Bruno Sudret

Thierry Yalamas

Emmanuel Noret

Pierre Willaume

Sensitivity Analysis of Nested Multiphysics Models Using Polynomial Chaos Expansions

Uncertainty quantification in computational mechanics has received much attention in the past few years. In this context, global sensitivity analysis (GSA) aims at quantifying which random input parameters (or combinations of parameters) are the most important in order to explain the variability of the model response. The present paper considers models of complex systems which are made of nested submodels, i.e. where input parameters of some submodels are obtained as the output of other submodels. A general framework is proposed in order to represent all the intermediate random variables (i.e. those variables that are obtained as response quantities of first-level submodels) by their polynomial chaos expansions. The coefficients of the various PC expansions are computed by regression and combined in order to obtain analytically the sensitivity (Sobol') indices. The crucial point is the representation of the joint PDF of intermediate input random variables, for which a Nataf transform is used. The approach is illustrated by sensivity analysis of a multiscale micromechanics-based model of concrete elastic properties.

INTRODUCTION

Uncertainty quantification in computational mechanics has received much attention in the past few years. In this context, global sensitivity analysis (GSA) aims at quantifying which random input parameters (or combinations of parameters) are the most important in order to explain the variability of the model response. Various measures of importance have been proposed including partial correlation coefficients (PCC), standard regression coefficients (SRC) and Sobol' indices (Saltelli et al., 1999). The latter, which are usually computed using Monte Carlo simulation, allow the analyst to rank efficiently the input random variables whatever the complexity of the model.

In this paper we are interested in multiscale modelling in which several hierarchical models are nested, i.e. the response of one model corresponds to the input of another model and so on. This type of nested models may be represented by a network, in which each node is a model (having its own input and output vector) and some connections with the other nodes. They appear for instance when micromechanical models of materials at various scales are considered (i.e. a homogenized material property obtained at one scale is used as the input parameter of a homogenization procedure at the next scale). The concept of nested models appears more generally in all mutiphysics models. When considering nested models from the point of view of sensitivity analysis, it is desirable to establish importance measures (such as Sobol' indices) at various levels, i.e. the sensitivity of any response quantity at any level with respect to each variable of lower level (which may be itself the response of a lower level model).

Recently, sensitivity analysis based on Sobol' indices has been combined with polynomial chaos (PC) expansions, showing a great accuracy and computational efficiency [START_REF] Berveiller | Stochastic finite elements: a non intrusive approach by regression[END_REF](Sudret, , 2008)). PC expansions have been introduced in stochastic computational mechanics in the early 90's by [START_REF] Ghanem | Stochastic finite elements -A spectral approach[END_REF], based on the representation of stochastic processes due to Wiener (1938). Recently, non intrusive computational techniques have allowed one to compute the coefficients of the PC series expansions from a set of evaluations of the deterministic model under consideration, namely projection [START_REF] Ghiocel | Stochastic finite element analysis of seismic soil-structure interaction[END_REF] or regression [START_REF] Berveiller | Stochastic finite elements: a non intrusive approach by regression[END_REF], see a review in Sudret (2007).

In this paper, the technique developed in Sudret ( 2008) is extended to multiscale nested models. The difficulty lies in the representation of the intermediate random variables in a suitable manner to carry out a further PC expansion. The Nataf transform used in structural reliability is introduced for this purpose.

The basics of polynomial chaos expansions and global sensitivity analysis are recalled in Section 2 and 3 respectively. The adaptation of PC expansions for nested models is detailed in Section 4. The Nataf approximation is introduced. The application example is then presented in Section 5.

POLYNOMIAL CHAOS EXPANSIONS

2.1 Mathematical framework Consider a mechanical system described by a numerical model M which can be analytical or more generally algorithmic (e.g. a finite element model). Suppose that this model has M uncertain input parameters which are represented by independent random variables (X 1 , . . . , X M ) gathered in a random vector X of prescribed joint probability density function f X (x). Hence the model response denoted by Y = M(X) is also random. For the sake of simplicity, Y is assumed to be scalar throughout the paper (in case of a vector response Y , the following derivations hold componentwise). Provided that the random variable Y has a finite variance, it may be recast as follows (Soize and Ghanem, 2004):

Y = M(X) = α∈ N M a α ψ α (X) (1)
This expansion is referred to as the polynomial chaos (PC) representation of Y . The a α 's are unknown deterministic coefficients and the ψ α 's are multivariate polynomials which are orthonormal with respect to the joint PDF f X of the input random vector X, i.e. E [ψ α (X)ψ β (X)] = 1 if α = β and 0 otherwise. For instance, if X is a standard normal random vector, the ψ α are normalized multivariate Hermite polynomials.

Estimation of the polynomial chaos coefficients

The PC coefficients can be estimated using a non intrusive regression scheme [START_REF] Berveiller | Stochastic finite elements: a non intrusive approach by regression[END_REF]Sudret, 2007). This method requires the choice of a truncation of the PC i.e. a non empty finite set A = {α 0 , . . . , α P -1 } ⊂ N M which contains the multi-indices of the retained basis polynomials ψ α 0 , . . . , ψ α P -1 . The corresponding PC approximation is denoted by

Y A ≡ M A (X) = α∈A a α ψ α (X)
which rewrites Y A = a T ψ(X), by introducing the vector notation:

a = {a α 0 , . . . , a α P -1 } T (2) ψ(X) = {ψ α 0 (X), . . . , ψ α P -1 (X)} T (3)
Let us consider a set of realizations of X denoted by X = {x (1) , . . . , x (N ) } and referred to as the experimental design. Let us denote by Y the associated set of model response, say Y = {M(x (1) ), . . . , M(x (N ) )}. The unknown coefficients a may be computed by performing a leastsquare minimization [START_REF] Berveiller | Stochastic finite elements: a non intrusive approach by regression[END_REF], i.e. by minimizing the mean-square truncation error

1/N N i=1 M(x (i) ) -M A (x (i) ) 2 .
Using the above notation the solution reads:

â = (Ψ T Ψ) -1 Ψ T Y (4)
where Ψ is a N × P matrix such that Ψ ij = ψ α j (x (i) ), i = 1, . . . , N, j = 0, . . . , P -1. The size N of the ED must be greater than P to make this problem well posed.

Post-processing of the PC coefficients

Once the PC coefficients have been computed, it is straightforward to compute quantities of interest such as the probability density function and the statistical moments of the response quantity Y . Indeed, due to the orthonormality of the PC basis, the mean value of the model response is the first coefficient of the expansion, while the variance is obtained by the sum of the squares of the coefficients :

E [Y ] = â0 Var [Y ] = α∈A\{0} â2 α (5)
In order to obtain a graphical representation of the response PDF, the series expansion may be simulated using Monte Carlo simulation. This yields a sample set of response quantities, say {y (i) , i = 1, . . . , n}. From this set, an histogram may be built. Smoother representations may be obtained using kernel smoothing techniques, see e.g. Wand and Jones (1995):

fY (y) = 1 n h K n i=1 K y -y (i) h K (6)
In this expression, K(x) is a suitable positive function called kernel, and h K is the bandwith parameter.

Well-known kernels are the Gaussian kernel (which is the standard normal PDF) and the Epanechnikov kernel

K E (x) = 3 4 (1 -x 2 ) 1 |x|≤1 .
The bandwith parameter is selected according to the choice of the kernel and the sample size n. In the present case, the MC simulation of the PC expansion (which is analytical and polynomial) is rather inexpensive and usually negligible with respect to the model runs that were previously required to obtain the PC coefficients. Thus a large sample may be used for the kernel approximation, e.g. n = 1, 000, 000. In case of such large samples, the obtained kernel density is independent of the choice of the kernel function.

3 GLOBAL SENSITIVITY ANALYSIS Global sensitivity analysis aims at quantifying which input parameters (or group of parameters) have a great influence on the response variability. Among various methods (see a review in Saltelli et al. (1999), variance decomposition (also known as ANOVA) methods have shown a great accuracy. The idea behind is to decompose the variance of the model response Var [M(X)] into contributions of each input parameter or combinations thereof. The Sobol' decomposition (Sobol', 1993) allows one to formulate the problem. Suppose the input parameters are defined on the M -dimensional unit cube [0, 1] M after some suitable transformation. The Sobol' decomposition of M(x) into summands of increasing dimension reads:

M(x) = M 0 + M i=1 M i (x i )+ 1≤i<j≤M M ij (x i , x j ) + • • • + M 12...M (x) (7)
where M 0 is a constant and where it is imposed that the integral of each summand M i 1 ...is (x i 1 , . . . , x is ) over any of its arguments is zero, i.e. :

1 0 M i 1 ...is (x i 1 , . . . , x is ) dx i k = 0 for 1 ≤ k ≤ s (8)
The classical properties of this decomposition are the following. The constant M 0 is the mean value of the function: M 0 = [0,1] M M(x) dx where dx stands for dx 1 . . . dx M for the sake of simplicity. Due to Eq.( 8), the summands are orthogonal to each other in the following sense:

[0,1] M M i 1 ...is (x i 1 , . . . , x is )M j 1 ...jt (x j 1 , . . . , x jt ) dx = 0 for {i 1 , . . . , i s } = {j 1 , . . . , j t } (9)
With the above assumptions, the decomposition in Eq.( 7) is unique.

Consider now that the input parameters are independent random variables uniformly distributed over [0, 1]:

X = {X 1 , . . . , X M } T , X i ∼ U(0, 1) (10)
As a consequence, the model response Y = M(X) is a random variable, whose variance D (also called total variance in the literature on global sensitivity) reads:

D = Var [M(X)] = [0,1] M M 2 (x) dx -M 2 0 (11)
By integrating the square of Eq.( 7) and by using ( 9), it is possible to decompose the total variance (11) as follows:

D = M i=1 D i + 1≤i<j≤M D ij + • • • + D 12...M (12)
where the partial variances appearing in the above expansion read:

D i 1 ...is = [0,1] s M 2 i 1 ...is (x i 1 , . . . , x is ) dx i 1 . . . dx is (13)
The Sobol' indices are defined as follows :

S i 1 ...is = D i 1 ...is /D (14)
Each index S i 1 ...is is a sensitivity measure describing which amount of the total variance is due to the uncertainties in the set of input parameters {i 1 , . . . , i s }. The first order indices S i give the influence of each parameter taken alone whereas the higher order indices account for possible mixed influence of various parameters.

PC EXPANSIONS AND NESTED MODELS

In many engineering problems, complex systems are modelled by a series of sub-models, which are linked to one another: the response quantities of some models are input to other submodels, thus the term "nested models" (see Figure 1).

Definition of nested models

Let us consider a complex model defined as a series of submodels {M q , q = 1, . . . , Q}. Each submodel has an input vector which can be split into two parts:

• a subvector denoted by X q , which corresponds to basic random variables for which a probabilistic model is available, say a joint probability density function f Xq (x q ).

• a subvector denoted by Z q , which corresponds to intermediate random variables, i.e. variables that are response quantities of other submodel(s).

Let us denote by Y q the response random vector of the q-th submodel. The following notation is used:

Y q = M q (X q , Z q ) (15)
Note that the basic input random vectors {X q , X r , q = r} are not necessarily independent (i.e. thay may have some common components). Similarly, random vectors X q , Z r may also be dependent, e.g. when Z r is obtained as the response of a model M r which also depends on X q or a subvector of it.

4.2 PC expansion of the first-level model response A so-called first-level submodel is a model M q (X q ) which only depends on a basic random vector X q with prescribed joint PDF f Xq (x q ). The input random vector is first approximated using the so-called Nataf transform [START_REF] Nataf | Détermination des distributions dont les marges sont données[END_REF] whose definition is recalled in the next paragraph. The Nataf transform denoted by X q = T (ξ) allows one to represent the input random vector in terms of independent standard normal variables. The transform is in general an approximation based on the margins and the linear correlation between the components. It is however exact when a) the components are independent, b) X q is a Gaussian vector or c) X q is defined by the set of margins and a Gaussian copula density (Nelsen (1999)), see [START_REF] Lebrun | An innovating analysis of the Nataf transformation from the copula viewpoint[END_REF] for details).

In all cases, it is the model function M q composed with the Nataf transform T that is represented by its PC expansion, namely:

Y q ≡ M q (T (ξ)) ≈ α∈A a q α ψ α (ξ) (16)

Nataf transform

The Nataf transform [START_REF] Nataf | Détermination des distributions dont les marges sont données[END_REF] has been introduced in structural reliability analysis in order to transform a random vector with prescribed marginals and correlation coefficients into a standard normal random vector, which is required e.g. in FORM analysis (Der [START_REF] Der Kiureghian | Structural reliability under incomplete probability information[END_REF][START_REF] Ditlevsen | Structural reliability methods[END_REF]. The basic equations are now recalled.

• Suppose a random vector X has prescribed marginal distributions, say F X i (x i ), i = 1, . . . , M and correlation matrix ρ. Each component is first transformed into a standard normal variable ξ i , i = 1, . . . , M :

ξ i = Φ -1 (F X i (x i )) (17)
• Then it is assumed that Ξ = {ξ 1 , . . . , ξ M } T forms a standard normal correlated random vector whose joint PDF reads:

f Ξ (ξ) = ϕ M (ξ; ρ) = (2 π) -M/2 exp - 1 2 ξ T • ρ-1 • ξ (18)
The joint PDF of X is thus approximated by the Nataf distribution:

f X (x) = f X 1 (x 1 ) . . . f X M (x M ) ϕ(ξ 1 ) . . . ϕ(ξ M ) ϕ M (ξ, ρ) (19)
The so-called pseudo-correlation matrix ρ is computed componentwise by the consistency equation, which ensures that the correlation coefficient between any two components {X i , X j } of X, as computed from ( 19) is the input correlation coefficient ρ ij :

ρ ij = R 2 x i -µ X i σ X i x j -µ X j σ X j ϕ 2 (ξ i , ξ j , ρij ) dξ i dξ j (20)
Due to the burden of such a computation in the general case, values of ρij /ρ ij have been tabulated for various couples of distributions (f X i (x i ), f X j (x j )) (Der [START_REF] Der Kiureghian | Structural reliability under incomplete probability information[END_REF].

PC expansion of the next-level model response

Let us now consider a so-called intermediate submodel, i.e. a submodel M q (X q , Z q ) which depends on the response of other submodels through the intermediate random vector Z q .

As mentioned above, the joint PDF of Z q is not prescribed by the analyst, as it results from the propagation of some basic random vectors through other submodels. In the current framework, each intermediate random vector is thus represented as a set of polynomial chaos coefficients in a suitable PC basis.

In order to be able to cast the intermediate variables in a suitable way with respect to polynomial chaos expansions, the following strategy is adopted:

• the intermediate vectors, whose PC expansion is known, is sampled using crude Monte Carlo simulation or Latin Hypercube Sampling. Let us denote by Z ≡ {z (i) , i = 1, . . . , n} the sample set and {z (i) j , i = 1, . . . , n} the sample set of the j-th component of Z q . The size of Z may be large (e.g. 10 5 -10 6 ) since the function to be simulated is analytical.

• For each component of Z q , say Z q,j , the empirical PDF and CDF is estimated by kernel smoothing techniques (Wand and Jones, 1995) from the corresponding sample set. For instance, using a Gaussian kernel function ϕ(.), the marginal PDF of Z q,j may be estimated by:

fZ q,j (y) = 1 n h K n i=1 ϕ y -z (i) j h K (21)
and the related CDF reads:

FZ q,j (y) = 1 n n i=1 Φ y -z (i) j h K (22)
where Φ(.) is the standard normal CDF.

• The Pearson correlation matrix of the sample set Z is computed, say R q :

R q k,l = 1 n σk σl n i=1 (z (i) k -zk ) (z (i) l -zl ) (23)
where {z k , zl } and {σ k , σl } are the empirical mean values and standard deviations of the samples of Z q,k and Z q,l .

• The Nataf transform of vector Z q into a standard normal random vector is computed numerically using the marginal CDF obtained by kernel smoothing ( 22) and the correlation matrix R q (23). The pseudo-correlation matrix is computed numerically by solving Eq.( 20).

5 APPLICATION EXAMPLE : HOMOGENIZA-TION OF CONCRETE PROPERTIES In this section we aim at illustrating the above methodology on the modelling of the hydration of cement-based materials. From a deterministic point of view the hydration phenomena is modelled with a multiscale micromechanics-based approach develop by [START_REF] Bernard | A multiscale micromechanics-hydration model for the early-age elastic properties of cement-based materials[END_REF]. The aim here is not to analyze the quantitative results on the example but to demonstrate a possible application of sensitivity analysis for nested models.

Micromechanics-hydration model

The modelling of the microstructure of concrete is broken up into four elementary levels (Figure 2). Two homogenization schemes are successively used: the self-consistent scheme for level II and the Mori-Tanaka scheme [START_REF] Mori | Average stress in matrix and average elastic energy of materials with misfitting inclusion[END_REF] for all the other levels.

Level 0 The C-S-H matrix forms at early ages by hydration of C 3 S and C 2 S. The characteristic length of this level varies between 10 -8 and 10 -6 m. At this level, at least two types of C-S-H exist with different mechanical properties and volume fractions. Level I The characteristic length of this level varies between 10 -6 and 10 -4 m. At this level, cement paste is composed from the C-S-H matrix with unhydrated cement products, large portlandite crystals, aluminates and macro-porosity. Once the C-S-H phase percolates, the composite cement paste consists in inclusions embedded into a C-S-H matrix.

Model 2: Self-consistent homogenization scheme The input parameters of this second model are: Young's modulus and Poisson's ratio of hydrates (aluminates and CH) and their respective volume fraction, the volume fraction of water and void and the homogenized Young's modulus and Poisson's ratio resulting from Model #1 (E I , ν I ). The output of this model are the homogenized Young's modulus and Poisson's ratio of the cement paste (E II , ν II ).

Level III In this level, mortar is considered. It is made of a cement paste matrix, sand particle inclusions and an interfacial transition zone (ITZ). This ITZ is not taken into account in this study due to lack of data on its characteristics. The characteristic length of this level varies between 10 -3 and 10 -2 m.

Model 3: Mori-Tanaka homogenization scheme

The input parameters of the third model are the Young's modulus and Poisson's ratio of sand and its volume fraction, and the homogenized Young's modulus and Poisson's ratio of the cement paste (E II , ν II ). The output of this model are the homogenized Young's modulus and Poisson's ratio of the mortar (E III , ν III ).

Level IV In this level, concrete, which is made of a mortar matrix, granular inclusions and an interfacial transition zone (ITZ), is considered. This ITZ is not take into account in this study due to lack of data on its characteristics. The characteristic length of this level varies between 10 -2 and 10 -1 m.

Model 4: Mori-Tanaka homogenization scheme

The input parameters of the fourth model are the Young's modulus and Poisson's ratio of agregates and their volume fraction, and the homogenized Young's modulus and Poisson's ratio of the mortar (E III , ν III ). The output of this model are the homogenized Young's modulus and Poisson's ratio of concrete (E IV , ν IV ).

Probabilistic model

The probabilistic model is summarized in Table 3. Four independent lognormal random variables are introduced. The parameters of the distribution have been selected based on the values of ranges of variation given in [START_REF] Bernard | A multiscale micromechanics-hydration model for the early-age elastic properties of cement-based materials[END_REF]. This probabilistic model may be certainly improved by discussions with concrete experts. Nevertheless it is sufficiently accurate to provide a significant application example.

Uncertainty results

Two types of information are obtained from this simulation, namely the statistical moments of intermediate and final output parameters, and global sensitivity results through Sobol' indices.

The statistical moments of the various intermediate quantities (Young's moduli and Poisson's ratios of C-S-H mix, cement paste and mortar) and the last-level output (Young's modulus and Poisson's ratio of concrete) are given in Table 2. In the first two columns, the moments are obtained from the "nested" approach, in which each submodel is expanded as a PC series and the coefficients are post-processed according to Eq.( 5). In the last two columns ("single model"), the final output is considered as a unique function of all the basic input parameters, and the associated PC expansion is derived. This calculation shows that the obtained mean and standard deviation are equal to those obtained through the nested approach: from this point of view, the nested use of PC expansions does not influence the quality of the lastlevel results. The robustness of the nested approach is thus validated by this comparison.

The Sobol' indices of the various intermediate and last-level output quantities with respect to their respective input parameters are gathered in Table 3. The very end product of the proposed methodology appears in this table: indeed, it is possible to calculate sensitivity indices of intermediate response quantities vs. intermediate variables, which could not be achieved by considering a single model.

The aim of this short example is not to give a detailed interpretation of the results in terms of concrete formulations. Nevertheless, one can notice that parameter E b (high density C-S-H Young's modulus) has a strong influence on the final result. It may be confirmed with a deeper analysis including a more relevant probabilistic model.

CONCLUSION

The paper presents a framework for sensitivity analysis of complex models made of nested submodels. The proposed methodology is based on polynomial chaos expansions of the various submodels. The important step is the way intermediate random variables, which are output variables of some given submodels, may be transformed into standard variables in order to expand the next-level model response quantities. The Nataf transform is used for this purpose.

It is believed that the approximation introduced is of second order in the context of sensitivity analysis. However, this point should be further investigated. It may be possible to use full generalized chaos expan-sions that are consistent with a completely general joint PDF of input parameters. However, this would be far more complex since the basis of expansion is no more polynomial.(Soize and Ghanem, 2004;Sudret, 2007).

Furthermore, the interpretation of the Sobol' indices computed by this approach may be also carried out carefully: for the intermediate models, the input variables are possibly correlated, and then transform through the Nataf approximation into standard normal variables. Rigorously speaking, the indices computed as the post-processing of the chaos coefficients refer to those standard normal variables and not directly to the underlying physical variables. A rigourous interpretation would require the study of sensitivity indices with respect to correlated variables, which is somehow an open question, although some approaches have been recently proposed (Xu and Gertner, 2008).

Model 1 :

 1 Mori-Tanaka homogenization scheme The input parameters of this model are the Young's modulus and Poisson's ratio of high density and low density C-S-H (E a , E b , ν a , ν b ) and the high density C-S-H volume fraction denoted by f b . The output of this model are the homogenized Young's modulus and Poisson's ratio of the C-S-H mix (E I , ν I ).

Figure 2 :

 2 Figure 2: Multiscale microstructure of cement-based material[START_REF] Bernard | A multiscale micromechanics-hydration model for the early-age elastic properties of cement-based materials[END_REF]

Table 1 :

 1 Probabilistic description of the input parameters

	Material properties	Probabilistic model

Max Type of PDF Mean St. Dev. 0 Low density C-S-H Young's Modulus E a (GPa) 21.7 19.I (from level-I homogenization) Aluminate hydrate Poisson's ratio ν I (from level-I homogenization) III (from level-III homogenization) Mortar Poisson's ratio ν III (from level-III homogenization)

Table 2 :

 2 Mean value and standard deviation of the elastic properties resulting from each step of homogenization: single model vs. nested models Nested models Single model Mean Std. dev. Mean Std. dev.

	E I 35.517 1.037
	ν I	0.111	0.011
	E II 26.548 0.657
	ν II	0.169	0.006
	E III 33.975 0.474
	ν III 0.111	0.007
	E IV 34.947 0.152 34.949 0.152
	ν IV	0.132	0.005	0.132	0.005
	Table 3: Sobol' indices of the various levels
		Level Index definition Value
		I	E I /E a	0.02
			E I /E b	0.98
			ν I /E a	0.43
			ν I /E b	0.57
		II E II /E I	0.97
			E II /ν I	0.00
			E II /E 1	0.03
			E II /ν 1	0.00
			ν II /E I	0.06
			ν II /ν I	0.26
			ν II /E 1	0.00
			ν II /ν 1	0.68
		III E III /E II	1.00
			E III /ν II	0.00
			ν III /E II	0.85
			ν III /ν II	0.15
		IV E IV /E III	0.54
			E IV /ν III	0.46
			ν IV /E III	0.88
			ν IV /ν III	0.12

Nelsen, R. (1999). An introduction to copulas, Volume 139 of Lecture Notes in Statistics. Springer. Saltelli, A., S. Tarantola, and K. Chan (1999)


Petro Kolosov

Polynomial identities involving Pascal's triangle rows

Keywords: 2010 Mathematics Subject Classification. 26E70, 05A30 Binomial coefficients, Binomial theorem, Pascal's triangle, Binomial sums, Binomial distribution, Binomial identities

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

POLYNOMIAL IDENTITIES INVOLVING PASCAL'S TRIANGLE ROWS

PETRO KOLOSOV

Abstract. In this short report we consider the famous binomial identity

2 n = n k=0 n k
Based on it, the following binomial identities are derived

m n = n k=0 k j=0 n k k j (-1) k-j m j , m n = n k=0 k j=0 n j n -j k -j (-1) k-j m j ,
where n k are binomial coefficients and (m, n) are non-negative integers.

Introduction

We start from the famous relation about row sums of the Pascal triangle, that is

2 n = n k=0 n k , (1.1)
where n k = n! k!(n-k)! are binomial coefficients [START_REF] Ronald L Graham | Concrete mathematics: a foundation for computer science[END_REF]. Identity (1.1) is straightforward because the Pascal's triangle is n/k 0 1 2 3 4 5 6 7 8 0 1 1 1 1 2 1 2 1 3 1 3 3 1 4 1 4 6 4 1 5 1 5 10 10 5

Now we can notice that

3 n = n k=0 n k • 2 k (1.2)
Continue similarly we can generalize the equations (1.1), (1.2) as follows

2 n = n k=0 n k • 1 k 3 n = n k=0 n k • 2 k 4 n = n k=0 n k • 3 k . . . m n = n k=0 n k • (m -1) k Obviously, it
m n = n k=0 k j=0 n k k j (-1) k-j m j (1.3)
where (m, n) are non-negative integers.

Proof. Recall the induction over m, let be a base case m = 2, hereby

2 n = n k=0 n k (2 -1) k (1.4)
Reviewing an equation (1.4) we can see that

(2 + 1 m=3 ) n = n k=0 n k • ((2 -1) + 1 m-1 ) k (1.5) Continue similarly it is straightforward that m n = n k=0 n k • (m -1) k .
However, we are able to expand the part (m -1) k by means of Binomial theorem [START_REF]Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF], that is

(m -1) k = k j=0 k j (-1) k-j m j = k j=0 k j (-1) k m k-j
So that now we are able to merge both results

m n = n k=0 n k • (m -1) k and (m -1) k = k j=0 k j (-1) k-j m j = k j=0 k j (-1) k m k-j to receive m n = n k=0 n k • (m -1) k = n k=0 n k k j=0 k j (-1) k-j m j = n k=0 k j=0 n k k j (-1) k-j m j = n k=0 k j=0 n k k j (-1) k m k-j
Theorem (1.1) may be verified using Mathematica command PolynomialIdentity[m, n] at [START_REF] Kolosov | Polynomial identities involving Pascal's triangle rows[END_REF]. This completes the proof. □ Moreover, by means of the binomial identity [[Gro16], Chapter 4]

n k k j = n j n -j k -j
The polynomial m n is identical to

m n = n k=0 k j=0 n j n -j k -j (-1) k-j m j = n k=0 k j=0 n j n -j k -j (-1) k m k-j

Conclusions

The following binomial identities are derived

m n = n k=0 k j=0 n k k j (-1) k-j m j = n k=0 k j=0 n k k j (-1) k m k-j m n = n k=0 k j=0 n j n -j k -j (-1) k-j m j = n k=0 k j=0 n j n -j k -j (-1) k m k-j
Moreover, above results are verified by means of specified Mathematica scripts available at github.com/kolosovpetro/PolynomialIdentitiesInvolvingPascalsTriangleRows.

Verification of the results

Main results of this paper may be verified using Mathematica scripts from [START_REF] Kolosov | Polynomial identities involving Pascal's triangle rows[END_REF] as follows

  is simply a form of the Binomial theorem (m + 1) n = n k=0 n k m k . Therefore, we conclude this version of the Binomial theorem as Theorem 1.1. (Binomial theorem.) The following identity involving polynomial m n holds

  k -j (-1) k m k-j

Table 2 .

 2 Triangle generated by the function n k • 2 k . Can be reproduced using Mathematica function GeneratePascalLikeTriangle[2, 8] at [Kol22]. Sequence A013609 in OEIS [Slo64].

					1
	6 1 6 15 20 15 6 1
	7 1 7 21 35 35 21 7 1
	8 1 8 28 56 70 56 21 8 1
	Table 1. Pascal's triangle [CG96]. Each k-th term of n-th row is n k •1 k . Sequence
	A007318 in OEIS [Slo64].
	Consider a generating function such as f 2 (n, k) = n k • 2 k . The function f 2 (n, k) generates
	the following Pascal-like triangle
	n/k 0 1	2	3	4	5	6	7	8
	0 1
	1 1 2
	2 1 4	4
	3 1 6 12	8
	4 1 8 24 32	16
	5 1 10 40 80	80	32
	6 1 12 60 160 240 192	64
	7 1 14 84 280 560 672 448 128
	8 1 16 112 448 1120 1792 1792 1024 256


EHESS Pascal Engel

THE DOXASTIC ZOO

Keywords: Belief, doxastic attitudes, propositional attitudes, norms, judgment

The doxastic zoo contains many animals : belief, acceptance, belief in, belief that, certainty, conjecture, guess, conviction, denial, disbelief in, disbelief that, judgment, commitment, etc. It also contains belief's "strange bedfellows": credences, partial beliefs, tacit beliefs, subdoxastic states, creedal feelings, feelings of knowing, in-beween believings, pathological beliefs, phobias, aliefs , delusions, biases, besires. How to order the zoo? I propose to distinguish doxastic attitudes from non-doxastic epistemic attitudes. The criterion is the existence of correctness conditions. Most bedfellows do not have such normative conditions. So they should not be allowed in the Doxastic Club.

Introduction : the menagerie of belief

The task of classifying the species of belief seems to elude us. The very idea of asking what one believes at a given time, or how many beliefs a person has, or even whether someone believes rather than is in another state with respect to a given content, seems not to make real sense. Nevertheless we ascribe beliefs to ourselves and to others, we talk of acquiring or losing them, and we give them a number of properties, such as being silly or wise, rational or not, justified or not. Some philosophers tell us that belief is such simple and unanalysable notion that it cannot be defined. 2 Others, like Wittgenstein, suggest that it is so diverse that the best that we can say is that it consists in a motley of psychological states, 1 Versions of this article were presented in 2012 at the SIFA conference in Alghero, and in the Brasilian conference on analytic philosophy in Fortaleza. I thank my friends and participants for their invitations and comments. The nature of belief and judgment are topics on which Eva Picardi wrote in various ways. I learnt a lot from her. It has been a great luck for me to come to know her.. 2 Hume, Inquiry (II) Mill (Mill 1843, book I, ch. v, § 1, :88), Brentano (1874, I, 64) and Stout (1896: 99) take belief and judgment to be primitive and undefinable. Cf [START_REF] Van De Schaar | George Stout and the Origins of analytic Philosophy[END_REF] which bear with each other only family resemblances. But, as J.N. Findlay (1962: 94) remarked, this kind of answer is no answer, since it attempts to dispel our sense of unity inadequately brought out in a range of cases, without showing this sense to be ill founded."Belief" may not be defined, but it indicates a place in the map of the mind, and we can at least indicate the role of this state among other states. I cannot here give a full picture of the family tree of belief, but at least hope to give a profile of this animal within the doxastic zoo.

Part of the difficulty comes from the fact that belief does not stand on its own among other mental states. Many psychological attitudes are related to belief, in the sense that they are often species of belief, or involve the possession of belief. In the most general sense, belief involves assent, or holding-true a proposition. Beliefs vary depending on the nature of the expression or vehicle. Believing may be a mere disposition to act, with no conscious or verbal expression. It may be merely tacit. It may be an inner conscious assent in the mind without verbal expression. It may be expressed in an act of linguistic assertion. It may be occurrent or long-standing. It may vary in degrees, ranging from merely subjective opinion, as a conjecture or hypothesis, a mere guessing, or a firm conviction. These degrees may depend on the weight of the evidence that subjects claim to have, in which case the beliefs are held with more or less confidence or certainty. But can these degrees of confidence be quantified? Are there degrees of beliefs and credences? Beliefs can also be associated to voluntary acts, as in judgments, decisions as acceptances, commitments, takings for granted, postulations and presupposition, anticipations or presumptions. They can be more or less firm, such as doubts, conjectures and convictions. There are attitudes related to beliefs or based on beliefs, although distinct from them, such as noticings, perceivings, intentions, regrets, hopes, expectations, or indeed judging. All of these attitudes may be related to different epistemic sources and variously to knowledge, to different emotions, they can be more or less firm, more or less long-lived. Some may be so deeply entrenched, such as "hinge propositions" that many philosophers do not take them as beliefs. All of these kinds of belief-like attitude I shall call beliefs' cousins. In what sense are they beliefs, and if they are, how do they relate to this genus? What is the shape of the family, and how remote are the cousins? Such questions have resurfaced very often in the history of philosophy, but they have become even more complex when contemporary epistemology and psychology have investigated various kinds of mental states, which one might call belief's "strange bedfellows" [START_REF] Bayne | Belief's Strange Bedfellows[END_REF]. First, there are the "tacit beliefs" [START_REF] Lycan | Tacit Belief[END_REF], which that one has not never thought about but that one is disposed to believe when told, such as "Kant's earlobe is saller than the Sea of Tranquility".

There are "subdoxastic states" (Stich 1986) , which are informations, and perhaps representations, processed in our perceptual or memory systems when we perform various cognitive tasks (such as sketches in visual processing, or "mental models" in reasoning) but which are not properly believed. There are delusions, or pathological beliefs, such as Capgras'syndrom, where people "believe" that their friends or family have been replaced by impostors, or Cotard's delusion, where people "believe" that they are dead [START_REF] Davies | Pathologies of Belief[END_REF]Stone 1992, Bayne andPacherie 2006). There are the "feelings of knowing" [START_REF] Koriat | Knowing and the feeling of knowing[END_REF], feelings of familiarity, of "déjà-vu" or of strangeness. Finally, there are the so-called aliefs [START_REF] Gendler | Alief and Belief[END_REF], which include phobias, and various emotion-induced feelings and representations, and biases which are supposed to influence our attitudes and our reasoning about various subject matters (Brownstein and Saul 2015) . In each of these cases, there is a debate as to whether such states are beliefs at all. These issues loom large in contemporary philosophy of mind and epistemology, and I cannot hope to solve these problems case by case, or to go into the widely discussed topics of the relations between belief an emotion, belief and inference, belief and mental representation, belief and the will, which stand behind these issues5 . Nor do I hope to produce a full taxonomy of belief-attitudes or of the doxastic realm. My main aim here is to discuss the methodology of the philosophy of belief, one of the main problems of which is to establish a classification of beliefs and other related attitudes. I shall try to give a criterion to individuate beliefs and locate them with respect to other epistemic attitudes.

What's in a belief, that ink may character?

Let us try to give the usual profile of belief. This should not be a definition, but a set of characteristics which are typically ascribed to beliefs. These are the following:

(i) Representational content: Beliefs are attitudes with a (linguistic or not) content (ii) Direction of fit: beliefs "aim at truth" and are supposed to be adapted to the world (ii) Causal or motivational profile: beliefs are dispositions to act as if their contents were true (iv) Reason and evidence: beliefs are supposed to have reasons, which consist in evidence (v) Involuntariness: belief are not (normally) under the control of the will (vi) Degree: beliefs have degrees, in being more or less certain (vi) Inferential integration: beliefs are supposed to be inferentially coherent (vi) Immediate access: most beliefs can be in principle consciously accessed (vii) Context independence: believing is not relative to context (vii) Emotional impact: beliefs have at least some causal relations with emotions (viii) Phenomenology: some beliefs at least have a certain phenomenology.

These traits have to be spelled out through various theories of belief, which conflict on crucial points. Thus dispositional of functionalist theories emphasise the dispositional and causal features of beliefs, which entail that they are not necessarily conscious and that they are necessarily action-oriented, whereas "Cartesian" theories stress this very feature and say that the practical properties of belief are not primary. Some philosophers deny that beliefs have degrees, while others take the latter to be degrees of subjective probability. Some take belief contents to be semantic, meaning that they are necessarily linguistic. Some hold that beliefs can be voluntary. Not all theorists agree that beliefs have a specific connection to emotions and have a phenomenology. In spite of these differences, everyone agrees that a least some beliefs possess some of these features. Moreover these features apply mostly to rational beliefs, those which are supposed to be true, consciously accessible, held for reasons, and inferentially coherent, whereas the strange bedfellows are most of the time irrational beliefs, if they are beliefs at all. Some, perhaps most, beliefs arise from causes which have nothing to do with the evidence or reasons that one has for them. Some people are prone to wishful thinking and do not care for having true beliefs. Most religious beliefs are not held for reasons having to do with evidence or coherence. This raises the question whether these eight features can really characterize beliefs in general. As soon as one raises such questions, one is tempted to locate the characteristics under another heading: acceptances, commitments, credences, etc., which in turn differ from credences and dispositional beliefs. Are these modes of belief or distinctive kinds of doxastic states? A mere taxonomy, however, cannot solve a philosophical problem.

Our problem is to individuate beliefs and to locate them within the broader territory of propositional attitudes, and most specifically of epistemic attitudes. The latter are attitudes aimed at truth and that depend upon evidence. In this respect, not only belief, but also perception, judgment, inference, supposition, some cases of imagination, and knowledge itself can count as epistemic attitudes. But there is a problem about the frontiers of the epistemic domain . Not only is belief sometimes hard to distinguish from other attitudes, but it is also central to other psychological states, which often presuppose the possession of beliefs.

Perceiving as an experience is not believing, but it often depends on beliefs and leads to beliefs. To be surprised is to discover that P when one believed that not P. Imagining is not believing but is hard to understand without the existence of belief. Can one have desires without having beliefs about what one desires? Most attitudes and emotions presuppose belief: if one regrets something, one believes that it is the case and desires that it had not been the case, , if one is angry at someone, one believes that someone did something offensive, , etc. Belief is as much characterized by its causal and logical links with other attitudes as it is characterisable as a single and isolated state.

So why not characterize belief in functional terms instead of trying to define belief, by describing it as the state which is the conjunction of the platitudes (i)-(viii), conjoined in a Ramsey sentence? It would look like this:

( S) such that ( X) ( p) (i) X is
in some state which represents p (ii) If X believes that p x is disposed to act as if p (iii) X believes that p on the basis of evidence (iv) If X believes that p, p is not under the control of X's will (v) if X believes that p and q can be inferred from p X is disposed to believe that q (vi) If x believes that p, x is disposed to be in some emotional state (vii) X believes that (s)he believes p (viii) If X believes that p, there is a certain phenomenology.

These are the most common features of belief, and its cousins and bedfellows instantiate them in most, but not all cases. The minimal core of beliefhood would be the representational property (i) and causal functional property (ii): belief is a kind of contentful state which has certain causes and certain effects. This very wide characterization would certainly allow us to include among beliefs the so-called "aliefs", such as the kind of vertigo which one often experiences in elevators with glass floors, epistemic feelings such as déjà vu, as well as attitudes like acceptance, where one can take a proposition for granted without believing it.

Such states share some of the characteristics of the general profile (i)-(viii), but not all: they involve representations, have a causal profile, a phenomenology and an emotional tone, but they are not in general associated with a reflective mode or with a rational and evidential role.

But this functionalist strategy yields too loose a categorization. Virtually every state which involves a belief, either as a cause or as a consequence, becomes a belief. Thus attitudes such as hope, regret and a number of emotions also involve the minimal core of representation cum causal profile, but it would be wrong to include these within beliefs. If, however, we cut our doxastic cake too narrowly and exclude from the doxastic club all but the states which conform to the (i)-(viii) features, we risk chauvinism. Moreover, we should not multiply kinds of beliefs beyond necessity. Consider cases of negligence or forgetfulness. I take my broken watch for repair at the watchmaker. A few moments after leaving the shop, I want to know what time it is, and consult my empty wrist: I forgot that I had left my watch for repair. Do I believe that my watch is on my wrist or not? [START_REF] Maloney | It's Hard to believe[END_REF] suggests that there are two kinds of beliefs, those which are dispositional and action-oriented and not necessarily conscious ("A-Beliefs") and those which are consciously entertained or accessible ("C-Beliefs"). A very simple-minded reaction would be to say that I believed that my watch was on repair, but forgot it, hence temporarily ceased to believe it, or revised my belief. Is it necessary here to postulate two kinds of beliefs? It is always costly to posit too many entities, but such examples are enough to show that among the features (i)-(viii) one can privilege the dispositional and motivational ones, or those which stress the conscious, linguistically manifestable aspects. The philosophy of belief in this sense is not exceptional in leading to an opposition between the functionalist third-person perspective on the mind and the first-person "Cartesian" perspective. If we privilege the former, we take belief to be mostly a dispositional/ causal profile, and take accessible beliefs to be merely epiphenomenal. Thus we are able to include many states which, like aliefs, delusions and epistemic feelings, are unconscious or semi-conscious, but we run the risk of excluding those beliefs which are the products of our commitments and of our "judgment sensitive attitudes" (Scanlon 1998) 8 .

Alternatively, we could envisage a third category: Dennett (1978) talks of "opinions", Sperber (1985) talks of "quasi-beliefs", and Schwitzgebel (2001) talks of "in-between believings".

These might be located in between the conscious and dispositional ones. Dennett has a distinction between beliefs as well entrenched dispositions, and "opinions" which are merely verbal and short-lived assents to a sentence. Of many of the contents which float in our minds we are unsure whether they are beliefs or not, in the sense of being apt to be expressed as assertions. As Bernard Williams says: "It is far from being true that every thought swimming around in one's mind is already the content of a belief as opposed to some other mental state such as a guess, a fancy, or (very importantly) a wish. […] in many …cases, it is not merely that we do not know what we believe (though this is of course often true), but that a given content has not come to be a belief at all." (Williams 2002: 82) So the category of "in between believings" is very large indeed. It contains all our shortlived assents, our tacit beliefs and it can include aliefs and other epistemic feelings or emotions. The problem with the strategy of dubbing these beliefs "in-between" is not that such phenomena aren't real, but that it is difficult to suppose that there is a specific category of intermediate believings. Aren't in one sense all believings "in between" believings? But if we make this move, we simply rehearse the functionalist strategy: for the intermediary beliefs are those which possess all the features of belief which are action-oriented. How to draw the line? Compare with colours (Zimmermann 2008). We can improve our colour scheme by locating an intermediary colour, say purple, between blue and indigo. But we cannot improve the scheme by adding a colour dubbed "in between blue and red". If we apply this strategy to belief we are led to ask whether the state of mind a person is in is a belief. We could improve the scheme by showing that there are genuine psychological states, presumably physical, which underpin the so-called "in-between believings". The notion of "in-between believing" at best deflates our attempt to taxonomise the varieties of beliefs, and at worst has eliminativist consequences. For it acknowledges that most of what we call "belief" does not fall within one category or another, and is at best a mere instrument of prediction at the service of an "intentional stance". The only way in which we could remove this fuzziness is by locating the neurophysical states responsible for the entities of our folk psychology. We would encounter the same sort of problems if we used a distinct scheme, such as the one inspired by the famous "two minds" or " two systems" view of cognition advocated (Kahneman 2010) Should we say that there are two kinds of beliefs, those which are the products of System 1 and those which are the products of system 2? The same problem would arise if we managed to locate beliefs within three familiar divisions of the brain: procedural with the cortex cerebellum and striatum, declarative with the hippocampus, emotional with the amygdala. Should we say that dispositional beliefs are in the first, conscious and asserted beliefs in the second, and that all the strange bedfellows, such as aliefs and feelings, are in the third? Or should we say that there is only one kind, belief, which spreads over the three brain systems? Such divisions do not help.

We find similar difficulties when trying to add a new kind of animal to our doxastic zoo, credences. Credences are not beliefs, in the sense of full beliefs, but degrees of belief or partial beliefs, the states in which we are when we are not fully confident that a certain proposition is true. According the Bayesians, these degrees are degrees of probability. Notice that, for Bayesians, all beliefs are essentially probabilistic. Probability is not in the content of some beliefs, but in the attitude of belief itself. It is not that we believe that something is, say, 0.5, or 0.7 probable. It is that we have a 0.5 or 0.7 credence in it happening. Every belief is thus a probabilistic belief (even if the credence happens to be 1 or 0). But the same problem as with in-between belief arises: should we say that full belief and full disbelief are a degree of belief 1 and 0, that half beliefs are not beliefs? Or should we say that there are full or allout beliefs which do not have degrees on the one hand and credences or degrees of belief on the other hand [START_REF] Kaplan | Decision theory as Philosophy[END_REF])? These difficulties are well known. But for our purposes here, which have to do with descriptive and psychological adequacy, the main difficulty is that

Bayesianism is implausible if it amounts to the reductive claim that all beliefs, and all the members of the doxastic family, have degrees which are degrees of probability. Bayesianism may be defended as a theory of what rational belief ought to be, but not as a descriptive theory. A lot of psychological work shows that humans are not good Bayesians, and that full beliefs cannot be reduced to partial beliefs (Holton 2015). If we want to preserve the idea that credences are a specific kind of belief, we have to say that they refer to one kind of belief only, perhaps a certain kind of dispositional belief, whereas our flat-out or full beliefs constitute another kind. 10

These considerations favor a pluralistic or layered view of belief and its cousins and bedfellows, each forming a specific kind. Chauvinism, which reduces belief to one privileged dimension would leave out too much. The deflationary functionalist strategy leads to the view that there are no real differences between beliefs proper and in-between beliefs. It unifies the domain of belief-attitudes at the cost of blurring the categories: we lack any proper way of individuating beliefs and distinguishing them from their subspecies. Nor can we classify beliefs into dualistic schemes, such as those which have a definite dispositional profile in opposition to those which are conscious or accessible to reflection, or between full beliefs and credences. For these divisions are not exhaustive. We need a better criterion.

A normative account of belief

The account of belief I propose belongs is "essentialist" [START_REF] Hazlett | A Luxury of the Understanding[END_REF] The correctness condition of an attitude is the condition in which it is appropriate or "fit" to its object. This is related to the familiar idea of the "directions of fit" of attitudes, but it is distinct: directions of fit are supposed to be either mind-to-world (epistemic attitudes) or world-to-mind (conative attitudes), but within attitudes of each kind there can be distinct kinds of objects to which they are fit. Each kind of attitude has a certain typical object, to which it is supposed to adapt or which is adapted to it. This object is neither the type of intentional or propositional content of the attitude nor its token content (e.g. beliefs as a type have an intentional content, which in a particular occasion may be such or such a proposition).

Intentional content is not individuative of the attitudes, for the same intentional content can be the object of different attitudes. Thus I can believe that Trump is president, desire that Trump is president, be horrified that Trump is president, regret that Trump is president, etc. And attitudes are not individuated by their intentional content since a single attitude, say desire, can have many different intentional contents. The correctness thesis says that there are types of content which are appropriate to each attitude. It is better formulated in terms of the traditional distinction between the formal object of an attitude and its intentional object. 13The formal object of an attitude is the kind of object to which typically the attitude is directed to. It contrasts with its particular object, which is the specific object of the attitude in a given circumstance. This idea is more familiar in the case of emotions. The formal objects of emotions are in general evaluative properties. Thus the formal object of fear is what is fearable, the formal object of love is what is lovable, etc. The particular object of fear could be on a given occasion an earthquake, a monster or a spider. The particular object of love could be Juliet or Romeo or my cat Felix. If we extend this to other attitudes we would say that the formal object of desire is what is desirable, and that the particular objects of desire could be a beautiful person or a big amount of money. The objects may be individuals, propositions or states of affairs. Thus one can be afraid that an earthquake arrives, or that a spider is on the wall, or desire that Juliet loves one. The obvious problem with this proposal is that it looks trivial or false, since there does not seem to be a class of things which objectively have these evaluative properties. Some spiders are worthy of fear, others not, some people or animals are worthy of love, others not. But let us leave that difficulty aside, and let us see how it can apply to epistemic attitudes. The formal object of belief is not the believable, since any proposition can be in principle believed (except perhaps paradoxical ones, such as "This proposition is believed by no one"). Rather, it is truth, and truth makes the attitude properly epistemic. The fact that truth is the formal object of an attitude should not be confused with the fact that a number of attitudes which are not epistemic can be expressed as having a propositional content. Thus conative attitudes have this featureto desire that P is often to desire that P is true, to hope that P is to hope that P is trueand a number of attitudes or emotions such as imagining, fearing, can take propositional complements. Neither is the fact that the formal object of an epistemic attitude is truth to be confused with its direction of fit.

Why? Because the fact that one can construe attitude verbs with a propositional complement, hence with contents susceptible of being true or false, does not imply that the direction of fit and the formal object of these attitudes is truth. Although imagining involves a relation to propositionsone can imagine that one is an Oxford don, or that China invades the U.Sis it not clear that the aim of imagining is truth, i.e., that one imagines that P with the aim of accepting P only if it is true [START_REF] Velleman | On the aim of Belief[END_REF]. One can perfectly imagine or hope situations one does not believe or imagine to be true: the point of imagination and of hope is to be able to do so. The potentially propositional nature of the content of such attitudes entails that the predicate is true can automatically be affixed to these contents. But it does not entail that their formal object is truth. The formal object of an attitude is the object to which the attitude is "fitted" in the sense of being the appropriate or correct one, for this attitude. Correctness is the feature an attitude has when it is fit to its formal object. But the latter is not the direction of fit. We should distinguish the correctness condition from the satisfaction condition. In the case of belief and other epistemic attitudes, the satisfaction condition is that the belief is true or that the corresponding state of affairs obtains. It is, as [START_REF] Mulligan | Intentionality, Knowledge and Abtract Objects[END_REF] puts it, because the belief is true that it is satisfied. In the case of a desire, or a hope, or a wish, the satisfaction is that desire, the hope or the wish be satisfied or realized. In the case of conjectures, these are satisfied when they are probable. But the belief, or the desire, may be satisfied by sheer luck:

for instance my belief that 225+333 = 558 may be satisfied by pure guess. The correctness condition says more: it says that if one has a belief P it ought to be true if it is to fulfill it satisfaction condition. So the satisfaction condition involves necessarily a normative term. It

does not matter at this point whether this term is a deontic or an evaluative one, and what kind of guidance it can give to believers15 . But it has to capture the idea that the attitude requires a certain kind of object and that the object has to be appropriate to the kind of attitude. So this formal object is a normative property.

Contrast now belief with different attitudes. Some attitudes, like conjectures, hypotheses, suppositions, assumptions 16 and possibly guesses, do not have truth as their formal object and as what fulfills their correctness condition: their formal object is what is probable. This is very plausibly the case for credences, or states of confidence. Conative attitudes have distinctive formal objects, which are plausibly the good, or the appearance (the guise) of the good. 17 Interrogative attitudes such as questions have as formal object what is question-worthy or questionable, doubts have as formal objects what is doubt-worthy or doubtable.

Emotions have value properties as formal objects, and are correct when these objects are appropriate [START_REF] Teroni | Emotions and Formal objects[END_REF]. I shall not here attempt to offer a list. Let us consider now belief.

Why is it special among the attitudes? The formal object of belief, as an epistemic attitude, is truth. Other epistemic attitudes, such as perceivings, noticings, discoverings, surprises, some kinds of suppositions and hypotheses, and indeed knowledge can have truth as their formal object. Surprises are plausibly epistemic emotions, which arise from a contrast between what one believes to be true and what one expected to be true. But the correctness of a belief is specific. It does not consist simply in the fact that it is "aimed at truth" in the sense that it has a direction of fit. It consists in the fact that, as Williams says, it is a "fatal" flaw for a belief to be false: something is wrong, in the sense that the belief cannot be held or maintained if it is show to be false. Belief is the only attitude which is correct only if true. The correctness condition involves the nature of the success of the attitude. A belief is successful if true, and it fails if false. In a sense truth as the correctness or success condition seems to apply as well to suppositions, guesses, or surprises. A guess which is true is indeed successful, as well as a supposition. But one can make a good guess which does not hit truth, and which is only almost true. And one can suppose successfully without one's supposition being true. One can be surprised at something which turns out to be false. The success of a guess or of a supposition does not depend crucially on its being true, but on the way in which evidence has been collected. Indeed the best suppositions are those which are true, but a supposition which turns out to be false is not necessarily wrong. Imagining has an intentional object, and to imagine that P is to imagine that P is true, but the success of an act of imagination does not depend on it delivering true outputs. The correctness condition of supposing consists in respecting a different norm from truth: evidence. Evidence is indeed evidence for truth, but supposing, can be successful even when the truth target is not hit. Similarly for guessing: a true guess is indeed successful, but it is not simply correct when true. Thus in a quiz it is as important to guess quickly as to guess truly [START_REF] Owens | Does Belief have an aim ?[END_REF]. Belief, in contrast, is correct if and only if true. There is no room for balancing the aim of truth, or the correctness condition of being true, with other aims 18 . This sets belief apart from other epistemic attitudes. A question might arise, however, about knowledge. Isn't its correctness condition truth? Indeed it is, but its correctness condition is automatically fulfilled: knowledge is factive, since it entails the 18 Engel 2013, 2013 a , contra, Reisner 2008. truth of the content. This is not the case for belief. Thus, knowledge and belief share their correctness condition, although in the case of knowledge this condition cannot fail to obtain

The doxastic taxonomy revisited

The taxonomy I propose does not amount to a theory of belief, but it presupposes it. In the first place, I reject eliminativism: I take most states admitted by common sense psychology to be real. I am less sure, however, that the states which philosophers have designated as "inbetween" beliefs or "aliefs" belong to the doxastic zoo, although these may be close of some species (for alief, close to beliefs and to emotions). Second, I agree with the functionalists, pragmatists and dispositionalists about belief that belief is at least a disposition to act and is in part individuated by its causal role. Belief is certainly a complex, multitrack disposition [START_REF] Ryle | The Concept of Mind[END_REF]., But this does not place it apart from other dispositional states, such as epistemic feelings and epistemic emotions, such as surprise, wonder, or feelings of familiarity. Nor does the dispositional theory allow us to differentiate belief from such strange bedfellows as "aliefs", tacit beliefs, and biases on the one hand, for all these states have a behavioral and dispositional profile. The dispositionalist view does not allow us to distinguish belief from the higher-order judgmental attitudes such as suppositions, conjectures, acceptances, commitments, because these states are clearly not simply dispositions to act. The criterion I have proposed is: belief is the only attitude whose correctness condition is truth. This separates off belief from states which are not attitudes and most of its strange bedfellows.

There are no correctness conditions for epistemic feelings, such as feelings of familiarity, feelings of knowing or feelings of déjà vu [START_REF] Dokic | Seeds of Knowledge[END_REF]): these can be successful, and make us able to locate in memory the content we try to retrieve, but they can also fail to deliver any content. There is no rule or norm for their success, or any condition which could constitute what it is for such states to be correct. They are not attitudes at all, since it is not clear that they have a propositional content. They certainly are based on some kind of evidence, but it is always unclear where this evidence lies for the agent, and to what use it can be put to infer anything. It is not even clear what their contents are, and in this sense they do not have formal objects. They do not have reasons, and cannot be held for a reason. Their content in some cases is exhausted by their phenomenology, and we can therefore be skeptical that there be a "cognitive phenomenology" in their case [START_REF] Bayne | Cognitive Phenomenology[END_REF]. One may doubt that aliefs and tacit beliefs be kinds of belief, in spite of the fact that they may involve or be related to beliefs(ditto for biases). I would also reject delusions as clear cases of beliefs, although I would be prepared to take them into an intermediary status (Bayne and Pacherie 2005) because they do not have correctness conditions. By disqualifying most of the bedfellows as beliefs and as attitudes and by excluding them from the doxastic club, it seems that I take as the criterion of an attitude the fact that these are under the agent's control and that one can be responsible of the attitude that one takes. This is true of some attitudes, those of the judgmental kind, but not of all attitudes. Emotions, by most criteria, are attitudes which have correctness conditions and formal objects. But in most cases we do not control our emotions and are not responsible for them. Epistemic emotions, such as surprise and wonder, are not attitudes for which we are responsible, or which involve any kind of agency. But like all emotions, including non-epistemic ones, they are based on beliefs which are their cognitive bases, even though they are not beliefs, because they do not have truth as their correctness condition. Nor can we include hinges within beliefs. They do not have truth and correctness conditions, are not subject to epistemic norms, and in this sense are not attitudes at all [START_REF] Engel | Epistemic Norms and the Limits of Epistemology[END_REF].

In driving a wedge between beliefs and their strange bedfellows, it seems that I exclude credences from beliefhood. Bayesians would disagree. Actually it is arguable that credences share a lot with beliefs in that they have correctness conditions , obey norms [START_REF] Buchak | Beliefs, Credences and Noms[END_REF] and are sensitive to evidence and to principles of rationality. Thus, they are not strange bedfellows, but close cousins. Yet, by definition their correctness condition is not truth, but high probability. According to the normativity and correctness criterion proposed here, only judgmental attitudes or attitudes which are judgement-sensitive are beliefs properly so called: judgments, epistemic acceptances and commitments all qualify, because they all can be taken as premises for one's reasoning. These may or may not be voluntary in a strong sense: for some acceptances may be only implicit. But they are all categorical or all out, and do not have degrees. They are all subject to a correctness norm, which is truth. Some acceptances are pragmatic, contextual, and voluntary [START_REF] Cohen | An Essay on Belief and Acceptance[END_REF]. In this sense they are not beliefs. But some kinds of acceptance, those which have an epistemic aim, such as hypothesizing, or taking certain things for granted, are clearly judgmental attitudes. On the basis of these A lot more should be said to refine these categories. To classify a state under the heading of belief implies that i) it has a causal role, iii) is non voluntary iii) is sensitive to reasons and to evidence, and most of all is an attitude which has truth as its correctness conditions. Some beliefs can be purely dispositional in the sense that they are revealed in behavior (including verbal) and are not reflectively conscious. Others are judgmental, in the sense that they are the object of a conscious and all-out assent (not necessarily verbal). They can be voluntary, not because they can be willed for non-epistemic reasons (such as pleasure or comfort), but because in holding them one commits oneself to them, and to the inferences to which they lead. Some of them are parts of rational plans and strategies, such as epistemic acceptances.

Credences enjoy an intermediary status, being both very close to dispositional belief by being in large part non reflective, attitudinal, and normative. But their correctness condition, probability, is not truth, hence they are not beliefs. Pragmatic acceptances are attitudes with correctness conditions, aimed not at truth, but at utility. There can be also intermediary epistemic attitudes, such as conjectures, suppositions, expectations, presumptions and hypotheses, with correctness conditions, but not clearly truth oriented. Hypotheses are closer to epistemic acceptances, and presumptions closer to dispositional beliefs. But they are not judgmental in the sense that a thinker does not commit himself to them as he usually does in the case of judgments. All other states fail, in one way or another, to be attitudes, to have correctness conditions, and to be evidence oriented. They are all epistemically loaded, as they involve the processing of information. They all have a causal and behavioral profile, but they do not involve the kind of assent, reason and normative basis that goes with belief. They can, phrase, constitute "seeds of knowledge", but they are not in the business of knowing [START_REF] Dokic | Seeds of Knowledge[END_REF].

I have not included knowledge among these attitudes because knowledge is, strictly speaking, neither a species of belief nor an attitude, although it entails belief. Knowledge entails truth, but it would be wrong to say that truth is its correctness condition . Knowledge is not only justified or warranted belief, but justified and warranted belief entailing truth. It is actually the norm of belief, what belief aspires to. Knowledge is the ideal of belief., I shall content myself to indicate how it relates to the nature of belief, when we compare it to other attitudes. Our attitudes, such as belief, desire, hope, etc., are associated to various presuppositions about the kinds of judgements, inferential relations, causal profiles, and strategies that they involve. These presuppositions are in many ways contingent and can vary, depending on social, historical, or cultural factors. They are in various ways tied to our nature, and can depend on all kinds of individuals' circumstances: some people are afraid of spiders, many believe and desire weird things. These contingencies, however, do not affect the essence of belief. Each attitude has an ideal profile, one which it ought to have. It is particularly the case for belief: it is an attitude which is associated, implicitly or not, in a believer, to what he takes himself to believe, to what he considers that he ought to believe. I have called this feature here "correctness condition", but one could call it the ideal of belief.

This ideal gives to belief the primacy within the doxastic zoo. Ideals are limits, and one can be more or less far from the ideal. In the case of belief with respect to other epistemic and doxastic attitudes, the purest form of the ideal is judgement (Findlay 1962). The closer a belief kind of state is to this ideal, the closer it is to the core of believing. The farther it is from this ideal, the less it is part of the doxastic realm.

  or irrational, as justified or not, as well confirmed or not, as good or bad to have, as obligatory or not. Mere feelings, experiences, dispositions, or mental episodes, are not evaluated as good or bad: one can only say that one has them, or that one tends to have them in many circumstances. The same would be true about sentiments in the moral domain: if these were mere psychological states of approval, we could not say that they are good or bad, rational or not, evaluable as right or wrong. The same applies to belief. It involves a normative essence. States that conform to one or the other of the features (i)-(viii), may resemble belief and sometimes are called "beliefs", but if they are merely defined in causal terms, they will be called "beliefs" only by courtesy. Without the normative dimension, a mental state, however close to belief, is not a belief. This is not to deny that belief has a causal and natural profile. Beliefs can have all sorts or causes and effects and are psychological states in the mind. But qua beliefs, they are evaluable. A state which one could not characterise both in causal and in normative terms would not be a belief..

	(i)	All attitudes have specific correctness conditions
	(ii)	Belief is the only attitude whose correctness condition is truth.

: it takes belief to have a distinctive nature, which allows us to set this attitude apart from other doxastic attitudes and from the bedfellows. The main argument in favour of this view is that the list of platitudes (i)-(viii) leaves out the normative dimension of belief. We evaluate beliefs as

10

Leitgeb 2017 defends this "independence option ".

rational

11

What kind of normative properties are distinctive of belief? Evaluative or axiological ones, such as good or bad, valuable or not valuable? Or deontic ones, such as right or wrong, correct or incorrect? Some philosophers prefer the concept of reason. There are a number of debates on this issue. One way to avoid these debates is to use a generic notion, that of correctness.The normative properties relevant for attitudes are correctness properties, and that such properties involve the presence of specific norms. Belief is governed by a norm of truth: a belief is correct if and only if it is true. The exact form of the norm, and the kind of guidance which it is supposed to give us is a matter of discussion. 12 This norm is unique to belief, and sorts it out from the other attitudes and belief-like states. I defend two claims:

I have often discussed these issues , e.g in[START_REF] Engel | Believing, accepting, and holding true[END_REF][START_REF] Engel | Dispositional Belief, Assent and Acceptance[END_REF][START_REF] Engel | Believing and Accepting[END_REF]Engel , 2013, to appear , to appear

A number of writers have emphasized this dimension of belief:[START_REF] Cohen | An Essay on Belief and Acceptance[END_REF], Hieronymi 2007[START_REF] Gilbert | Joint Commitment[END_REF][START_REF] Holton | Intention as a Model for Belief[END_REF]

Contra see Bennett 1991 and Papineau's 2013.

See Bykvist & Hattiangadi 2007, Mc Hugh 2010,

The distinction is a medieval one, taken up by Brentano and his disciples, reintroduced by[START_REF] Kenny | Action, Emotion and Will[END_REF] See also De Sousa 2005[START_REF] Teroni | Emotions and Formal objects[END_REF][START_REF] Mulligan | Intentionality, Knowledge and Abtract Objects[END_REF]

See Glüer and Wikforss 2009, Steglish-Pedersen 2010.

On assumptions see[START_REF] Meinong | Ueber Annahmen[END_REF]

Velleman 2000, Deonna and[START_REF] Lauria | The nature of desire[END_REF]


Zhaofeng Han
email: zhaofeng.han@insa-lyon.fr

Thibaut Chaise

Cyril Mauger

Thomas Elguedj

Nicolas Boisson

Mahmoud El Hajem

Numerical simulation of a water cavitation jet peening

Keywords: Surface treatment, water cavitation peening, cavitating jet, numerical simulation

Shot peening is commonly used as an effective method to extend the service life of parts and reduce the residual tensile stresses. The main principle of this surface treatment method is to introduce compressive residual stresses into parts, thereby preventing the crack-propagation. Water cavitation peening (WCP) is a recent technology that allows generating high-intensity pressure waves and micro-jet at the surface of parts via a cavitating jet. However, it is a challenge to predict the pressure intensity at the treatment surface and therefore being able to make predictive modelization of the process. To anticipate the pressure distribution around the cavitating jet, a multiple steps approach has been developed. First, through the study of a single cavitation bubble, understand the pressure pulse generated by a single bubble during its collapse process. The second step is to establish a numerical simulation of the cavitating jet near the nozzle with ANSYS Fluent. Accordingly, the cavitation area and the vapor volume fraction at different section of cavitating jet will be predicted. Then, the mechanical loading at the surface of parts will be defined by these sources. The second part of the approach is here studied. The simulation results will be compared with experimental high-speed camera imaging of the cavitating jet.

Introduction

Shot peening is a surface treatment process that reduces residual stresses from the manufacturing of a part by applying compressive residual stressed on the surface. This process delays the propagation of cracks and increases the corrosion resistance and service life of the part [START_REF] Farrahi | Effect of shot peening on residual stress and ratigue life of a spring steel[END_REF][START_REF] Torres | An evaluation of shot peening, residual stress and stress relaxation on the fatigue life of AISI 4340 steel[END_REF][START_REF] Tirosh | Extended fatigue life by shot peening process via shakedown analysis[END_REF]. Conventional shot peening consists in projecting shots onto the surface of the part. Therefore, it can contaminate the surface of the part and its environment during processing. For some parts with complex geometry, conventional shot peening does not allow to reach all surfaces. To overcome these problems, water cavitation peening (WCP) [START_REF] Soyama | High-speed observatios of the cavitation cloud around a high-speed submerged water jet[END_REF][START_REF] Soyama | Use of Cavitating jet for Introducing Compressive Residual Stress[END_REF] has been developed. The part to be treated is placed in an enclosure filled with water. A high-speed water jet send to the part. The velocity difference between the surrounding water and the jet induces significant shear stresses which will give rise to the appearance of strongly vortex structures, and produce a lowpressure area [START_REF] Sou | Numerical simulation of incipient cavitation flow in a nozzle of fuel injector[END_REF]. Moreover, cavitation bubble clouds can be generated in the low-pressure area. The simulation of the submerged water jet is an important step to anticipate the pressure distribution generated by the cavitation jet [START_REF] Sonde | Modeling of cavitation peening : Jet, bubble growth and collapse, micro-jet and residual stresses[END_REF].

The cavitation peening process can be simulated macroscopically. This study uses ANSYS Fluent to achieve the simulation. For cavitation flow, it is composed by water and vapor, in the process of cavitation, the two phases will also be accompanied by mass transfer. In this simulation, the mixture model [START_REF] Bowmen | Theory of Mixtures[END_REF] has been used for multiphase flow, the realizable k-ε model [START_REF] Shih | A new k-ε Eddy-Viscosity Model for high Reynolds Number Turbulent Flow -Model Development and Validation[END_REF] simulates the turbulence and the Singhal et al. May 10-13, 2021, Daejon, Korea * Corresponding Author: zhaofeng.han@insa-lyon.fr model [START_REF] Singhal | Mathematical basis and validation of the full cavitation model[END_REF] (also called full cavitation model) simulates the mass transfer by cavitation between the water and the vapor. In addition, it is assumed that between the two phases and at the inner wall of nozzle a noslip condition is satisfied

Model establishment

Figure 1 describes the geometry of the simulated nozzle. Due to the axi-symmetry of the considered problem, only a slice of the model is shown. This geometry of the nozzle is obtained by high-precision measurement of the real nozzle used in the experiment. In order to optimize the calculation time, full-scale modeling of the water tank is not performed here. After studying the convergence of the computational domain in detail, we chose to model an area of 50  10 mm in size. The red border in the left of Figure 1 is the inlet of the system. The boundary condition here is a pressure-inlet, according to the actual pressure of the experiments, the pressure inlet is 30 MPa. In addition, the blue boundary represents the environmental pressure expressed as a pressure-outlet with a value of 0 MPa. The inner wall of the nozzle, represented in green in Fig. 1, satisfies the no-slip boundary condition and the outer wall of nozzle, in black, satisfies the slip boundary condition. The orange boundary in Fig. 1 corresponds to the condition of axi-symmetry. ANSYS-ICEM has been used to create computational domains and structured grids. Finally, the entire computational domain is divided into 197 040 uniformly structured 2D dimensional grids with a maximum size of 50 µm × 50 µm. In this simulation, the turbulence intensity is 5% for both pressure-inlet and pressure-outlet.

Numerical simulation results and analyses

The establishment of the cavitation flow is divided into two steps here. The first step is to solve the mixture model and the realizable k-ε model to obtain a jet flow without cavitation used as the initial stage. After it is stabilised, the full cavitation model (Singhal et al. [START_REF] Singhal | Mathematical basis and validation of the full cavitation model[END_REF]) is added to simulate the cavitation phenomenon and start the transient calculation. Figure 2 shows the velocity distribution of the underwater water jet in contour map and the vorticity magnitude in path-line diagram at 5.65 ms. It can be seen that due to the sudden reduction in the diameter of the nozzle, the water flow is accelerated rapidly under the pressure, and will reach a maximum speed of 246 m/s inside the nozzle. When the water flow sprays out of the nozzle, under the double action of inertial forces and pressure, the water flow will continue to rush forward. However, the still water in the surrounding environment will have a certain resistance to the May 10-13, 2021, Daejon, Korea * Corresponding Author: zhaofeng.han@insa-lyon.fr water jet, thus preventing the water flux from continuing to move, and a huge shear force is generated at the contact surface of the water jet and the environmental fluid. In addition to the velocity distribution, the distribution of vapor volume fraction is also studied. Figure 3 shows the distribution of vapor volume fraction at 5.65 ms. For the cavitation inside the nozzle, the highspeed water jet located in the center of the nozzle and the boundary layer of the nozzle generate a huge shear force, and a small range of vortices are generated at the wall of the beginning of the nozzle. This process is accompanied by the generation of a large number of cavitation bubble and these bubble cores will expand and accumulate rapidly, thereby forming cavitation bubble cloud, the maximum volume fraction of vapor reaches 100 %. In the expansion area of the nozzle outlet, there will be a back flow near the inner wall of the nozzle, and under the impact of the high-speed cavitation flow, a second low pressure zone appears due to the strong shear force between the back flow and the cavitation flow. This process will be amplified to a certain extent cavitation area. After that, the volume fraction of vapor gradually decreases with the flow of the water jet, and disappears quickly at the nozzle outlet. For more detail of cavitation jet, Figure 4 (a) shows the curve of vapor volume fraction on the axis of symmetry. To understand the fluctuation of cavitation flow, four characteristic surfaces were added to the model, and the physical parameters on the 4 surfaces were tracked over time. The 4 surfaces are the nozzle "throat", the center of nozzle, nozzle outlet and the center of water tank. The average value of vapor volume fraction of these 4 surfaces over time has been shown in Fig. 4 (b), the average fluctuation period is 0.43 ms. This period will be compared with experimental data in the future.

In the experiment, we can only observe the presence or absence of cavitation areas through high-speed cameras, so it is difficult to determine the cavitation areas by the volume fraction of vapor. On the experimental images, the light rays are diffracted by the liquid-vapor interface, which leads to shadow areas on the camera sensor. Even for very low vapor volume fractions, this diffract phenomenon still produces. In order to facilitate the comparison with the experimental data in the future, the area with the vapor volume fraction of 3%, 4% and 5% is marked in the form of white contour lines in Fig. 3. In this study, 5% vapor volume fraction is regarded as the visible cavitation area in the experiment, so the length of cavitation jet (from nozzle outlet to contour line of 5%) is about 5 mm in this simulation. May 10-13, 2021, Daejon, Korea * Corresponding Author: zhaofeng.han@insa-lyon.fr

Conclusions

A model of cavitating jet with a geometry of real nozzle structure is established. The distribution of velocity, vorticity, the volume fraction of vapor inside and near the nozzle and the fluctuation of cavitation flow has been analyzed in detail. The average fluctuation period is 0.43 ms, and the length of cavitation jet with 5 % vapor volume fraction is about 5 mm. The current results show the potential of predicting the cavitation area and then predict the pressure of the cavitating jet through the model.

In the future, the results of numerical simulation will be compared with experimental results to verify the accuracy of the model.

Figure 1 .

 1 Figure 1. Schematic representation of the model geometry used to simulate cavitation jet. The inner wall (green border) hypothesis satisfies the no-slip boundary condition and the outer wall (black border) satisfies the slip boundary condition.

Figure 2 .

 2 Figure 2. The upper half of the image is the velocity distribution represented by the contour map and the vorticity magnitude is represented by the path-line diagram in the lower part at 5.65 ms.

Figure 3 .

 3 Figure 3. Distribution of volume fraction of vapor at 5.65 ms. The white contour lines from outside to inside represent the contours with volume fraction of vapor 3%, 4% and 5% respectively.

Figure 4 .

 4 Figure 4. (a) Distribution curve of volume fraction of vapor on the axis of symmetry at 5.65 ms; (b) The curve of the average value of vapor volume fraction on the characteristic surface over time. The average fluctuation period is 0.43 ms.

Acknowledgments: This work was supported by the project MoCaPee of the Institut Carnot Ingénierie@Lyon.


Regnault Gabriel
email: gabriel.regnault@insa-lyon.fr

Mauger Cyril

Blanc-Benon Philippe

Inserra Claude

Gabriel Regnault

Experimental observation of micrometric bubble pair oscillations and induced effects

Keywords: Bubble pair oscillations, shape modes, microstreaming

The study of an acoustic bubble pair is a first step in the understanding of mechanisms ruling cavitation clouds and groups of contrast agents. Such bubbles interact through acoustic coupling but can also be influenced by the fluid flows that they generate. Here, this fluid interaction is evidenced for a micro-metric bubble pair. Conditions on shape instabilities needed to obtain such interactions are shown using experimental observations of free bubble induced fluid flows.

Introduction

The acoustic bubble has been studied for more than a century. First considered as a purely theoretical object, it has now resonance in many applicative fields. One can cite, for example, the use of coated microbubbles in acoustic imaging [START_REF] Tanter | Ultrafast imaging in biomedical ultrasound[END_REF], in bubble-mediated therapies [START_REF] Marmottant | Controlled vesicule deformation and lysis by single oscillating bubbles[END_REF], but also for industrial applications [START_REF] Chahine | Modeling of surface cleaning by cavitation bubble dynamics and collapse[END_REF], in which more violent processes can be used. In all these applications, microbubbles allow one to focus the acoustic energy from the characteristic length of the sound wave-length, to the much smaller one of the bubble (two orders of magnitude smaller in water). But in all those kinds of applications, bubbles do not often come alone, but rather gathered in small groups, for coated bubbles, or in large clouds for cavitation bubbles. Such configurations imply that bubbles are close to each other. When acoustically excited, they will start oscillating. Along with these oscillations comes individual sound emission: each bubble acts as a sound source and radiates in its vicinity as a secondary (compared to the primary external one) sound source. Each bubble interacts with the other ones through this acoustic emission. Many effects highlighting these interactions have been reported in literature. They range from change in the resonance frequency [START_REF] Combriat | Acoustic interaction between 3D fabricated cubic bubbles[END_REF] up to oscillatory behavior modifications [START_REF] Liang | Investigations on dynamics of interacting cavitation bubbles in strong acoustic fields[END_REF]. Also, the change in the local pressure gradient around two closely-spaced bubbles has been evidenced as a cause of interaction force, called the secondary Bjerknes force, which makes the two bubbles attract or repel each other if they respectively oscillate in or out of phase.

While the acoustic interaction has been widely studied, only few studies focused on the case of bubble-induced fluid interaction. Such phenomena occur when bubbles are either translating or oscillating non-spherically. Few papers have reported experimental observations of these flows, for wallattached bubbles [START_REF] Tho | Cavitation microstreaming patterns in single and multiple bubble systems[END_REF], while theoretical approaches [START_REF] Doinikov | Acoustic microstreaming produced by nonspherical oscillations of a gas bubble. I. Case of modes 0 and m[END_REF] or numerical simulations [START_REF] Jalal | Interaction of Spherical Particles Owing to Steady Streaming Induced by Ultrasound[END_REF] consider the case of free bubbles in an infinite medium. The induced fluid flows can come either from scattering of the spherical wave radiated by one bubble on the other one [START_REF] Doinikov | Microstreaming generated by two acoustically induced gas bubbles[END_REF] or directly by non-spherical oscillations. It is a slow mean flow, whose origin lies in the attenuation of the wave scattered by a bubble in the surrounding viscous fluid. May 10-13, 2021, Daejon, Korea

In this paper, we report observations of fluid interactions in the case of an oscillating bubble pair in water, kept far away from boundaries. We focus here on fluid flows, while a previous study on the interaction force has already been published [START_REF] Regnault | Secondary radiation force between two closely spaced acoustic bubbles[END_REF]. Bubble pair induced fluid flows are shown and discussed in relation to bubble oscillating behaviors.

Materials and methods

The experimental setup used for this work is displayed in Figure 1. A transparent plastic tank (PMMA) of 8-cm side is filled with filtered water. Bubbles are created one by one, using pressurized air supplied through a thin capillary of a 20-µm inner diameter. Multiple consecutive bubbles are created inside the tank. They all rise up due to buoyancy. The bubbles size ranges from 70 to 120 µm. In order to trap them, a 1 MHz standing wave is generated inside the tank using a transducer (Sofranel IDMF018, 1 MHz nominal frequency) situated on top of the water tank. Since the bubbles are much larger than the resonant size of a bubble at 1 MHz, they are trapped due to the primary Bjerknes force at pressure anti-nodes of the standing wave [START_REF] Leighton | Primary Bjerknes forces[END_REF]. The inter-bubble distance corresponds to half the wavelength of the 1 MHz field, following the theory of primary radiation force [START_REF] Leighton | Primary Bjerknes forces[END_REF]: d ~ 750 µm. Once trapped, only a pair of bubbles situated on two consecutive nodes is kept.

The bubble oscillations are initiated by emitting in the tank a low frequency wave, at 31.9 kHz, using a Langevin Transducer (SinapTec®, 31 kHz nominal frequency) situated below the tank. The bubble oscillations and positions are recorded with a CMOS camera (Vision Research, V12.1) at a cadence of 130 kfps. The observation is achieved using backlight illumination. This way, the bubble contour within the focal plane of the camera can be extracted. Further explanations on the method can be found in [START_REF] Regnault | Secondary radiation force between two closely spaced acoustic bubbles[END_REF]. Microstreaming measurement can also be achieved using this set-up. As shown in literature, for free bubbles in an unbounded fluid, spherical oscillations are not sufficient to generate microstreaming. One thus has to trigger shape modes in order to observe such fluid flows. Here, the shape instabilities are triggered either by increasing the low-frequency pressure amplitude for sufficiently large bubbles or by making them grow, through rectified diffusion, at constant but large pressure amplitude. Once at least one bubble is non-spherically oscillating, the fluid flow is observed using fluorescent tracers (R700 * Corresponding Author: Gabriel Regnault, gabriel.regnault@insa-lyon.fr ThermoFisher, 0,71µm in diameter) enlightened by a continuous wave laser sheet of 150µm in depth. The camera used for the bubble dynamics observation is as well used to assess microstreaming, but at lower frame rates, within the range of [500-1000] fps. A recording session consists in consecutive observation of bubble dynamics and induced fluid flows, in order to allow linking their interface motion to microstreaming.

Results

As predicted, no microstreaming is observed if both bubbles are only spherically oscillating. Such oscillations lead only to acoustic coupling. The inter-bubble distance is too large here to enable the creation of cavitation microstreaming through scattering of the wave emitted by each bubble on the other one. Only with the triggering of non-spherical oscillations come fluid flows and possible fluid interactions between the bubbles.

In Figure 2, particular cases of the two-bubble microstreaming that has been observed are shown. The bubble contours and positions are shown on left of every sub-figure while the fluid pattern is on the right. The fluid flows are obtained by superimposing hundreds of consecutive images. Here, it is shown that shape modes and bubble induced fluid flow do not always come with fluid interaction. Evidence of such cases is displayed in Figure 2.a, b, c. In case a), the inter-bubble distance is 1.5 mm, twice the distance between two consecutive nodes. Both bubbles oscillate on the main mode 4. The fluid patterns are confined close to the bubble interface and do not connect to each other. A background mean flow is observed as well, but it only slightly affects the pattern extension around the bubbles. In case b), the two bubbles oscillate as well on mode 4 but with an inter-distance of 750 µm. The fluid pattern is formed by 8 lobes, also near the bubbles interfaces. No fluid interaction is observed. In case c), only one bubble * Corresponding Author: Gabriel Regnault, gabriel.regnault@insa-lyon.fr d,e) For some cases, the fluid motions can reach the other bubble and even interact, leading to recirculation zones between the two bubbles. May 10-13, 2021, Daejon, Korea undergoes a shape mode, of order 5. The microstreaming that it generates does not reach the bottom bubble (which remains spherical here) even if the inter-bubble distance is decreased, due to secondary Bjerknes force [START_REF] Regnault | Secondary radiation force between two closely spaced acoustic bubbles[END_REF].

Although the previous cases are not, the following two indeed evidence microstreaming interaction. The shape modes here, in cases d) and e), are quite intriguing. They belong to the zonal instabilities and not to the sectoral ones. For the sake of brevity, no discussion on this point is provided here. Such modes have almost a shape close to axisymmetric mode 2 in side view within the focal plane. Their associated fluid flow looks like a large cross (see Figure 2.d), a pattern which is confirmed by recently published theories [START_REF] Inserra | Acoustic microstreaming produced by nonspherical oscillations of a gas bubble. III. Case of selfinteracting modes n-n[END_REF][START_REF] Spelman | Arbitrary axisymmetric steady streaming: Flow, force and propulsion[END_REF]. When both the bubbles generates microstreaming, it leads in this case to modifications of the pattern, with re-circulation zones in the area between the two bubbles.

Conclusions

Fluid flow interaction between two non-spherically oscillating bubbles is not straightforward to achieve. Conditions on the mode order, and possibly its kind, must be reached. The global flow is enriched with re-circulation zones, which can be of great interest for bubble induced therapies since shear stress in these areas is increased. Further results must be obtained to enrich these first observations.

Figure 1 .

 1 Figure 1. The experimental set-up. a) First the bubbles are created and observed by backlight illumination at 130 kfps. b) Fluorescent tracers injected in the water are enlightened by a laser sheet to observe microstreaming, using the same camera.

Figure 2 .

 2 Figure 2. Bubble pair induced fluid flow patterns and associated snapshots of the bubbles interfaces. a, b, c) No fluid interaction is observed. Microstreaming remains confined close to the bubbles. d, e) For some cases, the fluid motions can reach the other bubble and even interact, leading to recirculation zones between the two bubbles.

* Corresponding Author: Gabriel Regnault, gabriel.regnault@insa-lyon.fr

This work was supported by the LabEx CeLyA of the University of Lyon (ANR-10-LABX-0060 / ANR-11-IDEX-0007).