L Gallimard
email: laurent.gallimard@u-paris10.fr

Error bounds for the reliability index in finite element reliability analysis

Keywords: structural reliability, finite element analysis, error bounds, goal-oriented error estimation, reliability index, FORM approximation

This work presents an extension of the goal-oriented error estimation techniques to the reliability analysis of a linear elastic structure. We use a first-order reliability method in conjunction with a finite element analysis (FEA) to compute the failure probability of the structure. In such a situation the output of interest that is computed from the FEA is the reliability index . The accuracy of this output, and thus of the reliability analysis, depends, in particular, on the accuracy of the FEA. In this paper, upper and lower bounds of the reliability index are proposed, as well as simple bounds of the failure probability. An application to linear fracture mechanics is presented.

INTRODUCTION

Many applications in structural analysis require taking into account stochastic properties of material, geometry or loads. Different methods based on the probability theory have been developed to quantify the effect of data uncertainties. First-order reliability method (FORM) [START_REF] Ditlevsen | Structural Reliability Methods[END_REF][START_REF] Lemaire | Fiabilité Des Structures[END_REF] is commonly used in reliability analysis. It is based on the computation of a reliability index [START_REF] Hasofer | An exact and invariant first order reliability format[END_REF], followed by an estimation of the failure probability. In many cases, the structural response cannot be expressed explicitly and the reliability analysis needs to be coupled with a finite element method. This coupling implies difficulties, such as prohibitive computational effort. Moreover, the development of a finite element discretized model starting from a continuous mechanical model leads to a partial loss of the information contained in the continuous model and, thus, to the introduction of discretization errors. As a consequence, the finite element analysis (FEA) introduces an approximation error in the computation of the reliability index that depends on the mesh used. The control of this discretization error is then necessary in order to minimize the computation effort, as well to assess the quality of the computed reliability index.

Methods have been developed over many years to evaluate the global quality of FEA [START_REF] Babuska | A posteriori estimates for the finite element method[END_REF][START_REF] Ladevèze | Error estimate procedure in the finite element method and application[END_REF][START_REF] Zienkiewicz | A simple error estimator and adaptive procedure for practical engineering analysis[END_REF]. For linear problems, all of these methods provide a global energy-based estimate of the discretization error. When the dimensioning criteria involve particular outputs of the FEA (stresses, displacements, intensity factors, etc.) this information is insufficient for dimensioning purposes. The development of error estimators for such quantities was initiated in the 1980s [START_REF] Babuska | The post-processing approach of the finite element method. Part II: the calculation of the stress intensity factor[END_REF][START_REF] Kelly | Procedures for residual equilibration and local error estimation in the finite element method[END_REF]. Recently, numerous works have been published that provide error estimates and bounds for several quantities of interest [START_REF] Rannacher | A feedback approach to error control in finite element methods: application to linear elasticity[END_REF][START_REF] Peraire | Bounds for linear-functional outputs of coercive partial differential equations: local indicators and adaptive refinement[END_REF][START_REF] Prudhomme | On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors[END_REF][START_REF] Ladevèze | Local error estimators for finite element analysis[END_REF][START_REF] Ohnimus | Local error estimates of fem for displacements and stresses in linear elasticity by solving local neumann problems[END_REF][START_REF] Prudhomme | Practical methods for a posteriori error estimation in engineering applications[END_REF][START_REF] Gallimard | Error estimation of stress intensity factors for mixed-mode crack[END_REF][START_REF] Gallimard | Evaluation of the local quality of the von mises's stress and l2-norm of the stress[END_REF]. However, these works have been developed for deterministic FEA and little research has been carried out on the verification of probabilistic models [START_REF] Deb | Solution of stochastics partial differential equations using Galerkin finite element techniques[END_REF][START_REF] Ladeveze | Verification of stochastic models in uncertain environments using the constitutive relation error method[END_REF].

In this paper, we consider the case of a linear elastic structure where the limit-state function G is written as: G = I c -I h , where I c is the resistance of the structure and I h is a particular output evaluated by the FEA. The calculation of the failure probability is performed by the FORM approximation. In such a computation three sources of errors are introduced:

• an algorithm error due to the gradient-based algorithm that is used to compute the reliability index, • a geometric error due to the FORM approximation, • a discretization error due to the computation of I h by an FEA.

As far we know, only the first two points have been addressed in the literature (see [START_REF] Zhang | Two improved algorithms for reliability analysis[END_REF][START_REF] Wang | Safety index calculation using intervening variables for structural reliability analysis[END_REF][START_REF] Der Kiureghian | Multiple design points in first and second-order reliability[END_REF][START_REF] Mitteau | Error evaluation for the computation of failure probability in static structural reliability problems[END_REF]). The objective of this work is to study the effect of the finite element discretization on the computation of the reliability index and to propose upper and lower bounds for this reliability index, which take into account the approximation errors introduced by the FEA.

The paper is organized as follows: In Section 2, we present the basis of the reliability analysis. In Section 3, we describe the problem to be solved. Section 4 is devoted to the presentation of a method to obtain bounds of the reliability index, as well as bounds of the FORM failure probability. Finally, Section 5 presents an application to linear fracture mechanics.

PRINCIPLE OF RELIABILITY ANALYSIS

In this section, we describe the problem of reliability assessment and we introduce the reliability index [START_REF] Ditlevsen | Structural Reliability Methods[END_REF][START_REF] Mohamed | Finite element and reliability: a method for compound variables-application on a cracked heating system[END_REF][START_REF] Pendola | Combination of finite element and reliability methods in nonlinear fracture mechanics[END_REF]. Structural reliability aims at computing the probability of failure of a mechanical system by accounting for uncertainties arising in a model description (geometry, material properties) or the environmental data (prescribed displacement and external forces).

Let m be the vector of the model parameters on which there are some uncertainties, and let N be the size of m. This vector is modelled by a random vector X and f X (x) is the probability density function. The structure failure state is defined by a limit-state function G such that • G(x)<0 is a failure state for the structure, • G(x) = 0 is the limit state, • G(x)>0 is a safe state for the structure.

The random variable defined by Z = G(X) is called the safety margin of the structure. According to the definition of G, the system fails when this margin is lower or equal to zero. The failure probability P f is then given by:

P f = G(x) 0 f X (x) dx (1)
The evaluation of the integral defined by Equation ( 1) is not easy because it represents a very small quantity and because the probability density function is not usually well known because of the scarcity of the statistical data. A classical way to evaluate this integral is to use the first-or the second-order reliability methods FORM/SORM [START_REF] Ditlevsen | Structural Reliability Methods[END_REF]. They are based on the computation of a reliability index, followed by an estimation of the failure probability.

In [START_REF] Hasofer | An exact and invariant first order reliability format[END_REF], the authors propose to define the limit-state function G in the space of independent standard Gaussian variables instead of the space of the physical variables. The transformation from the physical variables X to the normal variables Y is given by

X = T (Y) ( 2
)
This transformation is called the isoprobabilistic transformation which transforms the probability integral into

P f = G(x) 0 f X (x) dx = G(T (y)) 0 Y (y) dy (3)
where Y (y) is the n-dimensional standard normal density with normal components. The limit-state function in the space of standard independent variables Y is written as:

G(x) = G(T (y)) ≡ H (y) (4)
In this space, the Hasofer-Lind reliability index , is defined as the minimum distance between the origin of the space O and the surface of the limit state

= min H(y) 0 y T y (5)
The solution of this problem gives the value of as well as the coordinates y * i of a point P * in the space called the design point. The FORM failure probability is given by

P f,FORM = (-) (6)
where is the cumulative distribution function of a standard Gaussian variable.

THE PROBLEM TO BE SOLVED

The elastic model

Let us consider a 2D elastic structure defined in a domain bounded by * . The external actions on the structure are represented by a prescribed displacement u d over a subset D of the boundary, a surface force density T defined over N = * -D , and a body force density b defined in . The Hooke's operator of the material is denoted by K. Thus, the problem can be formulated as: Find a displacement field u and a stress field r defined in which verify:

• the kinematic constraints:

u ∈ U and u | D = u d (7)
• the equilibrium equations:

r ∈ S and ∀u * ∈ U 0 -r : e(u * ) d + b•u * d + N T•u * d = 0 (8)
• the constitutive relation:

r = Ke(u) (9)
U is the space in which the displacement field is being sought, S = L 2 [ ] 3 the space of the stresses, U 0 the space of the fields in U which are zero on D , and e(u) denotes the linearized deformation associated with the displacement:

[e(u)] ij = 1 2 (u i, j +u j,i ) (10)

Computation of the reliability index

In a classical deterministic case, the objective of the computation is to verify that a quantity of interest denoted I (u), depending on the solution of the mechanical problem u, is inferior to a critical value I c . In a stochastic computation we assume that the data (prescribed displacement and forces, geometry) and the critical value I c are no longer deterministic and can be described by stochastic functions. The solution of the model problem becomes a stochastic displacement field denoted u(X). The structure failure state is defined by a limit-state function G(X) = I c (X)-I (u(X)) such that

• I (u(x))> I c (x) is a failure state for the structure,

• I (u(x)) = I c (x) is the limit state,

• I (u(x))< I c (x) is a safe state for the structure.

The objective of the computation becomes to compute the probability of failure of the structure by using the FORM approximation, and the computation of the reliability index . In the space of the independent standard Gaussian variables Y, the failure domain is given by

D = {y; H (y) 0} with H (y) = I c (y)-I (u(y)) (11)
and computing the reliability index consist in solving the following minimization problem: find such that = min y∈D y T y (12)

Finite element analysis and approximation errors

The Galerkin finite element method provides an approximation u h (Y) to u(Y) in a finite element space U h ⊂ U. The finite element approximation transforms the failure domain

D h = {y; H h (y) 0} with H h (y) = I c (y)-I (u h (y)) (13)
and the minimization problem becomes: find h such that

h = min y∈D h y T y (14)
Two errors are introduced in the computation of the reliability index

• an error due to the resolution of the optimization problem under constraint [START_REF] Prudhomme | Practical methods for a posteriori error estimation in engineering applications[END_REF],

• an error due to the finite element discretization.

In this paper, we focus on the error introduced by the FEA and we assume that the optimization problem is solved exactly. The optimization problem described in Figure 1 is solved by an improved version of the classical HLRF algorithm [START_REF] Rackwitz | Structural reliability under random load sequences[END_REF], the iHLRF algorithm developed in [START_REF] Zhang | Two improved algorithms for reliability analysis[END_REF][START_REF] Der Kiureghian | Efficients algorithms for second order reliability analysis[END_REF][START_REF] Haukaas | Strategies for finding the design point in non-linear finite element reliability analysis[END_REF] (see [START_REF] Der Kiureghian | Multiple design points in first and second-order reliability[END_REF][START_REF] Der Kiureghian | Efficients algorithms for second order reliability analysis[END_REF][START_REF] Haukaas | Strategies for finding the design point in non-linear finite element reliability analysis[END_REF] for the detailed studies on the convergence of the optimization problem).

BOUNDS OF THE RELIABILITY INDEX

In this section, upper and lower bounds for the reliability index are presented. These bounds allow computing the bounds of the failure probability obtained by the FORM method. The computation of these bounds assume that the quantity of interest I (u) is a linear functional.

Bounds of the reliability index

Let us assume that we are able to compute lower and upper bounds of the quantity of interest I (u(y)), from the finite element solution and the data, such that

I -(u h (y)) I (u(y)) I + (u h (y)) (15)
We define a reliability index low h associated with the upper bound

I + (u h (y)) low h = min y∈D + h y T y (16)
where D + h is the failure domain defined by

D + h = {y; H + h (y) = I c (y)-I + (u h (y)) 0} (17)
and a reliability index upp h associated with the lower bound

I -(u h (y)) upp h = min y∈D - h y T y (18)
where D - h is the failure domain defined by

D - h = {y; H - h (y) = I c (y)-I -(u h (y)) 0} (19)
Equation ( 15) leads to

I c (y)-I + (u h (y)) I c (y)-I (u(y)) I c (y)-I -(u h (y)) (20)
and hence to

D - h ⊂ D ⊂ D + h (21)
From ( 21) we deduce the following bounds for the reliability index :

low h upp h ( 22
)
Let us define the relative error on the reliability index

ε( h ) = -h h ( 23
)
Equation ( 22) leads to the following error bounds:

low h h -1 ε( h ) upp h h -1 (24)
An estimation of this error is computed by As (-) is a monotonous function, the computation of lower and upper bounds for the reliability index leads to simple bounds of the FORM failure probability due to the finite element approximation are computed by solving successively two minimization problems: a minimization problem M + related to the upper bound of the quantity of interest, and a minimization problem M -related to the lower bound of the quantity of interest. These minimization problems are solved by the iHLRF algorithm [START_REF] Zhang | Two improved algorithms for reliability analysis[END_REF]. At each step, the optimization algorithm requires the construction of an upper bound of the quantity of interest I + (u h (y)) (or a lower bound I -(u h (y))), as described in Figure 2. These quantities can be related to the bounds of the discretization error in the quantity of interest. Let us introduce + h (y) and - h (y) such that

ε( h ) = max low h h -1 , upp h h -1 (25)
P low f,FORM P f,FORM P upp f,FORM (26)
- h (y) = I (u h (y))-I -(u h (y)) + h (y) = I + (u h (y))-I (u h (y)) (27)
By introducing - h (y) and + h (y) in Equation ( 15), one obtains

-- h (y) I (u(y))-I (u h (y)) + + h (y) ( 28
)
where I (u(y))-I (u h (y)) is the discretization error in the quantity of interest. The following section is devoted to the computation of the bounds - h (y) and + h (y).

Estimation of the outputs of interest

In this section, we briefly recall the techniques developed in the literature for the case where the quantity of interest is a linear functional I of the displacement

R : e(u(y)) d + f•u(y) d (29) I(u(y))= C
where R and f are the given extractors which define the output of interest (an example of the output of interest is given in Section 5). We refer the reader to References [START_REF] Prudhomme | On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors[END_REF][START_REF] Ladevèze | Local error estimators for finite element analysis[END_REF][START_REF] Ohnimus | Local error estimates of fem for displacements and stresses in linear elasticity by solving local neumann problems[END_REF][START_REF] Prudhomme | Practical methods for a posteriori error estimation in engineering applications[END_REF] for a detailed description of this approach, and to [START_REF] Gallimard | Evaluation of the local quality of the von mises's stress and l2-norm of the stress[END_REF][START_REF] Strouboulis | A posteriori estimation and adaptive control of the error in the quantity of interest. Part 1: a posteriori estimation of the error in the von Mises stress and the stress intensity factor[END_REF][START_REF] Beckers | An optimal control approach to a posteriori error estimation in finite element method[END_REF][START_REF] Heintz | On adaptive strategies and error control in fracture mechanics[END_REF][START_REF] Ruter | Goal-oriented a posteriori error in linear elastic fracture mechanics[END_REF] for the extension to non-linear quantities of interest. For the sake of clarity we will omit the reference to the space of the random parameters Y. Let us introduce the classical discretization error in Equation ( 28)

e h = u-u h ( 30
)
due to the linearity assumption, one has

I (e h ) = I (u)-I (u h ) (31)
thus, the estimate of |I (u)-I (u h )| is equivalent to the estimate of |I (e h )|. Following [START_REF] Ladevèze | Local error estimators for finite element analysis[END_REF], we consider the following auxiliary problem: find u aux ∈ U 0 and r aux = Ke(u aux ) such that:

Ke(u * ) : e(u aux ) d = I (u * ) ∀u * ∈ U 0 ( 32
)
One obtains the following relation for the error estimate:

I (u)-I (u h ) = I (e h ) = Ke(e h ) : e(e aux h ) d ( 33
)
where e aux h = u aux -u aux h is the discretization error of the auxiliary problem. Following [START_REF] Prudhomme | Practical methods for a posteriori error estimation in engineering applications[END_REF], the error in the quantity of interest can be written as

I (u)-I (u h ) = Ke(e h ) : e(e aux h ) d = 1 4 u + -u + h 2 u - 1 4 u --u - h 2 u ( 34
)
where u u = Ke(u) : e(u) d , u ± = su±s -1 u aux and u ± h = su h ±s -1 u aux h . The scaling factor s is chosen so that se h u and s -1 e aux In order to determine bounds for the error in the quantity of interest, Equation [START_REF] Tada | The Stress Analysis of Crack Handbook (2nd edn)[END_REF] indicates that it is sufficient to bound the energy norm of u + -u + h and u --u - h . Methods to obtain bounds on the error in the energy norm are now available in the literature [START_REF] Rannacher | A feedback approach to error control in finite element methods: application to linear elasticity[END_REF][START_REF] Peraire | Bounds for linear-functional outputs of coercive partial differential equations: local indicators and adaptive refinement[END_REF][START_REF] Prudhomme | On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors[END_REF]. In this paper, the methodology proposed in [START_REF] Gallimard | A constitutive relation error estimator based on traction-free recovery of the equilibrated stress[END_REF] which is based on the concept of constitutive relation error estimator [START_REF] Ladevèze | Error estimate procedure in the finite element method and application[END_REF] is used to build bounds of these errors. This method leads to lower bounds ẽ± h and upper bounds ê± h such that (we refer the reader to [START_REF] Gallimard | A constitutive relation error estimator based on traction-free recovery of the equilibrated stress[END_REF] for more details)

ẽ+ h u + -u + h u ê+ h ẽ- h u --u - h u ê- h (35)
Once these bounds are computed, the classical upper and lower bounds are easily derived from Equations ( 34) and (35)

-- h I (u)-I (u h ) + h ( 36
)
where

- h = 1 4 ( ê- h ) 2 -1 4 (ẽ + h ) 2 and + h = 1 4 (ê + h ) 2 -1 4 (ẽ - h ) 2
The bounds of the output of interest are directly computed from I h , ẽ± h , ê±

h I + (u h ) = I (u h )+ 1 4 ( ê+ h ) 2 -1 4 ( ẽ- h ) 2 and I -(u h ) = I (u h )-1 4 (ê - h ) 2 + 1 4 (ẽ + h ) 2 (37)

APPLICATION TO FRACTURE MECHANIC

The output of interest I in the following problems is the mode I stress intensity factor (SIF) K I computed at the crack tip in a cracked structure represented in Figure 3. An extraction operator is used to compute the SIF K I . This extraction operator is based on a modification of Stern's contour integral [START_REF] Stern | A contour integral computation of mixed-mode stress intensity factors[END_REF] proposed in [START_REF] Gallimard | Error estimation of stress intensity factors for mixed-mode crack[END_REF]. The calculation of the SIF consists in an integral around an arbitrary crown 2 around the tip of the crack (Figure 4), and leads to K I without introducing any errors except for those due to the calculation of u

K I = I (u) = 2 (Ke( ũI )-rI ) : e(u) d - 2 rI grad()•u d ( 38
)
where rI and ũI are singular analytical solutions computed at the crack tip and a function such that (M) = 1 on C 1 and (M) = 0 on C 2 (for further details see [START_REF] Gallimard | Error estimation of stress intensity factors for mixed-mode crack[END_REF][START_REF] Stern | A contour integral computation of mixed-mode stress intensity factors[END_REF]). The approximate value computed by an FEA I h will be denoted K Ih with

K Ih = I (u h ) = 2 (Ke( ũI h )-rI ) : e(u h ) d - 2 rI grad()•u h d (39)
The limit state of the structure is given by a failure criterion which is the risk that the SIF K I reaches the toughness of the material

K Ic G = K Ic -K I 0 (40)
The studied structures are plates in plane stress with a finite crack shown in Figures 5 and6. The deterministic parameters of the model are the Young modulus E = 210 000 MPa, the Poisson ratio = 0.25, L = 32 mm, w = 8 mm. The random variables of the model are the crack length a, the toughness of the material K Ic in the two examples, the loading t 1 in example 1, the loading t 2 and the angle of the crack in example 2,

• the crack length a is supposed to have a lognormal distribution with mean m a = 2 mm and a standard deviation a = 0.4, • the material toughness K Ic is supposed to follow a normal distribution with a mean value of m K = 4 MPamm 1/2 and a standard deviation of K = 0.4, • the tensile load t 1 is supposed to follow a normal distribution with a mean value of m t 1 = 1 Mpa and a standard deviation of t 1 = 0.1, • the shear load t 2 is supposed to follow a normal distribution with a mean value of m t 2 = 0.5 Mpa and a standard deviation of t 2 = 0.1, • the angle of the crack is supposed to follow a normal distribution with a mean value of m = 5 • and a standard deviation of = 5 • . These variables are easily transformed in standard normal variables u a , u t , u K and u

• a = exp( a u a + a ), • t = t u t +m t , • K Ic = K u K +m K , • = u +m
where a and a are the parameters of the lognormal distribution given by a = log(m a / 1+(m a / a ) 2 ) = 0.674 and a = log(1+(m a / a ) 2 ) = 0.198. The informations on the reliability models are summarized in Table I.

X (a, t 1 , K I c ) (a, , t 2 , K I c ) G K I c -K I (a, t 1 ) K I c -K I (a, , t 2 ) Y (u a , u t , u K ) (u a , u , u t , u K ) H K u K +m K K u K +m K -K I (exp( a u a + a ), t u t +m t )
-K I (exp( a u a + a ), u +m , t u t +m t )

Figure 7. Initial meshes (DoF: 234, 326, 670, 1250, 1846).

We use 6-node triangular elements to perform the finite element computations, and we study the evolution of the results as a function of the mesh size. The five meshes used are represented in Figure 7.

Remark

It must be noticed that as we need to remesh the structure on each iterations of the iHLRF algorithm where we modify the crack length or its angle, we use in fact different meshes during the computation of h . On the examples studied, the number of elements of these different meshes varies little (less than 10%). We choose, as a reference for the number of degree of freedom, the mesh built with a equal to its mean value and equal to 0.

Example 1: plate submitted to a tensile load

For this problem, analytical results are available for K I in [START_REF] Tada | The Stress Analysis of Crack Handbook (2nd edn)[END_REF] (Equation (41)) with less than 0.1% error:

K I = (1.-0.025 2 +0.06 4 ) sec 2 t √ a ( 41
)
where =a/w. We perform first a deterministic computation with the random variables set to their mean values. Equation (41) leads to a reference value for the SIF K I,ref = 2.575. The evolution of the FE SIF as well as the computed lower bound K - Ih and upper bound K + Ih are reported in Table II. It is worth noting that the deterministic computation gives SIFs that are smaller than the critical value. In this example, the computed bounds are very sharp, the lower bound and the upper bound are a better approximation of the reference value than the finite element solution. The fact that the finite element solution is outside the bounds is not inconsistent, as the relation (36) only implies that K - Ih and K + Ih bound the reference solution, but not necessarily the FE solution. For the reliability computations, the initial values of the standard normal variables u a , u t and u K are set to zero. We can obtain a reference value for , using the iHLRF algorithm with K I computed from Equation (41). At convergence the reliability index is ref = 2.356, and we will use this value as a reference value. The convergence of the iHLRF algorithm is shown in Figure 8 for the five meshes and for the analytical solution. It can be seen that for this simple problem the algorithm converges very rapidly.

The bounds of the reliability index as a function of the mesh size are reported in Table III. In this example, the gap between the computed bounds upp h and low h decreases rapidly. It must be noticed that, as in the deterministic computation of the bounds of K I , the finite element evaluation of may be outside the computed bounds. However, ref is inside the bounds low h and upp h . The computation of the relative error estimate ε( h ) [START_REF] Rackwitz | Structural reliability under random load sequences[END_REF] gives a valuable information on the accuracy of the reliability computation. It can be compared with the reference error on h : |ε( h )| (Equation ( 23)), by computing an effectivity index = ε( h )/|ε( h )|. This effectivity index seems to be sufficiently sharp with values ranging from 1.32 to 1.68.

Following Equation ( 6), the reference failure probability obtained with a FORM approximation is P f,ref = (ref ) = 9.2×10 -3 . The failure bounds derived from Equation ( 26) are presented in Table IV. It can be observed that, although the finite element probability of failure converge to the reference failure probability, the failure probability computed on the coarsest meshes is clearly inferior to the reference value and can lead to an underestimation of the risk. On the other hand, the computation of the bounds gives a good estimation of the failure probability on the third mesh (670 DoF) with an upper bound equal to P upp f,h = 9.5×10 -3 .

Example 2: plate with an inclined crack

The loading of the second problem involves a combination of modes I and II at the tip of the crack. We have computed a reference solution using the iHLRF algorithm with a refined mesh (59690 Dof). In this case we obtain ref = 2.548. The convergence of the iHLRF algorithm is shown in Figure 9 for the five meshes and for the reference solution. The iHLRF algorithm converges rapidly when the mesh is sufficiently refined.

In Table V, we study the evolution of the bounds of the reliability index as a function of the mesh. Although the gap upp h -low h is greater than in Example 1, the bounds become very accurate as soon as the mesh is sufficiently refined (less than three percent of error for the 670 DoF mesh). The computation of the error estimate ε( h ) and of the reference error |ε( h )| shows a good behaviour of the estimation as the effectivity index is ranging from 1.04 to 1.23.

In example 2, the reference failure probability is P f,ref = (-2.548) = 5.5×10 -3 . The bounds obtained by the FORM approximation are presented in Table VI. As in example 1 (see Table IV), the FEA leads to an underestimation of the failure probability when the mesh is not sufficiently refined. We obtain rapidly a good approximation of the upper bound of the failure probability with P upp f,h = 5.6×10 -3 on the second mesh (326 DoF), however, on this mesh the lower bound is very small P low f,h = 4.0×10 -3 , and a reliability analysis with a more refined mesh is necessary to obtain less than 10% of uncertainty on the computation of failure probability with P f = 5.2±0.3×10 -3 on the third mesh (670 DoF).

CONCLUSIONS

A method has been presented to recover lower and upper bounds for the computation of a reliability index in a finite element reliability analysis. This method is based on goal-oriented error estimation techniques that have been initially developed for deterministic computations. These bounds can be used to assess the quality of the finite element meshes used during a reliability analysis. They can also be used to derive bounds of the failure probability when a FORM approximation is used. Numerical experiments in linear elastic fracture mechanics show that the computed bounds are sufficiently sharp to control the error introduced by a finite element approximation in a reliability analysis. These bounds will allow to develop mesh adaptative strategy during a finite element reliability analysis.

Figure 1 .

 1 Figure 1. Computation of the reliability index.

Figure 2 .

 2 Figure 2. Computation of the bounds of the reliability index: * = + or * = -.

  with P low f,FORM = (-upp h ), P upp f,FORM = (-low h ) and P f,FORM = (-). The bounds low h and upp h

Figure 3 .

 3 Figure 3. The structure .

Figure 4 .

 4 Figure 4. Zoom around the crack's tip.

Figure 5 .

 5 Figure 5. Example 1 (tensile load).

Figure 6 .

 6 Figure 6. Example 2 (shear load).

Figure 8 .

 8 Figure 8. Convergence of the reliability index evaluation by an iHLRF algorithm (tensile load).

Figure 9 .

 9 Figure 9. Convergence of the reliability index evaluation by an iHLRF algorithm (inclined crack).

Table I .

 I Reliability models.

	Example 1	Example 2

Table II .

 II Example 1 (tensile load): Deterministic computation K I,ref = 2.575.

	DoF		K Ih		K + Ih		K -Ih
	234		2.509	2.613		2.509
	326		2.522	2.592		2.540
	670		2.560	2.581		2.566
	1250		2.567	2.577		2.570
	1846		2.570	2.577		2.572
		2.8
					analytical solution
					234 DoF
		2.7			326 DoF
					670 DoF
		2.6			1250 DoF 1846 DoF
	Beta	2.5
		2.4
		2.3
		2.2	1	2	3	4	5
					Iteration

Table III .

 III Example 1 (tensile load): Reliability computation ( ref = 2.356).

	DoF	h	upp h	low h	upp h -low h	|ε( h )| (%)	ε( h ) (%)
	234	2.531	2.510	2.238	0.272	6.89	11.58	1.68
	326	2.453	2.417	2.326	0.091	3.93	5.18	1.32
	670	2.385	2.373	2.345	0.028	1.19	1.66	1.40
	1250	2.377	2.366	2.349	0.017	0.87	1.19	1.37
	1846	2.372	2.363	2.350	0.013	0.47	0.74	1.56

Table IV .

 IV Example 1 (tensile load): Failure probability bounds-P f,ref = 9.2×10 -3 .

	DoF		P f,h			P low f,h		P	upp f,h
	234 326 670 1250 1846		5.7×10 -3 7.1×10 -3 8.5×10 -3 8.7×10 -3 8.8×10 -3	6.0×10 -3 7.8×10 -3 8.8×10 -3 9.0×10 -3 9.1×10 -3		1.26×10 -2 1.00×10 -2 9.5×10 -3 9.4×10 -3 9.4×10 -3
		3
		2.8
		2.6
	Beta	2.4 2.2			Reference solution (59690 DoF) 234 DoF
					326 DoF
		2			670 DoF
					1250 DoF
		1.8			1846 DoF
		1.6	0	2	4	6	8	10
					Iteration

Table V .

 V Example 2 (inclined crack): Reliability computation ( ref = 2.548).

	DoF	h	upp h	low h	upp h -low h	|ε( h )| (%)	ε( h ) (%)
	234	2.821	2.723	2.485	0.238	9.68	11.91	1.23
	326	2.759	2.650	2.538	0.112	7.65	8.01	1.05
	670	2.626	2.586	2.541	0.045	2.97	3.24	1.09
	1250	2.616	2.586	2.545	0.041	2.60	2.71	1.04
	1846	2.587	2.567	2.539	0.030	1.51	1.86	1.23

Table VI .

 VI Example 2 (inclined crack): Failure probability bounds-P f,ref = 5.5×10 -3 .

	DoF	P f,h	P low f,h	P	upp f,h
	234 326 670 1250 1846	2.4×10 -3 2.9×10 -3 4.3×10 -3 4.4×10 -3 5.1×10 -3	3.2×10 -3 4.0×10 -3 4.9×10 -3 4.9×10 -3 5.1×10 -3	6.5×10 -3 5.6×10 -3 5.5×10 -3 5.5×10 -3 5.6×10 -3

ACKNOWLEDGEMENTS

This work was supported by the Publishing Arts Research Council under the grant number 98-1846389.


Au Docteur

Lydie Abaléa

Je remercie le jury :

Au Professeur Jean-Yves Le Reste, je vous remercie de me faire l'honneur de présider ce jury de thèse.

Au Professeur Bernard Le Floc 'h, je vous remercie de participer à ce jury de thèse.

Au Docteur Patrice Nable, je vous remercie d'avoir dirigé cette thèse, de m'avoir proposé de participer à ce travail de recherche. Soyez assuré de ma reconnaissance et de tout mon respect.

La statisticienne, Gatineau Florence, pour son aide et son travail

MAITRES DE CONFERENCES DES UNIVERSITÉS -PRATICIENS HOSPITALIERS

MONOT Alain Français

RIOU Morgan Anglais

Je remercie ma tutrice :

Le Docteur Catherine Bourillet, de m'avoir accompagné dans mon 3 e cycle avec sérieux et sympathie.

Je remercie mes proches :

Tout d'abord, à ma mère, pour ton affection, le soutien que tu m'as apporté.

A mon frère, Mathieu, pour son soutien et pour sa relecture avisée.

A mes grands-parents, Louis et Annik, pour votre générosité, d'avoir toujours cru en moi et m'avoir soutenu pendant toutes ses années.

A ma famille, mes cousins, mes cousines, mes oncles et tantes. Au « bon docteur Marcel » pour m'avoir encouragé à choisir la médecine générale après l'ECN. Pour le plaisir de passer des moments en famille.

A mes amies, à Elodie, Emma et en particulier à Marion et Marine pour m'avoir supportée durant toutes ses années, soutenu et écoutée. Merci pour votre joie de vivre.

A mes co-internes

Et je remercie :

Les médecins, infirmiers, aides-soignants et personnels paramédicaux rencontrés pendant mes études.

A tous ceux qui ont fait de moi, le médecin que je suis aujourd'hui. Merci A patient is considered "depressive" if her/his mean HSCL-25 score is greater than or equal to 1.75. In the HSCL-group one in 16 patients have performed PSE-9 while in the HSCL + group, the ratio of 1 in 2.

Results

Patients included met the inclusion criteria. After removed duplicates and wrongly inclusion, 1126 patients were included among 1134. The factor analysis showed that HSCL-25 tool is a one-dimensional tool -which combined an anxiety and a depression dimensions -with a Cronbach alpha of 0.93.

Conclusion

The HSCL-25 scale has a high eigenvalue. It was a one-dimensional tool which combined items according anxiety and depression. This reliable tool will provide survey in primary care daily practice.

Table of content

BACKGROUND

Depression is one of the most common mental illness in France (1,2) and Europe [START_REF] Ayuso-Mateos | Depressive disorders in Europe: prevalence figures from the ODIN study[END_REF][START_REF] Prevalence | Severity, and Unmet Need for Treatment of Mental Disorders in the World Health Organization World Mental Health Surveys[END_REF]. This syndromic disease combine multiple symptoms as sadness, loss of interest, loss of pleasure, feelings of guilt or low self-esteem, sleep or appetite disorders, fatigue, lack of concentration… and somatic disorders(5-7). In France and many other countries, depression is defined according to the diagnostic criteria of ICD-10 and DSM-V.

According to the World Health Organization (WHO) depression affects 350 million people in the world [START_REF] Oms | La dépression[END_REF]. The prevalence varies between the different countries in Europe [START_REF] Inpes | prévalence, recours au soin et sentiments d'information de la population[END_REF][START_REF]Data and statistics[END_REF][START_REF] King | Prevalence of common mental disorders in general practice attendees across Europe[END_REF]. The prevalence of depression is twice as high for women as for men.

Depression affects more than 3 million people in France. According to a survey published in 2012 of National Institute of Prevention and Education for Health (INPES), the prevalence is estimated to be between 5 and 12% of the whole population [START_REF] Sapinho | Mesure de l'épisode dépressif majeur en population générale: apports et limites des outils[END_REF]. Currently, nearly 8 million

French people have experienced or will experience depression during their lifetime (11,12).

And 7.5% of French people aged between 15 and 85 years old suffered from depression in the past 12 months (13).

Depression take a significant impact on emotional, social and occupational life (1). It is a major risk factor for suicide [START_REF] Lejoyeux | Prevalence and risk factors of suicide and attempted suicide[END_REF][START_REF] Aouba | La mortalité par suicide en France en[END_REF]. France is one of the European countries with the highest rate of suicide mortality [START_REF]Tentatives de suicide et suicides / Données de surveillance par pathologie / Santé mentale / Maladies chroniques et traumatismes / Dossiers thématiques / Accueil Tentatives de suicide et suicides / Données de surveillance par pathologie / Santé mentale / Maladies chroniques et traumatismes / Dossiers thématiques[END_REF]. Each year, more than 10,400 people die from suicide, e.g. one in fifty deaths in France [START_REF] Beck | Tentatives de suicide et pensées suicidaires en France en[END_REF].

Its management is multimodal, combining or not drugs treatments and psychotherapeutics activities.

In most European countries, General practitioners (GPs) are the port of call to diagnose and manage depressed patients [START_REF] Alonso | Psychotropic drug utilization in Europe: results from the European Study of the Epidemiology of Mental Disorders (ESEMeD) project[END_REF][START_REF] Dezetter | Factors Associated With Use of Psychiatrists and Nonpsychiatrist Providers by ESEMeD Respondents in Six European Countries[END_REF][START_REF] Ani | Depression symptomatology and diagnosis: discordance between patients and physicians in primary care settings[END_REF]. General practitioners are the first consulted, before psychiatrists and psychologists [START_REF] Starfield | Global health, equity, and primary care[END_REF][START_REF] Ho | The Efficacy and Cost-Effectiveness of Stepped Care Prevention and Treatment for Depressive and/or Anxiety Disorders: A Systematic Review and Meta-Analysis[END_REF]. GPs are present in 67% of the care paths and in half of the cases, they are the only one to take care [START_REF] Mitchell | Clinical diagnosis of depression in primary care: a metaanalysis[END_REF][START_REF] Norton | Prévalence des troubles psychiatriques en médecine générale selon le patient health questionnaire : adéquation avec la détection par le médecin et le traitement prescrit[END_REF][START_REF] Ferrari | The burden attributable to mental and substance use disorders as risk factors for suicide: findings from the Global Burden of Disease Study 2010[END_REF][START_REF] Bonté | Episode dépressif caractérisé de l'adulte : prise en charge en premier recours. Recommandation de bonne pratique[END_REF][START_REF] Lépine | Depression in the community: the first pan-European study DEPRES (Depression Research in European Society)[END_REF].

Diagnosing major depression has a high specificity but low sensitivity in general practice [START_REF] Mitchell | Clinical diagnosis of depression in primary care: a metaanalysis[END_REF]. For the past 25 years, it is known to academic psychiatry and primary care that only 10-50% of depressed patients receive adequate treatment because of the difficulty and the inability to recognize depression. Less than one person in four suffering from depression consults a doctor, and is therefore diagnosed and treated appropriately [START_REF] Ferrari | The burden attributable to mental and substance use disorders as risk factors for suicide: findings from the Global Burden of Disease Study 2010[END_REF][START_REF] Bagby | The Hamilton Depression Rating Scale: Has the Gold Standard Become a Lead Weight?[END_REF][START_REF]Conférence de consensus : la crise suicidaire : reconnaître et prendre en charge[END_REF]. This may be related to the patient's reluctance to express symptoms and the variability in the symptomatology of depression [START_REF] Jorm | Mental health literacy. Public knowledge and beliefs about mental disorders[END_REF]. But another obstacle in depression recognition is that traditional psychiatric interviewing is not appropriate in primary care settings because it takes too long [START_REF] Mathers | Projections of Global Mortality and Burden of Disease from 2002 to[END_REF][START_REF] Sharp | Screening for depression across the lifespan: a review of measures for use in primary care settings[END_REF][START_REF] Sartorius | An international study of psychological problems in primary care. Preliminary report from the World Health Organization Collaborative Project on "Psychological Problems in General Health Care[END_REF][START_REF] Reynolds | Depression and Aging: A Look to the Future[END_REF][START_REF] Eisinger | Syndrome dépressif[END_REF].

The European General Practice Research Network (EGPRN), member of the World Organization of National Colleges, Academies and Academic Associations of General Practitioners / Family Physicians (WONCA), was established to promote and develop research in general medicine and primary care and to initiate and coordinate international research. In 2014, a systematic review of the literature validated 7 diagnostic tools for depression in primary care, comparing them to a psychiatric clinical examination according to DSM major depressive criteria [START_REF] Nabbe | Systematic literrature review : what validated tools are used to screen or diagnose depression in general practice ? European General Practice Research Network[END_REF][START_REF] Nabbe | Which DSM validated tools for diagnosing depression are usable in primary care research? A systematic literature review[END_REF]. After a consensus method (RAND / UCLA), the "Hopkins Symptom Checklist in 25 items" (HSCL-25) taking the most combined efficiency, reliability and ease to use [START_REF] Nabbe | Systematic literrature review : what validated tools are used to screen or diagnose depression in general practice ? European General Practice Research Network[END_REF][START_REF] Nabbe | Which DSM validated tools for diagnosing depression are usable in primary care research? A systematic literature review[END_REF][START_REF] Beck-R | Quel est l'outil de diagnostic de la dépression le plus approprié chez le patient adulte, en médecine générale en Europe, selon ses critère d'efficacité, de reproductivité et[END_REF][START_REF]Consensis d'expets sur la traduction en Français d'une échelle d'auto-évaluation de la dépression[END_REF][START_REF] Hoeymans | Measuring mental health of the Dutch population: a comparison of the GHQ-12 and the MHI-5[END_REF][START_REF] Nabbe | Depression and multimorbidity in family Medicine: systematic literature review : what valided tools are used to screen or diagnose depression in general practice ? Study FPDM[END_REF]. The third phase, in December 2014, was the round translation of the HSCL-25 scale in French, according to a forward/backward Delphi procedure translation followed by a cultural check, in order to keep linguistic and meaning homogeneity from the original version.

The HSCL-25 is a self-rating scale on the existence and severity of both anxiety and depression symptoms during the preceding week, used to identify psychiatric illness in The study focused the result of HSCL-25 on the score greater than or equal to 1.75.

According to the accuracy features of the HSCL-25, a patient is recognized depressed if his score is greater than or equal to 1.75. Conversely, if his score is strictly below the threshold of 1.75, the patient will not be recognized as depressed.

This part of the study focused the results on the internal validation and dimensionality of the French HSCL-25 scale in primary care daily practice.

Inclusion criteria

Adult patients older or equal than 18 years old (Legal age of majority in France), without age limit. Patients had to give their written ethical consent for participation in the study. Patient had to complete the HSCL-25 self-questionnaire and submit it to the investigating.

Exclusion criteria

In order to avoid puerperal depression in which the management is different, women with a reported pregnancy were not included in the study [START_REF]Troubles émotionnels et psychiques des mères. Le vécu des femmes et des couples, leurs besoins[END_REF]. As a puerperal depression needs a specific management. Adults consulting for a medical certificate were excluded as the research was focused on adults consulting for at least a medical condition in primary care.

Psychotic patients were excluded. Patients requiring immediate acute care were excluded

Location of study

The recruitment was carried out in primary care, in the surgeries of the investigating practitioners belonging to the network of internship masters (MSU) of the University Department of General Medicine (DUMG) of Brest, included in the EA SPURBO Network.

The population was according to the centres: urban, semi-rural and rural. The study was carried out in the North Brittany

In the waiting room patients had a leaflet explaining the study, an HSCL-25 scale and a consent form. The recruitment was made either spontaneously by the participants after reading the explanatory notice, or on proposal and after information by the GP investigator after information in the waiting room. Two groups have been defined: § a HSCL+ group § a HSCL-group.

A stratified randomization was performed in each group, with: § with a step on one out of two (1/2) in the HSCL+ group § with a step on one out of sixteen (1/16) in the HSCL-group.

The selected randomized patients were offered an interview for PSE-9. § If the patient had a HSCL+, the PSE-9 was performed within one week of inclusion. § If the patient had a HSCL-, the PSE-9 was performed within one month after inclusion.

For patients not randomly selected, the study was terminated. But all patients with a score of ≥ 1.75 were informed by the investigating physician, usually as potentially depressed, to initiate the necessary care with them GPs, according to the ethical principle and the ethical consent form.

According to Nettelbladt et al., the expected difference is of the order of 60% against 15%, in all cases at least 50% against 20%. With these assumptions and for a power of 90%, it was necessary to have measured at least 88 PSE-9 in HSCL+ group and 44 PSE-9 in HSCL-.group for randomised patients [START_REF] Nettelbladt | Test characteristics of the Hopkins Symptom Check List-25 (HSCL-25) in Sweden, using the Present State Examination (PSE-9) as a caseness criterion[END_REF].

This required the recruitment of 880 patients without counting the follow-up patients, the frequency of which should not exceed 20%. 1100 patients had to be included in order to take account of follow-up patients. They were divided between the three semi-rural, urban and rural centres.

Method of analysis

Statistical analysis was carried out by statisticians

All patients were included in the analysis, except incorrectly included and duplicates (only the first consultation was taken into account).

The final analysis of the data was to be carried out after the database freeze at the end of the data review meeting.

The data had to be analyzed by the Data Management Unit (UGD) of the Brest CHRU.

The statistical analyzes had to be carried out with SAS software version 9.4 and R version 3.2.0.

The tests had to be carried out at the risk of first species alpha 5%.

The number of subjects was re-evaluated during the study due to the unexpected distribution of patients in both groups. The analyzes concerning the validation of the questionnaire were added: dimensionality and internal consistency.

A factor analysis was carried out to determine the number of dimensions of the HSCL-25 scale and the distribution of the items in each of these dimensions (main component analysis, factor analysis with varimax rotation) and a study of the internal consistency of Each dimension obtained (calculation of the alpha coefficient of Cronbach).

RESULTS

The study team consisted of 2 physician researchers, 3 Gps trained in the psychiatric examination using the PSE-9, 1 psychiatrist, 1 statistician, 20 general practitioners (divided into 3 geographical areas: urban, semi-rural and Rurales), 1 Data Manager and 1 Research

Coordinator.

Duration of the study

The inclusion period was 20 weeks. The duration of participation for each patient was 1 week.

The total duration of the study was 12 months. The study was conducted between 2015 and 2016.

Flow chart

The number of patients included, having passed HSCL-25, who were randomized to the PSE-9 group or not, and having passed the PSE-9, are shown in the flow chart (Figure 1). Wrongly included: n=2 Duplicates : n=6 1134 patients were selected: 2 patients were wrongly included and 6 were duplicates.

Among 1126 patients, a group of 886 patients HSCL-and a group of 240 patients HSCL+ were included.

Stratified randomisation:

According a pitch of 1/16, 55 patients were randomised in the HSCL-group.

According a pitch of 1/2, 118 patients were randomised in the HSCL+ group.

Population included and sample randomized

The age and sex of included population is described in table 2 After randomization:

55 patients in the HSCL-25-group had to pass the PSE-9.

• 46 patients performed the PSE-9, and 9 were follow-up.

• 6 were positive to PSE-9 and 40 negative.

118 patients in the HSCL+ group had to pass the PSE-9.

• 96 patients performed the PSE-9, and 22 were follow-up.

• 67 were positive to PSE-9 and 29 were negative.

Results are presented in the contingency table in Table 4. Dimensional structure and internal consistency A factor analysis to determine the number of dimensions and the distribution of items on these dimensions have been carried out.

The results of the eigenvalue diagram are presented in Figure 2. It is confirmed that the french HSCL-25 is a one-dimensional tool.

Study of the internal consistency of each dimension obtained

Table presents the distribution that could be obtained if two dimensions were retained (as described in the literature: depression and anxiety). Note a different distribution of items between our study and the literature.

According to the factor analysis, a group 1 (corresponding to a "depression" dimension) is retained with items 11-19 and 21-24. And a group 2 (corresponding to an "anxiety" dimension) with items 1-10, 20 and 25.

In the literature, items related to anxiety are items of 1-10 and items of depression of 11-25. Table 6 shows the analysis that contains the value of the Cronbach alpha index.

The value is 0.93 in a one-dimensional analysis. This result was upper the minimum threshold of 0.7 (Nunnaly, 1978). This 0.7 tag is arbitrary, but widely accepted by the scientific community. Consequently, we can say that for this HSCL-25 scale composed of 25 items, we obtain a satisfactory internal coherence.

DISCUSSION

The HSCL-25 is a scale on the existence of both anxiety and depression symptoms.

This survey studied the dimensionality and internal validity of the French version.

This depression diagnostic tool is confirmed as a one-dimensional scale. The reproducibility of the entire scale, according to Cronbach's alpha was 0.93. This high result showed her high reproducibility in daily practice for outpatients; especially if the two subscales depression and anxiety are combined to converge towards the same objective.

The HSCL 25 is now the second validated self administered questionnaire in French to diagnose depression in primary care. The 4 DSQ was the first one but with two pitfalls: no Cronbach's alpha calculation for its reliability in French and no link with the gold standard diagnosis in psychiatry (DSM V). ( 46)

In primary care research, in daily practice, the multiplicity of users requires the use of tools with high stability.

Dimension

Cronbach's alpha

dimension only

Item axis 1 0,93 In the literature, there are other one-dimensional scales adapted for primary care. In the twodimensional analysis, the Cronbach alpha index obtained in group 1 "depression" (items 11-19 and 21-24) is 0.89 (compared to 0.9 in the literature). The results in group 2 "anxiety"

(items 1-10, 20 and 25) have a Cronbach coefficient at 0.87 (compared to 0.85 in the literature).

According to the psychometric properties, elements strongly correlated with each other probably indicate the measure of the same construct and this is exactly what we mean when we say of a scale that it is internally consistent. Moreover, this also refers to the idea that this scale measures a construct that is one-dimensional. Having a one-dimensional measurement increases the Cronbach's alpha.

This scale is coherent (or homogeneous) because the elements (depression and anxiety) converge towards the same result, which increases Cronbach's alpha.

The Patient Satisfaction with Psychotropic (PASAP) scale has been shown to have good psychometric properties in a large bipolar population and therefore appears adequate to assess the satisfaction of the treatment. The main component analysis was in favor of unidimensionality and a Cronbach's alpha coefficient of 0.85. Its short length and good acceptability make it suitable for clinical research [START_REF] Nordon | Patient satisfaction with psychotropic drugs: Validation of the PAtient SAtisfaction with Psychotropic (PASAP) scale in patients with bipolar disorder[END_REF].

A study on pathological gambling among young people aged 15 to 19 was carried out in Italy using the SOGS-RA (Sychometric Evaluation of the South Oaks Gambling Screen: Revised for Adolescents) tool [START_REF] Colasante | Italian Adolescent Gambling Behaviour: Psychometric Evaluation of the South Oaks Gambling Screen: Revised for Adolescents (SOGS-RA) Among a Sample of Italian Students[END_REF]. The one-dimensional principal component of SOGS-RA was accepted, and with an acceptable internal consistency with a Cronbach alpha of 0.78.

The CLDSES (cholesterol-lowering diet self-efficacy scale) study of initial self-efficacy of cholesterol-lowering diets shows good psychometric properties with an internal coherence of 0.95 and factor analysis in favor of unidimensionality of the subscales [START_REF] Burke | Evaluation of the shortened cholesterol-lowering diet self-efficacy scale[END_REF].

When creating a scale, there are two possibilities: either it will be created to deal with a single theme e.g. one-dimensional or multidimensional.

The results depend therefore on how the scale is constructed and the goal sought. There seems to be no consensus on the unidimensional or multidimensional character of the scales, this depends on the character sought in the scale. In our case, we could have distinguished a "depression" dimension and an "anxiety" dimension, but the two themes are close enough to form a single dimension.

The literature finds the opposite of the bi-dimensional scales used in primary care.

The WHOQOL-Brief study showed a 5-domains model (psychological, physical, social relations, environment and level of independence) with adequate internal coherence. This study confirmed the multidimensional nature of the study [START_REF] Oliveira | Toward an understanding of the quality of life construct: Validity and reliability of the WHOQOL-Bref in a psychiatric sample[END_REF].

The OPAS (Ocular Pain Assessment Survey) study was a survey on the assessment of eye pain, specifically designed to measure eye pain and quality of life. The intensity of eye pain was the primary outcome measure, and the secondary outcome measures were interference with quality of life, aggravating factors, associated factors, intensity of associated non-ocular pain and self-reported symptomatic relief. The factor analysis retained 6 subscales confirming multidimensionality. The coefficient Cronbach's α was 0.83. OPAS is a reliable and responsive tool with strong psychometric and diagnostic properties in multidimensional quantification of the intensity of corneal and ocular surface pain and quality of life [START_REF] Qazi | Validity and Reliability of a Novel Ocular Pain Assessment Survey (OPAS) in Quantifying and Monitoring Corneal and Ocular Surface Pain[END_REF].

The Hopkins questionnaire was originally developed for non-traumatized populations but was later adapted and widely used for the assessment of traumatized populations [START_REF] De Fouchier | Validation of a French adaptation of the Harvard Trauma Questionnaire among torture survivors from sub-Saharan African countries[END_REF]. HSCL-25 has been used to identify psychiatric disorders in primary care, family planning services, refugee populations and migrant populations [START_REF] Syed | Concordance between Hopkins Symptom Checklist (HSCL-10) and Pakistan Anxiety and Depression Questionnaire (PADQ), in a rural self-motivated population in Pakistan[END_REF].

Currently, the Hopkins symptoms checklist is widely used in primary care in various situations, such as psychological trauma after natural disasters, terrorist acts or among refugees for example [START_REF] Schock | Impact of new traumatic or stressful life events on pre-existing PTSD in traumatized refugees: results of a longitudinal study[END_REF].

One can cite the study "Mental health and psychosocial problems in the aftermath of the The tool is regularly used on the refugee population [START_REF] De Fouchier | Validation of a French adaptation of the Harvard Trauma Questionnaire among torture survivors from sub-Saharan African countries[END_REF][START_REF] Schock | Impact of new traumatic or stressful life events on pre-existing PTSD in traumatized refugees: results of a longitudinal study[END_REF][START_REF] Sonne | Psychosocial predictors of treatment outcome for trauma-affected refugees[END_REF][START_REF] Sonne | Treatment of trauma-affected refugees with venlafaxine versus sertraline combined with psychotherapy -a randomised study[END_REF][START_REF] Lavik | Mental disorder among refugees and the impact of persecution and exile: some findings from an out-patient population[END_REF]. The study "Mental disorder among refugees and the impact of persecution and exile: some findings from an outpatient population" [START_REF] Lavik | Mental disorder among refugees and the impact of persecution and exile: some findings from an out-patient population[END_REF] showed in particular the use of HSCL-25 refugees constitute a population at risk of mental disorders. Past traumatic stresses and current existence in exile are independent risk factors.

limitations of the study

The study population was multicentric, coming from rural, semi-rural and urban areas.

Concerning the sampling data, there was a double validation by the data manager and the research coordinator. Initially, all calculations were made on a theoretical prevalence of depression of 7% in the French adult population, but the prevalence obtained in the study was 21% in general practice. This observed difference in prevalence can be explained by overrecruitment, which can be explained by the White's diagram. Based on a population of 1000 adults exposed to a health problem in the month, there were 750 people who could experience a health problem. Referring to the classical definition of primary care, only when a person consults with a health care practitioner begins primary care, about 250 of the 750 people with a health condition, or a quarter Population of departure [START_REF] Budowski | Comment former les futurs généralistes ? De la difficulté pour les généralistes de nombreux pays à enseigner dans les écoles ou les facultés de médecine[END_REF]. This phenomena can explains the focusing of health problems in daily practice surgeries

The difference is all the more important, as depressive syndrome is complex to diagnose.

Britain has the highest rate of suicide in France (62), it is very likely that this is correlated with a depression rate greater than in the general French population.

The data manager to verify that all HSCL-25 questionnaires were complete limited selection bias. The psychiatrist checked that all the PSE-9 questionnaires were complete.

For Information bias, all data were collected (sampling data, HSCL-25 and PSE-9) were compiled into a computer database. At the end of the study, all information was checked one last time and the database was frozen for statistical work.

CONCLUSION

The dimensional structure was studied, the eigenvalue diagram showed that the HSCL-25 is a one-dimensional tool.

A factorial analysis was carried out to test the internal coherence of each dimension.

According to the factor analysis, a group 1 (corresponding to a "depression" dimension) is retained with elements 11-19 and 21-24. And a group 2 (corresponding to an "anxiety" dimension) with elements 1-10, 20 and 25. The literature defines 2 different groups: an "anxiety" group from 1 to 10 and a "depression" group from 11 to 25 In the analysis of the Cronbach alpha index, the value retained was 0.93 in a one-dimensional analysis. The internal consistency of the scale is satisfactory. This scale is coherent, it has a high level of homogeneity.

These results validate the HSCL-25 scale compared to PSE-9 in its French version, in primary care for its reliability. As the HSCL-25 was selected for its efficiency, reliability and ergonomics this result will be useful in everyday practice in general practice in France.

The HSCL 25 is the second validated French self administered questionnaire to diagnose depression in general practice (according to its reliability). It is the only one that has achieved a full reliability study. It could be widely used in primary care. The great usability of this tool will allow research in daily practice, for outpatients.

Introduction

  Le diagnostic de dépression est difficile en ambulatoire. Les outils diagnostiques sont rarement utilisés par les médecins généralistes. La Hopkins Checklist Symptom en 25 questions (HSCL-25) est un outil clinique validé contre l'examen psychiatrique basé (selon les critères du DSM). Après une RAND/UCLA, la HSCL-25 a été sélectionnés comme le meilleur choix pour ses qualités combinées d'efficacité, de fiabilité et d'ergonomie. Elle a été traduite selon une procédure aller/retour basé sur méthode Delphi. La dernière phase de l'étude avait pour objectif de comparer les versions françaises de la HSCL-25 au « Patient State Examination version 9 »(PSE-9). Cette étude a pour objectif la validation interne et l'étude de la dimensionnalité de la HSCL-25 dans sa version française en pratique courante. Méthode Etude quantitative de comparaison entre HSCL-25 et PSE-9, sur plusieurs centres médicaux, pour des patients adultes, après consentement, considérés ou non dépressifs par les 2 outils. Un patient est considéré comme « dépressif » si le score moyen HSCL-25 est supérieur ou égal à 1,75. Dans le groupe HSCL+, 1 patient sur 2 a réalisé le PSE-9 dans la semaine qui suivait le test HSCL-25. Dans le groupe HSCL-, 1 patient sur 16 a réalisé le PSE-9 dans un délai de 1 mois. Résultats Les patients inclus répondaient aux critères d'inclusion. L'étude de la consistance interne selon une analyse factorielle montre une dimension « anxiété » et une dimension « dépression ». Ces dimensions sont combinées pour donner un outil unidimensionnel. L'alpha de Cronbach est élevé à 0,93. Conclusion La HSCL-25 a été traduite selon une procédure garantissant sa stabilité. Son étude de validation en pratique courante trouve une haute valeur propre et une grande stabilité. Fiable et ergonomique, elle peut être utilisé aussi bien en recherche qu'en pratique courante dans les cabinets de Médecine Générale. Introduction Diagnosis of depression is difficult and diagnostic tools are rarely used by GPs. The Hopkins Checklist Symptom in 25 items (HSCL-25) was retained to fill this gap. It is a clinical tool, validated against a psychiatric examination according to DSM major depression criteria. Following a RAND / UCLA, the HSCL-25 has been selected as the most efficient, reliable and ergonomic tool combined. The HSCL-25 has been translated into French using a forward/backward translation according a Delphi procedure. The last phase consisted in comparing HSCL-25 scale against Patient State Examination-9 version (PSE-9), it has confirmed the internal validation and dimensionality of the French HSCL-25 version in primary care. Method Adults' patients who completed ethical consent were selected. Patients lived in Finistère (France), in 2015.

  BACKGROUND .................................................................................................................... METHOD ............................................................................................................................... Inclusion criteria .............................................................................................................................. Exclusion criteria ............................................................................................................................. Location of study ............................................................................................................................. Number of patients .......................................................................................................................... Method of analysis ...........................................................................................................................

  primary care. It includes 25 items: 10 about anxiety, 15 about depression. Each item is rated on a 4-point Likert scale. The points are totalled and divided by[START_REF] Ferrari | The burden attributable to mental and substance use disorders as risk factors for suicide: findings from the Global Burden of Disease Study 2010[END_REF]. A 1-to-4 score is obtained. The patient is considered as a "probable psychiatric case" if the mean rating on the HSCL-25 is ≥ 1.55. A cut-off value of ≥ 1.75 is generally used for the diagnosis of major depression, defined as "a case, in need of treatment"[START_REF] Derogatis | The Hopkins Symptom Checklist (HSCL): A Self Report Symptom Inventory[END_REF]. The Present State Examination in its 9 th version (PSE-9), or standardized psychiatric examination, is a comprehensive, systematized, semi-directed clinical interview. The PSE was developed in England by Wing et al. It explores, in 140 items, a large part of the psychiatric semiology (43). It is a standardized questionnaire realised by a psychiatrist or a professional specially trained to the psychiatric examination, which evaluates the mental state. In Sweden, in 1993 a study of Nettlebladt & al aimed to evaluate the effectiveness of the HSCL-25 scale in primary care daily practice. A survey of patients consulting GPs has been conducted in order to compare HSCL-25 against PSE-9. It reported a concordance of 86.7% (44). This scale is not used by French GPs. A French survey based on the study of Nettlebladt & al has been decided.to determinate the external and internal validity of the French HSCL-25 version. This part of the study was focused on the internal validation and the dimensionality of the French HSCL-25 version. METHOD The EA SPURBO (primary care, public health, Brest Cancer Registry in Western Brittany) research network team carried out a quantitative cross-validation study of HSCL-25 against PSE-9, in an adult population, in daily practice, in medical surgery. It was a multi-centered, rural and urban, comparative and non-inferiority study, with 2 groups. The objective was the validation and characterisation of the French HSCL-25 scale version according to external and internal features. In order to compare the HSCL-25 and the PSE-9 scoring, a specials investigators GPs team were training by a combined Gps and psychiatrist researchers team, to use HSCL-25 and PSE-9.

  Number of patients Two types of patient were possible: § Patient HSCL-25 score ≥1.75 or HSCL+ or positive, § Patient HSCL-25 score <1.75 or HSCL-or negative.

Figure 1

 1 Figure 1: Flow Chart

Figure 2 :

 2 Figure 2: Diagram of eigenvalues

Table 7 :

 7 The Cronbach Alpha Coefficient French HSCL-25 version

  Nepal earthquakes: findings from a representative cluster sample survey"[START_REF] Kane | Mental health and psychosocial problems in the aftermath of the Nepal earthquakes: findings from a representative cluster sample survey[END_REF] carried out after the two major earthquakes in 2015 in Nepal. The purpose of this study was to examine the frequency of common mental health and psychosocial problems and their correlates as a result of earthquakes. The results were selected from several qualitative tools including only the Hopkins Symptom Checklist-25 for depression and anxiety. Tools also include the search for Post-Traumatic Stress Disorder (PTSD-Civilian Checklist), The consumption of dangerous alcohol (AUDIT-C), Symptoms of severe psychological distress (WHO / UNHCR Symptom Assessment Program), Suicidal Idea (Composite International Diagnostic Interview), Perceived needs (scale of perceived humanitarian needs (HESPER)), and functional impairment (scale developed locally). The overall results suggest that there were significant levels of psychological distress. The results highlight the importance of appropriate prevention strategies to reduce the risk of distressing disorder in post-disaster mental health care systems. The study "Validity of the Hopkins Symptom Checklist-25 amongst HIV-positive pregnant women in Tanzania" (56) was designed to validate the detection of depression in pregnant women with HIV in Tanzania by the Hopkins-25 (HSCL-25). The internal consistency of HSCL-25 (alpha 0.93) was adequate. HSCL-25 has demonstrated utility as a diagnostic tool for depression in primary care in Dar es Salaam, but the results are discussed because it has shown an inability to measure the severity of symptoms in this African cultural context and in developing countries where HIV is widespread. The French translation procedure was careful to use a method that preserved linguistic and meaning homogeneity, combining a forwardbackward translation by Delphi procedure and a cultural control by consensus group. With this method, avoiding discrepancies related to the biases of language and culture and context has preserved the homogeneity from the original. The HSCL-25 was adapted in the same way for French or English outpatients. HSCL-25 was used in a longitudinal study "A Longitudinal Study of Psychological Distress and Exposure to Trauma Reminders After Terrorism" (57) on psychological distress 2.5 years after the terrorist attack on Utoya Island, (Norway), among survivors.

  Confusion bias was limited as the self-administered questionnaire was completed by every patient without external influence. All responses collected during the PSE-9 interview were retrospectively monitored by the psychiatrist in a group meeting with the 3 reviewers.General practitioners are faced with a difficulty in diagnosing depression by the multitude of symptoms. Screening is difficult because the depressive syndrome is masked by somatic complaints and emotional difficulties patients are poorly expressed. The Hamilton scale was little used because it was not adapted to a common practice in general practice. The HSCL-25 scale, divided into 2 sub-sections related to the presence and intensity of anxiety and depressive symptoms, will allow a self-completed questionnaire in 5-10 minutes by patients to screen for Patients suffering from psychiatry, in primary care. The efficacy, reliability and ergonomic qualities of HSCL-25 retained in phase 2 of the study will enable daily use in general practice. The use of HSCL-25 will allow a more important diagnosis in general medicine, which is one of the most common diseases (under diagnosed) in France. The main risk of depression is suicide. Aided by an adapted tool will allow the appropriate diagnosis and management of depressive patients. The applications of the HSCL-25 are multiple with an application for research in general practice, because her high reproducibility and onedimensionality is focused on anxious and depressed outpatients. Her ease to use is an advantage, it increases the feasibility of survey but also It can be use in a perspective of teaching way.

DE 2 ère CLASSE

	DELLUC Aurélien	Médecine interne
	DE VRIES Philipe	Chirurgie infantile
	HILLION Sophie	Immunologie
	LE BERRE Rozenn	Maladies infectieuses-Maladies tropicales
	LE GAC Gérald	Génétique
	LODDE Brice	Médecine et santé au travail
	QUERELLOU Solène	Biophysique et Médecine nucléaire
	SEIZEUR Romuald	Anatomie-Neurochirurgie
	MAITRES DE CONFERENCES -CHAIRE INSERM
	MIGNEN Olivier	Physiologie
	MAITRES DE CONFERENCES
	AMOUROUX Rémy	Psychologie
	HAXAIRE Claudie	Sociologie -Démographie
	LANCIEN Frédéric	Physiologie
	LE CORRE Rozenn	Biologie cellulaire
	MONTIER Tristan	Biochimie et biologie moléculaire
	MORIN Vincent	Electronique et Informatique

MAITRES DE CONFERENCES ASSOCIES MI-TEMPS

	BARRAINE Pierre	Médecine Générale
	NABBE Patrice	Médecine Générale
	CHIRON Benoît	Médecine Générale
	AGREGES DU SECOND DEGRE

.......................................................................................................... Duration of the study ....................................................................................................................... Flow chart ......................................................................................................................................... Population included and sample randomized ............................................................................... Dimensional structure and internal consistency ...........................................................................

	A factor analysis to determine the number of dimensions and the distribution of items on these
	dimensions .....................................................................................................................................
	Study of the internal consistency of each dimension obtained .....................................................

RESULTS .....................

DISCUSSION ......................................................................................................................... CONCLUSION ...................................................................................................................... BIBLIOGRAPHY ..................................................................................................................

Table 1 :

 1

	ITEMS N°	HSCL-2 ORIGINAL VERSION	FRANCE
		Choose the best answer for how you felt over the past week	Veuillez choisir la réponse qui décrit le mieux comment globalement vous vous sentiez toute la semaine dernière
		Being scared for no reason	Vous avez peur sans raison
		Feeling fearful	Vous vous sentez effrayé
		Faintness	Vous avez une sensation d'étourdisse-
			ment
		Nervousness	Vous vous sentez nerveux
		Heart racing	Vous avez l'impression que votre coeur bat anormalement vite
		Trembling	Vous avez la sensation de trembler
		Feeling tense	Vous vous sentez tendu
		Headache	Vous avez des maux de tête
		Feeling panic	Vous vous sentez paniqué
		Feeling restless	Vous vous sentez agité
		Feeling low in energy	Vous manquez d'énergie
		Blaming oneself	Vous ressentez une sensation de culpabilité
		Crying easily	Vous pleurez facilement
		Losing sexual interest	Vous ressentez un désintérêt pour la vie sexuelle
		Feeling lonely	Vous avez une sensation de solitude
		Feeling hopeless	Vous vous sentez désespéré
		Feeling blue	Vous avez le cafard
		Thinking of ending one's life	Vous avez pensé à mettre fin à votre vie
		Feeling trapped	Vous vous sentez pris au piège
		Worrying too much	Vous vous inquiétez trop
		Feeling no interest	Plus rien ne vous intéresse
		Feeling that everything is an effort	Tout est un effort pour vous
		Worthless feeling	Vous avez le sentiment d'être bon à rien
		Poor appetite	Vous avez perdu l'appétit
		Sleep disturbance	Votre sommeil est perturbé

Table 2 :

 2 Age and sex of included population

	Variable	Class	Overall	Group	Group	P *
			Population	HSCL-	HSCL+
	Age	N	1126	886	240	<0.001
		Mean +/-	55,62 +/-	56,60 +/-	51,98 +/-
		SD	18,4	18,6	17,0
		Median	59	61	53
		(q1-q3)	(42 -70)	(42-72)	(38 -66)
		Min-max	18-94	18-94	19-91
	Sex	Male	452 (40%) 390 (44%) 62 (26%)	<0.001
		Female	674 (60%) 496 (56%) 178 (74%)
	SD: Standard deviation
	*obtained by Student t test for quantitative variables and Chi² test for qualitative variables
	The median age was 59 years old (42-70). The patients were aged between 18 and 94 years
	(table 2).

Table 4 :

 4 Contingency table for HSCl-25 and PSE-9

	PSE-9
	« Depressive » « Not depressed » TOTAL

Table 5 :

 5 Weight of items on each dimension according to factor analysis

		1 factor selected	2 factors selected
	ITEM	Axis 1	Axis 1 (Depression) Axis 2
		(Depression	and	(Anxiety)
		Anxiety combined)
	Item 1	0.54	0.19	0.60
	Item 2	0.58	0.25	0.58
	Item 3	0.50	0.17	0.56
	Item 4	0.69	0.27	0.73
	Item 5	0.51	0.08	0.67
	Item 6	0.53	0.19	0.57
	Item 7	0.72	0.36	0.67
	Item 8	0.48	0.22	0.47
	Item 9	0.71	0.38	0.63
	Item	0.67	0.38	0.57
	Item	0.60	0.57	0.27
	Item	0.69	0.58	0.39
	Item	0.56	0.43	0.36
	Item	0.45	0.46	0.16
	Item	0.67	0.64	0.30
	Item	0.73	0.70	0.33
	Item	0.75	0.69	0.35
	Item	0.55	0.61	0.15
	Item	0.68	0.65	0.30
	Item	0.68	0.39	0.58
	Item	0.68	0.75	0.19
	Item	0.63	0.71	0.16
	Item	0.68	0.69	0.25
	Item	0.54	0.48	0.28
	Item	0.59	0.36	0.47
	In blue: items related to depression	In red: items related to anxiety

Table 6 :

 6 The Cronbach Alpha Coefficient French HSCL-25 version

	Dimension

Remerciements

Quelle est la validation interne

Adults' patients who completed ethical consent were selected. Patients lived in Finistère (France), in 2015.

A patient is considered "depressive" if her/his mean HSCL-25 score is greater than or equal to 1.75. In the HSCLgroup one in 16 patients have performed PSE-9 while in the HSCL + group, the ratio of 1 in 2.

Patients included met the inclusion criteria. After removed duplicates and wrongly inclusion, 1126 patients were included among 1134. The factor analysis showed that HSCL-25 tool is a one-dimensional tool -which combined an anxiety and a depression dimensions -with a Cronbach alpha of 0.93.

The HSCL-25 scale has a high eigenvalue. It was a one-dimensional tool which combined items according anxiety and depression. This reliable tool will provide survey in primary care daily practice.

MOTS CLES