Philippe Michel
email: philippe.michel@ec-lyon.fr

Bhargav Kumar Kakumani

ASYMPTOTIC BEHAVIOR OF SOLUTION OF A NONLINEAR RENEWAL EQUATION WITH DIFFUSION

Keywords: Nonlinear renewal equation, steady state, convergence AMS subject classifications. (MSC2010) 92D25, 35A01, 35A02, 35B40

In this paper, we consider the nonlocal nonlinear renewal equation with diffusion under nonnegative initial data. Under some assumptions on the birth and death rates we prove the existence and uniqueness of the nonlinear renewal equation with diffusion and its corresponding adjoint equation also. In the next part, we prove the convergence of the solution to its steady state as time tends to infinity using the generalized relative entropy inequality and Poincaré Writinger type inequality.

1. Introduction. The structured mathematical models (age structured and size structured) describing the behavior of the cell population have been studied in [START_REF] Arino | A survey of cell population dynamics[END_REF][START_REF] Basse | A mathematical model for analysis of the cell cycle in cell lines derived from human tumors[END_REF][START_REF] Goldbeter | Biochemical oscillators and cellular rhythms[END_REF] (and the references within). According to the biologists, the matter of which sites are active on various chromosomes determines the true age of a biological entity [START_REF] Brewer | The age-dependent eigenfunctions of certain Kolmogorov equations of engineering, economics, and biology[END_REF]. This true age is a multidimensional variable and can be determined by time since birth. We are mainly concerned about the population and not on the individuals, hence we assume that average aging in the population is measured from time since birth (renewal). Because of lots of sources of variation in the vector valued age of individuals, the population as a whole diffuse in population age variable. Let u(t, x) be the population density of cells at time t having age x. Assume that B, d are birth and death rates respectively. We are interested to study the dynamics of the following renewal equation with diffusion.

(1.1)

          
u t (t, x) + u x (t, x) + d(x, S 1 (t))u(t, x) = Cu xx (t, x), t > 0, x > 0, u(t, 0) -Cu x (t, 0) = ∞ 0 B(x, S 2 (t))u(t, x)dx, t > 0, u(0, .) = u 0 (.) > 0,

u 0 ∈ L 1 (R + ) ∩ L 2 (R + ),
where

(1.2) S i (t) = ∞ 0 ψ i (x)u(t, x)dx, ∀i = 1, 2.
Where C is a positive constant of diffusion, u 0 is the initial age distribution, ψ 1 , ψ 2 are the competition weight and S(t) := (S 1 (t), S 2 (t)) is the weighted population which depends on the competition weight ψ i and population density u. The steady state equation of (1.1) is given by

(1.3)              U ′ (x) + d(x, S1 )U (x) = CU ′′ (x), x > 0, U (0) -CU ′ (0) = ∞ 0 B(x, S2 )U (x)dx, ∞ 0 U (x)dx < ∞, Si = ∞ 0 ψ i (x)U (x)dx.
and its corresponding adjoint equation reads as (1.4)

           -φ ′ (x) + d(x, S1 )φ(x) = Cφ ′′ (x) + φ(0)B(x, S2 ), x > 0, φ ′ (0) = 0, ∞ 0 φ(x)U (x)dx = 1.
Throughout the paper, we assume that the functions d, B, ψ are nonnegative and continuous. Further we assume that there exists L > 0 such that for all x, S ā, Sb we have

(1.5) |B(x, S ā) -B(x, Sb)| ≤ L|S ā -Sb|, |d(x, S ā) -d(x, Sb)| ≤ L|S ā -Sb|, (1.6) ∂ ∂S 1 d(., .) > 0, ∂ ∂S 2 B(., .) < 0, (1.7) 0 ≤ ψ(x) ≤ ψ M
where ψ M is a positive constant. Equation (1.1) with C = 0 is popularly known as McKendrick-Von Foerster (MV) equation (see [START_REF] Cushing | An Introduction to Structured Population Dynamics[END_REF][START_REF] Thieme | Mathematics in Population Biology[END_REF]). There are several mathematicians who worked on the stability estimates and longtime behavior of the MV equation ( [START_REF] Cushing | An Introduction to Structured Population Dynamics[END_REF][START_REF] Iannelli | Mathematical Theory of Age-Structured Population Dynamics[END_REF][START_REF] Webb | Theory of non-linear age-dependent population dynamics[END_REF] and the references therein) or MV -like (see [START_REF] Iannelli | Two-sex age structured dynamics in a fixed sex-ratio population[END_REF][START_REF] Kakumani | Extinction and blow-up phenomena in a nonlinear gender structured population model[END_REF] for instance). In [START_REF] Perthame | Nonlinear renewal equations[END_REF][START_REF] Tumuluri | Steady state analysis of a non-linear renewal equation[END_REF] the authors have discussed the existence and uniqueness of a weak solution and have also proved the linear stability around the nontrivial steady state of the nonlinear renewal equation. In order to determine the longtime behavior of the solution, generalized relative entropy (GRE) plays a crucial role. The asymptotic behavior of the solution to the nonlinear MV equation has been studied in [START_REF] Michel | General entropy equations for structured population models and scattering[END_REF][START_REF] Michel | General relative entropy inequality: an illustrations on growth models[END_REF][START_REF] Perthame | Transport Equations in Biology[END_REF] using the GRE method. The well studied semi group theory techniques can also be applied to prove the existence, uniqueness results (see for instance [START_REF] Benabdallah | Approximation by Diffusion of Renewal Equations[END_REF][START_REF] Iannelli | Approximation of a population dynamics model by parabolic regularization[END_REF]).

The linear version of equation (1.1) with C = 1 has been studied in [START_REF] Abdellaoui | Decay solution for the renewal equation with diffusion[END_REF]. Touaoula et. al., proved the existence and uniqueness of a weak solution. They have used Poincaré Writinger's type inequality to prove the exponential decay of the solution for large times to a steady state. In [START_REF] Michel | Asymptotic behavior for a class of the renewal nonlinear equation with diffusion[END_REF], Michel et. al., considered the nonlinearity in the boundary term in equation (1.1) and proved the convergence of the solution towards the steady state problem. In [START_REF] Kakumani | On a nonlinear renewal equation with diffusion[END_REF], Kakumani et. al., proved the existence and uniqueness of a weak solution with S 1 = S 2 and they have also proved the longtime behavior is some particular cases. The aim of this paper is to extend the global convergence results of [START_REF] Michel | General entropy equations for structured population models and scattering[END_REF][START_REF] Michel | General relative entropy inequality: an illustrations on growth models[END_REF][START_REF] Perthame | Transport Equations in Biology[END_REF] (using the entropy method) to a more general class of nonlocal nonlinear diffusive MV equation than those seen in [START_REF] Abdellaoui | Decay solution for the renewal equation with diffusion[END_REF][START_REF] Kakumani | On a nonlinear renewal equation with diffusion[END_REF][START_REF] Michel | Asymptotic behavior for a class of the renewal nonlinear equation with diffusion[END_REF]. To do that, we first prove the existence and uniqueness of the solution of the equation in section 2. We also prove the existence and uniqueness of the corresponding steady state equation in the same section. Then, in section 3, we prove the asymptotic behavior of the solution using the GRE result. Finally, in section 4, we present some numerical simulations and we conclude.

2. Existence and uniqueness. In this section we prove existence and uniqueness result of solution to (1.1)-(1.2), (1.3) and (1.4). We use the same definition of weak solution and we follow the similar arguments which are used in [START_REF] Kakumani | On a nonlinear renewal equation with diffusion[END_REF] to prove the existence and uniqueness result to (1.1)-(1.2). We start with the following a priori estimate of u.

Lemma 2.1. Assume that S(.) ∈ L ∞ loc (R + ) × L ∞ loc (R + )
, then there exists a unique weak solution u ∈ C R + ; L 1 (R + ) ∩L 2 loc (R + ; W 1,2 (R + )) which solves (1.1). Moreover, we have u ≥ 0, and

(2.1) ∞ 0 |u(t, x)|dx ≤ e ||(B-d)+||∞t ∞ 0 |u 0 (x)|dx. Theorem 2.2. Assume (1.5)-(1.7), then there is a unique weak solution u ∈ C R + ; L 1 (R + ) ∩ L 2 loc (R + ; W 1,2 (R + )) solving (1.1) -(1.
2). Proof. To prove the existence and uniqueness of (1.1)-(1.2) we use banach fixed point theorem. First, we consider the banach space X := C([0, T ],

∞ ) and T is chosen later. Let X + be the set of all nonnegative continuous functions on [0, T ], let

ϑ := ||(B -d) + || ∞ . Let S : [0, T ] × [0, T ] → R 2 be a continuous function. Then we have u ∈ C [0, T ]; L 1 (R + ) which is a solution of (2.2)        u t (t, x) + u x (t, x) + d x, S 1 (t) u(t, x) = Cu xx , t ∈ (0, T ), x > 0, u(t, 0) -Cu x (t, 0) = ∞ 0 B x, S 2 (t) u(t, x)dx, t ∈ (0, T ), u(0, x) = u 0 (x), x > 0.
Now define a map Γ :

X + × X + → X + × X + by S(t) -→ ∞ 0 ψ 1 (x)u(t, x)dx, ∞ 0 ψ 2 (x)u(t, x)dx .
Note that solutions of (1.1) -(1.2) are precisely fixed points of the map Γ. Therefore, it is enough to prove that Γ is a contraction map. Let u 1 (t, x), u 2 (t, x) be solutions of (2.2) corresponding to S a (t) := (S a

1 (t), S a 2 (t)), S b (t) := (S b 1 (t), S b 2 (t)) respectively. Then ũ := u 1 -u 2 satisfies            ũt + ũx + d(x, S a 1 (t))ũ + [d x, S a 1 (t) -d x, S b 1 (t) ]u 2 = C ũxx , t ∈ [0, T ], x > 0, ũ(t, 0) -C ũx (t, 0) = ∞ 0 B x, S a 2 (t) ũ(t, x) + [B x, S a 2 (t) -B x, S b 2 (t) ]u 2 dx, ũ(0, x) = 0.
Multiplying the above equation by sgn(ũ) and integrating in x and use (1.5) to get

d dt ∞ 0 |ũ(t, x)|dx ≤ ∞ 0 B x, S a 2 (t) |ũ|(t, x) + B x, S a 2 (t) -B x, S b 2 (t) u 2 (t, x) dx - ∞ 0 d x, S a 1 (t) |ũ|(t, x)dx + ∞ 0 d x, S a 1 (t) -d x, S b 1 (t) u 2 (t, x)dx ≤ ∞ 0 B x, S a 2 (t) |ũ|(t, x) + L|S a 2 (t) -S b 2 (t)|u 2 (t, x) dx - ∞ 0 d x, S a 1 (t) |ũ|(t, x)dx + L|S a 1 (t) -S b 1 (t)| ∞ 0 u 2 (t, x)dx ≤ ϑ ∞ 0 |ũ(t, x)|dx + L|S a (t) -S b (t)| ∞ 0 u 2 (t, x)dx (where |S a (t) -S b (t)| := |S a 1 (t) -S b 1 (t)| + |S a 2 (t) -S b 2 (t)|) ≤ ϑ ∞ 0 |ũ(t, x)|dx + L sup 0≤t≤T |S a (t) -S b (t)| |u 0 | L 1 e ϑt dx.
By Gronwall's inequality we get

∞ 0 |ũ(t, x)|dx ≤ Lt sup 0≤t≤T |S a (t) -S b (t)| |u 0 | L 1 e ϑt ,
and hence

sup 0≤t≤T ∞ 0 |ũ(t, x)|dx ≤ LT sup 0≤t≤T |S a (t) -S b (t)| |u 0 | L 1 e ϑT .
Therefore from the definition of Γ and using (1.7) we obtain

sup 0≤t≤T |Γ(S a ) -Γ(S b )| ≤ ψ M sup 0≤t≤T ∞ 0 |ũ(t, x)|dx ≤ Lψ M T sup 0≤t≤T |S a (t) -S b (t)| |u 0 | L 1 e ϑT .
Now choose T such that Γ is a strict contraction map in a Banach space X + × X + . Since T does not depend on the iteration process, we repeat the same process for [T, 2T ], [2T, 3T ], .... to obtain that there exists a unique solution u to (1.1)

-(1.2) in C R + ; L 1 (R + ) . Therefore we get that S i (.) is continuous and hence S i (.) ∈ L ∞ loc (R + ). Hence there exists a unique weak solution u ∈ C R + ; L 1 (R + ) ∩ L 2 loc R + ; W 1,2 (R + ) to the system (1.1)-(1.
2). Now we prove the existence and uniqueness of (1.3) and (1.4). First we observe that for a given S, we have to consider the associated eigenvalue problem of (1.3) and (1.4). Proposition 2.3. Assume (1.6) and further assume that

(2.3) m ≤ ψ 1 (x) ψ 2 (x) ≤ M, 0 < m ≤ M < ∞, and
(2.4) B(x, 0) -d(x, 0) > 0 B(x, 0) -d(x, ∞) < 0,
are satisfied then there exists a solution to (1.3)- (1.4). Moreover, if m = M = 1 then we have directly uniqueness of the solution.

Before we prove the Proposition 2.3, we prove some lemmas which are helpful to prove this proposition. We notice that for a given S there exists (λ S , U S , φ S ) solution to the eigenproblem (see [START_REF] Abdellaoui | Decay solution for the renewal equation with diffusion[END_REF] for details),

(2.5)

                   ∂ x U S = C∆U S -d(x, S1 )U S -λ S U S , U S (0) -CU ′ S (0) = B(x, S2 )U S (x)dx, U S ∈ W 1,2 (R + ), -∂ x φ S = C∆φ S -d(x, S1 )φ S + φ S (0)B(x, S2 ) -λ S φ S , φ S ∈ W 1,2 (R + ), φ ′ S (0) = 0 and φ S U S (x)dx = 1.
Lemma 2.4. Assume (1.6) then we have

(2.6) ∂ ∂ S1 λ ∂ ∂ S2 λ = -( ∂ ∂ S1 d)U S φ S dx φ S (0) ( ∂ ∂ S2 B)U S dx < 0 0 .
Proof. The proof goes in similar lines that are given in [START_REF] Michel | Optimal proliferation rate in a cell division model[END_REF][START_REF] Michel | Fitness optimization in a cell division model[END_REF]. Therefore we skip the proof.

Lemma 2.5. Assume (1.6), λ (0,0) > 0, λ (0,∞) < 0 and (2.3) then there exists a solution to (1.3).

Proof. Using that λ (0,0) > 0, λ (0,∞) < 0 and the decay (2.6), we have the existence of Γ decreasing regular function defined on [0, S *

1 [ (with S * 1 ∈ [0, ∞]) so that { S = ( S1 , S2 ) : λ S = 0} = {( S1 , Γ( S1 )) : S1 ∈ R + } ⊂ R 2 + (1-dimension manifold). Let Ũ( S1, S2) = S1 U ( S1, S2) U ( S1, S2) (x)ψ 1 (x)dx , and
β : S1 → Ũ( S1, S2) (x)ψ 2 (x)dx,
then by assumption (2.3), we have that m S1 ≤ β( S1 ) ≤ M S1 , for all S1 , therefore there exists S+ 1 and S-

1 such that Γ( S+ 1 ) = M S+ 1 , Γ( S- 1 ) = m S- 1 .
Moreover, for S1 small enough ( S1 , β( S1 )) belongs to the connex set of boundary

{( S1 , M S1 ) : S1 ∈ [0, S+ 1 ]}∪{( S1 , m S1 ) : S1 ∈ [0, S- 1 ]}∪{( S1 , Γ( S1 )), S1 ∈ [ S+ 1 , S- 1 },
and for S1 large enough ( S1 , β( S1 )) belongs to the connex set of boundary (see Figure 1)

{( S1 , M S1 ) : S1 > S+ 1 } ∪ {( S1 , m S1 ) : S1 > S- 1 } ∪ {( S1 , Γ( S1 )), S1 ∈ [ S+ 1 , S- 1 }.
Therefore, there exists ( S1 , S2 ) such that Ũ(

S 1 S 2 β( S1 ) M S1 Γ( S1 ) m S1
Fig. 1: S1 → Γ( S1 ) and S1 → β( S1 ). We see there exists S1 such that β( S1 ) = Γ( S1 ) Remark 2.1. It is easy to check that the Proposition 2.3 is an immediate consequence of Lemma 2.5. Notice that using (2.4), we have λ (0,0) > 0, λ (0,∞) < 0 are satisfied. Uniqueness of U : Let U solution of the eigenproblem given by the Proposition 2.3 and V an another positive solution to

V ′ + d(x, V ψ)V = CV ′′ , V (0) -V ′ (0) = B(x, V ψ)V (x)dx
with V ψ = U ψ. Then there exists V , φ, λ solution to the eigenproblem

V ′ + d(x, V ψ) V = C V ′′ -λ V , V (0) -V ′ (0) = B(x, V ψ) V (x)dx -φ′ + d(x, V ψ) φ = C φ′′ -λ φ + B(x, V ψ)φ(0), φ′ (0) = 0 with λ = 0 (since ∂ ∂S λ S < 0)
. Therefore by integration, we have λ V φ = 0 and V = 0.

3. Asymptotic Behavior. In this section, we prove that the solution of (1.1) converges to the solution of (1.3) for large time. To prove this convergence we need additional hypothesis on φ, B and d, which are given below.

Definition 3.1. Let

f = u/U, dν(x) = U (x)φ(x)dx, and
f ν := f (t, x)dν(x) = u(t, x)φ(x)dx
The first assumption consists in the comparison between ψ i and φ : (3.1)

kψ i (x) ≤ φ(x) ≤ Kψ i (x), 0 < k ≤ 1 ≤ K < ∞ and ψ i (x)U (x)dx = 1, i = 1, 2.
Therefore we chose Si = 1, i = 1, 2. Further, the vital rates should be of the form:

(3.2) B(x, S 1 ) = B(x, S 2 ) x ∈ (0, x 1 ], B(x) x ∈ (x 1 , ∞). (3.3) d(x, S 1 ) = d(x, S 1 ) x ∈ (0, x 2 ], d(x) x ∈ (x 2 , ∞).
for a fixed x 1 , x 2 in (0, ∞). Remark 3.1. Under the assumptions (3.2) and (3.3), we have Poincaré Writinger type inequality (see [START_REF] Abdellaoui | Decay solution for the renewal equation with diffusion[END_REF] for an unbounded domain) i.e.,

∞ 0 (f -f ν ) 2 dν ≤ C 3 ∞ 0 ∂f ∂x 2 dν,
where C 3 depends on C, max B, min B, max d, min d. Moreover, as in previous work (see [START_REF] Kakumani | On a nonlinear renewal equation with diffusion[END_REF]) we add the assumption (to prove the convergence of the linear case)

(3.4) ∃C 0 , C 1 , C 2 s.t. B(x, .) ≥ C 0 φ(x)/φ(0), sup S2 | ∂B ∂S 2 |(x) < C 1 φ(x)/φ(0) ∀x ∈ R + and sup x,S1 | ∂d ∂S 1 | < C 2 ,
and finally, we need that for a given diffusion constant C, the variation of B and d are small enough, i.e.,

r := max 2C 1 √ C 3 K k √ C 0 C , 2C 2 K Ck (1 -( 2C2C3 K kC ) 2 ) , 2C 2 Ck K2 + 2C 1 √ C 3 K2 k √ C 0 C 2 < 1 2 , (3.5)
where K := max(K, f ν (t = 0)). Theorem 3.2. Under assumption (3.1)-(3.5) we have

u(t, .) → U (.), as t → ∞ in L 2 dν (R + ).
Proof of the main Theorem 3.2. We first state the GRE computation (see [START_REF] Kakumani | On a nonlinear renewal equation with diffusion[END_REF][START_REF] Michel | General Relative Entropy in a nonlinear McKendrick model[END_REF][START_REF] Michel | General relative entropy inequality: an illustrations on growth models[END_REF]).

Lemma 3.3. Assume that u solves (1.1) then we have

(3.6) d dt f ν = [d(x, 1) -d(x, S 1 (t))]f (t, x)dν(x) -φ(0) f [B(x, 1) -B(x, S 2 (t))]U (x)dx, and
(3.7) d dt (f -f ν ) 2 (t, x)dν(x) = -2C (f ′ ) 2 dν(x) -φ(0) [f (t, x) -f (t, 0)] 2 B(x, 1)U (x)dx -2φ(0)[f (t, 0) -f ν ] f [B(x, 1) -B(x, S 2 (t))]U (x)dx -2 f [d(x, S 1 (t)) -d(x, 1)](f -f ν )dν(x).
Proof. By computation (multiplying equation (1.1) by φ and integrating with respect to x), we find that

d dt f (t, x)dν(x) = [-∂ x u + C∆u -d(x, S 1 (t))u]φ(x)dx = [∂ x φ(x) + C∆φ(x) -d(x, S 1 (t))φ(x)]u(t, x)dx + φ(0) B(x, S 2 (t))u(t, x)dx = [d(x, 1) -d(x, S 1 (t))]f (t, x)dν(x) -φ(0) f [B(x, 1) -B(x, S 2 (t))]U (x)dx,
and for the computation of (3.7) refer [START_REF] Perthame | Transport Equations in Biology[END_REF] (or) see appendix (Lemma 5.2). Lemma 3.4. Assume (3.1) then for all

f ∈ L 1 dν (R + , R + ) ∩ W 1,2 dν (R + , R + ) we have (3.8) k f (x)ψ i (x)U (x)dx ≤ f ν ≤ K f (x)ψ i (x)U (x)dx, (3.9) k (f ′2 )ψ i (x)U (x)dx ≤ (f ′ ) 2 dν(x) ≤ K (f ′2 )ψ i (x)U (x)dx.
Proof. The proof is direct consequence under the assumption (3.1). Lemma 3.5. Assume (1.6) then we have

(3.10) min(k, f ν (t = 0)) ≤ f ν ≤ max(K, f ν (t = 0)), ∀t ≥ 0. (3.11) min(k, f ν (t = 0))/K ≤ f (t, x)ψ i (x)dx ≤ max(K, f ν (t = 0))/k, ∀t ≥ 0. (3.12) lim sup t→∞ | f ν -1| ≤ max i lim sup t→∞ | f ν -S i |. Proof. Using (3.8), if f ν ≥ K then S i ≥ 1 (resp. if f ν ≤ k then S i ≤ 1
) and by equation (3.6) and (1.6) we have f ν is decreasing (respectively f ν is increasing). Now, using Lemma 3.4 we find the inequalities (3.10) and (3.11). In order to prove the inequality (3.12), define ǫ i (t) := f ν -S i (t) then

d dt f ν = [d(x, 1) -d(x, f ν -ǫ 1 (t))]f (t, x)dν(x) -φ(0) f [B(x, 1) -B(x, f ν -ǫ 2 (t))]U (x)dx = ((1 + ǫ 1 (t)) -f ν ) d(x,1)-d(x, f ν-ǫ1(t)) ((1+ǫ1(t))-f ν ) f (t, x)dν(x) +((1 + ǫ 2 (t)) -f ν ) B(x,1)-B(x, f ν -ǫ2(t)) ((1+ǫ2(t))-f ν ) f (t, x)U (x)dx.
therefore, we have

d dt | f ν -1| = -(|1 -f ν | ± ǫ 1 (t)) d(x, 1) -d(x, f ν -ǫ 1 (t)) ((1 + ǫ 1 (t)) -f ν ) f dν(x) -(|1 -f ν | ± ǫ 2 (t)) B(x, 1) -B(x, f ν -ǫ 2 (t)) ((1 + ǫ 2 (t)) -f ν ) f (t, x)U (x)dx,
and we find that (3.12) is satisfied. Hence the lemma is proved. Proposition 3.6. Assume (3.1) -(3.4) then we have that, for

C 2 < Ck 2C3 K , lim sup t→∞ | f ν -S i | ≤ r max i lim sup t→∞ |S i -1|.
Proof. Using (3.1) and Cauchy-Schwarz inequality we find that

d dt (f -f ν ) 2 (t, x)dν(x) ≤ -2C C3 (f -f ν ) 2 (t, x)dν(x) -φ(0) [f (t, x) -f (t, 0)] 2 B(x, 1)U (x)dx +2φ(0)( |f (t, 0) -f (t, x)| 2 dν(x)) 1/2 C 1 |S 2 -1| f |B(x,1)-B(x,S2(t))| C1 U (x)dx +2C 2 ( f 2 dν(x)) 1/2 |S 1 -1|( (f -f ν ) 2 (t, x)dν(x)) 1/2 ≤ -2C C3 (f -f ν ) 2 (t, x)dν(x) -C 0 [f (t, x) -f (t, 0)] 2 dν(x) +2C 1 ( |f (t, 0) -f (t, x)| 2 dν(x)) 1/2 |S 2 -1| K +2C 2 ( f 2 dν(x)) 1/2 |S 1 -1|( (f -f ν ) 2 (t, x)dν(x)) 1/2 ≤ -C C3 (f -f ν ) 2 (t, x)dν(x) -C0C C3 [f (t, x) -f (t, 0)] 2 dν(x) (f -f ν ) 2 (t, x)dν(x) +2C 1 ( |f (t, 0) -f (t, x)| 2 dν(x)) 1/2 |S 2 -1| K +2C 2 ( f 2 dν(x)) 1/2 |S 1 -1|( (f -f ν ) 2 (t, x)dν(x)) 1/2 ,
and define w(t

) := (f -f ν )(t, .) 2 L 2 dν
, then we have

d dt w(t) ≤ - C 0 C C 3 |f (t, 0) -f (t, x)| 2 dν(x) 1/2 [(w(t)) 1/2 - 2C 1 √ C 3 K √ C 0 C |S 2 -1|] - C C 3 (w(t)) 1/2 [(w(t)) 1/2 - 2C 2 C 3 ( f 2 dν(x)) 1/2 C |S 1 -1|].
Using the similar argument in Lemma 3.5, we get lim sup

t→∞ w(t) ≤ max 2C 1 √ C 3 K √ C 0 C lim sup t→∞ |S 2 -1|, lim sup t→∞ 2C 2 C 3 ( f 2 dν(x)) 1/2 C |S 1 -1| .
Thus, we find that lim sup

t→∞ (f -f ν )(t, .) 2 L 2 dν ≤ max lim sup t→∞ 2C 2 C 3 K kC 2 f (t, .) 2 L 2 dν , 2C 1 √ C 3 K2 k √ C 0 C 2 ,
and for

C 2 < Ck 2C3 K , we have lim sup t→∞ f L 2 dν ≤ max K (1 -( 2C2C3 K kC ) 2 ) , K2 + 2C 1 √ C 3 K2 k √ C 0 C 2 , therefore lim sup t→∞ w(t) ≤ max 2C 1 √ C 3 K √ C 0 C lim sup t→∞ |S 2 -1|, 2C 2 K C (1 -( 2C2C3 K kC ) 2 ) lim sup t→∞ |S 1 -1|, 2C 2 C K2 + 2C 1 √ C 3 K2 k √ C 0 C 2 |S 1 -1| .
Since we have

| f ν -f ψiU | = | f -f ν ψiU | ≤ 1 k (f -f ν ) 2 ν , we find that lim sup t→∞ | f ν -S i | ≤ r max i lim sup t→∞ |S i -1| is satisfied. Proof. (of Theorem 3.2)
To prove this theorem it is enough to prove that (ff ν ) 2 (t, x) L 2 dν = 0, as t → ∞. Using Lemmas 3.3-3.5 and Proposition (3.6), we have

lim sup t→∞ | f ν -1| ≤ lim sup t→∞ | f ν -S i | ≤ r lim sup t→∞ |S i -1|,
and finally lim sup Lemma 3.6). Therefore S i → t→∞ 1, f ν → t→∞ 1 and finally we have

t→∞ |S i -1| ≤ lim sup t→∞ | f ν -1| + lim sup t→∞ | f ν -S i | ≤ 2r lim sup t→∞ |S i -1|, with r < 1/2 (see
lim sup t→∞ (f -f ν ) 2 (t, x) L 2 dν = 0.
Hence the theorem is proved.

4. Numerical Simulations. In this section, we present some examples to illustrate the result of Section 3. We have considered finite difference schemes with uniform step size to perform our numerical simulations. Let ∆t and ∆x be the step size for time and age respectively. We have used ∆t = 0.004 and ∆x = 0.1 for our numerical simulations.

Example 1 (convergence of the solution to the steady state) Assume that the vital rates, initial data and the competition weights are given by

d(x, S) = (1 + e -x )S, B(x, S) = 2 + e -x-S , u 0 (x) = e -x , ψ 1 = ψ 2 = 4.1376χ [0,22] .
In Figure 2, we compare the normalized solution of (1.1)-(1.2) with the steady state (normalized) at t = 20 and the absolute error plot is also given. We observe that the normalized solution of (1.1)-(1.2) is in good agreement to the normalized vector of its steady state. In Figure 4, we plot the solution of (1.1)-(1.2) at x = 10 and we notice that the solution converges for a longtime. We have also given the plots of (U, φ) in Figure 3.

Example 2 (behavior of S(.) when C or ψ i 's varies) In this example, we assume that the birth and death rates can be written as a product of two terms: one depends of the age x and the other depends on the weighted population S. The vital rates and initial data are given by:

d(x, S) = 1 + x 10 S 10 + 0.1 , B(x, S) = 10e -80(x-2) 2 -1.1S , u 0 (x) = e -x .
In Figures 5 and6, we have given the plot of S(.) with C = 0.15 and C = 0.00001 using ψ 1 = ψ 2 = 1χ [0,22] , we notice that the oscillations will disappear as C increases. We also observe that the S(.) converges for large value of the diffusion constant in large time, where as it does not converge for small values of the diffusion constant. Contrary, for the fixed diffusion constant C = 0.00001 from the Figures ( 6)-( 9), we notice that the S(.) also converges for large time when ψ i varies (keeping supp(ψ i ) fixed).

Conclusions.

We have proved the existence and uniqueness result for a nonlocal nonlinear renewal equation with diffusion and its corresponding steady state equation also. We have used the fixed point techniques to prove the results. We have established the convergence of the solution to its steady state for large time. We took the help of GRE inequality and the Poincaré Writinger type inequality for unbounded domain. The assumptions on the vital rates i.e., assumptions (3.2)-(3.4) are strong to prove the convergence of the solution, we would try to minimize the assumptions on the birth and death rates to prove the convergence part. Some numerical simulations are given to make sure that the solution converges. These simulations shows that the normalized solution of (1.1)-(1.2) converges to the normalized vector of its steady state for large time. We have also given an example where the S(.) oscillates and does not converge for the small values of the diffusion constant C and the oscillations vanishes and converges when C is large or ψ i 's are moving towards +∞ (keeping supp(ψ i ) fixed). We would like to work on the bifurcation analysis of the diffusion constant C, local convergence and linear stability around the steady state of the solution of the equation (1.1)-(1.2). Proof. First we consider the case of bounded domain, i.e.,

               U ′ R (x) + d(x, S1 )U R (x) = CU ′′ R (x), x ∈ (0, R), U R (0) -CU ′ R (0) = R 0 B(x, S2 )U R (x)dx, R 0 U R (x)dx = 1, and U R (R) = 0. (5.1)
and its adjoint equation is given by (5.2)

           -φ ′ R (x) + d(x, S1 )φ R (x) = Cφ ′′ R (x) + φ R (0)B(x, S2 ), x ∈ (0, R), φ ′ R (0) = 0, R 0 φ R (x)U (x)dx = 1, and φ R (R) = 0.
Denote U R , φ R be the solutions of (5.1) and (5.2) respectively. Existence of U R , φ R can be obtained by using the method of subsolution and supersolution (see [START_REF] Pao | Nonlinear Parabolic and Elliptic equations[END_REF]).

Step 1: In this step, we prove the existence of solution to (1.3). Multiply U R on both sides to equation (5.1) and integrate from 0 to R and use integration by parts to get

C R 0 |U ′ R | 2 dx + R 0 d(x, S1 )|U R | 2 dx ≤ 1 2 B 2 M .
C R 0 |U ′ R | 2 dx + d m R 0 |U R | 2 dx ≤ C R 0 |U ′ R | 2 dx + R 0 d(x, S1 )|U R | 2 dx ≤ 1 2 B 2 M .
Hence, U R ∈ W 1,2 (R + ). Set U R (x) = 0 in x > R, then there exists a subsequence (still denote them by U R ) we get the existence of U which is the solution of (1.3) such that U R ⇀ U in W 1,2 (R + ). Uniqueness follows from the standard arguments (see Section 2).

Step 2: In this step, we prove the existence of solution to (1.4). Multiply e where C 3 = -Cφ(0)H u(t,0) U(0) U x (0).

Now we consider

C 1 + C 2 = H ′ u(t,0) U(0) φ(0) -Cu x (t, 0) + u(t,0) U(0) CU x (0) = H ′ u(t,0) U(0) φ(0) -u(t, 0) +

Fig. 2 :Fig. 3 : 3 . 5 . 1 .

 23351 Fig. 2: left:Solution of (1.1)-(1.2) at t = 20. Right: Absolute error.

Fig. 4 :Fig. 5 :Fig. 6 :Fig. 7 :

 4567 Fig. 4: Solution of (1.1)-(1.2) at x = 10.

Fig. 8 :Fig. 9 :

 89 Fig.8: C = 0.00001, ψ 1 = ψ 2 = 1χ[START_REF] Abdellaoui | Decay solution for the renewal equation with diffusion[END_REF][START_REF] Tumuluri | Steady state analysis of a non-linear renewal equation[END_REF] .

0 B 2 R 0 (

 020 equation with g(x) and integrate from 0 to R to get , S1 )φ R (x)e x g(x)dx = φ R (0, S2 )e x g(x)dx , where g solves the differential equation(5.3) -(e x c g ′ (x)) ′ = U R (x), , x ∈ (0, R), g(R) = 0, g ′ (0) = 1.It is easy to observe that ex c g(x) ≤ U R (x). Using (1.7) we get C+dm C+BM ≤ φ R (0) ≤ 1 + dM C . Multiplying φ R (x)U R (x) to equation (5.2) and integrate from 0 to R and use integration by parts to getC S1 )(φ R (x)) 2 U R (x)dx = φ R (0) R (x, S2 )φ R (x)U R (x)dx -R (x)) 2 U R (x)dx + d m φ R (x)) 2 U R (x)dx ≤ φ R (0)B M ≤ (1 + d M C )B M .

∞ 0 B 1 U

 01 (x, S 2 )u(x)dx +u(t, 0) 1x, S 2 )u(x)dx -u(t,0) U(0) ∞ 0 B(x, S2 )U (x)dx .Now we considerL + C 3 = H u(t,0) U(0) φ(0)U (0) -Cφ(0)H u(t,0) U(0) U x (0) = H u(t,0) U(0) φ(0) ∞ 0 B(x, S2 )u(x)dx.Hence after rewriting we getd dt ∞ 0 H(u/U )U φdx = CD H dif f (u) + D H ren (u) + Remainder.Now take H(x) = x 2 and use (3.6) to get (3.7).

Acknowledgements. Bhargav Kumar Kakumani has been supported by Institut Camille Jordan and CSIR.

It is easy to show that φ R ∈ L ∞ (R + ) (see [START_REF] Abdellaoui | Decay solution for the renewal equation with diffusion[END_REF]). By similar analysis as in Step 1, we get the existence of φ which is the solution of (1.4). From the Lebesgue theorem we get ∞ 0 U (x)φ(x)dx = 1. Remark 5.1. In equation (5.1), even if

∞ the proof still works by modifying the constants accordingly.

Lemma 5.2. Let u, U, φ be solution to equation (1.1), (1.3) and (1.4) respectively. Let the entropy is defined by

where H is a convex function. Then we have

where the entropy dissipation due to diffusion and the renewal terms are

then we get

where

First we Consider

where C 1 = -CH ′ u(t,0) U(0) φ(0)u x (t, 0). Now consider

where C 2 = Cφ(0)H ′ u(t,0) U(0) u(t,0) U(0) U x (0). Adding I 1 and I 2 to get

Now consider


Single-mode LD (a) 1 W 1 100 MW.cm -2 .sr -1

Broad area LD (b) 7 W 1 x 6 110 MW.cm -2 .sr -1

Tapered LD (c) 12 W 1 x 1.2 1000 MW.cm -2 .sr -1

ϕ 1 (t) ϕ 2 (t) ϕ N (t)
Current driver

Phase measurement

Optimization algorithm

Active control of the N currents is mandatory to ensure a stable phase relationship between them

External cavity

The external cavity is designed to favour the collective operation of the emitters by inducing a coupling between them.

External cavity light from one emitter is reflected into the others

The laser array operates as a sampled gain medium in a single cavity • Scaling to large number of emitters is still challenging.

• Active vs passive ? Electronic vs optic ?

• Detailed analysis of the physics of passively phaselocked lasers still needed.

• Careful design & optimization of the CBC architecture in regard with the devices.

• New results in BRIDLE expected !

  measurement of the power and beam quality of a laser source [unit : W.m -2 .sr -1 ]! The brighter the better … ability to focus a high power on a small area with a low NA

•Ferdinand=Ferdinand

  (a) SCOWL -Donnelly et al, IEEE JQE 39, 2 (2003) (b) P. Crump, BRIDLE (c) Fiebig et al. Elec. Lett, 44, p1253 (2008) output power of single-mode devices increase the power density on facets damages & deterioration of lasers ↗ output power by combining parallel laser sources while maintaining the beam quality Spectral beam combining B SC = N x B 1 = superposition of ≠ laser lines with grating / dichroïc mirrors / vol Bragg gratings ↗ output power ↗ emission width W L & ↘ divergence θ ↗ spatial brightness + narrow Δλ amplification of a single-frequency laser beam in a tapered amplifier in a Mach-Zehnder interferometer configuration : reference amplified We observe highly-contrasted fringes on the combined port the seed and amplified beams are phase-locked the coherence between them is ≥ 96 % amplification of a single-frequency laser beam in a tapered amplifier in a Mach-Zehnder interferometer configuration : reference amplified The fringes shift with the currents in the ridge & taper section, because of thermally-induced change of the optical path in the amplifier Phase-shift with ridge current 0.025-0.037 π/mA Amplification of a single-frequency laser beam in multiple amplifiers in //

FerdinandFerdinand

  LD amplified in a YbDFA Division of the seed beam and injection in the 47 amplifiers Creedon et al, "High efficiency coherent beam combining of semiconductor optical amplifiers" , Optics Letters 37, 23 p5006 (2012) Ferdinand-Braun-Institut -BERLIN -23 th January 2015 Active phase-locking of a SCOWL array Creedon et al, "High efficiency coherent beam combining of semiconductor optical amplifiers" , Optics Letters 37, 23 p5006 (2012) Optimization of the amplifier driving currents to maximize the combined power (udate rate = 6 kHz) ! Demonstration of the active phase-locking & coherent beam combining of 47 semiconductor amplifiers ! Total output power = 40 W with η CBC = 87% : B ~ 2.5 GW.cm -2 .sr -1 B SC = 25 x B 1 Yb doped fibers amplifiers : 1.93 kW combined 79 % efficiency, M 2 = 1.1LOCSET active feedbackRedmond et al., "Diffractive coherent combining of a 2.5 kW fiber laser array into a 1.9 kW Gaussian beam," Opt. Lett. 37, 2832-2834 et al., "Coherent beam combining of two femtosecond fiber chirped-pulse amplifiers," Opt.Lett. 36, 621-623 (2011)

  The external cavity favours constructive interferences between the multiple beams.

FerdinandFerdinand

  Minimum losses in the laser cavity for constructive interferences on BS in the P arm : passive phase-locking & coherent combining of the two lasers et al, Optics Letters 36, 3810 (2011) Minimum losses in the laser cavity for constructive interferences in the 0 th order of the PG : passive phase-locking & coherent combining PG cavity : Talbot effect tiled aperture T.Y. Fan, Laser beam combining for high-power, high-radiance sources, µm, 2w = 30 µm, λ = 976 nm 0 W. H. F. Talbot "Facts relating to optical science" Philos. Mag. 9 (1836) Near-field diffraction effect observed for a grating illuminated by monochromatic light : ! self-images (E, ϕ) at multiples of effect used in an external cavity to maximize the coupling between emitters : maximum back reflection of light for L ext = Z T /4 Near-field propagation of 10 in-phase Gaussian lasers, demonstrating the self-imaging Talbot effect. et al, Appl. Phys. Lett. 93, 211102 (2008) et al, Appl. Phys. Lett. 93, 211102 (2008) F' Measurement of the far-field profile I = 4.4 x I th theory experiment Spectral locking of each laser diodes Narrow linewidth (Δλ < 0.1 nm) Laser threshold I th = 0.9 A P max = 1.7 W @ 4 A (4 x I th ) Operation in the in-phase mode Highly-contrasted fringes in the FF V ≥ 80% Diffraction of the N = 10 coherent beams on a phase grating within 1 direction ⇒ coherent superposition of the emitters in the near-field plane NFʻ • Passive phase-locking of diode lasers in the Talbot external cavity + Passive conversion of the complex pattern in a Gaussian-like mode combining, ed .Brignon (Wiley -VCH) • New interest in coherent beam combining techniques in the laser community (fiber, solid-state, diodes …) • Better understanding of the limits • High-brightness CBC laser sources have been demonstrated Special thanks to P. Georges, M. Hanna, D. Paboeuf, L. Leveque, G. Schimmel, S. Janicot, I. Doyen


Daouda Sawadogo
email: daouda.sawadogo@univ-lr.fr

Ronan Champagnat
email: ronan.champagnat@univ-lr.fr

Pascal Estraillier
email: pascal.estraillier@univ-lr.fr

Adaptive Digital Resource Modeling as Service Provider and Consumer

Keywords: digital resource modeling, usability, interoperability, adaptation, learning platform I

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

environment specification, we are aimed at providing a digital resource model that is easy to manage and integrate into user's environment. This means that each digital resource should be delivered and used according to user's characteristics and environment context. This is all the more true when it comes to digital learning environments such as LMS (Learning Management System), MOOC (Massive Open and Online Courses), PLE (Personal Learning Environment), RBLE (Resource-Based Learning Environment), which raise new challenges in digital resource modeling that should comply with them. For example, Personal Learning Environments are modern tools built upon the concept of how users (learners in this context) can organize their digital resources in a personal environment. But digital resources can become quickly complex and difficult to adapt to users' needs.

To answer these research questions, we introduce our approach, which relies on the experiences learned from software development area. Particularly, we consider API-based approach to build software as services instead of web frameworks, since today, applications are built by consuming and producing APIs [START_REF] Anuff | APIs are Different than Integration[END_REF]. Some research works [START_REF] Green | An adaptable personal learning environment for e-learning and e-assessment[END_REF] [5] [START_REF] Chatti | Toward a Personal Learning Environment Framework[END_REF] tries to propose architectures and mechanisms to create learning environments, which are adaptive to their users' requirements and profiles. In [START_REF] Kiy | An Adaptive Personal Learning Environment Architecture[END_REF] the authors proposed a PLE that adapts itself to the needs and preferences of each user. In [START_REF] Brusilovsky | User models for adaptive hypermedia and adaptive educational systems[END_REF] the authors propose adaptive mechanisms that can adapt themselves to the context of each user model in adaptive educational systems. These studies are mainly focused on the learning tools and less interested on the digital resource integration across users' environments. Hence, literature studies on PLE are mainly on the personalized tools that allow a user organizing, choosing, sharing their data and personalizing his/her environment. Thus, PLE architecture contains three important parts: The user model, which is the main component of the system, the digital resources model and the tools that are used to manage or process the resources.

II. RESOURCE-BASED PLATFORM CHARACTERISTICS

Digital resources are an essential component of PLE. The authors in [START_REF] Hill | Teaching and learning in digital environments: The resurgence of resource-based learning[END_REF] proved that the potential of Resource-Based Learning Environments (RBLEs) for instruction and learning is considerable. Resources are generally distributed across different sites, such as libraries, classrooms, and learning environments. They claimed that in large measure, this situation evolved in response to pragmatic concerns such as the cost of duplication and assumptions related to resource usage (e.g., documents copyright). However, widespread distribution without the benefit of a comprehensive retrieval system has complicated the location and the use of such resources for learning. Packaging has also a complicated resource use. For example, the packaging design for a specific context makes it difficult to use its content in other learning contexts. Consequently, it often limits the access and use of its contents for other contexts. They authors keep demonstrating that the process used to integrate multiple resources into a coherent learning environment has not been well established. Indeed, this paper tries to provide to a user a suitable digital resource model for multiple resources integration and usage enhancement.

Table I summarizes existing digital resource models and the last line highlights our proposal characteristics developed in this paper. In Table I, we notice that the digital resource initiatives are incompatible with PLE and do not fulfill our most criterions. That means that digital resources are increasingly produced, but the operation of using them are not optimized. For example, MERLOT (Multimedia Education Resource for Learning and Online Teaching) contains about 62 584 learning materials [START_REF] Merlot | Material Results -MERLOT II[END_REF], GLOBE (Global Learning Objects Brokering Exchange) contains about 1.2 million of learning objects [START_REF] Ochoa | On the use of learning object metadata: The GLOBE experience[END_REF] and FUN (France Numeric University) contains about 33 807 pedagogical resources [START_REF] Fun | 33807 pedagogic digital resources[END_REF]. Most of these digital resource repositories are difficult to integrate into user's personal learning environment. For example, in [START_REF] Costagliola | Scorm run-time environment as a service[END_REF], the authors identified the problem of SCORM (Sharable Content Object Reference Model) in its model, it is difficult to be used in LMS. Our goal is to propose a model that improves digital resource usability in terms of interoperability and collaboration in users' platforms.

An evaluation of FUN repository showed that respectively 23% of students and 28% of teachers involved in the questionnaire use the resource of FUN repository [START_REF] Boyer | Thematic digital universities : Evaluation[END_REF]. Thus, the problems identified are as follows: -Collaboration issues in user environment; -Adaptation problem to the user needs; -Production oriented based model not a usability-based model. Therefore, these issues have raised new requirements in the multiple digital resources modeling. In [START_REF] Brown | The Next Generation Digital Learning Environment: A Report on Research[END_REF], the authors stated that it requires new architectures and five domains of core functionality such as interoperability and integration, personalization, analytics, collaboration and accessibility for New Generation Digital Learning Environment (NGDLE).

The contributions of this research study are as follows: Defining a methodology of digital resource modeling that will provide the best user experience and enhance resource usability in terms of retrieving, interoperability, adaptation and collaboration. In our study, we define an interactive and adaptive system as a situation-based scenario system [START_REF] Trillaud | SITUATION-BASED SCENARIOS FOR E-LEARNING[END_REF]. A "situation" is a component of the system where actors interact using local resources in a specific context to achieve one or more common objectives. The system contains three main components as shown in Fig. 1, the user model, the resource model and the process management model. In this system, the user is involved in a situation and he/she uses resources to achieve his/her objective. The management of these digital resources in a situation can help the user retrieve the appropriate resources and integrate and use it into his/her situation.

III. BACKGROUND

A. Study context

A situation is described with IMS-LD metadata [START_REF] Berlanga | IMS LD reusable elements for adaptive learning designs[END_REF] and allows the system to capture the context of user activity. The user model is a representation of information about an individual user that is essential for an adaptive system to provide the adaptation function, [START_REF] Brusilovsky | User models for adaptive hypermedia and adaptive educational systems[END_REF]. The resource model is the representation of information about each resource that is essential to help the user selecting it and integrating it into an activity. The process management represents the mechanism, which is associated to each user in a given situation to help him/her use the resource and manage his/her environment.

In this context our objective consists in helping each user in a situation to easily integrate and use a digital resource in his/her environment to achieve the situation goals. To achieve the aforementioned objective, we propose in this paper a new resource usability-driven method for digital resource modeling. That will enable a resource discoverability, interoperability, adaptation and collaboration in any environment. To do so, we adopt two approaches: (i) the design of digital resource as RESTful-based [START_REF] Richardson | RESTful Web APIs[END_REF] services. A digital resource is a component that provides services to its users through producing and consuming services; (ii) integration of digital resources as Minimum Viable Product (MVP) according to Platform Thinking approach [START_REF] Choudary | Home -Platform Thinking Labs[END_REF]. Digital resource has to be considered as the basic component of digital resource management-based environment. A Minimum Viable Product is the smallest thing you can build that delivers customer value [START_REF] Choudary | Home -Platform Thinking Labs[END_REF]. An adaptive digital resource (ADR) [START_REF] Sawadogo | Adaptive Digital Resource Modelling for Interactive System[END_REF] is composed as shown in Fig. 2 by: metadata describing the resources, tools for manipulating the resources in the environment, rules to verify the resources's ownership to adapt it according to user characteristics. The "service" class is the new added class that gives an interface to third-party applications to consume the ADR exposed services. Each digital resource in the environment includes: -Document: the set of files which composes a resource; -Metadata: to describe the resource's content (IEEE-LOM [START_REF] Barker | What is IEEE Learning Object Metadata/IMS Learning Resource Metadata?[END_REF] application profile is used); -Operators: define the operations on resources and the operations between the elements contained in a resource. Operators for a resource are defined as: 1) Composition; 2) Aggregation; 3) Fusion; 4) Filtering. -Rules: define the rules to manipulate a resource; -Services: define access points for other applications. The component used to adapt the execution and to control rules in each situation are: the user profile (described in [START_REF] Sawadogo | User profile modelling for digital resource management systems[END_REF]); the resource profile (the metadata of the resource IEEE-LOM model); the process profile; the contexts (by using IMS-LD model [START_REF] Berlanga | IMS LD reusable elements for adaptive learning designs[END_REF]). Our method consists in modeling an interoperable and collaborative digital resource ensuring discoverability functions and usability capacity, which is easy to integrate into users' platforms.

B. Adaptive Digital Resource core architecture

IV. METHOD FOR DIGITAL RESOURCE MODELING AND

Interoperability means that the resource needs to be well described, context free, retrievable, usable, and technology independence. RESTful-based services are a good answer to implement these characteristics. Collaboration means that the digital resource needs to be produced, evaluated, shared and consumed by diverse stakeholders in the platform. Platform thinking approach can be used to implement these characteristics. The following section details our method to design digital resource which fulfills our characteristics (Table I).

A. ADR as RESTful-based service

REST stands for REpresentational State Transfer [START_REF] Fielding | Principled Design of the Modern Web Architecture[END_REF], which is an architectural style for networked hypermedia applications. It is primarily used to build Web services that are lightweight, maintainable, and scalable. A service based on REST is called a RESTful service. REST is not dependent on any protocol, but almost every RESTful service uses HTTP as its underlying application protocol.

More than a decade after its introduction, REST has become one of the most important technologies for Web applications. Its importance is likely to continue growing quickly as all technologies move towards an API orientation. Every major development language now includes frameworks for building RESTful Web services [START_REF] Vaqqas | RESTful Web Services: A Tutorial[END_REF]. As such, it is easy for developers and architects to build RESTful services.

In RESTful service every system uses resources. These resources can be pictures, video files, web pages, business information, or anything that can be represented in a computer-based system. In our context, the resource is each ADR involved in the user environment. The purpose of a service is to provide a window to its clients so that they can access these resources. Service architects and developers need this service to be easy to implement, maintainable, extensible, and scalable. RESTful design promises these characteristics and even more. In general, RESTful services have many properties and features described in [START_REF] Vaqqas | RESTful Web Services: A Tutorial[END_REF] and [START_REF] Varanasi | Introduction to REST[END_REF]. Within these properties and features our proposal uses:

1) Representation: The focus of a RESTful service is on resources and how to provide access to these resources. A resource can easily be thought of as an object as in OOP (Object-Oriented Programming). It consists in identifying the resources and determining how they are related to each other. As illustrated in Fig. 2 ADR is the representation of resource model in the platform. The new component added to this model is the service class, which describes the RESTful API service delivered by each digital resource. After identifying the resource, the next step is to define a way to represent these resources in our system. To do so, we make use of JSONLD (JavaScript Object Notation for Linked Data) [START_REF]JSON-LD -JSON for Linking Data[END_REF] data format. JSON-LD is a subset of the key-value based JSON document format that provides a way of describing machine-readable information with semantic context. JSON is used as a base data format for search engines such as Elastic Search [START_REF] Elastic | Elastic • Revealing Insights from Data (Formerly Elasticsearch)[END_REF] and NoSQL data store which is used by our platform architecture. JSON-LD is also designed to lower the barrier for data publishers who wish to provide "Linked Data" so that concepts and entities can be identified with certainty [START_REF] Katz | Implementing Transitive Credit with JSON-LD[END_REF].

2) Addressing resource: REST requires each resource to have at least one URI. Each resource is identified by its id. According to JSONLD format the id is in URL format. This URL has following format: "@id": http://prise.univ-lr.fr/resource/1821. RESTful systems may have a uniform interface.

3) Stateless and caching: A RESTful service is stateless and does not maintain the application states for any client. A request cannot be dependent on a past request and a service treats each request independently. We use Trace-based system [START_REF] Settouti | A trace-based system for technology-enhanced learning systems personalisation[END_REF] to implement stateful services in our architecture, since we need a mechanism to maintain the resource context of use or to maintain user service contexts. Caching is the principle of storing the generated results and using the results stored instead of generating them repeatedly if the same request arrives later. This can be done on the client side or server side and it is a way of enhancing the service performance. The trace-based system is used for that purpose.

4) Link between resources: A resource representation can contain links to other resources. In a situation execution, users sometimes request for additional information (or other similar resources) or some resources require complementary information. In [START_REF] Sawadogo | Connected Digital Resource Management in Personal Learning Environment[END_REF] we propose a method to connect digital resource based on similarity algorithm and software configuration approach.

5) Documenting: RESTful services do not necessarily require a document to help clients discovering them. Thanks to URIs, links, and a uniform interface, it is extremely simple to discover RESTful services at runtime. But documenting an API for customers can foster API's adoption [START_REF] Varanasi | Documenting REST Services[END_REF]. We used swagger metadata to create an interactive REST API documentation that enables documentation to be synchronized with any changes made in our RESTful services. The documentation generally contains information about the API such as API version, title, description, license, etc.

Finally, The advantage of web service approach is to provide an environment that allows each service to be independent of the technology and gives methods to encapsulate and to adapt digital resource usability in user's environment.

It enhances interoperability and discoverability. It gives to the user a capacity to: -Identify: what is the resource? And what is its configuration? -Control: how to configure the resource to be used in the situation context? -Adaptation: in the executive environment, how to adapt the resource to the user's context? -Interaction: what interaction feature may be used to interact with another resource? -Maintenance: how to maintain the resource state or save the last used configuration for reusability? The Fig. 3(a) presents the main components of an API-based service in the system. Each API may contain a description metadata (for example using swagger metadata), a set of methods to be requested, a set of arguments exploited to adapt the results needed and a specific behavior when the service is called.

B. ADR integration as platform thinking MVP

Platform thinking is one of the most innovative approaches that help developer maximize digital ecosystem design. "A platform is a plug-and-play business model that allows multiple participants (producers and consumers) to connect to it, interact with each other and create and exchange value." [START_REF] Stagars | Platform Thinking for Startup Success[END_REF] The goal of a platform is to enable interactions between producers and consumers.

Interaction is a fundamental unit of analysis on a platform. Some platforms may have multiple participants connecting to them and many different types of interactions.

MVP [START_REF] Stagars | The Lean Startup Changed Everything[END_REF] is the core interaction of Resource-based platform. ADR producers and consumers perform repeatedly set of actions to gain value out of the platform. These actions are illustrated in the Fig. 3(b). In this figure, ADR, stakeholders create, use, adapt and collaborate as a set of actions:

-Create: In every interaction, there is, at least, one producer (as a role in the platform) who creates the resource; -Use: In every interaction, there is, at least, one consumer, who consumes the resource; -Adapt: To encourage usability the resource can be refined, re-used and personalized by the consumer; -Collaborate: To encourage good quality and quantity of resource creation, as well as to ensure that the right consumers are using the right resources. In collaboration actions, the consumers can share, comment, evaluate or review resources. Indeed, our proposed adaptive resource model is assumed to be reusable in any situation context [START_REF] Sawadogo | Adaptive Digital Resource Modelling for Interactive System[END_REF]. In order to integrate multiple resources into a coherent learning environment [START_REF] Kong | A study of building a resource-based learning environment with the inquiry learning approach: Knowledge of family trees[END_REF], it is necessary to adapt each resource to allow interaction within them. In [START_REF] Sawadogo | Connected Digital Resource Management in Personal Learning Environment[END_REF] a method has been proposed to connect digital resource. To extend this proposal to the integration of these resources into the user platform, our method consists in: 1) Creating and configuring the resources, which are suitable to be used by a consumer in a situation. Each resource should meet the RESTful requirements for interoperability issues; 2) Enabling collaboration between users and the resources; 3) Updating and maintaining each resource in the system for reusability and personalization feature. This third step consists in maintaining the resource life cycle. Each resource is reconfigured and updated to take into account the usage experience. Our method allows the integration of multiple digital resources in a situation-based scenario or in a situation. Interoperable, adaptive and collaborative digital resource helps users efficiently achieve the situation objectives. In our model, the configuration process consists in activating or deactivating the right "operator" or handling "rules" or "services" to make the interaction useful for the consumer request. The general architecture of our proposition is illustrated in Fig. 4. In the platform architecture, we adopted RESTful API micro-service topology [35] to allow the platform model to perform RESTful API-based communication with the internal components of the platform and also with third-party applications.

V. EXPERIMENTATION AND RESULTS

We have experimented our method in a platform called PRISE 1 for PeRsonal Interactive research Smart Environment that allows researchers to manage their digital resources. In PRISE, each digital resource is represented as a reading report. A reading report is an object that is created by users of PRISE to describe and save their digital resource in the platform. It contains the resource bibliography, the user summary and review and also the community comments. PRISE platform offers:

-Digital resource management with standard operations of reading report such as create, read, update, delete (CRUD); -Social networks that include our model of researcher's profile (IMS-LIP extension [START_REF] Berlanga | IMS LD reusable elements for adaptive learning designs[END_REF]); -Situation management that allows a user to perform an activity. The resource model and the platform model allow resources to be retrieved and integrated into the situation execution:

• To enable collaboration among the users around the resources; • To automatically adapt resource presentation with user's profile. For example, if the user prefers French and the resource supports both French and English versions, the system will deliver the French version to the user; • To automatically activate or deactivate the resource options according to arguments of the web service. In this experimentation, we built digital resource web services with DunglasApiBundle [START_REF] Dunglas | API Bundle -API Platform[END_REF]. DunglasApiBundle is an easy to use and powerful system to create hypermedia-driven REST APIs [START_REF] Wikipedia | HATEOAS -Wikipedia, the free encyclopedia[END_REF]. It is a component of the API platform framework and it can be used as a standalone bundle for the Symfony framework. It embraces JSON for Linked Data (JSON-LD) and Hydra Core vocabulary web standards [START_REF] Lanthaler | Creating 3rd Generation Web APIs with Hydra[END_REF].

In a reading report the user can perform one of the following requests: -GET readingReport: returns all information about the mentioned reading report readingReport. Consumers can also define what information they are seeking for, it could be the list of comments, the review information, the summary information or the bibliographic 1 http://prise.univ-lr.fr information; -POST readingReport: allows producers to post a readingReport or to update existing reading report information; -DELETE readingReport: allows producers to delete readingReport; -PUT readingReport: allows consumers or producers to update readingReport information. We have evaluated the platform with 12 participants and 278 reading reports. The participants spent about four weeks using the platform to perform their daily activities such as summarizing their readings, writing articles, searching for relevant reading reports, etc. After that, we asked participants to evaluate the platform through an online form (survey). The results of the survey are illustrated in the graphics of Fig. 5 and Fig. 6(a,b,c). They show that 75% of the participants agreed that the resource representation in PRISE is useful for them, and 83.33% of them also agreed that the PRISE resource model allows collaboration between resources and users (producers and consumers), but only 41.67% agreed that it connects consumers and producers.

Another evaluation carried out concerns participants' feedback about using PRISE digital resource model. 67% are interested in the function that enables third party service for the digital resource such as connecting the resource with easychair service (to allow the resource to be automatically sent to easychair or HAL or shareLatex with their grants). 33% agreed to enable calling external services in the resource such as translator services to adapt the digital resource language to translate the resource in their native language.

B. Collaboration enhancement

Digital resources represent a good opportunity to enhance collaboration among users sharing same interests. In the PRISE experiment, 83% of participants of the platform agreed that digital resources allow users to collaborate (Fig. 6(a)). The collaboration is implemented through resource evaluation, sharing and comment. We think that collaboration is the main function since many participants expressed a high interest in it in the PRISE digital resource model, which enhances resource quality.

C. Adaptation and reusability enhancement

The resource design model we proposed enables adaptation mechanism for the user. RESTful is stateless which means that each digital resource can be contextualized according to the user characteristics and reused in any situation. According to feedback received from the participants. 75% are interested in filtering the content of each resource according to their interest and expertise. 58,33% are interesting in getting additional or complementary detailed information associated to the resource. All of the participants agreed with the adaptation characteristic (fig. 6(c)). In RBLE, PLE, LMS, MOOC, etc. the main component, which interests users, is the resource management and use. For example, MOOC systems are mainly short video-based resource that involves users in a learning activity to get knowledge through these resources.

To our knowledge, few works have been carried out in the field. In [START_REF] Hill | Teaching and learning in digital environments: The resurgence of resource-based learning[END_REF], the authors argued that the process used to integrate multiple resources into a coherent learning environment has not been well established. Learners and designers are accustomed to using resources for specific purposes. Users are not generally accustomed to reuse resources developed for one purpose for another [START_REF] Doiron | Partners in Learning: Students, Teachers, and the School Library[END_REF]. Research is needed to demonstrate how resources can be linked to meet various needs. Strategies to assist learners in analyzing and interpreting resources in terms of their validity and reliability are also needed [START_REF] Fitzgerald | The cognitive process of information evaluation: A collective case study[END_REF]. Our objective was to well define this process to integrate multiple resources into a coherent learning environment based on the digital resource retrieval and interoperability enhancement.

Learning Object Metadata initiative published in 2002 [START_REF] Ims | IMS Meta-data Best Practice Guide for IEEE 1484.12.1-2002 Standard for Learning Object Metadata[END_REF] is one of the widest adopted standards for educational resource description. IEEE-LOM gives the opportunity for education resource producers to describe their resources. But for their retrieval, we need to build search engines such as MERLOT or GLOBE. The main issue of these systems lies in the lack of web service that allows consumers to integrate the resources retrieved into the user personal environment. Therefore, IEEE-LOM repositories do not really use the potential of web service capacities and do not express the value of education resources in terms of interoperability and collaboration. They only propose means to search resources based on keywords.

In [START_REF] Adl | SCORM 2004 4th Edition Run-Time Environment {RTE}, SCORM 2004. Advanced Distributed Learning[END_REF], the SCORM Run-Time Environment (RTE) represents an opportunity for education resource designers and developers to launch Learning Object in SCORM compliant system. This model enhances learning object interoperability in different LMS. But, the SCORM model is difficult to adopt or to implement and use [START_REF] Costagliola | Scorm run-time environment as a service[END_REF]. The authors in [START_REF] Costagliola | Scorm run-time environment as a service[END_REF], proposed to externalize the SCORM RTE functionalities and to offer them as service according to Service Oriented Architecture (SOA) model. Despite this proposal, SCORM model is limited to LMS and faces many difficulties to be adopted by developers.

Our proposal to use RESTful approach, which is largely adopted by developers, can help to solve digital resource compliance with LMS.

As well-known, LTI [START_REF]IMS Learning Tools Interoperability[END_REF][44] and xAPI [45][46] use API model to respectively connect learning tools and tracking learning activity data. These interesting works prove the interest of web service API in enhancing interoperability between platforms, but they do not focus on digital resource interoperability. Thus, our approach mainly aims to enhance resource usability in terms of interoperability (retrieving and integration), collaboration and adaptation and in much more to create a system as a platform where digital resources produce value for both producers and consumers. This approach can enhance resource usability, resources repositories adoption by their users and resource integration in PLE and RBLE.

VII. CONCLUSION

The exponential growth of digital learning resource production raises challenges concerning their usability and interoperability. This research work aims to prove that well digital resource modeling can optimize and enhance their integration and use in users' platforms.

We found that modeling digital resource with RESTful-based API methodology and platform thinking approach actually enhance digital resource usability capability in terms of retrieval, interoperability, adaptation and collaboration. Thus, in the context of learning environment, the contribution value enhances PLE and RBLE capability to integrate multiple digital resources into the user platform. We plan to extend digital resource adaptation mechanisms to enhance resource usability in terms of digital resources personalization.

Fig. 1 .

 1 Fig. 1. Digital resource management architecture

Fig. 2 .

 2 Fig. 2. Adaptive Digital Resource diagram class

Fig. 3 .

 3 Fig. 3. Adaptive Digital Resource API architecture and features

Fig. 4 .

 4 Fig. 4. PRISE API REST-based micro-services architecture

Fig. 5 .

 5 Fig. 5. Digital resource feature modeling evaluation in PRISE

Fig. 7 .

 7 Fig. 7. Interoperability evaluation

  Adaptive Digital Resource Modeling as Service Provider and Consumer

	Daouda Sawadogo, Ronan Champagnat, and Pascal Estraillier

TABLE I :

 I COMPARISON OF DIGITAL LEARNING RESOURCE MODEL CHARACTERISTICS

	Project	Resource model	Discovery	Interoperability	Adaptation	Collaboration	Execution Environment
	IEEE-LOM	Accessibility-based	+	-	-	-	LMS, LO Repository
	ADL-SCORM	Interoperability-based	-	+	+	-	LMS
	OER	Reusable-based	-	+	+	-	LMS OER repository
	EDUPUB	eBook resource-based	-	+	-	-	Reading system
	Our Proposal	Service-oriented based	+	+	+	+	Platform-based environments

LOM: Learning Object Metadata; SCORM: Sharable Content Object Reference Model; OER: Open Educational Resources; EDUPUB:EDUcation PUBlishing; LMS: Learning Management System; LO: Learning Object; DR:Digital Resource

Ronan Champagnat obtained his Ph.D. in computer science at LAAS-CNRS laboratory in Toulouse (France) in the dependability domain. Since 1998, he is an Associate Professor at University of La Rochelle within the L3i Laboratory. His current work deals with Interactive Adaptive Systems where the system adapts its execution to users interactions and behaviors. He explores the application of system storytelling and scenario structuration technics in order to improve application scenario structuring and adaptation in interactive systems.

Pascal Estraillier is a full professor at the Computer Sciences department and the research Laboratory L3i "Informatics, Image, Interactions" at the University of La Rochelle. He is also a scientific adviser in the directorate "National Management of Research and Innovation" at the French Ministry of Research, in charge of ICT area. His researches concern the architecture of software components in distributed and cooperative systems. He applies its results on Multi-agent Paradigm and uses formal specification theories in order to validate the behavior and the interactions between components and to manage interoperability constraints. The results are mainly applied to the games, serious game and e-learning domains in many research projects funded by the E.U, the French A.N.R, the regional council, and several companies (France Telecom, XD-Production,…).


B Mantisi

Generation of polycrystalline material at the atomic scale

Polycrystalline structure plays an important role in the macroscopic properties of a solid material. In this paper we propose a new code to generate a polycrystalline material at the atomic scale. Our polycrystalline systems are based on the random generation from initial germs. From a mechanical point of view, it brings new features as for example ductility of 3D polycrystalline aluminum. By using molecular dynamics simulations, we also show the eect of temperature on a geological material, olivine, one of the most abundant silicate mineral of the Earth upper mantle. The importance of porous media is also considered (e.g. metals).

Introduction

The importance of atomic dislocations on the mechanical behaviour of solid materials is well known and the study of polycrystalline materials is important in that respect. Polycrystalline materials are common in nature. For example, in metallurgy, grain boundaries will generate new macroscopic behaviours compare with a mono-crystalline phase. In geophysics, the polycrystallinity of rocks is a rule. Moreover the Earth mantle being submitted to extreme conditions of temperature and pressure, the inuence of the grain boundaries on mechanical and transport properties of rocky mantle is then important to know. A fruitful way to approach the pressure and temperature dependence of these materials is to use molecular dynamics (MD) simulations.

MD simulations are widely used in many elds as mechanics, physics, geology and biology. The main ingredient is the force eld (FF), which governs the interaction between particles. The validity of the simulation results depends directly on the accuracy of the FF. Furthermore, for polycrystalline materials the initial conguration of the atoms in the simulation cell has to be as realistic as possible. In general, molecular dynamics codes of the literature give the possibility to build only mono-crystalline structures [START_REF] Plimpton | Fast Parallel Algorithms for Short-Range Molecular Dynamics[END_REF]. Some other codes were developed to build more complex crystalline structures made of proteins [START_REF] Martinez | Packmol: A package for building initial congurations for molecular dynamics simulations[END_REF] or polymers [START_REF] Mainge | Dendrimer building toolkit: Model building and characterization of various dendrimer architecture[END_REF]. But, concerning polycrystalline material, users face up to a dicult task. The challenge is to built a polycrystalline structure including a part of randomness. An obvious way would be to proceed as in a real experiment: i.e. cooling a liquid slow enough to let appear and grow mono crystalline germs. This method works well for metals simulated by MD [START_REF] Shibuta | A molecular dynamics study of cooling rate during solidication of metal nanoparticles[END_REF] for which single symmetries (fcc or bcc) facilitate the growth from a small nucleus. However, the success of the method is potential dependent and does not work necessarily for every kind of materials (e.g. silicates). Another method consists in building crystalline cells with a Voronoi tessellation [5] [6], and assigns to each crystalline area a random orientation. The main advantages of this method is that it is independent of the FF, it is scalable to dierent sizes and applicable to dierent kind of simulations (MD and nite element methods for example). However, all grains are separated by planes dened a priori, whereas some roughness is expected at grain boundaries. Moreover, no porosity can be introduced in that way to model polycrystalline structures. Moreover, atomistic simulation of polycrystalline materials has shown already nice results [START_REF] Schiøtz | Atomic-scale modeling of plastic deformation of nanocrystalline copper[END_REF] and therefore a code especially devoted to build such initial structure would be a good starting point. In this context, we propose to build a polycrystalline material from the growth of mono-crystals which is stopped when such crystals join each other within the simulation box. The code produces at the atomic scale a random conguration representative of a polycrystalline system. Here, randomness means a random distribution of initial germs with random orientation and random growth. Generated polycrystals are an as-sembly of several perfect monocrystals, i.e. defects are not introduced inside a grain. In section II we present the basic ideas of the method and how the algorithm works. In section III some details of the code are given. In section IV a rst example presents the eect of polycrystallinity on the plasticity of aluminum. The second example deals with the partial melting of olivine, a silicate mineral of the Earth upper mantle. We conclude and give some perspectives in section V.

Description of the code

The code uses a geometrical criterion based on the unit cell structure (UCS) of the material under investigation. Figure 1 sketches the process discussed below. The algorithm follows a growth process, which implies to consider two kinds of chemical entities (An entity is the unit cell of the considered crystal or a subunit of this unit cell). Chemical entities that do exist (RE for real entities) and chemical entities that can potentially exist (VE for virtual entities) as rst neighbours of REs. The initial state is a distribution of REs (germs) to which we assign a random location and a random orientation in the simulation box. These germs are the rst RE in the box. A virtual grid mapped on the crystal symmetry is associated with each germ. The second step is the crystal growth of these initial germs from their associated grid. For that, VE are dened as nearest neighbour of RE on their grid, and then algorithm pick up randomly a VE and declares it as RE. The new RE implies rst to delete VEs (i.e VEs from other germs) closer than the minimum distance between entities dened from the UCS and second to declare new VE following the grid of the last dened RE. The process is stopped when the grids are lled taking into account that entities of dierent grids can not overlap each others. An important task of the code is to deal with the congurations where entities belonging to dierent grains become close to each other. The growth process is very ecient since all random pick up are accepted, because VE have been dened such that they do not overlap with any other RE. To avoid boundary eects during the simulation, the code considers periodic boundary conditions when writing RE and VE. Some accelerating features, discussed in the next part, have been implemented, giving then access to large system sizes in an acceptable generating time. The main accelerating feature is a subdivision of Instead of computing all distances and checking a possible overlap with all atoms in the box, the computation deals with a small number of surrounded boxes (more precisely dened in the following). We plot in gure 2 the time it takes to generate system sizes reported on x-axis for 3-dimension and 2-dimension systems respectively. We note that for a 2 dimensional system, it takes less than one minute on a Intel Xeon E5-2650 @ 2.6GHz with gfortran compiler and just one core, to generate a conguration composed of 200 000 atoms. Details on the code Grain growth. Figure 3 presents the two dimensional growth of 20 mono atomic germs with a cfc geometry in a 200x200 box units (1 box unit = atomic diameter). The separation between grain is imposed to be 2 box units. Since the algorithm is not a real growth process because temperature and then diusion of species are not considered. Concerning the grains statistics evolution, the bigger the grains, the higher the number of VE around have to be considered, i.e more chance they have to be larger than others. By contrast, some mono-crystals are trapped, because they are closely surrounded by other mono-crystals precluding to them any further growth. The density of germs and the space between them when the simulation starts are thus very important.

Code optimisation. The code has been written to generate many dierent congurations of several hundred of thousand of atoms. Any loop on all atoms at each building step would be a lost of time. We reduce the computational time using sub boxes paving the simulation cell with sizes of a few angstroms. Then all atoms or entities are labelled in a sub box and the check for non-overlapping is done using nearest neighbour boxes. This is the main feature which renders the code so ecient (A two dimensional box containing around 200 000 atoms is generated in less than one minute). The size of small boxes is related to the nearest neighbour distances within the unit cell of the crystal under investigation. This method applies to any crystal structure.

Adjustable parameters. The user can easily change the dimension of the box (2 or 3 dimensions), the size of the box, the number of grains, and the space between grains. The space between grains is equal or greater than the typical distance between two atomic species of the crystal unit cell. Whatever is the space imposed between the grains (porosity), the nal conguration needs to be minimized in energy with an adequate FF. The code does not provide a minimization method, since existing molecular dynamics codes do it already pretty well. In the forthcoming, we will use LAMMPS for minimizing boxgenerated structures. During the minimization process, the free surfaces minimize their area, porosity is restructured (Figure 5). As we will see later, porosity at the nano-scale may bring new features in the behaviour of materials under extreme conditions.

Crystal orientation.

The germ orientation is chosen at random when the simulation starts. In 3D, for each germ we associate a unit vector built from two random points in a box, which is a rotation axis to which we associate a random rotation angle. In 2D, z-axis perpendicular to the system plane is the rotation axis, the rotation angle is chosen randomly. The user can specify the x-y axis correspondence with the lattice parameters, and then the corresponding lattice rotation axis . For example, for a, b and c lattice vector, one can specify that a is along the z axis, and b, c along x,y axis respectively. A cyclic permutation is done here. Note that in 2d systems, the lattice vector along z will be the thickness of the sample. L x ,L y and L z box edges are specied, then the code will take as L z the integer times the lattice vector closer than the requested values(For instance, if | c| along z is 4Å, and the requested value is L z =19Å, then the nal value will be

L z = 5 • | c| = 20Å).
Periodic boundary conditions. We choose to build the system with periodic boundary conditions (PBC). If one atom goes out from the box, then it position is shifted by the box size in the opposite direction. Thus, the system can be seen as an innite continuum of grains periodically replicated. However, because of PBC, a grain may growth, cross the boundary and met itself. Since the initial orientation of the grain is random and not correlated with the size of the box, the probability to have a perfect match when both parts met each others is close to zero. To avoid this it is recommended to start at least with 6-8 grains for 2D simulations and 10-12 grains for 3D simulations. We split the unit cell in four parts, each containing the correct stoichiometry of Mg 2 -SiO 4 . To do this and keep local geometries entities (SiO 4 tetrahedra) intact, we move oxygen atoms out of the box considering others unit cells around (arrows on gure 4a). The scheme of the code is in considering the center of mass of the dened sub unit cell structure to i) ll the box, and to ii) verify that there is no atomic overlap between the selected VE and surrounding REs. For mono-atomic structures such as fcc aluminum for example, there is no need to care of this. This code was primarily designed to study multi component materials constitutive of the Earth mantle (e.g. silicate minerals). But, it can also help for the study of metallic alloys that are mechanically loaded, therefore serving for the study of mixed alloys and their diusion of atomic species or as a starting point for kinetic Monte-Carlo simulations devoted to grain growth evolution.

Examples method of calculation

For molecular dynamics simulations and energy minimization we used the LAMMPS code [START_REF] Plimpton | Fast Parallel Algorithms for Short-Range Molecular Dynamics[END_REF]. The centrosymmetric coecient (Cs) is used to characterize the local order and highlights grain boundaries. This coecient is computed for each atoms i over its N rst neighbours j as follows:

Cs i = | N j ( r i -r j ) | (1)
and is equal to zero if the atoms around are perfectly ordered. The number of neighbours N is dependent of the crystal geometry we want to characterize (e.g. for fcc N = 12). The mechanical behaviour is studied by stress-strain curves: the strain is imposed and the stress is computed from the Irving-Kirkwood formula [START_REF] Irving | The statistical mechanical theory of transport processes. IV. The equation of hydrodynamics[END_REF]:

σ = 1 V N i N j =i m i v i • v i + r ij • F ij (2)
where i and j are labelled particles, V the volume of the sample, N the total number of particles, v i the particule's i velocity, r ij the distance vector between particle i and j, and F ij the force between particles i and j deriving from the associated interaction potential. Note, that this tensor is dened as symmetric since in the rst part (kinetic) the velocities can commute, and concerning the second part the two vectors are collinear. This feature is in fact resulting from the action-reaction law and then applicable to multi-body interactions [START_REF] Admal | A unied interpretation of stress in molecular systems[END_REF].

Results

We rst present the example of an aluminum sample under mechanical load, and next the one of a polycrystalline olivine one submitted to high temperature and pressure. Because the FF used for aluminum is at very short range, our computational resources allow us to reach much bigger sample sizes than for silicate minerals where Coulombic interaction are present. Stress-strain curve: from fragility to ductility.

We build a 3 dimensional aluminum polycrystal in a cubic box (19.3x19.3x19.3nm) containing 403 076 atoms. Figure 5 shows the simulation box after energy Figure 5: [Color Online] The box contains 403 076 aluminum atoms and sizes 19.3nm edge. 80 grains are generated and the space between grains is non zero. After energy minimization we clearly obtain voids which are found between grains. The color code represent the centrosymmetric parameter. This picture has been generated using AtomEye [10].

minimization. The FF used here is computed from the Embedded Atom Method (EAM) [START_REF] Daw | Semiempirical, quantum mechanical calculation of hydrogen embrittlement in metals[END_REF] [START_REF] Daw | Embedded-atom method: Derivation and application to impurities, surfaces and other defects in metal[END_REF] for Aluminum and tabulated by Mishin & al [START_REF] Mishin | Interatomic potentials for monoatomic metals from experimental data and ab initio calculations[END_REF]. The values are interpolated using spline function. Grain boundaries and free surfaces are characterised by Cs i = 0. The deformation was performed using minimization algorithm to avoid any shock wave due to the high strain rate (which is usual in classical MD due to the time scale); it gives access to the mechanical energy and does not take into account any thermal activation. Then the system is extended along the x direction in keeping constant the y,z section of the box. The strain is applied by imposing a small displacement to atoms with a deformation step of = ∆x/L 0 = 10 -4 . It is followed by a minimization of the local energy using a conjugated gradient method.

We compare the respective behaviour of a monocrystalline aluminum (lattice vector aligned along the box edge), a non-porous polycrystalline aluminum and a 1 % porous polycrystalline aluminum. The stress-strain curve is represented in gure 6 by plotting the calculated Von Mises equivalent stress as function of the applied strain. The mono-crystalline sample is fragile but can sustain a high stress before breaking, here at = 0.19. Concerning the polycrystalline system, one clearly notices a change from a fragile behaviour to a ductile one, and, indeed, the material ows under strain leading a much lower value of the yield stress. It is also well known that porosity plays a crucial role in the mechanical behaviour. Void growth is a topic largely developed in analytical mechanics (e.g. Gurson model [START_REF] Gurson | Continuum theory of ductile rupture by void nucleation and growth: Part I, yield criteria and ow rules for porous ductile media[END_REF]), and has recently been observed from molecular dynamics studies [START_REF] Potirniche | A molecular dynamics study of void growth and coalescence in single crystal nickel[END_REF] [16] [17] [START_REF] Seppala | Onset of Void Coalescence during Dynamic Fracture of Ductile Metals[END_REF]. Voids bring a new degree of freedom to redistribute energy. In fact, pores imply free surfaces which can store energy. In absence of free surfaces, the energy accumulates in the bulk and is released suddenly, creating cracks at the nanoscale. Figure 6 conrms this general tendency because the stress-strain curve of the porous material presents a larger ductility than the non-porous samples. Using the polycrystalline generation code, we thus clearly show the importance of the granular aspect of matter at the nanoscale on the macroscopic mechanical behaviour.

Example 2: Shear strain and grain boundary correlation. We build a pseudo 2 dimensional aluminum sample with sizes 79.7 × 79.7 × 2.0 nm devoted to a shear strain along the x-axis. The result of the grain growth code is presented on gure 7 with a color code associated to the local stress (xx component of the stress tensor) for dierent shear strains. This color code clearly shows the grain boundaries. Figure 7a presents a conguration which has been minimized using a conjugate gradient method. We use, as previously the EAM FF for describing the interactions between Al atoms. Following gures 7b,c,d and e represent the system for dierent shear strains. Several responses to the strain are observed here. First, defect appear and move but generally collapse in a grain boundary. Second, some cells tilt out from their original xy plane and this is one elementary mechanism for releasing stress. Third, grains on the upper side of the sample divide into smaller grains as long as the deformation occurs. This feature is quite unexpected and results from the displacement of aligned oscillating stress values similar to a dislocation. The orientation of crystals is not initially dened here, and one can observe the dislocation propagation and their interaction with other defects or even grain boundaries in the most general case. Moreover, when the sample is followed in time, one can note several dynamical features. Grain boundaries move, i.e. atoms belonging to a crystal and close to another grain can switch their orientation to be align along the neighbouring grain. We also clearly notice dislocation-like motion inside a grain. This example can be studied in more details by computing the associated Burger vector [20], characterizing grains size evolution during deformation or defect displacements.

Example 3:

Silicate minerals of Earth's mantle : Polycrystalline olivine. Here we investigate the temperature eect on grain boundaries and diusion located in channels between grains generated by the code. Channels in polycrystalline rocks are of considerable interest for the geophysical community, as they tend to dominate the electrical conductivity [START_REF] Yoshino | Electrical conductivity of mantle minerals: role of water in conductivity anomalies[END_REF], and can be seen as an additional degrees of freedom.

Olivine is one of the most abundant constituent of the Earth's upper mantle. It is a silicate, whose unit cell composition is M g 2x F e 2(1-x) SiO 4 . The presence of Mg and Fe cations leads to a solidus and a liquidus boundary in the phase diagram, where solid and liquid co-exist. Evaluation of the electrical conductivity has been done experimentally [START_REF] Pommier | Experimental investigation of the electrical behavior of olivine during partial melting under pressure and application to the lunar mantle[END_REF] [START_REF] Baba | Mantle dynamics beneath the East Pacic Rise at 17 • : Insights from the Mantle Electromagnetic and Tomography (MELT) experiment[END_REF] and unexpectedly the electrical conductivity, coming from the ionic transport, is high for a solid below the solidus. Molecular simulations can clearly bring a new insight in this eld, in particular if the grain distribution can be fairly reproduced. This can be achieved with our code.

We build a pseudo 2 dimensional polycrystalline olivine in a cubic box of 59x59 nm edge, 2.0 nm thick and containing around 679 000 atoms (Figure 4b). We use the accurate FF of Duls et al. (2016) for describing the interaction in silicates. Once the polycrystal is generated, the potential energy is minimized using a conjugate gradient method. Next, the sample is equilibrated by MD at 300K during 0.5 ns, 800K during 0.5 ns, 1500K during 3.0 ns and nally at 1700K during 3.0 ns. The pressure is xed at 1GPa. We compute the total displacement of the atoms at 1700K during 3ns. The histogram of displacement is presented in gure 8a. Two types of displacement can be identify, one corresponding to the atomic motion within grains, and the other one associated with the melt (here melt is used for grain boundaries). The dichotomy is clearly visible in Figure 8a with a threshold at 1.7 Å, a distance which is similar to the Si-O bond length (1.6 Å). Figure 8b represents the simulation cell and highlights atoms which displace more than the threshold distance (1.7 Å). One notices that Atomic displacement are generally very small except at grain boundaries where a true diusion can take place along grains. This nding supports phenomenological models attempting to estimate the electrical conductivity of polycrystals from basic geometries (planes, tubes, spheres...). But this is not in the scope of the present study.

Conclusion

In this article we have presented a code aimed to generate polycrystalline materials of various chemical composition and unit cell structure. This code generates the growth in 2 or 3D of a polycrystal from an initial distribution of randomly oriented crystalline germs. Moreover, it allows to control either space between grains or the porosity of the system (i.e. unit defaults or bigger voids). Two model systems are investigated for illustration, a sample of polycrystalline aluminum sheared and extended and a silicate mineral (olivine) at high temperature. In most of the MD simulation studies of the literature only mono-crystals are considered although it is well known that polycrystallinity and grain boundaries deeply aect the mechanical and thermodynamic properties [7] [21]. An accurate description of solid materials accounting for polycrystallinity seems to be crucial to link microscopic behaviour to macroscopic ones. We are condent that the code presented here may be fruitful to build complex material at the nanoscale.

Figure 1 :

 1 Figure 1: [Color Online] a. One VE is selected, b. and declared as RE. New VE are then dened. c. Another VE is selected and declared as RE. If it overlap with existing VE, the latter are suppressed. If New VE overlap with existing RE they are not created.

Figure 2 :

 2 Figure 2: [Color Online] Time eciency to generate polycrystalline systems. a. is for 3 dimensional systems and b.is for 2 dimensions. Labels on pictures refer to the number of atoms created. All computations were made on a Intel Xeon E5-2650 @ 2.6GHz core with gfortran compiler.

  Stoichiometry conservation. In order to keep the stoichiometry of the total sample constant, one needs to rst characterize entities in the Unit Cell and be aware of possible atomistic local structures (e.g. SiO 4 tetrahedron). For example the chemical composition of Olivine is Mg 2 -SiO 4 , the unit cell contains 28 atoms (4 Si, 16 O and 8 Mg) (gure 4a).

Figure 4 :

 4 Figure 4: [Color Online] a. Olivine unit cell. Atoms grouped together with a dashed line are entities used to build crystal, which preserve neutrality and stoichiometry. Arrows point out atoms which are outside of the unit cell. b. Sample of Olivine, San Carlos type, at 1700K and 1GPa, which sizes 59 × 59 × 2 nm.

Figure 6 :

 6 Figure 6: [Color Online] The stress-strain curves are plotted for an extension deformation of alluminium samples: a crystal of 16 nm size (red), a polycrystalline with 80 grains of 20 nm (green), a polycrystalline with 80 grains and voids of 20 nm (blue).

Figure 8 :

 8 Figure 8: [Color Online] a. histogram of displacements of each atoms after 3ns at 1700K and 1GPa. b. Atoms which displaced more than 0.17nm. The region in white are grains.

Figure 3 :

 3 Figure 3: [Color Online] Growth of a model polycrystalline material in a plane. The unit cell is a cubic network.Colours are randomly aected for each grains. Separation between grains is an initial specication of at least 2 units here.

Figure 7 :

 7 Figure 7: [Color Online]Aluminum polycrystalline sample which sizes 79.7 × 79.7 × 2.0 nm. Each picture represents dierent shear loads. The color map is related to the xx stress tensor component for each atom. It emphasizes defaults, dislocations and grain boundaries. The two rsts pictures are for 0 and 0.05 strain. The middle picture is at 0.15 strain and labels on it point out defaults, dislocations and tilted network. Finally pictures on bottom are from left to right 0.2 and 0.25 strain.

Acknowledgements

The author gratefully acknowledge B. Guillot, N. Sator and M. Micoulaut for fruitful discussions and revision of the manuscript. This work was done with the nancial support of the ERC grant n • 279790.