El Houssaine Quenjel

Patrick Perre
email: patrick.perre@centralesupelec.fr

Efficient prediction of the thermal conductivity of wood from its microscopic morphology

Keywords: thermal conductivity, homogenization, REV, image processing, tomography

In this work, we combined the homogenization and finite volume methods to predict the solid fraction and the effective thermal conductivity from 3D real morphologies of wood, namely spruce and poplar. High resolution scans performed by a nano-tomograph, together with image processing are two steps of great importance to obtain the digital representation of the real morphology suitable for computation. These tools allow the generation of the 3D mesh of the thresholded sample. The stationary diffusion model is directly considered to gain in performance. Numerical results revealed that several minutes of CPU time are enough to predict the values of the thermal conductivity on the representative volumes. Compared to some of our recent works, the present methodology is not only efficient, but also more accurate.

Introduction

There is an increasing interest in using insulation materials for building constructions to optimize the energy consumption [START_REF]Wood handbook -Wood as an engineering material[END_REF]. Wood-based materials exhibit good features for this purpose. To understand the wood characteristics and its role in the improvement of the energy efficiency, experimental tests are often conducted. Taking advantage of tomography technology together with high performance computing, numerical prediction is also an appropriate way to estimate macroscopic properties, notably the effective conductivities [START_REF] Delgado | A critical review of hygrothermal models used in porous building materials[END_REF][START_REF] Van Belleghem | Benchmark experiments for moisture transfer modelling in air and porous materials[END_REF][START_REF] Woloszyn | Tools for performance simulation of heat, air and moisture conditions of whole buildings[END_REF].

Synchrotron facilities enable a suitable representation of the wood anatomy [START_REF] Brodersen | Visualizing wood anatomy in three dimensions with high-resolution X-ray microtomography (μCT)-a review[END_REF]. However, lab-equipment tomographs are nowadays able to reproduce the digital interpretation of the material structure. Scans of high resolution provide a good description of the sample with no degradation [START_REF] Badel | Predicting oak wood properties using X-ray inspection: representation, homogenisation and localisation. Part I: Digital X-ray imaging and representation by finite elements[END_REF][START_REF] Baensch | Damage evolution in wood: synchrotron radiation micro-computed tomography (SRμCT) as a complementary tool for interpreting acoustic emission (AE) behavior[END_REF][START_REF] Bucur | Nondestructive characterization and imaging of wood[END_REF][START_REF] Lux | 3D characterization of wood based fibrous materials: an application[END_REF][START_REF] Standfest | Determination of the microstructure of an adhesive-bonded medium density fiberboard (MDF) using 3D sub-micrometer computer tomography[END_REF][START_REF] Zauner | Failure and failure mechanisms of wood during longitudinal compression monitored by synchrotron micro-computed tomography[END_REF]. From the 3D scans, image processing is required for thresholding and to extract objective information such as the distribution and the partition of the phases, quantifying density, or the orientation of the cell walls.

Then, the morphology is used to proceed to homogenization, using suitable computational methods, for estimating the macroscopic properties of wood. For instance, finite elements were investigated in [START_REF] Hunt | Two-dimensional finite heat transfer model of softwood. Part I. Effective thermal conductivity[END_REF][START_REF] Van Belleghem | Benchmark experiments for moisture transfer modelling in air and porous materials[END_REF][START_REF] Sova | Effective thermal conductivity models applied to wood briquettes[END_REF] for the computation of the thermal conductivity. Finite volume approximation was carried out in [START_REF] Badel | Predicting oak wood properties using X-ray inspection: representation, homogenisation and localisation. Part I: Digital X-ray imaging and representation by finite elements[END_REF][START_REF] Perré | Determination of the material property variations across the growth ring of softwood for use in a heterogeneous drying model. part 2. use of homogenisation to predict bound liquid diffusivity and thermal conductivity[END_REF] to predict liquid diffusivity and thermal conductivity. The work established in [START_REF] Louërat | Heat and moisture diffusion in spruce and wood panels computed from 3D morphologies using the lattice boltzmann method[END_REF] was devoted to the lattice Boltzmann meshless scheme enabling the simulation of heat and moisture diffusion in spruce and wood panels. In our recent paper [START_REF] Quenjel | Computation of the effective thermal conductivity from 3d real morphologies of wood[END_REF] we developed an explicit finite volume approach allowing the prediction of the macroscopic thermal conductivity of wood. Besides the REV size constraint, working at the pore scale requires tiny time steps, which makes the convergence towards the steady state too slow. To mitigate this issue, a volume reduction strategy was also proposed following the longitudinal direction. The idea yields convenient results. However. It is based on a stronger assumption adapted to small set of materials whose phases are in parallel in the longitudinal direction.

The core point of the present contribution is to perform computations using the stationary model instead of the transient one. Its mains assets are summarized as follows : (i) no reduction in the volume size is required; (ii) taking into account larger REVs is now possible (iii) spectacular gains in terms of the computational cost are reported; (iv) accuracy is ensured. Consequently, this methodology provides a better comprise to predict the macroscopic thermal conductivity efficiently and accurately.

Materials and methods

Sample preparation and tomography

The wood samples come from well-identified boards available in our laboratory. They were dried outside before storing them in air-dry conditions. The size of the spruce and poplar species was adjusted to fit small cylindrical shapes of the order of millimeters. This was done with the help of a wood-turning machine. Such a cut is mandatory to achieve the high resolution. The diameter and the height of the cylinders were 5.64 mm and 3.11 mm, respectively, for spruce and poplar. All these samples are scanned at a micrometer resolution thanks to a lab X-ray nanotomograph (UltraTom by RX-solutions). The obtained 3D image of the sample is given by a heavy stack of 2D images following the longitudinal direction. We denote by (R), (T) and (L) the natural material directions, referred to as radial, tangential and longitudinal. The scanned samples are illustrated in Figures 12. Precisely, the resolution is 3 µm for spruce and poplar.

Morphology segmentation

The image processing procedure was implemented in Python. High-resolution scans lead to large sets of data. We follow a practical idea to extract the representative elementary volume (REV) within any complex geometry. The 3D morphology is a series of 2D images. The REV is chosen by specifying its center and size. One then crops the original 3D image and applies the global segmentation methodology thanks to Otsu's automatic thresholding technique [START_REF] Otsu | A threshold selection method from gray-level histograms[END_REF]. Figures 34. exhibit the results of the binary morphologies. Two phases are considered. The assigned code 255 (0) stands for the gaseous (solid) phase. The 3D mesh of the morphology is also generated at this stage. It can be seen that the cell walls are more aligned in the (R,T) section for spruce than poplar. This would have an impact on the macroscopic property.

A word on the homogenization

Homogenization is a mathematical method, which consists of predicting macroscopic properties from the microscopic structure of materials [START_REF] Hornung | Miscible displacement[END_REF][START_REF] Suquet | Elements of homogenization for inelastic solid mechanics[END_REF]. In this work, we focus on a stationary diffusion equation to derive the effective thermal conductivity. Let Ω be the 3D parallelepiped Ω= [ x 0 ,x m ] × [ y 0 ,y n ] × [ z 0 ,z p ] extracted from the original morphology. Consider the problem -div ( α ∇ u)=0, [START_REF]Wood handbook -Wood as an engineering material[END_REF] where u accounts for temperature, and the parameter α=λ 0 / ( ρc p ) is the local thermal coefficient. The number λ 0 indicates the local thermal conductivity, ρ refers to density, and c p informs the heat capacity. The boundary condition is given by u Dir =1 on the face x=x 0 , and u Dir =0 on the opposite face x=x m . The remaining boundary is adiabatic. In the numerical section, the factor ρc p can be set to unity. The local thermal coefficient λ 0 is fixed to 0.5 (W/m.K) in the case of the solid phase and to 0.023 (W/m.K) for the gaseous phase. The problem ( 1) is solved owing to the finite volume method on the mesh of Ω. To deduce λ t of the REV material, one evaluates the ratio of the x-flux average ⟨ F x ⟩ to the imposed gradient δu / L x ; that is λ t = ⟨ F x ⟩ L x /δu .

2 | 6

The finite volume approximation

Let N Ω be the total number of degrees of freedom. The mesh Γ= { K i ,i= 1,… ,N Ω } is a partition of Ω with control volumes. Let x i denote the mass center of K i . The set of faces of K i is denoted by Ε i . Each interface σ ij =K i K j is shared by two volumes K i and K j . The latter volume is identified to an edge if the interface is located on the boundary. The unit normal to σ ij from K i to K j is expressed by n ij . The two-point approximation of the flux assumes the orthogonality condition on the mesh. This means that the line connecting x i to x j is orthogonal to σ ij . This condition is fulfilled on hexahedral meshes in 3D generated from the morphology.

Integrating the main model (1) on K i and using the divergence theorem yields

-∫ K i div (α ( x )∇ u ( x )) dx=-∑ σ ij ∈ Ε i ∫ σ ij α (s ) ∇u (s ) ⋅ n ij ds= 0 .
The physical quantities are supposed constants by cells. Then, the flux is approximated in only one direction provided by the unit normal vector to the face because the other ones are orthogonal to n ij . As a consequence, the numerical scheme reads

∑ σ ij ∈ Ε i τ ij ( u i -u j ) =0, for all i= 1, …,N Ω , u j =u j Dir , if σ ij ⊂ ∂ Ω ,
where τ ij refers to the transmissibility coefficient

τ ij =α ij | σ ij | / ‖ x i -x j ‖ .
The coefficient α ij is the harmonic average of α i =α ( x i ) and α j =α ( x j ) .

Computational results

The objective of this section is to show the ability and the efficiency of the proposed finite volume scheme to predict the macroscopic thermal conductivity. The numerical scheme is implemented in Fortran. The linear solver is based on the conjugate gradient method.

Solid fraction

The data of the scanned samples are extremely big. Their size order are 1800×1900×1150 voxels, which makes the image processing as well as the macroscopic property computations inaccessible. Then, we are led to deal with embedded volumes. For this purpose, form the initial image, we extract the REV under the form of hexahedral subset. Its center is denoted by (xc,,yc,zc). The length, width and height are respectively given by In other words, the REV writes

REV = [ x c -Δx,x c +Δx ] × [ y c -Δy,y c +Δy ] × [ z c -Δz,z c +Δz ] .
The center of the REV is fixed to (1155, 870, 555) in the case of spruce while the center of the REV is set to (940, 940, 555) for poplar. To determine the considered dimensions, we use an increasing uniform sequence of the REV sizes containing six elements. For l=1, … ,6 we consider Δx l =Δy l =16 l, Δz l =2Δx l -1. 1 records the results on the solid fraction of the two samples. It is observed that the first volumes are not representative in the case of poplar because they are too small. Also, the solid fraction occupies 32% within the whole space for spruce and 38% for poplar. 2 depicts the results on the macroscopic thermal conductivity for spruce and poplar. They are listed following the orthotropic axes of the material and the REV size. Compared to the work [START_REF] Quenjel | Computation of the effective thermal conductivity from 3d real morphologies of wood[END_REF], the values are quite similar, especially when the volume size is increasing. Indeed, we found that the predictive values are λ R t =0.11W/m.K, λ T t =0.11W/m.K and λ L t =0.16 W/m.K for spruce. In the case of poplar, λ R t =0.14W/m.K, λ T t =0.12 W/m.K and λ L t =0.2W/m.K. This confirm that the present approach preserves the property ranges. On the other hand, we checked that the obtained values are bounded in their physical ranges [START_REF] Siau | Transport processes in wood[END_REF]. The larger (lower) bound corresponds to phases placed in parallel (series). It is also possible to dissociate the effect of the solid fraction from the thermal conductivity by introducing mixtures laws, see [START_REF] Quenjel | Computation of the effective thermal conductivity from 3d real morphologies of wood[END_REF] for more details.

Convergence of the thermal conductivity

Performance of the numerical method

We are interested in the efficiency of the proposed numerical method and its accuracy to predict the macroscopic thermal conductivity. This can be quantified by the performance of the solver. In Table 3, for each REV size, we display the required CPU time in minutes as well as the maximum of the errors, over the orthotropic directions, committed in the computation of λ t . The latter was considered as the stopping criterion fixed to 2 × 10 -2 in the previous works [START_REF] Louërat | Heat and moisture diffusion in spruce and wood panels computed from 3D morphologies using the lattice boltzmann method[END_REF][START_REF] Quenjel | Computation of the effective thermal conductivity from 3d real morphologies of wood[END_REF]. More importantly, it is worth underling that the last test on the volume REV6 required exorbitant CPU time following the strategy of [START_REF] Quenjel | Computation of the effective thermal conductivity from 3d real morphologies of wood[END_REF] whereas only few minutes are needed now to get same outcomes. The results approve that the current methodology is more accurate and very cheap. This renders it a good candidate to handle large volumes at different positions.

Conclusion

In this paper we made use of the homogenization and finite volume methods to compute the thermal conductivity of spruce and poplar species in their orthotropic directions. The samples are scanned with a lab Xray nanotomograph to obtain the 3D morphology. The latter was treated using image processing tools to retrieve the distribution of the phases and to generate the mesh. These data become inputs of the computational algorithm. The stationary diffusion problem is solved directly without passing through the transient regime. The goal is to save the computational cost and to enable large-size mesh to be treated. Numerical evidences were reported to address the effect of the representative elementary volume (REV) on the macroscopic property.

In future contributions, we outlook to provide a deep compassion between the predictive approach and the experimental measurements for several wood species including more than two phases. This is for instance the case of materials including fibers and binders. Because the wood is highly anisotropic, another interesting avenue is to study the impact of the cell walls orientation on the macroscopic property.

Acknowledgments: this study was carried out in the Centre Européen de Biotechnologie et de Bioéconomie (CEBB), supported by the Région Grand Est, Département de la Marne, Greater Reims and the European Union. In particular, the authors would like to thank the Département de la Marne, Greater Reims, Région Grand Est and the European Union along with the European Regional Development Fund (ERDF Champagne Ardenne 2014-2020) for their financial support of the Chair of Biotechnology of CentraleSupélec.

Nomenclature

Figure 2 :

 2 Figure 2: scanned sample of poplar. Figure 1: scanned sample of spruce.

Figure 1 :

 1 Figure 2: scanned sample of poplar. Figure 1: scanned sample of spruce.

Figure 3 :

 3 Figure 3: thresholded volume of spruce. 101×101×255 voxels.Figure 4: thresholded volume of poplar. 128×128×255 voxels.

Figure 4 :

 4 Figure 3: thresholded volume of spruce. 101×101×255 voxels.Figure 4: thresholded volume of poplar. 128×128×255 voxels.

Table 1 :

 1 solid fraction in terms of the REV size.

	REV number	Size (voxels)	ϵ s Spruce	ϵ s Poplar
	REV1	32 × 32 × 63	0.3272	0.4858
	REV2	64 × 64 × 127	0.3049	0.4278
	REV3	96 × 96 × 191	0.3275	0.3818
	REV4	128 × 128 × 255	0.3139	0.3854
	REV5	160 × 160 × 319	0.3236	0.3905
	REV6	192 × 192 × 383	0.3183	0.3889
	Table

Table 2 :

 2 Thermal conductivity for each species in terms of the REV size.

	Sample	λ	t (W/m.K)	REV1	REV2	REV3	REV4	REV5	REV6
	Spruce		t λ R λ T t	0.1160 0.1194	0.1115 0.1045	0.1131 0.1155	0.1133 0.1119	0.1124 0.1139	0.1109 0.1122
			λ L t	0.1706	0.1641	0.1714	0.1691	0.1705	0.1680
	Poplar		t λ R λ T t	0.1767 0.1676	0.1580 0.1333	0.1330 0.1226	0.1402 0.1158	0.1415 0.1212	0.1425 0.1186
			λ L t	0.2487	0.2216	0.1990	0.1999	0.2024	0.2020
	Table

Table 3 :

 3 CPU time in minutes and the maximum value of residuals.

		Spruce	Poplar
	REV number CPU time (mins)	Max residuals CPU time (mins) Max residuals
	REV1	3.37 × 10 -2	4.42 × 10 -4	1.74 × 10 -2	4.56 × 10 -4
	REV2	1.71 × 10 -1	1.10 × 10 -3	1.89 × 10 -1	1.19 × 10 -3
	REV3	7.50 × 10 -1	3.59 × 10 -4	8.24 × 10 -1	4.15 × 10 -5
	REV4	2.23	2.78 × 10 -4	2.50	3.02 × 10 -4
	REV5	5.44	5.29 × 10 -4	5.48	1.23 × 10 -4
	REV6	10.21	1.78 × 10 -5	10.55	7.33 × 10 -5


Romain Boulandet

Mark Allado

Cédric Pinhède

Emmanuel Friot

Renaud Côte

Philippe Herzog

Simulation of a hybrid (passive/active) acoustic measurement room

Keywords: semi-anechoic chamber, active control of scattered sound field, acoustic testing

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Introduction

An anechoic chamber is an experimental room whose walls absorb sound waves, reproducing free field conditions ideal for acoustic testing. It is the ideal place to characterize microphones and loudspeakers or low background noise emissions because the room does not cause reflections that could disrupt measurements. Basically, the acoustic energy emitted by the sound source under test is transmitted through the air in the form of pressure waves. As the waves travel through the absorbing material, they cause mechanical losses via the conversion of part of the sound energy into heat, resulting in acoustic attenuation. A wedge-shaped geometry further allows for a gradual change in the acoustic impedance of the transmission media, causing sound waves to be more effectively absorbed by the material. Current anechoic chambers typically provide accurate low-frequency acoustic measurements down to 70 -100 Hz depending on wedge length and material characteristics. Below this cut-off frequency, however, these rooms amplify sound excessively when subjected to dynamic excitation having a frequency close to one of its so-called natural frequencies [START_REF] Kuttruff | Room acoustics[END_REF][START_REF] Morse | Theoretical acoustics[END_REF]. Measuring the sound level of a source would therefore be distorted at very low frequencies. This phenomenon of resonance is due to the presence of a constructive interference between incident and reflected waves. This creates standing waves, i.e. waves that bounce back and forth between the room walls and whose wavelengthor a multiple thereofcoincides with the room's dimensions. The alternative is to use active technology to minimize the pressure field scattered by the walls. The strategy is based on the principle that a unique linear operator maps the total acoustic field close to the walls to the field scattered by these walls, whatever the direct acoustic field [3] [4] [5]. This "scattering operator" can be first identified from off-line measurements involving a known direct pressure field and then used to calculate the scattered pressure field generated by an unknown direct field. A set of secondary sources distributed around the room is finally used to minimize this scattered sound field, as for an "ordinary" total sound field picked up by microphones.

In this paper, we present the implementation of a digital twin used to optimize the acoustic performance of such a hybrid measurement room by understanding its overall behavior, including its sensitivity to tuning and design parameters. We describe the modeling and calculation steps involved in cancelling the scattered pressure field, and present preliminary results showing the expected performance.

Modeling of the Measurement Room

This section presents the modeling of the active control strategy in a lightly damped room. Figure 9 shows a view of the planned hybrid (passive/active) measurement room. A set of secondary sources (loudspeakers placed close to the walls) is controlled from the pressure field captured by an array of microphones (in red) delimiting a volume where we seek to minimize the contribution of the pressure field diffracted by the room. Additional microphones (in blue) are used to observe the effectiveness of the control in relation to the source to be characterized.

Analytical modeling of a rectangular room

For a monopole point source and considering lightly absorbing walls, there is an analytical solution in three dimensions for the pressure at any point in the room. The pressure 𝑝(𝒓) can be expressed in terms of the Green's function 𝐺(𝒓, 𝒓 0 ), which is constructed from undamped room mode shapes evaluated at the receiver 𝒓 and source 𝒓 0 positions and a frequency-dependent damping term 𝜉 𝑛 . The expressions are:

𝑝(𝑟) = 𝑗𝜌 0 𝜔𝑞 0 𝐺(𝒓, 𝒓 0 ) (1)
where 𝜌 0 is the density of air, 𝜔 is the driving angular frequency, and,

𝐺(𝒓, 𝒓 0 ) = ∑ 𝜓 𝑛 𝑡 (𝒓)ψ 𝑛 (𝒓 0 ) (𝑘 𝑛 2 -𝑘 2 + 2𝑗ξ 𝑛 𝑘 𝑛 𝑘) ∫ ψ 𝑛 2 d𝑉
𝑉 𝑛 [START_REF] Morse | Theoretical acoustics[END_REF] where 𝜓 𝑛 are the eigenfunctions representing the mode shapes as cosine functions, 𝑓 𝑛 = 𝑐𝑘 𝑛 2𝜋 ⁄ are the corresponding eigenfrequencies, and 𝑉 is the room volume. The Green's function represents a triple summation over modes in the three orthogonal cartesian directions, with indices 𝑛 representing the different modes. More details on the analytical description can be found in [START_REF] Morse | Theoretical acoustics[END_REF].

Scattered pressure field control strategy

Let 𝒑𝑠𝑐𝑎 be the pressure field scattered by the room at the minimization points when the primary source is operating alone. The vector 𝒖 of control signals driving the secondary sources is determined by minimizing at each frequency:

𝐽 = ‖𝒑𝑠𝑐𝑎 -𝑪𝒖 ‖ 2 (3)
where 𝑪 is the system transfer matrix between the control sources and the resulting pressure at the minimization points. More details on analytical modelling can be found in [START_REF] Pinhède | Towards an active semi-anechoic room: simulations and first measurements[END_REF].

Determining the scattered pressure field

In this section, we present the implementation of the control described above in a room as a twostage process. The first step is to determine off-line all the transfer functions between the secondary sources located in the measurement zone and the minimization points distributed around it. Next, the total pressure field 𝒑 𝑡𝑜𝑡 is calculated in the same way but considering now the rectangular room, as shown in Figure 3. Then, the scattered pressure field in Eq. ( 3) is derived as

𝒑𝑠𝑐𝑎 = 𝒑𝑡𝑜𝑡 -𝒑𝑑𝑖𝑟 (4)
Successive activation of the sources and saving of results is done from a MATLAB code using LiveLink™ for MATLAB®.

Determining the system transfer matrix

The second step is to compute the system transfer matrix 𝑪 in Eq. ( 3). The 54 secondary sources shown in Figure 4 are activated one after the other as explained above. The normal acceleration is specified in the model as

𝑎 = 𝑗𝜔𝑞 0 𝑆 ⁄ ( 5
)
where 𝑆 = 0.0133 m 2 is the area of the pistons. The resulting pressure field is evaluated at the 62 minimization points (see

Simulation Results

Simulation parameters

In what follows, we consider a rectangular room measuring 4.5 × 3.65 × 3 m 3 . In the simulation, we used a constant specific acoustic impedance of 8 • 10 4 Pa.s/m for the walls (corresponding to an absorption coefficient of approximately 2% with a purely real impedance) and a sound hard boundary the floor. The flow rate imposed by the test source is 10 -4 m 3 /s. The simulations are run using the Pressure Acoustics interface available in the Acoustics module, in the frequency range 20 Hz to 200 Hz in steps of 1 Hz. The mesh is controlled by the physics of the model, considering the maximum wavelength of interest. The computation time required to solve the simulation, which includes 32'000 domain elements, is around 1500 s on a desktop computer for 181 frequency steps.

Room frequency response

The plot in Figure 5 shows the results obtained for the direct response, the total response in the room with control on/off, and the contribution of the scattered pressure field. This is the frequency response between the monopole point source and the observation point indicated by red and yellow markers in Figure 3, respectively. As can be seen in Figure 5, the room modal response with control off shows peaks and dips corresponding to the pressure anti-nodes and pressure nodes at the observation point, respectively. Note that the total pressure field 𝒑𝑡𝑜𝑡 (off) is mainly due to the scattered pressure field 𝒑𝑠𝑐𝑎. With control on, on the other hand, the total pressure field 𝒑𝑡𝑜𝑡 (on) tends towards the direct pressure field response of a monopole point source which would be placed at the same distance from a sound hard floor. This significantly attenuates the pressure field scattered by the room. However, this is no longer the case when the frequency exceeds around 180 Hz. This is linked to number of sources and microphones, whose spacing becomes greater than the wavelength.

Total pressure field with control off

The plot in Figure 6 shows the pressure distribution in the room with control off at three different frequencies. These correspond to the first mode in the x-and y-directions and the first tangential mode in the xz-plan, respectively, and align well with the 1 st , 2 nd and 5 th peaks on the plot in Figure 5. Pressure antinodes are shown in red and pressure nodes in blue/green. Although the calculated frequency step is relatively coarse, we find the expected modal shapes for room modes (100), ( 010) and (101), respectively.

Total pressure field with control on

The graphs shown in Figure 7 illustrate the pressure distribution when the control is switched on. It can be seen that the pressure distribution in the room no longer follows the structure of the uncontrolled room modes (Figure 6). We can no longer distinguish either the pressure nodes or antinodes. On the other hand, we can see how the secondary sources work to minimize the pressure field diffracted by the room as a function of their positions relative to the test source. Figure 8 shows the cross-sectional surfaces in the 𝑦𝑧plane of the test source, the position of which is indicated by a red spot near the center of the slice.

As can be seen in Figure 8, the total sound pressure tends towards the free field response of the test source in a large volume around it. Near walls, on the other hand, higher sound pressure levels are due to the radiation of secondary sources to minimize the scattered pressure field. The results also show how the control effort applied to the secondary sources depends on the shape of the room modes to be controlled.

Characterization of the demo room

In what follows, we present results obtained in the demo room currently under construction at LMA (see Figure 9). The room is rectangular and 5.33×4.22×2.73 m3 in size. The walls and ceiling are made of double layered plasterboard (2×13 mm) and the floor is painted concrete. The access door is made of solid wood and the frame is fitted with seals. As can be seen in Figure 9, the loudspeakers and microphones of the active control system are positioned using a truss structure, which is also intended to support the panels of absorbent materials. Excitation is provided by a test source consisting of two push-pull loudspeakers (not shown in Figure 9) placed on the floor in one corner, with the measurement microphone in the opposite corner. Figure 10 shows the frequency response of the source flow rate measured in the room shown in Figure 9, defined using an interpolation function in the model. More details on the sound source can be found in [START_REF] Pinhède | Design and measurement of a reference source at lower frequencies[END_REF]. Figure 11 shows the computed frequency response in a corner of the room in terms of sound pressure level (SPL), compared with experimental measurement in the demo room described above.

In the model, the wall impedance is specified using a frequency-dependent absorption coefficient obtained from measurements in the room shown in Figure 9. Furthermore, excitation is applied using a dual monopole point source whose output depends on frequency and is derived from measurements taken in the real room. Simulation is performed over the frequency range 20 Hz to 140 Hz, in steps of 0.125 Hz. As can be seen from Figure 11, there is a good agreement between the experimental and numerical results. As expected, the room modes are far enough apart that the bandwidths of the modes do not overlap. The room modes are more damped compared with the response shown in Figure 5, which considers walls with lower damping. The discrepancies between measured and calculated data can partly be attributed to the presence of elements not considered in the model (truss structure for example), as well as to approximations in the positioning of the measurement and excitation points. These preliminary numerical results, recalibrated from experimental data, provide a more accurate picture of the actual response of an actively controlled room.

Conclusion

In this paper, we present the implementation of a digital twin of a hybrid (passive/active) acoustic measurement room. The approach of combining a COMSOL Multiphysics model using the Pressure Acoustics, Frequency Domain interface and LiveLink™ for MATLAB® to run simulations is proving effective in obtaining relevant results, partly derived from recalibration from experimental data. This gives us the room eigenmodes, adjusted for frequency and damping. To do this, we used the data measured in the real room under construction to adjust the wall impedance conditions and the flow rate of the test source in the model. This preliminary step provides a solid basis for studying the control strategy in a relatively well-controlled environment. For example, it becomes possible to study the influence of the placement and directional characteristics of the test source or the number and arrangement of secondary sources on the overall performance of the active control. The model allows, among other things, to study the effect of a regular or irregular distribution of secondary sources and/or minimization points, to avoid problems of singularities in the control matrices.

Having the digital twin of the actively controlled room makes it possible to understand its overall behavior, including its sensitivity to tuning and design parameters. This allows control configurations to be quickly tested before considering their practical implementation in the real room. The next steps will be to include in the model the layer of absorbing material around the perimeter of the room (walls and ceiling) and to replace the simple piston geometry by a more realistic loudspeaker driver using a lumped model represented by an Electric Circuit physics. This would also make it possible to assess the power consumption required by the active control system with a view to optimizing it. This information provides a useful resource for improving the design and tuning parameters of tomorrow's active anechoic chamber, and therefore the technology they help to power.

Figure 1 .

 1 Figure 1. View of the planned hybrid (passive/active) acoustic measurement room.

Figure 2 et 3

 23 Figure show the distribution of 72 monopole point sources and 62 minimization points allowing all transfer functions to be identified.

Figure 2 .

 2 Figure 2. Geometry for the calculation of the direct pressure field where the 72 monopole point sources are regularly distributed in the blue box and the 62 minimization points are distributed all around.

Figure 3 .

 3 Figure 3. Geometry for the calculation of the total pressure field in the room where the 72 monopole point sources are distributed throughout the volume to be controlled (blue box) and the 62 minimization points are distributed all around.

Figure 4 )

 4 , giving the transfer matrix 𝑪 with dimensions 62 × 54 × 181 over the frequency range 20 Hz to 200 Hz in steps of 1 Hz. The command 𝒖 driving the 54 control sources is finally obtained by solving Eq. (3).

Figure 4 .

 4 Figure 4. Geometry to determine the system transfer matrix where the 54 secondary sources are modeled as flush-mounted circular pistons and the 62 minimization points are distributed all around.

Figure 5 .

 5 Figure 5. Room frequency response with control switched off/on

Figure 6 .

 6 Figure 6. Total sound pressure level with the control off at 38 Hz, 47 Hz and 69 Hz.

Figure 7 .

 7 Figure 7. Total sound pressure level with the control on at 38 Hz, 47 Hz and 69 Hz.

Figure 8 .

 8 Figure 8. Slice plot showing how pressure is distributed inside the room with control the on at 38 Hz, 47 Hz and 69 Hz, in the 𝑦𝑧-plane of the test source.

Figure 9 .

 9 Figure 9. Photo of the actual demonstrator currently under construction at LMA.

Figure 10 .

 10 Figure 10. Measured frequency response function of the sound source output.

Figure 11 .

 11 Figure 11. Frequency response function of the acoustic measurement room (with the active system switched off and without absorbing materials).

Acknowledgements

This work was supported as part of the France 2030 investment plan and the Aix-Marseille Université -A*MIDEX Excellence Initiative (AMX-19-IET-010).