Q Yin
email: qi.yin@utc.fr

F Druesne

P Lardeur
email: pascal.lardeur@utc.fr

Probabilistic free vibration analysis of composite structures using the Modal Stability Procedure

As shown by experimental results, the manufacturing of composite structures induces a quite important variability of the mechanical behavior. In this paper, the Modal Stability Procedure (MSP) based on one nominal finite element analysis and a fast Monte Carlo Simulation (MCS) is developed for probabilistic free vibration analysis of laminated composite structures modeled by finite elements. Material properties (elastic properties, densities…) and physical properties (thicknesses and fiber orientations) are considered as uncertain parameters and are represented by random variables. The variability of natural frequencies is evaluated with the MSP. Two examples: an eight-layer composite square plate and a stiffened ten-layer composite rectangular plate, are studied. The results are compared with those obtained by the direct MCS, considered as a reference, and those presented in the literature. The comparison shows that the MSP provides quite accurate results with high computational efficiency. Moreover, an error indicator which is able to approximately evaluate the error level due to the modal stability assumption is proposed.

Introduction

Nowadays composite structures are largely used in various industrial sectors. However, the uncertainties in material and physical properties of composite laminated structures lead to a noticeable variability of responses. Figure 1 shows the frequency response functions (FRFs), observed experimentally with a laser scanning vibrometer, of 8 theoretically identical composite plates. The results highlight a high variability level of FRFs, up to 40dB. Uncertainties in composite structures result in significant variability of dynamic response. Therefore, it is necessary to model the input variability to study its effects on dynamic analysis of composite structures in order to understand and control the phenomenon of output variability. Probabilistic analysis, which allows to predict the variability of response, has been applied in dynamic analysis of composite structures for several years. The Monte Carlo Simulation (MCS) is a robust method and often used as a reference. With the direct MCS, a large number of random trials are performed, generally leading to several thousands of finite element analyses. This method is used to study the variability of dynamic response for a composite truss by Allegri et al. [START_REF] Allegri | Stochastic analysis of the vibrations of an uncertain composite truss for space applications[END_REF] and for thin-walled composite beams by Piovan et al. [START_REF] Piovan | Dynamics of thin-walled composite beams: Analysis of parametric uncertainties[END_REF]. However, this method is quite expensive in computational cost, several fast methods have thus been developed. The perturbation method is a largely used fast probabilistic approach. Using this method, Singh et al. investigate the influence of uncertain material properties on the variability of natural frequencies for composite plates [START_REF] Singh | Natural frequencies of composite plates with random material properties using higher-order shear deformation theory[END_REF], spherical panels [START_REF] Singh | Free vibration of laminated spherical panels with random material properties[END_REF] and cylindrical panels [START_REF] Singh | Free vibration of composite cylindrical panels with random material properties[END_REF]. Tripathi et al. [START_REF] Tripathi | Free vibration of laminated composite conical shells with random material properties[END_REF] develop the perturbation method for conical shells. The variability of natural frequencies and FRFs for composite plates [START_REF] Dey | Thermal uncertainty quantification in frequency responses of laminated composite plates[END_REF][START_REF] Dey | Stochastic free vibration analysis of angle-ply composite plates -A RS-HDMR approach[END_REF][START_REF] Dey | Bottom up surrogate based approach for stochastic frequency response analysis of laminated composite plates[END_REF] and shallow doubly curved shells [START_REF] Dey | Stochastic free vibration analyses of composite shallow doubly curved shells -A Kriging model approach[END_REF] is studied by Dey et al. using methods based on design of experiments. Chakraborty et al. [START_REF] Chakraborty | Stochastic free vibration analysis of laminated composite plates using polynomial correlated function expansion[END_REF] apply polynomial correlated function expansion to study the variability of natural frequencies for composite plates with uncertain material properties, thicknesses and fiber orientations.

However, for laminated composite structures research is still in progress to develop methods which are compatible with any standard finite element software and are able to fast evaluate the variability of response with large numbers of random variables, high levels of input variability and large size models. In statics, 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] have proposed the Certain Generalized Stresses Method (CGSM). In this study, the Modal Stability Procedure (MSP), based on a mechanical assumption of modal stability, is therefore developed for probabilistic free vibration analysis of laminated composite structures. This method has been applied to study the variability of vibration response for homogeneous [START_REF] Arnoult | The modal stability procedure for dynamic and linear finite element analysis with variability[END_REF][START_REF] Druesne | Fast methods based on modal stability procedure to evaluate natural frequency variability for industrial shell-type structures[END_REF] and sandwich [START_REF] Druesne | Variability of dynamic responses of frequency dependent visco-elastic sandwich beams with material and physical properties modeled by spatial random fields[END_REF] structures. In this paper, the variability of natural frequencies for laminated composite structures is evaluated with the MSP, taking into account uncertain material and physical properties. Two examples are treated: an eight-layer composite square plate and a stiffened ten-layer composite rectangular plate. The accuracy of results is verified by comparing with those obtained by the direct MCS. The computational cost and an error indicator are also investigated.

Modal Stability Procedure

Presentation of the MSP

The MSP, a numerical method which allows to study the variability of natural frequencies and FRFs, has been applied in different contexts [START_REF] Arnoult | The modal stability procedure for dynamic and linear finite element analysis with variability[END_REF][START_REF] Druesne | Fast methods based on modal stability procedure to evaluate natural frequency variability for industrial shell-type structures[END_REF][START_REF] Druesne | Variability of dynamic responses of frequency dependent visco-elastic sandwich beams with material and physical properties modeled by spatial random fields[END_REF]. The studies presented in the literature show that the MSP is able to provide very satisfactory results. In this paper it is developed to study the variability of the natural frequencies of laminated composite structures. This method is based on a mechanical assumption: the mode shapes of the structure are considered independent of input parameters uncertainty.

For modal probabilistic analysis, the MSP requires only one single finite element analysis in the nominal configuration which is compatible with any standard finite element software, Abaqus [START_REF]Abaqus Analysis User's Guide version 6.14[END_REF] is used in this study. The mode shapes and modal strains, obtained by the nominal finite element analysis, are used in metamodels allowing the calculation of natural frequencies or FRFs. Once the nominal information has been obtained, a vector of random variables is generated to represent input uncertain parameters. Then a fast MCS is performed using the MSP formulation, which allows to evaluate natural frequencies or FRFs. The variability (mean value, standard deviation, coefficient of variation and distribution) of natural frequencies or FRFs is subsequently obtained. This simulation is of course much less expensive than the direct MCS. The flowchart of the MSP is given in figure 2.

MSP formulation for natural frequency variability

In this paper the MSP is developed to study the variability of natural frequencies of laminated composite plates using a multilayered composite shell theory with transverse shear effects. The structures may be modelled with triangular or quadrilateral shell finite elements. Thanks to the modal stability assumption, a formulation of natural frequency can be written. First of all, the eigenvalue problem in dynamic analysis is given by:

 i i i KM    (1)
where K and M are respectively the stiffness and mass matrix, i  and i  are respectively the eigenvector and the eigenvalue for the ith mode. Then the expression of the Rayleigh quotient is used in modal analysis and the angular frequency ω can be written as:

2  K M    (2)
In our study the subscript "0" is used to define the nominal case and the subscript "p" is used to define the perturbed case. The perturbed eigenvector p  can be expressed as:

0    pp    (3)
According to the MSP assumption,  p  is considered 0. Taking into account Equations ( 2) and (3), the perturbed angular frequency p  is written as:

        00 00 2 00 00          T T p p p p p T T p p p p K K M M            (4)
where the input variability is introduced into the stiffness matrix p K and the mass matrix p M . Considering that the structure is modeled by finite elements, elementary matrices are used and Equation (4) can therefore be transformed into:

00 1 2 00 1      e e n T , j p , j , j j p n T , j p , j , j j k m    (5)
where p , j k and p , j m are respectively the perturbed stiffness and mass matrix for the jth element. 0,j  is the nominal elementary eigenvector for the jth element. In Equation ( 5), the denominator is proportional to the kinetic energy and the numerator expresses the internal strain energy. Our objective is to prevent the calculation of m may not be available with a standard finite element software. So the numerator is developed using the expression of internal energy in each element:

1 2 j elem int lm, j lm, j V dV      (6)
  0 0 0 0 0 0 0 0 0 0 1 2       p , j elem T T T T T int , j p , j , j , j p , j , j , j p , j , j , j p , j , j , j CS ,p , j , j S e A e e B B e D A dS        (7)
where

p , j
S is the area of the jth element, 0,j e , 0,j  and 0,j  are respectively membrane strains, curvatures and transverse shear strains for the jth element. They are considered to be nominal due to the MSP assumption and obtained by one finite element analysis. A are respectively the membrane, membrane-bending coupling, bending stiffness matrix and corrected transverse shear stiffness matrix for the jth element. These matrices contain the input variability.

For the denominator of Equation ( 5), p , j m is considered as a concentrated mass matrix and is written as: According to Equations ( 7) and (8), Equation ( 5) can be transformed into:

1    l n p, j p, j ,k p, j ,k p, j ,k d k m S h I / n  (8)
  0 0 0 0 0 0 0 0 0 0 1 2 00 11           e p ,
I S h / n           (9)
In Equation [START_REF] Dey | Bottom up surrogate based approach for stochastic frequency response analysis of laminated composite plates[END_REF] it is observed that an integration must be evaluated. A Gauss numerical integration is used.

In our study, the strains are assumed constant and are calculated at the center of each element, which leads to a single Gauss point. Equation ( 9) then becomes:

  0 0 0 0 0 0 0 0 0 0 1 2
I A h / n                    ( 10
)
where w is the weighting coefficient and detJ is the determinant of the Jacobian matrix. Equation ( 10) is assumed to be valid for any random value of uncertain parameters. It is used by a fast MCS to evaluate perturbed natural frequencies for each trial with the expression 2  f  . The statistical quantities such as mean value, standard deviation, coefficient of variation and distribution are then calculated.

3 Eight-layer composite square plate

Presentation of the example

The first example, an eight-layer E-glass/epoxy composite square plate (300mm  300mm), has been studied by Chakraborty et al. [START_REF] Chakraborty | Stochastic free vibration analysis of laminated composite plates using polynomial correlated function expansion[END_REF] with polynomial correlated function expansion. The stacking sequence is asymmetric [45/-45]4. This plate is relatively thin with a ratio between the thickness and the length 0 012  h / l .

. All the edges of the plate are considered to be clamped. Based on the convergence study, a 16 [START_REF]Abaqus Analysis User's Guide version 6.14[END_REF]  mesh with 256 S4R elements (see figure 3 which shows the first mode shape) is used for the probabilistic analysis. The material properties (density , elastic moduli

Variability of natural frequencies

In this example, the variability of the first four natural frequencies is evaluated by a fast MCS using MSP formulation and a direct MCS. The latter is used as the reference. Figure 4 gives the mean value   f  and the standard deviation   f  obtained by both methods with 10000 trials, as well as those presented by Chakraborty et al. [START_REF] Chakraborty | Stochastic free vibration analysis of laminated composite plates using polynomial correlated function expansion[END_REF]. All the uncertain parameters are taken into account. It can be observed that the results obtained with the three approaches are very close for all the four natural frequencies. The error between MSP and direct MCS is lower than 0.7% for the mean value and 1.1% for the standard deviation.

So the MSP provides quite satisfactory results. The distribution of natural frequencies is also investigated. The probability density functions of the first four natural frequencies is shown in figure 5. It can be observed that the curves of MSP and direct MCS are very close, with a small shift. The difference between the results obtained by MSP and Chakraborty et al. [START_REF] Chakraborty | Stochastic free vibration analysis of laminated composite plates using polynomial correlated function expansion[END_REF] is more considerable for modes 2 to 4. The authors think that this is due to the fact that Chakraborty et al. [START_REF] Chakraborty | Stochastic free vibration analysis of laminated composite plates using polynomial correlated function expansion[END_REF] used a coarse mesh. Results shown in figures 4 and 5 prove the computational accuracy of the MSP. Then each type of uncertain parameters (material properties, thicknesses and fiber orientations) is taken into account individually in order to highlight their influence on the variability of natural frequencies. Figure 6 gives the coefficient of variation of natural frequencies   c.o.v. f evaluated by the MSP for each type of uncertain parameters as well as all of them. It can be observed that the output variability depends very little on the mode number. Taking into account all uncertain parameters leads to the highest variability. Uncertain material properties bring a higher variability than uncertain thickness. However, the output variability is nearly 0 when fiber orientations are uncertain. The results show that the variability of natural frequencies is more sensitive to uncertain material properties and thicknesses.

Error indicator

In our study, the results calculated by the direct MCS are used as the reference to validate the proposed method. But its cost is very high, especially for large size structures because a great number of finite element analyses are performed in this method. The error indicator is therefore a quite interesting challenge, which allows to quickly predict the error level. The objective of the indicator is to estimate the error level of the results obtained by the proposed method MSP compared with the reference, without performing a direct MCS with a large number of trials. In this example, the variability of natural frequencies is evaluated by MSP and direct MCS with 10 or 10000 trials. The errors between these two methods are calculated for 10 trials using the same input data and compared with exact errors, which are those obtained with 10000 trials. The results are given in figure 7. For the mean value, the errors for 10 and 10000 trials are almost the same, which means 10 trials can predict the error level of mean value. For the standard deviation, the difference between the errors obtained with 10 and 10000 trials is higher but remains low. Therefore 10 trials are sufficient to give a good estimation of the error level of standard deviation. According to the above results, an error indicator using 10 trials is able to approximately assess the error level.

4 Stiffened ten-layer composite rectangular plate

Presentation of the example

The second example concerns a stiffened rectangular plate (figure 8) which finds its interest in many sectors in particular aeronautic and naval industry. A primary plate (210mm 460mm  2mm) is made of 10 layers of taffeta glass tissue and polyester. The tissue is laid in only one direction, which leads to a stacking sequence [0]10. A stiffener in form of inverted "T" is positioned on this plate, with base 50mm, height 25mm and thickness 4mm. It consists of two "L" parts which are made of 10 layers of the same material as the primary plate. The geometric shape of this stiffened plate is shown in figure 8. The primary plate and the stiffener are manufactured by the infusion process and assembled by adhesive bonding.  GG4.1GPa and   1.79g/cm 3 . Three groups of uncertain parameters are introduced and lead to four cases: in case 1, the material properties are considered as uncertain with input coefficient of variation   uncertain parameters are taken into account. These levels of input variability are high. For one given layer, the nine parameters are assumed to be independent. The parameters of different layers are either dependent, which means 9 random variables in total, or independent, which means up to 270 random variables considering that there are three parts and each of them has ten layers. The variability of the first ten frequencies is studied by MSP with 10000 trials. A mesh with 5704 elements is used here following a convergence study. Figure 9 shows the first two modes shapes.

Variability of natural frequencies

In this study, the MSP with 10000 trials is used to evaluate the variability of the first ten natural frequencies. Moreover, the MSP and direct MCS using 10 trials are also applied as an error indicator. Figure 10 gives the coefficient of variation of natural frequencies in case 1 with dependent or independent uncertain material properties, as well as the errors provided by the error indicator. In can be observed that the errors are very low, which means that the MSP provides accurate results. The variability level for each mode is moderate. The output variability is comprised between 9% and 11% with dependent uncertain parameters, which is lower than the input variability. With independent uncertain parameters, the output variability is much smaller, with a value comprised between 2.5% and 3.5%. The output variability levels are lower than the input variability levels, due to a compensation phenomenon which is more pronounced when the parameters are independent.

Dependent (7 variables) Independent (210 variables)

Figure 10: Variability of natural frequencies with uncertain material properties (case 1)

The results of case 2 are shown in figure 11. It can be observed that the results obtained by MSP and direct MCS are quite close. When the uncertain thicknesses are dependent, the variability of natural frequencies is nearly constant with a value comprised between 14% and 15%, which is close to the input variability. This output variability is higher than in case 1. With independent uncertain thicknesses, the variability of each frequency is comprised between 3% and 4%.

Dependent (1 variable) Independent (30 variables)

Figure 11: Variability of natural frequencies with uncertain thicknesses (case 2)

The results of case 3 are given in figure 12. In this case it can be observed that the output variability levels are very small and the errors between MSP and direct MCS are relatively large. The output variability levels are less than 1% with dependent parameters and less than 0.3% with independent parameters. In this context, the influence of fiber orientation uncertainty can be neglected considering the low output variability, compared with uncertain material properties and thicknesses.

Dependent (1 variable) Independent (30 variables)

Figure 12: Variability of natural frequencies with uncertain fiber orientations (case 3)

At last the variability of natural frequencies is studied with all the uncertain parameters. The results are given in figure 13. It can be observed that the errors between the MSP and direct MCS are very small. The output variability of the first ten modes is comprised between 4% and 6% with independent uncertain parameters. While all uncertain parameters are dependent, the input variability leads to a quite large output variability, reaching 18%.

Dependent (9 variables) Independent (270 variables)

Figure 13: Variability of natural frequencies with all types of uncertain parameters (case 4)

Previous work proves that the MSP is able to provide accurate results, but the computational time is an important issue as well. This method consists of two parts: a nominal finite element analysis, which allows to obtain nominal mode shapes and modal strains, and a post-processing where 10000 MCS trials are performed with Matlab. The cost of the MSP is compared with the direct MCS, which leads to 10000 finite element analyses in our study. In order to better compare these two methods, the cost is evaluated by counting floating operations (flops). While the number of layers has a weak effect on the cost, an example of an eight-layer composite plate is studied.

The numbers of flops of one MSP trial and one direct MCS trial are compared. For one MSP trial, the number of flops is obtained by counting flops of each algebraic operation [START_REF] Druesne | Fast methods based on modal stability procedure to evaluate natural frequency variability for industrial shell-type structures[END_REF]. For one direct MCS trial, namely one finite element free vibration analysis, the number of flops is in the order of Figure 14 shows the number of flops of one MSP trial and one direct MCS trial for evaluating the variability of one mode. It is observed that the cost of direct MCS is always over 100 times bigger than the cost of MSP. Moreover, the number of flops of the direct MCS increases faster than the MSP with the number of elements, which means that the efficiency of the MSP is more competitive for a large size model.

Conclusion

In this paper, the Modal Stability Procedure (MSP) has been developed to study the variability of natural frequencies of laminated composite structures. Material properties (densities, elastic and shear moduli…) and physical properties (thicknesses, fiber orientations) are considered as uncertain parameters. The method is based on the mechanical assumption that the mode shapes are weakly sensitive to uncertain parameters. Thanks to this assumption, modal element strains and mode shapes are extracted following one finite element calculation in the nominal configuration. Then a fast MCS is carried out using a metamodel allowing the calculation of natural frequencies.

A first example concerning a laminated square plate and a second example concerning a stiffened laminated rectangular plate are studied in this paper. The variability of natural frequencies is evaluated. The results calculated by the MSP are compared with those obtained by the direct MCS, considered as a reference, and those presented in the literature. The comparison shows that the MSP provides quite accurate results. In these examples, the variability of material properties and thicknesses has a more important effect on the output variability, while uncertain fiber orientations lead to a very small output variability. An error indicator using 10 trials has been proposed, which is able to predict the error level between MSP and direct MCS without performing a large number of finite element analyses. Compared with the direct MCS, the MSP is quite competitive in computational cost, in particular for large size models.

Figure 1 :

 1 Figure 1: FRFs variability observed experimentally of 8 theoretically identical composite plates

Figure 2 :

 2 Figure 2: Flowchart of the MSP



  the strains for the jth element. For a laminated composite structure, elem int  can also be expressed as:

  density and thickness for kth layer of jth element, d n is the number of nodes in the element, I is an identity matrix whose size is 6 d n × 6 d n because there are six degrees of freedom per node. In matrix I the translational inertia terms are unitary while the rotational inertia terms are assumed to be zero.

Figure 4 :

 4 Figure 4: Variability of natural frequencies (Hz)

Figure 5 :

 5 Figure 5: Probability density of natural frequencies (Hz)

Figure 6 :

 6 Figure 6: Influence of different types of uncertain parameters on the variability of natural frequencies

Figure 7 :

 7 Figure 7: Error indicator on the variability of natural frequencies

Figure 8 :

 8 Figure 8: Photograph and geometric shape of the stiffened plate

Figure 9 :

 9 Figure 9: First two mode shapes of the stiffened composite plate

  of degrees of freedom and b the half-band of global stiffness matrix.

Figure 14 :

 14 Figure 14: Number of flops of one MSP trial and one direct MCS trial

Figure 15 :

 15 Figure 15: Number of equivalent finite element analyses for the MSP

Table 1 :

 1 are 23 independent uncertain parameters in total. The mean value, the standard deviation, the coefficient of variation and the type of distribution of these uncertain parameters are given in table1. The variability of the first four natural frequencies is studied. Description of uncertain parameters

	1 E and 2 E , Poisson's ratio 12  ,

i h and ply orientations i  are considered as random. As a result, there Figure 3: First mode shape of the square composite plate Number Uncertain parameter Mean Standard deviation Coefficient of variation Distribution 1 1 E (MPa)


Q Yin
email: qi.yin@utc.fr

F Druesne

P Lardeur
email: pascal.lardeur@utc.fr

PROBABILISTIC ANALYSIS OF COMPOSITE STRUCTURES USING THE CGSM

Keywords: Composite structures, Probabilistic analysis, Finite elements, Random material properties

In this paper, the Certain Generalized Stresses Method (CGSM) is developed for probabilistic analysis of laminated composite structures modeled by finite elements. The material and physical properties are considered as random parameters and are represented by random variables or fields. These random fields are discretized by Karhunen-Loève expansions and the local average method. The CGSM is based on the assumption that the generalized stresses are stable even with the presence of uncertain parameters. Thanks to this assumption, it is possible to calculate the strain energy of the structure, for any combination of the uncertain parameters. Then, an explicit expression of the displacement at a point is obtained by using the Castigliano's theorem. The mean value, the standard deviation and the distribution of the displacement can be evaluated by a Monte Carlo Simulation (MCS) using this expression. For calculating the variability of a displacement, only one nominal finite element analysis with two load cases are required, then a post-processing stage is performed. In addition to the displacements, the variability of strains, stresses and failure criteria can also be evaluated. Two examples are studied: the first example is an eight-layer asymmetric bending plate under a uniformly distributed load; the second example is an eight-layer symmetric bending shell under nine concentrated loads. The results are compared with those obtained by the direct MC simulation, considered as a reference, and those presented in the literature. The comparison shows that the CGSM + MCS approach provides quite accurate results and highlights the high computational efficiency of the method proposed.

INTRODUCTION

Today composite laminated structures are widely used in various industrial sectors like energy, aerospace and transport, because of their excellent mechanical properties. At the same time, a high variability of material and physical properties is also induced by their manufacturing process. Therefore the study of uncertainty of composite structures is a very interesting and important challenge.

So far, many studies have been investigated to take into account the uncertainties of composite structures in numerical models. The variability of stresses of a perforated plate is investigated by Van Vinckenroy and de Wilde [START_REF] Van Vinckenroy | The use of Monte Carlo techniques in statistical finite element methods for the determination of the structural behavior of composite materials structural components[END_REF] using the Monte Carlo Simulation (MCS). Kamiński and Leśniak [START_REF] Kamiński | Homogenization of metallic fiber-reinforced composites under st ochastic ageing[END_REF] study the homogenization of metallic fiber-reinforced composites using a perturbation method. Chen and Soares [START_REF] Chen | Spectral stochastic finite element analysis for laminated composite plates[END_REF] employ a spectral method by to evaluate the variability of displacements of a plate under a concentrated load. Noh [START_REF] Noh | Stochastic finite element analysis of composites plates considering spatial randomnes s of material properties and their correlations[END_REF] investigate the influence of the variability of elastic and shear moduli using a weighted integral method. A perturbation method is used by António and Hoffbauer [START_REF] António | From local to global importance measures of uncertainty pro pagation in composite structures[END_REF] for the probabilistic analysis of the displacements and the failure criteria of a composite shell. For the probabilistic failure analysis of laminated composite plates, the perturbation method is respectively employed by Salim et al. [START_REF] Salim | Buckling of laminated plates with random material cha racteristics[END_REF], Onkar et al. [START_REF] Onkar | Generalized buckling analysis of laminated plates with random material properties using stochastic finite elements[END_REF][START_REF] Onkar | Probabilistic failure of laminated composite plates using the stochastic finite element method[END_REF], and Lal et al. [START_REF] Lal | Stochastic nonlinear failure analysis of laminated composite pl ates under compressive transverse loading[END_REF].

However, these existing methods are not satisfactory. They may not be able to afford accurate results when the input variability is high, and the computation time may be very high, like the MCS. So an economical and reliable method, taking into account the variability of composite laminated structures in finite element calculations, is needed. In this article, a non-intrusive method, the Certain Generalized Stresses Method (CGSM), is developed. This method is firstly proposed by Lardeur et al. [START_REF] Lardeur | La méthode MEGC pour le calcul de dispersion du comportement s tatique des treillis[END_REF][START_REF] Lardeur | Une méthode rapide pour la prise en compte des incertit udes dans les structures minces ; application aux treillis[END_REF][START_REF] Lardeur | Application de la méthode MEGC aux treillis de barres hyperstatiques[END_REF][START_REF] Lardeur | The Certain Generalized Stresses Meth od for the static finite element analysis of bar and beam trusses with variability[END_REF] for the probabilistic analysis of isotropic bars and beams. Mahjudin et al. [START_REF] Mahjudin | La méthode CGSM pour l'analyse statique d es plaques avec variabilité[END_REF] develop this method for homogenous and isotropic plates.

Based on a mechanical assumption, the CGSM can efficiently provide accurate results, even if the level of variability and the number of variables are large. Moreover, it is able to take into account the variability of not only material properties (elastic moduli...) but also physical properties (plate thickness...). In order to calculate the statistical quantities (mean value, standard variation, distribution…) of a displacement with the CGSM, a MCS is performed on an explicit expression of displacement. This method can also be employed to evaluate the variability of strains, stresses and failure criteria.

In the next section, the random variables and fields are presented. Section 3 gives the process of the CGSM, as well as the CGSM formulations for composite plates and shells. In Section 4, two numerical examples are studied with various uncertain parameters. For verifying the numerical accuracy and the computational efficiency of the CGSM, the results are compared with those obtained by the direct MCS and presented in the literature. Conclusions are given in the last section.

RANDOM VARIABLES AND FIELDS

For a parametric approach, the uncertain parameters are represented by random variables or random fields. Random variables mean that uncertain parameters are uniform for each finite elements. Random fields mean that uncertain parameters are different for each finite elements, and the variation depends on the distance of elements.

A random variable of an uncertain parameter E(x) can be obtained by:

( ) ( )(1 C.V. ) Ex μ x ξ  (1)
where C.V. and μ(x) are respectively the coefficient of variation and the mean value of the uncertain parameter, ξ is a random variable which is defined by a truncated Gaussian distribution law.

According to Ghanem and Spanos [START_REF] Ghanem | Stochastic finite elements: a spectral approach[END_REF], the random field can be discretized by the Karhunen-Loève expansion. A random field ( , )

Ex  can be represented by:

1 ( , ) )) ) ( ( ( m ii i Exθ ξ x μ x λ f x         ( 2
)
where m is the number of terms retained, λi and fi(x) are respectively the eigenvalues and eigenvectors of the covariance matrix [Cov], () ξθ  is a vector of random variables which is defined by a truncated Gaussian distribution law. The Eq. ( 2) can be written as:

( , ) ( ) [ ] ( ) Exθ μ x L ξ θ     (3) with 1/2 [ ] [ ] L Cov  (4)
In order to get [L], two methods can be used to decompose the covariance matrix [Cov]: the Cholesky decomposition or the singular value decomposition. These two methods generally lead to close results. Here the local average method is employed to get [Cov]. This method is proposed by Vanmarcke and Grigoriu [START_REF] Vanmarcke | Stochastic finite element analysis of simple beams[END_REF] and developed by Zhu et al. [START_REF] Zhu | Stochastic FEM based on local averages of random vector fi elds[END_REF]. In this method, the random fields are obtained by an integration over the elements. The covariance matrix between two elements is thus described:

2 [ ] exp ij ij ij ij i j AA i j x y xy σ Cov dA dA AA λλ          (5)
where i A and j A are respectively the area of the elements i and j, Δ ij x and Δ ij y are respectively the distances between two arbitrary points of the elements i and j in the x and y direction.

CERTAIN GENERALIZED STRESSES METHOD

The CGSM has been employed by Lardeur et al. [START_REF] Lardeur | La méthode MEGC pour le calcul de dispersion du comportement s tatique des treillis[END_REF][START_REF] Lardeur | Une méthode rapide pour la prise en compte des incertit udes dans les structures minces ; application aux treillis[END_REF][START_REF] Lardeur | Application de la méthode MEGC aux treillis de barres hyperstatiques[END_REF][START_REF] Lardeur | The Certain Generalized Stresses Meth od for the static finite element analysis of bar and beam trusses with variability[END_REF][START_REF] Mahjudin | La méthode CGSM pour l'analyse statique d es plaques avec variabilité[END_REF] for the probabilistic analysis of isotropic and homogeneous structures. In this section, it is developed for bending multilayered composite plates and shells with variability of material and physical properties. The assumption and the process of the CGSM are recalled. Then the CGSM formulations are presented.

Presentation of the CGSM

The CGSM is based on a mechanical assumption that the generalized stresses are constant even in presence of uncertain parameters. The global organization chart of the CGSM is given in Figure 2. In order to calculate the variability of a displacement, firstly, a finite element analysis with two load cases is performed with a standard software for calculating the generalized stresses. Thanks to the CGSM assumption, the internal strain energy can be expressed for any combination of uncertain parameters. A simplified expression of the displacement is obtained by using Castigliano's theorem. Then a MCS using this expression is performed. In this process, the variability of material properties and/or physical properties is taken into account. This process allowed evaluating the mean value, the standard deviation and the distribution of a displacement. In addition to the displacements, the variability of strains, stresses and failure criteria can also be evaluated by the CGSM.

CGSM formulations for thin composite plates and shells

For studying the variability of the displacement at point P with the CGSM, a semi-analytical expression of the displacement is firstly obtained by using Castigliano's theorem:

int π U F    ( 6
)
where F is the load applied in the direction of displacement and πint is the internal strain energy. For a thin composite plate and shell, the transverse shear can be ignored and πint can be expressed as:

1 [ ]{ } 2 int V π ε Q ε dV     (7)
with V the volume of the structure, [Q] the stiffness matrix and {ε} the plane strain vector. {ε} can also be written as:

{ } { } { } ε e κ  (8)
with {e} membrane strains and {κ} curvatures. Then Eq. ( 7) can be transformed into:

1 ( [ ]{ }+ [ ]{ }+ [ ]{ }+ [ ]{ }) 2 int V π e Q e e Q κ κ Q e k Q κ dV           (9)
Then the multilayered plate and shell theory is used (see for example [START_REF] Reddy | Mechanics of laminated composite plates and shells: theory and analysis[END_REF]). The axial forces {N} and the moments {M} can be obtained as follows:

{ } [ ]{ } [ ]{ } N A e B κ  (10) { } [ ]{ } [ ]{ } M B e D κ  (11)
with the membrane stiffness matrix

/2 1 /2 1 [ ]= [ ] [ ] ( ) n t k k k t k A Q dz Q z z   
   [START_REF] Lardeur | Application de la méthode MEGC aux treillis de barres hyperstatiques[END_REF] the membrane-bending coupling stiffness matrix

/2 22 1 /2 1 1 [ ]= [ ] [ ] ( ) 2 n t k k k t k B z Q dz Q z z   
   [START_REF] Lardeur | The Certain Generalized Stresses Meth od for the static finite element analysis of bar and beam trusses with variability[END_REF] the bending stiffness matrix

/2 2 3 3 1 /2 1 1 [ ]= [ ] [ ] ( ) 3 n t k k k t k D z Q dz Q z z       (14)
where t denotes the total thickness, k denotes the layer number, According to the relationship between the strains and the generalized stresses, {N} and {M} can be introduced into Eq. ( 9):

1 2 3 4 1 ( [ ]{ }+ [ ]{ }+ [ ]{ }+ [ ]{ }) 2 int A π N S N M S N N S M M S M dA           (15)
with [S1] the membrane flexibility matrix, [S2] and [S3] the membrane-bending coupling flexibility matrix, [S4] the bending flexibility matrix. The structure is modeled by finite elements, therefore Eq. ( 15) can be written as

1 2 3 4 1 1 ( [ ] { } + [ ] { } + [ ] { } + [ ] { } ) 2 e i n int i i i i i i i i i i i i i A i π N S N M S N N S M M S M dA             ( 16
)
where ne is the number of finite elements, {N}i and {M}i are respectively the axial forces and the moments in element i, Ai is the area of element i.

In order to use Castigliano's theorem, the load F should be introduced in the expression of πint. {N}i and {M}i are therefore decomposed into two contributions:

{ } { } { } i i i N N' F N"  ( 17
) { } { } { } i i i M M' F M"  ( 18
)
where F is the force applied at point P in the direction of interest; {N′} and {M′} are respectively the axial forces and the moments in element i, due to the forces applied on the whole structure except at point P in the direction of interest; {N′′} and {M′′} are respectively the axial forces and the moments in element i, due to a unitary force applied at point P in the direction of interest. These generalized stresses are considered as constants throughout an element and independent of uncertain parameters. They can be calculated by a nominal finite element analysis with two load cases.

Thanks to Eqs. ( 17) and ( 18), F is introduced into Eq. ( 16). Then Castigliano's theorem is employed and a simplified expression of displacement for thin composite plates and shells can be obtained:

1 2 3 4 1 1 ( + ) 2 e n i i i i i i U A P P P P      (19) with 1 1 1 1 [ ] { } + [ ] { } +2 [ ] { } i i i i i i i i i i P N" S N' N' S N" F N" S N"        2 2 2 2 [ ] { } + [ ] { } +2 [ ] { } i i i i i i i i i i P M" S N' M' S N" F M" S N"        3 3 3 3 [ ] { } + [ ] { } +2 [ ] { } i i i i i i i i i i P N" S M' N' S M" F N" S M"        4 4 4 4 [ ] { } + [ ] { } +2 [ ] { } i i i i i i i i i i P M" S M' M' S M" F M" S M"       
Eq. ( 19) is assumed to be valid for any random value of the material and physical parameters. In thin composite plate and shell, this expression is used for each trial of MCS.

NUMERICAL EXAMPLES

In this section, the CGSM is applied to the static probabilistic analysis of multilayered composite plates and shells. Two examples found in the literature are studied: the first example is an eight-layer asymmetric bending plate under a uniformly distributed load; the second example is an eight-layer symmetric bending shell under nine concentrated loads. The variability of displacements and failure criteria is evaluated. The results obtained by the CGSM and the direct MCS, as well as those presented in the literature, are compared in order to demonstrate the numerical accuracy and the computational efficiency of the CGSM.

Eight-layer bending composite plate

The first example, described in Figure 2, is a square composite laminate plate (20m×20m) under a uniformly distributed load q=1Pa. This example is firstly studied by Noh [START_REF] Noh | Stochastic finite element analysis of composites plates considering spatial randomnes s of material properties and their correlations[END_REF] with the weighted integral method. The plate is simply supported and asymmetrically stacked with a stacking sequence [0/90]4. The thickness of this plate is h=0.1m. The material is Graphite/Epoxy (T300/934) with material properties: longitudinal elastic modulus E1=131GPa, transversal elastic modulus E2=10.34GPa, Poisson's ratio υ12=0.22, shear modulus G12=6.895GPa, tensile strength in fiber direction XT=1778MPa and in transverse direction YT=55.2MPa, compressive strength in fiber direction XC=1731MPa and in transverse direction YC=294MPa, shear strength SXY=101.2MPa. The uncertain parameters are E1, E2, G12, XT, YT, XC, YC and SXY. All these uncertain parameters are dependent and represented by Gaussian random fields with the local average method. The input coefficient of variation is 10% and the correlation length (λx=λy=λ) varies from 0.001L to 100L. The variability (coefficient of variation C.V.) of displacement and Tsai-Wu failure index at center is studied. The direct MCS, as well as the MCS using the CGSM formulation (Eq. ( 19)), are performed with 7000 samples. Because of the geometrical symmetry, a quarter of the example plate is modeled by four-node shell elements. Using the convergence study, an optimal mesh with 36 elements (6×6) is used.

The results obtained by the CGSM and the direct MCS, as well as those presented by Noh with the weighted integral method, are given in Figure 3. It can be observed that the results obtained by the three methods are always very close. The curves of the CGSM and the direct MCS are almost coincident. The coefficient of variation of displacement and Tsai-Wu failure index at center increases with the correlation length. When the correlation length is very small (0.001L), the coefficient of variation of response tends to 0. When the correlation length is large (100L), the output coefficient of variation tends to the input coefficient of variation value 10%. Considering that all the uncertain parameters are dependent, this result is expected.

Eight-layer bending composite shell

The second example, studied by António and Hoffbauer [START_REF] António | From local to global importance measures of uncertainty pro pagation in composite structures[END_REF] with a perturbation method, is a clamped cylindrical composite shell under nine vertical loads Pk=7000N (Figure 4). The stacking sequence of this eight-layer structure is [+α/+α /-α /-α]s with the total thickness h=0.02m. The ply angle α, which is the angle between the fiber direction and the x-axis, varies from 0° to 90°. The material is E-glass/Epoxy with material properties: E1=38.6GPa, transversal elastic modulus E2=8.27GPa, Poisson's ratio υ12=0.26, shear modulus G12=4.14GPa, tensile strength in fiber direction XT=1062MPa and in transverse direction YT=31MPa, compressive strength in fiber direction XC=610MPa and in transverse direction YC=118MPa, shear strength SXY=72MPa. The structure is modeled by 32 four-node shell elements (4×8). In this example, the variability of maximum displacement and strength ratio (reciprocal of Tsai-Wu failure index) is evaluated for various α. There are two cases of uncertain parameters: in case 1, the material properties E1, E2, YT and SXY are uncertain; in case 2, the thickness h is uncertain. The parameters are independent and represented by random variables with a Gaussian distribution. As shown in Figure 4, the shell is divided in four macro-elements and for each macro-elements there is one laminate. The uncertain parameters are uniform for each laminate and independent between different laminate. Therefore there are 16 random variables in case 1 and 4 random variables in case 2 for each trial MCS. The input coefficient of variation is 6%. The direct MCS, as well as the MCS using the CGSM formulation (Eq. ( 19)), are performed with 10000 samples.

Figure 5 shows results in case 1, where the material properties are uncertain. The results obtained by the CGSM and the direct MCS, as well as those presented by António and Hoffbauer with a perturbation method, are quite close. The coefficient of variation of displacement decrease at first and then increase along the ply angle domain, and it is always lower than the input variability (6%). The minimum value is obtained with α=40°, which is about 2%. For strength ratio, it is observed that the variability is divided into two intervals [0°, 60°] and [70°, 90°]. In each interval, the coefficient of variation is almost the same. The maximum value is about 7.1%, which is a little lager than the input variability. Results in case 2, obtained by three methods, are given in Figure 6. In this case the thickness of structure is uncertain. It can be observed that curves obtained are very close. There is a little difference between the curves of CGSM and direct MCS, but the error is very small (less than 3%). The uncertainty of thickness has a more important influence on the variability of response than material properties. The maximum coefficient of variability is about 16% for displacement and 10% for strength ratio. Moreover, the variation of variability is small, along the ply angle domain.

CONCLUSIONS

In this paper the CGSM is developed for the static probabilistic analysis of multilayered composite plate and shell taking into account random material and physical properties. Based on the CGSM assumption, an explicit expression of displacement is obtained by using the Castigliano's theorem. Two academic examples, an eight-layer asymmetric bending plate under a uniformly distributed load and an eight-layer symmetric bending shell under nine concentrated load, are studied. The variability of displacement and Tsai-Wu failure criterion is evaluated by the CGSM, taking into account the variability of material and physical properties.

The results calculated by the CGSM are compared with those obtained by the direct MCS, considered as a reference, and those presented in the literature. The CGSM provides quite accurate results for the studied examples. In addition, this method has obvious advantage in computational cost. Only one finite element analysis with two load cases is required for evaluating the variability of a displacement.

Figure 1 :

 1 Figure 1: Global organization chart of the CGSM.

1 kz

 1  and k z are respectively the top and bottom z coordinates of kth ply.

Figure 2 :

 2 Figure 2: Eight-layer bending composite laminate plate under a distributed load.

Figure 3 :

 3 Figure 3: Variability of displacement and Tsai-Wu failure index at center.

Figure 2 :

 2 Figure 2: Eight-layer bending composite laminate plate under a distributed load.

Figure 5 :

 5 Figure 5: Variability of maximum displacement and strength ration in case 1.

Figure 6 :

 6 Figure 6: Variability of maximum displacement and strength ratio in case 2.

ACKNOWLEDGEMENTS

This work was financed by the China Scholarship Council (CSC). The financial support is gratefully acknowledged.


Junjie Hao

Haochen Liu

Wenda Zhang

Fenghuan Zhao

Rui Lu

Bing Xu

Ziming Zhou

Kai Wang

Marie-Helene Delville

Compact stable Quantum Dots via Amide-Mediated Synthesis of PMO-Based Multifunctional Ligand

Keywords: quantum dots, PMO, on-chip, light emitting diode Quantum dots, PMO, on-chip, light emitting diode

We present a new class of polymeric ligands for quantum dots (QDs) stabilization, to yield compact stable particles (~100-200 nm diameter), and high quantum yields (> 47 %). The uniform and tunable size of these polymer-coated QDs also allows us to demonstrate their utility in achieving bright and efficient WLEDs with superior stability.

Introduction

Quantum dots (QDs), due to their wavelength tunability, have been widely used in light emitting device (LEDs)-based displays [START_REF] Dai | Solution-processed, highperformance light-emitting diodes based on quantum dots[END_REF][START_REF] Jun | Bright and Stable Alloy Core/Multishell Quantum Dots[END_REF][START_REF] Sun | multicoloured light-emitting diodes based on quantum dots[END_REF][START_REF] Chen | Highly Efficient and Stable Luminescence from Microbeans Integrated with Cd-Free Quantum Dots for White-Light-Emitting Diodes[END_REF][START_REF] Chen | Effects of Nano-TiO2 Particles on Conversion Efficiency of Quantum Dots Light Converting Nanocomposites[END_REF][START_REF] Bin | Targeting Cooling for Quantum Dots in White QDs-LEDs by Hexagonal Boron Nitride Platelets with Electrostatic Bonding[END_REF] , solar cells [START_REF] Liu | Scattering enhanced quantum dots based luminescent solar concentrators by silica microparticles[END_REF][START_REF] Meinardi | Large-area luminescent solar concentrators based on /`Stokes-shift-engineered/' nanocrystals in a mass-polymerized PMMA matrix[END_REF][START_REF] Jasieniak | Solution-Processed Sintered Nanocrystal Solar Cells via Layerby-Layer Assembly[END_REF] and as fluorescent biomarkers [START_REF] Liu | Compact Biocompatible Quantum Dots via RAFT-Mediated Synthesis of Imidazole-Based Random Copolymer Ligand[END_REF][START_REF] Gao | Highly Bright and Compact Alloyed Quantum Rods with Near Infrared Emitting: a Potential Multifunctional Nanoplatform for Multimodal Imaging In Vivo[END_REF][START_REF] Liu | Quantum Dots with Phenylboronic Acid Tags for Specific Labeling of Sialic Acids on Living Cells[END_REF] , etc. To obtain QD based light emitting diodes (QD-LED), in most cases, high quality QDs with specific emission wavelength are uniformly dispersed in a polymer matrix to form a light converting layer. This composite sheet is then coated on LEDs to achieve desired light performance [1-3, 13, 14] . However, up to now a major barrier toward the widespread use of QDs has been the presence of a trade-off among three desirable QD properties for LEDs: high QY, uniform size, and high stability (both over time and in a harsh environment), in particular, the abscission of QDs' outer ligands, and the deterioration of QDs by permeate oxygen and moisture [START_REF] Lu | 32-3: Stability Enhancement of Light Emitting Diode Based on Quantum Dots through Atomic Layer Deposition[END_REF][START_REF] Kim | Color-converting bilayered composite plate of quantum-dot-polymer for high-color rendering white light-emitting diode[END_REF][START_REF] Jong-Hoon | White lighting device from composite films embedded with hydrophilic Cu(In, Ga)S 2 /ZnS and hydrophobic InP/ZnS quantum dots[END_REF][START_REF] Kim | Quantum Dot/Siloxane Composite Film Exceptionally Stable against Oxidation under Heat and Moisture[J][END_REF][START_REF] Kim | Distinguished Student Paper: Quantum Dot/Siloxane Composite Film Exceptionally Stable Against Heat and Moisture[END_REF] . Such a problem is commonly found in hightemperature processes of LED packaging (e.g. silicone curing). The principle behind such a phenomenon would be given by the incompatibility between the hydrophobic surface of the QDs and that of encapsulant in WLED (e.g. silicone, or other polymers) [START_REF] Ziegler | Silica-Coated InP/ZnS Nanocrystals as Converter Material in White LEDs[END_REF] . Consequently, it is energetically preferred for QDs to be deteriorated on surface and aggregate in host matrix, which gives the undesired quenching of photoluminescence.

In order to improve the stability of QDs, amphiphilic polymers have been widely used in cell imaging [START_REF] Liu | Compact Biocompatible Quantum Dots via RAFT-Mediated Synthesis of Imidazole-Based Random Copolymer Ligand[END_REF][START_REF] Liu | Quantum Dots with Phenylboronic Acid Tags for Specific Labeling of Sialic Acids on Living Cells[END_REF][START_REF] Medintz | Quantum dot bioconjugates for imaging, labelling and sensing[END_REF][START_REF] Duan | Quantum Dots with Multivalent and Compact Polymer Coatings for Efficient Fluorescence Resonance Energy Transfer and Self-Assembled Biotagging[END_REF][START_REF] Zhang | Click-Functionalized Compact Quantum Dots Protected by Multidentate-Imidazole Ligands: Conjugation-Ready Nanotags for Living-Virus Labeling and Imaging[J[END_REF][START_REF] Hu | A fast synthesis of near-infrared emitting CdTe/CdSe quantum dots with small hydrodynamic diameter for in vivo imaging probes[END_REF] , and various functionalized polymers were used to modify quantum dots. Water soluble quantum dots were even fabricated by encapsulating the QDs into (poly(maleic acid-altoctadecene)) [START_REF] Cho | Highly Fluorescent and Stable Quantum Dot-Polymer-Layered Double Hydroxide Composites[END_REF][START_REF] Kumar | Recent advances and remaining challenges for polymeric nanocomposites in healthcare applications[END_REF] . The polymer-encapsulated QDs were electrostatically assembled to form QD-polymer-LDH (layered double hydroxide) composites, so as to multi-protect the QD surface from the environment by both the polymer and the LDH nanosheets. However, such an approach would not necessarily protect the QDs from oxygen and moisture, hence still imposing an obstacle and unravelling the full potential of stability improvement.

In this work, we report this simple method using the polymeric ligands to yield the compact egg-structure particles. To address the fundamental problem of the QDs ligands instability in the traditional solid state QDs systems, the polymer employed here is a stable multifunctional poly(maleic anhydride-alt-1-octadecene) (PMO) which is expected to confine both quantum dots and ligands. The stable PMO-modified QDs effectively demonstrated a high quantum yield (> 47%), a low non-specific binding, as well as controlled and adjustable size (from 100 nm to 1 μm). The incorporation of PMO-modified QDs into functionalized silica shell is also demonstrated [START_REF] Liu | Highly stable quantum dots with silica-poly(EGDMA-co-MAA) synergistic protection and the preliminary application in immunoassay[END_REF][START_REF] Hu | Silica-Polymer Dual Layer-Encapsulated Quantum Dots with Remarkable Stability[END_REF] . For details, the QDs coupled with excess ligands were used as egg-yolk, the functional PMO as eggwhite, and the functional silica as egg-shell, respectively. The uniform and tunable size of these polymer-coated QDs also allow us to demonstrate their utility in achieving bright and efficient WLEDs with superior stability. And the as-prepared high quality QDs composites have great potential to be used in high performance displays with wide color gamut.

Experiments

Preparation of CdSe/ZnSe/ZnS (Core/Shell/Shell) Quantum Dots (QDs):

All the chemicals were of analytical reagent grade and were used without further purification. The pure water was purchased from Wahaha. The CdSe bare nanocrystals and core/shell/shell quantum dots were synthesized via our previous "TOP-assisted SILAR" method [START_REF] Hao | 12.3: Core-Shell Quantum Dots Synthesized by Using Tri-n-Octylphosphine-Assisted Method for High-Color-Saturation Displays[END_REF][START_REF] Hao | A facile route to synthesize CdSe/ZnS thick-shell quantum dots with precisely controlled green emission properties: towards QDs based LED applications[END_REF][START_REF] Ren | Prepare core-multishell CdSe/ZnS nanocrystals with pure color and controlled emission by tri-n-octylphosphine-assisted method[END_REF] .

Preparation of Monoethanolamine functionalized poly(maleic anhydride-alt-1-octadecene) (PMOM):

The amphiphilic polymer PMOM was prepared through the amide reaction by using monoethanolamine and PMO. The hydrophily can be adjusted by the monoethanolamine to carbonyl functions ratio (from 0% to 100%).

Preparation of Water-Soluble Polymer-Encapsulated CdSe/ZnSe/ZnS QDs:

The quantum dots with free ligands were used for the phasetransfer process. The QDs were encapsulated by the amphiphilic polymer to transfer them into water without exchanging initial surface ligands. Typically, 100 mg PMOM was dissolved in chloroform solution at room temperature. The chloroform solution containing 1 mg QDs coupled with 5 mg ligand was added to the polymer solution (QDs: polymer= 1:100). The solution was vigorous stirred for 1 h at room temperature, and then, 10 times of water was introduced into the mixture. The chloroform was removed slowly. The particle size of PMOM-QDs composites can be adjusted by the monoethanolamine ratio in PMOM, which can be easily changed from 100 nm to 1 μm.

Formation of the egg-structure composites particles:

APTES ((3-aminopropyl)triethoxysilane) was used as the silane coupling agent to form the egg-shell. Typically, APTES coupled with EDC/NHS was introduced into the PMOM-QDs mixture solution. After the coupling reaction completion, TEOS and NH4OH•H2O was used to form the thick silica egg-shell.

Results and discussion

In this work, we have successfully synthesized egg-structure composite particles. The QDs were encapsulated by the amphiphilic polymers to transfer them into water without exchanging initial surface ligands. As illustrated in Fig. 1a, the multivalent polymer ligand PMOM with alkane and carboxyl groups was synthesized by reacting poly(maleic anhydride-alt-1octadecene) (PMO) (Mn= 30,000-50,000) with monoethanolamine. The high reactivity of anhydride with primary amines facilitated the quantitatively controlled reaction. The ratio of modified maleic anhydride groups can be adjusted from 0% to 100%. In an optimized structure, about 80% of the maleic anhydride groups were chemically modified. The synthesis process of PMOM coated QDs is presented graphically in Fig. 1b. After the chloroform was evaporated from the mixed system, the uniformly sized particles were dispersed in aqueous solution. The single particle contains multiple quantum dots with free ligands, the polymers here employ the multifunctional PMOM to confine quantum dots and ligands. And then the hard shell of silica can be formed gradually, as shown in Fig. 1c, using (3-aminopropyl)triethoxysilane (APTES) as the silane coupling agent to form the egg-shell. TEOS was then used as the shell material for further growth. Finally, as can be seen in Fig. 1d, the high luminescent QDs and free ligands can be first confined inside the flexible polymer corona, and then be protected by the silica shell. The stable egg-structure particles were achieved by coupling the two kinds of protective layer. The fluorescence spectra of core-shell QDs synthesized by TOP-SILAR method is shown in Fig. 2. Comparing the emissions in Fig. 2, we can see that little PL shift (3 nm) was observed between the PL spectra of QDs in chloroform solution (black line) and PMO/PMOM modified QDs in aqueous solution (red and blue lines). The inset in Fig. 2 shows the corresponding PL QYs. The QY ratio of PMOM-QDs (y:x=1:0) aqueous solution (47%) and QDs chloroform solution (63%) was about 75%, indicative of a relatively high utilization and low degree of self-absorption. And such characteristics are expected to cater the fabrication of high efficiency QLED. While the PMO-QDs (y:x=0:1) sample exhibits a very low QY (23%), which may be due to the poor dispersion of PMO in water and the large size of the composite particles. Fig. 3 shows the picture of the PMOM-modified QDs composite particles with different ratios of monoethanolamine. The left part represents 100% anhydride amide reaction (PMOM), and the right part represents 0% reaction (PMO). We can see from Fig. 3a-b, the 100% reaction samples can form a uniform and stable microemulsion solution, while in PMO-QDs system the droplet cannot disperse evenly. After the chloroform volatile completely, as shown in Fig. 3c-d, obvious Mie scattering can be seen in the right part, which corresponding to the PMO-QDs sample. This can also be confirmed by TEM, as shown in Figure 4. The particle sizes are 100 nm and 1 μm for these two samples, respectively. As shown in the inset of TEM images, the nanoparticles are dispersed very well, which provide that quantum dots and ligands can be coated together in the vesicles in our method. We can easily achieve different sizes of QDs-polymer composite particles, and the size can be adjustable controlled from 100 nm to 1 μm. Due to the rich active carboxyl group on the surface of particles (as shown in Fig. 1c), the uniform particles can be further reacted with silane coupling agent and form the silica egg-shell.

Conclusion

In summary, we have described detailed synthetic methods for the highly luminescent, uniform size and modifiable egg-structure particles. The absolute PL QY can reach up to 47%, and the size can be controlled and adjustable from 100 nm to 1 μm. This method provides a strong advantage for delicate control of optical properties because it can apply to any type of QDs whatever their pre-existing coating. In principle, this method allows composite fabrications using QDs covering a wide range of sizes and chemical nature. And the as prepared stable polymer-QDs particles have great potential to be used in high performance displays with wide color gamut.

Figure 1 .

 1 Figure 1. Schematic diagrams illustrating the process for preparing the egg-structure composite particles. (a). Schematic illustration of the synthesis of PMOM ligands, the molar ratio of y:x can be changed from 0:1 to 1:0 by adjusting the of monoethanolamine; (b). The procedure to prepare PMOM coated QDs; (c). Approach for the fabrication of silica coated egg-structure composite particles, the hard protective layer can be obtained by the dense growth of silica; (d). Schematic diagram of different positions in egg-structure particle.

Figure 2 .

 2 Figure 2. The PL emission of diluted QDs chloroform solution (black line), PMOM modified QDs aqueous solution (red line) and PMO modified QDs aqueous solution (blue line). Inset: comparison of PL QYs of CdSe/ZnS QDs in chloroform solution versus PMO/PMOM modified QDs in aqueous solution.

Figure 3 .

 3 Figure 3. The images show the vials containing modified-QDs particles, the left part represents (PMOM), and the right part represents (PMO). (a) The mixture of modified, QDs, ligands, chloroform, and water; (b) Hand shaking, and then standing for 10 minutes; (c) Magnetic stirring and slow evaporation of chloroform, dispersion in aqueous solution; (d) The laser beam passes through the solution.

Figure 4 .

 4 Figure 4. TEM images of PMOM-QDs particles as shown in Figure 3. (a) PMOM-QDs; (b) PMO-QDs.

Acknowledgements

The work is supported by National Key Research and Development Program of China administrated by the Ministry of Science and Technology of China (No. 2016YFB0401702), National Natural Science Foundation of China (No. 61674074 and 61405089), Development and Reform Commission of Shenzhen Project (No. [2017]1395), Shenzhen Peacock Team Project (No. KQTD2016030111203005), Shenzhen Key Laboratory for Advanced quantum dot Displays and Lighting (No.ZDSYS201707281632549), Guangdong University Key Laboratory for Advanced Quantum Dot Displays and Lighting (No. 2017KSYS007), Distinguished Young Scholar of National Natural Science Foundation of Guangdong (No. 2017B030306010), Zhonghuan Quantum Co., Ltd (18YFZCGX00580), and also thanks the start-up fund from Southern University of Science and Technology..


J L Dulong

Frédéric Druesne

P Villon

A Priori Reduction method and A Priori Hyper-Reduction method for a non linear problem

The A Priori Hyper-Reduction (APHR) method was developed for few years by D.Ryckelynck [START_REF] Ryckelynck | A priori hyperreduction method: an adaptive approach[END_REF]. Based on an extremely reduced finite element model, this method aims to perform extremely fast finite element calculations. This work aims to study some aspects of its numerical behavior.

Methods

In statics, the standard Finite Element Method (FEref) results in a system of equations which is solved by a Newton Raphson (NR) procedure. After calculating the tangent stiffness matrix KT and the equation residual r, the displacement increment du is obtained by the linear system K T du=r .

A first way to reduced the system size [START_REF] Krysl | Dimensional Model Reduction in Non-linear Finite Element Dynamics of Solids and Structures[END_REF] is to project the displacement on a base Φ, known a priori : u= a . The reduced displacement a is now calculated by equation also projected on this base (1), following a Galerkin approach. This method will be called A Priori Reduction (APR).

 T K T  da= T r (1)
The previous equation results in a reduced stiffness matrix, which can be inverted rapidly, but a large matrix product is necessary between KT and Φ . To reduce this last cost, Ryckelynck has proposed [START_REF] Ryckelynck | A priori hyperreduction method: an adaptive approach[END_REF] to assemble KT only on few degrees of freedoms, choosen correctly, and determined by a sparse identity matrix P. This method was called APHR. The resulting iterative equation (2) comes from the minimization of the residual following [START_REF] Lefevre | Different formulations for model reduction to simulate the crush of a mechanical part[END_REF].

 T K T T P K T  da= T K T T P r (2)

Results

Some tests were led on a short embedded beam, loaded at its extremity (figure 1a). The mechanical problem is non linear due to neohookean material, and large displacements. Calculations was led with FEAP software. An important point is that an adaptative time increment was used allowing the method to increase the time increment if possible.

Figure 2a shows that time CPU increases when loading force increases. Mainly, for low force, APR and APHR are faster than FEref, but the speed up is reduce (about 1.6 ). On the contrary, for high loading force, FEref curve is characterized by a slope change : for maximum force, the speed up of APR and APHR is about 5. The change in FEref behavior is due to limit in NR iterations (figure 2b) : if 5 FEref iterations cannot solve the system, then time increment has to be reduced. On the contrary, APR and APHR have the possibility to increase the number of iterations to reach solution. And then, time increment can stay large : it can even increase. This behavior can be explained by the fact than, with FEref method, only one search direction is used to solved the system, determined by KT. With APR and APHR, several search directions are used, determined by base Φ size.

Comparing the two A Priori methods, APHR is slightly faster than APR but speed up is low. Indeed, time to assemble and calculate reduced stiffness matrix is decreased compared to APR, but a larger base Φ is required to compensate for the lack of information due to reduced mesh (figure 1b). So the time to adapt the base becomes preponderant during resolution.

Conclusion

The speed up due to APHR compared to APR is low. The main part of the speed up is due to the numerical behavior of FEref limited in number of iterations. The A Priori methods are superior when non linearity is large.

Figure 1 :Figure 2 :

 12 Figure 1: Complete (a) and reduced (b) meshes

ECCM 2010 IV European Conference on Computational Mechanics Palais des Congrès, Paris, France, May 16-21, 2010 A Priori Reduction method and A Priori Hyper-Reduction method for a non linear problem

J.L. Dulong, F. Druesne, P. Villon Laboratoire Roberval, Université de Technologie de Compiègne, France, {jean-luc.dulong, frederic.druesne, pierre.villon}@utc.fr


Pauline Seguinot

Ying Su

Isabelle Sanchez

Audrey Bloem

Pascale Brial

Anne Julien Ortiz

José-Manuel M Guillamon

Carole Camarasa

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Impact of nitrogen sources on the fermentative kinetic and aroma compounds production by non-Saccharomyces yeasts in alcoholic fermentation

Determination of characteristic parameters : -Tlag : lag phase (h) -Rmax : maximal fermentation rate (g/L/h) -Trmax : time to reach Rmax (h) • SC 09 : exhaustion of most of the nitrogen sources except Ser (15%), Lys (11%), His (85%), Glu (9%) and Cys (39%).

• TD 291 : complete consumption of Phe, Glu, Arg and Ala. Limited ability (residual nitrogen > 80%) to assimilate His and Gly. Higher consumption of Cys, Glu ans Ser than SC 09. • MP 346 : consumption between 47 and 98% of initial nitrogen. Higher consumption of Cys, His and Ser than SC 09.

 Correlation between the amount of consumed nitrogen and fermentation performances. In particular, inability to sustain fermentation on poorly consumed nitrogen sources.

Percentage of residual nitrogen at the end of fermentation :

Fermentation profiles and kinetics parameters depend on both the strain and the nitrogen source

Efficiency of nitrogen sources to support wine fermentation

++ : best nitrogen sources + : average sources + : poor sources. In bold, nitrogen sources in the same category for the 3 strains

• In contrast to MP 346, TD 291 and SC 09 are able to achieve fermentation using most of amino acids as sole nitrogen source

• Many similarities in nitrogen sources preferences between non-Saccharomyces strains and S. cerevisiae Over the past 50 years, the completion of wine fermentation and the formation of undesirable off-compounds have been mainly controlled through the use of selected starter strains of S. cerevisiae. Little attention was paid to the non-Saccharomyces yeasts, predominant in grape juice but rapidly outcompeted by S. cerevisiae during fermentation. However, the potential of some of these species, thank to their ability to produce fermentative aromas, to improve the sensory quality of wines is now recognized even if the metabolic specificities of non-Saccharomyces yeasts remain ill-characterised. To fill this shortcoming that is a hindrance to an efficient use of non-Saccharomyces strains during wine fermentation, the aim of this study was to investigate to impact of the nature of nitrogen source on the fermentation performances and the production of volatile molecules of 2 non-Saccharomyces species.

Distinct fermentative profiles were observed depending on the nitrogen source and the yeast strain. Yeasts had their own preferences in term of nitrogen sources that have shown diverse efficiency to sustain fermentative activity. Analysis of aroma compounds production according to the nitrogen source revealed a redistribution of carbon and nitrogen thought the entire metabolism network, as the presence of a sole amino acids allow the production of all aroma compounds and not only those directly produced by the amino acid. Differences between the species have been highlighted, especially concerning the production of acetate esters. Nitrogen consumption is consistent with fermentation kinetics. These results provide new insights for the management of co-cultures of non-Saccharomyces and S. cerevisiae, emphasizing the importance of managing the nitrogen nutrition.

• Overall, increase in the production of compounds derived from the nitrogen source

• Low ability of TD 291 to produce acetate esters except phenylethyl acetate

• In contrast, with MP 346 increase in the production of acetate esters except phenylethyl acetate. Low producer of acids and their ethyl esters derivatives

Impact of the nitrogen source on the production of some aroma through Ehrlich pathway

Percentage differences between the production on complete medium and on sole nitrogen source medium

• Huge increase in volatiles molecules that directly derive from α-keto-isovalerate.

•  Similar responses to the nature of the nitrogen sources for the 3 species  However, some species specificities, likely related to their genetic background Aromatic profiles of TD 291 in fermentation on sole nitrogen source (normalized concentrations) :

• Most of studied aroma compounds are produced irrespective of the nitrogen source  Compounds synthetized from MCC metabolites , regulated by total nitrogen availability

  : INRA, Universite Montpellier, Montpellier SupAgro, 34060, Montpellier, France 2 Lallemand SAS, 31700, Blagnac, France 3 Food Biotechnology Department, Instituto de Agroquímica y Tecnología de Alimentos, IATA-CSIC, Paterna, Valencia Spain 4 UMR MISTEA: INRA, Montpellier SupAgro, 34060, Montpellier, France Macrowine 2018 -Zaragoza Material and Methods Study of 3 yeasts stains, naturally present on grapes :  Torulaspora delbrueckii (TD 291)  Metschnikowia pulcherrima (MP 346)  Saccharomyces cerevisiae (SC 09) Fermentation in synthetic media with only one nitrogen source (200 mgN/L) among 27 : Ade, Ala, Arg, Asp, Asn, Cit, Cys, Cyt, GABA, Gln, Glu, Gly, His, Ile, Leu, Lys, Met, NH4, Orn, Phe, Pro, Ser, Thr, Trp, Tyr, Urea, Val Measure of concentrations of 35 aroma compounds at the end of fermentation by GCMS Residual nitrogen at the end of fermentation determined by NOPA analysis (amino acids) or enzymatic kit (NH 4 )

  Moderate increase of volatile compounds not derivated from valine metabolism, as propanol  Impact of MCC Percentage difference between the production of aroma compounds by TD 291 in complete medium vs on medium with only valine : Production of volatile compounds during fermentation using a sole nitrogen source Offline determination of metabolites Fermentation in 15 mL-tubes -1 AA per media PhenoFerm : automatic weighting device  Point each 4 h per tube to accurate determination of fermentation rate His, Lys, Cys : unable to support wine fermentation Incomplete fermentation Low fermentation rate Long lag phase SC 09 and TD 291 efficiently ferment sugars and most of the used nitrogen sources Complete fermentation High fermentation rates Long lag phase MP 346 : low performances regardless the nitrogen source Incomplete fermentation Low fermentation rates Short lag phase Principal component analysis of fermentation parameters : lag phase (Tlag), maximal fermentation rate (Rmax), and time to reach the Rmax (Trmax).

•

  Some differences :  Phe supports more efficiently TD 291 and SC 09 fermentation than MP 346 fermentation  Trp can not be used by MP 346 to carry out fermentation unlike SC 09 and TD 291  Lys sustains a moderate fermentation by TD 291, but not for SC 09 and MP 346


Pauline Seguinot

Isabelle Sanchez

Audrey Bloem

Anne Julien Ortiz

Carole Camarasa

Impact of nitrogen sources on the fermentative kinetic of non -Saccharomyces yeasts

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

Application for co-culture management

• Representation of S. cerevisiae lag phase minus non-Saccharomyces strains lag phase = when the difference is > 0, SC lag phase is longer than the lag phase of the other strain  Under most conditions, shorter lag phase for non-Saccharomyces  Non-Saccharomyces start to consume nitrogen before S. cerevisiae = competition for nitrogen in sequential or co-inoculation ?

• Representation of the probability of incomplete fermentation (final CO 2 production < 90 g/L)

 Over the past 50 years, wine fermentation has been mainly controlled through the use of selected starter cultures of S. cerevisiae, to ensure the completion of fermentation and to avoid undesirable off-compounds. The non-Saccharomyces yeasts, predominant in grape juice, are rapidly outcompeted by S. cerevisiae during the process because of their poor adaptation to increasing concentrations of ethanol and the lack of oxygen. Even though these species were used to be considered as spoilage microorganisms for a long time, their potential to improve the sensory quality of wines is now recognized. However, the metabolic specificities of non-Saccharomyces yeasts remain ill-characterised, in particular their ability to catabolize the nitrogen compounds. It is thus critical to fill this shortcoming, nitrogen assimilation playing a key role in the fermentation process and the production of aroma compounds. Therefore, the aim of this study is to investigate the effect of various nitrogen sources on the fermentative kinetic of four non-Saccharomyces yeasts to determine their preferences in comparison with S. cerevisiae , in order to manage and optimize the implementation of these strains in co-culture with S. cerevisiae.

Distinct fermentative profiles were observed depending on the nitrogen source and the yeast strain. Yeasts had their own preferences in term of nitrogen sources, which have shown diverse efficiency to sustain fermentative activity. It would be interesting to complete these results with the analysis of nitrogen impact on yeast growth, to provide a comprehensive overview of the capabilities of non-Saccharomyces yeasts to efficiently metabolize nitrogen compounds.

Overall, for most nitrogen sources, non-Saccharomyces yeasts exhibit a shorter lag phase than S.cerevisiae and a lower fermentation rate. These results provide new insight for the management of sequential or co-inoculation of non-Saccharomyces and S. cerevisiae, emphasizing the importance of managing the nitrogen nutrition throughout the process.

Material and methods

++ : best nitrogen sources + : average sources + : poor sources. In bold, nitrogen sources in the same category for the 5 strains

  MP and MF are not able to complete the fermentation regardless of the nitrogen source  TD is less effective than SC, but is able to complete the fermentation in many conditions = consumption of nutrients in the medium (sugar, nitrogen, …)  Competition between TD and SC in co-culture ? Nitrogen sources classification and efficiency IMPACT OF NITROGEN SOURCES ON THE FERMENTATIVE KINETIC OF NON-SACCHAROMYCES YEASTS SEGUINOT P. a,b , SANCHEZ I. c , BLOEM A. a , ORTIZ-JULIEN A. b , CAMARASA C. a 33rd International Specialised Symposium on Yeast, June 26-29 2017, Cork, Ireland a : UMR SPO, INRA, SupAgroM, Université de Montpellier, Montpellier, France b : Lallemand SAS, Blagnac, France c : UMR MISTEA, INRA, SupAgroM, Montpellier, France

••

  Same classification for the three Metschnikowia • SC 09 and TD 291 have mostly the same preferences • Torulaspora delbrueckii (TD 291)  Metschnikowia pulcherrima (MP 346)  Metschnikowia fructicola (strain A and B) (MFa and MFb)  Saccharomyces cerevisiae (SC 09) Fermentation in synthetic media with only one nitrogen source among 27 : Blue circle : no fermentation • Orange circle : MP 346 and MF a et b , short lag phase, low R max • Green circle : TD 291 and SC 09, longer lag phase, higher R max Different fermentation profiles depending on the strain and the nitrogen source Incidence of the nitrogen source :  histidine and cysteine = no fermentation  arginine, glutamine, glutamate, ammonium = efficiently used regardless of the strain Incidence of the strain = two groups of strains having the same profile :  SC and TD : best adapted for alcoholic fermentation, high R max , complete the fermentation most of the time  MF and MP : low R max , do not complete the fermentation but short lag phase Individuals are grouped depending on their fermentation kinetic profiles in 7 clusters (fermentation with 200 mg/L of YAN , mean values of maximal fermentation rate (R max ) and lag phase (T lag ) are indicated in boxes, H: high, M: medium, L: low, FC: completion of fermentation, red: MFa, blue: MFb, green: MP, purple: SC, orange: TD) Cluster 3 R max : M (0.546 g/L/h) T lag : H (80.5 profiles PCA of characteristic parameters measured for 5 strains grown on sole nitrogen source Efficiency of nitrogen sources depending on the strain Impact of nitrogen nature on the lag phase Capacity of strains to complete the fermentation