Martin Bretzner

Renaud Lopes

Ray Mccarthy

Delphine Corseaux

Florent Auger

Gillian Gunning

Nicolas Beauval

Antonino Bongiovanni

Meryem Tardivel

Charlotte Cordonnier

Florent Delphine Corseaux D E F

Charlo Tte Cordonnier

Jean- Pierre Pruvo

Sophie Susen D E F

Xavier Leclerc

Grégory Kuchcinski

Texture parameters of R2* maps are correlated with iron concentration and red blood cells count in clot analogs: A 7-T micro-MRI study

Keywords:

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Introduction

Acute ischemic stroke treatment (AIS) using intravenous thrombolysis (IV t-Pa) and/or mechanical thrombectomy (MT) aims at a quick recanalization of the occluded artery.

Despite the improvement of the reperfusion drugs and devices, flow restoration fails to be achieved in 29% of patients resulting in unfavorable clinical outcome [START_REF] Goyal | Endovascular thrombectomy after large-vessel ischaemic stroke: a meta-analysis of individual patient data from five randomised trials[END_REF]. Clots responsible for intracranial occlusions remained unavailable for examination until the recent proliferation of histopathological studies, secondary to MT, provided new insights about their composition.

They tend to be heterogeneous and are often classified by the relative composition of two main components, red blood cells and fibrin, the quantities of each are inversely correlated [START_REF] De Meyer | Analyses of thrombi in acute ischemic stroke: a consensus statement on current knowledge and future directions[END_REF]. These constituents have been associated with procedural outcome: RBC-rich clots being easier to retrieve during mechanical thrombectomy and more sensitive to intravenous fibrinolysis [START_REF] Brouwer | Clot pathophysiology: why is it clinically important?[END_REF]. Other thrombus components that play major roles in thrombosis and clot make-up are: platelets, von Willebrand factor (VWF), neutrophil extracellular traps (NETs) and leucocytes. In line with these data, additional reperfusion strategies are being developed. New devices tailored for challenging fibrin-rich clots or clots that are prone to fragmentation are being benchmarked [START_REF] Fennell | What to do about fibrin rich 'tough clots'? Comparing the Solitaire stent retriever with a novel geometric clot extractor in an in vitro stroke model[END_REF]. New drugs, such as ADAMTS-13, a VWF specific protease, or DNAse1 targeting NETs, have been proposed to increase the performance of IV t-Pa [START_REF] Ducroux | Thrombus neutrophil extracellular traps content impair tPA-induced thrombolysis in acute ischemic stroke[END_REF], [START_REF] Denorme | ADAMTS13-mediated thrombolysis of t-PA-resistant occlusions in ischemic stroke in mice[END_REF]. However, to be of most benefit, real-time selection of the optimal recanalization strategy based on histology is needed. As such, histological composition of the clot should be assessed at the time of the diagnosis, during pre-therapeutic imaging. The blooming artifact in MRI, has been attributed to a predominant RBC fraction within clots. However, quantitative methods able to accurately predict the RBC concentration are lacking. R2* relaxometry is a non-invasive method capable of quantifying the MR signal drop related to the presence of paramagnetic substances and R2* values have been linearly correlated with brain iron concentration in post-mortem studies [START_REF] Langkammer | Quantitative MR Imaging of brain iron: a postmortem validation study[END_REF]. RBC are iron-rich cells because of the high iron content in hemoglobin, an iron binding protein. Thus, we hypothesized that R2* relaxometry could quantify iron content within clot analogs and that iron content is correlated with RBC concentration.

Beyond RBC percentage, the study of the spatial distribution of the clot's constituents could provide some clue about its mechanical property or its nature. Texture analysis of MR images is an emerging field in radiology that enables automatic high-throughput extraction of quantitative features of an image that are invisible to the naked eye. Results show promise in predicting histological phenotypes of tumors [START_REF] Aerts | Decoding tumour phenotype by noninvasive imaging using a quantitative radiomics approach[END_REF], [START_REF] Wu | Exploratory study to identify radiomics classifiers for lung cancer histology[END_REF]. Although mostly evaluated in the field of oncology, this approach could help to distinguish histological subtypes of clots.

The aim of the study was to analyze texture parameters of R2* maps of clot analogs on a 7-T micro-MRI and to evaluate their relationship with RBC and iron concentration.

Material and methods

Material

Twelve clot analogs were generated using a previously described methodology to obtain a wide range of RBC concentrations [START_REF] Duffy | Novel methodology to replicate clot analogs with diverse composition in acute ischemic stroke[END_REF]. Briefly, clot analogs of increasing RBC concentrations were formed under static conditions in standing vials using ovine venous blood.

MR acquisitions

All MR acquisitions were performed on a 7-T micro-MR system with a single channel surface coil (BioSpec Brucker, Billerica, Massachusetts, US). A 3D multi-echo gradient-echo sequence was performed using the following parameters: FOV = 16 × 16 × 16 mm; matrix = 128 × 128 × 128, voxel = 0.125 × 0.125 × 0.125 mm; TEs = 7, 14, 21, 28 ms; TR = 50 ms; flip angle = 15°; number of excitations = 7; total acquisition time = 95 min.

All clots were fully immerged into separate vials filled with PBS (phosphate buffer saline 10%) at room temperature (18-22 °C) and leaned on a dedicated support.

Immediately after MR acquisitions, clots were cut into two parts. The first part was frozen at -80 °C for later iron quantification. The second part was fixed into paraformaldehyde for histopathological analysis.

MR imaging analysis

Texture analysis

First-order and second-order statistical parameters derived from the histograms were calculated using the pyradiomics toolbox [START_REF] Griethuysen | Computational radiomics system to decode the radiographic phenotype[END_REF]. First order parameters described the histograms of voxel intensities using common metrics such as mean, median, variance, skewness, kurtosis, uniformity, entropy, total energy and root mean squared. The second order parameters described the relationship between intensity and spatial distribution. In order to reduce the number of parameters, we analyzed texture using only a single methodology. There are no guidelines regarding the method to use. We hypothesized that GLRLM would be the best suited to evaluate the distribution of RBC along fibrin fibers.

Iron quantification

Clot analogs were accurately weighted and mineralized 25 min at 100 °C with 7.7 mol/L nitric acid solution in ultrapure water, before further iron concentration determination by graphite furnace electrothermal atomic absorption spectrometry (AAnalyst 800, Perkin Elmer, Waltham, USA).

Histopathological analysis

All the samples were fixed in paraformaldehyde, embedded in paraffin and then cut. Slices were stained using Martius Yellow (RBC) and Scarlet Red (Fibrin) then scanned with a digital slide scanner (Axio ScanZ1, Zeiss, Oberkochen, Allemagne). Erythrocytes were automatically counted using a homemade macro on ImageJ based on local maxima filtering [START_REF] Schindelin | Fiji: an open-source platform for biological-image analysis[END_REF]. Performance of the automatic count macro was assessed against manual labelling of RBC on a validation subset of 10 microscope fields with an excellent correlation coefficient (r 2 = 0.91).

Statistical analysis

Statistical analyses were performed using SPSS

Results

Clots main characteristics are presented in Table 1. RBC count was strongly correlated with iron concentration within clots (r = 0.87, P < .001) as shown in Fig. PCA performed on texture parameters demonstrated that two components were necessary to explain 81% of the variance of the data. The addition of a third component marginally improved the explained variance. The loading plot of the PCA (Fig. 3) showed that 2 families of features could be identified and that second-order parameters provided different information compared to first-order parameters. The first 2 components of the PCA explain 81% of the variance. First order features GLRLM features significantly correlated with iron concentration or RBC count are marked with a full dot.

Discussion

Using experimental stroke clot analogs, we confirmed the hypothesis that R2* values and texture parameters were linearly correlated with iron content and RBC count. RBC rich clots were associated with higher R2* values and more coarse R2* maps.

This result is in accordance with recent publications of Bourcier et al. [START_REF] Bourcier | MRI quantitative T2* mapping to predict dominant composition of in vitro thrombus[END_REF] and Spencer et al. [START_REF] Christiansen | Drangova Simultaneous R2* and quantitative susceptibility mapping measurement enables differentiation of thrombus hematocrit and age: an in vitro study at 3[END_REF] that showed R2* values could accurately identify RBC rich clots on clinical 1.5 Tesla and 3 Tesla MRI, respectively. The main strength of our study is the use of a 7-Tesla magnetic field. As the main magnetic field increased, the T2* relaxation time would be shorter, and the susceptibility effect of paramagnetic substances would be increased. Another benefit is the increase of the spatial resolution. The blooming artifact on T2* or SWI images is a well-known imaging feature of vessel occlusion in ischemic stroke and has been previously suspected to reflect RBC-rich clots but experimental validation of such hypothesis was lacking [START_REF] Kim | Park Histologic analysis of retrieved clots in acute ischemic stroke: correlation with stroke etiology and gradient-echo MRI[END_REF]. Unlike the quantitative approach we propose, it relies on a binary qualitative evaluation and is not able to depict the broad range of RBC concentrations as revealed by histological studies of clots retrieved during endovascular procedures [START_REF] De Meyer | Analyses of thrombi in acute ischemic stroke: a consensus statement on current knowledge and future directions[END_REF]. R2* mapping also has the advantage of overcoming the variability related to the settings of the acquisition parameters such as the echo time [START_REF] Carneiro | MRI relaxometry: methods and applications[END_REF].

Histopathological studies also revealed the intra-clot heterogeneity and complexity of the clots retrieved during MT. Beyond clot's main composition (RBC rich vs. RBC poor), the texture analyses performed in this study provide insights about the spatial distribution of the clot's constituents. Although it could seem confusing that RBC rich clots showed both a higher 'uniformity' and a more coarse texture on R2* maps, the mathematical meaning is not contradictory. First order parameters such as 'uniformity' are calculated from the image histograms and explore intensity values while second order features (GLRLM) provide information about the spatial distribution of those values. A higher 'uniformity' means the pixels tend to show similar intensity values through the entire clot and a more 'coarse' texture as defined by run length means that the clot consists in larger areas with similar gray level value. In this study, R2* texture analyses showed more long gray level runs in RBC-rich clots and more short gray level runs in fibrin rich clots. According to our findings, clot analogs rich in RBC seemed to be comprised of larger patches of homogeneous texture that could suggest the presence of larger islets of RBC and/or scarcer mesh of fibrin. Among second order features, principal component analysis highlighted 'low gray level run emphasis' as a potential discriminator of clot subtype. This parameter was higher in fibrin rich clots and indicated a higher prevalence of contiguous pixels with low intensity values. Fibrin rich clots have been reported to increase friction and thus render retrieval more challenging [START_REF] Gunning | Clot friction variation with fibrin content; implications for resistance to thrombectomy[END_REF]. Some thrombectomy devices targeted to these challenging clots are being benchmarked and will soon be available for use in clinical practice. As such, prior knowledge of the clot subtype could help plan the procedure and speed up reperfusion. The percentage of RBC and fibrin in clots affects its mechanical properties and response to reperfusion therapy, but another process plays an important part: the clot contraction [START_REF] Kim | Quantitative structural mechanobiology of platelet-driven blood clot contraction[END_REF]. This natural process occurs over time as platelets and fibrin fibers form a dense meshwork compressing the erythrocytes into tightly-packed arrays of polyhedrocytes making the clot stiffer and impeding the exchange rate with the external environment [START_REF] Cines | Clot contraction: compression of erythrocytes into tightly packed polyhedra and redistribution of platelets and fibrin[END_REF]. In a recently published study, Tutwiler et al. reported contraction to hamper the rate of fibrinolysis by a 4fold factor [START_REF] Tutwiler | Blood clot contraction differentially modulates internal and external fibrinolysis[END_REF]. In the present study we found that fibrin rich clots were significantly smaller than other clots, which could express a more contracted state. Assessing the contracted vs non-contracted state of thrombi at the acute phase of stroke remains for now inaccessible but texture analysis of clot's MRI images could bring some answers in the future.

Our study has several limitations. First, it was designed as a proof of concept study on ex vivo stroke clot analogs, which were more homogeneous than in vivo human stroke thrombi.

Although texture parameters did add some value to the analysis, the association with the RBC count remained stronger with R2* histogram features. Validation of our results on heterogeneous clot analogs and/or real life thrombi is mandatory. High magnetic field MRI enabled us to acquire images with a 0.125-microns spatial resolution, which is still far from achievable in clinical routine brain MRI. However, experimental studies like this one give direction for further clinical application. Secondly, only 12 clot analogs were analyzed, further validation on a larger sample of human thrombi is needed. Finally, we did not normalize the quantitative features extracted nor tested different neighborhood window sizes, this could hamper results reproducibility.

Conclusions

Texture analysis of MRI R2* maps can accurately quantify the red blood cell count and iron content of in vitro stroke clot analogs. This brings out new neuroimaging biomarkers to improve treatment strategy in acute ischemic stroke.

  [START_REF] Laridan | Neutrophil extracellular traps in ischemic stroke thrombi[END_REF].0 software (SPSS Inc., Chicago, IL, USA). Quantitative parameters were expressed as median (interquartile range). Three clusters of clots were identified: poor clots with low RBC count (< 0.02 RBC/μm 2 ) and very low iron concentration (< 10 μg/g), intermediate clots with intermediate-to-high RBC count (0.07-0.11 RBC/μm 2 ) and intermediate iron concentration (200-500 μg/g), and rich clots with high RBC count (> 0.10 RBC/μm 2 ) and high iron concentration (> 700 μg/g). Characteristics were compared in the three groups using the Kruskal-Wallis test. Correlations between R2* parameters and (a) iron concentration and (b) RBC count were assessed using Pearson's correlation coefficients (r). Statistical testing was done with α risk levels of 0.05 and 0.01. In order to deal with type 1 error in multiple testing the false discovery rate was calculated and P values were adjusted using the Benjamini and Hochberg method. Data was explored using principal component analysis (PCA) to understand the information provided by the different texture parameters.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Dot plot of iron concentration by RBC count in clot analogs. RBC count was linearly correlated to iron concentration (r = 0.87, P < .001). Three clusters were visually observed: poor clots (blue) with low RBC count (< 0.02 RBC/μm 2 ) and very low iron concentration

Fig. 3 .

 3 Fig. 3. Loading plot of the principal component analysis performed on texture parameters.

Table 1 .

 1 Characteristics of the clots analogs (n = 12).

	Characteristics		All (n = 12)	clots	Clot concentration
	Empty Cell		Empty Cell	Poor (n = 3)	Intermediate (n = 4)	Rich (n = 5)	Adj P
	Volume (mm 3 )		202.7 (110.0-256.1)		91.2 (75.0-96.0)	202.7 253.8) (170.0-	250.1 258.0) (217.6-	.04
			30.97		26.63	29.80	32.89
	Mean R2* (s -1 )		(27.03-		(25.35-	(27.75-	(31.60-	.03
			32.57)		27.16)	31.27)	33.12)
	RBC (10 5 cells/mm 2 )	concentration	9.49 (1.81-11.20)	0 (0-0)	8.03 (7.25-9.50)	11.20 11.45) (11.19-	.02
	Iron concentration (μg/g)	(69.3-854.4) 420.6	2.7

1. Illustrative examples of poor, intermediate and rich clots with corresponding R2* histograms are presented in Fig. 2.

Table 2 .

 2 Higher RBC count and iron concentration were significantly correlated with:

 •a global positive shift of the R2* histogram with higher '10th percentile', 'median', 'mean' and '90th percentile';  •an increase of the global magnitude of voxel values with higher 'total energy' and 'root mean squared';  •a greater uniformity of the voxel values with higher 'uniformity' and lower 'entropy'.

Table 2 .

 2 Correlation between texture parameters derived from R2* histograms, iron concentration and RBC count (n = 12). are adjusted the for the false discovery rate using the Benjamini & Hochberg method. **: Result is significant at the 0.01 level. *: Result is significant at the 0.05 level.

	Texture parameters	Iron concentration	RBC concentration

  Denorme et al. reported VWF to be inversely correlated to the RBC content in human stroke clots retrieved during MT and showed that the addition of ADAMTS13 to IV t-Pa could reduce the infarct size in a stroke mice model[START_REF] Denorme | ADAMTS13-mediated thrombolysis of t-PA-resistant occlusions in ischemic stroke in mice[END_REF]. Boeckh-Behrens et al. found leucocytes to be inversely correlated to the RBC fraction of the clot, and Laridan et al. reported NETs to be co-localized in leucocyte rich portions of human stroke thrombi and showed that adding DNAse1 to IV t-Pa could dissolve stroke clots faster ex vivo[START_REF] Laridan | Neutrophil extracellular traps in ischemic stroke thrombi[END_REF],[START_REF] Boeckh-Behrens | The impact of histological clot composition in embolic stroke[END_REF]. Imaging studies of the stroke clot at the acute

	phase could help by personalizing thrombolysis through adding ADAMTS13 and/or DNAse1
	to IV t-Pa, specifically targeting VWF and NETs, improving reperfusion and therefore
	clinical outcome.

As MRI performance improves, radiologists are able to see more and more hints of the clot histopathology. While our clot analogs were quite homogeneous, real stroke clots are complex, Duffy et al. studied the fragments retrieved per pass of mechanical thrombectomy of stroke patients and reported the first 2 passes to mainly retrieve RBC rich fragments while fibrin rich fragments were more difficult to remove

[START_REF] Duffy | Per-Pass analysis of thrombus composition in patients with acute ischemic stroke undergoing mechanical thrombectomy[END_REF]

. Texture analysis of real stroke clots would be relevant because of this mixed layout and could help with selection of the most appropriate thrombectomy device prior to the first attempt.

Funding

This research did not receive any specific grants from funding agencies in the public, commercial or not-for-profit sectors.


Matthieu Cordonnier

Nicolas Keriven

Nicolas Tremblay

Samuel Vaiter

Convergence of Graph Neural Networks with generic aggregation functions on random graphs

On s'intéresse au comportement limite des Réseaux de Neurones sur Graphes appliqués aux grands graphes aléatoires. On démontre que sous certaines hypothèses de régularité, un GNN converge vers un homologue «continu». L'originalité de ce travail est de considérer des fonctions d'agrégation abstraites et générales tandis que les études antérieures traitent de cas particuliers. On quantifie les convergences à l'aide de bornes non asymptotiques en probabilité basées sur l'inégalité de McDiarmid pour des agrégations ayant une régularité de type lipschitzienne. Le cas du maximum, non inclus dans cette catégorie, est traité à part.

Abstract -We study the limit behavior of GNNs on large random graphs. We consider GNNs with a generic aggregation function and show that under mild regularity conditions, they converge to a "continuous" counterpart. We provide some non asymptotic bounds with high probability for this convergence which encompass several cases of aggregation such as, the mean, or the maximum.

Introduction

A classical approach to study the properties of Graph Neural Networks [START_REF] Michael | Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges[END_REF] (GNNs) is to compare them to the Weisfeiler-Lehman test for the graph isomorphism problem [START_REF] Xu | How Powerful are Graph Neural Networks[END_REF]. Nevertheless, this approach becomes questionable for very large graphs, where we would rather focus on global tendencies. The latter are traditionally modelled with random graphs [START_REF] Goldenberg | A survey of statistical network models[END_REF]: since the early Erdös Rényi model, they have become classical tools in statistics, statistical learning and statistical physics, for instance to study clustering problems [START_REF] Lei | Consistency of spectral clustering in stochastic block models[END_REF] or limits of large graphs [START_REF] Lovász | Large Networks and Graph Limits[END_REF] and their properties. In the case of GNNs, this approach has shed light on their generalization properties [START_REF] Maskey | Stability and Generalization Capabilities of Message Passing Graph Neural Networks[END_REF] or stability to deformations [START_REF] Keriven | Convergence and Stability of Graph Convolutional Networks on Large Random Graphs[END_REF][START_REF] Ruiz | Graphon Signal Processing[END_REF]. In this paper, our purpose is to examine whether a GNN on a large random graph is close to a "continuous" limit on the random graph model. We aim to address Message-Passing GNN (MPGNN) with generic aggregation function, including non-smooth ones like max, while previous work [START_REF] Keriven | Convergence and Stability of Graph Convolutional Networks on Large Random Graphs[END_REF][START_REF] Maskey | Stability and Generalization Capabilities of Message Passing Graph Neural Networks[END_REF][START_REF] Ruiz | Graphon Neural Networks and the Transferability of Graph Neural Networks[END_REF] focused on particular cases such as Graph Convolutional Network or degree normalized mean. We give sufficient conditions under which the discrete model is a good approximation of the continuous analogue, and provide non asymptotic deviation bounds. Depending on the regularity of the aggregation function, we obtain different rates of convergence. Due to space constraints, we refer the reader to the full version of the paper [START_REF] Cordonnier | Convergence of Graph Neural Networks With Generic Aggregation Functions on Random Graphs[END_REF] for the mathematical proofs.

Notations and definitions

We fix a positive integer d and X a compact subset of R d , endowed with the infinite norm x ∞ = max i |x i | as well as its Borel sigma algebra. Except when specified differently, all topological concepts will be considered relatively to • ∞ . This work was partially supported by the French National Research Agency in the framework of the « France 2030 » program (ANR-15-IDEX-0002), the LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01), and the ANR grants GRANDMA (ANR-21-CE23-0006), GRANOLA (ANR-21-CE48-0009) and GRAVA (ANR-18-CE40-0005).

The space of continuous functions from X to R p is written C(X , R p ), and it is also equipped with the supremum norm

f ∞ = sup x∈X f (x) ∞ . The group of permutations of {1, . . . , n} is denoted as S n . If x = (x 1 , . . . , x n ) is an n-tuple and σ an element of S n , we define the n-tuple σ • x as σ • x = (x σ -1 (1) , . . . , x σ -1 (n) ).
The set of bijections φ of X such that both φ and φ -1 are measurable is a group for the composition of functions. We call this group the group of automorphisms of X and denote it as Aut(X ). We denote as P(X ) the set of probability measures on X . For a measure P ∈ P(X ) and a bijection φ ∈ Aut(X ), the push forward measure of P through φ is defined as φ # P (A) = P (φ -1 (A)) for all open sets A. Since this makes Aut(X ) act on the set of probability measures, we also use the notation φ • P = φ # P , which is standard for a (left) group action. For the same reason, we shall use the notation φ

• f = f • φ -1 and φ • W = W (φ -1 (•), φ -1 (•))
whenever f is a measurable function on X and W is a bivariate measurable function on X × X . Sets are represented between braces {•}, whereas multisets, sets in which an element is allowed to appear twice or more, are represented by double braces { {•} }. If M and M are two multisets of the same size, say n, containing elements from a metric space (E, d), we define their distance by:

δ(M, M ) = min σ∈Sn max xi∈M,x i ∈M d(x i , x σ(i) ). (1)
We define the sampling operator the following way. If f : E → E and X = (x 1 , . . . , x n ) ∈ E n :

S X f = (f (x 1 ), . . . , f (x n )) ∈ E n . (2)
Graph. An undirected weighted graph G with n vertices is defined by a triplet (V, E, w), where V = {v 1 , . . . , v n } is a finite set called the set of vertices (or nodes) and E is the set of edges. The set of neighbors of a vertex v i in G is referred to as N (i). The weight function w assigns a nonnegative number to each edge. It is often represented by a symmetric function w : V 2 → R + and the abbreviation w i,j is used to denote the weight w(v i , v j ) = w(v j , v i ) where {v i , v j } ∈ E. The set of graphs defined on the vertex set V is denoted as G(V ).

Graph signal. Given a graph G ∈ G(V ), where |V | = n, a signal on G is a map from the set of vertices V to R d that assigns a d-dimensional vector z i to each vertex v i . The images from all vertices are stacked into a tensor Z of size n × d. Abusing notations, we may not distinguish between the map and its image Z, the latter being also named the signal.

Random Graph Model.

A random graph model is a couple (W, P ) where P is a probability measure on X and W : X 2 → [0, 1] is a similarity kernel, i.e. a symmetric measurable function. We generate a random graph of size n from (W, P ) as follows:

X 1 , . . . , X n iid ∼ P, w i,j = w j,i = W (X i , X j ) . (3)
When convenient, we will use the short notation X = (X 1 , . . . , X n ) for the tuple of the vertices of a random graph. We call G n (W, P ) the distribution from which random graphs with n nodes are drawn. We bring the reader's attention to the fact that in the above definition, a random graph is always fully connected and edge may have a weight equal to zero. A common model [START_REF] Keriven | Convergence and Stability of Graph Convolutional Networks on Large Random Graphs[END_REF] is to add a Bernoulli distribution to the connectivity, which is not done here for the sake of simplicity.

Graph and Random Graph Model isomorphism. Two graphs

G 1 and G 2 in G(V ) are said to be isomorphic if there is a permutation σ ∈ S n such that E 2 := {{v σ -1 (i) , v σ -1 (j) }| {v i , v j } ∈ E 1 } and w 1 (v i , v j ) = w 2 (v σ -1 (i) , v σ -1 (j)
). This permutation is called a graph isomorphism. In this case we note

G 2 = σ • G 1 . Moreover, if Z is a signal on G 1 and σ ∈ S n , σ • Z is an isomorphic signal on the graph σ • G 1 .
Two probability measures P 1 and P 2 on X are said isomorphic if there is some φ in Aut(X ) such that P 2 = φ • P 1 . Similarly, two random graph models (W 1 , P 1 ) and (W 2 , P 2 ) on X are said to be isomorphic if there is a φ in Aut(X ) such that (W 2 , P 2 ) = (φ • W 1 , φ • P 1 ), in this case, we will note (W 2 , P 2 ) = φ • (W 1 , P 1 ).

MPGNN

A L-layer MPGNN iteratively propagates a signal over a graph. At each step, the current representations of every node's neighbors are gathered, transformed, and combined to update the node's representation. Let G ∈ G(V ) with |V | = n, and Z = Z (0) ∈ R n×d0 be a signal on it. There are L operators (F (l) ) 1≤l≤L such that, at each layer, the update Z (l+1) of the signal is computed node-wise by:

z (l+1) i = F (l+1) z (l) i , { { z (l) j , w i,j } } j∈N (i) ∈ R d l+1 . ( 4
)
The F (l) are referred to as aggregations. They are the core of a MPGNN and their fundamental property is to be invariant to the permutation of the neighbors. As a result, the full network is equivariant to permutations of the graph, that is, consistent with graph isomorphism.

Proposition 3.1. Call Θ G (Z) = Z (L) . For all Z ∈ R n×d0 , σ ∈ S n , we have Θ σ•G (σ • Z) = σ • Θ G (Z).
Typically, the aggregation first transforms the z (l) j through a learnable transformation ψ (l+1) , then combine them using a weighted mean (in a broad sense, such as arithmetic mean, point-wise maximum, etc..) with weights

c (l+1) i,j = c (l+1) z (l) i , z (l) j , w ij (5)
such that (4) can be rewritten as

z (l+1) i = { { ψ (l+1) (z (l) j ), c (l+1) i,j } } j∈N (i) .
Examples.

1. Convolution:

z (l+1) i = 1 |N (i)| j∈N (i) w i,j ψ (l+1) (z (l) j ). 2. Degree Normalized Convolution: z (l+1) i = j∈N (i) wi,j k∈N (i) w i,k ψ (l+1) (z (l) j ).
3. Graph Attention (GAT): the c (l) from ( 5) are learnable:

z (l+1) i = j∈N (i) c (l+1) i,j k∈N (v i ) c (l+1) i,k ψ (l+1) (z (l) j ).
4. Max Convolution: z

(l+1) i = max j∈N (i) w i,j ψ (l+1) (z (l) j ).

Continuous MPGNN

As we will see in the next section, when the number of nodes grows, MPGNNs will often converge to "continuous models" on random graphs. Let (W, P ) be a random graph model. A Llayer continuous MPGNN (cMPGNN) propagates a function over the latent space X , using operators F (l+1) P that take a vector and a function as input and output a vector. For an input f = f (0) ∈ C(X , R d0 ), f (l+1) is computed by:

f (l+1) (x) = F (l+1) P f (l) (x), f (l) , W (x, •) ∈ R d l+1 .
(6) For notational convenience, we overload the notations F (l) p as

F (l) P (f, W ) : x → F (l) P (f (x), (f, W (x, •)). (7)
Such, Eq. ( 6) now writes as

f (l+1) (x) = F (l+1) P (f (l) , W )(x). (8)
Naturally, we also demand the cMPGNN to be equivariant to random graph model isomorphisms. To that extent, we need the following assumption on the operators F (l)

P : Assumption 4.1. There is a subgroup H ⊂ Aut(X ) such that ∀1 ≤ l ≤ L, ∀f ∈ C(X , R d l ), ∀φ ∈ H: F (l) φ•P (φ • f, φ • W ) = φ • F (l) P (f, W ).
Ass. 4.1 is inspired by the classical change of variable formula in Lebesgue integration: for any φ bijective and measurable and any measurable map f , f dP = φ • f d(φ • P ).

φ ∈ H, Θ φ•(W,P ) (φ • f ) = φ • Θ W,P (f ).
Ideally, one would like H = Aut(X ). However, we will see with Ex. 4 that this is not always possible and one may have to restrict equivariance to a subgroup of Aut(X ).

Examples (continuous equivalents of 1, 2, 3 and 4). a. Convolution:

f (l+1) (x) = y W (x, y)ψ (l+1) (f (l) (y))dP.
b. Degree Normalized Convolution:

f (l+1) (x) = y W (x,y) t W (x,t)dP (t) ψ (l+1) (f (l) (y))dP. c. GAT: f (l+1) (x) = y c (l+1) (x,y,W (x,y)) t c (l+1) (x,t,W (x,t))dP (t) ψ (l+1) f (l) (y) dP. d. Max Convolution: f (l+1) (x) = ess sup P W (x, .)ψ (l+1) (f (l) (.)).

Construction of the cMPGNN

Let (W, P ) be a random graph model and f ∈ C(X , R d ).

In this subsection, we consider a single layer of a MPGNN applied on a random graph G n ∼ G n (W, P ) and input matrix S X f as node features. We present a canonical way of defining a limit corresponding cMPGNN layer on (W, P ) with input map f , under some convergence assumptions that will be satisfied in our examples. Since there is no multi layers in this section, we drop the superscript indexation.

Definition 5.1. Let F be a MPGNN layer and (W, P ) a random graph model. For f ∈ C(X , R d ) we define the sequence

F P,n (f, W ) : x → E X1,...,Xn [F (f (x), { {(f (X k ), W (x, X k ))} } 1≤k≤n )] (9)
where the expected value is taken over all the X 1 , . . . , X n iid ∼ P . Let F be an operator of the form (8) taking value in C(X , R d ) and suppose we have a non-trivial subgroup H ⊂ Aut(X ) such that for any f ∈ C(X , R d ), for any φ ∈ H,

F φ•P,n (φ • f, φ • W ) • ∞ ---→ F φ•P (φ • f, φ • W ). ( 10
)
Then we say that F is the continuous counterpart of F .

This F is a good candidate to be a cMPGNN. Indeed, it satisfies 4.1 on the φ for which it is well defined. 3-c. Call V (x, y) = c(f (x), f (y), W (x, y)), if ψ is bounded and α ≤ V ≤ β a. s for some 0 < α ≤ β, then c. is the continuous counterpart of 3. for the full H = Aut(X ).

4-d. Suppose that W and ψ, are continuous and that the measure P is strictly positive on X i.e, any nonvoid relative open of X has a strictly positive measure by P . Then d. is the continuous counterpart of 4. for H = Hom(X ): the subgroup of Aut(X ) made of the φ ∈ Aut(X ) that are homeomorphisms.

The rates of convergence are given in the next section.

Convergence of MPGNN to cMPGNN

Let (W, P ) be a random graph model, (G n ) n≥1 be a sequence of random graphs drawn from G n (W, P ). We consider a MPGNN (F (l) ) 1≤l≤L and its continuous counterparts (F (l) ) 1≤l≤L . For an f ∈ C(X , R d0 ), does the MPGNN on G n with input signal S X f actually converge to the cMPGNN on (W, P ) with input signal f ? If yes, at which speed? In this section we provide non asymptotic bounds with high probability to quantify this convergence.

Our main theorem states that, under mild regularity condition, with high probability, (S X (f ))

(L) i is close to f (L) (X i ).
To compare the output of this signal through the discrete network to the output of f through the continuous counterpart, we shall use the following Maximum Absolute Error (MAE):

MAE(f ) = max i (S X (f )) (L) i -f (L) (X i ) ∞ .

Bounded differences method

Our main result relies on the so called McDiarmid concentration inequality, which relies on the following property. Definition 6.1 (Bounded Differences Property). Let f : E n → R p be a function of n variables. We say that f has the bounded differences property if there exist n nonnegative constants c 1 , . . . , c n such that for any 1 ≤ i ≤ n: f (x 1 . . . , x i , . . . , x n ) -f (x 1 , . . . , x i , . . . , x n ) ∞ ≤ c i , (11) for all x 1 , . . . , x i , x i , . . . , x n .

Fix, x 1 ∈ X , we are interested at the bounded differences of

F (l) (f (l-1) (x 1 ), { {(f (l-1) (x k ), W (x 1 , x k ))} } k≥2 ) (12)
as a map of the n -1 variables x 2 , . . . , x n . If c 1 , . . . , c n satisfy (11), since (12) is invariant to the permutation of x 2 , . . . , x n , they can be taken all equal. We call 12) is continuous as a function of x 1 , it is also bounded by compactness, we define

D (l) n (x 1 ) = c 1 = • • • = c n . Moreover, since (
D (l) n = sup x1∈X D (l) n (x 1 ) (13)
Since for all l, F (l) is the continuous counterpart of F (l) , using the notations of Def. 5.1, we let (a

(l)
n ) be a sequence of positive reals such that a (l) n → 0 and for all n F (l)

P,n (f, W ) -F (l) P (f, W ) ∞ ≤ a (l) n . (14)
Next, we suppose, that (13), (14) and Def. 5.1 are satisfied. Plus that the F (l) have some "Lipschitz-like" smoothness.

Assumption 6.1.(i) For all 1 ≤ l ≤ L, we endow R d l-1 × [0, 1] with the norm (y, t) 1 = y ∞ + |t| and call δ 1 the corresponding distance on multisets as defined in (1). Let x, x ∈ R d l-1 and M, M be two multisets of same cardinal n containing elements of R d l-1 × [0, 1], then there exist two constants µ (l) ≥ 0 and λ (l) F,n > 0 such that:

F (l) (x, M) -F (l) (x , M ) ∞ ≤ µ (l) F x -x ∞ + λ (l) F,n δ 1 (M, M ). (15)
(ii) The sequence (λ

(l) F,n ) is bounded over n.
(iii) We have some D

n and a (l) n as defined in (13) and ( 14). (iv) The F (l) are the continuous counterparts of the F (l)) . Theorem 6.1. Under Ass. 6.1 for any ρ > 0,

MAE X (f ) LD n n ln n2 L d max ρ + La n-1 . ( 16
)
with probability at least 1 -ρ. In (16),

D n = max l D (l) n , d max = max l d l , a n = max l a (l)
n and hides some multiplicative constants which do not depend on n.

This bound suggests that if the bounded differences are sharp enough, typically

D n = o(1/ √ n ln n), the MAE is small. Corollary 6.1.1. If D n = o(1/ √ n ln n) then MAE X (f ) con- verges to zero in probability.

This table sums up the results on the examples.

Example

D n a n Cvrg by Th. 6.1

1-a O(1/n) 0 2-b O(1/n) O(1/ √ n) 3-c O(1/n) O(1/ √ n) 4-d
Ω(1) -

The case of max

It turns out that MPGNNs with max aggregation do not have sharp bounded differences. Nevertheless, we provide a bound for its convergence based on other concentration inequalities. Definition 6.2 (Volume retaining property). We say that the probability space (X , P ) has the (r 0 , κ)-volume retaining property if for any r ≥ r 0 and for any x ∈ X ,

P (B(x, r) ∩ X ) ≥ κm(B(x, r)) (17)
Where B(x, r) is the ball of center x and radius r and m is the classical d-dimensional Lebesgue measure in R d

Clearly, volume-retention implies strict positiveness of the measure. This property ensures that the measure of a relative small ball centered in a point of X is at least a portion of the volume of that ball in R d . As an example, it is easy to check that ([0, 1] d , m) has the (1, 1/2 d )-volume retaining property. For a volume retaining probability space, we obtain a new concentration inequality Lemma 6.1. Let g : X 2 → R p be λ g Lipschitz and (X , P ) have the (r, κ)-volume retaining property for some r, κ > 0. Then for any ρ > 0, for any random variables X 1 , . . . , X n iid ∼ P , with probability at least 1 -ρ: Armed with this lemma, we are ready to state the non asymptotic bound for a MPGNN with max aggregation: Theorem 6.2. For a Max Conv. MPGNN, assume that (X , P ) has the (r, κ)-volume retaining property. Then for any ρ > 0, we have with probability at least 1 -ρ,

M AE(f ) L 1 n -1 ln 2 L-1 nd max ρ 1/d . (18)
This bound indicates a different behaviour when the aggregation is a maximum. It depends on the input space's dimension and decreases significantly slower than (16) for a large d.

Proposition 4 . 1 .

 41 Call Θ W,P (f ) = f (L) . Under Ass. 4.1, for any f and any

Proposition 5 . 1 .

 51 Let F be the continuous counterpart of F as defined in Def. 5.1. Then it satisfies Ass. 4.1 for any φ ∈ H. We prove that the continuous equivalents of our previous examples 1, 2, 3 and 4 are respectively a, b, c and d: Examples. 1-a. a. is the continuous counterpart of 1. for H = Aut(X ) 2-b. If ψ is bounded and W ≥ α for some α > 0. Then b. is the continuous counterpart of 2. for the full H = Aut(X ).

Figure 1 :

 1 Figure 1: Log-scaled illustration of (18) for d = 2, 3, 5. The dashed line represents (1/n) 1/d and the plain line the MAE. L = 3, X = [0, 1] d with the Lebesgue measure, W (x, y) = exp(-x -y 2 ) and f = 1.


H Noel

T Martin

V Aubin

G Dezanneau
email: guilhem.dezanneau@centralesupelec.fr

An alternative method for the analysis of Resonant Ultrasound Spectra of isotropic materials

Keywords: Resonant Ultrasound Spectroscopy, elastic coefficients, Young's Modulus, Poisson's ratio, isotropic

Resonant Ultrasound Spectroscopy is known as one of the most accurate techniques to determine materials' elastic properties. Its use has been limited until now since the extraction of elastic coefficients from resonance spectra requires some expertise. Recent efforts have been made to automatise the analysis and favour the widespread use of this method.

Advanced mathematical tools have indeed been developed to produce robust exploitation of resonance spectra, mainly based on probabilistic approaches. Within this dynamics, we propose here a method that allows obtaining in a very efficient way the elastic coefficients of isotropic materials. The method is applicable to complete or incomplete resonance spectra.

Our approach exploits the subtleties of resonance spectra in isotropic materials, the main one being that resonance frequencies depend in a complex manner on the Poisson's ratio  but are proportional to the square root of Young's Modulus E. Starting from a set of resonance frequencies, from sample dimensions and density, our method is able to determine the real E and  coefficients quickly and without making any hypothesis on their initial value. We also demonstrate that our approach is valid when the resonance spectrum presents some missing frequencies -which can occur in real cases -or when these frequencies are affected by experimental error. Finally, we demonstrate in two real cases presented in previous articles that our approach is robust and rapid to determine accurate elastic coefficients.

Introduction

Elasticity, which correlates the matter displacements to the applied stress within a solid, appears as the first of the solid mechanical properties. It can be measured by many techniques such as compression tests [START_REF] Gholami | Mechanical and elastic properties of transversely isotropic slate[END_REF][START_REF] Hild | Multiscale displacement field measurements of compressed mineral-wool samples by digital image correlation[END_REF][START_REF] Kuhn | Mechanical Testing and Evaluation[END_REF], Impulse Excitation Technique [START_REF] Swarnakar | Determination of the Si Young's Modulus between room and melt temperature using the impulse excitation technique[END_REF][START_REF] Heritage | Impulse Excitation Technique for Dynamic Flexural Measurements at Moderate Temperature[END_REF], nano-indentation [START_REF] Li | A Review of Nanoindentation Continuous Stiffness Measurement Technique and Its Applications[END_REF][START_REF] Fujikane | Nanoindentation examination of Yttria-Stabilized Zirconia (YSZ) Crystal[END_REF][START_REF] Patel | Extracting single-crystal elastic constants from polycrystalline samples using spherical nanoindentation and orientation measurements[END_REF], Time-of-flight [START_REF] Sturtevant | Evaluating the Effectiveness of the Transmission Line Model in Pulse-Echo Couplant Layer Corrections[END_REF][START_REF] Periyannan | Simultaneous moduli measurement of elastic materials at elevated temperatures using an ultrasonic waveguide method[END_REF][START_REF] Bahloul | Young's modulus evolution at high temperature of SiC refractory castables[END_REF] or Resonant Ultrasound Spectroscopy (RUS) [START_REF] Migliori | Resonant ultrasound spectroscopic techniques for measurement of the elastic moduli of solids[END_REF][START_REF] Migliori | Implementation of a modern resonant ultrasound spectroscopy system for the measurement of the elastic moduli of small solid specimens[END_REF][START_REF] Maynard | Resonant ultrasound spectroscopy[END_REF][START_REF] Sedlak | Determination of all 21 independent elastic coefficients of generally anisotropic solids by resonant ultrasound spectroscopy: benchmark examples[END_REF][START_REF]Ultrasonic Spectroscopy -Applications in Condensed Matter Physics and Materials Science[END_REF][START_REF] Balakirev | Resonant ultrasound spectroscopy: The essential toolbox[END_REF]. This last technique has many advantages over the other ones: a very high accuracy, no need for multiple measures for the whole elastic tensor determination, and no need for a big sample [START_REF] Migliori | Resonant ultrasound spectroscopic techniques for measurement of the elastic moduli of solids[END_REF]. It has also been used in a wide range of thermodynamic conditions such as low temperature [START_REF] Balakirev | Resonant ultrasound spectroscopy: The essential toolbox[END_REF],

high temperature [START_REF] Li | High Temperature Resonant Ultrasound Spectroscopy: A Review[END_REF][START_REF] Goto | Apparatus for measuring elastic constants of single crystals by a resonance technique up to 1825 K[END_REF], under electric or magnetic fields [START_REF] Petculescu | Magnetic field dependence of galfenol elastic properties[END_REF][START_REF] Schiemer | Magnetic field and in situ stress dependence of elastic behavior in EuTiO3 from resonant ultrasound spectroscopy[END_REF][START_REF] Darling | Electric field effects in RUS measurements[END_REF] or even in the case of samples showing high ultrasonic attenuation [START_REF] Wilen | Acoustic resonance spectroscopy of soft solids[END_REF]. Despite these significant advantages, the use of RUS has been limited to a few groups all over the world: Los Alamos (USA) with 104 publications, Oak Ridge National Laboratories (USA) with 54 publications, M. Carpenter's group at Cambridge (U.K.) with 95 publications, Osaka University (Japan) with 51 publications, Ruhr University of Bochum (Germany) with 20 publications, M. Landa's group (Czech Republic) with 40 publications, to name some of the main actors in the field. It reflects the fact that a degree of expertise is needed to perform RUS measurements and analysis. The measurement by itself is indeed somewhat tricky, even if all the optimised experimental details were presented in previous articles [START_REF] Migliori | Resonant ultrasound spectroscopic techniques for measurement of the elastic moduli of solids[END_REF][START_REF] Migliori | Implementation of a modern resonant ultrasound spectroscopy system for the measurement of the elastic moduli of small solid specimens[END_REF][START_REF] Maynard | Resonant ultrasound spectroscopy[END_REF][START_REF] Zadler | Resonant ultrasound spectroscopy: theory and application[END_REF]. These articles reported on the preparation of a good sample for RUS (shape and size), on the requirement concerning electronics to acquire RUS spectra or on potential sources of errors like the pressure on the sample during acquisition.

Commercial devices were also proposed, which help to minimise the source of errors [26,27].

Considering this, the main difficulty during a RUS characterisation would lie in analysing acquired spectra. The least-square non-linear optimisation usually leads to the correct values of elastic coefficients once a correct association of modes is made and the potential absence of modes is identified. The approach becomes hard to apply on materials with unknown elastic properties or complex elastic tensors, as in low-symmetry single crystals. This latter case represents a significant difficulty since as much as 21 elastic coefficients must potentially be retrieved if no knowledge is available about sample symmetry. Some methodologies were proposed either based on a step-by-step strategy coupled with complementary experiments [START_REF] Sedlak | Determination of all 21 independent elastic coefficients of generally anisotropic solids by resonant ultrasound spectroscopy: benchmark examples[END_REF][START_REF] Farzbod | Using eigenmodes to perform the inverse problem associated with resonant ultrasound spectroscopy[END_REF][START_REF] Ogi | Complete mode identification for resonant ultrasound spectroscopy[END_REF] or on user-independent advanced mathematical analyses, such as the Bayesian approach or the use of genetic algorithms [START_REF] Bernard | Bayesian normal modes identification and estimation of elastic coefficients in resonant ultrasound spectroscopy[END_REF][START_REF] Bales | Bayesian inference of elastic properties with resonant ultrasound spectroscopy[END_REF][32]. Even if isotropic materials' elastic properties are by far easier to analyse, the limited use of the RUS technique, even in this case, underlines the expertise or time needed to analyse resonance spectra.

In the present work, we present an alternative approach applicable to isotropic materials only, leading to a fast and correct determination of the elastic coefficients. We expect our approach to make the RUS characterisation more accessible to scientists and engineers, following previous works aiming to popularise the RUS approach [START_REF] Balakirev | Resonant ultrasound spectroscopy: The essential toolbox[END_REF]. In the first part, we describe the hypothesis, approaches and mathematical tools that we used for the RUS analysis. We then apply our approach to 100 virtual isotropic rectangular parallelepiped samples chosen with random size, density and elastic coefficients. We demonstrate that it allows determining the elastic coefficients in the case of resonance spectra containing all the expected modes in a given frequency range and in the case of missing modes. We also show that we still obtain correct values for the elastic coefficients even if some statistical error is present in the spectrum resonance frequencies to be fitted. Finally, we apply our approach to two real examples that were presented in the literature as representative cases, and we show once more the efficiency of the approach.

II. Calculations details

II.1. Theoretical approach

Our approach is adapted to the analysis of samples with some given characteristics:

-Samples are expected to be isotropic.

-Samples are supposed to be prepared accordingly to an assumed geometry [START_REF] Migliori | Resonant ultrasound spectroscopic techniques for measurement of the elastic moduli of solids[END_REF][START_REF] Migliori | Implementation of a modern resonant ultrasound spectroscopy system for the measurement of the elastic moduli of small solid specimens[END_REF][START_REF] Maynard | Resonant ultrasound spectroscopy[END_REF]. In particular, care should be taken to prepare samples with a well-defined parallelepiped shape and suitable dimensions to avoid overlapping modes.

-Ultrasonic resonance spectra should be acquired with no loading on the sample.

After considering as a requirement that high-quality resonance spectra are obtained, our approach uses the fact that a suitable representation of the elastic tensor allows simplifying the problem deeply. Indeed, the elastic tensor for an isotropic material is represented as: It can also be represented as:

𝐶
𝐶 𝑖𝑗 = 𝐸 (1+𝜐)(1-2𝜐) ( 1 -𝜐 𝑣 𝑣 0 0 0 𝑣 1 -𝜐 𝑣 0 0 0 𝑣 𝑣 1 -𝜐 0 0 0 0 0 0 1-𝑣 2 0 0 0 0 0 0 1-𝑣 2 0 0 0 0 0 0 1-𝑣 2 )
, Where E and  are Young's Modulus and Poisson's ratio, respectively. From the second form, it is clear that the Cij elastic coefficients are a complex function of but are all proportional to Young's modulus E. This means that resonance frequencies are proportional to the square root of E. Due to this form of the elastic tensor, isotropic materials with the same Poisson's ratios will present resonance spectra with the same resonance frequencies sequence. To illustrate this, we show in Figure 1, the calculated resonance spectra (standard procedure explained in the following) for three different Poisson's ratios and three different Young's moduli for an isotropic sample of a given size and density. From this comparison, it is evident that the value of E only affects the scale of the frequency axis, leaving the pattern of the spectrum the same, this pattern being controlled by  (see vertical graphs in Figure 1). The problem of elastic coefficient extraction in isotropic materials can be drastically simplified through our approach. Once spectra are calculated for a given E0 value and a range of  Poisson's ratios, any spectrum corresponding to any (E, ) couple can be deduced from the (E0, ) spectrum by multiplying all its frequencies by (E/E0) 1/2 . We can calculate spectra for a vast range of (E, ) efficiently and compare them to the original known spectrum. After this comparison, one can determine Edet and det the best optimised elastic coefficients. This is the basis of our optimisation scheme, represented in Figure 2. The fundaments and procedure for the Resonant Ultrasound Spectroscopy forward model calculations have made the subject of many articles. We thus refer the reader to the previous articles mentioned in the introduction, particularly those of Migliori's group [START_REF] Migliori | Resonant ultrasound spectroscopic techniques for measurement of the elastic moduli of solids[END_REF], which constitute a complete and reliable source to better understand the ins and outs of RUS principle practice.

For the forward model, the evaluation of vibration modes of elastic objects with free boundaries can be simplified using the Rayleigh-Ritz method, allowing to calculate natural vibration frequencies. This approach supposes that the Lagrangian of a system is stationary with respect to small perturbations in the eigenfunctions. This requirement that Lagrangian is stationary reduces to finding the eigenvalues of the following equation:

𝐊𝒂 = 𝜔 2 𝑴𝒂 [1]
Where K is the stiffness matrix, M is the mass matrix, and a is a vector of displacements in three dimensions (see ref.

12-17 and 28-31 for details). The interesting point is that the matrix K is proportional to the elastic constants Cij, i.e. proportional to Young's Modulus in the case of isotropic materials. By factorising E in the previous equation, one can easily demonstrate that the resonance frequencies of a given sample are directly proportional to Young's Modulus square root. This is the main property that we used here to fasten the calculations since the spectra for any (E, ) couple can be deduced from the spectrum (E0, ) by simply multiplying all frequencies by √ 𝐸 𝐸 0 ⁄ .

Along the years, several significant advances were made to facilitate the calculation of the forward model [START_REF] Demarest | Cube resonance method to determine the elastic constants of solids[END_REF][START_REF] Ohno | Determination of elastic constants of trigonal crystals by the rectangular parallelepiped resonance method[END_REF][START_REF] Visscher | On the normal modes of free vibration of inhomogeneous and anisotropic elastic objects[END_REF], and RUS calculation codes were subsequently made available to the scientific community, for instance, from the Los Alamos National Laboratory [36], and the Physical Acoustic Lab (Auckland, New-Zealand) [37]. We used the "RUS.m" Matlab code developed by Matt Fig [START_REF] Figg | Resonant Ultrasound Spectroscopy (RUS). Matlab Central[END_REF], adapted to our specific purpose. This code is essentially a forward model which allows obtaining the resonance frequencies of a given sample with known dimensions, density and elastic coefficients. The virtual samples (see below) spectra were obtained by assuming a n = 20 polynomial order for the forward model; the calculated spectra used during elastic coefficient extraction were obtained considering a n = 12 polynomial order.

Based on the scheme presented above, a more detailed calculation procedure is the following:

-We first consider a known spectrum SK associated with a real or virtual sample with a known size and density. This spectrum, typically containing 30-40 frequencies, can be obtained from experiments or from calculations and is the one from which we want to extract the elastic coefficients.

-We used the forward model to calculate the spectrum for the same sample (size, density, shape) fixing E 0 = 100 GPa and  varying from 0.1 to 0.4. In our case, we calculated 601 spectra (SC) with thus a 0.0005 step in .

-We obtained the theoretical spectra for all E in a given range by multiplying the frequencies obtained in the previous step by √ 𝐸 𝐸 0

. In our case, we used a 0.1 GPa step in E.

-We calculate a distance measure between the known spectrum and all the calculated spectra (601 x n). In our case, we simply associated the frequencies of the known spectrum to the closest frequencies in the calculated spectra and sum all the absolutes values of differences in frequencies to obtain a distance measure:

𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒(𝑆 𝑇 , 𝑆 𝐶 ) = ∑ 𝑚𝑖𝑛 ∀𝑓 𝑗 ∈𝑆 𝐶 ∀𝑓 𝑖 ∈𝑆 𝐾 |𝑓 𝑖 -𝑓 𝑗 | [2]
As a result, we obtain a couple (Edet, det) presenting the smaller distance between the known and the calculated spectra. The proposed distance measure which compares sets of different sizes is similar to the Hausdorff distance [START_REF] Gardner | IEEE Conference on Computer Vision and Pattern Recognition (CVPR)[END_REF].

II.3. Virtual and real samples

To validate our approach, we input the dimensions and density of the "virtual sample," chosen of a rectangular parallelepiped (RP) geometry. It is a virtual sample for the purpose of the proof of concept worked out here (random dimensions are used), equivalent to the investigated sample in a real experiment, where the mass and dimensions would be measured directly. We considered 100 virtual samples having different sizes and elastic properties. These virtual samples are rectangular parallelepipeds with dimensions L1, L2 and L3 in the range 0.5 -5.5 cm (the absolute size has actually not much importance on the final result), their density between 2000 and 7000 kg/m 3 , and their elastic coefficients E and  in the range 100 -400 GPa and 0.1 -0.4 respectively. We represent, in Table 1, ten of those samples. We also report in Table 1

III. Results

III.1. Results for virtual test-samples

In addition to the results shown in Table 1, we show in Figure 3 the results of our analysis for samples having perfect RUS spectra i.e. calculated with the forward model using a high polynomial order (n = 20), thus having all the resonance modes present and determined with great precision. Figure 3 plots the Poisson's ratio and Young's Modulus as deduced from our procedure and compared to those of 200 virtual samples. This figure shows a perfect agreement between the determined values and the known ones. As a whole, the calculated and known values differ by less than 0.1% for E and less than 0.5% for  , a precision here only limited by the precision we used for both E and  variables. Concretely we used steps of 0.1

GPa for E screening, and 0.0005 for  screening.

III.2. Results for virtual test-samples with altered spectra

III.2.1. Influence of missing modes on elastic coefficients determination

The results presented above confirm the method's potential interest since it leads to the determination of correct elastic coefficients without any initial guess value and required action. We consider in this part RUS spectra in which some resonance modes are missing. We have artificially removed a given percentage of modes (from 0% to 70%) from the original virtual spectra. The removed frequencies may include the first frequency, which is always considered as important experimentally due to its pure shear character.

The analysis of missing modes' influence was performed in an exhaustive way using 20 different combinations for each percentage of missing modes explored for 50 virtual samples, representing more than seven thousand cases. In the following graph, we present the ratio between the determined values and the known elastic properties as a function of the percentage of missing peaks. Even for a very high percentage of missing modes, our approach gives determined values of E close to the known value, the error being always less than 0.5% and 99% of the time less than 0.2%. The error on  determination is more important reaching around 2% for some samples and a high percentage of missing modes. For 30% of missing modes, the percentage of solutions determined for which the error on  remains within 0.5% is greater than 95%. These results are particularly interesting since the approach allows determining the elastic coefficients even with incomplete spectra, without the need for assuming it or without the need to evaluate the number of missing modes.

II.2.2. Influence of noise in the spectrum

In this part, we introduced some errors artificially on frequency measures in the original known resonance spectra. These errors are set to follow a normal distribution and are supposed to reflect potential sources of error coming from a poor sampling of frequencies during RUS spectrum acquisition. It is not expected to mimic bad acquisition conditions like those due to pressure on the sample, non-homogeneous dimensions, etc... which would induce non-random frequency shifts.

III.3. Study of real samples

As reported in Table 1, we obtained good determined values for both E and  for these two real examples. Through our optimisation scheme, Edet and EKnown (resp. det and Known) differ by ~0.2% (resp. ~0.3%) for sample 1 and by ~0.1% (resp. ~0.5%) for sample 2. We did not have to make any assumption on the first mode, to presuppose any elastic coefficients, or to propose a specific mode association to make the analysis. Besides, our approach allows representing the solution as a map, which may help apprehend better the solution's uniqueness. In Figure 6, we represent the inverse of the distance (Equation 2) between calculated and known spectra. This map shows that the proposed solution is the only one giving acceptable values, the rest of the (E, ) variable space presenting a poor agreement with target data. A zoom on this map, close to the determined elastic coefficients region, provides a natural path toward error estimation. Following an approach similar to the one proposed by Migliori et al. [START_REF] Migliori | Resonant ultrasound spectroscopic techniques for measurement of the elastic moduli of solids[END_REF], A 2D-ellipsoid in the parameter space surrounding the minimum of distance between calculated and known spectra can be determined corresponding to a 2% increase in the distance measure. This ellipsoid may provide a realistic error estimate for both E and v parameters. Following this approach, the error estimation for Edet for samples 11 and 12 is smaller than 0.1% in both cases, and the error estimation for detfor samples 11 and 12 is 0.3% and 0.6%, respectively.

IV. Discussion

The approach presented here exploits the elastic tensor's unique properties in isotropic materials, particularly the fact that we can factorise Young's Modulus in the expression of Cij, making the calculation of resonance modes for a wide range of E and  coefficients more straightforward. The approach does not pretend to provide new conceptual tools in the field of Resonant Ultrasound Spectroscopy. It simply proposes some method to efficiently obtain elastic coefficients without any expertise needed, any identification of the peaks to specific resonance modes and any assumption on E and  values. It is nevertheless brute-force in that it explores the whole variable space to find the determined solution.

In practice, the extraction of elastic coefficients following our scheme, i.e. scanning a wide range of E and , can last for some tens of seconds. For instance, an optimisation for E ranging from 100 to 400 GPa with a 0.1 GPa resolution and with  from 0.1 to 0.4 (with a 0.0005 step), would typically last 90 s on a personal computer. We have to mention here that all the results presented here were obtained with a code that was not specially optimised for speed. It could perform faster if the writing of elastic matrices during resonance mode calculation and interpolation of frequencies were optimised.

Our approach is also interesting in that it allows obtaining a solution map, which further gives a visual representation of the solution. This representation could help reveal potential concurrent solutions in case of unreasonable determined values (Edet,det) or even be used as proof of isotropicity. Our approach for elastic coefficient determination is only adapted to the RUS analysis of isotropic samples. Contrary to some previous approaches, which suggest obtaining guess values for C44 by considering the shear character of the very first mode, the approach proposed here makes no hypothesis on elastic coefficients. Besides, the fact that we can manage till 30% of missing modes without significant degradation of the results could be interesting when dealing with incomplete spectra. Thus our approach, even if conceptually very simple, is potentially very useful for non-expert users or for in situ characterisations, during which it is often difficult to control the absence of resonance modes. Indeed, while several experiments can easily be performed at room temperature to obtain a complete set of experimental resonance frequencies, such an approach is hard to apply on samples studied in situ, for instance, as a function of temperature or of a magnetic field. Obviously, our approach aims not to enter into competition with more advanced or general methodologies based on probabilistic approaches, which are especially powerful in determining elastic coefficients of samples with lower symmetry, i.e. for polycrystalline samples with texture or for single crystals.

V. Conclusion

This work proposes an alternative approach to analyse the Resonant Ultrasound Spectra of elastically isotropic samples to determine their two independent elastic coefficients. The procedure is straightforward, robust and efficient and can be implemented easily. Elastic coefficients can be obtained in a few tens of seconds with high accuracy without prior knowledge of the sample's elastic behaviour. It is also reliable when some resonance modes are absent in the original spectrum, which drastically simplifies the measure for the interpretation of in situ RUS measurements.

Figure 1 .

 1 Figure 1. Resonance spectra of samples with size L1=2 mm, L2= 3 mm, L3 = 5 mm, d = 2962

Figure 2 .

 2 Figure 2. Explored scheme for the elastic coefficients extraction

Figure 3 .

 3 Figure 3. Ratio between determined and known elastic coefficients for virtual samples:

Figure 4 .

 4 Figure 4. Ratio between determined and known elastic coefficients for virtual samples as a

Figure 5 .

 5 Figure 5. Ratio between determined and known values of elastic coefficients for virtual

Figure 6 .

 6 Figure 6. Map of the inverse of the distance between calculated and target spectra for a

b Samples dimensions and density elastic coefficients (known) elastic coefficients (determined) L1 (cm) L2 (cm) L3 (cm) Density (kg/m 3 ) EK (GPa) K Edet (GPa)

	10	1.095 2.992	5.299	3119	202.12	0.275	202.12	0.276
	11	1.192 1.093	0.986	2788	71.51	0.3381	71.5	0.3390
	12	0.775 0.905	1.319	4402	118.6	0.3209	118.3	0.3192
		a 1-10 : virtual samples
		b 11, 12 : real samples taken from [29-31]
								det
	1	4.573 5.029	1.135	2487	374.01	0.289	373.8	0.289
	2	1.892 3.234	5.287	6852	389.46	0.147	389.36	0.147
	3	5.289 2.927	4.501	6578	142.56	0.226	142.55	0.226
	4	4.461 5.297	3.779	6670	110.71	0.355	110.73	0.354
	5	3.894 4.289	4.215	2856	217.66	0.297	217.64	0.296
	6	4.030 0.659	1.884	6117	113.85	0.129	113.85	0.130
	7	3.974 2.085	5.251	3907	110.33	0.232	110.32	0.231
	8	4.327 4.475	4.434	5231	246.92	0.233	246.88	0.233
	9	4.047 4.273	1.880	2813	303.91	0.296	303.89	0.296

two real samples taken as good examples for applying an original RUS analysis procedure from previous articles

[START_REF] Ogi | Complete mode identification for resonant ultrasound spectroscopy[END_REF][START_REF] Bernard | Bayesian normal modes identification and estimation of elastic coefficients in resonant ultrasound spectroscopy[END_REF][START_REF] Bales | Bayesian inference of elastic properties with resonant ultrasound spectroscopy[END_REF]

.

Sample a,

Table 1 .

 1 Virtual (n° 1-10) and real (n° 11 and 12) samples characteristics including the dimensions of the rectangular parallelepiped samples (L1, L2, L3), the density and elastic isotropic coefficients EK and K. Results of the elastic coefficients extraction procedure.

Funding sources : This work has benefited from the financial support of the LabEx LaSIPS (ANR-10-LABX-0032-LaSIPS) managed by the French National Research Agency under the "Investissements d'avenir" program (ANR-11-IDEX-0003) through the MECATMO project. Part of this work was performed using HPC resources from the "Mésocentre" and « RUCHE » computing centers of

Data availability statement

The raw/processed data required to reproduce these findings cannot be shared at this time due to technical or time limitations.

In this section, we discuss the case of two real samples, which were presented in previous articles as good examples to illustrate the difficulty to identify normal modes and obtain the resulting elastic constants in isotropic solids.

Real sample 1

The first one numbered 11 in Table 1 corresponds to a polycrystalline Al alloy, presented first by Ogi et al. [START_REF] Ogi | Complete mode identification for resonant ultrasound spectroscopy[END_REF]. This sample was used to demonstrate the interest of using laser-assisted RUS to identify the symmetry of resonance modes and thus facilitate modes association. The same sample was further used by S. Bernard et al. [START_REF] Bernard | Bayesian normal modes identification and estimation of elastic coefficients in resonant ultrasound spectroscopy[END_REF] to demonstrate the Bayesian-based approach's efficiency for the determination of elastic coefficients. In the original article, the authors mentioned: "We tried to deduce the Cij using the usual RUS approach with the initial set of Cij used to compute the displacements … pairing the closest resonance frequencies from the measurements and calculations. The inverse calculation failed to converge, indicating that the usual RUS method is sensitive to the initial guesses ". These authors then succeeded in refining elastic constants only after identifying vibration modes from displacement measurements through an interferometry method, taking benefit from the fact that "the displacement distribution patterns are insensitive to the elastic constants ".

Real sample 2

The second example, n°12 in Table 1, is a polycrystalline Ti-6Al-4V alloy with fine grains. This sample was used to demonstrate the Bayesian inference efficiency to determine elastic coefficients and prove the isotropic character of the sample [START_REF] Bales | Bayesian inference of elastic properties with resonant ultrasound spectroscopy[END_REF]. This sample did not make the object of elastic coefficients determination from the standard procedure, by which we can only compare with the results obtained from the Bayesian approach.