe ikasinγ(xcosθ+ysinθ) (25)
Hv (k a sinγcosθ, k a sinγsinθ, k, ω)dγdθ

2 |φ pp (k, ω)| dk,
the rectangular method for the numerical integration in Eqs. ( 25) and ( 21), the NNI becomes

I N v (x, ω) = ρ a ωk 3 a (4π 2 ) 3 k∈Ωt θ∈[0,2π] γ∈[0, π 2 ] sinγ √ cosγe ikasinγ(xcosθ+ysinθ) (26) Hv (k a sinγcosθ, k a sinγsinθ, k, ω)δγδθ 2 |φ pp (k, ω)| δk, I N p (x, ω) = 1 (4π 2 ) 3 ρ a ω k∈Ωt k∈Ωa 4 k 2 a -k2 x -k2 y Hp ( kx , ky , k, ω)e i kx δ k 2 ( 27
)
|φ pp (k, ω)| δk.
Ω t is a truncated wavenumber domain and δγ, δθ, δk and δ k are the increments in the numerical integration. For the ADF excitation, since the normalized CSD function φADF pp (k x , k y , ω) is null for the wavenumbers larger than the acoustic wavenumber, the truncated wavenumber domain is basically the acoustic circle Ω a .

It is also noteworthy that the NNI formulae expressed by Eqs. ( 26) and ( 27) can be used for both infinite planar structure and finite plate in an infinite baffle. To compute the NNI, one requires determination of the sensitivity functions. The sensitivity functions can be either calculated analytically or numerically. For example, the finite element method can be employed to obtain the sensitivity functions. In the following section, the NNI formulation is applied to a finite baffled panel for which the sensitivity functions are analytically determined.

Application to Rectangular Baffled Panels

A rectangular baffled panel excited by a stochastic pressure field is shown in Figure 1. The spatial average of the auto spectrum density (ASD) of the panel velocity is given by [30-32] where H vs (x, k, ω) is the sensitivity function of the panel velocity excited by a unit wall plane wave. The spatial average of the ASD of the panel velocity is given by

S vv (x, ω) = 1 4π 2 ∞ |H vs (x, k, ω)| 2 φ pp (k, ω)dk, (28)
V 2 = 1 A A S vv (x, ω)dA, (29)
A = L x L y is the panel surface area and L x , L y are the panel length and width in the x and y directions, respectively. Eqs. ( 28) and ( 29) can be evaluated using rectangular method as described in Ref [START_REF] Karimi | A hybrid numerical approach to predict the vibrational responses of panels excited by a turbulent boundary layer[END_REF] The ASD of the radiated pressure from the panel excited by the stochastic field is also given by

S pp (x, ω) = 1 4π 2 ∞ |H p (x, k, ω)| 2 φ pp (k, ω)dk, (30)
assuming that CSD of the stochastic field is known, it can be seen from the equations in Sections 2-4 that to evaluate Π rad , I act , I N , S vv and S pp , the sensitivity functions of panel velocity, normal fluid particle velocity and radiated pressure have to be known. In what follows, determination of these sensitivity functions are discussed.

Determination of the Sensitivity Functions

For a simply supported rectangular panel excited by a unit wall plane wave, the sensitivity function H vs (x, k, ω) corresponding to the velocity at point x is given by [START_REF] Karimi | A hybrid numerical approach to predict the vibrational responses of panels excited by a turbulent boundary layer[END_REF]

H vs (x, k, ω) = iω M m=1 N n=1 ψ mn (k)ϕ mn (x) Ω(ω 2 mn -ω 2 + iηωω mn ) , (31)
Ω = ρ s hL x L y /4 is the modal mass. The modal frequencies are given by

ω mn = D ρ s h mπ L x 2 + nπ L y 2 , (32)
where D = Eh 3 /(12(1-ν 2 )) is the flexural rigidity, E is the Young's modulus and ν is Poisson's ratio. The modal forces ψ mn are calculated by integration over the panel surface as follows

ψ mn (k) = A ϕ mn (x)e -i(kxx+kyy) dA = I x m (k x )I y n (k y ), (33)
where ϕ mn (x) are the panel mode shapes given by

ϕ mn (x) = sin mπx L x sin nπy L y , (34)
and

{I r s (k r )|(r, s) = (x, m) ∨ (y, n)} =        sπ L r (-1) s e -i(krLr) -1 k 2 r - sπ L r 2 , k r = sπ L r 1 2 iL r , otherwise        .( 35
)
At the interface between the panel and the acoustic domain, the structural velocity v s is equal to fluid particle velocity v in the normal direction, that is H v (x, k, ω) = H vs (x, k, ω). As such, the spectral sensitivity function of normal fluid particle velocity Hv ( k, k, ω) can be obtained analytically using a Fourier transform as follows

Hv ( k, k, ω) = ∞ H v (x, k, ω)e -i kx dx = M m=1 N n=1 a mn ( k, ω)ψ mn (k), (36)
where

a mn ( k, ω) = iω ψ mn ( k) Ω(ω 2 mn -ω 2 + iηωω mn ) , (37)
and ψ mn and I r s are given by Eqs. ( 33)- [START_REF] Maxit | Simulation of the pressure field beneath a turbulent boundary layer using realizations of uncorrelated wall plane waves[END_REF].

Since Hp ( k, k, ω) is related to Hv ( k, k, ω) by Eq. ( 6), to obtain H p (x, k, ω), one can compute the inverse Fourier transform of Eq. ( 6). However, in order to avoid an additional inverse Fourier transform we used an alternative approach based on the Lyamshev reciprocity principle [33; 34]. Figure 2 illustrates the Lyamshev reciprocity principle for a baffled panel.

According to Lyamshev reciprocity principle, the ratio of the pressure at point x over the applied normal force at point x is equal to the ratio of the normal velocity of the panel at point x over the volume velocity Q v of a monopole source placed at point x, that is, where

H p/F (x, x , ω) = H v/Qv (x , x, ω), ( 38
) 𝑥 𝑦 𝑧 𝑥 𝑦 𝑧 ( ) F x  ( ) p x ( ) v Q x ( ) v x 
H v/Qv (x , x, ω) = iω M m=1 N n=1 F mn (x)ϕ mn (x ) Ω(ω 2 mn -ω 2 + iηωω mn ) , (39)
and

F mn (x) = A p(x, x , ω)ϕ mn (x )dx , (40)
where p(x, x , ω) is the acoustic pressure generated by a monopole source and is given by

p(x, x , ω) = iρ a ωQ v 2πr e -ikar , r = |x -x | . ( 41
)
The sensitivity function of the radiated pressure is given by

H p (x, k, ω) = ∞ H p/F (x, x , ω)e -ikx dx , (42)
substituting Eqs. ( 38)-( 39) into Eq. ( 42), the sensitivity function H p (x, k, ω) can be written as follows

H p (x, k, ω) = M m=1 N n=1 a mn (k, ω)F mn (x), ( 43
)
where F mn (x) is given by Eq. ( 40) and can be numerically computed using 240 rectangular method. 3.17 Damping loss factor, η 0.005

Results and Discussion

A rectangular baffled panel with simply-supported boundary conditions is considered. The dimensions and material properties of the panel are listed in Table 1. The fluid density and kinematic viscosity were set to 1.225 kg/m 3 and 1.511 × 10 -5 m 2 /s, respectively.

Modeling TBL and ADF Excitations

The surface contributions of the panel to the radiated sound power under two different stochastic excitations, namely TBL and ADF are examined.

The CSD of the stochastic field can be expressed in terms of the ASD function Ψ pp (ω) and the normalized CSD function of the stochastic field φpp (k, ω) as follows [35; 36]

φ pp (k, ω) = Ψ pp (ω) φpp (k, ω). (44)
Eq. ( 44) can be used to evaluate the CSD of both the ADF and TBL excitations. A unity ASD is assumed for both excitations. The normalized CSD functions given in Appendix A were also used to evaluate the TBL and ADF excitations, respectively. For TBL excitation, it is assumed that the TBL is stationary, homogeneous and fully developed over the panel surface. Moreover, it is assumed the vibration of the panel does not alter the wall pressure field (WPF). The Mellen model described in Appendix A was used to evaluate the CSD function of the WPF [START_REF] Mellen | Wave-vector filter analysis of turbulent flow[END_REF]. The TBL parameters were estimated based on theoretical formula for a flat panel from literature and are given in Table 2 [START_REF] Karimi | A hybrid numerical approach to predict the vibrational responses of panels excited by a turbulent boundary layer[END_REF]. The convective velocity U c was approximated as follows [32; 38] U c ∼ = U ∞ (0.59 + 0.3e -0.89δ * ω/U∞ ), (45) where U ∞ is the free flow velocity and δ * is boundary layer displacement thickness.

Determination of Cut-off Wavenumbers and Wavenumber Resolutions

It has previously been reported when a panel is excited by a TBL, the effect of convected ridge can be neglected for frequencies well above the aerodynamic frequency [26; 32]. Therefore, to predict the vibroacoustic response of the panel the cut-off wavenumber can be defined based on the flexural wavenumber. This is due to the filtering effect of the structure. In this study, it was confirmed that the same criterion can be used to evaluate the NNI. One can plot the forcing function and sensitivity function to illustrate the filtering effect. To do this, Eq. ( 22) can be further written in a compact form as follows

I N v (x, ω) = 1 4π 2 ∞ HN (x, k, ω) 2 φ pp (k, ω)dk, (46)
where HN (x, k, ω) is the NNI sensitivity function given by

HN (x, k, ω) = ρ a ωk 3 a 4π 2 θ∈[0,2π] γ∈[0, π 2 ] sinγ √ cosγe ikasinγ(xcosθ+ysinθ) (47) Hv (k a sinγcosθ, k a sinγsinθ, k, ω)δγδθ
wavenumber k c = ω/U c , that is, f c = U 2 c
ρ s h/D/(2π) [START_REF] Marchetto | Experimental prediction of the vibration response of panels under a turbulent boundary layer excitation from sensitivity functions[END_REF]. For the parameters chosen here and at a flow speed of 40 m/s, f c =29 Hz. It can be seen from both figures that except at very low frequencies the spectral levels of the velocity and the sound power of the panel under the ADF excitation are significantly higher than those for the panel excited by the TBL (a unity ASD of the stochastic field was assumed for both excitations). Further, the shape and trend of the panel velocity response under the TBL excitation is very similar to that under the ADF excitation. However, a different behavior for the radiated sound power can be observed in Figure 5. The radiated sound power between resonance frequencies for the ADF excitation is relatively flat whilst the sound power at those frequencies form a curved shape in the spectra for the TBL excitation. Figure 6 shows the radiation efficiency of the panel for both the ADF and TBL excitations. The radiation efficiency of a panel is given by [START_REF] Dingguo | Sound power radiated from rectangular plates[END_REF]

σ = Π rad Aρ a c a V 2 , (48)
vertical lines in Figure 6 indicate the resonance frequencies of the panel, the mode number for each resonance frequency has also been shown ((m, n) mode means an m mode in the x-direction and an n mode in the y-direction). It can be observed from Figure 6 that at very low frequencies the radiation efficiency of the panel is independent of the excitation force, and at higher frequencies the radiation efficiency of the panel under the ADF excitation is generally higher than that of the panel excited by the TBL, particularly at non-resonance frequency, the ADF excited panel efficiently radiates sound to the acoustic domain. At resonance frequency, the radiation frequency is almost the same for both excitations. and8 for the panel under the TBL and ADF excitations, respectively. It can be observed that regardless of excitation, at each frequency (particularly at the resonance frequencies) the map of S vv is very similar to that of S pp . This is not surprising as S pp was evaluated on the surface of the panel, and the sensitivity functions of velocity and pressure have similar characteristic and are related to each other by Eq. ( 6). Figures 7 and8 show that the active normal intensity I act of the panel excited by the ADF is higher than that under the TBL excitation, this is consistent with the sound power results presented in Figure 5. Further, it can be seen that the maps of I act for both excitations are very similar and the patterns at the resonance frequencies are highly dominated by the mode shapes. It should be noted that since normal fluid particle velocity is zero over the baffle (outside the panel surface), the active normal intensity is also zero everywhere on the baffle. Therefore, plotting I act over the panel surface shows the total intensity pattern, and the total radiated sound power can be evaluated by taking the integral of I act over the panel surface. However, the NNI is not necessarily zero on the baffle. To obtain the total sound power from the NNI, its entire distribution over the infinite boundary surface has to be considered as indicated by Eq. ( 13). Hence, the whole NNI distributions are plotted over a large boundary surface at z = 0 for the selected frequencies as shown in Figures 9 and 10 As can be seen in Figure 9(c) almost the whole area of the panel under the ADF excitation is contributing to the radiated sound. At this frequency a high radiation is expected, this is consistent with the results in Figure 6 where the radiation efficiency of the panel is close to 100 % (i.e. σ = 1) at 630 Hz . From the maps of the NNI at the peaks of the radiation efficiency

Conclusions

The non-negative intensity was analytically formulated in wavenumber domain for planar structures subject to random excitations. To calculate the NNI, the CSD of the stochastic field and either the sensitivity function of pressure or normal fluid particle velocity were required. The proposed formulation can be used for both infinite planar structure and finite plate in an infinite baffle. The NNI was used to quantify the regions on a simply supported baffled panel excited by the TBL and ADF which radiate energy to the far field. Comparing maps of the ASD of the pressure and panel velocity, and active intensity with those of the NNI at different frequencies revealed that the NNI is a powerful tool to identify hot spots on the panel surface which contribute to the sound power. It was also found that the NNI distribution is dependent on the excitation type as well as on the frequency of excitation. It was shown that the more the NNI distribution is concentrated within the panel surface, the higher the radiation efficiency becomes. In other word, high radiation efficiency can be achieved if the most area of the panel contributes to the radiated sound power, and this can be identified using the NNI.

The ADF model

The normalised CSD function of the ADF in the wavenumber-frequency space is given by [START_REF] Marchetto | Vibroacoustic response of panels under diffuse acoustic field excitation from sensitivity functions and reciprocity principles[END_REF].

φADF pp (k x , k y , ω) =    2π k a k 2 a -k 2 x -k 2 y , k 2 a > k 2 x + k 2 y 0, k 2 a ≤ k 2 x + k 2 y    , (A.2)

  one in terms of pressure sensitivity function and the other one based on the sensitivity function of normal fluid particle velocity. Both formulae are dependent on the CSD function of the stochastic field. The two formulae are given by

Figure 1 :

 1 Figure 1: A baffled panel under stochastic excitation.

Figure 2 :

 2 Figure 2: Illustration of the Lyamshev reciprocity principle for a baffled panel.

Figure 3 (

 3 Figure 3(a) presents a map of the NNI sensitivity function at (x, y) = (0.4 m,0.4 m) and for k y = 0. The black dashed lines correspond to the panel flexural wavenumbers. It can be seen that the sensitivity function reaches its maximum values at wavenumbers smaller than or close to the flexural wavenumbers. However, for the wavenumbers larger than the flexural wavenumbers the magnitude of the function is still considerable, particularly

Figure 3 : 2 (

 32 Figure 3: Maps of the (a) NNI sensitivity functions HN (x, k, ω)2

5. 3 .

 3 Figures 4 and 5 respectively present the spatial average of the ASD of the panel velocity and the radiated sound power of the panel under the TBL and ADF excitations. The TBL excitation strongly excites the structure at the aerodynamic coincidence frequency, f c , which occurs when the flexural wavenumber given by k p = (ω ρ s h/D) 1/2 is equal to the convective

Figure 4 :

 4 Figure 4: Predicted mean quadratic velocity spectra for the TBL and ADF excitations (dB ref. 1 (m/s) 2 /Hz).

Figure 5 :

 5 Figure 5: Predicted acoustic power of the panel under the TBL and ADF excitations (dB ref. 1 × 10 -12 (W)).

Figure 6 :

 6 Figure 6: Radiation efficiency of the panel under the TBL and ADF excitations.

5. 4 .

 4 The NNI Calculation To identify the surface contributions of the panel to the radiated sound power under the ADF and TBL excitations, the NNI has been computed at four discrete resonance frequencies of 177 Hz, 307 Hz, 691 Hz and 924 Hz as well as at two non-resonance frequencies of 630 Hz and 700 Hz. The maps of S vv , S pp , I act and I N at the selected frequencies are presented in Figures 7

For

  Figures 7 and 8, in addition to the corner modes which effectively generate supersonic waves to the far field for both excitations at 691 Hz, there is a hot spot in the middle of the panel for the ADF excitation which radiates energy to the far field. Results in Figures 7 and 8 revealed that the NNI distribution depends on the excitation type and frequency.

  Figures 9 and 10 can be considered as an image of the excitation sources viewed by the acoustic domain. For instance, Figure9shows that at 177 Hz the size of each hot spot is around 1 m which corresponds to the half acoustic wavelength. Hence, the spatial resolution of the NNI is directly related to

(

  results are not shown here), it was confirmed that concentration of the NNI distribution within the panel surface results in high radiation efficiency of the panel under the ADF/TBL excitations. The formulation derived here can be applied to identify hot spots of a structure under stochastic excitations. Further, it can give an insight into the radiation efficiency of the structure based on the NNI distribution over the structural-acoustic boundary surface.

Figure 7 :

 7 Figure 7: Maps of S vv , S pp , I act and I N for the panel under the TBL excitation at a flow velocity of U ∞ = 40 m/s and at selected frequencies of (a) 177 Hz, (b) 307 Hz, (c) 630 Hz, (d) 691 Hz, (e) 700 Hz and (f) 924 Hz.

Figure 8 :

 8 Figure 8: Maps of S vv , S pp , I act and I N for the panel under ADF excitation at a flow velocity of U ∞ = 40 m/s and at selected frequencies of (a) 177 Hz, (b) 307 Hz, (c) 630 Hz, (d) 691 Hz, (e) 700 Hz and (f) 924 Hz.

Figure 9 :

 9 Figure 9: Comparison of the NNI between the panel under ADF excitation (left column) and under TBL excitation (right column) over a large surface at z = 0 for selected frequencies of (a) 177 Hz, (b) 307 Hz, (c) 630 Hz.

Figure 10 :

 10 Figure 10: Comparison of the NNI between the panel under ADF excitation (left column) and under TBL excitation (right column) over a large surface at z = 0 for selected frequencies of (a) 691 Hz, (b) 700 Hz and (c) 924 Hz.

Table 1 .

 1 Dimensions and material properties of the panel

	Parameter	Value
	Young's modulus, E (GPa)	70
	Poisson's ratio, ν	0.3
	Mass density, ρ s (kg/m 3 )	2700
	Length, L x (mm)	480
	Width, L y (mm)	420
	Thickness, h s (mm)

Table 2 .

 2 TBL parameters at a flow speed of 40 m/s

	Parameter

Acknowledgments

This research was supported by the Australian Government through the Australian Research Council's Discovery Early Career Project funding scheme (project DE190101412).

Appendix A: The normalized CSD function of TBL and ADF excitations

The Mellen model

The Mellen normalized wavenumber-frequency model is given by [START_REF] Mellen | Wave-vector filter analysis of turbulent flow[END_REF] φTBL

where k c = ω/U c , α x = 0.1 and α y = 0.77.


Guillaume Bergler

Carole Camarasa

Anne Ortiz-Julien

Benoît Colonna-Ceccaldi

Audrey Bloem

Differences in the volatile fingerprints and fermentation performances of non-Saccharomyces and Saccharomyces wine yeasts

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

Introduction

Experimental conditions

These data provide a wide picture of the diversity of the fermentation performances of 41 enological strains, and their ability to produce some unusual aromas.

It could be interesting to compare these production with those obtained using synthetic medium, in order to remove all the potential precursors present in the natural grape must. This will allow to assess the intrinsic ability of wine yeasts to produce unusual aromas. Furthermore, the characterization of the other fermentative aromas (higher alcohols, esters and fatty acids) will provide an overview of the volatile profile of yeasts after fermentation. This knowledge will be further exploited to control the aroma fingerprint of wines.



  High production of terpen (except T2) by L30 High production of T1 and T8 by C. stellata but limited formation of other compounds Important diversity between strains for BD, with the highest production by L11 Significant

*

  Differentiation Differences among Saccharomyces strains mostly due to BD and LAC NB : Results after fermentation on Chardonnay  Impact of both the strain and the medium composition on kinetic variables  Significant interaction strain / medium, with a more pronounced strain effect on Chardonnay Study of the Central Carbon Metabolism (CCM)  More strains able to complete the fermentation on Ugni Blanc  Important production of glycerol on both cultivars by C. stellata and S. bacillaris, -31 rd May 2018 -ZARAGOZA C, D, E, F : Violin plots coupled to box-plots A B Value divided by 10 ; ** Value divided by 50  Similarity of production profiles of CCM metabolites between Chardonnay and Ugni


Hervé Morel
email: herve.morel@insa-lyon.fr

Dominique Bergogne

Dominique Planson

Bruno Allard

Régis Meuret
email: regis.meuret@hispano-suiza-sa.com

New Applications in Power Electronics Based on SiC Power Devices

Keywords: SiC devices, JFET, Power Electronics, High Temperature, High Voltage

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

Introduction

Nowadays some silicon carbide, SiC, power devices are commercialized like Schottky diodes. Moreover power JFETs are available as industrial samples. The two latter devices enable to build most of power electronic converters. One main issue is for which applications silicon carbide devices can advantageously replace classical silicon power semiconductor devices ?

The first class of applications for SiC power devices are applications for which the use of silicon power devices are not possible. The second class of applications for SiC devices correspond to systems where mass reduction is a big issue, i.e. for embedded systems like aeronautic systems.

Fig. 1 illustrates the first case of applications. The figure yields the thermal breakdown limit versus the breakdown voltage of an infinite plane junction in various semiconductor. These are theoretical limits. Spots correspond to industrial silicon power devices. It is clear that there are very near from the theoretical limit. The particular case of the CoolMOS is because it is not a junction but a superjunction. Concerning high voltage devices the thermal limit is too a major constraint for silicon devices. So, clearly, applications becoming possible because of new SiC power devices are high temperature and/or high voltage applications. However, most of recent studies have shown that the use of SiC devices does not consist in a simple replacement of silicon devices by silicon carbide devices. Indeed a new design of the new power electronic systems taking advantage of the properties of the new SiC power devices are required.

High Temperature Applications

Several recent studies address high temperature applications. For instance the More Electric Aircraft, MEA, demands high temperature power electronics for electrification of actuators mounted on the aircraft engine. In such applications, the ambient temperature, i.e. the engine temperature, may reach 200°C. Due to self heating of the power semiconductor devices, the "junction" temperature may reach 250°C even 300°C. Silicon power device are not applicable because the voltage rate is about 1200V and thermal breakdown is reached as shown in Fig. 1.

Several studies have shown the operation of an inverter cooled at 300°C. The industrial goal is 200°C of ambient temperature.

Remaining questions are,

• the reliability of the system, i.e. mainly at the packaging level,

• the converter structure (diode-less or not) to optimize efficiency and reliability,

• the driver structure, including protection operation (taking advantage of the high limit of SiC thermal breakdown, short-circuit and avalanche robustness of power device are very good).

• high temperature passive devices Several studies in the world are under development with different similar objectives. Most of them are based on SiC JFET because there are almost available at the industrial level. However research devices like MOSFET, and high temperature Schottky diodes, or other wide-band semiconductor like GaN may contribute to medium or long term alternatives to the current studies.

High Voltage Applications

Such applications are medium terms applications because high voltage devices are not ready for industrial applications. However 10 kV demonstrators have been developed, 15kV demonstrators are under development even 30kV and further may be soon obtained.

The interest of such devices are enormous because they enable to • replace heavy 50 Hz high voltage electrical transformers by high voltage power electronics.

• replace AC-current distribution networks by DC-current distribution networks and eliminating numerous associated problems like instability (wind-power farm ...) even blackout. That enables an easier integration of renewable energy sources (Photovoltaic, Wing power ...) and remove the need of 50Hz electric transformers.

• improve the high voltage system protection with very fast active protection systems

The main issues of these new applications are,

• availability of high bipolar diodes and switches (BJT, Thyristors, ...)

• electrical insulation of the high voltage converters (packaging, passivation, ...)

• thermal cooling of the high voltage converters • highly insulated driver for high voltage converters even multi-level converters.

Power Electronics in the Embedded Systems

The mass reduction and medium high temperature are applications for SiC devices, but in such a case the interest of SiC devices have to be proved even from the economic point of view, because silicon alternatives exist. However, the use of high junction temperature to reduce the mass of the thermal cooling system is a way to be explored with the reliability of such new systems.

Fig. 1 :

 1 Fig. 1: Temperature limit versus voltage limit of semiconductor devices


Tarek Ben Salah

Damien Risaletto

Christophe Raynaud
email: christophe.raynaud@insa-lyon.fr

Kamel Besbes

Dominique Bergogne

Dominique Planson

Hervé Morel

Determination of ambipolar lifetime and epilayer thickness of 5 kV SiC bipolar devices by switching transient studies

Introduction

A new experimental set-up is developed and validated to characterize high voltage diodes in switch transient regime. Parameters extracted from DMTVCA (Depletion Mode Transient Voltage Current Analysis) and OCVD (Open Circuit Voltage Decay) techniques, that are the ambipolar lifetime, the epilayer thickness, the doping thickness and the device area, are validated in a buck converter on a resistive load. The experimental set-up enable to measure nice current and voltage transient characteristics, i.e. without noise and high parasitic wiring influence. Experimental results are compared to device simulations and a good agreement is found.

Description of the samples

High voltage 4H-SiC bipolar diodes have been realized by AMPERE laboratory. A complete description and static characterization in forward and reverse bias is given in [1]. N-type 4H-SiC epilayer from Cree Research of 40 µm ± 25% thickness and a doping level of 1.1×10 15 cm -3 (± 100%) grown on a N+-type substrate has been used for fabrication of high voltage bipolar diodes. Emitter has been realized by implantation of Al and results in a P+ region doped at 4×10 19 cm -3 on ~0.45 µm. Junction Termination Extension (JTE) have been realized by implantation of Al (2×10 17 cm -3 on ~0.45 µm). Different diameters of the emitter have been realized. Biggest diodes (∅ 1 mm) show a forward current ~300 mA at 5 V, but low breakdown voltage (~700 V) in regard to the theoretical value (~5-6 kV). For simulation, the device simulator DESSIS™ enables mixed-mode simulation and is used to simulate the diode recovery within the switching-cell. .

Experimental procedure and results

The main idea of our extraction procedure is a decoupling between high level injection and space charge region phenomena.

DMTVCA technique

The DMTVCA technique [2] is based on the unclamped inductive switching circuit. It is used to study the blocking of the diodes from an initial state with a null current. DMTVCA technique is an alternative to classical C-V measurements, but it does not have voltage limitations. Parameters extracted from these figures are the effective area S = 1.25 mm², the doping level of the epilayer N D = 7.65×10 14 cm -3 , and the width of the epilayer W = 42 µm.

OCVD technique

A second more classical technique, the OCVD technique [3] is used to extract the ambipolar lifetime τ. This technique consists in switching the diode from a forward state to zero volt and to measure the subsequent transient decrease of the current. The decrease of the voltage is related to τ. To achieve a higher accuracy, experimental characteristics are compared and fitted with calculated characteristics using finite-element simulation software (DESSIS).

The estimation of the ambipolar lifetime τ is based on an optimization process where experimental and simulation data are compared. From these measurements, τ is found in the range 41 -46 ns.

Model validation in switching cell

The validity of these 4 parameters, identified from DMTVCA and OCVD techniques, is highly dependent on the agreement between simulation and experimental data. This agreement can be further validated once the diode is measured in the high injection level. The validation process needs a comparison between simulation and measurements in a circuit closed to real working conditions, i.e. a converter.

Conclusion

For the first time, a new OCVD technique has been applied to SiC devices. A good correlation between measurements and simulations has been verified. The final paper will describe the complete parameter extraction process and will detail the switching cell.

Figure 1 :

 1 Figure 1: DMTVCA current (left) and voltage (right) measurements for the SiC bipolar diode.

Figure 2 :

 2 Figure 2: current (left) and voltage (right) measurements for the SiC bipolar diode in the OCVD circuit.

Figure 3 :

 3 Figure 3: Comparison between simulated and measured current (left) and voltage (right) for the SiC bipolar diode in the resistive switching cell.