Benoit Goutorbe

Anne-Laure Abraham

Mahendra Mariadassou

Valentin Loux

Olivier Rué

Anne Plauzolles

Ghislain Bidaut

Philippe Halfon

Sophie Schbath

Shallow sequencing: a cost-effective and accurate alternative to WGS for taxonomic profiling ?

come   L'archive ouverte pluridisciplinaire

Shallow sequencing: a cost-effective and accurate alternative to WGS for taxonomic profiling ?

Benoit Goutorbe 1,2,3 , Anne-Laure Abraham 1 , Mahendra Mariadassou 1 , Valentin Loux 1,4 , Olivier Rué 1,4 , Anne Plauzolles 2 , Ghislain Bidaut 3 , Philippe Halfon 2 and Sophie Schbath

Context

In the rising research area of microbiota-associated health outcomes, clinical researchers have to deal with the critical choice of the analytic technique used to characterize patients' microbiota. This choice is usually binary, with metabarcoding, a low-cost and an efficient way to identify and quantify organisms present in an ecosystem (taxonomic profiles) which has a limited resolution and suffers from well known biases (amplification biases, variation in copy numbers, etc), and whole genome sequencing (WGS), which offers deep insights about both taxonomic and functional profiles but is much more expensive (about 10 times the cost of metabarcoding) and produces data massively more complex to analyse.

Due to a huge inter-patient variability, large cohorts are needed to extract reliable information. Thus, a large majority of studies are carried out using metabarcoding techniques. Shallow WGS (WGS at very low sequencing depth, down to 500 K reads/sample) is one of the techniques that could fit into this technological gap : a recently published paper [START_REF] Hillmann | Evaluating the Information Content of Shallow Shotgun Metagenomics[END_REF] demonstrated the huge potential of this approach but left many important questions unanswered.

Materials & Methods

Using several public human gut microbiota datasets, covering several diseases and health conditions [START_REF] Loomba | Gut Microbiome-Based Metagenomic Signature for Non-invasive Detection of Advanced Fibrosis in Human Nonalcoholic Fatty Liver Disease[END_REF][START_REF] Matson | The commensal microbiome is associated with anti-PD-1 efficacy in metastatic melanoma patients[END_REF][START_REF] Qin | Alterations of the human gut microbiome in liver cirrhosis[END_REF] we evaluated the information content of deep and shallow WGS. In order to do so, we subsampled in each dataset the reads at several sequencing depths, down to 10K reads/sample. Using Unified Human Gut Genome [START_REF] Almeida | A unified sequence catalogue of over 280,000 genomes obtained from the human gut microbiome[END_REF] as reference database, we compared two aligners (BWA MEM, bowtie2 ) and a k-mer based method (kraken2 ). We also focused on assessing how redistributing ambiguously mapped reads could enhance the profiles at critical sequencing depths. We assessed how low-depth sequencing degrades (i) the taxonomic profile and (ii) downstream statistical analysis (here biomarker discovery and diagnosis ability).

Results

Our preliminary results indicate that the taxonomic profiles seem to be robust at low sequencing depth. As expected, relatively abundant populations are less impacted than rare ones. Diagnosislike classifications, often focused on relatively abundant populations, don't seem to be impacted by sequencing depth. Further investigations are needed to identify in which contexts shallow sequencing is enough to characterize an ecosystem and discriminate patients regarding to clinical outcomes.

1

  Université Paris-Saclay, INRAE, MaIAGE, 78350, Jouy-en-Josas, France 2 Laboratoire Alphabio, 1 rue Melchior Guinot, 13003, Marseille, France 3 CRCM, Inserm, CNRS, Institut Paoli-Calmettes, Aix-Marseille Université, 13009, Marseille, France

	4 Université Paris-Saclay, INRAE, BioinfOmics, MIGALE bioinformatics facility, 78350, Jouy-en-Josas,
	France
	Corresponding author: benoit.goutorbe@inrae.fr


T Nguyen Van

I De Moraes

N M Dempsey

C Champeaux

Frédéric Dumas-Bouchiat
email: frederic.dumas-bouchiat@unilim.fr

Textured Nd-Fe-B hard magnetic thin films prepared by pulsed laser deposition with single alloy targets

Keywords: Nd-Fe-B, hard magnetic thin film, pulsed laser deposition (PLD). I

Nd-Fe-B thin films have been successfully developed by Pulsed Laser Deposition (PLD) onto silicon substrates. In order to explore the crucial role of target composition during PLD, five single alloy targets, representing specific atomic compositions (Nd/Fe atomic ratio varied in the range 0.18-0.47) have been tested and studied. In a determined deposition temperature range of 550°C-630°C, thin films made from a target with Nd/Fe=0.45 present the best hard magnetic properties, a quasi-single magnetic phase, both coercivity and remanence above 1 T and a square loop shape. FIB-EBS analyses associated with XRD, SEM and SQUID magnetic measurements indicate that the high values of both remanent magnetization and coercivity are ascribed to film texturation and to partial decoupling of the hard magnetic Nd2Fe14B grains by a secondary Nd-rich phase.

PLD performed with other target compositions and non-optimized deposition temperatures leads to poorer magnetic properties, owing to the presence of soft magnetic phase and non-textured or oxidized thin films.

Introduction

Hard magnetic materials represent a key component of macroscopic electro-mechanical devices (e.g. wind turbines, hybrid-electric vehicles, air-conditioners) [1] and hold enormous potential for use at the microscale (micro-systems dedicated to telecommunications, internet of things (IoT) and bio-technology) [START_REF] Cugat | Magnetic micro-actuators and systems (MAGMAS)[END_REF][START_REF] Pivetal | Micro-magnet arrays for specific single bacterial cell positioning[END_REF]. Today's high performance magnets are based on rare earth -transition metal intermetallic phases [1], and the highest energy densities are achieved for magnets based on the tetragonal Nd2Fe14B phase (maximum theoretical energy product (B.H)max = 512 kJ/m 3 ) [1,[START_REF] Sagawa | New material for permanent magnets on a base of Nd and Fe (invited)[END_REF][START_REF] Croat | Pr-Fe and Nd-Fe-based materials: A new class of highperformance permanent magnets (invited)[END_REF]. Theoretical modelling suggests that the energy product can be significantly enhanced in nanocomposites which intimately combine a high coercivity hard phase with a high magnetization soft phase [START_REF] Skomski | Giant energy product in nanostructured two-phase magnets[END_REF]. While both top-down [START_REF] Yadav | Mechanical Milling: a Top Down Approach for the Synthesis of Nanomaterials and Nanocomposites[END_REF] and bottom-up [START_REF] Hu | Structure and magnetic properties of bulk anisotropic SmCo5/α-Fe nanocomposite permanent magnets prepared via a bottom up approach[END_REF] techniques have been used to fabricate hard-soft nanocomposites, the reported energy products fall far short of predicted values. This is attributed to insufficient control of the dimensions of the soft magnetic phase, which according to modelling, should be no larger than twice the domain wall width, d, of the hard phase (d = 2.6, 3.9 and 4.5 nm for SmCo5, Nd2Fe14B and CoPt, respectively) [1]. Laser Vaporization Nanoparticle Sources (LVNS) have been used to prepare magnetic nanoparticles of diameter as low as 2 nm [START_REF] Dupuis | Intrinsic magnetic properties of bimetallic nanoparticles elaborated by cluster beam deposition[END_REF][START_REF] Alayan | Application of a static quadrupole deviator to the deposition of size-selected cluster ions from a laser vaporization source[END_REF][START_REF] Dupuis | Magnetic moments in chemically ordered massselected CoPt and FePt clusters[END_REF]. Reactors which combine a nanoparticle source with a film deposition source (e.g. Pulsed Laser Deposition (PLD) [START_REF] Gaudin | A dual nanosecond-pulsed laser setup for nanocomposite synthesis-Ag nanoparticles in Al2O3/VO2 matrix[END_REF][START_REF] Dumas-Bouchiat | Magnetic domains in Cocluster assembled films deposited by LECBD[END_REF][START_REF] Dumas-Bouchiat | Cobalt cluster-assembled thin films deposited by low energy cluster beam deposition: Structural and magnetic investigations of deposited layers[END_REF], or sputtering [START_REF] Fischer | Magnetic properties of iron cluster/chromium matrix nanocomposites[END_REF]) hold the potential to prepare hard-soft nanocomposites in which the size of soft nano-inclusions in a hard magnetic matrix can be restricted to below 10 nm. As a first step towards fabricating hard-soft nanocomposites with a reactor based on the association of a pulsed LVNS and a PLD system, here we report on the fabrication of NdFeB films by PLD. Literature reports on the fabrication of NdFeB films by PLD fall into two groups. The first deals with epitaxial growth using multiple targets (Nd, Fe, B, or the combination of one of the two), on single crystalline Al2O3 or MgO substrates [START_REF] Kwon | Effect of composition on phase formation, microstructure and magnetic properties of Nd-Fe-B thin films[END_REF][START_REF] Hannemann | Effect of Cr and Ta buffers on hard magnetic Nd2Fe14B films[END_REF][START_REF] Neu | Growth, microstructure, and magnetic properties of highly textured and highly coercive Nd-Fe-B films[END_REF]. The second deals with films deposited using a single composite Nd-Fe-B target on tantalum or silicon substrates [START_REF] Nakano | A method of preparing anisotropic Nd-Fe-B film magnets by pulsed laser deposition[END_REF][START_REF] Nakano | Relationship between target materials and various properties of PLD-made isotropic Nd-Fe-B films[END_REF][START_REF] Zheng | Properties of NdFeB film grown on silicon substrate by PLD under external magnetic field[END_REF][START_REF] Nakano | Nd-Fe-B Film Magnets With Thickness Above 100 µm Deposited on Si Substrates[END_REF]. In the latter case films were either deposited at elevated temperatures or at room temperature and then annealed to crystallize the hard magnetic Nd2Fe14B phase. Such films show isotropic magnetic properties, which limits the remanent magnetization values achievable, while out of plane anisotropy was achieved in the epitaxially grown films made with multiple targets. Though the use of multiple targets allows a fine tuning of the film composition, it requires ultra-high vacuum conditions in light of the susceptibility to oxidation of pure Nd and it adds complexity to the control of the PLD procedure. The use of a composite Nd-Fe-B target allows for simpler two-target PLD systems (Nd-Fe-B target and a buffer / capping layer target) and more tolerance on the base oxygen level, but testing of different target compositions is required to tune the film composition. It is known that Nd content in excess of that of the stoichiometric Nd2Fe14B phase is required to induce coercivity in NdFeB magnets in bulk or film form. The excess Nd leads to the formation of a non-magnetic grain boundary phase that serves to magnetically decouple neighboring Nd2Fe14B grains. Here we report on the fabrication of out-of-plane textured NdFeB films on thermally oxidized Si substrates by PLD with single composite alloy Nd-Fe-B targets.

II. Experimental details

A set of five composite targets were prepared by induction melting followed by casting into a Cu mould.

The overall Nd/Fe atomic ratio was varied from target to target (#1 = 0.47, #2 = 0.45, #3 = 0.42, #4 = 0.33, #5 = 0.18) while the B/Nd ratio was kept constant (~0.5). Ta/Nd-Fe-B (~150 nm)/Ta tri-layers were grown by PLD on (001) silicon substrates thermally oxidized to a depth of 100 nm (the nominal thickness of the Ta layers was 50 nm). The role of the Ta underlayer is to prevent substrate-film interdiffusion. Moreover, Ta was shown to promote out-of-plane (oop) epitaxial growth of Nd2Fe14B prepared by PLD or sputtering [START_REF] Hannemann | Effect of Cr and Ta buffers on hard magnetic Nd2Fe14B films[END_REF][START_REF] Neu | Growth, microstructure, and magnetic properties of highly textured and highly coercive Nd-Fe-B films[END_REF][START_REF] Dempsey | High performance hard magnetic NdFeB thick films for integration into micro-electro-mechanical systems[END_REF]. The role of the Ta capping layer is to protect the film from oxidation. The base pressure in the deposition chamber was about 5´10 -8 mbar, and PLD was performed with an excimer laser KrF (LightMachinery, IPEX-742, λ=248 nm, pulse duration 25 ns) operated at a fluence of about 6 J.cm -2 . The target-substrate distance was fixed at 3.3 cm and during deposition, the substrate was heated from the backside by a halogen lamp in the temperature range 500°C-700°C.

Compositional and structural characterization was performed in a Scanning Electron Microscope (SEM) (LEO Gemini 1530 VP) equipped with energy dispersive spectroscopy (EDS) and x-ray diffraction (XRD) in a q-2q configuration (Cu-Kα1 radiation-Bruker AXS D8 Advance). Cross-sectional SEM imaging was carried out on both cleaved and Focused Ion Beam (FIB) -cut (Ga ion beam, Crossbeam 550, Carl Zeiss Microscopy GmbH) samples. Magnetic hysteresis loops were measured at room temperature in a SQUID-VSM (Quantum Design MPMS 3) in a maximum applied field of 6.0 T. Atomic Force Microscopy (AFM) measurements were performed using a Dimension Icon AFM-BRUKER microscope. The microcantilever used was from AppNano (ACT probe, the nominal k = 34 N/m and fo = 300 kHz).

III. Results and Discussion

A first set of films were prepared by deposition on substrates maintained at a temperature of 600°C, which is just under the reported crystallization temperature of the Nd2Fe14B phase (630°C), and was shown to induce epitaxial out-of-plane oriented growth in NdFeB films prepared by PLD [START_REF] Kwon | Effect of composition on phase formation, microstructure and magnetic properties of Nd-Fe-B thin films[END_REF][START_REF] Hannemann | Effect of Cr and Ta buffers on hard magnetic Nd2Fe14B films[END_REF][START_REF] Neu | Growth, microstructure, and magnetic properties of highly textured and highly coercive Nd-Fe-B films[END_REF]. Erreur ! Source du renvoi introuvable. compares the average values of Nd/Fe atomic ratio, estimated from EDS analysis of typically ten thin films deposited from each composite target, with that measured on the targets themselves. The Nd/Fe ratio is systematically lower in the films than in the corresponding target, dropping from 0.47 in the Nd richest target to 0.44 (average value) in the Nd richest films and from 0.18 in the Nd poorest target to 0.11 (average value) in the Nd-poorest films. The dashed line in Erreur ! Source du renvoi introuvable. corresponds to the expected film composition if the deposition process were congruent. The drop in Nd/Fe ratio from target to film may be attributed to differences in both the melting point of the concerned elements (Nd (1016°C) versus Fe (1538°C)), and their relative atomic mass (Nd: 144.24 g versus Fe: 55.85 g), the latter potentially affecting their distribution in the plasma plume and possibly leading to a redistribution of elements at the surface of the target. During laser-target interactions and film deposition, the constituent elements may slightly migrate, inducing local changes of the composition, as known for PLD [START_REF] Toyserkani | Ultrashort pulsed laser surface texturing[END_REF]. It should be noted that the average Nd/Fe ratio in the Nd-poorest films is below the stoichiometric ratio of 0.14 of the Nd2Fe14B phase.

Erreur ! Source du renvoi introuvable.a presents XRD patterns of representative samples of the films deposited at 600°C using the different targets. In all patterns, the Si(400) and Ta(110 all have a Nd/Fe ratio higher than the Nd2Fe14B phase, is attributed to the presence of a Nd-rich paramagnetic phase that serves to magnetically decouple Nd2Fe14B grains. Relative variations in the coercivity values of these Nd-rich films may be attributed to differences in how the Nd-rich phase is distributed and the presence of secondary magnetic phases. While the sharpness of the hysteresis loop as it approaches coercivity is comparable in films made from targets #1 to #3, it is higher in the film made from target #4 and higher again in the Nd-poorest film made from target #5. The coercivity drops to just 0.2 T in the latter film, while the remanence ratio M0T/M6T drops to 0.2. These low values reflect the presence of alpha-Fe in this film, identified in XRD analysis and coherent with the Nd/Fe ratio of the film being less than that of the Nd2Fe14B phase.

)
An SEM image of the FIB-cut cross section of a film made from target #2, is shown in Erreur ! Source du renvoi introuvable.a. Bright regions in the Nd-Fe-B layer, (one such region is identified by an orange arrow), are attributed to the Nd-rich phase while the dark regions (e.g. the region identified by a white arrow), are attributed to the predominant Nd2Fe14B phase. Based on image analysis of a number of such images, the volume content of the Nd-rich phase is estimated to be about 30 %. This value agrees very well with a value of 33 %, deduced from the average Nd/Fe atomic ratio measured by EDS analysis (Erreur ! Source du renvoi introuvable.) and considering the film to be composed of Nd2Fe14B and pure Nd. An out-of-plane magnetic hysteresis loop of this sample is compared with an in-plane (ip) loop measured on the same sample (no demagnetizing field correction)

in Erreur ! Source du renvoi introuvable.b. In this case, absolute values of magnetization, based on an estimate of the sample volume are represented on the y-axis. Note that the sample volume is calculated based on measurements of the film thickness and surface area and on a rough estimate of the volume of droplets, which are characteristic of metallic films made by PLD [START_REF] Zheng | Properties of NdFeB film grown on silicon substrate by PLD under external magnetic field[END_REF][START_REF] Van De Riet | Reduction of droplet emission and target roughening in laser ablation and deposition of metals[END_REF]. Droplets are estimated to account for ~ 25 vol% of the overall deposit. Extrapolating the ip and oop hysteresis loops leads to an intercept at about 8 T, which is comparable to the magnetocrystalline anisotropy of the Nd2Fe14B phase (7.6 T). The lower value of remanence and higher value of high field slope of the ip measurement reflect the oop texture induced during film growth at 600°C. The error on the magnetization value, which is dominated by the error in volume estimation, is ~27 %. Indeed, an outof-plane saturation magnetization value closer to 1 T can be expected for a well textured NdFeB film in which the hard magnetic Nd2Fe14B phase accounts for 70 vol%. As the best magnetic properties were obtained for films produced with target #2, results reported hereafter concern films made with this target.

A set of films was made using target #2, in which the temperature of the substrate during deposition (Td) = 0.9 T [START_REF] Hannemann | Intrinsic and extrinsic properties of epitaxial Nd2Fe14B films[END_REF]) and alloy targets (µoHc = 0.72 T and µoMr = 0.9 T [START_REF] Nakano | A method of preparing anisotropic Nd-Fe-B film magnets by pulsed laser deposition[END_REF][START_REF] Nakano | Relationship between target materials and various properties of PLD-made isotropic Nd-Fe-B films[END_REF][START_REF] Nakano | Nd-Fe-B Film Magnets With Thickness Above 100 µm Deposited on Si Substrates[END_REF]). Note that rare earth based films are susceptible to oxidation and the PLD of elemental targets cited above was carried out under UHV (10 -9 mbar) conditions. High deposition rates help to suppress oxidation of films deposited from alloy targets in chambers with a relatively poor base vacuum level, which is the case for the triode sputtered (18 µm/h, 10 -6 mbar) and PLD with alloy target (40 µm/h, 10 -6 mbar) films cited above. It is worth noting that while the base pressure of the chamber used here is better (10 -7 mbar), the deposition rate is significantly lower (0.5 µm/h). Thus, the PLD system used here with an alloy target allows the fabrication of hard magnetic thin films and when combined with LNVS it has the potential to produce magnetic nanocomposites with soft inclusions of well controlled nm-size.

was
Plane-view SEM images of films showing distinct surface morphologies are shown in Erreur ! Source du renvoi introuvable.. The surface of the film deposited at Td =500°C is characterized by "islands" of diameter up to 300 nm on an underlying layer with grains of diameter < 100 nm (Erreur ! Source du renvoi introuvable.a).

The surface of the film deposited at Td of 630°C is characterized by a high density of grains of size approaching 1 µm (Erreur ! Source du renvoi introuvable.b, identified by white arrows) some of which are separated by much smaller underlying grains of size 50-100 nm (identified by orange arrows). In the case of the film deposited at 650°C, connected platelet-like grains of size greater than 1 µm, with the plane of the platelets sometimes tilted with respect to that of the Si substrate, are identified, together with smaller grains of size close to 200 nm and an under-layer of grains of size 50-100 nm (Erreur ! Source du renvoi introuvable.c). AFM analysis indicates that the rms surface roughness increases from ~10 nm, for the film deposited at 500°C, to a maximum ~43 nm at 630 °C and then slightly reduces to ~35 nm at 700°C (Erreur ! Source du renvoi introuvable.). The maximum in rms roughness for the film deposited at 630°C followed by a drop at higher deposition temperatures is coherent with the compact isolated grains and inter-connected island (leading to a flatter surface in the same area of measurement) microstructures, respectively. The observed coarsening of the microstructure with increased deposition temperature may facilitate magnetization reversal through domain wall motion, which could contribute to a reduction of coercivity in the films deposited at higher temperatures.

IV. Conclusions

150 nm hard magnetic thin films of Nd-Fe-B were fabricated by PLD using a single ternary alloy target.

The best magnetic properties were obtained for an Nd/Fe atomic target ratio of 0.45, a value far from that of 0.14 which corresponds to that of the Nd2Fe14B phase. Films deposited on silicon substrates with buffer and capping layers of Ta at a temperature in the range of 550°C-630°C present the best hard magnetic properties, i.e. oop texture, coercivity values of up to 1.3 T, single magnetic phase behavior and a square loop shape. A Nd-rich phase corresponding to around 30 vol% of such films was revealed by structural (XRD, SEM, FIB-EBS) characterization.

This phase, presumed to be paramagnetic, served to partially magnetically decouple Nd2Fe14B grains, leading to the high coercivity values. Remanent magnetization values estimated to be close to 1 T validate these materials developed by PLD as a promising candidate for model materials to study certain aspects of magnetization reversal systems and for thin film magnet applications.

  peaks of the substrate and the buffer and capping layers are identified. Peaks of the Nd2Fe14B phase are identified in all films, while reflections attributed to pure Nd (corresponding to (004) and (008) peaks) are identified in Nd-rich films (from target #1 -#3) and a peak of alpha-Fe (110) is identified in the Nd-poorest film (target #5). The relatively high intensities of the (00l) Nd2Fe14B reflections compared to other reflections from this phase indicate preferential out of plane orientation. SEM images of cleaved cross sections of these films are shown in Erreur ! Source du renvoi introuvable.b. The film made from the target with the lowest Nd/Fe ratio (target #5) is predominantly composed of very fine equiaxed grains. Increasing the Nd/Fe ratio leads to bigger grains which become more and more elongated in films made from targets #4 and #3 and typically columnar in films made from targets #2 and #1. Most of the columnar grains traverse the film and are separated by some smaller equiaxed grains. AFM analysis performed at the surface of films (#1-#5) confirmed data obtained by SEM concerning the variation in grain size. The surface roughness of films (areas free of droplets) quantified by the root mean square (rms) roughness values increase by a factor of ~8 globally following the Nd/Fe atomic ratio (from about 4 nm to 33 nm for films with Nd/Fe = 0.11 to 0.42, respectively ) (Erreur ! Source du renvoi introuvable.c), in agreement with the fact that the grain size also increases with Nd content. Out-of-plane magnetic hysteresis loops of representative samples of the films deposited at 600°C using the different targets are compared in Erreur ! Source du renvoi introuvable.d. Note that the loops are not corrected for demagnetizing field effects, and the magnetization values are normalized. The film made from target #2 has both the highest values of coercivity (1.1 T) and remanence ratio M0T/M6T (0.91). All other films show very obvious two-phase or multi-phase behaviour. The films made from targets #1, #3 and #4 show comparable values of remanence ratio M0T/M6T (0.75). The coercivity drops to 0.9 T in the film made from target #1 and to 0.75 T in the films made from targets #3 and #4. The development of coercivity in films produced from targets #1-#4, which

  varied in the range 500°C-700°C. An EDS estimation of the Nd/Fe ratio for this series of thin films indicated very little variation of the Nd/Fe ratio (0.42 ± 0.01) between the films. XRD patterns of these films are compared in Erreur ! Source du renvoi introuvable.a. The degree of oop orientation of the NdFeB layer, reflected in the relative intensities of the (00l) peaks compared to other peaks of the Nd2Fe14B phase, increases with increasing temperature up to 650°C. The intensities of Nd2Fe14B and Nd peaks are greatly reduced in the film deposited at 700°C, and peaks of Nd2O3 are identified in the film. Peaks attributed to Ta2O3 are identified in films deposited at 630°C and above. Normalized oop hysteresis loops of this set of films are compared in Erreur ! Source du renvoi introuvable.b and the evolution of coercivity and remanence with deposition temperature are plotted in Erreur ! Source du renvoi introuvable.c. Relatively large error bars are associated with the µoMr values, owing to the significant error in estimating the volume of these films, as discussed above. Films deposited between 550°C and630°C have coercivity values in excess of 1 T, and comparable overall loop shape. The significantly lower value of coercivity of 0.5 T of the film deposited at 500°C is attributed to poor distribution of the Nd-rich grain boundary phase, resulting in poor decoupling of the Nd2Fe14B grains. The drop in coercivity for films deposited at above 630°C is attributed to a combination of the evolution in microstructure (see below) and degradation of the film through oxidation. Initiation of oxidation at 650°C may not be visible using conventional XRD as it occurs over a very short range. The loop shape of the film deposited at 700°C is poorer than that deposited at 650°C, because of the greater extent of oxidation, as evidenced in the XRD patterns. This indicates that the base oxygen pressure of the PLD chamber used is too high to grow high quality Nd-Fe-B at temperatures of 650°C and above. A deposition temperature in the range 550°C-630°C favours both high coercivity and high remanence. The highest values of coercivity and remanence achieved in this study (µoHc = 1.3 T and µoMr > 1 T) are in the range of the best values reported for thin or thick films (e.g., Nd-Fe-B thick films fabricated by high deposition rate triode sputtering (µoHc = 1.6 T and µoMr = 1.4 T[START_REF] Dempsey | High performance hard magnetic NdFeB thick films for integration into micro-electro-mechanical systems[END_REF], µoHc = 2.7 T and µoMr = 1.15 T[START_REF] Dempsey | High-coercivity Nd-Fe-B thick films without heavy rare earth additions[END_REF]). The coercivity values are intermediate while the remanence values are higher compared to films prepared by PLD with elemental targets (µoHc = 2 T and µoMr

Figure 2 :

 2 Figure 2: Characterization of Ta/Nd-Fe-B(150 nm)/Ta films deposited at 600°C on Si/SiO2 substrates (a) XRD patterns of thin films made from the five different targets; (b) SEM images of the fractured cross sections of films from the five targets; (c) AFM rms surface roughness; and (d) Out-of-plane magnetic hysteresis loops (uncorrected for demagnetization field and normalised) of films deposited from the five targets.

Figure 3 :

 3 Figure 3: Characterization of the film made from target #2 deposited at 600°C on Si/SiO2. (a) FIB-EBS crosssectional images in which Nd2Fe14B grains elongated in the z direction (black regions, one of which is identified by a white arrow) are found together with a Nd rich phase (gray regions, one of which is identified by an orange arrow); and (b) in-plane (ip) and out-of-plane (oop) magnetic hysteresis loops (uncorrected for demagnetization field).

Figure 4 :

 4 Figure 4: The evolution in structural and magnetic properties of Ta/Nd-Fe-B/Ta thin films on Si/SiO2 substrates made from target #2 as a function of deposition temperatures-Td ranging from 500°C to 700°C. (a) XRD diffractograms of the thin films with an enlarged range from 25° to 35°; (b) oop magnetic hysteresis loops (uncorrected for demagnetization field and normalized); and (c) coercivity (µoHc) and remanent magnetization (µoMr) together with curves representing guides to the eye.

Figure 5 :

 5 Figure 5: Plane-view SEM images showing the distinct surface microstructures of Ta/Nd-Fe-B/Ta films deposited at Td (a) 500°C; (b) 630°C; and (c) 650°C. The scale bar in all images is 1µm.

Figure 6 :

 6 Figure 6: The AFM rms surface roughness of Ta/Nd-Fe-B/Ta films from target #2 as a function of deposition temperatures-Td.

Acknowledgements

The authors thank the French National Research Agency (ANR) for financial support provided by the project ANR-SHAMAN 16-CE09-0019 project. This work benefited from technical support from the Common Service of Material Characterizations (CARMALIM) of the Université de Limoges. The authors gratefully acknowledge Richard Haettel and Frederico Orlando Keller from Institut Néel for fabricating the induction melted targets.


Inga Tijunelyte

Rémi Malbec

Bayan Chami

Jean Cacheux

Christophe Dez

Thierry Leichlé

Pierre Cordelier

Aurélien Bancaud
email: abancaud@laas.fr

micro-RNA 21 detection with a limit of 2 pM in one minute using a size-accordable concentration module operated by electrohydrodynamic actuation

Keywords: Electrohydrodynamic migration, RNA/DNA concentration, miR-21, fine needle biopsy

We present a fluorimetry-based technology for micro-RNA-21 (miR-21) sensing based on the concentration of miR-molecular beacon (MB) complexes and flushing of unbound MB. This concentration module consists of a microfluidic channel with the shape of a funnel operated with electrohydrodynamic actuation. We report a limit of detection of 2 pM in less than one minute for miR-21 alone, and then demonstrate that miR-21 levels measured in fine needle biopsy samples from patients with pancreatic cancer correlate with the reference technique of reversetranscription polymerase chain reaction (RT-PCR). Altogether, this technology has promising clinical performances for the follow-up of patients with cancer.

Introduction

MicroRNAs (miRs) are single-stranded RNA molecules of 15-27 nucleotides that regulate gene expression at the post-translational level. miR concentration is profoundly altered in cancer, and this deregulation may participate to carcinogenesis (Calin and Croce, 2006;Pichler and Calin, 2015). Monitoring miRs levels thus constitutes a valuable prospect for the management of patients with cancer (Etheridge et al., 2011). Next-generation sequencing, which provides genome-wide miR expression levels, is essential for de novo identification of miR-based biomarkers. Yet, the processing of clinical samples, which contain very low material input, remains a challenge (Coenen-Stass et al., 2018). For targeted miRs quantification, the gold standard is Reverse-Transcriptase quantitative Polymerase Chain Reaction (RT-qPCR) and its miniaturized low background version (RT-digital droplet PCR, ddPCR; (Forero et al., 2019)), which rely on labor intensive protocols with multiple controls (Stein et al., 2017).

Alternatively, the race for fast technologies that do not rely on enzymatic amplification has not slowed down [START_REF] Cacheux | [END_REF]. Enzyme-free technologies for miR titration can coarsely be cast into two categories depending on whether detection occurs in bulk or on surfaces. Surface detection reaches high sensitivity but generally depends on sophisticated microfabrication strategies [START_REF] Bournet | 3 Laurent Palazzo; 4 Jérôme Cros; 5 Sarah Tubiana; 6[END_REF]. Bulk detection is readily achieved by fluorimetry, and most frequently relies on molecular beacons (MBs), which change conformation and become fluorescent upon hybridization to the target (Baker et al., 2011). This method is limited in sensitivity to 1 nM (Baker et al., 2011) and dynamic range due to the background signal from unreacted MBs (Garcia-Schwarz and Santiago, 2012). The limit of detection (LOD) has nevertheless been decreased to the fM level by replacing organic fluorophores with quantum dots (Su et al., 2014), or to 5 pM using more sophisticated signal transduction schemes based on endonucleases that provoke the release of a fluorophore engrafted to the probe and initially quenched by a gold nanoparticle [START_REF] Degliangeli | Absolute and Direct J[END_REF]. Furthermore, signal amplification schemes based on strand displacement reactions have been implemented in order to convert each miR hybridization event into a cascade of detectable readouts (Li et al., 2015;Wu et al., 2015).

These assays reached sub-pM to sub-fM LODs (Liang et al., 2017;Song et al., 2016;Wang et al., 2019;Wu et al., 2016), though at the expense of generally long time to result (Shin et al., 2019).

In another yet non-mutually exclusive direction, fast and sensitive detection characterized by a LOD of 5 pM in 3 minutes has been reported by concentrating miR-MB complexes with isotachophoresis (Bahga et al., 2013;[START_REF] Bercovici | Proc. Natl. Acad. Sci[END_REF][START_REF] Persat | [END_REF].

Concentration can also be performed with the µ-Laboratory for DNA analysis and separation (µLAS) technology, which is based on the combination of hydrodynamic transport and a counter electrophoretic force (Malbec et al., 2019;Milon et al., 2020;Ranchon et al., 2016).

Electrohydrodynamic actuation induces transverse migration oriented toward the channel walls.

Because transverse forces increase with hydrodynamic flow velocity, electrophoretic velocity, and DNA molecular weight (MW) (Chami et al., 2020(Chami et al., , 2018)), this technology enables DNA separation in a linear channel, i.e. with constant hydrodynamic and electrophoretic settings. In a microchannel with the shape of a funnel, the amplitude of transverse forces progressively increases as the flow velocity and electric field build up near the constriction (Fig. 1A). By adjusting the pressure and tension, hydrodynamic and electrophoretic forces (along the x-axis in Fig. 1A) can be balanced to define a position of null velocity. Upstream of this stagnation point, hydrodynamic forces prevail so that molecules continuously flow to and accumulate at this position. Here, we use µLAS for detection of the cancer-promoting miR-21 (Humeau et al., 2015;Pfeffer et al., 2015) based on the selective enrichment of miR-21:MB complexes and elimination of unbound MB due to their lower MW. In "ideal" conditions, i.e. with a mixture only containing the MB and miR-21, we report a LOD of 2 pM within 30 seconds. We then prove that miR-21 titration can be performed in complex samples containing high concentrations of uncomplimentary RNA, and establish that absolute miR-21 levels measured in fine-needle biopsy samples of patients with pancreatic cancer are correlated to those of RT-qPCR.

Materials and methods

Microfluidic chips and data acquisition

Microfluidic chips were fabricated following the same protocol as described in ( (Malbec et al., 2019), see Supplementary Material). The maximal and minimal channel widths were 800 and 5 µm, respectively, and its depth at the constriction was 2 µm (Fig. 1A). The chip was filled with a solution containing 22 mM of NaCl and 5% (m:v) 1.3 MDa polyvinylpyrrolidone (the viscosity and relaxation time of the solution were 31 mPa.s and 1.1 ms, respectively (Naillon et al., 2019)).

Actuation was operated using a 7 bar pressure controller (Fluigent) and a DC high voltage power supply (Labsmith). Videos were recorded with a Leica microscope equipped with a light engine for cyan excitation (Lumencore) and an sCMOS digital camera (Hamamatsu). We used a 20X air objective (Numerical Aperture of 0.8) and 4x4 binning (pixels of 1.3 µm). The inter-frame interval was set to 2 s with an exposure time of 0.2 s.

Fine needle biopsy samples

Fine needle biopsy material was obtained from the Bacap repository (The BACAP Consortium et al., 2018). Total RNA was extracted with miRNeasy Micro Kit (Qiagen). For miR reverse transcription, we used 10 ng of total RNA and miRCURY™ LNA™ miRNA RT Kit (Qiagen). Real-time PCR for miR-21 was performed using LNA™-enhanced microRNA qPCR primers (Qiagen) and the StepOne plus sequence detection system (Applied Biosystems).

Genomic sequences

miR-21C: 5'-(A) 35 -TAGCTTATCAGACTGATGTTGA-(A) 35 -3'. miR-21: 5'-UAGCUUAUCAGACUGAUGUUGA-3' MB: 5'-FAM-GCGCGTCAACATCAGTCTGATAAGCTACGCGC-BHQ1-3', FAM, and BHQ1 standing for 6-carboxyfluorescein and black-hole quencher 1.

Results and discussion

Electrohydrodynamic actuation settings for miR-21 detection

We first report the analysis of a stoichiometric solution of the miR-21 target (miR-21C) and a complementary MB at 100 nM in the microfluidic chip of 2 µm in height (Fig. 1A). We used a pressure of 2.5 bar and a counter electrophoretic force associated to a tension of 275 V.

The resulting maximum flow velocity and electric field were 20 mm/s and 17 kV/cm, respectively (see definition in Fig. 1A and details of the calculation in Supplementary Material). The corresponding Reynolds number and electrophoretic Peclet number were 10 -3 and 50, respectively, implying that the flow was laminar and electrophoresis occurred. We confirmed that the electric field did not induce temperature change, and that electrohydrodynamic actuation did not impair nucleic acid pairing (Supplementary Fig. S1). In Fig. 1B, we plot miR-21C:MB concentration kinetics. Within few seconds after application of the electric field, a fluorescence signal became measurable at the constriction (image 1 in Fig. 1B). The fluorescence intensity signal, as inferred from a Gaussian fit of the intensity signal along the symmetry axis of the chip (inset of Fig. 1B and red dashed arrow in the upper right micrograph), reached a plateau after ~40 s. As soon as the electric field was turned off, trapped molecules were flushed out from the concentrator (image 4), and the fluorescence signal dropped back to its initial level. These concentration and flush cycles could be repeated multiple times with a repeatability of ~10% for consecutive experiments on the same chip and 25% in between different chips (Supplementary Fig. S2). Notably, in comparison to a fragment of 1 kbp (Milon et al., 2020), the concentration of the miR-21C:MB complex of ~30 bp required flow velocities and electric fields enhanced by 20fold and 200-fold, respectively.

Selective concentration of miR-21:MB complex

To prove the selective concentration of the miR:MB complex and concomitantly the flushing of unbound MB, we performed one experiment with a "dual chip" containing two funnels in tandem operated with the same electrohydrodynamic settings, as initially designed for sizing expanded alleles in neurodegenerative diseases ( (Malbec et al., 2019), micrographs in Fig. 2A). This chip geometry allowed us to directly and instantaneously compare two mixtures composed of miR-21C and MB at concentrations of 1 and 100 nM or 10 and 100 nM (blue and black datasets in Fig. 2A, respectively). After one minute of concentration using and of 30 mm/s and 19 kV/cm, a strong and a faint signal were observed in the channels that coincided with the difference in miR-21C concentration (micrograph 1 in Fig. 2A). The ratio of the two signals of ~9.2 fold in fact nearly matched the 10-fold difference in solution concentration. As the electric field was increased to 24 kV/cm, we noted that concentrated miR-21C:MB complexes migrated in the counter-flow direction associated to a drop of the signal at the constriction (See supplementary Movie 1 and Fig. 2A). This response was readily explained by the build-up of electrophoretic forces that triggered a mode of transport dominated by electrophoresis. Then, the fluorescence signal at the constriction started to increase again with the same intensity in both channels. This even signal corresponded to the residual fluorescence of unbound MB, which are concentrated at higher electric fields due to their low MW.

Consequently, this experiment showed that MB or miR-21C:MB could be selectively concentrated by tuning electrohydrodynamic settings. Interestingly, the concentration pattern of unbound MB resembled that of a "moustache" (image 2 in Fig. 2A). This shape could readily be accounted for by the insufficient electric field to efficiently concentrate MB, as already described in [START_REF] Arca | [END_REF], and the difference between electric and hydrodynamic field profiles near the boundaries. Electrophoresis prevailed near the lateral walls, because the fluid flow field decreased more rapidly than the electric field (Stone et al., 2004). Conversely, the electric field was insufficient to stop molecules at the symmetry axis of the channel, where hydrodynamic forces were maximal. The resulting pattern of depletion at the center and accumulation at the walls was that of a moustache.

Detection of miR-21 with a LOD of 2 pM in 30 s

We subsequently aimed to optimize electrohydrodynamic actuation settings to enhance miR-21C:MB signal. First, using a constant electric field of 9.4 kV/cm, we increased the pressure by steps of 1 bar, each lasting 20 s, i.e. increased from 8 to 48 mm/s (upper red indications in Fig. 2B). This experiment showed a succession of linear regimes for the concentration kinetics, the first five slopes being roughly proportional to the flow rate (not shown). Hence, hydrodynamics expectedly constituted the source of convection to the concentration module, implying that high flow rates were beneficial to detection. We then performed a "mirror" experiment where was set to 70 mm/s and gradually increased from 29 to 41 kV/cm (Fig. 2C, see Supplementary Fig. S3 for the temporal variation of the signal). The maximum of the intensity profile along the symmetry axis of the channel increased with from 29 to 38 kV/cm, and the position of the peak shifted away from the constriction (Fig. 2C). The maximum intensity signal then dropped, showing that the modulation of the electric field was critical to obtain peaked miR-21:MB signals. Given that we could not apply more than 200 V without a high risk of damaging the chip due to oxide breakdown, optimal detection therefore consisted in increasing the flow rate to obtain the highest signal to noise ratio for a tension of 200 V.

We finally focused on miR-21 detection by preparing four mixtures with 100 nM of MB and miR-21 at concentrations spanning 0.1 to 100 nM. The samples were consecutively analyzed in one chip starting from the lowest miR-21 concentration. We set and to 16 mm/s and 15 kV/cm (these conditions were consistent with ones used for fine needle biopsy samples, see below). The miR-21:MB fluorescence signal was inferred from the peak of the intensity plot along the symmetry axis of the channel, as shown in the inset of Fig. 2D. The saturation of the signal occurred after 30 s for the four samples (Supplementary Fig. S4). The signal of 110 and the standard deviation of the noise of 4.3 were associated to a signal to noise ratio of 25 for the lowest miR-21 concentration of 0.1 nM (inset of Fig. 2D). We then estimated the LOD by plotting the signal as a function of miR-21 nominal concentration (Fig. 2D). The power-law response associated to an exponent of 0.7 could be extrapolated to a signal equal to two times the standard deviation of the background [START_REF] Nič | IUPAC Compendium of Chemical Terminology: Gold Book[END_REF], leading to a LOD of 2.0 pM. Notably, the comparison of different MB batches showed an unexplained variability in miR-21:MB fluorescence signal, which was characterized by different power-law scaling signatures but comparable LODs (Supplementary Fig. S5).

3.4 miR-21 detection in fine needle biopsy samples miR-21 detection in biological samples is challenging because large amounts of background RNA molecules with broad size distributions are present in the sample (see the size distribution of the RNA extracted from pancreas cancer cell lines in Supplementary Fig. S6). We thus aimed to investigate whether and how background RNA molecules could affect the detection capabilities of our technology. For this, we reconstituted two "model" mixtures using (i) one RNA ladder with 7 bands of 100 to 1000 nt, and (ii) total RNA extracted from yeast cell cultures, which predominantly contained two bands of 1800 and 3400 nt. These solutions were diluted in buffer at a final concentration of 5 ng/µL, i.e. a comparable concentration to that of fine needle biopsy samples (see below), and they were spiked with 1, 10, and 100 nM miR-21 with the MB at 100 nM. We also prepared control solutions without RNA background, and processed the resulting nine samples on the same chip. In Fig. 3A, we report the fluorescence signal at constant of 9.4 kV/cm and varying from 4 to 32 mm/s. In the control, fluorescence intensity increased linearly with pressure, whereas a plateau was reached in both reconstituted samples for greater than 16 mm/s (see micrographs in Supplementary Fig. S7). At of 12 mm/s (Fig. 3B), we noticed that the power law exponent relating the signal to miR-21 concentration remained in the same range of 0.5 to 0.7 with or without RNA background. Moreover, although slightly degraded to 3 pM with the yeast RNA background, the LOD remained very comparable in the three samples. Hence, we performed miR-21 titration experiments in five fine needle biopsy samples from patients with pancreatic cancer in the low pressure regime with and of 12 mm/s and 9.4 kV/cm. After determination of the total RNA concentration by absorbance spectroscopy, ranging from 150 to 400 ng/µL, we diluted the samples down to the same total RNA concentration of ~20 ng/µL (see values in the inset of Fig. 3C). The associated miR-21 levels spanned nearly two decades from 3 to 260 pM. The five samples were processed in

Conclusions

In summary, an enzyme-free miR sensing strategy based on the selective concentration of miR hybridized to MB has been established. The LOD of 2 pM in one minute outperforms conventional MB bulk detection, and compares well to other concentration-based high sensitivity miR biosensing technologies. Further, we proved that the simplicity and rapidity of this technology was relevant to measure miR-21 levels in fine needle biopsy patients' samples. In order to enhance the sensitivity of this technology, two synergetic and complementary developments can be envisioned. First, the concentration module performances can be improved by optimizing the microfluidic chip geometry. Second, strand displacement reactions to amplify the readout signal can be integrated ahead of the microfluidic concentration module in the detection process.

Figure 1 .

 1 Figure 1. µLAS technology for miR-21 sensing. (A) The panel shows a sketch and an optical micrograph of the microfluidic chip. The hydrodynamic flow field is oriented toward the funnel and electrophoretic forces in the counter-flow direction. The flow velocity and electric field are maximal at the tip of the apex. Viscoelastic forces are controlled by electrohydrodynamic settings, and adjusted to concentrate miR-21:MB complexes and flush unbound MB (linear and hairpin structures, respectively). (B) The graph shows the temporal evolution of the maximum fluorescence intensity in relative fluorescence units (RFU), as deduced from the peak of the intensity profile along the symmetry axis of the channel (plot in inset). Numbers in fluorescence micrographs correspond to highlighted points in the graph. The scale bar corresponds to 100 µm.

Figure 2 .

 2 Figure 2. Detection of miR-21 in ideal conditions. (A) The blue and black datasets represent the signal over time in each channel of the dual chip. The first 15 s correspond to the steady concentration pattern of miR-21C:MB at and of 19 kV/cm and 30 mm/s. The electric field is then increased to 24 kV/cm, and the intensity becomes comparable in both channels. The electric field is eventually turned off to flush the chip. (B) The graph shows the temporal variation of the peak fluorescence intensity using a stoichiometric miR-21-C:MB concentration of 100 nM. is set to 9.4 kV/cm and spans 0 to 48 mm/s, as indicated in the upper red panel. Numbers in fluorescence micrographs correspond to highlighted numbers in the graph. (C) The plot presents the fluorescence intensity profile along the symmetry axis of the channel. Experiment is performed with stoichiometric miR-21C:MB concentrations of 100 nM, is set to 7 cm/s, and ranges from 0 to 41 kV/cm. The four micrographs at the right are recorded at 34, 38, 40, 41 kV/cm after 20 s of concentration. (D) The graph on the left shows miR-21:MB fluorescence signal as a function of miR-21 nominal concentration. MB concentration, and are set to 100 nM, 15 kV/cm, and 16 mm/s, respectively. The plot in inset presents the intensity profile along the symmetry axis of the channel for 0.1 nM of miR-21. The four micrographs correspond to concentrations of miR-21 of 100, 10, 1, and 0.1 nM (from top to bottom). The scale bars represent 200 µm.

Acknowledgments:

The authors thank Moinamina Mze Ahmed for help in the experiments with fine needle biopsy samples, and Yves Dusabyinema for BioAnalyzer RNA analysis. This work was partly supported by the LAAS-CNRS micro and nanotechnologies platform, member of the French RENATECH network. R.M. and B.C. thank the Carnot institute and Ligue contre le Cancer, respectively, for their PhD fellowships, and J.C. for the funding of the 4 th year of PhD by Fondation ARC. This project was partly supported by the ANR μLAS (ANR-16-CE18-0028-01) and the ANR Biopulse

one chip, and the total time of analysis was ~120 minutes with five consecutive detection cycles of 1 minute followed by rinsing during 15 minutes. We obtained a detectable signal for the five samples, and the plot of miR-21 fluorescence signal as a function of miR-21 nominal concentration measured by RT-qPCR showed a correlation associated to a coefficient of 0.7 (Fig. 3C). Note that the signal of the negative control (MB alone, red line in Fig. 3C) recorded at the end of experiment was lower than the signal for every patient sample. Though refined measurements over a larger cohort would clarify the discrepancy between the reference of RT-qPCR and our data for some samples, especially the samples with a RT-qPCR concentration of 100 and 3 pM, we concluded that miR-21 titration can be operated in clinical samples.

Declaration of interest:

The authors declare that they have no known competing financial interests or personal relationships that have appeared to influence the work in this paper.

CRediT authorship contribution statement:

Inga Tijunelyte:


Romain Viala
email: romain.viala@itemm.fr

Vincent Placet

Scott Cogan

Model-based evidence of the dominance of the guitar brace design over material and climatic variability for dynamic behaviors

Keywords: Physics-based modelling, Virtual prototyping, Uncertainty quantification, Screening analyses, Guitar braces, Spruce tonewood

come

Model-based evidence of the dominance of the guitar brace design over material and climatic variability for dynamic behaviors

Introduction

Part of this work and results and figures have been published in the PhD of first author [START_REF] Viala | Towards a model-based decision support tool for stringed musical instrument making[END_REF], whose download link is given in reference. Acoustic guitars comprise a wide range of instrument types, prices and qualities. Different families of instruments exist each with specific orientations and shapes of the stiffening ribs known as bracing patterns. During the fabrication process, guitar makers adapt the bracing design to insure a desired static and dynamic behaviours of the soundboard. It is often alleged that the design of a guitar is more important than its material properties. This implies that the design choices made by a guitar maker to achieve a desired behavior go beyond the injunction to simply use the "best'" tonweoods. Moreover, some instrument makers have purposely used low quality wood to support this idea [START_REF] Gore | Wood for Guitars[END_REF]. Nevertheless, it remains a belief, shared by instrument makers and many musicians, that high grade woods and specific species are key factors involved in achieving a desired guitar sound [START_REF] Bielski | Nonlinear modelling in time domain numerical analysis of stringed instrument dynamics[END_REF].

Considering that current wood availability issues affect more and more wood species, it is reasonable to ask if this approach remains valid.

Musicians attribute a specific sound and timbre of the instrument depending on the wood species and its anatomical features. However, this attitude is not justified and generally belied by rigorous studies. In [START_REF] Carcagno | Effect of back wood choice on the perceived quality of steel-string acoustic guitars[END_REF], it was shown that musicians could not distinguish between guitars with backs made with different wood species. The material properties of the wood of a given specie vary widely, and an overlap may be possible with other species, which may explain why a specific wood specie may not be identified. Acoustic guitar dynamics have been the subject of many studies, based on analytical and experimental approaches but also numerical methods using physics-based models. The experimental approach has been used for decades to observe the resonance modes of guitar soundboards, either isolated or when coupled with the sides and the remaining parts. As an example, it has been used in a deterministic manner for the comparison of different guitar families in relation to their bracing patterns [START_REF] Caldersmith | Designing a guitar family[END_REF]. The experimental approach is also useful to study the global dynamics of the guitar and its radiated sound by dealing with a macro response of the instrument [START_REF] Richardson | Guitar making-the acoustician's tale[END_REF][START_REF] Perry | Sound Radiation Measuremets on Guitars and Other Stringed Musical Instruments[END_REF].

More instrument making based studies have also been performed. As an example, the different fabrication steps have been studied to evaluate their impact throughout the construction process, using modal testing [START_REF] Elejabarrieta | Evolution of the vibrational behavior of a guitar soundboard along successive construction phases by means of the modal analysis technique[END_REF] or both models and experiments [START_REF] Inta | The acoustics of the steel string guitar[END_REF]. In order to highlight the variability of the guitar behaviors, even when their geometries are identical, other experimental techniques have been used, such as studying the bridge admittance [START_REF] Mansour | Post-classification of nominally identical steel-string guitars using bridge admittances[END_REF] as detailed in [START_REF] Woodhouse | Interpreting the input admittance of violins and guitars[END_REF].

Similarly, the impact of the bridge has been investigated using experimental harmonic analysis, visualisation, and simulation techniques [START_REF] Torres | Influence of the bridge on the vibrations of the top plate of a classical guitar[END_REF].

These different approaches have proven useful for the study of the impact of the bars on the guitar body response, [START_REF] Domnica | A practical evaluation method of dynamical behaviour of classical guitar bodies[END_REF] and to compare them to nominally identical numerical models of the body, where the variability of the wood was not implemented [START_REF] Curtu | Modal Analysis of Different Types of Classical Guitar Bodies[END_REF]. Using the Chladni method, it has been shown that, for a given body shape, the differences in the body dynamics made of different braces increase with the frequency [START_REF] Curtu | Correlations between the plates' vibrations from the guitar's structure and the physical, mechanical and elastically characteristics of the composite materials[END_REF]. Experimentally, at low frequencies, the deformed shapes were similar, but later the different areas created by the braces exhibited very different behaviour. Moreover, it has also been observed that, for the perception of a sound and timbre, the damping and frequency of the modes may be less important than the effective mass and area of the considered modes [START_REF] Elejabarrieta | Evolution of the vibrational behavior of a guitar soundboard along successive construction phases by means of the modal analysis technique[END_REF][START_REF] Inta | The acoustics of the steel string guitar[END_REF]. Despite the fact of studying existing instruments, experimental studies are limited when considering the relationship between instrument makers geometrical choices and the measured dynamic features. The main reasons for these limitations are the wood variability, the irreversibility of the modifications, as well as the time and cost of such approaches. To overcome these limitations, physics-based models are now considered as a powerful method that enable the study of a parameter effect, where all else unchanged. Physics-based models of guitars have been developed for decades [START_REF] Chaigne | Numerical simulations of stringed instruments-today's situation and trends for the future[END_REF]. The evolution of computational power enabled more and more sophisticated models such as the complete processus of the production of sound, from the plucked guitar string to the radiated sound [START_REF] Bécache | Numerical simulation of a gui-tar[END_REF]. One of the most complex model, combining complete structure and fluid-structure interactions, has been developed in [START_REF] Derveaux | Modélisation numérique de la guitare acoustique[END_REF] and enabled the computation of the radiated sound around the instrument.

More recently, models have reached an even higher level of detail, and simulated the interactions and collisions that occur on several kinds of acoustic guitars, such as the viola Caipira [START_REF] Paiva | Analyse modale vibroacoustique de caisse de résonance de Viola Caipira[END_REF].

In addition to the computation of the modal basis of guitar soundboards, the models have been used to compute the bridge admittance of the guitar [START_REF] Elie | Macro parameters describing the mechanical behavior of classical guitars[END_REF]. Another feature, the complex frequency domain assurance criterion has been used to evaluate the impact of a non-invasive restoration [START_REF] Pérez | Experimental assessment of the effect of an eventual non-invasive intervention on a Torres guitar through vibration testing[END_REF] and validate a model [START_REF] Viala | Towards model-based approaches for musical instruments making : validation of the model of a Spanish guitar soundboard and characterization features proposal[END_REF].

As for the violin, methods have been proposed to modify the shape of the braces to tune specific modes, and the height of the braces has been considered as the most influential parameter [START_REF] Dumond | Can a brace be used to control the frequencies of a plate ?[END_REF]. Recently, in [START_REF] Boven | Dynamic Response optimization of an acoustic guitar[END_REF], a study has performed the optimization of the geometrical characteristics of guitar soundboards (thickness of the top and height of the braces) based on the sound pressure level output.

But such methods are only justified if the brace patterns have more impact on guitar behaviour than the full variability of wood.

Hence the importance of comparing the relative influences of wood variability and brace design on the vibratory behaviour of guitars.

Toward this end, physics-based models will be used to explore the modal behavior of a large number of soundboard characterisitics taking into account uncertainty in both the material properties (spruce tonewood and braces) and the climatic conditions (temperature and humidity). Therefore, stochastic simulations are needed to describe the probabilistic nature of the parameters of a wood specie and climate.

In the next section, the finite element models and analyses are described.

Then, the results are given and discussed. Finally, a conclusion sums up the key results of this study and gives perspectives for further studies.

Models and methods

Three guitar bracing patterns will be considered with all else unchanged.

-The classical nylon string guitar (C guitar ), has a soundboard made of spruce or western red cedar. The braces are perpendicular to the soundboard grain with bars on the top part and fan-like bars on the bottom part of the soundboard.

-The steel-string acoustic guitar (A guitar ) is generally built in different body types. One of the characteristics of this type of guitar is the cross shaped braces.

-The Selmer guitar (S guitar ) is a type of steel string guitar. This style of guitar generally has a small oval or "D" shaped sound-hole along with a cutaway.

The braces are glued perpendicular to the grain.

Soundboard geometry

The models possess different brace types but the same sound-hole, rosette, bridge and cutaway. The geometries considered here are inspired from traditional templates. The template for the classical guitar is provided by the book "classical guitar making" [START_REF] Bogdanovitch | Classical guitar making[END_REF], the template for the Selmer guitar brace is provided by François Charle [START_REF] Charle | Plan guitare Selmer Maccaferri[END_REF], a luthier in Paris. The steel string guitar style braces are inspired from the book "Build your own acoustic guitar" [START_REF] Kinkead | Build your own acoustic guitar[END_REF]. The geometries of the three soundboards are made with the computer-aided design (CAD) software SOLIDWORKS ® . Top and bottom view of the soundboards are represented in the figure 1. The side of the soundboards where the bridge, saddle and rosette are represented are the same for each soundboard and their material parameters are fixed. For the three soundboards considered, only the other sides where the braces are glued are different. The thickness of each soundboard without braces is equal to 3 mm. The guitars with classical, steel string and Selmer braces will be labeled C guitar , A guitar , and S guitar , respectively.

Meshing of the assembly

The CAD are imported into PATRAN ® and the volumes (generally up to 20) are meshed with tetrahedral elements with quadratic interpolation (TET10) to increase the number of degrees of freedom in the thickness and avoid bending issues of solid elements. The mesh shows coincident nodes at the numerous interfaces. At the end of the process, the number of elements depend on the brace types and are close to 58000 elements and 105000 nodes. Once the volumes are meshed, clamped boundary conditions are applied to the contour of the soundboard. This is evidently an idealization the soundboard is in reality glued on the sides and linings and some parts of the soundboard are removed to glue the bindings. Once the mesh and the boundary conditions are completed, the material properties are introduced.

Material properties

The material properties introduced to represent the spruce variability are given in the table 2 [START_REF] Viala | Simultaneous non-destructive identification of multiple elastic and damping properties of spruce tonewood to improve grading[END_REF]. The dependence of the material properties with respect to the relative humidity is taken from [START_REF] Viala | Towards a model-based decision support tool for stringed musical instrument making[END_REF]. The guitar soundboard and bars are entirely made of spruce wood which is treated as an orthotropic material. It is defined with 12 parameters, three Young's moduli E i , three Coulomb's moduli G ij = G ji and six Poisson's ratios. The Poisson's ratio are related by the equation

ν ij E i = νji E j .
The bridge and rosette are often made of rosewood, Dalbergia which represents a wide variety of species. The properties of the rosewood parts are given in the table 1 [START_REF] Sprossmann | Characterization of acoustic and mechanical properties of common tropical woods used in classical guitars[END_REF], [START_REF] Guitard | Modèles prévisionnels de comportement élastique tridimensionnel pour les bois feuillus et les bois résineux[END_REF], [START_REF] Rossing | The science of stirng Insturments[END_REF]. It is a denser wood than spruce but exhibits similar rigidity in the longitudinal direction, and twice the rigidity in radial direction. The saddle is made of polyoxymethylene (POM). The POM is defined as an isotropic material with Young's modulus, Poisson's ratio and specific gravity equal to 3.1 GPa, 0.35 and 1.42 respectively. Once the material parameters are introduced in the model, it is possible to estimate the mass of the three soundboards, which are reported in the table 3.

The mass of the three soundboards vary with a maximum difference of 58 g, over 20 % more compared to the lightest soundboard (classical braces). The braces may significantly increase the mass which tends to inhibit the radiation of the sound. Moreover, this higher mass is supposed to decrease the resonance frequencies, although the stiffening effect of the bars may compensate this effect, especially when oriented perpendicular to the grain braces. Once the models are complete, they are used to compute the eigenmodes of each brace configuration.

The prestress state is not taken into account in the model, nevertheless, it has been shown that prestressing a plate in vibration tends to increase or decrease the eigenfrequencies of the plate depending on the degree of plate curvature [START_REF] Mamou-Mani | Précontraintes et vibration des tables d'harmonie[END_REF]. The current comparison aims to highlight trends instead of exact vibratory behaviour of the soundboards, and thus, the models are simplified.

Eigensolution calculation

The solution of the eigenvalue problem yields the undamped model eigenvalues. The modal basis is computed in the [20 ; 2500 Hz] frequency band and leads to a number of modes comprised between 40 and 50. For each eigenmode, the deformed shapes are displayed and the bridge admittance is computed with the modal superposition method. A modal damping ratio based on the experimental values is used (ξ = 1.15%) [START_REF] Viala | Towards model-based approaches for musical instruments making : validation of the model of a Spanish guitar soundboard and characterization features proposal[END_REF] for the bridge admittance calculation. In this study, a stochastic analysis is performed based on the probabilistic definition of the material parameters. For each brace configuration, 1000 computations are performed and the features of interest are extracted.

Morris sensitivity analysis

A Morris sensitivity analysis is performed [START_REF] Morris | Factorial Sampling Plans for Preliminary Computational Experiments[END_REF] as described in [START_REF] Viala | Simultaneous non-destructive identification of multiple elastic and damping properties of spruce tonewood to improve grading[END_REF]. Five stiffnesses and the density of each component, in addition to the temperature and the relative humidity are considered. The implementation of the dependence of the material properties with respect to the climatic parameters is similar to that found in [START_REF] Viala | Simultaneous non-destructive identification of multiple elastic and damping properties of spruce tonewood to improve grading[END_REF], [START_REF] Viala | Towards a model-based decision support tool for stringed musical instrument making[END_REF]. The material parameters and their nomenclature, used in the results section, are given in the table 4. In total, 14 parameters are considered for the sensitivity analysis, and the Morris sensitivity analysis is performed with 20 trajectories, leading to 301 runs (14 parameters + 1 × 20 trajectories + 1).

Modal overlap factor

The modal overlap factor (MOF) has been calculated for the defined third octave band and for all of the soundboard simulations. The low frequency domain (L.F.) is generally considered for a MOF inferior to 0.3. The mid frequency domain is considered for a MOF comprised between 0.3 and 0.7 and the MOF is calculated according to eq. 1,

M OF = M d × f c × η (1)
with M d the modal density by third octave bands, f c the central frequency of the third octave band. The modal density is given by eq. 2,

M d = ∆N m ∆f (2)
where ∆N m represents the number of modes that exist in a specific bandwidth ∆f , the bandwidths is defined as third octave band here. η is the loss factor of the system in the bandwidth and is considered here equal to 2.3 %, which is twice the modal damping value measured on guitar soundboards in [START_REF] Viala | Towards model-based approaches for musical instruments making : validation of the model of a Spanish guitar soundboard and characterization features proposal[END_REF], [START_REF] Mansour | Post-classification of nominally identical steel-string guitars using bridge admittances[END_REF].

Out of plane bridge admittance

The bridge admittance is computed on the point shown in the figure 3. The out-of-plane displacement (Z) is considered for a given input force in both X and Z directions equal to 1 N, which represents the input force of a string exhibiting motion in the XZ plane.

Correlation coefficient

In addition, a correlation coefficient is proposed to evaluate the relation between the material/climatic parameters and the bridge admittance amplitude. This feature will be used to evaluate which parameter has an influence on the value of the bridge admittance, for each frequency step, and gives the effective domains of a parameter in regard with the vibratory response of a given structure. For two variables X i and X j , the correlation coefficient is given by the eq. 3.

C = Cov(X i , X j ) σ X i σ X j (3)
With Cov(X i , X j ) the covariance between the variables X i and X j , and σ X i and σ X j their respective standard deviation.

Results & discussion

In this section, the results obtained with the 1000 computations of each cases are given. Firstly, a basic comparison of the modal basis of the three cases with initial values are given to qualitatively compare the soundboards.

Deterministic results

The modal bases of each nominal configuration are given in this section. The [START_REF] Mihalcica | Frequency response evaluation of guitar bodies with different bracing systems[END_REF], A guitar [START_REF] Carcagno | Effect of back wood choice on the perceived quality of steel-string acoustic guitars[END_REF] and S guitar [START_REF] Lee | Measuring and Understanding the Gypsy guitar[END_REF] respectively. Nevertheless, the high variability in term of soundboard shape and thickness, sound hole size and brace modifications prevent from consolidated test-model comparisons. The soundboards exhibit dipoles in both the X and Y directions, not necessarily in the same order. As an example, the frequency of the dipole mode of S guitar in the X direction is much higher than the other cases, which can be easily linked to the bars mainly glued in the X direction, where the rigidity of the soundboard (radial direction) is much smaller than in the Y (longitudinal) direction. Besides these simple modes, it is seen that the deformed shapes become rapidly more complex and the modes of the different configurations are no longer comparable as shown by the Modal Assurance Criterion. The detailed MAC of the modal basis of each configuration compared two at a time is given in the table 5.

This table shows that, out a total of 50 modes computed for each case, only a few modes exhibit close deformed shapes in the low frequencies domain, which is coherent with the results of [START_REF] Skrodzka | Modal parameters of two incomplete and complete guitars differing in the bracing pattern of the soundboard[END_REF], [START_REF] Domnica | A practical evaluation method of dynamical behaviour of classical guitar bodies[END_REF], [START_REF] Curtu | Modal Analysis of Different Types of Classical Guitar Bodies[END_REF] and [START_REF] Curtu | Correlations between the plates' vibrations from the guitar's structure and the physical, mechanical and elastically characteristics of the composite materials[END_REF]. The matched eigenfrequencies error is very high and it is clearly seen that the vibratory behaviour of the soundboards is very different in the considered frequency domain and for the nominal designs. The bridge admittances are shown in the figure 7. These curves are constructed with the modal bases and thus are consistent with the eigenfrequencies and the eigemode shapes at the driving point used for the admittance synthesis. Considering the first peak related to mode 1, a large frequency discrepancy is observed. Nevertheless, above 400 Hertz, these differences are smaller and bridge admittances can be similar despite the differences observed in the eigenmode shapes. The bridge admittance is a local evaluation and does not reflect the complete shapes of the eigenmodes.

Morris screening analysis results

Figure 8 shows the results of the Morris sensitivity analyses for each brace configuration. These results highlight a small dominance of the soundboard properties over the brace properties, regardless of the brace pattern. The longitudinal specific modulus and densities of the soundboard and braces are the most important, and the impact of the relative humidity is relatively high. Then the specific radial modulus followed by the shear modulus in plane LR. Moreover, it is shown that the ranking of the parameters depends of the brace configuration. For the remainder of the stochastic study, only the longitudinal and radial Young's moduli, and LR shear modulus of the soundboard and braces have been considered as uncertain elastic parameters.

Modal overlap factor

The modal overlap factors for the three soundboard configurations are displayed in figure 9 and detailed in the table 6. This figure shows that the modal overlap factor values and evolutions are similar for each soundboard, even though these structures exhibit very different modal behaviour. The Selmer guitar exhibits nevertheless a smaller modal overlap factor between 1250 and 2500

Hz. The mid frequency domain is reached for a frequency value comprised between 800 and 1200 HZ, which is the limit of a modal point of view. The high frequency domain is reached for frequencies comprised between 1600 and 2000

Hz. Above these values, modal analysis is no longer relevant since modes can no longer be differentiated, and energy methods should be preferred [START_REF] Boutillon | Vibroacoustics of the piano soundboard : Reduced models, mobility synthesis, and acoustical radiation regime[END_REF].

Stochastic results

The results of the 1000 computations are shown using different formats : the frequency dispersion of the first ten modes of the guitar soundboards, the statistics and fuzzy-FRF of their admittance, and the correlation coefficient between the parameters and the value of the FRF for each frequency step. The table 7 indicates the dispersion in the eigenfrequencies of a soundboard given the variability in its material parameters. The RSD of the eigenfrequencies of each cases is close to ± 5.5 %, which means that up to 68 % of the computed eigenfrequencies are between this value. Hence, in order to ensure that a design leads to different dynamics, the difference in the eigenfrequencies should be higher than 5.5 %.

All the bridge admittances are summed up in the figure 10 Numerous other correlations with parameters can be observed for the different braces patterns, as a function of the frequency bands. The results of the Morris sensitivity analysis are correlated, that is to say, the properties of the soundboard are most influential along with the relative humidity and the temperature. For the S guitar , the braces seem to have a higher impact on the overall response.

For this configuration the braces are larger and oriented perpendicular to the grain direction, which is a direction where the soundboard is significantly less stiff. Another output is observed where the figure shows some fringes. These fringes highlight the fact that some frequencies admittances are not influenced by any of the input parameters separately, as the coefficient correlation is equal to zero for each parameter, which is associated with an eigenfrequency.

Discussion

A screening analysis has demonstrated that longitudinal specific moduli and densities of soundboard and braces have the main impact on eigenfrequencies of the clamped soundboards, followed by the relative humidity, which is consistent with wood selection habits of guitar makers. The results show that material variability affect the dynamic features of guitars, such as eigenfrequencies, eigenmodes shapes and bridge admittances shape. Moreover, it is observed that the relative humidity also has a strong impact on the same features.

Nevertheless, it has been shown that the guitar maker design choices have a stronger impact on the dynamics of the soundboards. Indeed, the few matched eigenfrequencies are shifted in a more important way than with material and climatic changes. This highlights the dominant impact of braces on the low frequencies, in particular on the matched eigenmodes. For higher frequencies, the mode shapes are too different and they can no longer be matched between the different brace configurations. Moreover, it has been highlighted that for the perception of a sound and timbre, the damping and frequency of the modes may be less important than the effective mass and the zones of the considered modes [START_REF] Elejabarrieta | Evolution of the vibrational behavior of a guitar soundboard along successive construction phases by means of the modal analysis technique[END_REF]. This result is confirmed by the stochastic analyses results, especially fuzzy-FRF of each brace pattern configuration. There is no compensation effect due to the material variability that can lead to a similar vibratory behaviour of the structure. Hence, the design choices of the guitar makers can be considered as more influential than the wood choices and climatic changes, considering dynamic features such as eigenfrequencies, eigenmode shapes and bridge admit-

tances. An important perspective to consolidate these findings is to perform perceptive tests, with sounds synthesized with the modal bases using a string model and verify that, whatever wood or climate variability, listeners would still clearly differentiate each type of brace. The correlation coefficient proposed here for material and climatic parameters is a key tool to highlight the link between the admittance amplitudes and the input parameters as a function of frequency. This post-processing tool, associated with the geometrical input parameters, may be used for further tuning of the guitar soundboard to reach a desired response in a reduced frequency band. The relative humidity appears to be important, but it has to be pointed out that the evolution of the material parameters of the wood when undergoing climatic changes are considered for a hygroscopic equilibrium state of the wood. In reality, these effects are not immediate and it is assumed that the wood stiffness is not fully modified by a climatic change. One of the main perspective of this work is to study small differences for each given brace pattern, by changing geometrical parameters of each brace one by one. Also, the impact of the different materials for the braces and soundboards (like other species or composites) with similar geometry needs to be investigated.

Conclusion

In this paper, a physics-based model has been used to perform stochastic analyses on guitar soundboards with different brace patterns, corresponding to classical, steel-string and Selmer guitars. The comparison of the influence of material and climatic variability and guitar maker design choices has been performed in the dynamic domain. The results have shown that brace patterns have a higher impact on the dynamic features of a soundboard than material and climatic variability. The choice of a brace pattern family leads to dynamic behaviours that are not comparable with those of another family. This effect is not clearly apparent when considering the average and upper and lower bounds of the bridge admittances, but the fuzzy-FRF representation proposed gives more interesting insight. The main conclusion of this paper is that the design choices of brace shapes have a dominant impact on soundboard dynamics and that variations in wood density and stiffness is a second order effect. From an organological point of view, this provides a partial explanation for the existence of different guitars subcategories. Theses results and the methods proposed herein can provide decision support tools for instrument making, taking into account both geometrical and material changes. Futhermore, this approach can provide a basis for the robust optimisation of the guitar bracing patterns in order to develop designs that reproduce specific dynamic behaviours, for example of a specific appreciated instrument, even though the material and climate conditions may vary.

Name

Description

Sb E L Soundboard E L (MPa) Sb E R Soundboard E R (MPa) Sb G LR Soundboard G LR (MPa) Sb G RT Soundboard G RT (MPa) Sb G T L Soundboard G T L (MPa) Sbρ Soundboard density (g cm -3
) A guitar , (c) S guitar .

Bars E L Bars E L (MPa) Bars E R Bars E R (MPa) Bars G LR Bars G LR (MPa) Bars G RT Bars G RT (MPa) Bars G T L Bars G T L (MPa) Barsρ Bars density (g cm -3 ) T Temperature ( • C) RH Relative humidity (%)
Sb E L ρ 28% Braces E L ρ 22% RH 13% Sb ρ 7% Braces ρ 12% Sb E R ρ 8% Sb G LR ρ 7% Sb G RT ρ 3% (a) Sb E L ρ 31% Braces E L ρ 17% RH 15% Sb ρ 7% Bracesρ 4% Sb E R ρ 11% Sb G LR ρ 6% Sb G RT ρ 4% Braces G T L ρ 5% (b) Sb E L ρ 23% Braces E L ρ 24% RH 14% Sb ρ 6% Braces ρ 10% Sb E R ρ 7% Sb G LR ρ 6% Sb G RT ρ 3% Braces G T L ρ 7% (c)

  three modal bases possess up to 50 modes in the [20 ; 2500 Hz] frequency band. The first eight modes of each family of soundboard brace patterns are displayed in the figures 4, 5, and 6. The deformed shapes are shown for comparison : the first mode of each soundboards is a monopole mode, which is an expected result, nevertheless, the zone associated with this mode differs for each brace family. Considering the frequency of the first mode, the average value is equal to 216, 169 and 291 Hz for C guitar , A guitar and S guitar , respectively, with a coefficient of variation comprised between 5 and 6 %. Although the studied soundboard, thickness shape (with a cutaway), bridge type and sound hole size is a mix of different type of guitars, the trends are similar for real soundboards, whose first monopole is close to 200, 188 and 250 Hz for C guitar

  which displays mean admittance and lower and upper limit for each brace pattern. Up to 350 Hz, the FRF shapes of each cases are very different. The A guitar exhibits the lowest eigenfrequency of the first mode and the lowest absolute dispersion([145 ;200], centered at 170 Hz). The C guitar exhibits intermediate frequency of the first mode and absolute dispersion ([180 ;250], centered at 215 Hz). The S guitar exhibits the highest frequency of the first mode, suggesting much higher stiffness (also considering its higher mass) and absolute dispersion([250 ;340] Hz, centered at 290 Hz). The relative dispersion of each FRF upper and lower limits are all close to 30 %. The difference between the mean FRFs decreases above 400 Hz. The C guitar and S guitar bridge admittances are quite similar between 400 and 800 Hz, in term of average level and lower and upper FRF limits. Considering these features, between 600 and 750 Hz, the A guitar levels are very different from the two other configurations. Despite these observations, above 400 Hz the bridge admittances seem to merge together.These results have to be compared with those given by the fuzzy-FRFs, displayed in the figure11. In this figure, the dispersion in both amplitude and frequency of the bridge admittance is given for the three bracing patterns. The impact of the first mode on the low frequency admittance of the bridge is clearly seen for each type of braces. The maximum corresponding admittances of each case reach a similar value, close to 0.3, 0.5 and 0.2 mm/N for C guitar , A guitar and S guitar , respectively. Above the frequency of the first mode, it is shown that the three bridge admittances are not superposed, even if the material properties and the climate conditions may strongly vary. The amplitudes are not similar above 300 Hz, and each configuration exhibits complex responses, which is highlighted by this probabilistic point of view.In order to study the influence of the previous parameters with respect to the frequency bands, the correlation coefficient is shown in figure12. These results show that, depending on the brace pattern, the correlation between each material and climatic parameters and bridge admittance sampling varies. The longitudinal stiffness of the soundboard and the bars is strongly correlated with bridge admittance up to 220, 160 and 290 Hz for C guitar , A guitar and S guitar , respectively. The RH is correlated with the FRF amplitude at 280 (which is also the case for E L of the bars) and 420 Hz especially, for C guitar . For A guitar , the RH is correlated with FRF magnitude especially in [220 ;340] (which is also the case for E L and E R of the soundboard) and [600 ;650] Hz frequency bands.

Figure 1 -

 1 Figure 1 -Computer aided designs of : (a) top view, (b) C guitar , (c) A guitar , (d) S guitar .

Figure 2 -

 2 Figure 2 -(a), dimensions of the soundboard ; (b), dimensions of the rosette ; (c), dimensions of the bridge.

Figure 3 -Figure 5 -

 35 Figure 3 -Bridge admittance evaluation and input force position and direction.

Figure 7 -

 7 Figure 7 -Nominal FRF with inital material and climatic parameters.

Figure 8 -

 8 Figure 8 -Material screening analysis on matched eigenfrequencies of : (a) C guitar , (b)

Figure 9 -

 9 Figure 9 -Modal overlap factor as a function of the third octave bands, for three different cases of guitar bars

Figure 10 -

 10 Figure 10 -Mean and upper and lower limits for each case of braces.

FrequencyFigure 11 -

 11 Figure 11 -Fuzzy FRF of the bridge admittance for : (a) C guitar , (b) A guitar , (c) S guitar .

Figure 12 -Frequency

 12 Figure 12 -Coefficient of correlation between bridge admittance and material parameters : (a) C guitar , (b) A guitar , (c) S guitar .

Table 1 -

 1 Material properties of indian rosewood for bridge and rosette[START_REF] Sprossmann | Characterization of acoustic and mechanical properties of common tropical woods used in classical guitars[END_REF][START_REF] Guitard | Modèles prévisionnels de comportement élastique tridimensionnel pour les bois feuillus et les bois résineux[END_REF][START_REF] Rossing | The science of stirng Insturments[END_REF].

	Rosewood (Dalbergia)
	Material parameter Value
	E L (GPa)	13.3
	E R (GPa)	1.7
	E T (GPa)	1.0
	ν LR	0.38
	ν RT	0.49
	ν T L	0.02
	G LR (GPa)	0.93
	G RT (GPa)	0.2
	G T L (GPa)	0.8
	d (-)	0.79

Table 2 -

 2 Material properties of spruce implemented in the models, italic from[START_REF] Viala | Simultaneous non-destructive identification of multiple elastic and damping properties of spruce tonewood to improve grading[END_REF] at M C = 10%. Remaining values from[START_REF] Guitard | Modèles prévisionnels de comportement élastique tridimensionnel pour les bois feuillus et les bois résineux[END_REF] and[START_REF] Brémaud | Anisotropy of wood vibrational properties : Dependence on grain angle and review of literature data[END_REF].

		Parameter	Spruce Min. value Max. value
	E L ρ (MPa g -1 cm -3 )	29000	20590	35380
	E R ρ (MPa g -1 cm -3 )	2280	1460	3810
	E T ρ (MPa g -1 cm -3 )	1480	1300	1660
		ν LR (-)	0.37	-	-
		ν RT (-)	0.48	-	-
		ν T L (-)	0.02	-	-
	G LR ρ	(MPa g -1 cm -3 )	1850	1295	2442
	G RT ρ	(MPa g -1 cm -3 )	100	74	150
	G T L ρ	(MPa g -1 cm -3 )	1910	1070	2750
	Density (g cm -3 )	0.44	0.39	0.51
	Relative humidity (%)	50	20	85
	Temperature ( • C)	21	15	35

Table 3 -

 3 Physical properties of the soundboard models.

	Parameter	C guitar	A guitar	S guitar
	Mass (g)	237	244	295
	Volume (cm 3 )	516	571	647

Table 4 -

 4 Name and description of the parameters considered for the analysis.

Table 5 -

 5 Matched eigenfrequencies error (MEE) and MAC value of the three cases with nominal values of the parameters.

		C guitar		S guitar
	Mode Frequency (Hz) Mode Frequency (Hz)	MEE (%)	MAC (%)
	1	214.6	1	288.7	34.5	98.9
	2	353.5	4	645.0	82.5	88.1
	3	400.3	2	403.5	0.8	89.9
		C guitar		A guitar
	Mode Frequency (Hz) Mode Frequency (Hz)	MEE. (%) MAC (%)
	1	214.6	1	167.5	-22.0	99.5
	2	353.5	2	407.4	15.3	87.7
	3	400.3	3	442.7	10.6	88.3
	4	501.0	4	512.3	2.3	65.0
	8	750.8	6	770.6	2.6	55.4
	10	844.6	8	873.5	3.4	52.5
	12	965.5	9	953.5	-1.2	58.7
		S guitar		A guitar
	Mode Frequency (Hz) Mode Frequency (Hz)	MEE (%)	MAC (%)
	1	288.7	1	167.5	-42.0	99.2
	2	403.5	3	442.7	9.7	73.0
	3	532.7	4	512.3	-3.8	76.9
	4	645.0	2	407.4	-36.8	84.2

Table 6 -

 6 Values of modal overlap factor (%) for corresponding third octaves bands for different cases of guitar bars and frequency domain.

	Third octave band (Hz)	C guitar (%)	A guitar (%)	S guitar (%)	Domain
	160	0	9	0	L.F.
	200	7	0	0	L.F.
	250	2	0	3	L.F.
	315	7	0	3	L.F.
	400	14	14	2	L.F.
	500	18	16	9	L.F.
	630	16	9	14	L.F.
	800	37	32	21	M.F./L.F.
	1000	39	34	28	M.F./L.F.
	1250	48	46	46	M.F.
	1600	83	96	70	H.F./M.F.
	2000	101	97	87	H.F.
	2500	152	143	124	H.F.

Table 7 -

 7 Frequency mean values and absolute and relative standard deviation (RSD) of the three cases for a normal distribution of the frequencies.

	Mode	C guitar		A guitar		S guitar
		µ and SD (Hz) RSD (%)	µ and SD (Hz) RSD (%)	µ and SD (Hz) RSD (%)
	Mode 1	215.9 ±11.3	±5.1	168.9 ±9.2	±5.5	291.1 ±16.5	±5.6
	Mode 2	355.9 ±19.5	±5.4	413.9 ±23.8	±5.7	406.1 ±22.3	±5.4
	Mode 3	403.4 ±21.3	±5.2	448 ±24.6	±5.5	534 ±28.8	±5.3
	Mode 4	502.5 ±27.3	±5.4	511.1 ±29.2	±5.7	658.1 ±40.0	±6.0
	Mode 5	541.5 ±29.1	±5.3	743.5 ±40.1	±5.4	731.9 ±40.6	±5.2
	Mode 6	653.9 ±37.8	±5.8	774.4 ±42.6	±5.5	757.7 ±42.3	±5.3
	Mode 7	737.1 ±40.0	±5.4	848.7±46.5	±5.5	930 ±51.1	±5.3
	Mode 8	750.8 ±42.8	±5.7	888.5 ±49.6	±5.6	960.9 ±55.2	±5.6
	Mode 9	832.9 ±49.6	±5.6	959.4 ±54.8	±5.7	1101 ±60.8	±5.4
	Mode 10	858.4 ±45.9	±5.3	1006 ±57.6	±5.7	1159 ±64.8	±5.5
	Avg :	-	±5.4	-	±5.6	-	±5.5

Acknowledgements

Funding : This work has been performed in the Framework of EUR EIPHI (ANR-17-EURE-0002).