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A STATIONARY PHASE TYPE ESTIMATE

T. ALAZARD, N. BURQ, AND C. ZUILY Abstract. The purpose of this note is to prove a stationary phase estimate well adapted to parameter dependent phases. In particular, no discussion is made on the positions (and behaviour) of critical points, no lower or upper bound on the gradient of the phase is assumed, and the dependence of the constants with respect to derivatives of the phase and symbols is explicit.

For a fixed phase, the stationnary phase lemma (and its simplified version, the stationary phase estimate) is a very well understood tool which provides very good estimates for oscillatory integrals of the type (1)

I φ,b (λ) = R d e iλΦ(ξ) b(ξ)dξ ⇒ |I Φ,b (λ)| ≤ Cλ -d 2
The method of proof is quite standard and follows the classical path:

(1) Using the non degeneracy of the hessian of the phase, one knows that the critical points are isolated, hence for a compactly supported symbol there are finitely many such critical points. (2) Away from the critical points, the non stationary estimates (obtained for example by integrating by parts N times with the operator

L = ∇ ξ Φ • ∇ ξ iλ|∇ ξ Φ| 2 ,
gives an estimate bounded by C N λ -N (3) Near each critical point, performing first a change of variables (the Morse Lemma) to reduce to the case where the phase is quadratic, and then and exact calculation in Fourier variables gives the estimate (1)

When d = 1, Van der Corput Lemma provides a very robust estimate.

However, in higher dimensions, the situation is less simple, in particular when considering parameter dependent phases (with parameters living in a non-compact domain), where

(1) even away from the critical points, ∇ ξ Φ can degenerate, (2) the determinant of the hessian can degenerate, (3) the number of critical points can blow-up.

In view of numerous applications (for example dispersion estimates for solutions to PDE's), a precise control of the behaviour, with respect to the phase and symbol, of the constant C in (1) is necessary. Many robust methods to prove (1) have been developped (see for example [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF][START_REF] Muscalu | Classical and Multilinear Harmonic Analysis[END_REF][START_REF] Luiz | Nikolay Tzvetkov Oscillatory integral estimates and global well-posedness for the 2D Boussinesq equation[END_REF][START_REF] Stein | Harmonic Analysis[END_REF]). However, it seems that none of these results gives an estimate directly applicable to general situations. This was the motivation for this note. Let

I(λ) = R d e iλΦ(ξ) b(ξ) dξ where Φ ∈ C ∞ (R d ) is a real phase, b ∈ C ∞ 0 (R d
) is a symbol. We shall set K = supp b and let V be a small open neighborhood of K. We shall assume that (2)

(i) M k := 2≤|α|≤k sup ξ∈V |D α ξ Φ(ξ)| < +∞, 2 ≤ k ≤ d + 2, (ii) N l := |α|≤l sup ξ∈K |D α ξ b(ξ)| < +∞, l ≤ d + 1, (iii) |det Hess Φ(ξ)| ≥ a 0 > 0, ∀ξ ∈ V,
where Hess Φ denotes the Hessian matrix of Φ.

Theorem 1. There exists a constant C such that, for all (Φ, b) satisfying assumptions [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF], and for all λ ≥ 1,

(3)

|I(λ)| ≤ C a 1+d 0 1 + M d 2 +d 2 d+2 N d+1 λ -d 2 .
Remarks 2. 1. We notice that no upper bound (nor lower bound) on ∇Φ is required. This is important in particular in the case were the phase Φ depends on parameters. For instance, in some cases the phase Φ is of the form Φ(x, y, ξ) = (xy) • ξ + φ(x, y, ξ) where x, y are in R d . In these case ∇Φ = xy + ∇φ and there is no natural upper nor lower bound for it.

2. Here is another example (see [START_REF] Alazard | Strichartz estimates and the Cauchy problem for the gravity water waves equations[END_REF]). Assume Φ(x, y, ξ) = (xy) • ξ + tθ(x, y, ξ) where t ∈ (0, T ) and x, y are in R d . Assume that Φ and b satisfiy (i), (ii) uniformly in (t, x, y) and that |det Hess θ| ≥ c > 0 where c depends only on the dimension d.

Then setting X = x t , Y = y t we write iλΦ = iλt (X -Y ) • ξ + θ(t, tX, tY, ξ) and we may apply Proposition 1 with a 0 = c and λ replaced by λt. We obtain an estimate of I(λ) by t -d 2 λ -d 2 as soon as t ≥ λ -1 . 3. The term a 1+d 0 in (3) can be written as a 0 a d 0 where the factor a d 0 comes from the possible occurence of a -d 0 critical points of the phase on the support of b. In the case where Φ has only one non degenerate critical point this term could be avoided. In this direction we have the following result.

Theorem 3. Assume that Φ and b satisfy the assumptions (2) and that the map

(4) V → R d , ξ → ∇Φ(ξ), is injective.
Then one can find C > 0 depending only on the dimension d such that

(5) |I(λ)| ≤ C a 0 1 + M d 2 d+2 N d+1 λ -d 2 .
Here are two examples where Theorem 3 appplies.

Examples 4. 1. Assume besides (2) that (6) Hess Φ(ξ)X, X > 0, ∀ξ ∈ V, ∀X ∈ R d .
then ( 4) is satisfied.

For simplicity we shall assume that the neighborhood V of supp b appearing in (2) (i) is convex. First of all, since the symmetric matrix Hess Φ is a non negative, its eigenvalues are non negative. It follows from the hypothesis (2) (iii) (see (11)) that

(7) Hess Φ(ξ)X, X ≥ a 0 (C d M 2 ) d-1 |X| 2 , ∀ξ ∈ V, ∀X ∈ R d . With ξ, η ∈ V we write ∇Φ(ξ) -∇Φ(η) = 1 0 d ds ∇Φ(sξ + (1 -s)η) ds, = 1 0 Hess Φ(sξ + (1 -s)η) • (ξ -η) ds.
It follows from (7) that

∇Φ(ξ) -∇Φ(η), ξ -η ≥ a 0 (C d M 2 ) d-1 |ξ -η| 2 from which we deduce that a 0 (C d M 2 ) d-1 |ξ -η| ≤ |∇Φ(ξ) -∇Φ(η)| which completes the proof.
2. Let A be a real, symmetric, non singular d × d matrix and Ψ be a smooth phase such that M d+2 (Ψ) < +∞. Set Φ(ξ) = 1 2 Aξ, ξ + εΨ(ξ). Then if ε is small enough the assumptions in Theorem 3 are satisfied.

Remark 5. Notice that the estimates (3), [START_REF] Muscalu | Classical and Multilinear Harmonic Analysis[END_REF] do not seem to be optimal with respect to the power of a 0 since according to the usual stationnary phase method one could expect to have a Actually it is sufficient to prove the following weaker inequality. Theorem 6. 1. Under the hypotheses of Theorem 1 there exists F : R + → R + non decreasing such that for every λ ≥ 1

(8) |I(λ)| ≤ F(M d+2 )N d+1 1 a 1+d 0 λ -d 2 .
2. Under the hypotheses of Theorem 3 there exists F : R + → R + non decreasing such that for every λ ≥ 1

(9) |I(λ)| ≤ F(M d+2 )N d+1 1 a 0 λ -d 2 .
Proof of Theorems 1 and 3 given Theorem 6. We assume that ( 5) is proved and our goal is to deduce that (3) holds with C = F(1). Set t = 1 + M d+2 and consider λ ≥ 1. Since tλ ≥ 1 we can apply (8) with (λ, Φ) replaced with (tλ, Φ(ξ)/t) to deduce that

R d e iλΦ(ξ) b(ξ) dξ = R d e itλ Φ(ξ) t b(ξ) dξ ≤ F M d+2 Φ t N d+1 1 (a 0 /t d ) 1+d (tλ) -d 2 ≤ F M d+2 (Φ) t N d+1 1 a 1+d 0 λ -d 2 t d 2 +d 2 ,
which yields the wanted estimate. The case 2. is analogue.

We are left with the proof of Theorem 6. We begin by some preliminaries. 0.1. Preliminaries. In that follows we shall denote by C d a positive constant depending only on the dimension d and by F a non decreasing function from R + to R + which can change from line to line.

Point 1. First of all we may assume that (10) λ

1 2 a 0 ≥ 1. Indeed if λ 1 2 a 0 ≤ 1 then 1 ≤ (λ 1
2 a 0 ) -d and we write

|I(λ)| ≤ b L 1 (R d ) ≤ b L 1 (R d ) a 0 a 0 1 (λ 1 2 a 0 ) d ≤ CM d 2 b L 1 (R d ) 1 a 1+d 0 λ -d 2
since by ( 2), (iii) we have a 0 ≤ CM d 2 . Point 2. Set H = HessΦ. By (i) the eingenvalues (λ j ) j=1,...,d of H are bounded by

C d M 2 . It follows from (iii) that (11) |λ j | ≥ a 0 (C d M 2 ) d-1 , 1 ≤ j ≤ d. Therefore (12) |H(ξ)X| ≥ a 0 (C d M 2 ) d-1 |X|, ∀ξ ∈ V, ∀X ∈ R d .
We shall use the Taylor formula Proof. Indeed by ( 13) and (12) if ξ, η ∈ B(ξ * , δ) we can write

|∇Φ(ξ) -∇Φ(η)| ≥ a 0 (C d M 2 ) d-1 - a 0 6(C d M 2 ) d-1 |ξ -η| ≥ 5a 0 6(C d M 2 ) d-1 |ξ -η|. Let (ξ * j ) j=1,...,J ⊂ supp b, such that supp b ⊂ ∪ J j=1 B(ξ * j , δ).
Taking a partition of unity (χ j ) and setting b j = χ j b we have ( 16)

I(λ) = J j=1 I j (λ), I j (λ) = R d e iλΦ(ξ) b j (ξ) dξ.
Notice that χ j can be taken of the form χ 0

ξ-ξ * j δ so that (17) |∂ α ξ χ j (ξ)| ≤ Cδ -|α| .
Notice also that J ≤ C d δ -d .

Lemma 8. Let i ∈ {1, . . . , d} and

A i = ∂ i Φ |∇Φ| 2 .
One can find F : R + → R + non decreasing such that

(18) |D α ξ A i (ξ)| ≤ F(M 1+|α| ) 1+|α| k=2 1 |∇Φ(ξ)| k , |α| ≥ 1.
Proof. We proceed by induction on |α|. A simple computation shows that (18) is true for |α| = 1. Assume it is true for |α| ≤ l and let |γ| = l + 1 ≥ 2. Differentiating |γ| times the equality

|∇Φ| 2 A i = ∂ i Φ we obtain (19) |∇Φ| 2 D γ ξ A i = (1) -(2) -(3), with (1) = D γ ξ ∂ i Φ (2) = |β|=1 γ β (D β ξ |∇Φ| 2 )D γ-β A i , (3) = 2≤|β|≤|γ| γ β (D β ξ |∇Φ| 2 )D γ-β A i . We have |(1)| ≤ M l+2 . By the induction, |(2)| ≤ CM 2 |∇Φ|F(M l+1 ) l+1 k=2 1 |∇Φ| k . Now for |β| ≥ 2 we have |γ|-|β| ≤ l-1. Since |D β ξ |∇Φ| 2 | ≤ C 2 M |β|+1 |∇Φ|+F(M |β| ) the induction shows that |(3)| ≤ CF(M l+2 )(1 + |∇Φ|) l k=2 1 |∇Φ| k .
Dividing both members of the first equation in ( 19) by |∇Φ| 2 we obtain eventually

|D γ ξ A i | ≤ F(M l+2 ) l+2 k=2 1 |∇Φ| k , |γ| = l + 1.
This completes the proof of (18). We shall also need the following result. Proof. The proof goes by induction on |α|. For |α| = 1 it is a simple computation. Assume this is true for 1

≤ |α| ≤ k let |γ| = k + 1 ≥ 2. Set F (ξ) = |∇Φ(ξ)|. Then we write ∂ γ ξ F (ξ)F (ξ) | = ∂ γ ξ d j=1 (∂ j Φ) 2 .
The right hand side is bounded by

F(M |γ|+1 )(1 + F (ξ))
. By the Leibniz formula the left hand side can be written as 2F (ξ)∂ γ ξ F (ξ) plus a finite sum of terms of the form

∂ γ 1 ξ F (ξ) ∂ γ 2 ξ F (ξ) where 1 ≤ |γ j | ≤ k and γ 1 + γ 2 = γ.
For these last terms we can use the induction and we obtain

F (ξ)∂ γ ξ F (ξ) ≤ F(M |γ|+1 )(1 + F (ξ) + F (ξ)| 2-|γ|
). Dividing both members by F (ξ) we obtain

|∂ γ ξ F (ξ)| ≤ F(M |γ|+1 ) 1 + 1 F (ξ) + F (ξ) 1-|γ| ≤ F(M |γ|+1 )F (ξ) 1-|γ| F (ξ) |γ|-1 + F (ξ) |γ|-2 + 1 .
Since |γ| ≥ 2 and F (ξ) ≤ 2 we obtain the desired result.

Lemma 10. Consider the vector field L = A • ∇ where

A i = ∂ i Φ |∇Φ| 2 . For any N ∈ N we have (21) ( t L) N = |α|≤N c α,N ∂ α , with |∂ β ξ c α,N | ≤ F(M N -|α|+|β|+1 ) 2N -|α|+|β| k=N 1 |∇Φ| k .
(Here we set M 1 = 1. It occurs when β = 0, |α| = N.)

Proof. Again we proceed by induction on N. For N = 1 we have c α,N = A i if |α| = 1 and c α,N = divA if |α| = 0. Then (21) follows immediately from (18). Assume that ( 21) is true up to the order N and let us prove it for N + 1. We write

( t L) N +1 = t L( t L) N = (∇ • A)( t L) N = |α|≤N d i=1 ∂ i (A i c α,N ∂ α ), = |α|≤N (div A)c α,N ∂ α + |α|≤N A • ∇c α,N ∂ α + |α|≤N d i=1 A i c α,N ∂ i ∂ α , = |γ|≤N +1 c γ,N +1 ∂ γ , where c 0,N +1 = (div A)c 0,N + A • ∇c 0,N , if |γ| = 0, c γ,N +1 = (div A)c γ,N + A • ∇c γ,N + A i c α,N |α| = |γ| -1, if 1 ≤ |γ| ≤ N, c γ,N +1 = A i c α,N , if ∂ γ = ∂ i ∂ α , |α| = N, if |γ| = N + 1.
We estimate now each coefficient. First of all ∂ β c 0,N +1 is a finite sum of terms of the form (∂

β 1 ∂ i A i )(∂ β 2 c 0,N ) and (∂ β 1 A i )(∂ β 2 ∂ i c 0,N ) with β = β 1 + β 2 . Using (18)
and the induction the first term is bounded by

F(M |β 1 |+2 ) |β 1 |+2 k=2 |∇Φ| -k F(M N +|β 2 |+1 ) 2N +|β 2 | l=N |∇Φ| -l .
Concerning the second term, if

β 1 = 0, β 2 = β it is bounded by 1 |∇φ| F(M N +|β|+2 ) 2N +|β|+1 l=N |∇Φ| -l ≤ F(M N +1+|β|+1 ) 2(N +1)+|β| l=N +1 |∇Φ| -l .
If β 1 = 0 it is bounded by

F(M |β 1 |+1 ) |β 1 |+1 k=2 |∇Φ| -k F(M N +|β 2 |+1 ) 2N +|β 2 |+1 l=N |∇Φ| -l . Since N + 2 ≤ k + l ≤ 2N + 2 + |β 1 | + |β 2 | = 2(N + 1) + |β| we see that ∂ β c 0,N +1
satisfies the estimate in (21) with N replaced by N + 1.

Let us look to the term ∂ β c γ,N +1 with |γ| = N + 1. This term is also a finite sum of terms of the form (∂

β 1 A i )(∂ β 2 c α,N ), |α| = |γ| -1. As above, if β 1 = 0, using (18)
and the induction it is bounded by

1 |∇φ| F(M N -|γ|+1+|β|+1 ) 2N -|γ|+1+|β| l=N |∇Φ| -l ≤ F(M N +1-|γ|+|β|+1 ) 2(N +1)-|γ|+|β| l=N +1 |∇Φ| -l .
If β 1 = 0 it is bounded by

F(M |β 1 |+1
)

|β 1 |+1 k=2 |∇Φ| -k F(M N -|γ|+1+|β 2 | ) 2N -|γ|+1+|β 2 l=N |∇Φ| -l .
Since N +2 ≤ k+l ≤ 2N +2-|γ|+|β| we see that ∂ β c γ,N +1 satisfies also the estimate in (21). The estimates of the other terms are similar and left to the reader.

0.2. Proof of Theorem 6. Case 1. Let ψ ∈ C ∞ 0 (R) be such that ψ(x) = 1 if |x| ≤ 1, ψ(x) = 0 if |x| ≥ 2.
With the notation in (16), j beeing fixed, we write (22)

I j (λ) = e iλΦ(ξ) ψ λ 1 2 |∇Φ(ξ)| χ j (ξ)b(ξ) dξ + e iλΦ(ξ )(1 -ψ λ 1 2 |∇Φ(ξ)| )χ j (ξ)b(ξ) dξ =: K j (λ) + L j (λ).
We shall use (see ( 15)) the fact that on the support of χ j the map ξ → ∇Φ(ξ) is injective. Let us estimate K j . We write

|K j (λ)| ≤ C d a 0 N 0 λ -d 2 .
To estimate L j we introduce the vector field

X = 1 iλ ∇Φ |∇Φ| 2 • ∇ which satisfies Xe iλΦ = e iλΦ .
Now with N ≥ 1 to be chosen we write

L j (λ) = e iλΦ(ξ) ( t X) N (1 -ψ λ 1 2 |∇Φ(ξ)| )χ j (ξ)b(ξ) dξ.
Since X = 1 iλ L we can use (21) and we obtain

|L j (λ)| ≤ C N α=α 1 +α 2 +α 3 |α|≤N S α,N , where S α,N = λ -N |c α,N | ∂ α 1 ξ 1 -ψ λ 1 2 |∇Φ(ξ)| ∂ α 2 ξ χ j (ξ) ∂ α 3 ξ b(ξ) dξ.
Our aim is to prove that with an appropriate choice of N we have

(24) |L j (λ)| ≤ F(M d+2 )N d+1 1 a 0 λ -d 2 .
Step 1. α 1 = 0. Here we integrate on the set |∇Φ(ξ)| ≥ λ -1 2 . We use (15), the bounds (17), ( 21), (2) (iii) and we make the change of variable η = ∇Φ(ξ); then dη = |det Hess Φ(ξ)| dξ then

|S α,N | ≤ λ -N a 0 δ -|α 2 | N |α 3 | F(M N +1 ) 2N -|α| k=N |η|≥λ -1 2 dη |η| k ≤ C d λ -N a 0 δ -|α 2 | N |α 3 | F(M N +1 ) 2N -|α| k=N +∞ λ -1 2 r d-1-k dr. Taking N = d + 1 since |α j | ≤ |α| ≤ N we see that |S α,N | ≤ λ -N a 0 δ -|α| N d+1 F(M d+2 )λ N -1 2 |α|-d 2 ≤ 1 a 0 N d+1 F(M d+2 )λ -d 2 1 (λ 1 2 δ) |α| .
Since by ( 14) δ is proportional to a 0 and by (10) we have assumed that λ

1 2 a 0 ≥ 1 we obtain eventually |S α,N | ≤ 1 a 0 N d+1 F(M d+2 )λ -d 2
Step 2. α 1 = 0. Here, since we differentiate ψ we are integrating on the set λ

-1 2 ≤ |∇Φ(ξ)| ≤ 2λ -1 2 ≤ 1.
We can therefore use (20). We have to estimate

S α,N = λ -N |c α,N | ∂ α 1 ξ ψ λ 1 2 |∇Φ(ξ)| ∂ α 2 ξ χ j (ξ) ∂ α 3 ξ b(ξ) dξ.
By the Faa-di-Bruno formula we have We prove now the second part of Theorem 6.

∂ α 1 ξ ψ λ 1 2 |∇Φ(ξ)| = 1≤|β|≤|α 1 | a α,β ψ (β) λ 1 2 |∇Φ(ξ)| s i=1 λ 1 2 ∂ l i ξ |∇Φ|
In that case it is not necessary to make a localization of I(λ) in small balls of size δ as in the first case.

Then as before we write

-1 2 0

 2 in the right hand side.

  (13) ∇Φ(ξ) = ∇Φ(η) + Hess Φ(η)(ξη) + R, |R| ≤ C ′ d M 3 |ξ -η| 2 . Lemma 7. Let δ > 0 be defined by (14) δ = a 0 12C ′ d M 3 (C d M 2 ) d-1 . where C d , C ′ d have been defined above. Let ξ * ∈ supp b Then, (15) on the ball B(ξ * , δ) the map ξ → ∇Φ(ξ) is injective.

Lemma 9 .

 9 On the set {ξ : 0 < |∇Φ(ξ)| ≤ 2} we have (20) |∂ α ξ |∇Φ(ξ)|| ≤ F(M |α|+1 )|∇Φ(ξ)| 1-|α| , |α| ≥ 1.

|K j (λ)| ≤ |ψ λ 1 2 1 2

 11 |∇Φ(ξ)| χ j (ξ)b(ξ)| dξ and we set η = λ ∇Φ(ξ) then dη = λ d 2 |det Hess Φ(ξ)| dξ. Then using (2) (iii) and the notations therein we obtain (23)

α 1 ξ ψ λ 1 2| λ |β| 2 ψ (β) λ 1 2 1 | λ |β| 2 λ -1 2 ≤|η|≤2λ - 1 2(λ 1 2 δ) -|α 2 | . Since λ 1 2a 1+d 0 F

 121121110 k iwhere a α,β are absolute constants, si=1 k i = β, s i=1 k i |l i | = |α 1 |. Using (20) we deduce that ∂ |∇Φ(ξ)| ≤ F(M |α 1 |+1 ) 1≤|β|≤|α 1 |∇Φ(ξ)| |∇Φ| |β|-|α 1 | .Using (21) and the fact that|∇Φ(ξ)| ≤ 1 we have |c α,N | ≤ F(M N +1 )|∇Φ(ξ)| |α|-2N . Eventually we have ∂ α 2 ξ χ j (ξ) ≤ C α δ -|α 2 | .Performing as above the change of variables η = ∇Φ(ξ) we will have|S α,N | ≤ λ -N δ -|α 2 | a 0 F(M N +1 )N |α| 1≤|β|≤|α |η| |α|-2N +|β|-|α 1 | dη ≤ 1 a 0 F(M N +1 )N |α| λ -d 2 δ ≥ F(M 3 ) we obtain |S| ≤ 1 a 0 F(M N +1 )N N λ -d 2 .Therefore (24) is proved. Using (22), (23), (24) since N = d + 1 we obtain|I j | ≤ 1 a 0 F(M d+2 )N d+1 λ -d 2 , 1 ≤ j ≤ J. Now since J ≤ C d δ -d ≤ F(M 3 )a -d 0 using (16) we obtain eventually |I(λ)| ≤ 1 (M d+2 )N d+1 λ -d 2which completes the proof of the first case of the theorem.

1 2 1 2

 11 ) = e iλΦ(ξ) ψ λ |∇Φ(ξ)| b(ξ) dξ + e iλΦ(ξ) (1ψ λ |∇Φ(ξ)| )b(ξ) dξ =: K(λ) + L(λ)and the final estimate follows from (23) and (24).


Mohamed-Zied Ouertani

Lilia Gzara-Yesilbas

Luc Lossent

ENGINEERING CHANGE PROCESS: STATE OF THE ART, A CASE STUDY AND PROPOSITION OF AN IMPACT ANALYSIS METHOD

Keywords: Engineering change management, impact analysis, PLM, product modelling, integration, CSP

The main challenge for product designers, especially of complex products, is to predict as finely as possible the consequence of the product change for its structure as well as its propagation overall the Technical Information System (TIS). Starting from the work already undertaken in this domain, we underline the lack of methods and tools of impact analysis. Our contribution is therefore to bring a method with the aim to facilitate the impact analysis in a multi TIS context. This method, based on a product model supported by a constraints resolution tool, will allow resolving the various constraints levered in an impact analysis problem.

Introduction

Each product implies a great number of design choices. The ideal, to maintain competitive advantages or to correspond to the technological innovations with the competitors, is to get the right design the first time. However, changes due to immature decisions, changes in customer requirements or evolutions of manufacturing processes are unavoidable. Thus, Engineering Change Management (ECM), taking a great part in the development of product, is a vital and a necessary process in industrial companies. According to Maull and al., industries that identified the process of ECM as a principal problem were promoted in the development of their process [START_REF] Maull | The role of the bill-of-materials as a CAD/CAPM interface and the key importance of engineering change control[END_REF]. This is all the more critical as product engineering proceeds in a simultaneous and a collaborative way. Indeed, in a mechanical design context, Numerical Command Control activity is done by the planning department in a concurrent way compared to the design department. A mechanical part coming from the design department continues evolving whereas the planning department has already achieved the associated machining program and sometimes launched machining process. This research studies ECM process and focuses on techniques likely to manage the impacts of engineering changes given on a part or on a machining process. The optics is to propose the first elements of a methodology to analyze the impacts of a mechanical part change on the machining process following: a change given to a part by the design department : design change an evolution of machining means given by the planning department : process change The first section of the paper presents the ECM process and various related research works (Section 2). Then, a special interest is given to impact analysis activity. A study is undertaken to understand the techniques suggested in the literature to support impact analysis (section 3). Starting from this state of the art and towards the problems faced by our industrial partner, an approach of impact analysis is proposed (section 4).

Engineering Change Management process: definition and related works

The ECM process can be defined as being the process to introduce modifications on a product in a projected and systematic way [START_REF] Rouibah | Change management in concurrent engineering from a parameter perspective[END_REF]. It aims at satisfying design constraints and objectives, facilitating the task of manufacture, eliminating a design conflict, meeting customer requirements or answering the legislative evolution as in the aircraft industry.

Several strategies were defined to control this process [3] [4], leading to several decompositions of this process. However, we can distinguish three main phases in the ECM process:

Engineering Change Request (ECR): requests for change are collected with the necessary information to process the treatment. EC treatment: a feasibility study is undertaken in order to evaluate the economic and technique feasibility of the requested change. The analysis of the EC impacts allows detecting the consequence on the product and the organization. Then, a decision is taken about the continuation of the process (abort or implementation of the change). Solution embodiment: the solution is physically implemented, documents are updated and concerned actors are informed.

Regarding this process, several research works were undertaken during the ten last years. These works can be classified into three categories such as represented in Figure1:

Researches about Change Management

Exploratory works

Organisational solutions Technical solutions

Management strategies

Process organisation

Traceability

Impact analysis Process performance

[5] [6] [7] [8] [9] [10] [3] [12] [13] [17] [2] [18] [19] [20] [21] [14] [15] [16] [3] [4] [11]
Figure1. The structure of existing research in ECM Exploratory works were interested to understand the ECM process and to extract the problems related to this process. Works aiming at proposing organisational solutions tried to rationalize the ECM process by proposing strategies to control it [START_REF] Fricke | Coping with changes -causes, findings and strategies[END_REF] [START_REF] Nicholls | Getting engineering changes under control[END_REF], defining elements to improve its performance [START_REF] Terwiesch | Managing the Process of Engineering Change Orders: The Case of the Climate Control System in Automobile Development[END_REF] [12] [START_REF] Collaine | Une méthode d'évaluation de l'impact des évolutions du produit sur le système de production et la performance de l'entreprise[END_REF] and proposing new organisations of the process to better coordinate the activities of EC treatment and to reduce delays [START_REF] Riviere | Performances in engineering changes management[END_REF] [15] [START_REF] Huang | Development of a web-based system for engineering change management[END_REF]. Works focalising to bring technical solutions concern the change traceability and their impacts analysis. Concerning traceability we quote the work of Cohen [START_REF] Cohen | a data approach to tracking and evaluating engineering changes[END_REF] which proposes a product model in EXPRESS format. As for impact analysis, several approaches and techniques were proposed, by components interaction modeling, impact probabilities assessment [START_REF] Rouibah | Change management in concurrent engineering from a parameter perspective[END_REF] [18] [START_REF] Clarkson | Predicting change management in complex design[END_REF], etc.

In this context of ECM rationalisation, a research work was undertaken by the CRAN laboratory within a French national project (IPPOP * ) to propose a workflow model supporting the management of conflicts appearing during cooperative design due to a product definition modification [START_REF] Gzara | Cooperative design of mechanical product: specification for a conflict management support system[END_REF]. Conflicts resolution is done by driving the decision process undertaken to modify product data further to a conflict. This process is based on a succession of popularisation and mediation phases (Cf. Figure 2) : Popularization: the actor facing a conflict explains the current problem with the choices currently made on the product. He then proves or argues his motivation to modify the current solution by clarifying the negative consequences of the current choices on the product (in terms of product performance, cost, technical feasibility, coherence with contractual conditions, etc.).

Mediation: this action aims at advocating a solution to resolve the problem encountered and in this way to avoid the unfortunate consequences that could be generated by the former solution. It comes from the actor who has just popularized the problem in order to propose the solution, or from another actor concerned by the conflict who would approve the solution proposed by another actor.

Figure 2. UML activity diagram of the conflict resolution process

During popularization phase, actors need a method to assess quickly the impacts of any technical choice made by one actor and then react in the best delays. The aim of this research is to give a method for impact analysis to support popularization phase. In the following, a state of the art related to techniques likely to manage impacts is given (Section 3). Then, the proposed method is presented (Section 4).

Research works dealing with impact analysis activity

Impact analysis can be defined as an activity for which a required EC is transformed into a set of changes induced on the product or its manufacturing process. This analysis stage allows evaluating the technical and financial feasibility of a requested change. During last years, * Integration of Product-Process-Organisation for engineering Performance improvement Project manager many research works have focalised on impact analysis techniques. These works can be classified according to two approaches: by components interaction modelling and constraint propagation and by impact probabilities assessment.

Impact analysis by components interaction modelling

A parameter-based approach was proposed by Rouibah and al. in [START_REF] Rouibah | Change management in concurrent engineering from a parameter perspective[END_REF] using elementary engineering decisions. In this research, the ECM is seeing as an engineering decisions-making process. These decisions change or determine engineering variables and the most elementary of these engineering variables are referred as parameters. Parameters can refer to dimensions, as well as forces and movements. Parameters often share complex relationships, which may be represented with mathematical equations, diagrams or tables. The capture of parameter relations during concurrent design results in a parameter network. Then, once a change need is declared, this change is expressed in terms of one or many parameter's values changes (a change list). As soon as the actors, concerned by the elaboration and the evolution of the parameter to modify, have agreed on the change list and have specified a new draft value, the parameter status will be "in change". The impact of this modification is then evaluated by propagating the modification through the parameter network. This method only makes participants aware of the existing relation and possible change propagation between the changed and neighboring parameters in the network. The actors concerned by the modification must jointly clarify whether these parameters are actually affected by the change. A new list of changes is then created, containing only the parameters impacted by the change. Then, a workflow to release parameter approves and distributes the parameters to change and their values to the concerned actors. Figure 3 Ollinger and al. in [START_REF] Ollinger | REDESIGNIT -A constraint based tool for managing design changes[END_REF] reconcile the problem from another point of view by proposing an approach by constraints resolution through the REDESIGNIT tool. This tool allows evaluation of proposals for redesign plans. These proposals describe which quantity should be changed as well as the nature of these changes (dimensions, tolerances, matter…). The approach uses a direct dependency graph that describes the various relations between quantities. This graph is built basing on the following concepts: Quantity: referring to both physical properties of the device components and descriptions of the device's operation.

Causal influences between quantities describing how a change to one quantity affects the other quantities. There are four types of causal influences: M+, M-, LIMITS (AS_UPPER_LIMITS, AS_LOWER_LIMITS). Constraints on quantities: a constraint is an expression of a design requirement placed upon a particular quantity (FIXED, MAXIMIZE, MINIMIZE, RANGE). A causal path is thus built. If a change is applied to the quantity at the start of the path, every quantity downstream in the path will be changed as well. Change propagation will take the form of a propagation tree (cf. Figure 4). Given a causal pair, the program takes the direction of change in the cause quantity and the type of causal influence involved finding the change in the affected quantity. The program also allows to detect the undesirable side effect of the modification and to calculate the cost and the benefit of such modification. However, the generated redesign plans are abstract; they do not specify numerical values for the quantities.

Impact analysis by probabilities assessment

Even it is sometimes difficult to quantify a change impact on the whole product components, it is possible to predict the risk to change them. Clarkson and al. [START_REF] Clarkson | Predicting change management in complex design[END_REF], through the development of a mathematical model, propose a method to predict the risk of change propagation in terms of likelihood and impact of change. This method uses:

A design structure matrix to model direct dependencies between all the product components. A likelihood matrix expressing the probability of modifying the component b if the component a is modified (l b,a ). An impact matrix expressing the impact of an EC of a on b (i b,a ). A risk matrix expressing the risk to impact b if we modify a (r b,a ), with r b,a = i b,a * l b,a .

A modification is also propagated in a transverse way between indirectly dependent components. The model uses the same design matrix to express the indirect dependencies by transitivity. The influence of the EC is then the sum of direct and indirect effects.

In a recent research work, Jarrat and al. [START_REF] Jarratt | Use of Monte Carlo Methods in the Prediction of Change Propagation[END_REF] extended the original technique proposed by Clarkson, by involving counting along propagation paths, through the implementation of Monte Carlo techniques to simulate change propagation, hence greatly increasing the flexibility of the method.

We finally quote HO and al. [START_REF] Ho | Progressive Engineering Changes in Multi-level Product Structure in MRP System[END_REF] who developed an analytical procedure to compute progressive probabilities of ECs. The probability of changing a part at one level is affected by the changes in its parent levels. Then, a sensitivity analysis is performed to see how the shape of product structure affects resultant progressive ECs for the lower-level parts in the product structure.

Resume

To conclude this state of the art, the impact analysis process refers in all studied works to a resolution of a set of constraints. The various tools and methods then proposed approach the problem from the existing links and the relations between the different components of a product. By identifying these relations and based on the constraints imposed by a component on another, the problem is resolved by change propagation. However, in the whole studied works the change propagation is manual, which is difficult to manage for complex products, such as planes. We then underline a lack for a structured and an automatic method of change propagation able to analyze and to propagate quickly the impacts through several TIS. In section 3, we will develop the problematic faced by our industrial partner and then give our proposition to assist impact analysis process.

4 Towards a systematic method for impact analysis

ECM problems faced by the industrial partner

In a manufacturer of electric motors, an engine is composed of a set of components developed and manufactured inside the organization, but also in cooperation with external partners. Adding to these development partners, various sub-contractors take part to achieve some activities of industrialization and manufacture. The ECM process is then leaded in an extended enterprise context. From the survey we have conducted in our industrial partner to analyse the ECM process implemented in the organization, we deduce that the treatment of an EC order mainly depends on the type of change, its origin and its urgency. Then, a variety of ECM processes exist in the enterprise. We present here a generic process representing the common part of the observed processes. (cf. Figure 5)

Approval of the modification

Modification approved

Constraints non checked

Figure 5. Generic process of engineering change management (industrial case)

Any modification of an article in the CAD system involves a modification index to distinguish the successive modifications. An article is managed by its code and its modification index in the various TIS of the enterprise. The propagation of the modification index in the product structure is done in the PDM system (Product Data Management), already specifying the various existing relations between articles. However, when change propagates, subcontractor uses the old version of the article (with an erroneous index). Indeed, the modification and the evolution of the article index are not taken into account in the CAMM (Computer-Aided Management and Manufacturing), the only Information System (IS) shared with the sub-contractors.

Impact analysis approach

Our proposal is based on the assistances that can be given to the configuration managers to facilitate the evaluation of EC impacts, by resolving the problem of product data coherence between the various TIS. The objective is to develop a methodology which proposes the possible configuration of the product and/or its manufacturing process following a modification given to a product element or its manufacturing process. Such a methodology is based on two phases:

-modelling the interactions (impacts) product-product and product-manufacturing process, -using a programming constraints solver which propagates the obtained constraints (interactions).

Phase 1: impacts modelling

To enhance impact analysis, two modelling mechanisms are used:

To specify dependences between concepts (which are implemented in the same IS or in several ISs), association classes are attached to the links between concepts (cf. Figure 6). Three kinds of dependences are distinguished: class/class (a class modification lead to a modification of another class), class/attribute (a class modification implies a modification of an attribute inside this class or in another class) and attribute/attribute (an attribute modification implies modification of another attribute). In these three cases, the modification of a concept can be on its existence or just on its instance. To integrate the various IS models, a "bridge" concept is introduced between each IS model thanks to links of correspondence between concepts belonging to several IS. These links can be of association, composition / aggregation or generalization / specification kind (cf. Figure 7). This approach was tested within the industrial partner. First each of the existing IS is formalized by adding dependency concepts to allow capitalization of constraints between product concepts. Then, ISs are integrated thanks to correspondence links between the various product or process concepts established in each IS, and several dependency concepts are attached to model the interactions between them. Thus a distributed product model is obtained (cf. Figure 8). Finally, various scripts were developed to implement the various concept thus introduced. occurring on a product element and the various dependencies inside the product (with the manufacturing process), the proposed methodology proposes a list of the various change impacts. It proposes also, thanks to a constraint satisfaction problem solver, the various new possible configurations.

Figure 8. An overview of the distributed product model to support impact analysis

Figure 10. Overview of the proposed impact analysis methodology 5 Conclusions and Perspectives

Evaluating the impacts of engineering changes during design process is a highly critical problem in industry and it is usually managed by the actors themselves, based on their expertise. A manual resolution of the various impacts is only possible if the product is very simple (composed of a few number of parts). In case of complex products, such as cars or aircrafts, automatic resolution is necessary to resolve the whole induced constraints. Then the proposed approach addresses specifically complex products. However, since actual CSP solvers are not efficient (in terms of calculation delays) when the number of constraints is very high, it is necessary to apply the proposed approach to sub-units of the product (in order to have a less number of constraints).

ECM process is usually a highly human process and the proposed automated approach to analyze impacts does not substitute human decision making processes. It assists engineers during design review meetings by quickly evaluating the impacts of each change given to the product and by treating the whole possible impacts (engineers may forget some impacts). Moreover, it allows capitalizing the various changes occurring on products and their impacts, so as to constitute a corporate memory that can be reused in future ECM projects. It even allows keeping trace of the various discussions held by engineers during decision making processes (if a workflow engine is associated to it). However, the proposed approach is based on a manual capture of the various dependences. It requires a big effort of impacts capitalization and capture. This capitalization can be enhanced by building a domain ontology for technical impacts which matches technical changes with their possible impacts, according to various design contexts. Moreover, in order to avoid manual capitalization and capture of dependencies, it will be interesting to propose a system which would deduce automatically the dependences by capitalizing the various decisions taken by the design actors in past experiments of impact analysis. The use of Case Based Reasoning techniques may enable such deductions.

Figure 3 .

 3 Figure3. A workflow to release parameters[START_REF] Rouibah | Change management in concurrent engineering from a parameter perspective[END_REF]

Figure 4 .

 4 Figure 4. Propagation tree for quantity "piston stroke"[START_REF] Ollinger | REDESIGNIT -A constraint based tool for managing design changes[END_REF]

Figure 6 .

 6 Figure 6. Specification of concepts dependencies

2 3Figure 7 .

 27 Figure 7. Specification of IS integration

The model obtained above allows any actor identifying:

the possible changes of a product or process concept (classes or attributes). the possible effects on the other concepts (in terms of changing of the existence or the instance of a class or an attribute).

Phase 2: impacts propagation

To facilitate the impact analysis, we propose to connect the TIS integrated model, obtained in phase 1 (cf. Figure 8), to a constraint satisfaction programming solver (CSP) which captures from the TIS, the various constraints (dependencies) between product and/or process concepts and provides a list of impacts for each EC introduced on the product or on its manufacturing process. To do that, a correspondence between a constraint satisfaction problem mathematical model and the TIS integrated model is done. In figure 9 the various established correspondences are given.

Figure 9. Data correspondence

Once the product characteristics x i (variables) to modify are initialised and the constraints c i (dependencies) are fixed, the programming constraints solver provides the solution R by making intersection of the various solution domains r i corresponding to the variables values satisfying each of the constraints c i . We give here an example to better explain these correspondences. We consider the product characteristics (dimension, durability, tolerance…) and the machining characteristics (tools, schedule…) as variables (x 1 .. x n ). Given two constraints (c 1 , c 2 ) whereas: c 1 : if dimension > XX, then hole machining is impossible. c 2 : product durability must be between a and b. The solution domain r 1 is then the dimensions satisfying the hole machining and r 2 is the dimensions leading to a product durability between [a,b]. Once all solution domains are defined, the final solution corresponds to domains intersection.

The ECM analyser has then the various impacts but also the new possible configurations of the product and/or its manufacturing process.

In figure 10, we give a global view of the proposed methodology to support impact analysis process. From the design actor's expertise, we capitalize the various change impacts by introducing the product-process dependencies in the different TIS. Then, given a change


Mohamed-Zied Ouertani
email: mohamed-zied.ouertani@cran.uhp-nancy.fr

Lilia Gzara-Yesilbas

TOWARDS AN INTEGRATED WORKFLOW FOR COOPERATIVE PROCESS MANAGEMENT

Cooperative design deals with the sharing of interests and resources amongst actors to reach a common purpose. This purpose lays on the development of products through the coordination between information sharing, interactions and knowledge. The research objective is to specify an information system which can support the set of knowledge involved in cooperative design situations with a special focus on conflict management situation. In previous works, a conflict management protocol was proposed and carried out in margin of the design process in which the conflict has appeared. However, several dependencies exist between the conflict management process, the design process and the involved product and organisation data. This paper deals with the notorious problem of workflows integration by proposing mechanisms allowing data integration and control flows during conflict management process.

INTRODUCTION 1.1 Context

The increasing competition and complexity of products and processes require new organizational concepts for product development. Collaborative design deals with the sharing of various interests and resources among various actors with the aim of reaching a common purpose. This purpose relies on the development of products using knowledge sharing and interactions with some coordination between these varied activities. The overall research objective is to provide a collaborative design environment which allows inter-skill collaboration to be coordinated by defining a workflow system knowledge management in a collaborative design situation. In a previous work [START_REF] Yesilbas | Towards a knowledge repository for collaborative design process: focus on conflict management[END_REF], the focus was on particular situation of conflict management and protocol formalization driven by actors to resolve conflicts. This procedure is supported by a workflow implemented within CO2MED tool [START_REF] Lombard | Design in collaboration: existing trends and application to the case of conflict handling with CO²MED software[END_REF].

The conflicts as stated by March and al. [START_REF] March | Organizations[END_REF], appear in the impossibility to making a decision with the usual process. They are common in a multi-actor, inter-person, inter-group or inter-organisational context. A conflict is detected as soon as a given resource (human, software…) encounters a problem or a data incoherence of the ongoing design activity. This incoherence concerns the different points of view: product (design objective), process (design development) or organization (set up to pilot and implement the process).

In the proposed conflict management protocol, conflicts occurring during the studied processes (within Alstom Power Conversions Company) are tackled according to the cybernetic boucle, defined by Le Moigne (Le [START_REF] Moigne | If you do believe that your industrial system really is complex, then …[END_REF]. Indeed, as soon as a conflict is detected, observed manually or automatically, the issue is to set up a process aiming at deciding, i.e. connecting a succession of activities ending up in decisional action in order to resolve the detected conflict; then to inform, that is to forward the selected solution to the concerned design activities. An UML class diagram structures the decision phase of the conflict management protocol and positions the proposed control within the design process (see Figure 1).

Design Process

Inform

Problem Formulation and Requirements Specification

Conflict resolution cannot be achieved by one single actor; it requires the gathering of different expertise areas. Conflict management is a collaborative activity that seeks the interference of many actors; the latters correspond to the people:

-who were involved in the realisation of the conflict source piece of data (process leading to the data). A conflict source piece of data is the result of one or more previous activities in the design process. The actor / actors responsible for its creation should be involved then to know the reason of the creation of this data (with the given value) and be invited to its modification (since they are the authors).

-who undertake activities dependent on the conflict source piece of data. In case this data has been modified, it is recommended to spread the modification impact onto these activities for better design coherence.

Thus, one of the main tasks of the conflict resolution protocol is not only to identify these actors in order to invite them to participate in the resolution process but also to propagate the changes that would be made on the conflict source data to the whole body of entities concerned.

Based on the fact that a piece of data is the result of the conversion of a collection of data (thanks to qualified conversion activities) supported by another data set (thanks to qualified control activities such as IDEF 1 datagrams), a conflict source data have several dependence links with the problem design data (product technical data or organizational data). It is necessary to note that the term data corresponds to any information manipulated during design process. In an information system context, data can be a class name, a class instance, an attribute name or an attribute instance.

Identifying the actors that would be involved in the conflict resolution process would lead to the identification of this data dependence network. This will be done through the data seized within the workflow that manages the design process and within the Information System that supports the life cycle of the product concerned by the conflict as a whole ( the associated PLM). Once the data dependant on the conflict source piece of data are identified, it is easy to recognize the design activities and hence the authors concerned by the dependant data.

A typical integration problem of the proposed conflict resolution workflow with the various design points of view; i.e. Product, Process and Organization is faced then. This integration should take place not only in an intra-project context (integration with the design process during which the conflict appeared) but also in an inter-project context (integration with the whole set of design processes within the enterprise (see Figure 2). In fact, in a Concurrent Engineering context, several interdependent processes are launched simultaneously. The resolution of a conflict might call into question previous decisions. Therefore, it is appropriate to integrate the whole set of processes based on these former choices to move forward. This is meant to propagate the changes that could be made and to involve the authors of these different processes in the resolution of the conflict.

Outline

The forthcoming parts of this paper develop the elements of the proposed method to set up the data dependence network and identify the conflict resolution associated authors. It is organized as follows. Section 2 structures the design process to allow its integration within the product and organization points of view. A process model formalizing the concepts to be managed in the conflict resolution workflow is then obtained. Section 3 presents the applied techniques to build up the data dependence network. This returns with a set of treatments on the data obtained through the workflow running. Finally, we discuss related works and conclude the paper.

INFORMATION STRUCTURING

In a context of Information System development, we structure the design process using the UML language. Among a set of modelling languages and techniques designed to specify various aspects of enterprise engineering projects, we choose UML since it is a standard language that can be used to generate computerexecutable models that encode key aspects of software engineering projects.

Based on the definition that a design process is composed of a succession of activities, the designed product is thus the result of a number of data set transformations. Indeed, an activity transforms one (or several) input data (necessary to its execution) into a set of output data (necessary to the execution of next activities). These data can be of various natures: product, process or organisational. Then, an activity consumes resources (human, software, methodological…) and is subjected to constraints (specifications to be respected, availability of the resource, timelines, quality…) to achieve its objective.

A piece of data is created by an activity as an output to be consumed by another activity as an input. In addition to the activities of data transformation, it is necessary to take into account the control activities as well. A piece of data can be controlled by 'control activities' and supported by storage mechanisms. Indeed, in order to be able to transform a piece of data, the actor bases himself on intermediate activities as a support to his main one. These control activities break into a set of data which are consulted during the execution of the main activity; like consulting norms and standards or consulting the results of previous works to double check the transformation result of the main activity.. Then, according to the nature of the activity consuming a piece a data (control or transformation), we deal with "consulted" data (in control activities) and "required" data (necessary to the execution of a transformation activity). Figure 3 illustrates the various concepts introduced above. Accordingly, the actor is induced to transform the data while the design process is occurring. Based on his business expertise and his "know-how", the actor accomplishes his transformation task creating, this way, a dependence link between input and output data. It is often impossible to extract the type of this dependence link from the activity itself since this knowledge is not being formalized; especially that different types of links are available, such as composition and interface links [START_REF] Maurino | La gestion des données techniques[END_REF] and point of view link [START_REF] Harani | Une approche multi-modèles pour la capitalisation des connaissances dans le domaine de la conception[END_REF], etc.

To extract this "know-how", a "justification" attribute is added to the class Activity. This attribute recovers the "know-how" adopted by the actor when transforming the data and thus the dependence links between data. In fact, while the activity is taking place, the actor should justify and argue the transformations made on the data in order to achieve his objective. This justification can be done according to various modes: with simple and common terms known by all, using equations and mathematical formulas, by associating appendices (abacus, standard,…), by attaching plans CAD, by working out mechanical calculations… Being in a context of collaborative design, actors have to exchange data during the design process. Subjected to increasingly short timelines, actors are obliged to exchange intermediate data and therefore non-validated. Thus, a concept of data state is integrated in the process model. Indeed, during its life cycle, a piece of data can have one of these various states: created, draft, enabled and validated. The activity using a non validated data should wait until this data is validated to be able to confirm its own results. Figure 4 presents the overall process model thus obtained. In this model, the process is seen as a sub-class of 'Activity' which is composed of other activities. These activities can be either decomposable (process) and break up, then, into other activities (which can be operations or processes as well), or elementary (operation) and, thus, non decomposable [START_REF] Gzara | Les patterns pour l'ingénierie des systèmes d'information produit[END_REF]. When a process is taking place, the connection between the activities composing it is done by means of And/Or/Exclusive-OR connectors.

The process model required to build up the data dependences network is then structured. Section 3 presents the mechanisms necessary to obtain the data dependence network.

DATA PROCESSING TO BUILD UP THE DATA DEPENDENCES NETWORK

In order to build up a data dependence network, two possible approaches are considered: data mining applied to the "justification" attribute and data requests on the proposed model in Figure 4 in case the actor omits (voluntarily or involuntarily) to complete the "justification" attribute.

Data Mining Approach

The process model presented in the section 2 enables recovering the "know-how" mobilized by the actors during the execution of an activity. Thanks to the "justification" attribute, it is possible to extract actors' argumentations when they transform data. This being the starting point to apply a data mining technique.

Various research works have used the data mining technique to extract information from the texts. This technique makes it possible to benefit from the extracted information in order to convey new knowledge.

With regards to data mining tool development, several tools were proposed in the literature, such as TAMIS: Text Analysis by Mining Interisting_ruleS [START_REF] Cherfi | Étude et réalisation d'un système d'extraction de connaissances à partir de textes[END_REF]). This tool is set up on syntactic rules and is based on a functional mining methodology along with a symbolic approach. This contribution provides the analyst with accurate, established and justified set of information extracted from a group of texts.

As for data mining usage in a design context, we particularly quote the work of Gardoni [START_REF] Gardoni | Maîtrise de l'information non structurée et capitalisation de savoir et savoir-faire en Ingénierie Intégrée[END_REF] fostering the exchange of unstructured -and hence uncontrolled -information. Gardoni proposes a concept called MICA (Interactive Messaging System for Aerospatiale Matra Concurrency), enabling the structuring, sharing and access of the exchanged information as well as the capitalization of the "know" and the "know-how". In this respect, textual mining techniques are being used i.e. the Sampler tool is used. This tool is based on an ascendant hierarchical automatic classification algorithm; the latter provides a decreasingly fine partitions leading to the creation of a cluster of words.

Data Request Approach

According to the culture and policy of companies, it is not always for granted that actors fulfil the "justification" attribute and thus obtain sensitive information about the "know-how" proceeded. To be successful, people must be involved and aware about the effectiveness of such approaches.

To remedy to this problem, we suggest an alternative approach that requires more treatment but presents a better results guarantee. In fact, when actors omit or refuse to fill in the "justification" attribute (by fear to yield their "know-how"), an alternative would be to elaborate a series of requests on the model instances obtained by:

-identifying the data inputs/outputs of a transformation activity -identifying the input data of the control activities deployed to support the transformation activity

The results of this request allow the identification of the intermediate data set based on the succession of the activities. In fact, based on the "workflows patterns" proposed on (Van der Aalst, 2003) the downstream activities and their input/output data are identified.

Network of Data Dependences

Taking up a hierarchical classification approach similar to the one used by Gardoni, clusters of data are generated using the approach of data clustering. They are represented in cartography of terms and associations between these terms. The algorithm being applied to a preset textual field, the link between these terms is nothing but the actors' justification text. This semantic association can thus be considered as dependence. Dependences between data are specified (see Figure 5). This data dependences network can be translated into IF/THEN rules for the whole of the transformation activities possible for a piece of data. This will make it possible to identify the impact of the transformation of a piece of data on the set of data which are connected to it.

To better illustrate the data dependences network, an example (a mixing machine); focusing on a subsystem: the vertical column and the rack, is presented below. The alternative design solution is to screw together the column and the rack. Thus, the fixing process is defined: screw/nut assembly. Hence, the manufacturing process is also defined: drilling with the appropriate cutting speed. Then, the actor has to define the material property according to the fixing and manufacturing processes. Therefore, the material dimensions hardness and treatment are also fixed. Figure 6 shows the data dependences network according to the design process in question.

CONCLUSION

In a conflict management process, it is necessary to identify the actors that would contribute in the conflict resolution. In this paper, a product data dependences based method has been suggested in order to identify these actors. Hence a process model has been elaborated based on the extensions made to the IDEF concepts. Actors' identification would basically mean the identification of the intermediate data that have contributed to the definition of the conflict source piece of data. Accordingly, a data dependences network is build up with a data mining technique applied to the process model. A query applied to the link activity / resource would allow the identification of the actor responsible for each activity (as modelled in the process model).

In terms of perspectives, future works consists in implementing the proposed methodology within a software tool and then in applying it on an industrial case. Actors' identification discussed in this paper constitutes the first stage in the conflict management process (initialisation step); the second one would be the conflict resolution itself (decision step). Indeed, once actors have been identified, they are gathered in a new process to negotiate and mediate the various possible solutions. For this end, they would be guided by the CO2MED workflow.

Figure 1 -

 1 Figure 1 -Conflict resolution protocol within design process

  in project A1, in which conflict appeared Activity in project A, Launching concurrent Projects (A1, A2 and A3) Activities in projects A, A1, A2 and A3 which are dependant on in project A1, in which conflict appeared Activity in project A, Launching concurrent Projects (A1, A2 and A3) Activities in projects A, A1, A2 and A3 which are dependant on

Figure 2 -

 2 Figure 2 -Integration of resolution process in inter-project context

Figure 3 -

 3 Figure 3 -Extended activity and data concepts

Figure 4 -

 4 Figure 4 -Process model

Figure 5 -

 5 Figure 5 -Dependences links between data As presented previously, data could be viewed as a class or an attribute. Thus, data dependences concern class existence/instance or attribute existence/instance. Three kinds of dependences are distinguished: -modification on a Class Existence &/or Instance lead to a modification of the Existence &/or Instance of another class. -modification on an Attribute Existence &/or Value lead to a modification of the Existence &/or Value of another Attribute. -modification on Existence &/or Instance lead to a modification of the Existence &/or Value of another Attribute and vice versa.This data dependences network can be translated into IF/THEN rules for the whole of the transformation activities possible for a piece of data. This will make it possible to identify the impact of the transformation of a piece of data on the set of data which are connected to it.To better illustrate the data dependences network, an example (a mixing machine); focusing on a subsystem: the vertical column and the rack, is presented below. The alternative design solution is to screw together the column and the rack. Thus, the fixing process is defined: screw/nut assembly. Hence, the manufacturing process is also defined: drilling with the appropriate cutting speed. Then, the actor has to define the material property according to the fixing and manufacturing processes. Therefore, the material dimensions hardness and treatment are also fixed. Figure6shows the data dependences network according to the design process in question.

Figure 6 -

 6 Figure 6 -Dependence data network of the column/rack assembly

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