Mahyunirsyah Mahjudin

Pascal Lardeur
email: pascal.lardeur@utc.fr

Frédéric Druesne

Irwan Katili

Extension of the Certain Generalized Stresses Method for the stochastic analysis of homogeneous and laminated shells

Keywords: stochastic, finite elements, shells, laminated structures, CGSM, Monte Carlo simulation

The paper presents an extension of the Certain Generalized Stresses Method (CGSM) for the static finite element analysis of homogeneous and laminated shells with variability. The basic assumption is that the generalized stresses do not depend on input parameters perturbation. The CGSM is a non-intrusive method that requires only one finite element analysis with some load cases to calculate the variability of mechanical quantities of interest. The uncertain input parameters are material and physical properties. Uniform random parameters as well as random fields are considered. The displacements statistical results: mean value, standard deviation and distribution are obtained by Monte Carlo simulations, using a semi-analytical formula. Two examples are treated: the Scordelis-Lo shell roof and an automotive windscreen. The method provides results of good quality and is very economical from a computational time point of view.

Introduction

Improvement of the predictive capability of numerical models due to uncertainty is currently a relevant challenge. Taking into account uncertainty, leading to stochastic finite element models, is one way to describe more closely the physical phenomena and so to increase this predictive capability, as stated by Oberkampf et al. [START_REF] Oberkampf | Verification, validation and predictive capability in computational engineering and physics[END_REF] in a paper dedicated to the verification and validation methodology. Recently much research has been carried out to develop different approaches and apply them in the context of structural mechanics. As it is well known, stochastic finite element models are usually classified in two categories: probabilistic approaches and possibilistic ones. In a probabilistic approach the input parameters are described by statistical distributions and the objective is to predict the statistics of output quantities. In possibilistic approaches, only the bounds of input parameters are defined and the objective is to calculate the bounds of the output quantities.

Taking into account uncertainty in finite element analysis of plate and shell structures, is currently a research issue. With regard to bending plates, a lot of methods have been developed, since the early contribution of Graham and Deodatis [START_REF] Graham | Variability response functions for stochastic plate bending problems[END_REF]. In particular for laminated plates, numerous approaches for various applications were developed, namely by Jeong and Shenoi [START_REF] Jeong | Probabilistic strength analysis of rectangular FRP plates using Monte Carlo simulation[END_REF], 2 António and Hoffbauer [START_REF] António | Uncertainty analysis based on sensitivity applied to angleply composite structures[END_REF], Onkar et al. [START_REF] Onkar | Probabilistic failure of laminated composite plates using the stochastic finite element method[END_REF], Chen and Soares [START_REF] Chen | Spectral stochastic finite element analysis for laminated composite plates[END_REF], Pandit et al. [START_REF] Pandit | Stochastic perturbation-based finite element for deflection statistics of soft core sandwich plate with random material properties[END_REF], Chandrashekhar and Ganguli [START_REF] Chandrashekhar | Nonlinear vibration analysis of composite laminated and sandwich plates with random material properties[END_REF], Noh and Park [START_REF] Noh | Response variability of laminate composite plates due to spatially random material parameter[END_REF], Sobey et al. [START_REF] Sobey | Monte Carlo reliability analysis of tophat stiffened composite plate structures under out of plane loading[END_REF], Li et al. [START_REF] Li | Stochastic thermal buckling analysis of laminated plates using perturbation technique[END_REF], Sepahvand [START_REF] Sepahvand | Spectral stochastic finite element vibration analysis of fiber-reinforced composites with random fiber orientation[END_REF], Yin et al. [START_REF] Yin | The Certain Generalized Stresses Method for static analysis of multilayered composite plates with variability of material and physical properties[END_REF], Grover et al. [START_REF] Grover | Influence of parametric uncertainties on the deflection statistics of general laminated composite and sandwich plates[END_REF], Tomar and Talha [START_REF] Tomar | Influence of material uncertainties on vibration and bending behaviour of skewed sandwich FGM plates[END_REF]. Argyris et al. [START_REF] Argyris | Stochastic finite element analysis of shells[END_REF] applied the weighted integral and local average methods to calculate stochastic stiffness matrices of shell structures. Then as for bending plates, stochastic analysis applied to shell structures leaded to numerous contributions. For laminated shells, Tripathi et al. [START_REF] Tripathi | Free vibration of laminated composite conical shells with random material properties[END_REF] applied the first order perturbation technique for the free vibration stochastic analysis of laminated conical shells. Lal et al. [START_REF] Lal | Stochastic post buckling analysis of laminated composite cylindrical shell panel subjected to hygrothermomechanical loading[END_REF] applied the perturbation technique to the geometrically nonlinear stochastic analysis of laminated cylindrical shells. Broggi and Schüeller [START_REF] Broggi | Efficient modeling of imperfections for buckling analysis of composite cylindrical shells[END_REF] used the Monte Carlo simulation for buckling analysis of cylindrical shells. Dey et al. [START_REF] Dey | Stochastic free vibration analyses of composite shallow doubly curved shells -A Kriging model approach[END_REF][START_REF] Dey | Rotational and ply-level uncertainty in response of composite shallow conical shells[END_REF] and Mukhopadhyay et al. [START_REF] Mukhopadhyay | On quantifying the effect of noise in surrogate based stochastic free vibration analysis of laminated composite shallow shells[END_REF] used metamodels to study the stochastic natural frequencies of shells. The same type of approach was developed by Dey et al. [START_REF] Dey | Stochastic dynamic stability analysis of composite curved panels subjected to non-uniform partial edge loading[END_REF] for the stochastic buckling loads of shells. Pouresmaeeli et al. [START_REF] Pouresmaeeli | Uncertainty propagation in vibrational characteristics of functionally graded carbon nanotube-reinforced composite shell panels[END_REF] used the non-deterministic interval analysis method to study the variability of natural frequencies of functionally graded composite shells. Recently Tomar et al. [START_REF] Tomar | State of the art of composite structures in non-deterministic framework: A review[END_REF] presented a state of the art of composite structures in non-deterministic framework. A lot of research contributions are cited in the paper.

Numerous research works concern the modeling of input variability parameters. Techniques and methods have been developed for the calculation of random variables and especially for random fields. First a probabilistic approach requires the choice of statistical laws. Numerous laws have been debated in the literature and used for applications. The Gaussian distribution is probably the most commonly used model for random phenomena. To prevent mathematical difficulties due to possible extreme values, truncated Gaussian laws are often used. In the context of random fields, several variants were developed for the calculation of the covariance matrix. One distinguishes between point discretization methods and average discretization ones. The well-known point methods are the midpoint method (Der Kiureghian and Ke [START_REF] Der Kiureghian | The stochastic finite element method in structural reliability[END_REF]), the shape function or interpolation method (Liu et al. [START_REF] Liu | Random field finite elements[END_REF][START_REF] Liu | Probabilistic finite elements for nonlinear structural dynamics[END_REF]), the integration point method (Brenner and Bucher [START_REF] Brenner | A contribution to the SFE-based reliability assessment of nonlinear structures under dynamic loading[END_REF]), the optimal linear estimation method (Li and Der Kiureghian [START_REF] Li | Optimal discretization of random fields[END_REF]). The average methods are the local average method (Vanmarcke et al. [START_REF] Vanmarcke | Stochastic finite element analysis of simple beams[END_REF][START_REF] Vanmarcke | Random fields and stochastic finite elements[END_REF]) and the weighted integral method (Deodatis [START_REF] Deodatis | Weighted integral method. I: stochastic stiffness matrix[END_REF], Deodatis and Shinozuka [START_REF] Deodatis | Weighted integral method. II: response variability and reliability[END_REF]). Random fields are generally represented by series discretization methods. Several approaches have been used for stochastic analysis: the singular value decomposition (Gerbrands [START_REF] Gerbrands | On the relationships between SVD, KLT and PCA[END_REF]), the Karhunen-Loève expansion (Loève [START_REF] Loève | Probability theory II[END_REF], Ghanem and Spanos [START_REF] Ghanem | Stochastic finite elements method : a spectral approach[END_REF]), the spectral representation (Shinozuka and Deodatis [START_REF] Shinozuka | Simulation of stochastic processes by spectral representation[END_REF], Grigoriu [START_REF] Grigoriu | On the spectral representation method in simulation[END_REF]), and the polynomial chaos expansion (Wiener [40], Sakamoto and Ghanem [START_REF] Sakamoto | Polynomial chaos decomposition for the simulation of non-Gaussian nonstationary stochastic processes[END_REF]). However, these existing methods have some limitations, in particular in the context of industrial applications. They are often very costly in computational time, some of them work well only if a limited number of random parameters is considered, some methods are valid only for low level variability. For industrial applications, it is important to propose non-intrusive approaches so that standard software (Abaqus, MSC/Nastran, Ansys…) can still be used. So developing an economical and reliable method accounting for the variability in the finite element calculations is still a challenge. In 2012, Lardeur et al. [START_REF] Lardeur | The Certain Generalized Stresses Method for the static finite element analysis of bar and beam trusses with variability[END_REF] propose the Certain Generalized Stresses Method (CGSM) for the probabilistic analysis of isotropic bars and beams. Mahjudin et al. [START_REF] Mahjudin | Stochastic finite element analysis of plates with the Certain Generalized Stresses Method[END_REF] and Yin et al. [START_REF] Yin | The Certain Generalized Stresses Method for static analysis of multilayered composite plates with variability of material and physical properties[END_REF] develop this method for homogenous plates and laminated composite plates respectively.

In this paper, the CGSM is developed for homogeneous and laminated shells with uncertain material and physical properties. These uncertain parameters are represented by random variables or random fields. In Section 2, after recalling the principle of the CGSM, the formulation is detailed for homogeneous and laminated shells. In Section 3, two examples with several variants concerning the definition of random input parameters are treated. A synthesis of the performances of the CGSM in computational time is given in Section 4. Some conclusions and perspectives are drawn in Section 5. Nomenclature is given in Appendix.

The Certain Generalized Stresses Method

Basic ideas and principle of the CGSM

The flowchart of the CGSM is shown in Fig. 1. The CGSM is based on the assumption that the generalized stresses are independent of the uncertain parameters. This assumption is strictly met for statically determinate structures or if the perturbation due to uncertainty is uniform throughout the structure, that is to say if all the terms of the stiffness matrix are multiplied by the same coefficient. In reality mechanical structures are generally statically indeterminate and the perturbation is generally not uniform. Previous studies [START_REF] Yin | The Certain Generalized Stresses Method for static analysis of multilayered composite plates with variability of material and physical properties[END_REF][START_REF] Lardeur | The Certain Generalized Stresses Method for the static finite element analysis of bar and beam trusses with variability[END_REF][START_REF] Mahjudin | Stochastic finite element analysis of plates with the Certain Generalized Stresses Method[END_REF] have shown that the CGSM leads to satisfactory results for bar and beam trusses as well as for homogeneous or laminated plates, even if the CGSM assumption is not exactly met. The objective of this study is to assess the relevance of the mechanical assumption described above, for shells. In this paper an error indicator is proposed in order to obtain an estimation of the possible error. Thanks to this CGSM assumption, only one finite element run with some load cases, in the nominal configuration, is necessary to calculate the generalized stresses. To calculate the variability of displacement at m nodes, as explained in detail in Section 2.2, one finite element analysis with m+1 load cases is needed. The displacement at a given node is evaluated using Castigliano's theorem. The derivative of the strain energy with respect to the possible force applied at the point of interest leads to a semi-analytical expression of the displacement at this point. By using this expression, a Monte Carlo simulation is performed to calculate the mean value, standard deviation and distribution of the displacement. For some types of structures [START_REF] Lardeur | The Certain Generalized Stresses Method for the static finite element analysis of bar and beam trusses with variability[END_REF], the mean value and the standard deviation can also be obtained analytically. The CGSM can also be used to evaluate the variability of strains, stresses and failure criteria [START_REF] Yin | The Certain Generalized Stresses Method for static analysis of multilayered composite plates with variability of material and physical properties[END_REF]. The CGSM is a non intrusive method and it is compatible with the use of any standard finite element software. In this study, Abaqus [START_REF]Abaqus Analysis User's Guide, Abaqus 6.14[END_REF] has been used to treat the examples presented in Section 3. Other examples, not presented here, have been treated with the DKMQ element [START_REF] Katili | Shear deformable shell element DKMQ24 for composite structures[END_REF]. In summary, the CGSM can be considered as a post-treatment of one standard finite element calculation. This is an advantage from the computational time point of view. It is worth noting that nominal analyses in Abaqus are performed with standard finite elements, without any modification of their formulation. For examples presented in Section 3, S4R shell finite elements are used. In the CGSM approach, these nominal analyses are exploited only to calculate the generalized stresses.

Verification and validation methodology has been applied as far as possible. Validation which involves experimental results is not considered in this paper. For the verification stage, mesh convergence is studied and the number of trials chosen for Monte Carlo simulations is justified. Moreover errors due to the CGSM assumption are systematically estimated through an error indicator. These issues are discussed further in next sections of the paper.

Formulation of CGSM for homogeneous and laminated shells

In this section, the formulation of the CGSM is given for homogeneous and laminated shells, in particular for the calculation of the variability of displacements. This formulation takes into account material and physical properties and some of these properties may be uncertain. These uncertain parameters may be modelled by random variables or random fields. The homogeneous case being a particular case, the formulation is described only for the general laminated case. The overall strain energy is obtained by combining membrane, bending and transverse shear strain energies. For the laminated case, it is assumed in this paper that each layer has an isotropic mechanical behavior.

The plane stress elasticity matrix for layer k is:

= 1 - 1 0 1 0 0 0 1 - 2
1
with the elasticity modulus for layer k and the Poisson's ratio for layer k;

The transverse shear elasticity matrix for layer k is:

= 1 0 0 1 2
with the shear modulus for layer k.

Using the thick plate theory equations [START_REF] Reddy | Mechanics of laminated composite plates and shells: theory and analysis[END_REF], the strain energy for laminated plates can be written as:

= 1 2 χ ! " ! ! ! # $ % & χ ' () 3
with = *+ , + -+ ,-. the in-plane strains, χ = /χ , χ -χ ,-0 the curvatures; = *1 ,2 1 -2 . the out-of-plane shear strains; A, B, D and S the generalized membrane stiffness matrix, membrane-bending coupling stiffness matrix, bending stiffness matrix and transverse shear stiffness matrix respectively. These matrices are defined as: The relation between the strains and the generalized stresses is given by:

& χ ' = ! " ! ! ! # $ 69 & ? @ A ' = B B ! B "B ! ! ! #B $ & ? @ A ' 5
with ? = *C , C -C ,-. the force resultants, @ = *D , D -D ,-. the bending moments and A = *E , E -. the transverse shear forces.

Reporting Eq. ( 5) into Eq. ( 3), one obtains:

= 1 2 ? F A B B ! B "B ! ! ! #B $ % & ? @ A ' () 6
The strain energy may be decomposed into several contributions:

membrane G = 1 2 ? B ? () 7
membrane-bending coupling GH = ? B @ ()

8 bending H = 1 2 @ "B % @ () 9
transverse shear 4 = 1 2 A #B % A () 10
Finally the strain energy can be written:

= 1 2 ? B ? % () + ? B @ % () + 1 2 @ "B @ % () + 1 2 A #B A % () 11
Considering that the domain is spatially discretized and the mesh contains n finite elements, Eq. ( 11) gives:

= 1 2 8 ? B ? + 2? B @ + @ "B @ + A #B A % J () :9 12
In order to apply the Castigliano's theorem for calculating the displacement K at a point P in a given direction, the forces in each element are decomposed into two contributions:

for the force resultants

C , = C , L + MC , LL 13
C -= C - L + MC - LL C ,-= C ,- L + MC ,- LL for the moments D , = D , L + MD , LL D -= D - L + MD - LL D ,-= D ,- L + MD ,-

LL

for the transverse shear forces

E , = E , L + ME , LL E -= E - L + ME - LL
where i is the element number; F is the load applied at point P in the direction of interest (F is a force or a moment if the variability of a displacement or a rotation must be assessed respectively); the subscripts

N , O indicate the local directions; C , L , C - L , C ,- L , D , L , D - L , D ,- L , E , L ,
C , L , C - L , C ,- L , D , L , D - L , D ,- L , E , L , E - L and C , LL , C - LL , C ,- LL , D , LL , D - LL , D ,- LL , E , LL , E - LL respectively.
Substituting Eq. ( 13) into Eq. ( 12) gives:

14

Our objective is to consider simple interpolations of generalized stresses over one element. These interpolations are independent of the initial formulation of the element, consequently the development and application of the CGSM does not require any information about the

1 2 8 Q R S C , L + MC , LL C - L + MC - LL C ,- L + MC ,- LL T )U 99 )U 9 0 )U 9 )U 0 0 0 )U << $ S C , L + MC , LL C - L + MC - LL C ,- L + MC ,- LL T % J :9 + 2 S C , L + MC , LL C - L + MC - LL C ,- L + MC ,- LL T VU 99 VU 9 0 VU 9 VU 0 0 0 VU << $ S D , L + MD , LL D - L + MD - LL D ,- L + MD ,- LL T + S D , L + MD , LL D - L + MD - LL D ,- L + MD ,- LL T WU 99 WU 9 0 WU 9 WU 0 0 0 WU << $ S D , L + MD , LL D - L + MD - LL D ,- L + MD ,- LL T + X E , L + ME , LL E - L + ME - LL Y Z [U 99 0 0 [U \ X E , L + ME , LL E - L + ME - LL Y ] ^ () =
formulation of finite elements used for the nominal analysis. For T3 and Q4 elements considered here, the generalized stresses are assumed to be constant over one element [START_REF] Lardeur | The Certain Generalized Stresses Method for the static finite element analysis of bar and beam trusses with variability[END_REF]. The values calculated at the center of each element are considered. The relevance of this choice, directly related to convergence characteristics, is discussed in Section 3 where examples are treated. Eq. ( 14) becomes:

15

To obtain the displacement at a point P in a given direction, the Castigliano's theorem is applied:

K = _M 16
The displacement contains the following contributions:

K = G _M + GH _M + H _M + 4 _M 17
or

K = K G + K GH + K H + K 4 18 = 8 ) 2 Q R S C , L + MC , LL C - L + MC - LL C ,- L + MC ,- LL T )U 99 )U 9 0 )U 9 )U 0 0 0 )U << $ S C , L + MC , LL C - L + MC - LL C ,- L + MC ,- LL T :9 + 2 S C , L + MC , LL C - L + MC - LL C ,- L + MC ,- LL T VU 99 VU 9 0 VU 9 VU 0 0 0 VU << $ S D , L + MD , LL D - L + MD - LL D ,- L + MD ,- LL T + S D , L + MD , LL D - L + MD - LL D ,- L + MD ,- LL T WU 99 WU 9 0 WU 9 WU 0 0 0 WU << $ S D , L + MD , LL D - L + MD - LL D ,- L + MD ,- LL T + X E , L + ME , LL E - L + ME - LL Y Z [U 99 0 0 [U \ X E , L + ME , LL E - L + ME - LL Y ] ^
where K G is the displacement due to the force resultants, K GH is the displacement due to coupling between the membrane and bending effects, K H is the displacement due to the bending moments, and K 4 is the displacement due to the transverse shear forces.

Eq. ( 15) is used to calculate the different terms of the displacement. Displacement due to the force resultants is:

K G = 8 ) 2 )U 99 C 9 + )U C + )U << C < + )U 9 C ` 19
where:

C 9 = 2C , J LL aC , J L + MC , J LL b C = 2C -J LL aC -J L + MC -J LL b C < = 2C ,-J LL aC ,-J L + MC ,-J LL b C `= 2aC , J L C -J LL + C -J L C , J LL + 2MC , J LL C -J LL b 20
Displacement due to coupling between the membrane and bending effects is:

K GH = 8 ) VU 99 DC 9 + VU DC + VU << DC < + VU 9 DC ` 21
where:

DC 9 = C , J L D , J LL + C , J LL D , J L + 2MC , J LL D , J LL DC = C -J L D -J LL + C -J LL D -J L + 2MC -J LL D -J LL DC < = C ,-J L D ,-J LL + C ,-J LL D ,-J L + 2MC ,-J LL D ,-J LL DC `= C -J L D , J LL + C -J LL D , J L + 2MC -J LL D , J LL + C , J L D -J LL + C , J LL D -J L + 2MC , J LL D -J LL 22
Displacement due to the bending moments is:

K H = 8 ) 2 WU 99 D 9 + WU D + WU << D < + WU 9 D ` 23
where:

D 9 = 2D , J LL aD , J L + MD , J LL b D = 2D -J LL aD -J L + MD -J LL b D < = 2D ,-J LL aD ,-J L + MD ,-J LL b D `= 2aD , J L D -J LL + D -J L D , J LL + 2MD , J LL D -J LL b 10
Displacement due to the transverse shear forces is:

K 4 = 8 ) 2 [U 99 E 9 + [U E 25
where:

E 9 = 2E , LL aE , L + ME , LL b E = 2E - LL cE - L + ME - LL d 26
The expression of K summarized in Eq. ( 18) and detailed in Eqs. ( 19) to ( 26) is used for each trial of the Monte Carlo simulation, to calculate the displacement variability of a point of a laminated shell. For the homogeneous shell case, the general formulation presented here can be simplified. Namely it can be used by ignoring displacement component due to the membranebending coupling contribution K GH .

During the Monte Carlo simulations using the semi-analytical formula defined by Eqs. ( 18) to [START_REF] Der Kiureghian | The stochastic finite element method in structural reliability[END_REF], AI, BI, DI and SI are random matrices. All other terms, in particular those which concern generalized stresses, are certain. Consequently for a given Monte Carlo trial, instead of running a finite element analysis, using the semi-analytical formula allows the calculation of a displacement. Calculating the displacement by the semi-analytical formula is much faster than running a finite element analysis. The gain due to the use of the semi-analytical formula is quantified in Section 4.

The decomposition of the generalized stresses described in Eq. ( 13) is specific to one given point. So, to calculate the variability of displacement at m nodes, one finite element analysis with m+1 load cases is needed. This finite element analysis is performed in the nominal configuration, so for the different load cases the stiffness matrix is always the same and consequently, this matrix has to be inverted only once.

It is worth noting that the CGSM does not use any assumption concerning loading. So this method is compatible with any type of load case, namely concentrated and distributed loads may be considered. As far as linear quasi-static analysis is used, time-varying load vector may be considered as well, if the components of the load vector vary proportionally. Indeed in this case the generalized stresses vary proportionally in the same way.

Uncertainty modeling

Uncertainty modeling is represented by random variables or random fields. In Section 3, two examples are presented. The first one (Section 3.1) is studied using random fields but in the second one, only random variables are used. The uncertain input parameters are the material (E, ν) and physical (h) properties. For random fields, a specific methodology must be applied to define the values of uncertain parameters in each finite element. The CGSM is compatible with different methods for the construction of random fields. First the covariance matrix must be calculated. For the application of the CGSM to composite plates, Yin et al. [START_REF] Yin | The Certain Generalized Stresses Method for static analysis of multilayered composite plates with variability of material and physical properties[END_REF] used the midpoint method and the local average method. For the example described in Section 3.1, the midpoint method proposed by Der Kiureghian and Ke [START_REF] Der Kiureghian | The stochastic finite element method in structural reliability[END_REF] is used. The random field is discretized at the centers of gravity of elements. The covariance matrix is defined by: efghi j = σ 1 -l exp -l 27 with l = ∆q j + ∆r j λ 28 where ∆q j and ∆r j are respectively the distances between the midpoints of elements i and j in the x and y direction, λ is the correlation length.

Then according to Ghanem and Spanos [START_REF] Ghanem | Stochastic finite elements method : a spectral approach[END_REF], the random field is calculated by the Karhunen-Loève expansion.

Influence of the spatial discretization

Spatial discretization is an important issue in the finite element analysis of structures, particularly in presence of uncertainties. As suggested and tested by Mahjudin et al. [START_REF] Mahjudin | Stochastic finite element analysis of plates with the Certain Generalized Stresses Method[END_REF], in this study it is assumed that an optimal mesh for the nominal calculation is also convenient for the variability evaluation. Therefore, the strategy adopted here is finding the optimal mesh for the nominal configuration and then exploit it to calculate the variability. The optimal mesh is identified thanks to a classical convergence study by increasing progressively the mesh refinement level.

Error indicator

Since the CGSM uses a mechanical assumption, it is important to quantify the possible errors due to this assumption. The CGSM can be validated by comparing statistical results with the direct Monte Carlo simulation, considered as the reference method. But the cost of direct Monte Carlo simulation is very high because a great number of finite element analyses is required for this method. The proposal of an error indicator for the CGSM is therefore interesting. The objective of the indicator is to quickly estimate the error level on statistical results obtained by the CGSM, compared with the direct Monte Carlo simulation. The error indicator proposed here is based on an approach proposed by Yin et al. [START_REF] Yin | Performances assessment of the Modal Stability Procedure for the probabilistic free vibration analysis of laminated composite structures[END_REF] requiring a small number of random trials.

The errors between the results obtained by the CGSM and the direct Monte Carlo simulation, for a large number of trials meeting the convergence condition, are considered as the reference errors. For a number of trials nt the error between the MSP and the direct Monte Carlo simulation is evaluated by:

ss = t uvwx6 t yxu t yxu 29
where t uvwx and t yxu are respectively the results obtained by the CGSM and the direct Monte Carlo simulation with >z trials using the same input data. These results can be the mean value, standard deviation or coefficient of variation of displacements. The objective is to verify that the errors evaluated with a small or large number (10000 trials are considered here, this number being generally considered as relevant for a good convergence level of a Monte Carlo simulation) of trials are close. Yin et al. [START_REF] Yin | Performances assessment of the Modal Stability Procedure for the probabilistic free vibration analysis of laminated composite structures[END_REF] showed that using 10 trials, the estimation of errors is quite acceptable. This error indicator is assessed in Section 3 for the two examples described.

Examples

Scordelis-Lo shell roof

Presentation of the example

The first example, presented in Figure 2, is the homogeneous and isotropic Scordelis-Lo shell roof. This example has already been treated by Stefanou and Papadrakakis [START_REF] Stefanou | Stochastic finite element analysis of shells with combined random material and geometric properties[END_REF]. The two longitudinal edges are free and the two circular edges are supported by rigid diaphragms. Along the rigid diaphragms it is assumed that the displacements in directions x, y, and z are zero. The roof is subjected to gravity loading with magnitude 4000 N/m 27) and several correlation lengths: λ = 0.004L to 8.55L, are considered. The random fields are applied on the whole structure. The direct Monte Carlo simulations and the CGSM are performed with 10000 trials. The vertical displacement variability at point C, located at the middle of the longitudinal edge, is calculated.

The results of the convergence study performed in the nominal configuration are reported in Table 1. In order to show a first validation in the nominal case, the results of a classical analysis with Abaqus are given, as well as the results obtained with the CGSM formulation (Eq. ( 18)). In the latter case first one Abaqus analysis has been necessary to extract the generalized stresses. It is considered here that a given mesh is satisfactory if the error compared to a reference mesh is less than 0.5%. One can observe that the convergence speed is similar for a classical analysis with Abaqus and for the CGSM. This is a hopeful performance of CGSM. For this example, these results justify the choice of the approximation made in Section 2.2 (Eq. ( 15)), considering that the generalized stresses are constant over one element. The regular 60x84 mesh, containing 5040 S4R shell finite elements (see Figure 3), which meets the convergence criterion, is exploited for the calculation of variability.

Displacement variability

The CGSM results are compared to reference solutions obtained by direct Monte Carlo simulations. In the first case, the random parameter is E and three correlation lengths are considered. For each correlation length, the mean value, the standard deviation and the coefficient of variation of displacement at point C are reported in Table 2. The mean displacement is quite stable over the whole correlation length range. However variability level of the displacement decreases for small correlation lengths. This is due to a compensation phenomenon. Errors are given in Figure 4. The errors calculated with 10000 trials as well as the values of the error estimator requiring only 10 trials are reported. The CGSM provides very accurate results for the whole correlation length range. The errors are close to zero for a large correlation length but slightly increase when the correlation length decreases. Maximal errors are observed on the standard deviation, for the smallest correlation length case. The error estimator leads to a very good approximation of errors, indeed the difference between the error and the error estimator values is always less than 1%. In the second case, the random parameter is h and again several correlation lengths are considered. Table 3 shows the mean value, the standard deviation and the coefficient of variation of displacement obtained by the CGSM and the direct Monte Carlo simulations. Globally, the tendencies are similar to case 1, but the variability level is about twice higher when h is random.

As for case 1, errors reported in Figure 5 increase for small values of the correlation length.

Considering the thickness h as a random parameter leads to slightly larger errors, compared to the situation where the elasticity modulus E is random. These results show that in this example the CGSM assumption is less valid when the random parameter is the thickness. However, in all cases the errors remain acceptable, in the context of a quick method. Again, the error estimator leads to a good approximation of error, indeed the difference between the error and the error estimator values is always less than 3%. For small correlation lengths, the variability level is very low. As already stated, this is due to a compensation phenomenon. This figure clearly highlights that the displacement variability level is significantly higher when the random parameter is the thickness, compared to the situation where the random parameter is the elasticity modulus.

Presentation of the example

The second example deals with a car windscreen, which is a sandwich structure. A vibration study with variability of this Renault windscreen was presented by Scigliano et al. [START_REF] Scigliano | Verification, validation and variability for the vibration study of a car windscreen modeled by finite elements[END_REF] in the context of the verification and validation methodology. This example was also treated by Arnoult et al. [START_REF] Arnoult | The modal stability procedure for dynamic and linear finite element analysis with variability[END_REF] who studied the numerical assessment of variability of natural frequencies. In the present study this structure is subjected to pressure loading with magnitude 1900 N/m 2 . The windscreen is assumed clamped along its whole boundary. The sandwich structure consists of 5 layers as shown in Figure 8. The external layers (1 and 5, blue in Figure 8) are made of glass, layers 2 and 4 (yellow in Figure 8) are made of a polymer: the polyvinyl butyl (PVB). Layer 3 (orange in Figure 8) is made of a specific so-called "acoustic" polymer with very small elasticity modulus. The material properties and variability levels shown in Table 4, were initially proposed by Arnoult et al. [START_REF] Arnoult | The modal stability procedure for dynamic and linear finite element analysis with variability[END_REF], based on a study performed by the Renault company. The variability of material properties and thicknesses are represented by random variables. Truncated Gaussian distributions are considered for material properties and thicknesses, as suggested by Arnoult et al. [START_REF] Arnoult | The modal stability procedure for dynamic and linear finite element analysis with variability[END_REF]. The variables are defined for each layer and all random parameters are independent.

Results are observed at point A where the displacement is maximal. The variability of the vertical displacement at point A is calculated by direct Monte Carlo simulations and the CGSM, which are both performed with 10000 trials.

The results of the convergence study in the nominal configuration are reported in Table 5. As for example 1, the values are given for a classical analysis with Abaqus, as well as for the CGSM formulation (Eq. ( 18)). As for the first example, one can observe that the convergence characteristics are similar for both the finite element approach and the CGSM one. These results confirm the justification of the approximation made in Section 2.2 (Eq. ( 15)), considering that the generalized stresses are constant over one element. The regular mesh containing 744 S4R shell finite elements (see Figure 9), which meets the convergence criterion, is exploited for the calculation of variability.

Displacement variability

The CGSM results are compared to a reference solution obtained by a direct Monte Carlo simulation. Three cases are treated: the random parameters are the elasticity moduli, the thicknesses and both these two types of parameters for cases 1, 2 and 3 respectively (see Table 4). In this example there is no random field, so the parameters are uniform over a given layer. Cases 1, 2, 3 lead to 5, 5 and 10 independent random parameters respectively. For each case, the mean value, the standard deviation and the coefficient of variation of displacements at point A are reported in Tables 6, 7 and 8. The output variability levels are rather small. Errors are given in Figure 10. The CGSM provides very accurate results in all cases. The error estimator leads to very good approximation of errors, indeed the difference between the error and the error estimator values is always less than 2%.

. In order to test the robustness of the CGSM, a complementary case, with a larger input variability level, has been treated. The elasticity moduli and the thicknesses are random. The data are identical to those reported in Table 4, except for the glass layers which have a large influence on the mechanical behavior of the windscreen. The coefficients of variation are now increased: CV(Ei) = CV(hi) = 10% for the glass layers. Table 9 shows that the output variability level is now very high. Figure 12 shows that even in this case, errors remain quite limited and the error estimator gives satisfactory results again. Figure 13 shows the probability density function of the displacement at point A. Again, the distribution obtained with the CGSM is very close to the reference distribution obtained with the direct Monte Carlo simulation. These results highlight that for this example, the CGSM assumption, that is to say the generalized stresses are certain, is quite valid. The CGSM is particularly precise in this example because for one given layer, the perturbation is uniformly distributed over the whole structure. Consequently, the generalized stresses are quite certain.

Synthesis of the performances of the CGSM in computational time

Before assessing the overall performance of the CGSM, for one trial of the Scordelis-Lo shell roof example, the net efficiency of the CGSM formula Eq. ( 18), compared with a classical finite element analysis, is measured. The results, presented in Table 10 for three different meshes, highlight the significant gain obtained thanks to the CGSM formula. The tests to assess the global performance of the CGSM are performed for the industrial example: the windscreen. Results are presented in Figure 14 for several mesh refinement levels and several numbers of trials. The acceleration factor is the ratio between the total time necessary for the direct Monte Carlo simulation and the total time necessary for the CGSM. The calculation of the error indicator is taken into account in the CGSM computational time. Figure 14 shows that the acceleration factor increases with the number of trials, which is a hopeful characteristic of the method. The acceleration factor remains relatively stable for different mesh refinement levels. For a number of trials classically considered and equal to 10000, the acceleration factor is comprised between 700 and 800, depending on the mesh refinement level. This result reveals the high efficiency of the CGSM as far as computational time is concerned. Two examples were treated, the Scordelis-Lo shell roof and an automotive windscreen. Satisfactory results were obtained for the mean value, the standard deviation and the probability density of displacements. In particular, the method is very powerful for the windscreen example.

For both examples and different cases tested, the error estimator of statistical quantities gives satisfactory results. From a computational time point of view, the CGSM is very efficient compared to the direct Monte Carlo simulation. The CGSM leads to a large reduction of the high computational cost that characterizes most of the existing approaches.

As a perspective, the CGSM will be assessed for laminated shells made of anisotropic materials.

Other industrial cases will also be treated. For situations where the CGSM leads to some errors, it would be interesting to try and improve its efficiency. For example, the multi-fidelity concept is promising from this point of view. Displacement at a point P in a given direction due to force resultants K GH Displacement at a point P in a given direction due to coupling between the membrane and bending effects K H

Displacement at a point P in a given direction due to bending moments K 4 Displacement at a point P in a given direction due to transverse shear forces efghi j

Terms of the covariance matrix l Defined in Eq. ( 28) ∆q j

Distance between the midpoints of elements i and j in the x direction

Figure 1 .

 1 Figure 1. Principle of the CGSM method for calculating variability

4 with

 4 >= the number of layers and = the transverse shear correction coefficient.

Figure 2 .

 2 Figure 2. Scordelis-Lo shell roof under gravity loading

Figure 3 .

 3 Figure 3. Scordelis-Lo shell roof under gravity loading -the 60x84 mesh (5185 nodes and 5040 S4R elements)

4 CV

 4 ) (m) 4.638 10 -2 4.621 10 -2 4.639 10 -2 4.621 10 -2 4.639 10 -2 4.643 10 -2 σ(U) (m) 1.185 10 -4 1.156 10 -4 2.029 10 -4 1.992 10 -4 4.749 10 -4 4.753 10 -

Figure 4 .

 4 Figure 4. Scordelis-Lo shell roof under gravity loading (E random, CV(E) = 10%) -errors on the statistical results of displacement at point C

Figure 5 .Figure 6

 56 Figure 5. Scordelis-Lo shell roof under gravity loading (h random, CV(h) = 10%) -errors on the statistical results of displacement at point C Figure 6 shows the evolution of the coefficient of variation of displacement at point C, when the correlation length varies. The variability of the displacement increases non-linearly with the correlation length.For small correlation lengths, the variability level is very low. As already stated, this is due to a compensation phenomenon. This figure clearly highlights that the displacement variability level is significantly higher when the random parameter is the thickness, compared to the situation where the random parameter is the elasticity modulus. Figure6shows

  Figure6shows the evolution of the coefficient of variation of displacement at point C, when the correlation length varies. The variability of the displacement increases non-linearly with the correlation length. For small correlation lengths, the variability level is very low. As already stated, this is due to a compensation phenomenon. This figure clearly highlights that the displacement variability level is significantly higher when the random parameter is the thickness, compared to the situation where the random parameter is the elasticity modulus. Figure6shows

Figure 6 .Figure 7 Figure 7 .

 677 Figure 6. Scordelis-Lo shell roof under gravity loading (E or h random, CV(E) = 10%, CV(h) = 10%) -variability of displacement at point C Figure 7 shows the probability density function of the displacement at point C for an intermediate correlation length value λ = 0.17L, with E or h considered as uncertain. The distributions obtained with the CGSM are very close to the reference distributions obtained with the direct Monte Carlo simulation. However, due to moderate errors when the thickness is the random parameter, some discrepancy appears in this case. The distributions are almost Gaussian.

Figure 8 .

 8 Figure 8. Car windscreen under pressure loading -stacking sequence

Figure 10 .case 3 :

 103 Figure 10. Car windscreen (cases 1, 2, 3) -errors on the statistical results of displacement at point A

Figure 11 .

 11 Figure 11. Car windscreen under pressure loading -distribution of displacement at point A (U0 = 0.1469 mm)

Figure 12 .Figure 13 .

 1213 Figure 12. Car windscreen (case 4) -errors on the statistical results of displacement at point A

Figure 14 .

 14 Figure 14. Car windscreen under pressure loading -acceleration factor of CGSM5. ConclusionsThis research focuses on the development and the assessment of the Certain Generalized Stresses Method for the probabilistic analysis of homogeneous and laminated shells. The variability input parameters which are elasticity moduli and thicknesses, are defined by probabilistic laws. Uniform random variables as well as random fields are considered. The outputs are statistical quantities, namely the mean value, standard deviation and probability density functions of the displacements. The CGSM statistical results are obtained by Monte Carlo simulations, using a semi-analytical formula. They are compared with the direct Monte Carlo simulation considered as a reference. An error indicator which allows calculating an estimation of the possible errors due to the CGSM assumption, is proposed. This error indicator requires 10 finite element analyses.

LLF

  Bending moments in element i due to loads on the whole structure except at point P in the direction of interest D , LL , D - LL , D ,- LL Bending moments in element i due to a unitary load applied at point P in the direction of interest A, E , , E - Transverse shear forces E , , E - Transverse shear forces in element i Transverse shear forces in element i due to loads on the whole structure except at point P in the direction of interest E , LL , E - LL Transverse shear forces in element i due to a unitary load applied at point P in the direction of interest C 9 to C ` Defined in equation Eq. (20) DC 9 to DC ` Defined in equation Eq. (22) D 9 to D ` Defined in equation Eq. (24) E 9 to E Defined in equation Eq. Load applied at point P in the direction of interest K G

∆r j Distance between the midpoints of elements i and j in the y direction

  results obtained by the CGSM and the direct Monte Carlo simulation for nt trials t uvwx Statistical result obtained by the CGSM with >z trials t yxu Statistical result obtained by the direct Monte Carlo simulation with >z trials m X Mean value of variable X

  2 . The geometric characteristics are: L = 15.2 m, R = 7.6 m, { = 40 } . Sensitivity study leads to the conclusion that the Poisson's ratio has no significant influence on the variability of displacements. Consequently the Poisson's ratio ν is deterministic and its value is 0.3. The random parameters are the elasticity modulus E

and the thickness h, represented by isotropic random fields. Gaussian distributions are considered for E and h. As suggested by Stefanou and Papadrakakis

[START_REF] Stefanou | Stochastic finite element analysis of shells with combined random material and geometric properties[END_REF]

, samples with non-physical parameter values are discarded. The mean values of these parameters are m(E) = 2.1x10 11 N/m 2 and m(h) = 0.76 m. The coefficients of variation CV(E) and CV(h) are equal to 10%. An isotropic exponential correlation function defined in Eq. (

Table 1 .

 1 Scordelis

		Mesh 40x56	Mesh 60x84	Mesh 80x112
		2240 elements	5040 elements	8960 elements
		Displacement	Error	Displacement	Error	Reference
		(m)	(%)	(m)	(%)	(m)
	Classical FE analysis CGSM (Eq. (18))	4.624 10 -2 4.618 10 -2	0.8 0.7	4.596 10 -2 4.592 10 -2	0.2 0.1	4.587 10 -2

-Lo shell roof under gravity loading -convergence study of nominal displacement at node C

Table 2 .

 2 Scordelis

		error indicator: 10 trials	error: 10000 trials
		3.0%
	CGSM erorr	0.5% 1.0% 1.5% 2.0% 2.5%
		0.0%
		mean value	standard	coefficient of
			deviation	variation
		correlation length λ = 0.004L
		2.0%
	CGSM erorr	0.4% 0.8% 1.2% 1.6%
		0.0%
		mean value	standard	coefficient of
			deviation	variation
		correlation length λ = 0.17L

-Lo shell roof under gravity loading (E random, CV(E) = 10%) -variability of displacement at point C error indicator: 10 trials error: 10000 trials

Table 3 .

 3 Scordelis

		error indicator: 10 trials	error: 10000 trials
		0.04%
	CGSM erorr	-0.08% -0.04% 0.00%
		-0.12%
		mean value	standard	coefficient of
			deviation	variation
		correlation length λ = 8.55L

-Lo shell roof under gravity loading (h random, CV(h) = 10%) -variability of displacement at point C

Table 4 .

 4 Car windscreen under pressure loadingmaterial properties, physical properties and variability levels

	Layers	Elasticity modulus (MPa)	Poisson's ratio	Thickness (mm)	CV(hi) (%)	CV(Ei) (%)
	Glass (1)	70000	0.215	2.1		2	2
	polymer PVB (2)	346	0.491	0.33		5	20
	acoustic polymer (3)	12	0.491	0.1		15	20
	polymer PVB (4)	346	0.491	0.33		5	20
	Glass (5)	70000	0.215	2.1		2	2
		Mesh 1			Mesh 2		Mesh 3
		192 elements		744 elements	13200 elements
	Displacement	Error	Displacement	Error	Reference
		(mm)	(%)		(mm)	(%)	(mm)
	Classical FE analysis CGSM (Eq. (18))	0.1474 0.1494	0.3 1.7		0.1467 0.1469	0.1 0.0	0.1469

Table 5 .

 5 Car windscreen under pressure loading -convergence study of nominal displacement at point A

Figure 9 .

 9 Car windscreen under pressure loading -finite element mesh with 744 S4R shell elements

Table 6 .

 6 Car windscreen under pressure loading (case 1: E random) -variability of displacement at point A

		Random parameters: hi
		CGSM	dMC
	m(U) (mm)	0.1470	0.1468
	σ(U) (mm)	2.285 10 -3	2.282 10 -3
	CV(U) (%)	1.55	1.55

Table 7 .

 7 Car windscreen under pressure loading (case 2: h random) -variability of displacement at point A

		Random parameters: Ei and hi
		CGSM	dMC
	m(U) (mm)	0.1471	0.1469
	σ(U) (mm)	3.051 10 -3	3.049 10 -3
	CV(U) (%)	2.07	2.08

Table 8 .

 8 Car windscreen under pressure loading (case 3: E and h random) -variability of displacement at point A

Table 9 .

 9 Car windscreen under pressure loading (case 4: E and h random, for glass CV(Ei) = CV(hi) = 10%) -variability of displacement at point A

Table 10 .

 10 Scordelis-Lo shell roof -comparison of computational time for one trial

	Mesh	CGSM formula	Finite element analysis	Ratio
		(seconds)	(seconds)
	2240 elements	18.6 10 -3	7.22	388
	5040 elements	5.24 10 -2	21.6	412
	8960 elements	0.124	53.5	431

Acknowledgements

The authors are grateful to the Directorate of Higher Education (DIKTI) of Ministry of Higher Education, Research and Technology of Indonesian Government and the Embassy of France in Indonesia for the financial support. The last author gratefully acknowledges research grant from Universitas Indonesia under Q1Q2 International Journal Publication Program No. NKB-0304/UN2.R3.1/HKP.05.00/2019.


MD Clotilde Gomond-Le Goff

MD Laura Vivalda

MD Silvia Foligno

MD Barbara Loi

MD Nadya Yousef

MD, PhD Daniele De Luca
email: dm.deluca@icloud.com

Effect of Different Probes and Expertise on the Interpretation Reliability of Point-of-Care Lung Ultrasound

Keywords: Lung ultrasound, agreement, diagnostic accuracy, probe, expertise, neonate Intraclass correlation coefficient, LU: Lung ultrasound, NICU: neonatal intensive care unit, RDS: respiratory distress syndrome, TTN: transient tachypnoea of the neonate, SAFE: Sonographic Assessment of liFe-threatening Events, ROC: Receiver operator characteristics

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

ABSTRACT -Background.The effect of different probes and operators' experience on reliability of lung ultrasound (LU) interpretation has not been investigated.We aim to study the effect of probes and operators' experience on the interpretation reliability of LU in critically ill neonates.Methods.Prospective, blind, cohort study enrolling patients with basic patterns ("B";"severe B";consolidation).Patients were scanned with micro-linear (15 MHz;L15), phased-array sectorial (6-12 MHz;S7) and micro-convex (8 MHz;C8) probes, in random order.Static images were acquired in high resolution, anonymized and included in a pictorial database in random sequences.Seventeen clinicians with different LU experience were asked to blindly assess the pictorial database. Inter-rater agreement and interpretation reliability were analyzed.Sub-analyses according to expertise and probe, and multivariate linear regression (including an "expertise*probe" interaction factor) were also performed.Results. The agreement tends to be lower and more heterogeneous for residents (ICC:0.82 (95%CI:0.74-0.9),p<0.001; I 2 :67%,p=0.04),and for fellows (ICC:0.93 (95%CI:0.9-0.97),p<0.001; I 2 :69%,p=0.04),especially when using non-linear probes,compared to senior physicians (ICC:0.95 (95%CI:0.93-0.96),p<0.001; I 2 :0%,p=0.433).AUC values were high for all probes (L15:0.96 (95%CI:0.93-0.99); C8:0.91 (95%CI:0.85-0.98); S7:0.86 (95%CI:0.82-0.91)) and physicians (senior:0.95 (95%CI:0.83-0.99);fellows:0.95 (95%CI:0.75-0.99);residents:0.86 (95%CI:0.5-0.99)).Worse reliability and a higher heterogeneity were found when the evaluation was performed by residents (AUC:0.9 (95%CI:0.85-0.94),p<0.01; I 2 :93.6%,p<0.001), than by fellows (AUC:0.99 (95%CI:0.9-0.999),p<0.001; I 2 :34.3%,p=0.09) and/or by senior physicians (AUC:0.99 (95%CI:0.9-0.999),p<0.001; I 2 :18%,p=0.236).The "expertise*probe" interaction factor was associated with lower ICC (St.β:-0.69;p<0.0001;adjusted R 2 :0.99) and AUC (St.β:-0.76;p<0.0001;adjusted R 2 :0.98).Conclusions.LU interpretation in neonates shows good inter-rater agreement and reliability,irrespective of the probe and raters' expertise.The use of non-linear probes by novice operators is associated with lowest agreement and reliability. BACKGROUND The use of lung ultrasound (LU) in neonatal critical care is rapidly expanding, both as descriptive and functional technique. In fact, LU contributes to the diagnosis of respiratory disorders [START_REF] Raimondi | Point-of-care lung ultrasound in neonatology: classification into descriptive and functional applications[END_REF] and guides procedures such as surfactant administration and chest tube placement. [START_REF] Raimondi | Point-of-care lung ultrasound in neonatology: classification into descriptive and functional applications[END_REF][START_REF] Raimondi | Lung Ultrasound for Diagnosing Pneumothorax in the Critically Ill Neonate[END_REF][START_REF] Raschetti | Echography-Guided Surfactant Therapy to Improve Timeliness of Surfactant Replacement: A Quality Improvement Project[END_REF][START_REF] Corsini | Lung Ultrasound for the Differential Diagnosis of Respiratory Distress in Neonates[END_REF] LU is based on a simple and clearly defined semiology that varies little with patients' age, making it an interesting tool in critical care medicine. [START_REF] Raimondi | Point-of-care lung ultrasound in neonatology: classification into descriptive and functional applications[END_REF][START_REF] Volpicelli | International evidence-based recommendations for point-of-care lung ultrasound[END_REF] However, neonates and infants, have less muscular, adipose and sub-cutaneous tissue and a smaller and still developing lung compared to adult patients. Thus, high frequency, linear probes are commonly used in these patients, instead of the low frequency, convex ones used for adult patients. [START_REF] Raimondi | Point-of-care lung ultrasound in neonatology: classification into descriptive and functional applications[END_REF][START_REF] Lichtenstein | BLUE-Protocol and FALLS-Protocol[END_REF] In any intensive care unit ultrasound machines are often used to scan multiple organs and, therefore, several probes with different shapes and frequencies may be available. The micro-linear, "hockey stick", high frequency probe, commonly used for vascular access placement, [START_REF] Lamperti | International evidence-based recommendations on ultrasound-guided vascular access[END_REF] provides an ultrasound beam only reaching relatively superficial tissues. Therefore, this probe seems particularly interesting for the neonatal population, since its size and frequency are well adapted to the small lung and thin chest wall of newborn infants.

The choice of probe is based on a number of factors, such as patients' size and tissue depth, but the operators' expertise may also influence the quality of a LU exam and might be considered in the probes' choice. To the best of our knowledge, the performance of different probes for LU in pediatric and neonatal critical care has not been formally and comprehensively investigated. Our aim is to study the effect of different probes and operators' experience on the interpretation reliability of LU in critically ill neonates with cardio-respiratory compromise. METHODS

Patients

This was a prospective, blind, cohort study enrolling neonates admitted to our neonatal intensive care unit (NICU) for cardio-respiratory compromise, during the first half of 2019.

According to our local protocol, these patients undergo a routine LU scan upon NICU admission, as a first-line imaging technique. [START_REF] Escourrou | Lung ultrasound decreased radiation exposure in preterm infants in a neonatal intensive care unit[END_REF] LU findings are combined with the clinical and biological data [START_REF] Raimondi | Clinical Data Are Essential to Validate Lung Ultrasound[END_REF] to refine the diagnosis of respiratory distress syndrome (RDS), [START_REF] Vergine | Lung Ultrasound Accuracy in Respiratory Distress Syndrome and Transient Tachypnea of the Newborn[END_REF] meconium aspiration syndrome, [START_REF] Piastra | Lung ultrasound findings in meconium aspiration syndrome[END_REF] transient tachypnoea of the neonate (TTN), [START_REF] Raimondi | A Multicenter Lung Ultrasound Study on Transient Tachypnea of the Neonate[END_REF] pneumonia, 1 neonatal acute respiratory distress syndrome, [START_REF] De Luca | The Montreux definition of neonatal ARDS: biological and clinical background behind the description of a new entity[END_REF] or to estimate lung aeration and follow-up patients' evolution aeration. [START_REF] Brat | Lung Ultrasonography Score to Evaluate Oxygenation and Surfactant Need in Neonates Treated With Continuous Positive Airway Pressure[END_REF][START_REF] Martino | Lung Ultrasound Score Predicts Surfactant Need in Extremely Preterm Neonates[END_REF] Multiple scans are performed according to the clinical context, e.g. in the case of an unfavorable, or unexpected clinical course, or to monitor the progression of chronic pulmonary insufficiency of prematurity (CPIP). [START_REF] Steinhorn | Chronic Pulmonary Insufficiency of Prematurity: Developing Optimal Endpoints for Drug Development[END_REF] Exclusion criteria were: 1) major congenital malformations or major chromosomal abnormalities; 2) need for thoracic surgery; 3) NICU stay less than 12h; 4) cardiac arrest or life-threatening conditions preventing the possibility to perform an ultrasound study with various probes in an unique examination (see below). The study was conducted in agreement with the modified Declaration of Helsinki. The study received approval from the local (SRLF) ethical board (n.16/58) and written informed consent was obtained from parents upon NICU admission.

Ultrasound study

Each enrolled patient underwent a LU scan indicated by the clinical context as per our clinical protocol. LU was performed by clinicians on duty, as all members of our team are trained for LU since 2014. [START_REF] Escourrou | Lung ultrasound decreased radiation exposure in preterm infants in a neonatal intensive care unit[END_REF] LU scans were performed following a standardized practice, [START_REF] Brat | Lung Ultrasonography Score to Evaluate Oxygenation and Surfactant Need in Neonates Treated With Continuous Positive Airway Pressure[END_REF] using a critical care ultrasound device (CX-50 ® , Philips Healthcare, Eindhoven, The Netherlands) and a micro-linear, "hockey stick", high frequency (15 MHz) probe (L15, Philips Healthcare, Eindhoven, The Netherlands). Scans were performed by automatically adjusting the gain using the dedicated software feature of our ultrasound machine; depth and focus were adjusted according to patients' size and the sign of interest. [START_REF] Gargani | How I do it: Lung ultrasound[END_REF] The following LU pathological patterns identified by the physician performing the scan at the bedside and independently confirmed by another senior colleague expert in LU (NY) served as "gold standard" for the study aim:

• B pattern (i.e.: an alveolar-interstitial pattern, defined as the presence of ≥3 wellspaced B lines);

• severe B pattern (i.e.: a severe alveolar-interstitial pattern, defined as the presence of crowded and coalescent B lines with or without subpleural consolidations with a size of ≤1 cm);

• consolidations (i.e.: subpleural echo-poor area or one with tissue-like echotexture with size >1 cm and irregular borders, which may also have mixed hypo-and hyper-echogenic zones representing bronchogram).

When any of these patterns was identified and confirmed, the patient was immediately recruited for the study. The corresponding static image was saved directly during the LU scan, and examination was immediately repeated by scanning the same chest area with two other probes (a phased-array sectorial (6-12 MHz) S7, and micro-convex (8 MHz) C8 probes, Philips Healthcare, Eindhoven, The Netherlands), in a random order by the same physician. All static images taken with the different probes were saved. No videoclip was saved. LU images and clinical data were then immediately stored in a dedicated and secured archive where the images were saved in high resolution (600 dpi).

Images evaluation

After the recruitment phase, LU pictures were anonymized and grouped in a pictorial database by mixing all the images taken from different patients and probes in a random order. Only the principal investigator (CGLG) was aware of the order and able to identify the images and patients. After two weeks, the pictorial database was submitted to the evaluation of physicians with different levels of LU expertise. There was no selection in the choice of physicians, that is, the whole NICU team participated. In detail, images were evaluated by 6 senior physicians (i.e. attending neonatologists with at least 5 years of LU experience), 5 fellows (i.e. junior pediatricians with at least 1 year of LU experience), and 6 residents (i.e. young trainees during their first years of pediatric residency who trained in LU for approximately 6 months under fellows' or senior physicians' supervision [START_REF] Martino | Lung Ultrasound Score Predicts Surfactant Need in Extremely Preterm Neonates[END_REF] ). All assessors received the pictorial database electronically with an individual e-mail where they were asked to classify images as B, severe B or consolidation pattern without access to any other data. The random order of pictures in the database was different for each evaluator and physicians were instructed not to discuss the images with colleagues.

Evaluations were anonymous and only the level of physicians' expertise (as senior, fellow, resident) was recorded.

Data collection and statistics

Clinical data and interpretations of LU images were collected by the principal investigator (CGLG) and descriptive statistics was used. Two types of analyses were performed: the inter-rater agreement and the interpretation reliability. Intraclass correlation coefficient (ICC) and its 95% confidence interval (95%CI) were used to evaluate the interrater absolute agreement (or conformity of interpretations). [START_REF] Mcgraw | Forming inferences about some intraclass correlation coefficients[END_REF] Receiver operator characteristics (ROC) analysis was used to evaluate the reliability of LU images interpretation; ROC analysis results were expressed as areas under the curve (AUC) with 95%CI and AUC were compared using Hanley's method. [START_REF] Hanley | The meaning and use of the area under a receiver operating characteristic (ROC) curve[END_REF] A sub-analysis according to both expertise and type of probe was performed and, for this task, ICC and AUC were reported in Forrest-like plots in order to improve their comparison and visual interpretation. [START_REF] Zhou | Statistical methods in diagnostic medicine[END_REF] We reported results with both the fixed and the DerSimonian-Laird (inverse variance method) random-effects models. Consistency was evaluated using the I 2 -values (that is, the variation in ICC or AUC attributable to heterogeneity), performing a χ 2 test for heterogeneity and considering I 2 >50% as indicative for heterogeneity. [START_REF] Tridente | Porcine vs bovine surfactant therapy for preterm neonates with RDS: systematic review with biological plausibility and pragmatic meta-analysis of respiratory outcomes[END_REF] Finally, a multivariate linear regression with enter technique was performed to study the effect of operators' expertise (coded as residents: 1; fellows: 2; senior: 3), probes (coded as C8: 1; S7: 2; L15: 3) and their "expertise*probe" interaction factor on ICC (considered as the ICC obtained between operators of same degree of experience for each probe) and AUC (considered as the AUC obtained by each physician with each probe), as dependent variables. Adjusted-R 2 values were used to evaluate goodness-of-fit.

A formal sample size calculation was unfeasible since no studies investigated the effect of both operators' expertise and probe on the interpretation of LU semiology in neonates. However, based on the data accumulated in our previous studies where LU was performed by different operators, [START_REF] Brat | Lung Ultrasonography Score to Evaluate Oxygenation and Surfactant Need in Neonates Treated With Continuous Positive Airway Pressure[END_REF][START_REF] Martino | Lung Ultrasound Score Predicts Surfactant Need in Extremely Preterm Neonates[END_REF] we hypothesized an ICC of at least 0.7 on 3 different observations (3 probes or degrees of expertise). We used a table specifically designed for sample size calculation of ICC [START_REF] Bonett | Sample size requirements for estimating intraclass correlations with desired precision[END_REF] with α-error=0.05, width of 0.2, a power of 80% and this gave a sample size of 68 observations to be evaluated.

Analyses were performed with SPSS 15.0 (SPSS Inc, Chicago, IL) and MedCalc 13.3 (MedCalc bvba, Ostend, Belgium) and p-values <0.05 were considered statistically significant.

RESULTS

Tab.1 shows basic population details. Thirty-seven patients were enrolled. The pictorial database consisted of 27, 19 and 28 B, severe B and consolidation patterns (i.e.: a total of 74 ultrasound observations), respectively. Thus, a total of 222 (i.e.: 74*3) pictures of these patterns were acquired using the three probes. Fig. 1 illustrates the study flow chart and image acquisition. All assessors evaluated the database in less than 10 minutes. LU findings were used to finally give an integrated diagnosis of RDS (n=8), TTN (n=11), CPIP (n=10), while eight patients were diagnosed with extrapulmonary disorders. All patients received either invasive or noninvasive ventilation and the type of respiratory support did not change during LU. The conformity of interpretations was very high in the whole dataset (ICC=0.96; 95%CI: 0.94-0.97; p<0.001), but it was also high regardless of the expertise or the type of probe, as shown in Tab.2.

Further sub-analysis according to both expertise and probes is shown in Fig. 2. The agreement tends to be lower and more heterogeneous for residents (ICC: 0.82 (95%CI: 0.74-0.9), p<0.001; I 2 : 67%, p=0.04), and, to a lesser extent, for fellows (ICC: 0.93 (95%CI: 0.9-0.97), p<0.001; I 2 : 69%, p=0.04), especially when using non-linear probes, compared to senior physicians (ICC: 0.95 (95%CI: 0.93-0.96), p<0.001; I 2 : 0%, p=0.433).

ROC analysis for the different probes showed high AUC values (L15: 0.96 (95%CI: 0.93-0.99); C8: 0.91 (95%CI: 0.85-0.98); S7: 0.86 (95%CI: 0.82-0.91)) and the AUC comparisons did not reveal any significant difference between the probes (L15 vs C8: p= 0.378; L15 vs S7: p=0.099; S7 vs C8: p=0.483).

ROC analysis for the different levels of expertise showed high AUC values (senior physicians: 0.95 (95%CI: 0.83-0.99); fellows: 0.95 (95%CI: 0.75-0.99); residents: 0.86 (95%CI: 0.5-0.99)). However, the AUC differed between residents and senior physicians (p<0.0001), between residents and fellows (p<0.0001) and between senior physicians and fellows (p=0.02).

A further sub-analysis according to both expertise and probes (Fig. 3) confirmed a slight worse reliability and a higher heterogeneity for the evaluation performed by residents (AUC: 0.9 (95%CI: 0.85-0.94), p<0.01; I 2 : 93.6%, p<0.001), than by fellows (AUC: 0.99 (95%CI: 0.9-0.999), p<0.001; I 2 : 34.3%, p=0.09) and/or by senior physicians (AUC: 0.99 (95%CI: 0.9-0.999), p<0.001; I 2 : 18%, p=0.236). This seems notably different when using non-linear probes (green and red diamonds).

These results are confirmed by multivariate analysis showing that the "expertise*probe" interaction factor is significantly associated with lower ICC (standardized β: -0.69; p<0.0001; adjusted R 2 : 0.99) and AUC (standardized β: -0.76; p<0.0001; adjusted R 2 : 0.98). DISCUSSION Our findings show that the evaluation of basic LU parenchymal signs has good inter-rater agreement and interpretation reliability, irrespective of raters' expertise and the type of probe in critically ill neonates. However, there is a tendency to lower agreement and reliability with decreasing clinicians' experience. Moreover, when non-linear probes are used by less experienced operators, the agreement and reliability are lowest, and the heterogeneity of interpretations increases.

These findings are novel and useful, since they suggest that any probe can be used for the basic evaluation of LU semiology in critically ill infants. This may be especially valuable when LU is used in the emergency setting, such as during management of lifethreatening events. A dedicated ultrasound protocol (the so-called Sonographic Assessment of liFe-threatening Events (SAFE) protocol 1 ) has been recently proposed for this purpose and this algorithm suggests a one-probe approach for rapidity and simplicity, leaving the choice of the probe to the operator. [START_REF] Raimondi | Point-of-care lung ultrasound in neonatology: classification into descriptive and functional applications[END_REF] The results of our study reinforce this suggestion.

Our findings are consistent with those of the few other neonatal LU studies published on this matter. In the study by Vergine et al, evaluation of LU images by an external reader blinded to any other data was accurate for the diagnosis of RDS and TTN. [START_REF] Vergine | Lung Ultrasound Accuracy in Respiratory Distress Syndrome and Transient Tachypnea of the Newborn[END_REF] More interestingly, Brusa et al, reported good inter-rater agreement for the diagnoses of RDS and TTN given by physicians of different levels of experience based on basic LU findings. [START_REF] Brusa | Neonatal Lung Sonography: Interobserver Agreement Between Physician Interpreters With Varying Levels of Experience[END_REF] Our findings are also consistent with those of Corsini et l who evaluated the time needed to give an ultrasound diagnosis by operators of different experience. [START_REF] Corsini | Lung Ultrasound for the Differential Diagnosis of Respiratory Distress in Neonates[END_REF] However, to the best of our knowledge, there are no studies comparing the use of different ultrasound probes for LU exams in pediatric populations. Our study adds new information which may have practical implications. For novices, a non-linear probe gives the lowest level of agreement and reliability, and should be avoided, life-threatening situations excepted, until operators achieve a sufficient experience. However, the nonlinear probes do not change ICC or AUC when they are used by more experienced physicians, such as fellows or senior physicians. Moreover, although a possible "combination effect" of lack of experience with the use of a non-linear probe is logically plausible and methodologically significant, the reduction in ICC and AUC is small. Therefore, this could not have clinical significance for the majority of patients in the daily practice.

There are few studies investigating the effect of the different probes and ultrasonographers' experience in adult critical care and emergency medicine. Interestingly, these papers show specular and consistent results, as convex probes resulted more accurate in adult patients [START_REF] Sperandeo | Assessment of ultrasound acoustic artifacts in patients with acute dyspnea: a multicenter study[END_REF][START_REF] Pivetta | Sources of Variability in the Detection of B-Lines, Using Lung Ultrasound[END_REF] and the operators' expertise also influenced LU accuracy. [START_REF] Sperandeo | Assessment of ultrasound acoustic artifacts in patients with acute dyspnea: a multicenter study[END_REF][START_REF] Pivetta | Sources of Variability in the Detection of B-Lines, Using Lung Ultrasound[END_REF] Thus, combining our results and those from these studies may suggest that linear probes should not be used by novices on adult patients, as this would represent the same combination (less suitable probe and lower expertise) that we found in our setting.

Conversely, we did not evaluate the inter-rater agreement and reliability sub-divided for different LU signs: this deserves a specific study as it seems that interpretation reliability for some LU findings is lower. [START_REF] Gravel | Interrater reliability of pediatric point-of-care lung ultrasound findings[END_REF] We acknowledge some study limitations. The evaluation of the LU was done with static LU images and we did not take any videoclip, albeit LU is dynamic exam. Although, in theory, this may have influenced our results, it made the study easier to conduct, as we could not electronically store videoclips for technical reasons. We focused on basic LU patterns, which are easy to identify on static images. Therefore, the assessment of more complex or refined ultrasound signs, such as those of pneumothorax, was not realized, as this is only possible with a dynamic study. Also, the study focused on pathological parenchymal patterns and did not evaluate the normal A-pattern, since this would have implied the distinction from pneumothorax. Moreover, we were more interested to verify the effect of varying probe/operators' expertise in a critical population of neonates with cardio-respiratory compromise, since in these patients an incorrect evaluation may theoretically lead to more severe consequences. The study was conducted in a setting with high ultrasound expertise, given our routine use of LU as the first-line chest imaging technique. Therefore, the levels of expertise represented in this study may be different from units where LU is less frequently used. Moreover, we cannot add any information on the amount of training needed for an operator to use a non-linear probe without reducing reliability: this would require a specific study on the training curve. LU scans were not compared to other imaging techniques such as computerized tomography or chest X-rays since they are rarely performed in NICU care or are less accurate than LU, 1 respectively. Systematic differences among observers are also likely to exist: in this context, ICC is a composite measure of intra-rater and inter-rater variability. [START_REF] Shrout | Intraclass correlations: uses in assessing rater reliability[END_REF] Other statistical tools could have been used but, in light of the current statistical literature, ICC seems the best analysis to be performed. [START_REF] Maclure | Misinterpretation and misuse of the kappa statistic A[END_REF] Moreover, our results are consistent with studies using other statistical tools to evaluate agreement. [START_REF] Brusa | Neonatal Lung Sonography: Interobserver Agreement Between Physician Interpreters With Varying Levels of Experience[END_REF][START_REF] Sperandeo | Assessment of ultrasound acoustic artifacts in patients with acute dyspnea: a multicenter study[END_REF][START_REF] Pivetta | Sources of Variability in the Detection of B-Lines, Using Lung Ultrasound[END_REF] A complete blinding to the assessors was unfeasible since LU images have different (squared or triangular) footprints, due to the use of different probes. As micro-linear probes are routinely used in our NICU, the expertise with this probe might have allowed a better image interpretation. It might also be possible that providers have subconsciously acquired higher quality images with this probe. Finally, to better assess diagnostic accuracy, each patient would need to be scanned not only with different probes, but also by clinicians with different expertise. This would have allowed to evaluate image acquisition, but would have represented a more onerous study design, thus we decided to mainly focus on reliability of image interpretation. These biases are unavoidable, but also fairly minor and should not dramatically undercut the validity of our findings. CONCLUSIONS The evaluation of basic LU semiology in critically ill neonates shows good inter-rater agreement and interpretation reliability, irrespective of the probe used and the level of raters' expertise. However, the use of non-linear probes by novice operators should be discouraged, as it is associated with the lowest agreement and reliability.

Figure 1 .

 1 Abbreviations: L15: micro-linear, "hockey stick", high frequency (15 MHz) probe; S7:

Table 1 . Basic characteristics of the studied population.

 1 Data are expressed as mean (standard deviation) or median [25 th -75 th percentile] or number (%), as appropriate. Apgar, SNAPPE-II and CRIB-II scores are dimensionless numbers. Abbreviations: SNAPPE-II: Score for Neonatal Acute Physiology Perinatal Extension-II; CRIB: Critical Risk Index for Babies-II; LU: Lung Ultrasound.

	Patients	37
	Gestational age (weeks) 32.6 (5.2)
	Birth weight (grams)	2081 (1183)
	Male sex	23 (62.2%)
	Caesarean section	14 (37.8%)
	5' Apgar score	9 [8-10]
	SNAPPE-II score	5 [5-13]
	CRIB-II score	8.5 [5.8-14.7]
	Postnatal age at LU (h)	1 [0-13.2]

2. Inter-rater agreement sub-analysis per different levels of expertise and probes.

  Upper, middle and lower panels depict results for residents, fellows and senior physicians, respectively. Squares are sized proportionally to the weight of that particular operator-probe pair; squares and horizontal lines represent the intra-class correlation coefficient and its 95% confidence interval, respectively. The location of the diamond and its width represent the pooled estimated effect size and its 95% confidence interval, respectively. Effect size is illustrated both under the fixed and random effect models; as heterogeneity is statistically significant for residents and fellows, only random effect model results should be considered. Abbreviations: ICC: intra-class correlation coefficient; L15: micro-linear, "hockey stick", high frequency (15 MHz) probe; S7: phased-array sectorial (12 MHz) probe; C8: micro-convex (8 MHz) probe. Figure 3.

Receiver Operator Characteristics sub-analysis per different levels of expertise and probes

  . Upper, middle and lower panels depict results for residents, fellows and senior physicians, respectively. Red, blue and green indicate results for C8, L15 and S7 probes, respectively. Colored diamonds are sized proportionally to the weight of that particular operator-probe pair. The location of the white diamond represents the pooled estimated effect size. Horizontal lines represent 95% confidence intervals. Pooled results are illustrated both under the fixed and random effect models; as heterogeneity is statistically significant for residents, only random effect model results should be considered. Abbreviations: ROC: Receiver Operator Characteristics; L15: micro-linear, "hockey stick", high frequency (15 MHz) probe; S7: phased-array sectorial(12 MHz) probe; C8: micro-convex (8 MHz) probe.

ACKNOWLEDGEMENTS

Authors are grateful to all their colleagues who participated in the evaluation of lung ultrasound pictorial database. CGLG collected and interpreted data and wrote the first manuscript draft; LV, SF and BL collected and interpreted the data; NY designed the study, contributed to data collection, interpretation and analysis; DDL conceived and designed the study, performed the analysis and interpreted the data. All coauthors critically review the manuscript for important intellectual content, approved the final manuscript and agreed to be accountable for it. The corresponding author (DDL) assumes full responsibility for the integrity of the submission as a whole, from inception to published article, including data and analysis.

Authors have no conflict of interest to declare. The study has no sponsor.