Jocelyne Erhel

Zoubida Mghazli

Mestapha Oumouni

An adaptive sparse grid method for elliptic PDEs with stochastic coefficients

Keywords: Elliptic PDEs with random coefficients, Stochastic collocation method, Anisotropic sparse grid, Adaptation method

The stochastic collocation method based on the anisotropic sparse grid has become a significant tool to solve partial differential equations with stochastic inputs. The aim is to seek a vector weight and a convenient level for the method. The classical approach uses a posteriori approach on the solution, yielding to an additional prohibitive cost with a large stochastic dimension.

In this work, we discuss an adaptive approach of this method to calculate the statistics of the solution. It is based on an adaptive approximation of the inverse diffusion parameter. We construct an efficient error indicator which is an upper bound of the error on the solution. In the case of unbounded variables, we use an appropriate error estimation to compute suitable weights of the method. Numerical examples are presented to confirm the efficiency of the approach and showing that the cost is considerably reduced without loss of accuracy.

Introduction

The Monte Carlo method [START_REF] Fishman | Monte Carlo: Concepts, Algorithms, and Applications[END_REF] is the most classical approach used to compute statistical quantities of interest depending on the solution of partial differential equation with stochastic inputs. It consists of solving M deterministic problems, where M is the number of independent and identically distributed simulations (iid) of the parameters. The main disadvantage of this approach is its slow convergence given by the order O( 11 solving problems of stochastic PDE [START_REF] Ghanem | Stochastic finite elements: a spectral approach[END_REF][START_REF] Babuŝka | A stochastic collocation method for elliptic partial differential equations with random input data[END_REF][START_REF] Babuŝka | Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation[END_REF]. They approach the response of the model as a stochastic function by a polynomial interpolation in the stochastic space and provide an exponentially convergent approximation when the solution of the problem is a smooth function with respect to the stochastic variable. However, they require solving a large deterministic problem and this limiting their application sometimes to the problem with small stochastic dimension.

Those methods suffer from the curse of dimensionality, since the computational cost increases exponentially with a stochastic dimension and sometimes those methods are impracticable for very large dimension. Therefore, when the stochastic dimension is moderately large, it is interesting to minimize the cost and the effort of the resolution. This can be done by carefully selecting the stochastic degrees of freedom.

In this work, we discuss the anisotropic sparse grid method with adaptive approach inspired from the works [START_REF] Babuŝka | A stochastic collocation method for elliptic partial differential equations with random input data[END_REF][START_REF] Nobile | A sparse grid stochastic collocation method for elliptic partial differential equations with random input data[END_REF][START_REF] Erhel | Calcul de l'espérance de la solution d'une EDP stochastique unidimensionnelle à l'aide d'une base réduite[END_REF]. In [START_REF] Erhel | Calcul de l'espérance de la solution d'une EDP stochastique unidimensionnelle à l'aide d'une base réduite[END_REF], we have introduced the Kahrunen-Loève expansion of the inverse of the diffusion parameter where we have used the random variables basis of this expansion to compute the average solution of one dimensional elliptic problem. Motivated by this work, we use this inverse of the diffusion to construct an error indicator to carefully choosing the parameter of the collocation method which are the level and the anisotropic weights. More precisely, we give an appropriate expansion of the indicator on the Lagrange polynomials basis, then we use this basis to construct the solution of the stochastic problem. We prove that this indicator satisfies an upper bound of the error. This process requires only a sequence of interpolation problems unlike the traditional approach which requires a large sequence of deterministic problems.

This paper is organized as follows. In the first section, we introduce the mathematical problem setting with a finite noise assumption. In section 2, we recall the stochastic collocation method based on the full tensor product and sparse grid interpolation. In Section 3, we introduce an indicator which will be used to determine the stochastic degrees of freedom of the collocation method. We prove that this indicator gives an upper bound of the error on the solution. Finally, in section 4, we present numerical examples to illustrate the efficiency of the proposed approach and compare with the traditional approach.

Problem formulation

Let (Ω, F, dP) be a complete probability space, where Ω is the space of elementary event, F ⊂ 2 Ω is the σ-algebra of events and P is the probability measure. Also, we consider a bounded domain D ⊂ R d , with a smooth boundary ∂D. The parameters k and f are two random fields on Ω × D. We focus on the following linear elliptic boundary value problem: find a stochastic function, u : Ω × D -→ R such that the following equation holds (a.s) in Ω:

(P s ) -div(k∇u) = f in Ω × D, u | ∂D = 0. (1)
We make the following assumptions on the random parameters k and f to keep the well-posedness of the problem (P s ).

Assumptions 1:

• H1: f belongs in L p (Ω) ⊗ L 2 (D), for all p ∈ [1, ∞[.
• H2: There exist k min , k max such that, for each ω ∈ Ω (a.s) we have

0 < k min (ω) ≤ k(ω, .) ≤ k max (ω) and k -1 min ∈ L p (Ω).
Let H 1 0 (D) be the subspace of H 1 (D) consisting of the functions with vanishing trace on ∂D, then we define the space L p (Ω) ⊗ H 1 0 (D),

L p (Ω) ⊗ H 1 0 (D) := v : Ω -→ H 1 0 (D); Ω v p H 1 0 (D) dP < ∞ .
To introduce the stochastic discretization, we first state some finite dimensional assumption on the parameters k and f (see [START_REF] Babuŝka | A stochastic collocation method for elliptic partial differential equations with random input data[END_REF][START_REF] Nobile | A sparse grid stochastic collocation method for elliptic partial differential equations with random input data[END_REF][START_REF] Nobile | An anisotropic sparse grid stochastic collocation method for elliptic partial differential equations with random input data[END_REF]).

Finite dimensional noise

In many practical problems, random coefficients are parameterized by a finite number of uncorrelated random variables, sometimes independent, as in the case of a truncated Karhunen-Loève or Fourier expansion [START_REF] Loève | Probability Theory I" I[END_REF][START_REF] Loève | Probability Theory[END_REF][START_REF] Babuŝka | Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation[END_REF]. This motivates us to assume that k and f are parameterized by N random variables {Y n } N n=1 . The number N is called the stochastic dimension.

Assumptions 2 (Finite dimensional noise)

The stochastic fields k and f used in the computations are parameterized by a random vector

Y = (Y 1 , ..., Y N ) with N components such as k(ω, x) = k(Y(ω), x) and f (ω, x) = f (Y(ω), x).
We denote Γ n := Y n (Ω) the image of the set Ω by Y n for each n = 1, . . . , N and Γ the image Y(Ω). We assume that the components of Y are independent and ρ n is the density function of each Y n . Then Y has a joint probability density ̺(y) = N n=1 ρ n (y n ), ∀y ∈ Γ and Γ := N n=1 Γ n .

Exemple1 (Karhunen-Loève (K-L) expansion)

The Karhunen-Loòve expansion [START_REF] Loève | Probability Theory I" I[END_REF][START_REF] Loève | Probability Theory[END_REF] allows to expand each random field k ∈ L 2 (Ω)⊗L 2 (D) in the series expansion given by the product of deterministic functions and a random variables:

k(ω, x) = E[k](x) + ∞ n=1 λ n φ n (x)Y n (ω), (2)
where {λ n } ∞ n=1 et {φ n } ∞ n=1 are respectively eigenvalues and eigenfunctions solution of the following eigenvalue problem:

D cov[k](x, z)φ(z)dz = λφ(x). Where cov[k](x, z) := E[k(., x)k(., z)] -E[k(., x)]E[k(., z)] and {Y n } ∞ n=1 are un- correlated random variables given by Y n = 1 √ λ n D (k(., x) -E[k(x)])φ n (x)dx.
We define the truncated (K-L) expansion k N by,

k N (ω, x) = E[k](x) + N n=1 λ n φ n (x)Y n (ω). (3)
Exemple2 (A nonlinear coefficient) In some cases, to ensure some properties for data, as the positivity, the parameter k is given by a nonlinear transformation of a Gaussian field G like that k = g(G), where g > 0 is non negative and smooth. A widely used example is a log-normal coefficient k(ω, x) = e G N , where G N is the (K-L) truncation of the Gaussian field G.

Strong formulation

The stochastic functions k and f are parameterized by the vector Y as in Assumption 2, then by Doob-Dynkin's lemma, the solution u of (P s ) can be also parameterized by Y such that u(ω, x) = u(Y(ω), x). Therefore, we introduce the strong deterministic problem obtained by projecting (P s ) in H 1 0 (D) for all y in Γ (a.e):

   Find u(y, .) ∈ H 1 0 (D) such that D k(y, .)∇u(y, x).∇v(x)dx = D f (y, x)v(x)dx ∀v ∈ H 1 0 (D). (4)
for all y in Γ (a.e), by Lax-Milgram theorem, the problem (4) is well posed and u(y, .) satisfies:

u(y, .)

H 1 0 (D) ≤ C D k -1 min (y) f (y, .) L 2 (D) , (5)
where C D is the Poincaré constant. Under Assumption (H1, H2), the solution u belongs in L p ̺ (Γ) ⊗ H 1 0 (D) for all p ≥ 1, since:

E[ u p H 1 0 (D) ] ≤ C p D E[k -p min f p L 2 (D) ] ≤ C p D E[(k -1 min ) 2p ]E[ f 2p L 2 (D) ] .

Stochastic collocation method

The stochastic collocation method has the same principle as Monte Carlo method. It gives realizations of u by solving (4) at deterministic knots {y j } j of Γ and construct an approximation u h,p of u in the tensor space W p,h := P p (Γ) ⊗ V h (D). Where V h (D) is the standard finite element space and P p (Γ) is the span of tensor product polynomials with degree at most p = (p 1 , ..., p N ). The method is fully parallelized and provide exponential convergence when k and f are smooth with respect to y. We introduce the semi-discrete approximation u h obtained by projecting (4) into V h (D) for (a.s) y ∈ Γ:

   Find u h (y, .) ∈ V h (D) such that D k(y, .)∇u h (y, x).∇v h (x)dx = D f (y, x)v h (x)dx, ∀v h ∈ V h (D). (6)
We solve (6) at suitable knots {y j } j of Γ, then we obtain the family {u h (y j , .)} j to construct an approximation u h,p by multi-dimensional polynomial interpolation . It can be done either by a full tensor product rule of one-dimensional interpolation or on the sparse grid given by the Smolyak algorithm [START_REF] Smolyak | Quadrature and Interpolation Formulas for Tensor Products of Certain Classes of Functions[END_REF][START_REF] Wasilkowski | Explicit cost bounds of algorithms for multivariate tensor product problems[END_REF].

Full tensor product interpolation

The full tensor product interpolation is given by a product of one-dimensional interpolation, where a high order is used in each dimension. For n = 1, ..., N , let i n ∈ N + a level of one-dimensional interpolation, and X in := {y n,1 , ..., y n,m(in) } is a set of m(i n ) collocation points in Γ n .

For v ∈ C 0 (Γ n ; V h (D)), we introduce a sequence of one-dimensional Lagrange interpolation operator

U in : C 0 (Γ n ; V h (D)) -→ P pn (Γ n )⊗V h (D) defined by: U in (v)(z, x) = m(in) j=1 v(y n,j , x)ℓ in n,j (z), (7)
where ℓ in j (z) =

where p = (p 1 , . . . , p N ), m(i) = (m(i 1 ), . . . , m(i N )) the tensor space P p (Γ) := ⊗ N n=1 P pn (Γ n ) and we construct the knots in Γ by y j = (y 1,j 1 , . . . , y n,jn , . . . , y N,j N ) where each y n,jn ∈ X in . For each i, j := (j 1 , . . . , j N ) and y = (y 1 , . . . , y n ) ∈ Γ, we set L i j (y) = ⊗ N n=1 ℓ in n,jn (y n ) the multidimensional Lagrange polynomial.

Standard sparse grid method

The Standard sparse grid method is based on the Smolyak algorithm [START_REF] Smolyak | Quadrature and Interpolation Formulas for Tensor Products of Certain Classes of Functions[END_REF][START_REF] Wasilkowski | Explicit cost bounds of algorithms for multivariate tensor product problems[END_REF] which provides an effective way to approach multivariate functions by a polynomial interpolation. It is given by linear combinations of product formulas [START_REF] Erhel | Calcul de l'espérance de la solution d'une EDP stochastique unidimensionnelle à l'aide d'une base réduite[END_REF] where in each dimension, a small order of interpolation is used, this provides a significant reduction of the full interpolation complexity and the curse of dimensionality is reduced.

For each n = 1, . . . , N , we consider the one-dimensional operator U in associated to the knots {y n,j } m(in) j=1 as given in [START_REF] Davis | Methods of Numerical Integration[END_REF], and let ∆ in = U in -U in-1 be an incremental operator with U 0 = 0, for w ∈ N, we consider the set of multi-level defined by

X(w, N ) := i ∈ N N , i ≥ 1, |i| := N n=1 i n ≤ w + N .
The Smolyak interpolation operator A(w, N ) with level w is given by the formula:

A(w, N )v = i∈X(w,N ) (△ i 1 ⊗ . . . ⊗ △ i N )v. (9)
An equivalent formula (see [START_REF] Wasilkowski | Explicit cost bounds of algorithms for multivariate tensor product problems[END_REF]) which is more practical than ( 9) is given by:

A(w, N )v = w+1≤|i|≤w+N (-1) w+N -|i| N -1 w + N -|i| U i 1 ⊗ . . . ⊗ U i N v. (10)
To compute the interpolation A(w, N )v given in [START_REF] Gerstner | Numerical integration using sparse grid[END_REF], we need to evaluate v at knots of the "sparse grid" H(w, N ) = w+1≤|i|≤w+N {X i 1 × . . . × X i N }. In the case when the knots are nested, the Smolyak interpolant becomes more interesting, since, a significant reduction of the effort can be gained. Several nested points can be used according to their degree of accuracy and their growing number. For example: Newton-Cotes, Clenshaw-Curtis, Gauss-Patterson [START_REF] Davis | Methods of Numerical Integration[END_REF][START_REF] Gerstner | Numerical integration using sparse grid[END_REF][START_REF] Nobile | A sparse grid stochastic collocation method for elliptic partial differential equations with random input data[END_REF][START_REF] Burkardt | SGMGA, Sparse Grid Mixed Growth Anisotropic Rules[END_REF].

The anisotropic sparse grid method

In the standard sparse grid method, all of the directions are equitably treated globally with the same order, since any permutation of an admissible multi-levels in X(w, N ) is also admissible. However, when the dimensions have different behaviors, the solution is highly anisotropic and the convergence rate can be deteriorated when we use this isotropic method. Thus, an anisotropy interpolation is needed to improve the convergence.

We follow closely [START_REF] Nobile | A sparse grid stochastic collocation method for elliptic partial differential equations with random input data[END_REF], to define the anisotropic sparse grid formula. We consider a suitable weights α 1 , . . . , α N , for each dimension, α = min 1≤,n≤N α n and the generalized sets of multi-levels:

X α (w, N ) := i ∈ N N , i ≥ 1, N n=1 (i n -1)α n ≤ wα and Y α (w, N ) = i ≥ 1 : wα < N n=1 α n i n ≤ wα + |α| with |α| = N n=1 α n .
Then the anisotropic sparse grid formula is given by:

A α (w, N )v = i∈Xα(w,N ) (△ i 1 ⊗ . . . ⊗ △ i N )v, (11)
An equivalent formula of ( 11) is given by the product rule [START_REF] Gerstner | Dimension-adaptive tensor-product quadrature[END_REF]:

A α (w, N )v = i∈Yα(w,N ) c α (i) U i 1 ⊗ . . . ⊗ U i N v, (12)
where the combining coefficients

c α (i) = j∈{0,1} N ,i+j∈Xα(w,N ) (-1) |j| .
The computation of A α (w, N )u requires to evaluate v on the sparse grid:

H α (w, N ) = i∈Yα(w,N ) {X i 1 × . . . × X i N }. (13)
The anisotropic weights {α n } N n=1 allows to approximate adaptively u. In the next section we discuss a posteriori strategy to compute them with a reduced cost.

Error estimation and adaptivity in the stochastic space

The computational cost increases exponentially according to the stochastic dimension, sometimes, it can be dramatic. Thus, it is interesting to seek an appropriate way to reduce the global computation cost by carefully selecting suitable stochastic degrees of freedom and without loss of accuracy by an efficient strategy of adaptation.

Motivation

In the traditional approaches, when we attempt to approximate u by the full tensor product, adaptivity in the stochastic space consists of to start with a free degree of interpolation and increase the degree until reaching the desired accuracy, as described in [START_REF] Babuŝka | Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation[END_REF][START_REF] Babuŝka | A stochastic collocation method for elliptic partial differential equations with random input data[END_REF]. This approach seems very expensive, since in each iteration, we solve a set of deterministic PDE problems, which increases with N . The same strategy is used to determine the weights {α} N n=1 and the level w of the formula

A α (w, N )u.
In what follows, we introduce an indicator λ to determine with a reduce cost, the suitable stochastic degree of freedom to compute u. The problem (P s ) is equivalent to the following system:

-∇u = k -1 p div(p) = f.
The first equation shows that each stochastic dimension of u and k -1 p have the same behavior. From the second equation we expect that p and f have also the same behavior with respect to the stochastic variable. Therefore, we propose to combine those parameters to obtain an indicator λ = k -1 f , which allows to identify with a small effort the behavior of different directions of u.

In [START_REF] Erhel | Calcul de l'espérance de la solution d'une EDP stochastique unidimensionnelle à l'aide d'une base réduite[END_REF], for one-dimensional elliptic problem, we have computed the Karhunen-Loève expansion for k -1 , and use its random variables as a basis to build a projection of u which gives a good approximation of the expectation E[u]. In this work, we propose to use a similar methodology to approximate u in L 2 (Γ). We expand the indicator λ on the multi-dimensional Lagrange polynomials basis defined in [START_REF] Erhel | Calcul de l'espérance de la solution d'une EDP stochastique unidimensionnelle à l'aide d'une base réduite[END_REF] or [START_REF] Gerstner | Numerical integration using sparse grid[END_REF][START_REF] Ghanem | Stochastic finite elements: a spectral approach[END_REF], then we use this basis to approximate the solution u.

The following lemma concerns an estimate of the error u -A i u L 2 (Γ;H 1 0 (D) on u when we approximate it by the full tensor interpolation [START_REF] Erhel | Calcul de l'espérance de la solution d'une EDP stochastique unidimensionnelle à l'aide d'une base réduite[END_REF]. This is the main result given in [START_REF] Babuŝka | A stochastic collocation method for elliptic partial differential equations with random input data[END_REF] showing that each direction contributes to the global error with a decay rate depending on the size of the region where u is analytic. Lemma 4.1 ([3]) Let u be the solution of (P s ), A i u its full interpolation defined in [START_REF] Erhel | Calcul de l'espérance de la solution d'une EDP stochastique unidimensionnelle à l'aide d'une base réduite[END_REF]. We suppose that, in each direction, u admits an analytic extension in the region

Σ n (Γ n ; τ n ) := {z ∈ C, dist(z, Γ n ) ≤ τ n }, with the size τ n > 0. Then, there exists C F > 0 is independent of each p n = m(i n ) -1, such that: u -A i u L 2 ̺ (Γ)⊗H 1 0 (D) ≤ C F N n=1 β n (p n )e -νnp θn n (14) • if Γ n is bounded , θ n = β n = 1 and ν n = log 2τ n |Γ n | (1 + 1 + |Γ n | 2 4τ 2 n ) . • if Γ n is unbounded , θ n = 1/2, β n = O( √ p n ) and ν n = τ n /2.
The next proposition shows that the error λ -A i λ

L 2 ̺ (Γ)⊗L 2 (D) is an upper bound of the error u -A i u L 2 ̺ (Γ)⊗H 1 0 (D) .
Proposition 4.1 Let u be the solution of (P s ), λ = k -1 f and A i u and A i λ their approximations by the formula [START_REF] Erhel | Calcul de l'espérance de la solution d'une EDP stochastique unidimensionnelle à l'aide d'une base réduite[END_REF]. Then we have the following estimate:

u -A i u L 2 ̺ (Γ;H 1 0 (D)) ≤ C λ -A i λ L 2 ̺ (Γ,L 2 (D)) .
Where the positive constant C depends on u and λ.

The estimate given in Lemma 4.1 shows that the interpolation A i converges roughly like O(e -νp ) where ν = min n ν n and p the order used in its associated dimension. In the other word, there is p > 0 such that:

C 1 e -νp ≤ u -A i u L 2 ̺ (Γ;H 1 0 (D)) ≤ C 2 e -νp for each p > p. Thus, to compare u -A i u L 2 ̺ (Γ;H 1 0 (D)) with λ -A i λ L 2 ̺ (Γ;H 1 0 (D))
it is enough to compare τ n of u with that of λ for each n = 1, . . . , N . To do this, we adopt the following notations: for each n = 1, . . . , N , we put Γ * n = N j =n Γ j , y = (y n , y * n ) where y n ∈ Γ n and y * n ∈ Γ * n . We also suppose that f is polynomial with respect to y.

Proof of the Proposition 4.1 1) First, we consider the case where k is linear according to y: k = b 0 + N n=1 b n y n . Since f is polynomial on y, it is enough to compare the analyticity region of u with that of k -1 . The function u(., y * n , x) as a function of the variable y n , can be extended analytically in the ball {|z -y n | < τ n (u)} and we have by d'Alembert's rule:

1 τ n (u) = lim sup m-→∞ √ k∂ m yn ∇u L 2 (D) m! √ k∂ m-1 yn ∇u L 2 (D) (m-1)!
.

Since, we can write:

k -1 (z, y * n , x) = k -1 (y n , y * n , x) 1 + (z -y n )b n (x) k(y n , y * n , x) , then k -1 (., y * n , x) is analyti- cally extended in the region {|z -y n | < τ n (k -1
)}, such that:

1 τ n (k -1 ) = b n k ∞ .
On the other hand, by deriving m-time the problem (4) with respect to y n we obtain:

∂ m yn D k∇u∇vdx = D m l=0 m l ∂ l yn k∂ m-l yn ∇u∇vdx = ∂ m yn D f vdx ∀v ∈ H 1 0 (D),
for l ≥ 2 we have ∂ l yn k = 0 and ∂ yn k = b n , then:

D k ∂ m yn ∇u m! ∇vdx = - D b n ∂ m-1 yn ∇u (m -1)! ∇vdx + D ∂ m yn f m! vdx. (15)
Taking v = ∂ m yn ∇u in [START_REF] Loève | Probability Theory[END_REF] and by Cauchy-Schwartz inequality we obtain:

√ k∂ m yn ∇u L 2 (D) m! ≤ b n k ∞ √ k∂ m-1 yn ∇u L 2 (D) (m -1)! + C D k min (y) ∂ m yn f L 2 (D) m! .
Since f is polynomial on y, we have lim sup m-→∞

∂ m yn f L 2 (D) m! = 0, thus lim sup m-→∞ √ k∂ m yn ∇u L 2 (D) m! √ k∂ m-1 yn ∇u L 2 (D) (m-1)! ≤ b n k ∞ .
Then we deduce that τ n (u) ≥ τ n (k -1 ). Therefore, there exists C > 0 such that:

||u -A i u|| L 2 ̺ (Γ;H 1 0 (D)) ≤ C||λ -A i λ|| L 2 ̺ (Γ;L 2 (D)) . 2)
We suppose now that k is non linear according to y as k(y, x) = g(b 0 (x) + N n=1 b n (x)y n ), where g is an analytic and strictly positive function. Observe that in this case, k satisfies the assumption ensuring the analyticity of u (see [START_REF] Babuŝka | A stochastic collocation method for elliptic partial differential equations with random input data[END_REF]). Let {V l } l a discretization of the set Γ, where for each element V l with the size h l , the following estimate holds:

k(y, x) ≈ N n=1 a n (y l , x)y n ∀y ∈ V l .
Where y l is the center of V l . Let A l i u be the full interpolation for the restriction of u in each element V l , then by the previous case the following estimate holds:

||u -A l i u|| L 2 ̺ (V l ;H 1 0 (D)) ≤ c||λ -A l i λ|| L 2 ̺ (V l ;L 2 (D)) . (16)
Furthermore, for all y in V l , if we set ψ l (y, •) = u(y, •) -A i u(y, •), we have:

A l i ψ l = A l i u -A l i (A i u) = A l i u -A i u, since A i u is polynomial.
According to the classical h p convergence (see [START_REF] Babuŝka | Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation[END_REF][START_REF] Ciarlet | The Finite Element Method for Elliptic Problems[END_REF]), with a small size h l , the following estimation holds,

ψ l -A l i ψ l L 2 ̺ (V l ;H 1 0 (D)) ≈ ĉh p l ψ l L 2 ̺ (V l ;H 1 0 (D)) ( 17
)
where p is the smallest order used in the directions of A l i . Therefore, we deduce

||u -A l i u|| L 2 ̺ (V l ;H 1 0 (D)) ≈ ĉh p l ||u -A i u|| L 2 ̺ (V l ;H 1 0 (D)) . (18)
Also by the same argument, we have:

||λ -A l i λ|| L 2 ̺ (V l ;H 1 0 (D)) ≈ ĉh p l ||λ -A i λ|| L 2 ̺ (V l ;L 2 (D)) . (19)
Combining ( 18) and ( 19) with ( 16) we obtain in each V l :

||u -A i u|| L 2 ̺ (V l ;H 1 0 (D)) ≤ C||λ -A i λ|| L 2 ̺ (V l ;L 2 (D))
. Then the estimation hold on Γ.

Remark 4.1 The result of the proposition is still true with Smolyak interpolation A α (N, w), since the accuracy of the formula A α (N, w) depends on the size of the analyticity region of the function. Moreover, when k is linear with respect to y, the constant C can decrease with total number of collocation points used by A α (w, N ). Remark 4.2 (Choice of the reference points) When we attempt to determine a suitable polynomial degree p n according to direction n of A i λ, we set the other directions fixed at reference point y * n and we seek the degree p n such that two successive approximations are close. The degree m(i n ) may depend on y * n , so it must be carefully chosen. For a random field discussed in Section 2.1, the reference point y * n = 0 is adequate, since there is no interaction between directions.

In the next section, as an application of Proposition 4.1, we use the indicator λ = k -1 f to determine an appropriate level w and a weight vector α to compute A α (w, N )u.

Adaptive sparse grid

The main ingredients of the collocation method on the anisotropic sparse grid is the weight vector α and the level w. The vector α takes into account the importance between the dimensions. The level w is the total order of the interpolation, it controls the cardinality of the knots of the grid. Then, when we determine suitable weights and a convenient level w with a small cost, the method becomes more efficient.

Determination of the anisotropic weights

The main idea proposed in [START_REF] Nobile | An anisotropic sparse grid stochastic collocation method for elliptic partial differential equations with random input data[END_REF] consists of choosing each anisotropic weight α n as the exponential rate of the convergence according the direction n i.e α n = ν n , where ν n is the rate given in Lemma 4.1. The global error committed by the interpolation

A i u can be split into N parts, u -A i u L 2 ̺ (Γ,H 1 0 (D)) ≤ N n=1 ε n .
Each part ε n is the error contribution in the direction n which is estimated by:

ε n ≈ c n e -νnpn , when Γ n is bounded , (20)
ε n ≈ O( √ p n )e -νn √ pn , when Γ n is unbounded , (21)
When we attempt to solve highly anisotropic problem, we obtain numerically (Section 5) a non optimal vector weight with (21), since the resulting sparse grid is quite dense. The anisotropic property of the weight can is reduced because the square √ • in (21).

In order to remedy to this situation when Γ is unbounded, we propose to use an estimate error as in (20). We suppose that the density ̺ decreases like a gaussian kernel, and each density ρ n of Y n satisfies the following estimate:

ρ n (z) ≤ C n e -δz 2 , C n > 0 et δ > 0, ∀z ∈ R. (22)
We denote by H p (t) ∈ P p (R) the normalized Hermite polynomials,

H p (t) = 1 p!2 p π 1 2
(-1) p e -t 2 d p dt p e -t 2 .

We recall that they form a complete orthonormal basis of the space L 2 µ (R), where µ(t) = e -t 2 . We note also h p (t) = µ(t)H p (t), the associated Hermite functions. The following proposition gives an estimate of the error committed by U m v for an entire function v satisfying (22). The proof is based on an estimate of the Fourier-Hermite coefficients (see [START_REF] Boyd | Asymptotic coefficients of Hermite function series[END_REF]) for an entire function decreasing as a gaussian kernel.

Proposition 4.2

We consider κ in L 2 ρ (R; L 2 (D)), U i (κ) its Lagrange interpolation defined by m(i) Gauss points. We suppose that ρ satisfies (22), for each t ∈ R, κ(t, .) have an entire extension in the complex plan, and there is ν > 0 such that

e -ν|z| κ(z) L 2 (D) ≤ C ν < ∞, ∀z ∈ C. ( 23
)
Then, there is a decay g > 0 and a constant C does not depend on m(i) such that:

κ(t, .) -U i (κ)(t, .) L 2 ρ (R;L 2 (D)) ≤ Ce -gm(i) . ( 24
)
Proof: We consider the function κ(t, x) = κ(w(t), x) where w(t) = √ 2t δ and the expansion of κ in Hermite polynomials basis:

κ(t, x) = ∞ p=0 κ p (x)H p (t) where κ p (x) = R κ(t, x)H p (t)e -t 2 dt. ( 25
)
We set F (t, x) := κ(t, x)e -t 2 2 , its Fourier-Hermite expansion is given by:

F (t, x) = ∞ p=0 F p (x)h p (t) where F p (x) = R F (t, x)h p (t)dt, (26)
The Hermite coefficients F p in (26) are the same as in (25). Indeed,

F p (x) = R F (t, x)h p (t)dt = R κ(t, x)e -t 2 H p (t)dt = κ p .
The function κ satisfies (23), then F decreases as a gaussian kernel, since

κ(t, x) e -t 2 /2 ≤ C ν e √ 2 ν δ t e -t 2 /2 ≤ C ν e -1 2 (t- √ 2 ν δ ) 2 e ν 2 δ 2 ≤ de -qt 2 ,
where d and q are positive, finite, and depending on ν and δ. Clearly F is an entire function and decrease as a gaussian kernel, therefore (see [START_REF] Boyd | Asymptotic coefficients of Hermite function series[END_REF]) the Fourier-Hermite coefficients F p decreases geometrically as:

F p ≤ Ke -gp ∀p ∈ N,
where the constant K > 0 does not depend on p and the decay g > 0.

The following error can be bounded (see [START_REF] Babuŝka | A stochastic collocation method for elliptic partial differential equations with random input data[END_REF], Lemma 7) as:

κ(t, .) -U m(i) (κ)(t, .) L 2 ρ (R;L 2 (D)) ≤ min v∈P m(i) ⊗L 2 (D) max t∈R κ -v L 2 (D) e -(tδ) 2 4 . (27)
Taking v = Π m(i) κ in (27) the truncated Hermite expansion of κ up the order m(i)-1, and let ṽ = Π m(i) κ that of κ. We deduce:

κ(t, .) -U m(i) (κ)(t, .) L 2 ρ (R;L 2 (D)) ≤ max z∈R κ -Π m(i) (κ) L 2 (D) e -(zδ) 2 4 ≤ max t∈R κ -Π m(i) (κ) L 2 (D) e -t 2 2 ≤ ∞ p=m(i) κ p L 2 (D) max t∈R H p (t)e -t 2 2 ≤ C 1 ∞ p=m(i) e -gp max t∈R |h p (t)|.
Knowing that max t∈R |h p (t)| ≤ 1 ( (see e.g. [START_REF] Boyd | Asymptotic coefficients of Hermite function series[END_REF]), we get,

κ(t, .) -U m(i) (κ)(t, .) L 2 ρ (R;L 2 (D)) ≤ C 1 ∞ p=m(i) e -gp ≤ C 2 e -gm(i) .
Remark 4.3 When k = e G follows a log-normal law, the indicator λ is an entire function. Thus, in the case of unbounded variables, we use the estimate (24) to compute numerically the rate convergence in each direction of λ. Numerical tests have shown that using estimate (24) yields a fast convergence of the formula A α (w, N ).

In order to determine the anisotropic weight, we use the bound given by the Proposition 4.2 for both bounded and unbounded set Γ, we use the algorithm given in [START_REF] Nobile | A sparse grid stochastic collocation method for elliptic partial differential equations with random input data[END_REF] for the indicator λ instead of the function u. The bound given in ( 20) and ( 24) is only true foe the Gaussian knots, then to obtain suitable weights , we use those knots with the growth rule of the order m(i n ) = i n for each dimension of λ.

Algorithm to determine α n

• We fix y * n = 0, as a reference point, i.e y j = 0 for all j = n.

• Let m( ĩn ) = ĩn be an integer such that λ(y n , 0) := U ĩn λ(y n , 0) is considered as a reference approximation according to the direction n.

• For each p = 1, . . . , ĩn -1, we compute the error: err n (p) = λ(y n , 0) -U p λ .

• We use a linear least squares algorithm to estimate α n and c n solutions of the following minimization problem:

min αn,cn log(err n ) -log(c n ) + P α n 2 ,
where err n is the vector with components of the error on λ and P is the vector with components the number of collocation points.

We observe that with this algorithm the computation cost is clearly reduced. Indeed to determine each α n with the indicator λ, we compute P n = mn( mn+1) 2 interpolation problems, while when we use u, we must solve P n of a large linear systems. The algorithm which compute the weights of u is given in [START_REF] Nobile | An anisotropic sparse grid stochastic collocation method for elliptic partial differential equations with random input data[END_REF], by the following steps:

• We fix y * n = 0, as a reference point, i.e y j = 0 for all j = n.

• Let m( ĩn ) = ĩn be an integer such that ũh (y n , 0) := U ĩn u h (y n , 0) is considered as a reference approximation of u h according to the direction n, where u h is the finite element approximation of u.

• For each p = 1, . . . , ĩn -1, we compute the error: err n (p) = ũh (y n , 0)-U p u h .

• We use a linear least squares algorithm to estimate α n and c n solutions of the following minimization problem:

min αn,cn log(err n ) -log(c n ) + P α n 2 ,
where err n is the vector with components of the error on u h and P is the vector with components the number of collocation points.

Remark 4.4 In the computation of the vector α, we use the growth rule m(i n ) = i n to compute the interpolation U in since we need many results on the error to obtain a good estimate of α using a linear squares method.

In practice, to accelerate the convergence of the Formula A α (w, N ), we use an exponential growth rule m(i n ) = 2 in-1 or at least, the linear growth rule m(i n ) = 2i n -1. Thus, most of the computation realized to determine the vector α for u can not be reused to compute A α (w, N )u, since only the knots of the grid H α (w, N ) are involved in the construction of A α (w, N )u. In particular, if the grid H α (w, N ) is given by Gaussian knots, which are non nested points, almost the computation realized to determine the vector α for u is not reuse to compute A α (w, N )u.

Determination of the level w for Smolyak formula

In the classical approach, when we attempt to compute A α (w, N )u, the appropriate level w satisfies A α (w, N )u -A α (w + 1, N )u ≈ ε, where ε is a desirable tolerance. The cost requires by A α (w +1, N )u is generally greater than the combined cost of all previous levels, it is given by a large number of deterministic problems.

Non-linear coefficient Motivated by Proposition 4.1, we use the indicator λ to determine the convenient level w for the formula A α (w, N ), since the error λ-A α (w, N )λ is an upper bound of the error u -A α (w, N )u . The strategy consists of choosing the level w ∈ N such that for a given desirable tolerance ε > 0, the interpolation A α (w, N )λ satisfies:

λ -A α (w, N )λ ≤ ε C , (28)
where the constant C is given as in Proposition 4.1. It is shown in [START_REF] Nobile | An anisotropic sparse grid stochastic collocation method for elliptic partial differential equations with random input data[END_REF] that the error committed by A α (w, N )u satisfies the following bound:

u -A α (w, N )u ≈ C 1 u M -ν 1 , (29)
where the decay rate ν 1 depends on the analyticity of u. We have a similar bound for the error committed by A α (w, N )λ.

λ -A α (w, N )λ ≈ C 1 λ M -ν 2 . (30)
Therefore, if we assume that ν 1 ≈ ν 2 in the case where k is non linear with respect to y, we can estimate the constant C as follows:

C ≈ u λ ≈ A α (0, N )u A α (0, N )λ . ( 31
)
The indicator λ allows us to determine a suitable w to compute A α (w, N )u. By this strategy we avoid great computations, since A α (w, N )λ requires only a sequence of the interpolation problems and we avoid computing A α (w + 1, N )u given by a sequence of PDE problems.

Linear coefficient

When the parameter k is linear with respect to y, the constant C given in Proposition 4.1 decrease with M w , the number of the collocation points with level w, because the size of the analyticity region of u is large than that of λ. The decay rates ν 1 and ν 2 as in (29,30) satisfies ν 1 ≥ ν 2 and depend on α min the smallest anisotropic weight vector α [START_REF] Nobile | An anisotropic sparse grid stochastic collocation method for elliptic partial differential equations with random input data[END_REF]. In this case we propose to estimate the constant C for each w as follows:

C = C w ≈ A α (0, N )u A α (0, N )λ M -µ w .
The decay rate µ > 0 is computed numerically by the following way, we estimate the weight vector α for λ and consider ñ such that α ñ = min 1≤n≤N α, we compute only the weight αñ for u according to direction n = ñ and then we set µ = α ñ -αñ .

Using the indicator λ to determine a suitable level w of the formula A α (w, N ), a significant reduction of the computational effort is kept. By this approach, the computation of the weight of anisotropy and the interpolation A α (w + 1, N ) required only an evaluation of λ on the collocation knots and the sparse grid H α (w + 1, N ), instead of solving a set of deterministic PDE problems.

Numerical Examples

This Section illustrates the convergence and effectiveness of the proposed adaptive approach to the stochastic linear elliptic problem, as described in Section 4. We consider two cases of the parameter k, non-linear and linear with respect to y. The computational results presented here are realized by Matlab software.

-div(k∇u) = f in Ω × D, u | ∂D = 0, (32)
with D :=]0; 1[ 2 and f is deterministic f (ω, x) = cos(x 1 ) sin(x 2 ), x = (x 1 , x 2 ) ∈ D.

Non-linear case

In the first example, we consider k N which is non linear with respect to the stochastic variable and with one-dimensional spatial dependence:

k N (ω, x) = 0.5 + exp Y 1 (ω) √ πl c 2 1/2 + N n=2 ζ n ϕ n (x)Y n (ω) , (33)
the reals (ζ n ) N n=1 are strictly positive and given by:

ζ n := √ πl c 1/2 exp -(⌊ n 2 ⌋πl c ) 2 8 , if n > 1
the functions {ϕ n } N n=1 are defined by:

ϕ n (x) :=        sin ⌊ n 2 ⌋πx 2 , if n even, cos ⌊ n 2 ⌋πx 2 , if n odd.
Where l c is the correlation length and {Y n } N n=1 are independent gaussian random variables with mean zero and unit variance. The parameter k N is a truncation of a log-normal field k = e G , where G is a Gaussian field with a Gaussian covariance function [START_REF] Nobile | A sparse grid stochastic collocation method for elliptic partial differential equations with random input data[END_REF][START_REF] Nobile | An anisotropic sparse grid stochastic collocation method for elliptic partial differential equations with random input data[END_REF]. The order N is strongly related to the size l c , such that a large value of l c corresponds to a small value of N , then the problem is strongly anisotropic and vice-versa [START_REF] Babuŝka | Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation[END_REF][START_REF] Nobile | A sparse grid stochastic collocation method for elliptic partial differential equations with random input data[END_REF][START_REF] Nobile | An anisotropic sparse grid stochastic collocation method for elliptic partial differential equations with random input data[END_REF].

To check that the proposed approach is effective for the anisotropic Smolyak method, we consider the case where l c = 1 2 with N = 11 and l c = 1 5 with N = 41. The finite element space of the spatial discretization is the span of piecewise linear functions over a uniform triangulation of D with nd = 2712 unknowns. The collocation points are the roots of the Hermite polynomials and Γ = R d .

We will approximate the solution u by the interpolation A α (w, N )u, then compute E[A α (w, N )u] and var[A(w, N )u] as an approximation of the mean value and the variance of the function u respectively.

First, we determine the suitable weight vector α for the indicator λ = k -1 f with a posteriori approach then we seek a convenient level w such that the interpolation A α (w, N )λ satisfies a given threshold for E[λ]. Therefore, we use this vector α and the level w to compute E[A α (w, N )u] and var[A(w, N )u].

To determine the vector α in this case of unbounded set Γ, we use the bound (24) given in Proposition 4.2 instead of the bound (21) proposed in [START_REF] Babuŝka | A stochastic collocation method for elliptic partial differential equations with random input data[END_REF][START_REF] Nobile | A sparse grid stochastic collocation method for elliptic partial differential equations with random input data[END_REF].

We denote α and α the weight vectors obtained by a posteriori information with the bound (24), respectively for λ and u. We also denote by α the weight vector for λ obtained using the bound (21).

Table 1 shows the first ten values of vectors α, α and α. We remark that the components of the vector α for the indicator λ are closely comparable with those of the vector α for u. However, α is completely different with α, since they are obtained with different one-dimensional bound. Figure 3 shows that the errors u -A α (w, N )u|| and C λ -A α (w, N )λ|| have the same behavior. We have taken A α (9, N ) and A α (7, N ) as a reference solution for both functions u and λ, respectively when l c = 1 2 and l c = 1 5 .

Linear case

In the second example we consider a diffusion parameter k N which is linear with respect to the stochastic variable y ∈ Γ where Γ is a bounded set in R N :

k(ω, x) = π 2 6 + N n=1 1 1 + n 2 cos(nπx 1 ) sin(nπx 2 )Y n (ω), x ∈ [0, 1] 2 .
The deterministic load function is given by f (ω, x) = 1 + sin(x 1 )x 2 Y 1 (ω). The variables Y 1 , . . . , Y N are independent and uniformly distributed in the box (-1, 1).

We take N = 51 and we compute α using a posteriori approach, we consider A α (9, N ) as a reference solution for both functions u and λ. Table 2 gives the First ten weights for λ = k -1 f and u. We can see that the components of α are slightly smaller than the components of α. This is consistent with the theory, since when k is linear with respect to y, the analyticity region of u is larger than that of λ (proof of Proposition 4.1).

Figure 5.2 (left) shows that #H α (w, N ) is slightly smaller than #H α(w, N ). This could be explained by the fact that numerically we have α min(α) ≥ α min(α) .

Conclusion

In this work we have discussed one strategy of an adaptive collocation method based on the sparse grid method to approximate an elliptic problem with random inputs and data. The approximation of the problem is given by an anisotropic collocation in the stochastic space and the Finite Element approximation in the physical space. The emphasis has been on to seek the suitable number of knots in each dimension and the level of the method. We have introduced an error indicator based on the inverse of the diffusion coefficient and we have shown that this indicator gives an upper bound of the error on the solution.

The strategy consists of using this indicator to determine a convenient weight vector and level of the anisotropic sparse grid formula. The classical approach is based on a posteriori information on the solution and yields an additional cost, which is given by the resolution of a large sequence linear systems. Here the adaptation approach requires only a set of interpolation problems providing a significant reduction in the overall computation.

The numerical examples considered here are given with two examples of diffusion parameters with bounded and unbounded random variables. The numerical results show that we obtain a closely comparable weight vector to that obtained by classical approach. Also, we have obtained an equivalent error on the solution. The proposed adaptive approach yields substantial computational savings in comparison with the conventional adaptive approach. An immediate next step is to study the impact of the spatial discretization on the approximation quality and then make a posteriori analysis in the physical space.

Figure 1

 1 Figure1gives the cardinality of the grid H α (w, N ) and H α(w, N ). We observe that the weights obtained by the error bound given in Proposition 4.2 are more convenient than those obtained using the error bound O(e - √ mg ), since we obtained

Figure 1 :

 1 Figure 1: #H α(w, N ) versus #H α (w, N ), (left l c = 0.5, right l c = 0.2).

Figure 2 :

 2 Figure 2: #H α (w, N ) versus #H α(w, N ), (left l c = 0.5, right l c = 0.2).

Proposition 4 .

 4 1 shows that the error C λ -A α (w, N )λ|| is an upper bound of the error u -A α (w, N )u||, thus we use this error to determine a convenient level w satisfying u -A α (w, N )u|| ≈ tol. The strategy consists of seeking w such that C λ -A α (w, N )λ|| ≈ tol, where the constant C is given in (31).

Figure 3 :

 3 Figure 3: Error C E[λ ref -A α (w, N )λ] versus error on E[u] (left l c = 0.5, right l c = 0.2).

Figure 5 . 2 (

 52 Figure 5.2 (right) compares the error on E[u] with C w E[λ ref -A α (w, N )λ] ,where C w decreases with M w = #H α (w, N ) the total number of points, we ap-

Figure 4 :

 4 Figure 4: Left: Error C w E[λ ref -A α (w, N )λ] 2 versus error on E[u]. Right: Sparse grid cardinality #H α (w, N ) versus #H α(w, N ).

Table 1 :

 1 The first eleventh values of the weights obtained by a posteriori approach.

	l c	α 1	α 2	α 3	α 4	α 5	α 6	α 7	α 8	α 9	α 10	α 11
	1/2 2.57 2.26 2.27 4.31 4.31 8.18 8.18 12.50 12.51 20.58 20.59
	1/5 3.13 2.55 2.55 2.88 2.88	3.4	3.4	4.34 4.35 5.92 5.92
	l c	α1	α2	α3	α4	α5	α6	α7	α8	α9	α10	α11
	1/2 2.57 2.36 2.09 3.58 3.51 8.31 7.04 11.73 10.8 21.48 21.77
	1/5 3.13 2.88 2.50 2.33 2.16 2.56 2.62 3.55 3.27 4.22 4.18
	l c	α1	α2	α3	α4	α5	α6	α7	α8	α9	α10	α11
	1/2 9.47 7.91 7.92 15.14 15.14 21.87 21.87 30.16 30.17 42.96 42.95
	1/5 10.98 9.25 9.27 10.24 10.24 12.03 12.04 15.34 15.34 16.17 16.17

Table 2 :

 2 for λ 3.89 5.77 7.19 8.26 9.11 9.87 10.48 11.01 11.48 11.9 α for u 4.24 6.44 8.00 9.32 10.19 10.92 11.53 12.06 12.53 12.96 The first ten values of the weights obtained by a posteriori approach.

	α 1	α 2	α 3	α 4	α 5	α 6	α 7	α 8	α 9	α 10
	α

m(in) r=1,r =j z-yn,r y n,j -yn,r is the Lagrange polynomial with degree p n = m(i n ) -1. We define the full tensor product interpolation on C 0 (Γ; V h (D)) by:A i v(y, x) := U i 1 ⊗ . . . ⊗ U i N v(y, x) = j≤m(i) v(y j , x)L i j (y),(8)


Hocine Dehmous
email: dehmoushocine@yahoo.fr

Moussa Karama
email: moussa.karama@enit.fr

Hélène Welemane
email: helene.welemane@enit.fr

Contribution of a micromechanics-based approach for reliability assessment

Keywords: Reliability, composite materials, multi-scale analysis, design, optimization

The paper intends to widely develop the use of a new methodology for the design and optimization of composite materials and structures. Based on the coupling of reliability methods and homogenization techniques, such approach allows the integration of uncertainties at different scales in the problem analysis (i.e. from the microscopic scale of the composite material components up to the macroscopic scale of the structure) and the investigation of their consequences in the failure prediction. The principles and implementation steps of such original method have already been described in details (Int. J. Mech. Sci. 53 (2011) 935-945; Eng. Fail. Analysis 18 (2011) 988-998). This work focuses on its application and significant progress allowed in the design phase for engineering composite materials. Illustrations are presented on the case of a civil engineering structure, namely the Laroin footbridge (France) with carbon epoxy stay cables, and highlight reliability-based innovations and exploitations regarding the structure optimization.

Introduction

Uncertainties on constituent properties, fibre distribution, structural geometry, manufacturing process parameters or loading conditions clearly affect the design and optimization of composites structures. The related variability of their mechanical performances requires large safety factors in deterministic structural calculations and induces a significant loss in the weight/resistance ratio. In this context, reliability-based analyses have been developed for a few years to provide a consistent and rational estimation of the risk by introducing the random character of design parameters (materials properties, loads, geometry, ..). The quantitative evaluation of reliability indicators (such as the failure probability) obtained by these methods leads then to an enriched representation of the composites behaviour that significantly helps for both the design and maintenance of structures, and more generally for the development of composite materials [START_REF] Ditlevsen | Structural reliability analyses[END_REF][START_REF] Lemaire | Structural reliability[END_REF].

Most of reliability-based approaches are primarily designed to establish the reliability of a structure as function of the applied load and macroscopic parameters such as geometry and material strengths (for instance [START_REF] Baxevanakis | Fracture statistics modeling of laminate composites[END_REF][START_REF] Frangopol | Reliability of fiber reinforced composite laminate plates[END_REF][START_REF] Liu | Strength reliability analysis of aluminium-carbon fiber/epoxy composite laminates[END_REF][START_REF] Motley | Influence of uncertainties on the response and reliability of self-adaptative composite rotors[END_REF][START_REF] Philippidis | Probabilistic failure prediction for the FRP composites[END_REF]). An original approach that simultaneously considers reliability and micromechanics has been recently proposed by the authors to improve this representation [START_REF] Dehmous | Multi-scale reliability analysis of composite structures -Application to the Laroin footbridge[END_REF][START_REF] Welemane | Reliability analysis and micromechanics : a coupled approach for composite failure prediction[END_REF]. For heterogeneous materials such as composites, homogenization techniques allow indeed the derivation of their overall behaviour from microscopic features (components behavior and morphology). At the same time, local stress or strain within components can also be obtained, which is particularly relevant for the consideration of a local failure criteria (see [START_REF] Gilormini | Syntheses: Mechanical properties of heterogeneous media: Which material for which model ? Which model for which material ? Model[END_REF][START_REF] Zaoui | Continuum micromechanics : survey[END_REF]). The association of reliability methods with micromechanics offers then a better insight into the modelling of uncertainties affecting composites, including especially the fluctuations at various scales (microscopic to macroscopic) and also a physical motivation for the definition of the material reliability. These two points are important issues to understand the propagation of variability from micro to larger scales [START_REF] Shaw | A critical reliability evaluation of fibre reinforced composite materials based on probabilistic micro and macro-mechanical analysis[END_REF] but, above all, to derive a robust framework for reliability evaluation [START_REF] Chiachio | Reliability in composites -A selective review and survey of current development[END_REF].

Interest of the coupled approach has been previously illustrated for material engineering through the identification and validation of micromechanical models [START_REF] Welemane | Reliability analysis and micromechanics : a coupled approach for composite failure prediction[END_REF]. The ability to derive multi-scale analyses has also been demonstrated [START_REF] Dehmous | Multi-scale reliability analysis of composite structures -Application to the Laroin footbridge[END_REF]. The present work intends to focus on the question of design and optimization, through an application case in civil engineering. The objective is here to highlight the innovative tools provided by the methodology in the context of reliability-based design, for instance to get the best structure under a reliability constraint or to improve the manufacturing process.

After a general recall of the methodology and description of the case study, we present in what follows the design interest for an engineering material optimization, based on the integration of uncertainties related to the components properties and composite morphology.

General background and case study

The originality of the approach suggested by the authors lies in the association of two classical approaches for the study of composite materials:

• Reliability methods that introduce in the mechanical modelling the uncertain character of design variables and allow the study of the related consequences on the structure response,

• Homogenization techniques which aim at determining from the microstructure of a Representative Volume Element (RVE) of the material its overall (effective) mechanical behaviour and the local response within components for a given macroscopic load.

Theoretical developments and implementation details can be found in [START_REF] Dehmous | Multi-scale reliability analysis of composite structures -Application to the Laroin footbridge[END_REF][START_REF] Welemane | Reliability analysis and micromechanics : a coupled approach for composite failure prediction[END_REF]. We intend here to recall the key points of such coupling and assumptions used for the case study.

Coupling of reliability methods and micromechanics

In the coupled approach, major stages encountered during any reliability study take into account the specific aspects of micromechanics:

1. First the selection of random variables {X i } i=1,N in order to capture most of inherent fluctuations involved in the problem, and at the same time to offer at this starting point a broad analysis of design parameters; for heterogeneous materials such as composites, microstructural parameters clearly influence the material effective behaviour and should therefore be considered as random variables; important requirements are the availability of the statistical representation of these random parameters and the limit size of the problem (number N ) that should be compatible with reasonable calculation time;

2. The choice of the failure scenario defined by a mathematical function G: this step includes both an adequate representation of the structure mechanical behaviour and a physical definition of the limit state (according either to strength achievement or serviceability); the ability provided by micromechanics to derive local stresses or strains induced by a macroscopic load is a major asset for a consistent definition of G in relation with the physical mechanisms involved ;

3. The estimation of probabilistic indicators (e.g. probability of failure P f or reliability index β): in the context of composite structures investigation, one often resorts to numerical approximation methods in view of their suitability with finite element simulations and their design interest through sensitivity analyses on random variables.

Generally speaking, note that this approach should be conducted several times in order to select the significant random parameters of the problem considered and to ensure therefore the computational efficiency of reliability calculations.

Application to the Laroin footbridge

The demonstration of the potential of such approach for composites design and optimization is practically supported by a case study in civil engineering. The considered structure is a pedestrian footbridge located in Laroin (Pyrénées Atlantiques, France, Fig. 1(a)). Built in 2002, this is the first structure in this country conceived with composite stay cables [START_REF] Geffroy | La passerelle de Laroin[END_REF]. It is composed of a steel deck, two steel reversed V-shaped pylons of 20.60 m height, one high-strength steel stending cable for each pylon and its single span of 110 m length is maintained at each side by eight composite stay cables. Each stay cable is composed of two or three strands of seven unidirectional cylindrical composite rods (Fig. 1(b)).

Composite rods are made of Torayca high-strength carbon fibres T700SC-12K and Bostik Findley epoxy resin Eponal 401. They have been manufactured by pultrusion by the Toray Carbon Fibers Europe company (mean fibre volume fraction f f of 67%). The mechanical behaviour of the composite material is described by means of the Mori-Tanaka formulation [START_REF] Benveniste | A new approach to the application of Mori-Tanaka's theory in composite materials[END_REF][START_REF] Mori | Average stress in matrix and average elastic energy of materials with misfitting inclusions[END_REF] with a slight adjustment to reflect more effectively the manufacturing process. Based on the experience of Toray Carbon Fibers Europe, it seems indeed that only a fraction of fibres in the composite actually contributes to the composite response (due for instance to fibres misalignment, impregnation defects, etc.). Accordingly, the fibre volume fraction f f appearing in the classical expressions of the Mori-Tanaka constitutive law has been changed in this way:

f f → f ′ f = p act × f f (1)
with p act ≤ 100% the amount of fibres really active from the mechanical point of view. This parameter, which characterizes the quality of the manufacturing process, has been identified on the pultrusion production line. In this study, reliability assessment corresponds to the mechanical strength achievement of these rods under monotonic tensile load. The failure scenario is defined to account for the microstructural origin of the failure, namely by the fibres brittle behaviour through the following limit state function:

G = σ F f (n) -⟨σ I ⟩ (f ) (2)
where σ F f (n) denotes the fibre axial failure strength (axial direction denoted by unit vector n) and ⟨σ I ⟩ (f ) the maximum principal value of the average local stress over the fibre phase. If the first term is a material data provided by manufacturers, the second is derived through the micromechanical model.

Reliability calculations have been carried out with the probabilistic code FERUM (Finite Element Reliability Using Matlab [START_REF] Haukaas | FERUM user's guide[END_REF]) by means of a direct coupling with the micromechanical model. FORM (First-Order Reliability Method [START_REF] Ditlevsen | Structural reliability analyses[END_REF][START_REF] Lemaire | Structural reliability[END_REF]) approximation method has been used since it exhibits a remarkable computational efficiency and provides valuable data for design issues (such as elasticities, see section 3.1).

Micromechanical-based reliability analysis

The present work aims at studying the reliability-based design of composite rods used for the stay cables of the Laroin footbridge, subjected to monotonic longitudinal tension.

Reference point

Based on the micromechanical model developed in [START_REF] Dehmous | Multi-scale reliability analysis of composite structures -Application to the Laroin footbridge[END_REF][START_REF] Welemane | Reliability analysis and micromechanics : a coupled approach for composite failure prediction[END_REF], the five random variables {X i } i=1,5 taken into account in this study concerns both micro and macro scale data of the problem:

• components mechanical properties : fibre yield strength σ F f (n), • manufacturing process parameters : fibre volume fraction f f , active fibre fraction p act , rod diameter φ,

• load condition : axial load F , since their variability clearly affects the composite reliability. It has been shown in [START_REF] Welemane | Reliability analysis and micromechanics : a coupled approach for composite failure prediction[END_REF] that components elastic properties (fibre and resin) could be chosen as deterministic due to their weak influence. Regarding the considered case study, manufacturers (of fibre materials and composite materials) have provided the statistical distribution of the first four parameters that follow a Normal law (mean value and standard deviation are respectively denoted by X i and S Xi ). We assume also a Normal distribution for the load and the study is centred around the operating point F = 76 kN that corresponds to reliability index β = 3.0514 (such security level is approved in civil engineering structures for which β should be greater than 3 or, equivalently, failure probability P f should be greater than 10 -3 ). All these data are detailed in Table 1 The sensitivity analysis of the FORM method provides elasticities {e ri } i=1,N according to the distribution parameters of each random variable (here the mean value r i = X i or the standard deviation r i = S Xi ) defined by:

e ri = r i β ∂β ∂r i (P * ), ∀i ∈ [1..N ] (3)
with P * the design point (most probable failure point) [START_REF] Lemaire | Structural reliability[END_REF]. Such analysis gives the global tendency regarding the origin of the scatter in the material response. This helps to discriminate variables that could be considered as deterministic (such as components elastic properties [START_REF] Welemane | Reliability analysis and micromechanics : a coupled approach for composite failure prediction[END_REF]) and to compare variables between them. As an illustration, figure 2 gives the reliability elasticities for the composite rod regarding respectively the variables mean value (fig. 2(a)) and their standard deviation (fig. 2(b)). We can distinguish strength variables with e ri ≥ 0 (respectively loading variables with e ri ≤ 0) for which an increase in their mean value improves (resp. alters) the structure reliability. Whatever the variable, an increase of the standard deviation obviously leads to an increase in the failure probability. Here, we note that the mean value of the geometric parameter φ and the deviation scatter on the fibre yield strength σ F f (n) play the most crucial role in the structure reliability. Yet, such analysis does not provide practical arguments for designers to take quantitative decisions regarding the composite structure. Our aim is then to develop in what follows the micromechanics-based analysis from another point of view in order to really quantify the influences detected before and help for design choices.

Evolution range

In this way, we need first to consider allowable evolutions of the distribution parameters of the different variables. In particular, these evolutions have to be chosen according to the potentialities of materials and manufacturing machines and should also be compatible with assumptions of the mechanical model. To illustrate the approach, we have considered in table 2 an acceptable evolution range for the mean value X i of each random variable (range ratio relative to their reference value is indicated for clearness). Without specific data, an extended and uniform range ratio has been taken for the standard deviation S Xi (S Xi = 0 corresponding to the deterministic case).

Several simulations have then been carried out in which one distribution parameter of each random variable evolves independently within its interval defined in table 2, the other being fixed to their value given in table 1. For each calculation configuration, explicit formulation provided by the Mori-Tanaka scheme and the small number

) (n F f σ f f φ act p F i r e (a) ) (n F f σ f f φ act p F i r e (b)
Fig. 2: Elasticities er i according to the distribution parameters of random variables at the reference point: mean value ri = Xi (a) and standard deviation ri = SX i (b).

Random variables mean value

X i standard deviation S Xi ([0, 150%] of S ref Xi ) Fibre yield strength (MPa) σ F f (n) [4384, 5356] [0, 243.5] (±10% of X ref i ) Fibre volume fraction (%) f f [64, 70] [0, 0.5] (±5% of X ref i ) Active fibre fraction (%) p act [90, 100] [0, 1] (±5.2% of X ref i ) Rod diameter (mm) φ [5.8, 6.2] [0, 0.05] (±3% of X ref i ) Axial load (kN) F [64.6, 87.4] [0, 3.8] (±15% of X ref i )
Tab. 2: Evolution range of the distribution parameters of random variables around the reference point.

of variables allow to derive a precise value of reliability index β with a reasonable number of calls to function G (around 50).

Results and discussion

Figure 3 shows the evolution of index β according to the mean value of each random variable. An increase (respectively decrease) of β with X i is obviously obtained for strength variables (resp. loading variables). For the present model and application case, we obtain a quasi linear evolution of β regarding mean values X i of random variables. In agreement with the model correction (1) and elasticities described on figure 2(a), we find a similar behaviour for f f and p act variables and a major influence of φ through the steepest slope. For designers, such result allows to define either changes of parameters or the maximum load amplitude to respect a given reliability level. For instance, the reliability index β = 3.0514 at the reference point can be raised to β = 3.5 (corresponding to an increase of 15%) either by an increase of 2% of the fibre yield strength (that is with σ F f (n) = 4970 MPa) or with an increase of 1% of the rod diameter (that is with φ = 6.07 mm). On the other hand, we note that a reliability level of β = 2 remains satisfied with an increase of 6% of the mean load (that is with F = 80.6 kN). Regarding the effect of the standard deviation, figure 4 quantitatively highlights the influence on the reliability of the uncertainties on variables. As expected, more (resp. less) the variables are reproducible, more reliability increases (resp. decreases). The tendency observed on figure 2(b) is also reinforced with a weak influence of the scatter on manufacturing process parameters (f f , p act and φ). This indicates where to put the effort to get better reliable structures, namely on the fibre strength σ F f (n). For the same mean value as the reference point, an highly reproducible σ F f (n) (with S Xi → 0) can increase the reliability index up to β = 4.4 (increase of 44%). The variability of the load F plays also an important role on reliability, but the control over it is much more difficult, especially for civil engineering cases. Moreover, the evolution of β regarding the standard deviation of these two variables is of polynomial form with a crossing of the curves at the reference point: a better (resp. worse) reproducibility of σ F f (n) induces a greater increase (decrease) in β than the load F . This confirms the main importance of the fibre strength in the composite performance. The micromechanical-based study allows also to implement quantitative cross-analyses between random variables. This provides to engineers various combined solutions for design changes. We illustrate on the figure 5 the coupled study of manufacturing process parameters (namely the fibre content and active fibre fraction) and material parameters (namely the fibre yield strength and fibre content). As shown of figure 5(a), the reliability index β can be improved to 3.5 either:

• by enhancing the fibre content (with f f = 68.5%) while keeping the present manufacturing process,

• or by keeping the fibre content and improving the manufacturing process (with p act = 100%).

Regarding material features investigated on figure 5(b), same reliability improvement can be obtained either :

• with higher strength fibre (σ F f (n) = 5180 MPa) and a reduced fibre content (f f = 64%),

• or with higher fibre content (f f = 70%) and lower strength fibre (σ F f (n) = 4770 MPa).

Obviously, such kind of cross-analysis can be done also on standard deviations of variables or even between mean value and deviation of different parameters according to the needs and possible actions of designers.

Conclusion

The development of composite materials and composite structures requires a convenient consideration of the stochastic variability of the data entering their mechanical behaviour. The combination of reliability analysis and micromechanics allows to adress such issue. The use of classical and well-known frameworks (reliability methods and homogenization techniques) associated with the broad integration of uncertainties at different scales (micro to macro) and on various features (constituents properties, microstructural morphology, geometry, load) leads to a relevant tool for structural applications. This paper has illustrated some innovations provided by this approach for the reliability assessment of composite materials. Through various examples, we have investigated the consequences of design choices for composites in a probabilistic context. The quantitative description of different solutions at different scale or regarding different parameters such as manufacturing process, geometry, materials properties indicates which designs are qualified to achieve a given reliability level and helps for the optimization of mechanical systems. From this basic results, further work will now be conducted in order to show others design applications provided by the approach, and especially to extend it at a structure scale through the global study of the Laroin footbridge.

Fig. 1 :

 1 Fig. 1: Laroin footbridge: Picture (a) and structure (b).

Fig. 3 :

 3 Fig. 3: Evolution of reliability index β with respect to the mean value Xi/X ref i for each random variable Xi.

Fig. 4 :

 4 Fig.4: Evolution of reliability index β with respect to the standard deviation SX i /S ref X i for each random variable Xi.

Fig. 5 :

 5 Fig. 5: Cross-analyses of the evolution of reliability index β with respect to the mean values of couple of parameters: (f f , pact) (a) and (σ F f (n), f f ) (b).

  and will be considered as the reference point in what follows.

	Random variables		ref. mean value X	ref i	ref. standard deviation S ref Xi
	Fibre yield strength (MPa) σ F f (n)	4870		162
	Fibre volume fraction (%)	f f	67		0.333
	Active fibre fraction (%)	p act	95		0.667
	Rod diameter (mm)	φ	6		0.03
	Axial load (kN)	F	76		2.533

Tab. 1: Random variables -Reference point.

Acknowledgments

Authors fully acknowledge the Toray Carbon Fibers Europe company (ex Soficar, based in Abidos, France) and Freyssinet company (based in Vélizy Villacoublay, France) for technical data and support regarding composite materials and Laroin footbrige structure.


N Viviès

M Haussener

H Welemane
email: helene.welemane@enit.fr

B Trajin

P E Vidal

Characterization of an integrated buck converter using infrared thermography

This study deals with new integrated systems for power electronics applications including wide-gap semiconductors. Integration of Silicon carbide (SiC) components provides for instance new perspectives with higher temperature operating points than conventional Silicon (Si) semiconductors. The present work intends to study an integrated buck converter composed of a Silicon IGBT (Insulated-Gate Bipolar Transistor) and a Silicon carbide diode. By means of infrared thermography, an analysis of the thermal disparities induced within such a hybrid assembly under various electrical loads is proposed, regarding especially the consequences of the electrical power transfer and the spatial distribution of the thermal field.

Introduction

The emergence of wide-gap semiconductors allows to design electronic power modules with high compactness and high power density. Indeed, the maturity level of these specific components such as those made of Silicon carbide (SiC) and Gallium nitride (GaN) has strongly increased. Nowadays, they may be used for high integrated industrial applications. In order to optimize integration, higher temperature operating points than for conventional Silicon (Si) semiconductors are considered [START_REF] Funaki | Power Conversion With SiC Devices at Extremely High Ambient Temperatures[END_REF]. Consequently of high temperature, new constrains appear and become critical for power electronics assemblies. Several studies aim at identifying failure modes or critical interfaces [START_REF] Ye | Failure modes and FEM analysis of power electronic packaging[END_REF][START_REF] Smet | Ageing and Failure Modes of IGBT Modules in High-Temperature Power Cycling[END_REF]. In this field of research, non-destructive characterizations allow to study the mechanisms of degradation without disturbance due to the measurement itself (eg optical microscopy, acoustic microscopy, X-ray, infrared (IR) thermography). Specifically, IR thermography is a convenient and efficient technique for the investigation of systems, even under service conditions. This method has been successfully used for experimental mechanics applications, either for the materials and structures health investigation [START_REF] Maldague | Theory and practice of infrared technology for non-destructive testing[END_REF] or the monitoring of dissipative phenomena [START_REF] Chrysochoos | Infrared thermography, a potential tool for analysing the material behaviour[END_REF][START_REF] Goidescu | Damage investigation in CFRP composites using full-field measurement techniques: combination of digital image stereo-correlation, infrared thermography and X-ray tomography[END_REF], and seems to be well adapted in the case of electronic power assemblies.

Given the difficulties to obtain full controlled SiC semiconductors at low prices, this technology seems to be intended for high voltage applications. On the contrary, uncontrolled SiC semiconductors are already easily accessible and allow to build efficient and optimized conversion functions [START_REF] Ouwerkerk | Efficiency improvement using a hybrid power module in 6.6 kW non-isolated on-vehicule charger[END_REF]. Joint use of a classical controlled Si switch and an uncontrolled SiC switch may induce disparities, including thermal ones, in such hybrid assemblies. In this way, the present work intends to study an integrated buck converter composed of a Silicon IGBT and of a Silicon carbide diode through infrared thermography. In part 2 the electric system and the infrared thermography device are described. A theoretical analysis of Joule losses in the different components of the assembly is then proposed in section 3. These results are compared to experimental measurements and discussed in the final part.

System description and experimental device

Electric system

In this study, the power electronic function considered is a buck converter composed of a controlled Si IGBT (1200V -25A) and of a SiC diode (1200V -30A). This specific integrated function is completed with an external command circuit and driver for the switchings of the IGBT. The electrical load of the buck converter consists in a resistance of value R=1Ω and an inductance of value L=3mH. On figure 1(a) is depicted the electric operating diagram and figure 1(b) shows the related power electronic assembly. In this preliminary study, the power electronic assembly is not encapsulated using a dielectric polymer or an external case. Consequently, conduction and dissipations effects between thermal sources (power electronic chips and wire bondings) and the IR camera are not considered. Power electronic chips are brazed using a Tin-Copper-Silver (Sn-Cu-Ag) alloy on a Silicon nitride Si3N4 substrate covered by Copper power tracks. A thin layer of Gold is added on the Copper tracks as a protection against corrosion.

Load current ripple

The switching frequency of the IGBT is set at

kHz T f 1 / 1 = =
so as to get a switching period lower than the time constant of the electrical load ms R L 3 / = = τ . In addition, it is assumed that the buck converter operates in continuous conduction. In this case, the amplitude of the current ripple in the load during a switching period in steady state operation is approximated by:

α Lf RI E I load load - = ∆ (1)
where E denotes the input voltage and load I the average load current. The operating point of the load is set constant in this study, by the means of an average current of A I load 8 = . The voltage of the power source E is then adapted to the duty cycle α in order to meet the operating point chosen for the load. The current ripple in the load (1) is expressed according to the duty cycle and average load current:

Lf I I load load ) ( α - = ∆ 1 (2)
Given the electrical parameters of the studied system, the maximum value of the ripple current is obtained when the duty cycle tends to 0. In this case, the ripple current is equal to

A I load 6 2. = ∆
. Thus, whatever the operating point considered and therefore the duty ratio controlling the IGBT, the chopper is always operating in continuous conduction.

Root Mean Square (RMS) current

Thermal Joule losses are determined by the value of the RMS current in the load. For a switching period, let us consider the current load ) (t i load as the summation of a constant part load I and an asymmetrical triangular oscillating component with an amplitude of load I ∆

: [ ] [ ]        ∈ - ∆ - - ∆ + ∆ + = ∈ ∆ + ∆ - = T T t t T I I I I t i T t t T I I I t i load load load load load load load load load , , ) ( ) ( , , ) ( α α α α α α 1 1 2 0 2 (3)
It is shown in that case that RMS value of the current rms load I , is expressed as follows:

2 2 2 2 2 2 0 2 , ) 1 ( ) ( 1 f L R I I dt t i T I load load T load rms load α - + = = ∫ (4)
Thus, the RMS current in the load can be approximated by its average value load I . The maximum error between the RMS current and the average approximation is then provided by:

Lf R I error load ) ( max α - = 1 (5)
Given the system parameters, the upper bound of the relative error between RMS current and its average approximation theoretically reaches 8.3% when 0 = α . However, an exact calculation shows that the relative error really obtained does not exceed 0.5% in our case (see figure 2). This study is consistent with the highest oscillations of the load current obtained when the duty cycle tends to 0. So it will be considered in the following that RMS load current is equal to the average load current.

Fig. 2. Relative error on RMS load current

Infrared thermography

Infrared thermography (IR) is an optical technique for measuring the surface thermal field of a structure. It consists in acquiring the two-dimensional thermal radiation (radiance) emitted by a surface, which depends in particular on the temperature of the material. In addition to its non contact procedure, IR differs from the classical thermography through the range of wavelengths measured. Obtaining the evolution of the temperature field versus time by a non-intrusive manner as well as the ease of implementation, maneuverability and speed of this technique have led to multiple applications. Compared to conventional techniques, it provides clear advantages in the case of an inaccessible area or a complex environment. This technique is therefore particularly suitable in this case when radiation derives from small parts having a non-zero electric potential and subjected to an electric current flow. The IR thermography system is composed of a radiation sensor and of a station dedicated to conditioning and formatting the signal, temperature transcription and data recording [START_REF] Pajani | Thermographie. Principes et mesures[END_REF] . Camera control and recording are managed by the Altair software, the calibration operation is automatic. In a first approach, the emissivities of the studied components were calibrated using a thermocouple to obtain a surface temperature close to the ambient temperature when the system is in thermal equilibrium without any electrical solicitation. The emissivity of the surface of the IGBT is set to 75 . 0 = ε , those of the diode and wire bonding to 7 . 0 = ε . The emissivity along wire bonding is assumed constant given the small variations of their surface inclination [START_REF] Gaussorgues | La thermographie infrarouge. Principes, technologie, applications[END_REF]. Tests were conducted at room temperature and all lights off. To eliminate as much as possible disruptive effects of the environment and highlight thermal changes, a relative analysis on data was performed by subtracting at every step the thermal image of the beginning of the test ( 0

T T T - = ∆
). Figures 3(a

Switching losses

For the IGBT component, it is necessary to take into account the energy dissipated during blocking off E (with slow rate by definition) and switching-on on E [START_REF] Sui Too | Caractérisation et modélisation de composants IGBT et diode PiN dans leur environnement thermique sévère lié aux applications aéronautiques[END_REF]. In a simplified way, it is considered that the current at the switching-on and blocking steps evolves linearly during the switching time (respectively on t and off t times) and that voltage across the IGBT is equal to the input voltage E , it is shown that the dissipated energy during switching can be expressed by [START_REF] Dalmasso | Electronique de puissance : commutation[END_REF]:

2 2 off load off on load on t I E E t I E E = = (7)
In the case of a power diode, switching losses only occur in the blocking step due to reverse-recovery time and associated dissipated energy. However, for SiC diodes, the switching time is very low and the reverse-recovery losses also decrease [START_REF] Risaletto | Caractérisation électrique en commutation de diodes haute tension en carbure de silicium[END_REF]. Thus, in this study, the dissipated energy related to the diode switching will only be considered at blocking step ( diode off E , ).

Total losses

Total Joule losses in semiconductors, defined by the sum of conduction and switching losses, are provided in Eq. [START_REF] Pajani | Thermographie. Principes et mesures[END_REF]. It is clear that Joule losses in the power components are linear with respect to the duty cycle and therefore with respect to the operating point of the buck converter. It may be noted that, for a given load current and for constant onstate resistances during operation, the losses in the IGBT (respectively in the diode) will increase (respectively decrease) linearly with respect to the duty cycle.

diode off load diode on diode off on load IGBT on IGBT E f I R JL E E f I R JL , , , ) ( ) (
+ - = + + = 2 2 1 α α (8)

Thermal losses in wire bonding

Studies dealing with the robustness of electronic power assemblies often point out that connectivity and especially wire bonding are critical elements for systems reliability [START_REF] Smet | Ageing and Failure Modes of IGBT Modules in High-Temperature Power Cycling[END_REF]. For these connection elements, optimizing their position [START_REF] Ishiko | Design concept for wire-bonding reliability improvement by optimizing position in power devices[END_REF], materials or welding techniques [START_REF] Hamidi | Reliability and lifetime evaluation of different wire bonding technologies for high power IGBT modules[END_REF] are key factors to improve the performance of an assembly. In the considered system, power connections are based on pure Aluminium wires of m µ 300 diameter and an average length of cm 5 1.

. For the connections between the IGBT emitter and the cathode of the diode and between the anode of the diode and the external connection, two parallel wires are used. Based on the geometric data and Aluminum properties, the wire bonding resistance is evaluated to Ω = m R bond [START_REF] Chrysochoos | Infrared thermography, a potential tool for analysing the material behaviour[END_REF] . Joule losses in these components can then be expressed by: JL -) the losses between the anode of the diode and the external connections (resp. between the IGBT emitter and the cathode of the diode). Again, it appears that the Joule losses in the wire bonding are linear with respect to the duty cycle α . Furthermore, since the wire section is constant, it can be assumed that their resistance per unit of length is constant. Thus, in first approximation, the Joule losses per unit of length should also be constant along the wire bonding.

Experimental results

Operating point

In order to be able to compare the various operating points defined by the duty cycle, the analysis focuses on the thermal steady state of the power electronic components, leading to apply the electrical load on the buck converter for about 30 minutes for each operating point as shown in figure 4 = for all operating points. Furthermore, the variation range of the duty cycle is selected to operate in continuous conduction. Finally, the input voltage is adapted for each duty cycle in order to meet the constraint on the load current. Table 1 summarizes the various operating points selected for this study.

Thermal data processing

Thermal measurements exhibit an inherent variability related to the camera accuracy and to the thermal ambient noise. Let us consider the relative surface temperature of the diode shown in figure 4. For a thermal steady state, the measured signal consists theoretically in a mean value (useful signal) with added noise. Figure 5 shows the histogram of the relative thermal signal measured. It may be noted that the distribution seems to follow a Normal law with standard deviation 066 0. = σ , which is consistent with the camera resolution and with experimental conditions.

To improve the accuracy of the thermal measurements, the temperature is averaged over 10 frames at the end of the steady state. This increases the reproducibility of the results by reducing the variance between multiple measurements with the same experimental conditions.

Thermal behaviour of semiconductors

Figure 6 presents the temperature increase in the steady state at the surfaces of the diode and the IGBT for duty cycles shown in Table 1. These results support the analysis done in Section 3. Maximum temperature variations evolve indeed linearly with respect to α , increasing for the IGBT and decreasing for the diode. Linear interpolation thus leads to: Although switching losses depend on the value of the switched voltage, here they can be considered as constant between different operating points. This is due to the low values of input voltages with respect to the ratings of the components in the studied system. Thus, the heating of the IGBT when the duty cycle tends to 0 (

C °→ = 3 . 31 lim 0 α )
corresponds to the heating induced by the switching losses since no current flows through the IGBT in this case. Similarly, the limit case where the diode is not submitted to any current (equivalently, when the duty cycle approaches 1) provides the thermal behavior related to switching losses within this component, that is

C °→ = 4 . 25 lim 0 α
. These results tend to show that the switching losses in the diode are lower than those in the IGBT. This stands in agreement with the uncontrolled characteristics of the diode switching and also with the technological difference between these two components.

Moreover, it seems that such a difference in the design also leads to distinct overheating observed in the range of operating points. Indeed, it appears in Figure 6 and in the expressions [START_REF] Sui Too | Caractérisation et modélisation de composants IGBT et diode PiN dans leur environnement thermique sévère lié aux applications aéronautiques[END_REF] that the absolute value of the increase in the heating rate of the diode is lower than the one of the IGBT. This is explained by the lower on-state resistance of the SiC components [START_REF] Risaletto | Caractérisation électrique en commutation de diodes haute tension en carbure de silicium[END_REF], which induces a reduction in conduction losses.

In the context of highly integrated power modules, it may thus be preferable to select power components of wide-gap SiC technology in order to limit heat loss and to reduce the volume of the associated cooling systems. Furthermore, infrared thermography provides a relevant tool for identifying the hot spots of the assembly according to the operating point and consequently for designing external heat sinks, regarding both their position and their volume.

Thermal behaviour of wire bonding

In this study, a special attention was also paid to the wire bonding between the anode of the diode and the external connections. Only the steady state is considered in order to separate two thermal effects: those due to the current flow within the wires, and those associated to the heating of the diode and with the thermal conductivity of the wires. It is assumed that the thermal steady-state is reached in the wires faster than in electronic power components. For a duty cycle 5 . 0 = α , the temperature rise at the diode surface is equal to C °5 .

7

. Figure 7 shows the geometric profile and the surface thermal field of the considered wires. It is noted that the thermal elevation of the wires is higher than the diode ones, which cannot therefore be regarded at this time as a heat source for the wires. The heating of the wire bonding can then mainly be attributed to the electric current flow. Fig. 7. Geometric profile and the surface relative thermal field of wire bonding between the anode of the diode and the external connections Furthermore, the spatial distribution of the thermal field clearly shows heterogeneities along the wire length. Especially, the increase in the temperature is much higher at the diode welding point and at large curvatures of the wire. On the contrary, dissipation by convection is maximum and thermal impact is lower when the wires are far from the substrate and when their curvature is smaller. Thermography allows to show that the wire bonding design exclusively based on Joule losses (9) is not sufficient since nonlinear localized phenomena occur at interfaces. More generally, it should be emphasized that this type of study is of crucial interest for the optimization of these systems as these high thermal heterogeneities can also lead to the damage of the wires and of the conduction paths in case of insufficient cooling.

Conclusion

This work has demonstrated the interest of a hybrid assembly between a Si controlled component and SiC un-controlled component within a buck converter. Infrared thermography has been used to characterize the surface temperature distribution of the assembly, both under active electrical constraints and without contact. The use of a SiC diode leads to a decrease in the overall Joule losses of the assembly, allowing a clear benefit in the design of the heat dissipation elements. Furthermore, thermal analysis allowed to assess the contribution to the dissipated energy during switching and conduction phases within each component. Regarding wire bonding, the proposed method clearly shows hot spots at their edges that should be taken into account for the design of both wires and systems of heat dissipation and cooling. Further works will deal with accurate electrical measurements in order to characterize the dissipated energy during switching of the IGBT and diode but also to specify more precisely the on-state resistance of the components. From the knowledge of the heat loss within the assembly, an equivalent electrical thermal model can be developed to identify and represent the thermal response of the assembly with electric current flow. Furthermore, the thermal distribution along wire bonding will be studied to derive design criteria. A generic model of loss in the power assembly may then be proposed and implemented to correlate the overall performance of the static conversion structure. These thermal studies will finally allow to examine the dissipation and cooling systems and to optimize power structures and their ageing under electro-thermal stresses.

Fig. 1 .

 1 Fig. 1. Electric operating diagram (a) and studied assembly (b)

  . The device used here is a FLIR Titanium SC7000 retrofitted camera with InSb sensors, a focal plane of 256 320 × pixels and a thermal resolution of mK 20 at C °30 . The spectral band corresponds to the interval [ ] ), the integration time is set equal to s µ 950 . For all tests, the acquisition frequency is equal to Hz 10

Fig. 3 . 3 . Thermal analysis of the buck converter 3 . 1 . Thermal losses in power electronic components 3 . 1 . 1 .

 3331311 Fig. 3. Experimental device (a) and illustration of a surface relative thermal field on the assembly (b)

  Section 2, the average load current is kept constant to A I load 8

Fig. 4 .

 4 Fig. 4. Thermal variation in semiconductors (duty cycle 5 . 0 = α

Fig. 5 .Fig. 6 .

 56 Fig. 5. Histogram of the relative temperature of the diode in the steady state

Table 1 .

 1 Input voltage according to the duty cycle

	Duty cycle α	Input voltage E
	0	.	2	40	V
	0	.	35	22	.	8	V
	0	.	5	16		V
	0	.	65	12	.	3	V
	0	.	8	10		V

Acknowledgements

Authors wish to thank Philippe Lasserre, head of operations for the PRIMES platform of Tarbes [15], which enabled the realization of integrated buck converter. In addition, this study was conducted with the support of a student project within the National Engineering School of Tarbes.