Cyril Bouvier

Laurent Imbert

Laurent Imbert Faster

Faster cofactorization with ECM using mixed representations

Faster cofactorization with ECM using mixed representations

  In this talk, we introduce a novel implementation of the elliptic curve factoring method specifically designed for medium-size integers such as those arising by billions in the cofactorization step of the Number Field Sieve. In this context, our algorithm requires fewer modular multiplications than any other publicly available implementation. The main ingredients are: the use of batches of primes, fast point tripling, optimal double-base decompositions and Lucas chains, and a good mix of Edwards and Montgomery representations.

	WRAC'H 2019
	Workshop on Randomness and Arithmetics for Cryptography on Hardware
	Invited Talk
	Laurent Imbert
	Joint work with Cyril Bouvier
	LIRMM, CNRS, Univ. Montpellier, France

Abstract.

Keywords: Elliptic curve method, cofactorization, double-base representation, twisted Edwards curve, Montgomery curve, CADO-NFS More info: http://eco.lirmm.net/double-base_ECM/


Giovanna Ilardi

Jean Vallès

Jean Vall Ès

EIGHT CUBES OF LINEAR FORMS IN P 6

come

Introduction

In a private communication Rosa Miró-Roig and Hoa Tran Quang informed us that we wrongly affirm in [4, Proposition 5.5] that the ideal I = (L 3 1 , . . . , L 3 8 ) fails the WLP in degree 3, as it was conjectured in [6, conjecture 6.6]. Indeed, computing explicitely the Hilbert functions of the Artinian ring A = C[x 0 , . . . , x 6 ]/(L 3 1 , . . . , L 3 8 ) and A/(L) where the L i 's and L are general linear forms, they observed that I has the WLP in any degree except in degree 5 meaning that the multiplication map ×L : A i → A i+1 has not maximal rank only when i = 5.

In this short note, we justify geometrically why I has the WLP in degree 3 and why it fails to have it in degree 5. For the first one, our argument is based on a famous result by Alexander and Hirshowitz [1] who give a list of general sets of double points in P n that do not impose independent conditions on hypersurfaces of fixed degree. For the second one, we show that the failure is due to the existence of a pencil of cubics in P 5 passing through 9 quadruple points in general position in P 5 .

According to Miró-Roig and Hoa Tran Quang degree 5 is the only degree where there is a failure of the WLP. Actually, if the failure can always be explained by a special geometric situation, having the WLP, since it is expected in general, is harder to prove. Indeed it would be necessary to have a theorem generalizing Alexander-Hirshowitz classification, that is a list of general set of multiple points, with multiplicity bigger than 2, that do not impose independent conditions on hypersurfaces which is far to be known. That's mainly why we do not propose a description in any degree.

WLP in degree three

Associated to I there is the so-called Syzygy sheaf K defined by: 0

----→ K ----→ O 8 P 6 (L 3 1 ,...,L 3 8 ) -------→ O P 6 (3) ----→ 0.
Let us recall that according to [START_REF] Brenner | Syzygy bundles on P 2 and the Weak Lefschetz Property[END_REF] we have

A d+i = H 1 (K(i))
. By [4, Lemma 5.2] there are no syzygies of degree ≤ 1, i.e. H 0 (K) = H 0 (K(1)) = 0. Then taking the cohomology of the exact sequence defining K we obtain the dimension of A 3 and A 4 . It occurs that the multiplication map ×L : A 3 → A 4 have maximal rank if and only if its cokernel has dimension exactly 78. According to [START_REF] Di Gennaro | Singular hypersurfaces characterizing the Lefschetz properties[END_REF] and in particular to [4, Theorem 5.1], this cokernel coincide with the vector space of quartic cones in P 6 with vertex at {L ∨ } and 8 double points at the {L ∨ i }'s. But having a double point for a cone means that the line joining the double point to the vertex is also double. Then the dimension of this vector space of cones is exactly the dimension of the space of quartics in P 5 with 8 double points which is expected to be 126 -6 × 8 = 126 -48 = 78. Let us point out that this expected dimension is actually the dimension since 8 double points in general position in P 5 impose independent conditions to the quartics because this is not one of the exceptional cases listed by Alexander and Hirshowitz in [1]. This proves that 1 Proposition. The ideal I = (L 3 1 , . . . , L 3 8 ) has the WLP in degree 3 where L 1 , . . . , L 8 are general linear forms on P 6 . Remark 1. Since the multiplication map ×L : A 3 → A 4 is injective we know, using [5, Proposition 2.1], that ×L : A i → A i+1 are also injective for i = 0, 1, 2.

Remark 2. The multiplication map ×L : A 4 → A 5 is also injective according to Rosa Miró-Roig and Hoa Tran Quang. If we want to apply the same technic than before we have to compute the dimension of the space of quintics in P 5 with 8 triple points in general position. But we don't know if 8 triple points in general position impose independent conditions on quintics and cannot conclude that the expected dimension is the true dimension.

Failure of WLP in degree five

We give now a geometric argument explaining why the map ×L : A 5 → A 6 is not injective. Let us begin by computing the dimension of A 5 and A 6 .

Shifting by 2 the exact sequence defining K and computing the cohomology, we observe first that, according to [4, Lemma 5.2], H 0 (K( 2)) = 0. This implies that dim(A 5 ) = h 1 (K(2)) = 238. Let us compute now dim(A 6 ). Shifting by 3 the exact sequence defining K and computing the cohomology, one finds:

0 ----→ H 0 (K(3)) ----→ C 672 ----→ C 924 ----→ A 6 ----→ 0.
The vector space H 0 (K(3)) of syzygies of degree 3 is not empty since it contains the Koszul relations, say L 3 j .L 3 i + (-L 3 i ).L 3 j = 0. These relations are independent which gives s := h 0 (K(3)) ≥ 8 2 = 28. Then dimA 6 = 924 -672 + s = 280 + (s -28) 1 .

Consequently the map ×L : A 5 → A 6 is injective if and only if the cokernel has dimension 42 + (s -28). According to [START_REF] Di Gennaro | Singular hypersurfaces characterizing the Lefschetz properties[END_REF]Theorem 5.1] and [START_REF] Di | Laplace equations, Lefschetz properties and line arrangements[END_REF]Theorem 13] the dimension of this cokernel is the sum of the dimension of the space of syzygies, that is s, and the dimension of the space of sextics cones in P 6 with a vertex at {L ∨ } and with 8 quadruple points at the {L ∨ i }'s. As we wrote before in section (2) these sextics correspond to sextics in P 5 with 8 quadruple points in general position. Its expected dimension is 14 which added to the s syzygies would give s + 14 = 42 + (s -28).

But a special situation occurs: through 9 double points in general position in P 5 there is a pencil of cubics, let say (C 1 , C 2 ). There are also 8 cubics through our 8 points among these 9, let say (C 1 , C 2 , . . . , C 8 ) since of course C 1 and C 2 belong to this linear system. Now the vector space

(C 2 1 , C 1 C 2 , . . . , C 1 C 8 , C 2 2 , C 2 C 3 , . . . , C 2 C 8 )
has dimension 15 and it consists in sextics with 8 quadruple points. This proves that the cokernel of A 5 → A 6 has dimension at least s + 15. This proves that the ideal I fails the WLP in degree 5.

Remark 3. We observe that 9 quadruple points in general position does not impose independent conditions on sextics of P 5 . Indeed if we compute directly there is no such sextics. But of course the existence of a pencil of cubics passing through these 9 double points give sextics with 9 quadruple points which proves that the dimension is strictly bigger than the expected one.


THEOREM WITH L r ESTIMATES.. 2019. Eric Amar An Andreotti-Grauert

Eric Amar

An Andreotti-Grauert theorem with L r estimates

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

1 Introduction.

Let ω be a ∂ closed (p, q) form in C n with compact support K := Supp ω and such that ω ∈ L r (C n ), the Lebesgue space in C n . Setting K in a ball B := B(0, R) with R big enough, we know, by a theorem of Ovrelid [Ovrelid, 1971], that we have a (p, q -1) form u ∈ L r (B) such that ∂u = ω. On the other hand we also know, at least when q < n, that there is a current v with compact support such that ∂v = ω, by a theorem of Andreotti-Grauert [START_REF] Andreotti | Théorèmes de finitude pour la cohomologie des espaces complexes[END_REF]. So a natural question is: may we have a solution u of ∂u = ω with compact support and in L r (C n ) ?

There is a work by H. Skoda [Skoda, 1976] who proved such a result. Let Ω be a strictly pseudo-convex bounded domain in C n with smooth boundary then in [Skoda, 1976, Corollaire p. 295], H. Skoda proved that if f is a (p, q)form with measure coefficients, q < n, ∂ closed and with compact support in Ω, then there is a solution U to the equation ∂U = f such that U L r (Ω) ≤ C(Ω, r) f 1 , for any r such that 1 < r < 2n+2 2n-1 and U has zero boundary values in the sense of Stokes formula. This means that essentially U has compact support and, because Ω is bounded, f 1 f L r (Ω) . So he got the answer for Ω strictly pseudo-convex and 1 < r < 2n+2 2n-1 .

We answered this question by the affirmative for any r ∈ [1, ∞] in a joint work with S. Mongodi [START_REF] Amar | On L r hypoellipticity of solutions with compact support of the Cauchy-Riemann equation[END_REF]] linearly by the "method of coronas". This method asks for extra L r conditions on derivatives of coefficients of ω, when q < n; we shall denote the set of ω verifying these conditions W r q (Ω), as in [START_REF] Amar | On L r hypoellipticity of solutions with compact support of the Cauchy-Riemann equation[END_REF].

1

The aim of this work is to extend this result to Stein manifolds and get rid of the extra L r conditions W r q (Ω). For it we use a completely different approach inspired by the Serre duality [Serre, 1955]. Because Hahn Banach theorem is used, this method is no longer constructive as in [START_REF] Amar | On L r hypoellipticity of solutions with compact support of the Cauchy-Riemann equation[END_REF].

The basic notion we shall use here is the following. Definition 1.1. Let X be a complex manifold equipped with a Borel σ-finite measure dm and Ω a domain in X; let r ∈ [1, ∞], we shall say that Ω is r regular if for any p, q ∈ {0, ..., n}, q ≥ 1, there is a constant C = C p,q (Ω) such that for any (p, q) form ω, ∂ closed in Ω and in L r (Ω, dm) there is a (p, q -1) form u ∈ L r (Ω, dm) such that ∂u = ω and u L r (Ω) ≤ C ω L r (Ω) .

We shall say that Ω is weakly r regular if for any compact set

K ⋐ Ω there are 3 open sets Ω 1 , Ω 2 , Ω 3 such that K ⋐ Ω 3 ⊂ Ω 2 ⊂ Ω 1 ⊂ Ω 0 := Ω and 3 constants C 1 , C 2 , C 3 such that: ∀j = 0, 1, 2, ∀p, q ∈ {0, ..., n}, q ≥ 1, ∀ω ∈ L r p,q (Ω j , dm), ∂ω = 0, ∃u ∈ L r p,q-1 (Ω j+1 , dm), ∂u = ω and u L r (Ωj+1) ≤ C j+1 ω L r (Ωj ) .
I.e. we have a 3 steps chain of resolution.

Of course the r regularity implies the weak r regularity, just taking

Ω 1 = Ω 2 = Ω 3 = Ω.
Examples of 2 regular domains are the bounded pseudo-convex domains by Hörmander [Hörmander, 1994].

Examples of r regular domains in C n are the bounded strictly pseudo-convex (s.p.c.) domains with smooth boundary by Ovrelid [Ovrelid, 1971]; the polydiscs in C n by Charpentier [Charpentier, 1980], finite transverse intersections of strictly pseudo-convex bounded domains in C n by Menini [Menini, 1997]. A generalisation of the results by Menini was done in the nice work of [START_REF] Ma | L p estimates for Cauchy-Riemann operator on qconvex intersections in C n[END_REF]: they treated also the case of intersection of q-convex sets.

Examples of r regular domains in a Stein manifold are the strictly pseudo-convex domains with smooth boundary [Amar, 2016]. (See the previous work for (0, 1) forms by N. Kerzman [Kerzman, 1971] and for all (p, q) forms by J-P. Demailly and C. Laurent [Demailly and Laurent-Thiébaut, 1987, Remarque 4, page 596], but here the manifold has to be equipped with a metric with null curvature. See also [Amar, 2017] for the case of intersection of q-convex sets in a Stein manifold).

Let X be a Stein manifold and Ω a domain in X, i.e. an open connected set in X. Let H p (Ω) be the set of all (p, 0) ∂ closed forms in Ω. If p = 0, H 0 (Ω) = H(Ω) is the set of holomorphic functions in Ω. If p > 0, we have, in a chart (ϕ, U ), h ∈ H p (Ω) ⇒ h(z) = |J|=p a J (z)dz J , where dz J := dz j1 ∧ • • • ∧ dz jp and the functions a J (z) are holomorphic in ϕ(U ) ⊂ C n .

We shall denote L r,c p,q (Ω) the set of (p, q)-forms in L r (Ω) with compact support in Ω. We also use the notation r ′ for the conjugate exponent of r, i.e. 1 r

+ 1 r ′ = 1.
Our main theorem is:

Theorem 1.2.
Let Ω be a weakly r ′ regular domain in a Stein manifold X. Then there is a C > 0 such that for any (p, q) form ω in L r,c (Ω), r > 1 with:

• if 1 ≤ q < n, ∂ω = 0; • if q = n, ∀V ⊂ Ω, Supp ω ⊂ V, ω ⊥ H n-p (V ); there is a (p, q -1) form u in L r,c (Ω) such that ∂u = ω as distributions and u L r (Ω) ≤ C ω L r (Ω) .
The notion of r regularity gives a good control of the support: if the support of the data ω is contained in Ω\C where Ω is a weakly r ′ regular domain and C is a weakly r regular domain, then the support of the solution u is contained in Ω\C ′ , where C ′ is any relatively compact domain in C, provided that q ≥ 2. One may observe that Ω\C is not Stein in general even if Ω is.

There is also a result of this kind for q = 1, see section 3.3.

In particular the support of the solution u is contained in the intersection of all the weakly r ′ regular domains containing the support of ω.

The idea is to solve ∂u = ω in a space L r (Ω) with a "big weight η outside" of the support of ω; this way we shall have a "small solution u outside" of the support of ω. Then, using a sequence of such weights going to infinity "outside" of the support of ω, we shall have a u zero "outside of the support" of ω.

Comparing to my previous work [Amar, 2012] the results here are improved and the proofs are much simpler by a systematic use of the Hodge * operator.

I am indebted to G. Tomassini who started my interest in this subject on precisely this kind of questions and also to S. Mongodi for a lot of discussions during the preparation of our joint paper [START_REF] Amar | On L r hypoellipticity of solutions with compact support of the Cauchy-Riemann equation[END_REF].

I want to thank C. Laurent for many instructive discussions on this subject. Finally I also thank the referee for his/her careful reading of the manuscript and the incisive questions he/she asked.

Duality.

We shall study a duality between currents inspired by the Serre duality [Serre, 1955]. Let X be a complex manifold of dimension n. We proceed now exactly as in Hörmander [Hörmander, 1994, p. 119], by introducing a hermitian metric on differential forms locally equivalent to the usual one on any analytic coordinates system. We define the "Lebesgue measure" still as in Hörmander's book [Hörmander, 1994, Section 5.2]: associated to this metric there is a volume measure dm and we take it for the Lebesgue measure on X. Moreover, because X is a complex manifold, it is canonically oriented.

Weighted L r spaces.

Let Ω be a domain in X. We denote also dm the volume form on X. We shall take the following notation from the book by C. Voisin [Voisin, 2002].

To a (p, q)-form α on Ω we associate its Hodge * (np, nq)-form * α. This gives us a pointwise scalar product and a pointwise modulus:

(α, β)dm := α ∧ * β; |α| 2 dm := α ∧ * α, (2.1)
because α ∧ * β is a (n, n)-form hence is a function time the volume form dm.

We are also given a scalar product α, β on (p, q)-forms such that Ω |α| 2 dm < ∞ and the link between these notions is given by [Voisin, 2002, Lemme 5.8, p. 119]:

α, β = Ω α ∧ * β. (2.2)
We shall define now L r p,q (Ω) to be the set of (p, q)-forms α defined on Ω such that

α r L r p,q (Ω) := Ω |α(z)| r dm(z) < ∞,
where |α| is defined by (2.1).

Lemma 2.1. Let η > 0 be a weight. If u is a (p, q)-current defined on (np, nq)-forms α in L r ′ (Ω, η) and such that

∀α ∈ L r ′ (n-p,n-q) (Ω, η), | u, * α | ≤ C α L r ′ (Ω,η) , then u L r p,q (Ω,η 1-r ) ≤ C.
Proof. We use the classical trick: set α := η 1/r ′ α; ũ := 1 η 1/r ′ u then we have

u, * α = Ω u ∧ α = Ω ũ ∧ α = ũ, * α and α L r ′ (Ω) = α L r ′ (Ω,η) . We notice that α L r ′ (Ω) = * α L r ′ (Ω) because we have ( * α, * α)dm = * α ∧ * * α but * * α = (-1) (p+q)(2n-p-q) α,
by [Voisin, 2002, Lemma 5.5], hence, because ( * α, * α) is positive, ( * α, * α) = |α| 2 . By use of the duality L r p,q (Ω) -L r ′ n-p,n-q (Ω), done in Lemma 4.3, we get

ũ L r p,q (Ω) = sup α∈L r ′ n-p,n-q (Ω), α =0 | ũ, * α | α L r ′ (Ω) . But ũ r L r p,q (Ω) := Ω |u| r η -r r ′ dm = Ω |u| r η 1-r dm = u r L r (Ω,η 1-r ) .
So we get

u L r p,q (Ω,η 1-r ) = sup * α∈L r ′ p,q (Ω,η), α =0 | u, * α | α L r ′ (Ω,η)
.

The proof is complete.

It may seem strange that we have such an estimate when the dual of L r ′ (Ω, η) is L r (Ω, η), but the reason is, of course, that in the duality current-form there is no weights.

The point here is that when η is small, η 1-r is big for r > 1.

3 Solution of the ∂ equation with compact support.

3.1 r regular domains.

As we have seen, examples of r regular domains in Stein manifolds are the relatively compact s.p.c. domains with smooth boundary.

To prove that a Stein manifold Ω is weakly r regular we shall need the following lemma.

Lemma 3.1. Let Ω be a Stein manifold. Then it contains an exhaustive sequence of open relatively compact strictly pseudo-convexs sets {D k } k∈N with C ∞ smooth boundary.

Proof. For the case of Ω pseudo-convex in C n , the proof was already done explicitely in the proof of [Henkin and Leiterer, 1984, Theorem 2.8.1, p. 86].

By Theorem 5.1.6 of Hörmander [Hörmander, 1994] there exists a C ∞ strictly plurisubharmonic (s.p.s.h.) exhausting function ϕ for Ω. Take K ⋐ Ω such that dϕ = 0 on K. Because ϕ is s.p.s.h. then K = ∅. Then we use the [Henkin and Leiterer, 1984, Lemma 2.12.2, p. 93], to get: ∀ǫ > 0, ∃ρ ǫ s.p.s.h. C ∞ -function on Ω such that: (i) ϕρ ǫ together with its first and second derivatives is less than ǫ on Ω. (ii) The set Crit(ρ ǫ ) := {z ∈ Ω :: dρ ǫ (z) = 0} is discrete in Ω. (In a formula, the notation :: means "such that".) (iii) ρ ǫ = ϕ on K. As stated in Lemma 2.12.2 if ϕ ∈ C 2 then ρ ǫ ∈ C 2 , but in fact the proof of this Lemma 2.12.2 gives ρ ǫ = ϕ + χ j , where χ j is C ∞ . (see p. 93 in [START_REF] Henkin | Theory of functions on complex manifolds[END_REF]). Hence ρ ǫ has the same C k regularity than ϕ. Fix ǫ > 0, then the function ρ := ρ ǫ is also a s.p.s.h. exhausting function for Ω, because, from -ǫ ≤ ϕρ ǫ ≤ ǫ, we get that, for any α ∈ R, {z ∈ Ω :: ρ ǫ (z) < α} ⊂ {z ∈ Ω :: ϕ(z) < ǫ + α} and, because ϕ is an exhausting function, this set is relatively compact in Ω.

Because the set of critical points of ρ is discrete in Ω, the same way as in the proof of [Henkin and Leiterer, 1984, Theorem 2.8.1, p. 86 Let ω ∈ L r p,q (Ω), ∂ω = 0, by [Amar, 2016], we can solve ∂u = ω in D k with u ∈ L r p,q-1 (D k ) and

u L r (D k ) ≤ C k ω L r (D k ) ≤ C k ω L r (Ω) .
Hence if Γ is a compact set in Ω, there is a D k such that Γ ⋐ D k and we can take

Ω 1 = Ω 2 = Ω 3 = D k .
This proves the weak r regularity of Ω.

The main result.

Let X be a Stein manifold and Ω a domain in X.

In order to simplify notation, we set the pairing for α a (p, q)-form and β a (n-p, n-q)-form: ≪ α, β ≫:

= Ω α ∧ β.
With this notation we also have α, β =≪ α, * β ≫ .

Let Ω be a weakly r ′ regular domain in X. We set K := Supp ω ⋐ Ω and, by the definition of the r ′ weak regularity, we get 3 open sets such that

K ⋐ Ω 3 ⊂ Ω 2 ⊂ Ω 1 ⊂ Ω 0 = Ω with: ∀j = 0, 1, 2, ∀p, q ∈ {0, ..., n}, q ≥ 1, ∀α ∈ L r p,q (Ω j ), ∂α = 0, ∃ϕ ∈ L r p,q-1 (Ω j+1 ), ∂ϕ = α. Set the weight η = η ǫ := 1 Ω1 (z) + ǫ1 Ω\Ω1 (z) for a fixed ǫ > 0. Let ω ∈ L r,c
p,q (Ω). Suppose moreover that ω is such that ∂ω = 0 if 1 ≤ q < n and for any open

V ⋐ Ω, Supp ω ⋐ V we have ω ⊥ H n-p (V ) ⇐⇒ ∀h ∈ H n-p (V ), ≪ ω, h ≫= 0 if q = n.
We shall use the following lemma, with the previous notation: Lemma 3.3. Let E be the set of (np, nq + 1) forms α ∈ L r ′ (Ω, η), ∂ closed in Ω. Let us define L ω on E as follows:

L ω (α) := (-1) p+q-1 ≪ ϕ, ω ≫, where ϕ ∈ L r ′ (Ω 1 ) is such that ∂ϕ = α in Ω 1 . Then the form L ω is well defined and linear.

Proof. Because ǫ > 0 we have α ∈ L r ′ (Ω, η) ⇒ α ∈ L r ′ (Ω) and the weak r ′ regularity of Ω gives a ϕ ∈ L r ′ (Ω 1 ) with ∂ϕ = α in Ω 1 .
Let us see that L ω is well defined.

• Suppose first that q < n. In order for L ω to be well defined we need

∀ϕ, ψ ∈ L r ′ (n-p,n-q) (Ω 1 ), ∂ϕ = ∂ψ = α ⇒≪ ϕ, ω ≫=≪ ψ, ω ≫ . This is meaningful because ω ∈ L r,c (Ω), r > 1, Supp ω ⋐ Ω 1 . Then we have ∂(ϕ -ψ) = 0 in Ω 1 , hence, because Ω is weakly r ′ regular, we can solve ∂ in L r ′ (Ω 2 ): ∃γ ∈ L r ′ (n-p,n-q-1) (Ω 2 ) :: ∂γ = (ϕ -ψ). So ≪ ϕ -ψ, ω ≫=≪ ∂γ, ω ≫= (-1) p+q-1 ≪ γ, ∂ω ≫= 0 because ω is compactly supported in Ω 2 and ∂ closed.
Hence L ω is well defined in that case.

• Suppose now that q = n. For ϕ, ψ (np, 0) forms in Ω 1 , such that ∂ϕ = ∂ψ = α, we need to have ≪ ϕ, ω ≫=≪ ψ, ω ≫ . But then ∂(ϕψ) = 0, which means that h := ϕψ is a ∂ closed (np, 0) form, hence h ∈ H n-p (Ω 1 ). Taking V = Ω 1 in the hypothesis ω ⊥ H n-p (V ), we get ≪ h, ω ≫= 0, and L ω is also well defined in that case.

It remains to see that L ω is linear.

• Suppose first that q < n. Let α = α 1 + α 2 , with α j ∈ L r ′ (Ω, η), ∂α j = 0, j = 1, 2; we have α = ∂ϕ, α 1 = ∂ϕ 1 and α 2 = ∂ϕ 2 , with ϕ, ϕ 1 , ϕ 2 in L r ′ (Ω 1 ) so, because ∂(ϕϕ 1ϕ 2 ) = 0, we have

ϕ = ϕ 1 + ϕ 2 + ∂ψ, with ψ in L r ′ (Ω 2 ), so L ω (α) = (-1) p+q-1 ≪ ϕ, ω ≫= (-1) p+q-1 ≪ ϕ 1 + ϕ 2 + ∂ψ, ω ≫= = L ω (α 1 ) + L ω (α 2 ) + (-1) p+q-1 ≪ ∂ψ, ω ≫, but again ≪ ∂ψ, ω ≫= 0, hence L ω (α) = L ω (α 1 ) + L ω (α 2 ).
The same for α = λα 1 .

• Suppose now that q = n. We have

L ω (α) := (-1) p+n-1 ≪ ϕ, ω ≫, where ϕ ∈ L r ′ (Ω 1 ) is such that ∂ϕ = α in Ω 1 . Let α = α 1 + α 2 , with α j ∈ L r ′ (Ω, η), ∂α j = 0, j = 1, 2; we have α = ∂ϕ, α 1 = ∂ϕ 1 and α 2 = ∂ϕ 2 , with ϕ, ϕ 1 , ϕ 2 in L r ′ (Ω 1 ) so, because ∂(ϕ -ϕ 1 -ϕ 2 ) = 0, we have ϕ -ϕ 1 -ϕ 2 is a (n -p, 0) ∂-closed form, hence: ϕ = ϕ 1 + ϕ 2 + h, with h ∈ H n-p (Ω 1 ). So L ω (α) = (-1) p+q-1 ≪ ϕ, ω ≫= (-1) p+q-1 ≪ ϕ 1 + ϕ 2 + h, ω ≫= = L ω (α 1 ) + L ω (α 2 ) + (-1) p+q-1 ≪ h, ω ≫ . Taking V = Ω 1 in the hypothesis ω ⊥ H n-p (V ), we get ≪ h, ω ≫= 0, hence L ω (α) = L ω (α 1 ) + L ω (α 2 ).
The same for α = λα 1 . The proof is complete.

Remark 3.4. If Ω is Stein, we can take the domain Ω 1 to be s.p.c. with C ∞ smooth boundary, hence also Stein. So because K := Supp ω ⊂ Ω 1 ⊂ Ω, the A(Ω 1 ) convex hull of K, KΩ1 is still in Ω 1 , and any holomorphic function in Ω 1 can be uniformly approximated on KΩ1 by holomorphic functions in Ω. Then for q = n instead of asking ω ⊥ H n-p (Ω 1 ) we need just ω ⊥ H n-p (Ω).

Theorem 3.5. Let Ω be a weakly r ′ regular domain and ω be a (p, q) form in L r,c (Ω), r > 1. Suppose that ω is such that:

• if 1 ≤ q < n, ∂ω = 0; • if q = n, ∀V ⊂ Ω, Supp ω ⊂ V, ω ⊥ H n-p (V )
. Then there is a C > 0 and a (p, q -1) form u in L r,c (Ω) such that ∂u = ω as distributions and u L r (Ω) ≤ C ω L r (Ω) .

Proof.

Because Ω is weakly r ′ regular there is a

Ω 1 ⊂ Ω, Ω 1 ⊃ Supp ω such that ∀α ∈ L r ′ (Ω), ∂α = 0, ∃ϕ ∈ L r ′ (Ω 1 ) :: ∂ϕ = α, ϕ L r ′ (Ω1) ≤ C 1 α L r ′ (Ω) .
There is a Ω 2 such that Supp ω ⋐ Ω 2 ⊂ Ω 1 ⊂ Ω with the same properties as Ω 1 .

Let us consider the weight η = η ǫ := 1 Ω1 (z) + ǫ1 Ω\Ω1 (z) for a fixed ǫ > 0 and the form L ω defined in Lemma 3.3. By Lemma 3.3 we have that L ω is a linear form on (np, nq + 1)-forms α ∈ L r ′ (Ω, η), ∂ closed in Ω.

If α is a (np, nq + 1)-form in L r ′ (Ω, η), then α is in L r ′ (Ω) because ǫ > 0. The weak r ′ regularity of Ω gives that there is a ϕ ∈ L r ′ (Ω 1 ) :: ∂ϕ = α which can be used to define L ω (α).

We have also that α ∈ L r ′ (Ω 1 ), ∂α = 0 in Ω 1 , hence, still with the weak r ′ regularity of Ω, we have

∃ψ ∈ L r ′ (Ω 2 ) :: ∂ψ = α, ψ L r ′ (Ω2) ≤ C 2 α L r ′ (Ω1) .
• For q < n, we have ∂(ϕψ) = αα = 0 on Ω 2 and, by the weak r ′ regularity of Ω, there is a

Ω 3 ⊂ Ω 2 , such that Supp ω ⊂ Ω 3 ⊂ Ω 2 , and a γ ∈ L r ′ (Ω 3 ), ∂γ = ϕ -ψ in Ω 3 . So we get ≪ ϕ -ψ, ω ≫=≪ ∂γ, ω ≫= (-1) p+q-1 ≪ γ, ∂ω ≫= 0, this is meaningful because Supp ω ⊂ Ω 3 . Hence L ω (α) =≪ ϕ, ω ≫=≪ ψ, ω ≫ . • For q = n, we still have ∂(ϕ -ψ) = α -α = 0 on Ω 2 , hence ϕ -ψ ∈ H p (Ω 2 ); this time we choose V = Ω 2 and the assumption gives ≪ ϕ -ψ, ω ≫= 0 hence again L ω (α) =≪ ϕ, ω ≫=≪ ψ, ω ≫ .
In any cases, by Hölder inequalities done in Lemma 4.1,

|L ω (α)| ≤ ω L r (Ω1) ψ L r ′ (Ω2) ≤ ω L r (Ω) ψ L r ′ (Ω2) .
But, by the weak r ′ regularity of Ω, there is a constant C 2 such that

ψ L r ′ (Ω2) ≤ C 2 α L r ′ (Ω1) .
Of course we have

α L r ′ (Ω1) ≤ α L r ′ (Ω, η) because η = 1 on Ω 1 , hence |L ω (α)| ≤ C 2 ω L r (Ω) α L r ′ (Ω, η) .
So we have that the norm of L ω is bounded on the subspace of ∂ closed forms in L r ′ (Ω, η) by C ω L r (Ω) which is independent of ǫ.

We apply the Hahn-Banach theorem to extend L ω with the same norm to all (n-p, n-q +1) forms in L r ′ (Ω, η). As in the Serre Duality Theorem [Serre, 1955, p. 20], this is one of the major ingredients in the proof.

This means, by the definition of currents, that there is a (p, q -1) current u which represents the extended form L ω : L ω (α) =≪ α, u ≫ . So if α := ∂ϕ with ϕ ∈ C ∞ c (Ω), we get L(α) =≪ α, u ≫=≪ ∂ϕ, u ≫= (-1) p+q-1 ≪ ϕ, ω ≫ hence ∂u = ω as distributions because ϕ is compactly supported. And we have:

sup α∈L r ′ (Ω,η), α =1 |≪ α, u ≫| ≤ C ω L r (Ω) .
By lemma 2.1 with the weight η, this implies

u L r (Ω,η 1-r ) ≤ C ω L r (Ω) because |≪ α, u ≫| = | α, * u | and, as already seen, u L r (Ω,η 1-r ) = * u L r (Ω,η 1-r ) = * u L r (Ω,η 1-r ) .
In particular u L r (Ω) ≤ C ω L r (Ω) because with ǫ < 1 and r > 1, we have η 1-r ≥ 1.

Now for ǫ > 0 with η ǫ (z) := 1 Ω1 (z) + ǫ1 Ω\Ω1 (z), let u ǫ ∈ L r (Ω, η 1-r ǫ
) be the previous solution, then

u ǫ r L r (Ω,η 1-r ǫ ) ≤ Ω |u ǫ | r η 1-r dm ≤ C r ω r L r (Ω) .
Replacing η by its value we get

Ω1 |u ǫ | r dm + Ω\Ω1 |u ǫ | r ǫ 1-r dm ≤ C r ω r L r (Ω) ⇒ ⇒ Ω\Ω1 |u ǫ | r ǫ 1-r dm ≤ C r ω r L r (Ω) hence Ω\Ω1 |u ǫ | r dm ≤ C r ǫ r-1 ω r L r (Ω) .
Because C and the norm of ω are independent of ǫ, we have that u ǫ L r (Ω) is uniformly bounded and r > 1 implies that L r p,q-1 (Ω) is a dual by Lemma 4.3, hence there is a sub-sequence {u ǫ k } k∈N of {u ǫ } which converges weakly, when ǫ k → 0, to a (p, q -1) form u in L r p,q-1 (Ω), still with u L r p,q-1 (Ω) ≤ C ω L r p,q (Ω) . Let us write u k := u ǫ k . To see that this form u is 0 a.e. on Ω\Ω 1 let us write the weak convergence:

∀α ∈ L r ′ p,q-1 (Ω), u k , α = Ω u k ∧ * α → u, α = Ω u ∧ * α.
As usual take α := u |u| 1 E where E := {|u| > 0} ∩ (Ω\Ω 1 ) then we get

Ω u ∧ * α = E |u| dm = lim k→∞ Ω u k ∧ * α = lim k→∞ E u k ∧ * u |u| .
Now we have, by Hölder inequalities:

E u k ∧ * u |u| ≤ u k L r (E) 1 E L r ′ (E) . But u k r L r (E) ≤ Ω\Ω1 |u k | r dm ≤ (ǫ k ) r-1 C ω L r (Ω) → 0, k → ∞ and 1 E L r ′ (E) = (m(E)) 1/r ′ . Hence E |u| dm = lim k→∞ E u k ∧ * u |u| ≤ ≤ lim k→∞ C r (m(E)) 1/r ′ (ǫ k ) r-1 ω r L r (Ω) = 0, so E |u| dm = 0 which implies m(E) = 0 because on E, |u| > 0.
Hence we get that the form u is 0 a.e. on Ω\Ω 1 .

So we proved

∀ϕ ∈ D n-p,n-q (Ω), (-1) p+q-1 ≪ ϕ, ω ≫=≪ ∂ϕ, u ǫ ≫→≪ ∂ϕ, u ≫ ⇒≪ ∂ϕ, u ≫= (-1) p+q-1 ≪ ϕ, ω ≫ hence ∂u = ω in the sense of distributions. The proof is complete.

Remark 3.6. As in remark 3.4 if Ω is Stein for q = n instead of asking ω ⊥ H p (Ω 2 ) we need just ω ⊥ H p (Ω).

Remark 3.7. The condition of orthogonality to H p (V ) in the case q = n is necessary: suppose there is a (p, n -1) current u such that ∂u = ω and u with compact support in an open set

V ⊂ Ω, then if h ∈ H p (V ), we have h ∈ H p (V ), ≪ ω, h ≫=≪ ∂u, h ≫= (-1) n+p ≪ u, ∂h ≫= 0,
because, u being compactly supported, there is no boundary term and

≪ ∂u, h ≫= (-1) n+p ≪ u, ∂h ≫ .
This kind of condition was already seen for extension of CR functions, see [Amar, 1991] and the references therein.

Finer control of the support.

Here we shall get a better control on the support of a solution.

which ends the proof of the lemma.

Lemma 4.2. Let α ∈ L r p,q (Ω) then α L r p,q (Ω) = sup β∈L r ′ p,q (Ω), β =0

| α, β | β L r ′ (Ω) .

Proof. We choose β := α |α| r-2 , then: On the other hand we have

β L r ′ (Ω) = α r/r ′ L r (Ω) = α r-1 L r (Ω) , so α L r (Ω) × β L r ′ (Ω) = α r L r (Ω) = α, β . Hence α L r (Ω) = | α, β | β L r ′ (Ω) .
A fortiori for any choice of β:

α L r (Ω) ≤ sup β∈L r ′ (Ω) | α, β | β L r ′ (Ω) .
To prove the other direction, we use the Hölder inequalities, Lemma 4.1:

∀β ∈ L r ′ p,q (Ω), | α, β | β L r ′ (Ω) ≤ α L r (Ω) .
The proof is complete. Now we are in position to state:

Lemma 4.3. The dual space of the Banach space L r p,q (Ω) is L r ′ n-p,n-q (Ω).

Proof. Suppose first that u ∈ L r ′ n-p,n-q (Ω). Then consider:

∀α ∈ L r p,q (Ω), L(α) :=

Ω α ∧ u = α, * u .
This is a linear form on L r p,q (Ω) and its norm, by definition, is

L = sup α∈L r (Ω) | α, * u | α L r (Ω)
.

By use of Lemma 4.2 we get L = * u L r ′ p,q (Ω) = u L r ′ n-p,n-q (Ω) . So we have L r p,q (Ω) * ⊃ L r ′ n-p,n-q (Ω) with the same norm.

Conversely take a continuous linear form L on L r p,q (Ω). We have, again by definition, that:

L = sup α∈L r (Ω)
|L(α)| α L r (Ω) .

Because D p,q (Ω) ⊂ L r p,q (Ω), L is a continuous linear form on D p,q (Ω), hence, by definition, L can be represented by a (np, nq)-current u. So we have: because D p,q (Ω) is dense in L r p,q (Ω). So we proved L r p,q (Ω) * ⊂ L r ′ n-p,n-q (Ω) with the same norm. The proof is complete.

  ], we can find a sequence c k ∈ R, c k → ∞, such that D k := {z ∈ Ω :: ρ(z) < c k } make an exhaustive sequence of open relatively compact sets in Ω, ∂ρ = 0 on ∂D k , hence D k is strictly pseudo-convex with C ∞ smooth boundary, and finally D k ր Ω. The proof is complete. Proposition 3.2. A Stein manifold Ω is weakly r regular. Proof. By Lemma 3.1 there is an exhaustive sequence of open relatively compact s.p.c. sets in Ω, {D k } k∈N with C ∞ smooth boundary.

  ∀α ∈ D p,q (Ω), L(α) := Ω α ∧ u = α, * u .Moreover we have, by Lemma 4.2,L = sup α∈Dp,q(Ω) | α, * ū | α L r (Ω) = * u L r ′ (Ω)

Theorem 3.8. Let Ω be a weakly r ′ regular domain in a Stein manifold X.

Suppose the (p, q) form ω is in L r,c (Ω, dm), ∂ω = 0, if q < n, and ω ⊥ H p (V ) for any V such that Supp ω ⊂ V, if q = n, with Supp ω ⊂ Ω\C, where C is a weakly r regular domain.

For any open relatively compact set U in C, there is a u ∈ L r,c (Ω, dm) such that ∂u = ω and with support in Ω\ Ū , provided that q ≥ 2.

Proof.

Let ω be a (p, q) form with compact support in Ω\C then there is a v ∈ L r p,q-1 (Ω), ∂v = ω, with compact support in Ω, by theorem 3.5 or, if Ω is a polydisc in C n and if ω ∈ W r q (Ω), by the theorem in [START_REF] Amar | On L r hypoellipticity of solutions with compact support of the Cauchy-Riemann equation[END_REF]. Because ω has compact support outside C we have ω = 0 in C; this means that ∂v = 0 in C. Because C is weakly r regular and q ≥ 2, we have

Let χ be a smooth function such that χ = 1 in U and χ = 0 near ∂C ′ ; then set u := v -∂(χh).

We have that u

hence ∂χ = 0 there. Finally ∂u = ∂v -∂2 (χh) = ω and we are done.

If Ω and C are, for instance, pseudo-convex in C n then Ω\C is no longer pseudo-convex in general, so this theorem improves actually the control of the support.

Remark 3.9. The correcting function h is given by kernels in the case of Stein domains, hence it is linear; if the primitive solution v is also linear in ω, then the solution u is linear too. This is the case in C n with the solution given in [START_REF] Amar | On L r hypoellipticity of solutions with compact support of the Cauchy-Riemann equation[END_REF]. This theorem cannot be true for q = 1 as shown by the following example: take a holomorphic function ϕ in the open unit ball B(0, 1) in C n such that it extends to no open ball of center 0 and radius > 1. For instance ϕ(z

. Moreover B(0, R) is strictly pseudo-convex hence r ′ regular, but there is no function u such that ∂u = ω and u zero near the origin because any solution u will be C.R. on ∂B(0, R) and by Hartog's phenomenon will extends holomorphically to B(0, R), hence cannot be identically null near 0.

Never the less in the case q = 1, we have: Theorem 3.10. Let Ω be a weakly r ′ regular domain in a Stein manifold X. Then for any (p, 1) form ω in L r,c (Ω), ∂ω = 0, with support in Ω 1 \C where Ω 1 is a weak r ′ regular domain in Ω and C is a domain such that C ⊂ Ω and C\Ω 1 = ∅; there is a u ∈ L r,c (Ω) such that ∂u = ω and with support in Ω\C.

Proof.

There is u ∈ L r p,0 (Ω 1 ) such that ∂u = ω with compact support in Ω 1 , by theorem 3.5.

We proved in [START_REF] Amar | On L r hypoellipticity of solutions with compact support of the Cauchy-Riemann equation[END_REF]:

Theorem 3.12. Let f ∈ O( Dn ) be a holomorphic function in a neighborhood of the closed unit polydisc in C n and set Z := f -1 (0). Then for any (0, q) form ω in L r (D n \Z) ∩ W r q (Ω), ∂ω = 0, with compact support in D n \Z, for any k ∈ N, we can find a (0, q -1)-form β ∈ L r,c (D n ) such that ∂(f k β) = ω. Equivalently we can find a (0, q -1)-form η = f k β such that η ∈ L r,c (D n ), η is 0 on Z up to order k and ∂η = ω.

And by Remark 6.3 of this paper, the solutions are given by a bounded linear operator.

The following corollary will generalise strongly this result but at the price that we have not the linearity, nor even the constructivity of the solution.

Corollary 3.13. Let Ω be a Stein manifold. Let f be a holomorphic function in Ω and set Z := f -1 (0). Then for any (p, q) form ω in L r,c (Ω\Z), ∂ω = 0, if 1 ≤ q < n, and ω ⊥ H p ( Ω\Z) if q = n, there is a (p, q -1) form u ∈ L r (Ω\Z) such that ∂u = ω and u has its support still in Ω\Z.

Proof.

We first show that Ω\Z is Stein. Because f = 0 in Ω\Z we have that ϕ

Because Ω is Stein we have, by Theorem 5.1.6 of Hörmander [Hörmander, 1994], a strictly plurisubharmonic exhausting function ρ in C ∞ (Ω). Now the function γ := ϕ + ρ is still strictly plurisubharmonic and C ∞ in Ω\Z. Now we shall prove:

So we can apply [Hörmander, 1994, Theorem 5.2.10, p. 127] to get that Ω\Z is a Stein manifold. Now we are in position to apply Theorem 3.5. Let ω be a (p, q) form in L r,c (Ω\Z), ∂ω = 0, if 1 ≤ q < n, and ω ⊥ H p ( Ω\Z) if q = n, Theorem 3.5 gives a (p, q -1) form u ∈ L r (Ω\Z) such that ∂u = ω and u has its compact support in Ω\Z. The proof is complete.

Remark 3.14. This leaves open the question to have a linear (or a constructive) solution to this problem even in the case of the polydisc.

Appendix

Here we shall prove certainly known results on the duality L r -L r ′ for (p, q)-forms in a complex manifold X. Because I was unable to find precise references for them, I prove them here.

Recall we have a pointwise scalar product and a pointwise modulus for (p, q)-forms in X: This gives Hölder inequalities for (p, q)-forms:

Lemma 4.1. (Hölder inequalities) Let α ∈ L r p,q (Ω) and β ∈ L r ′ p,q (Ω). We have | α, β | ≤ α L r (Ω) β L r ′ (Ω) .


Fernando Bastos
email: fjbastos@fe.up.pt

Elsa Caetano
email: ecaetano@fe.up.pt

Álvaro Cunha
email: acunha@fe.up.pt

Xavier Cespedes

Olivier Flamand
email: olivier.flamand@cstb.fr

IDENTIFICATION OF A SLIGHT VORTEX INDUCED VIBRATION EPISODE AT THE GRANDE RAVINE VIADUCT

Keywords: Full-scale monitoring, bridge aerodynamics, vortex-induced vibration

The work here presented results from the study of the monitored data collected during a 2-year period at the viaduct of Grande Ravine at the Reunion Island. Based on the records collected by a set of anemometers, pressure sensors and accelerometers, it was possible to establish a wind model for the Grande Ravine implantation site, to estimate force coefficients for the deck, and to identify modal properties (vibration frequencies, damping and modal configurations). From the analysis of the structural response, it was possible specifically to identify a case of very slight vortex induced vibration, which is described in detail in this paper.

THE GRANDE RAVINE VIADUCT: STRUCTURE AND MONITORING SYSTEM

The permanent concerns of the scientific and engineering community with the structural health and safety of bridges, due to the increase of audacious design solutions involving longer and longer spans and very slender structural systems, have led to the instrumentation of a significant number of bridges with continuous monitoring systems. Many of these structures are located in zones with high susceptibility to the occurrence of extreme events, such as cyclones, demanding an adequate characterisation of loads and an accurate assessment of the corresponding response. The Grande Ravine viaduct is a slender girder bridge crossing a volcanic breach of 320m width and 170m depth, at the Reunion Island. The steel orthotropic deck is 288m long and 22.7m width and is supported by two concrete struts, inclined at 20° with horizontal, fixed in a concrete abutment funded in solid rock (Fig. 1).

The viaduct is located in an area frequently affected by tropical cyclones and, due to a particular concern regarding the understanding of the corresponding wind response, the Designer SETEC tpi decided to install a continuous monitoring system. This system includes 5 anemometers (Fig 1b), 14 pressure sensors distributed along the contour of two cross sections (Fig. 1c and1d), 6 temperature sensors, and 6 accelerometers, arranged in 3 pairs in transversal and vertical direction of 3 cross sections (Fig. 1b). Collaboration between the Designer, ViBest/FEUP and CSTB was established in order to process and analyse the recorded data, aiming at validating the wind studies carried out during the design stage.

Structural Response: Identification of modal parameters based on full-scale measurements

The signals synchronously measured by the different sensors were sampled at 40Hz and stored in a computer located at the bridge site. The raw data files were then downloaded via FTP and stored in a computer at FEUP (Porto). Subsequently, these files were subjected to a pre-processing treatment including the removal of spikes and possible offsets introduced by power supply or the measurement chain, and a decimation to 20Hz, followed by a packaging into 1-hour data files for further processing.

The application of an automated version of p-LSCF identification algorithm [START_REF] Magalhães | Operational Modal Analysis For Testing And Monitoring of Bridges And Special Structures[END_REF] to the acceleration data collected at the Grande Ravine viaduct during a 2-year period allowed to obtain a very reliable characterisation of the time evolution of the structure modal parameters. Fig. 2 shows the evolution of the bridge first 11 natural frequencies during the period under analysis. Table 1 presents the mean, minimum and maximum values of the first 11 frequencies and corresponding identified damping ratios. The modal configurations of the first two vibration modes are presented in Fig. 3. A more detailed study of the time variation of modal properties can be found in [START_REF] Bastos | Aerodynamic Behaviour of Long Span Structure[END_REF]. Table 1 -Results of the p-LSCD method for the monitoring period under analysis

IDENTIFICATION OF ONE EPISODE OF VORTEX INDUCED VIBRATION

This section focuses on the study of vortex-induced vibrations (VIV) with emphasis to phenomena of this nature captured during the continuous monitoring period of the viaduct of Grande Ravine. This type of phenomena occurs as consequence of harmonic excitation of the first vertical (or torsional) vibration mode of the deck due to the release of vortices in the wake of the section at a frequency close to a fundamental frequency of the structure. Given the geometry of the section of the deck, and due to the inclined walls and to the proximity of frequencies of the first horizontal and vertical modes, vortex induced vibrations can also be observed associated with the horizontal mode, although with a much more discrete response.

Generally, VIV episodes are of very short duration, stemming from the combination of a number of factors, which do not easily coexist. Apart from this fact, the observation of VIV is rare, since structures are studied in order to ensure the smallest possible impact on their users daily life, ie, to prevent the manifestation of such events with amplitudes perceptible to the users or that incur into structural fatigue. The most important factors for the occurrence of VIV phenomena are the range of potentially excitable wind speeds (directly proportional to the shedding frequency) and the ideal conditions of atmospheric turbulence in order to maintain the eddy flow structures.

VIV episodes are governed by the frequencies with which the vortices are released in the wake of the flow [START_REF] Cremona | Comportement au vent des ponts[END_REF]. These frequencies are related with the Strouhal number, a dimensionless quantity which varies from body to body depending on the velocity and on the geometry of the body immersed in the flow [START_REF] Cremona | Comportement au vent des ponts[END_REF][START_REF] Simiu | Wind Effects on Structures: Fundamental and Applications to Design[END_REF].

The occurrence of VIV is widely described in the literature [START_REF] Cremona | Comportement au vent des ponts[END_REF][START_REF] Simiu | Wind Effects on Structures: Fundamental and Applications to Design[END_REF][START_REF] Fujino | Wind Resistant Design of Bridge in Japan: Developements and practices[END_REF], standing out some famous cases of bridges, namely the Great Belt Bridge or even the well-known case of Tacoma Bridge where vibrations were initiated by of vortex shedding at low wind speed and rapidly converged to a flutter phenomenon.

In this context, the investigation of such phenomena requires the definition of a set of strategies that lead to the detection of events with characteristics compatible with those required for the occurrence of vortex-induced vibrations. In order to identify these phenomena, a first criterion has been established based on the observation of the correlation between the recorded acceleration amplitudes (through the rms value) and the mean wind velocity. Accordingly, the definition of the data sampling period for which the statistical properties are calculated is of extreme importance: it should not be set too short, so as not to suffer the effects of traffic influence on the viaduct and, at the same time, it should not be too long to be influenced by the turbulence of the global flow. Thereby the interval of 30 seconds was defined as the experimental sampling period to observe this phenomenon. On the other hand, based on the wind tunnel estimates obtained for the Strouhal number [START_REF] Bastos | Aerodynamic Behaviour of Long Span Structure[END_REF], knowing the range of possible velocities for the occurrence of this phenomenon, the events contained in the range of speed conducing to higher probability of occurrence of VIV phenomena were selected. Fig. 4 shows the variation of horizontal and vertical vibration amplitude (rms values) recorded by the accelerometers with the mean wind velocity based on averaging intervals of 30s.

The observation of this figure shows a cloud greatly influenced by traffic on the bridge. However, a set of points stand out from the overall, especially in the range of the highest mean wind velocity and with the same order of magnitude expected for the occurrence of such events.

Characterisation of VIV phenomena

The analysis of time series corresponding to the points marked in Fig. 4 allows to observe the building of a low amplitude harmonic vibration. This is evident from the observation of Fig. 5 displaying the horizontal and vertical acceleration time series at the viaduct mid-span.

The time interval analysed corresponds to a Sunday between 17:34:27 and 17:37:02, coinciding with a period of low traffic occurrence. The peaks observed in the spectral estimates of the horizontal and vertical acceleration signals correspond to the excitation of the 1st vertical vibration mode of the deck. Response in the 1st horizontal mode can also be observed in Fig. 7a), although with a slightly smaller amplitude than that associated with the 1st vertical vibration mode. The remaining vibration modes have a negligible contribution to the structural response.

It is also observed that during the period of analysis no traffic existed on the viaduct, and thereby the records of the acceleration have no response due to the vehicle passage and particularly the impulse due to the entrance of vehicles on the viaduct. These facts evidence the importance of characterising the wind synchronously with the presented events. Fig. 8 shows the time series of the wind direction, and the horizontal and vertical velocity components for the analysed time interval. The data shown in Fig. 8 evidence a very stable pattern, ie, quasi-constant horizontal velocity and direction. Table 2 summarises the statistical properties of the signals illustrated in Fig. 8. The velocity values shown in this Table are in the potentially excitable range of 15m/s to 19 m/s identified from wind tunnel tests in the context of the estimation of the Strouhal number [START_REF] Bastos | Aerodynamic Behaviour of Long Span Structure[END_REF]. The analysed time histories present lower turbulence intensity than that estimated in the context of characterisation of the atmospheric and aerodynamic properties of Grande Ravine viaduct site [START_REF] Bastos | Aerodynamic Behaviour of Long Span Structure[END_REF]. For very high wind conditions in which the pattern of atmospheric behaviour tends to homogeneity, turbulence intensity values may be very similar to the values observed for these time series.

It is also noted that during the observed period wind has ascending incidence and direction from the sea, almost normal to the viaduct. These characteristics are similar to those observed for the occurrence of higher intensity wind in the characterisation of local wind properties. The observation of Figs. 11 and12 shows that the signal recorded by the pressure sensor P3, positioned in the wake of the cross section, presents a strong harmonic component characterised by a frequency associated to the release of vortices in the wake of the section. The rate of vortex release is close to the frequency of the first vertical mode of the deck. In the case of the signal measured at the pressure sensor P10, positioned under incident wind, the reported harmonic component is almost imperceptible.

Still in the context of the analysis of the aerodynamic forces generated in the section contour, Fig. 13 shows the spectral estimates of the aerodynamic forces Fx and Fz derived from the integration of the pressures along the contour of the viaduct cross section [START_REF] Bastos | Aerodynamic Behaviour of Long Span Structure[END_REF]. The graph representing the spectral estimate of the lift force clearly shows a peak associated to vortex shedding. With respect to the spectral estimate of the drag force, a very tenuous peak can be observed, being this component of aerodynamic force essentially dominated by incident turbulence. This peak results from the pressure integration along the inclined walls of the cross section.

Estimation of the Strouhal number from full-scale measurements

At this moment, it becomes pertinent to estimate the Strouhal number (St) based on the frequency (n) estimated from the peak found in Fig. 13, the mean wind velocity (U) and the reference dimension B (width) of the deck, according to the equation (St = n.B/U ). Considering the values presented in Table 2, estimates of St range from 0.206 to 0.213. These estimates, summarised in Table 3, are very similar to those based on wind tunnel tests. Although extremely close, the differences can be justified by the different turbulence conditions between the prototype and the model. During the occurrence of VIV events, the increase of coherence between time series of pressure measured in contiguous points is expected. This occurs due to the synchronization of vortices released from the surface of the cross sections and structural natural frequency. This type of occurrence was described by several authors from tests conducted in wind tunnel [START_REF] Simiu | Wind Effects on Structures: Fundamental and Applications to Design[END_REF][START_REF] Dyrbye | Wind Loads on Structures[END_REF] and also reported in the context of the analysis of the wind tunnel test results [START_REF] Bastos | Aerodynamic Behaviour of Long Span Structure[END_REF]. In order to test this hypothesis for the full-scale approach, Fig. 14 shows the estimates of coherence functions between pressures measured in contiguous sensors P10-P14 (incident wind) and P3-P11 (wake flow) (see location in Fig. 1c and1d). 1.

The observation of this figure evidences a very tenuous increase of coherence in the case of the sensors positioned under direct influence of the wind (P10 and P14). In the case of sensors placed on the wake of flow (P3 and P11) notorious increase is observed.

FINAL REMARKS

Considering the results above presented for the overall pattern described in literature, it can be concluded that a vortex induced vibration event has occurred.

It is noted that, during the conception studies, it was observed that the Grande Ravine cross section, was susceptible to developing vortex shedding. This observation was first evidenced from the shape of the lift coefficient graph, presenting a negative slope, and subsequently confirmed by the configuration of the H1 aerodynamic derivative associated to the vertical damping component developed as consequence of aerodynamic forces on the deck, which presents a positive portion [START_REF] Bastos | Aerodynamic Behaviour of Long Span Structure[END_REF]. This observation was registered for low reduced velocities, with an intensity of velocities of the same order of those in which the vortex induced vibration was observed for the prototype.

The occurrence of vortex induced vibration requires the simultaneous presence of a number of factors, which probability of occurrence by themselves is very low. For this reason, the difficulty in finding the phenomena observed in this case study is a result of the decisions regarding the deck geometry taken during the design stage, in order to avoid the occurrence of vibratory phenomena with aerodynamic character. It should be noted that the constant high intensity traffic observed and the topographic condition are factors that can destabilise the possible mechanism of formation of these vortices, by increasing the turbulence near the walls of the viaduct cross section. In fact, it is very difficult to find the simultaneous condition of low turbulence intensity, low traffic on the viaduct, constant incidence and direction and, most important, wind intensity in the favourable range, in accordance with the Strouhal number, for the occurrence of harmonic vibrations due to vortex shedding synchronised with the natural frequencies of the structure.

Fig. 1 -

 1 Fig. 1-Scheme of instrumentation: (a) distribution of sensors along the viaduct profile; (b) pressure sensors in section 1, +1,7m from mid-span; (c) pressure sensors in section 2, +14,2m from mid-span.

Figure 2 -

 2 Figure 2 -Variation of the viaduct first 11 natural frequencies identified with the p-LSCF method in the 2-years monitoring period after grouping of modes.

Figure 3 -

 3 Figure 3 -Modal configurations of the first two vibration modes: a) 1st transversal and b) 1st vertical (in blue numerical, red ambient vibration test and green continuous monitoring).

Figure 4 -

 4 Figure 4 -Variation of rms values of (a) horizontal and (b) vertical acceleration with mean wind velocity for averaging intervals of 30 seconds.

Figure 5 -

 5 Figure 5 -Time-histories of (a) horizontal and (b) vertical acceleration during the period under analysis.

Fig. 6

 6 Fig. 6 shows a zoom of the signals presented in previous Fig. 5. This graph denotes a pattern of response purely dominated by a singular frequency, which is confirmed by observation of the spectral estimates of the acceleration signals shown in Fig. 7.

Figure 6 -

 6 Figure 6 -Zoom of time-history signals of (a) horizontal and (b) vertical acceleration of figure 7.57.

Figure 7 -

 7 Figure 7 -Spectral estimates of time-histories of (a) horizontal and (b) vertical acceleration.

Figure 8 -

 8 Figure 8 -Time-histories of wind (a) direction and velocity in (b) horizontal and (c) vertical directions during the period of 235 seconds.

Fig. 9

 9 Fig.9presents the spectral estimates of the velocity time series shown in Fig.8. These estimates have a similar configuration to those obtained in the detailed study of wind properties characterisation[START_REF] Bastos | Aerodynamic Behaviour of Long Span Structure[END_REF]. However, slightly higher values of turbulence scales are observed in the present records. Aiming at characterising a possible vibration episode induced by vortex shedding, the pressure signals recorded at the contour of the deck section during the observation period were analysed. Fig.10shows the temporal signals at the sensors P10 and P3, positioned under direct incident wind and in the wake of the cross section, respectively (see Fig.1 c) d)). A zoom of such records in a narrow time interval is shown in Fig.11. Spectral estimates of these records are presented in Fig.12.

Figure 9 -

 9 Figure 9 -Spectral estimates of (a) longitudinal and (b) vertical components of wind velocity during the period under analysis. In blue the von Karman theoretical spectral function.

Figure 10 -

 10 Figure 10 -Time-histories of pressure measured at sensors (a) P10 and (b) P3 during the time interval under analysis.

Figure 11 -Figure 12 -

 1112 Figure 11 -Time-histories of pressure measured at sensors (a) P10 and (b) P3 during the time interval under analysis

Figure 13 -

 13 Figure 13 -Spectral estimates of the estimated (a) drag and (b) lift forces during the period under analysis.

Figure 14 -

 14 Figure 14 -Coherence estimates between pressure sensor signals (a) P10-P14 (in yellow) and P3-P11 (in orange) recorded during the period under analysis. Theoretical decay exponential coherence function (in blue) of longitudinal wind componentbased on the parameters of Table1.

Table 2 -

 2 Statistical

parameters from wind time histories presented in Figure

8

, corresponding to a time interval of 215seconds.

Table 3 -

 3 Strouhal number: estimates based on wind tunnel and full-scale measurements. Paper presented at EACWE 2017, 7th European-African Conference on Wind Engineering, July 4-7, 2017, Liège, BEL

Acknowledgements

The present work has been developed in the context of a collaboration established between the University of Porto and the office SETEC.tpi, responsible for the design and installation of the monitoring system at the viaduct site. Additional cooperation has been established with CSTB. This work was financially supported by: Projects POCI-01-0145-FEDER-007457 -CONSTRUCT -Institute of R&D In Structures and Construction and PTDC/ECM-EST/2110/2014 -Dyncatline , funded by FEDER funds through COMPETE2020 -Programa Operacional Competitividade e Internacionalização (POCI)and by national funds through FCT -Fundação para a Ciência e a Tecnologia; The first author acknowledges the PhD scholarship (SFRH/BD/46126/2008).


Olivier Flamand
email: olivier.flamand@cstb.fr

Jean-Christophe Thomas Gem
email: jean-christophe.thomas@univ-nantes.fr

INFLATABLE WALLS FOR WIND TUNNELS

Introduction

Wind tunnels are usually designed for a prerequisite purpose. Wind tunnels for aeronautical studies aim at reproducing very low turbulence flow at high speed, what they attain through a settling chamber and funnelling effect with a contraction ratio of the section by 8 to more, BLWT are designed for making a boundary layer grow, what means a long flat floor upstream the model and industrial wind tunnel are designed with a continuous flat floor allowing forklifts and other handling vehicles to travel inside. It is therefore difficult to change the initial destination of a wind tunnel, pulling down walls and rebuilding is costly. In order to overcome this issue, an innovative device has been developed for the Jules Verne large climatic wind tunnel, with the aim to modulate its section, opening it to new markets. This device allows a fast and cheap adaptation of the size and shape of the wind tunnel section. Experience of the wind tunnel team with inflatable structures, initially related to modelling of mechanical behaviour of standalone structures under wind loads , was used for taking the challenge of the design, dimensioning and validation of new devices, to be used for changing the shape of the existing wind tunnel.

ORIGINAL WIND TUNNEL

The dynamic loop of the Jules Verne climatic wind tunnel was built more than 20 years ago as very stiff reinforced concrete structure [START_REF] Gandemer | La soufflerie climatique "Jules Verne[END_REF]. It lies inside a rectangle 85m long and 55m wide. Its floor is fully flat for letting forklifts circulate and is able to bear heavy loads (2 tons/square meter). It is made of 3 successive testing chambers 15x10m, 6x5m and 8x8m, each one being specialized in a particular kind of testing. The evolution of the shape from one testing chamber to the following one is done by converging or diverging walls and ceiling with a maximum slope of 7°, in order to avoid the flow detach from the walls.

The wind tunnel is powered with six fans, rotating at constant speed with blade pitch adjustable, each connected to an asynchronous electric motor of 520 kW. Because the blade pitch can be adjusted very quickly, it is possible to reproduce bursts. The maximum speed is 75 m/s. Considering today's needs, it is assumed that some tests performed in this wind tunnel could be achieved more efficiently in reduced size section. For instance the 15x10m section could be reduced in height to provide a 15m wide x 4m high section suitable for large aeroelastic bridge models. It is also planned to reduce the width of this 15x10m large section to a narrower section, 3m wide x 10m high, for testing lighting poles. This reflexion around section change is part of a larger project consisting in an improvement of the wind tunnel shape by changing one part of the circuit. These changes in the size and shape of the test section following the propellers have been checked by CFD modelling and by skin pressure and PIV measurements inside two reduced size models, at a scale 1/10 and 1/36. The shape of the converging walls was first improved in terms of efficiency, avoiding detached flow along the walls and reducing losses in the closed loop circuit, in the sake of reducing the power consumption of the facility.

Loads applied on these converging walls were measured in wind tunnel by pressure taps and calculated in the CFD modelling from the calculated pressures. Both approaches converged and gave a first evaluation of the loads these walls will have to withstand, in order to help in designing these structures. Among other technical solutions it was considered to design the convergent with inflatable structures.

WHY AN INFLATABLE STRUCTURE?

The wind tunnel team had some occasions in the recent past to test inflatable structures in the high speed section and was favourably impressed by the structural performances of such lightweight items. These tests, achieved in the scope of a PhD thesis [START_REF] Gandemer | La soufflerie climatique "Jules Verne[END_REF], were also an occasion to collaborate with team TRUST of the laboratory GeM, at Nantes University, which is specialized in the studies of inflatable structures, usually conducting analytical studies, finite element simulations and experiments on such inflatable structures.

Using the internal air pressure inside beams made of light coated fabrics for conferring a relevant stiffness to a building is not really a modern concept but is clearly underused. It is mostly applied for temporary structures, because it is easy to transport and the erection of the building is self propelled. No foundation is required, no crane or boom is needed, risk of accident to work is reduced at its minimum. An emblematic example of inflatable building is the mobile hangar that protects Solar Impulse airplane (figure 4). It should be noted that if foundations are not necessary, anchorages must be ensured in strong winds. One great advantages of these structure is that when the event ends, the temporary building is easily and quickly folded with a minimal impact on environment. Adapting the shape of the wind tunnel cross section to a narrower of a lower ceiling test chamber can be considered as a temporary evolution of the inner shape of the facility. The testing period usually last from one day to some weeks, after what the wind tunnel must come back to its initial configuration and the temporary wall dismounted and stored. One of the major benefit from inflatable structure it the ratio of its size in use to its size when stored. Because the fabrics is thin and easy to fold, when deflated an inflatable structure is very easy to pack away and can potentially be kept inside the wind tunnel, when solid temporary structures are the same volume when they are unemployed than when they are in use and require an outside solution for their storage.

Figure 5: experiments with inflated structures inside the Jules Verne wind tunnel, studying its collapsing with wind

They must be some drawbacks for such a structure, explaining why its use is not so common. The design team made a short list of them :

-Fragility of the fabrics. Is it possible a piece of hard material flies in wind and cut the membrane, making the temporary wall collapse during the experience? Of course this potential occurrence exists, but consequences would not be dramatic. Because the wind tunnel original walls are still in place, disappearance of one temporary wall will not change substantially the behaviour of the whole building. Consequences on the propellers themselves are of limited extend. The cost of fixing the tear is very limited. Repairing an inflatable structure can be achieved on site, quickly and easily. -Additional cost for the powering of the blower that is used for pressurizing the inflated structure?

Compared to the wind tunnel power, this is peanuts. The blower can be, at its maximum power, needing 1kW, when the wind tunnel needs 3 000 kW. -Risk of explosion and resulting blast? Actually, inside pressure of inflated structures is quite low and this is not an issue. -Will an inflated wall stay rigid enough for resisting to the pressures applied on it by the wind? This is one of the questions we aimed at answering. -Is there a risk that the inflated structure "beats" with wind and that this oscillation impacts the quality of the flow? This is also a question that this study was done for addressing.

Experience of the wind tunnel team with inflatable structures, initially related to modelling of mechanical behaviour of standalone structures under wind loads, was used for taking the challenge of the design, dimensioning and validation of new devices, to be used for changing the shape of the existing wind tunnel.

MODELLING OF AN INFLATABLE STRUCTURE

Construction of Inflatable Structures

Construction material:

Inflatable structures are generally made of flexible materials, which are relied together by different types of lying: gluing, sewing, or high frequency welding. In the most common applications of inflatables, these materials are of two types: films such as those used on inflatable cushions use in architecture, or textiles made of warp and weft yarns, interconnected by a coating which makes it possible to ensure the water or air tightness; They can be covered with different protective layers depending on the effects, properties or functions expected (UV protection, special roughness, etc.).

Stiffness of structure:

Textiles have almost zero flexural rigidity. The stiffness of the structures is ensured by a pre-tensioning of the fabrics, which may be done with of a mechanical device (by cables, for example, for tensioned membranes like the one used for roofs), or by a pneumatic device. This pretension is bi-axial in almost all cases. The state of pretension is paramount, since it ensures the integrity of the structure. One of the key criteria for the behaviour of textiles is the appearance of wrinkles. This corresponds to a local buckling, and is one of the key points of the dimensioning of these structures.

The family of pneumatic structures is divided into three groups: mono-membrane structures (only one membrane, simply supported by air), air-inflated structures, and hybrid structures. We are interested in this study to air-inflated structures. It is necessary to distinguish between the inflation step and the in-service step. The inflation step makes it possible to tension the fabric, and to obtain the initial geometry of the structure before deformation dues to other external loads. To understand the principle of operation of these specific structures, let's consider one of its simplest example: an inflatable beam (see figure 7). During the pre-tensioning step, the constituent material is subjected to longitudinal stresses and axial stresses. The figure presents the evolution of the axial stresses. The inflation inducts a uniform distribution of stresses inside the section of the beam. If the beam is submitted to bending loads, for example for a bi-applied beam, the flexural stresses (in black on figure) are superimposed on the pretension stresses (in red). This causes an increase in the axial stress in the lower part of this beam, and decrease in the stress in the upper part. Thus, by increasing the bending load, it is possible to cancel the axial stress, which causes the occurrence of a wrinkle. Note that the apparition of wrinkle is not crucial for the behaviour of the structure, since the beam can still withstand higher loads. But it can be considered as a design criterion often for aesthetic or psychological considerations. In the case where it is desired to use an inflatable structure to modify an air flow in a wind tunnel, it is necessary to ascertain the surface condition of the structure used, and this criterion of nonappearance of a wrinkle will also be important.

Designing inflatable structures

What are the key points of the design? This depends on the use of the structure. The criteria are generally based on quantification of the state of stresses in the material and on the displacements. There are maximal values of the state of constraints not to be exceeded with respect to the resistance limits of the fabrics and the connections, and the minimum values which avoids occurrence of a wrinkle, or a collapse. The second type of criterion concerns the deformations of the structure, and is based on its maximum displacements. It is required, for example, that the maximum displacement of a point of the structure is less than a predefined percentage of a dimension of the structure (width, height, length for example).

In some specific cases, attention will also be paid to the vibrations of the structures, to see if there are risks of resonance or not.

For the considered convergent, the proposed design criteria are:

-Positive tension state in the fabrics (no wrinkle appearance that can alter the flow) -Tension not exceeding the strength of the connections and fabrics -Minimal movements of the upper skin of the convergent -Avoid vibration modes.

Modeling

For inflatable structures, analytical or numerical studies can be carried out. There are analytical formulas for inflatable beams, but for more complex structures such as the convergent, or inflatable buildings, finite element software can be used. For this purpose, the structure is modelled by considering the material as a membrane, since the thickness of the fabric is of a very small dimension with respect to the other dimensions, and since there is no flexural rigidity of the material when it is not pre-stressed. The elements chosen for the numerical model are in our case membranes elements of type T6 or Q8. Since these elements have no flexural rigidity, it is necessary to perform a fictitious pretension at the first increment, which is cancelled after the first increment has passed. Concerning the materials, there is a great amount of work currently in the scientific community to define which model to use for the textile materials used in the membrane structures. The simplest model is the isotropic material (same properties in all directions), for which one will define a Young's modulus and a shear modulus. This model can be used for films for example, or for pre-sizing steps. To get closer to the physical reality of the material, an orthotropic model is preferred, considering the warp and weft directions, aligning them with the directions of orthotropy of the In this case, it is necessary to define two Young's moduli in the two orthotropy directions, a Poisson coefficient, and a shear modulus. This model is widely used for the design of textile structures, even if it does not fully account for the non-linearities observed during bi-axial tensile experiments in laboratories, nor for the viscoelasticity inherent of this type of materials. For the inflatable convergent, both models will be used.

The proposed calculation is a calculation in large transformations, using the total Lagrangian formulation. Thus, the second-order terms are taken into account in the deformation tensor, and the calculation is brought back to the initial configuration (non-pressurized). This means that the material coefficients used for the simulations are the coefficients corresponding to the natural state of the fabric, ie for the non-pressurized structure. The pressure is considered as an external load applied on the membrane. It is a follower force which means that the forces dues to the inside pressure remains always orthogonal to the membrane. External loading (here, wind loading) can be introduced in the form of point forces, or load densities. The boundary conditions are introduced at the nodes, and allow to model the connections of the structure with its environment.

Calculations and expected results

The simulations proposed are simulations showing the first results concerning the mesh of a structure similar to the convergent. We use for this a code developed in the GeM laboratory, which was originally planned for simpler structures than the one we are interested in here, and whose computational capabilities will soon be increased.

Figure 8 shows the mesh that has been used for this example: it is a structure of the same type that the one used in the wind tunnel, made of 6 partitions (against 10 for the convergent). For symmetry reasons, only half the structure is studied, by imposing that the middle partition stays in a vertical plane. Concerning the boundary conditions, it is considered that the nodes of the lower part remain in a plane, that the points of the leading edge remain on a line, and that a point of this line is fixed.

Resulting inflation:

Figure 9 presents the results of the inflation. This is an important step of the design because it allows to pre-define the final shape of the structure. In fact, this corresponds to a step usually used in the design of membrane structures: form-finding. What is the shape in the natural configurations (without any pressure inside) that will allow to get the shape which is required for the specific function for which it has been built?

EXPERIMENTAL SETUP

A model of an inflatable convergent was built at a scale 1/3 and tested in wind tunnel up to high wind speed. This model was equipped with pressure sensors glued on its surface. It was tested in a horizontal position and in a vertical arrangement in the high speed section of the wind tunnel which is 6m wide and 5m high. The model was 5m wide in order to fit vertically inside the test section and side spacers of 0.5m were added for the tests in horizontal position (figure 10). The actual shape of the convergent was checked in place at rest by manual measurement. Three rows of 20 pressure taps were located at the surface of the convergent model, all connected to two pressure scanners of 32 channels. The inside pressure was measured too and was varied by acting on the blower. Static pressure was measured by a Pitot tube located in the same section as the model.

DYNAMIC BEHAVIOUR OF THE CONVERGENT WITH WIND

The major unknown in the final convergent design is the dynamic behaviour of the inflated fabrics. Will it rattle or throb in the wind? The stability of the inflated structure at various wind speeds is a fluidstructure interaction problem which can be solved by a complex modelization that was not affordable in this project. This is the reason why measurements in wind tunnel have been used with FE modelling of the inflated cushion for a simplified checking of mode shapes frequencies.

The 1/3 model of the convergent was equipped with reflective tape targets in 15 locations, for the measurement of the structure vibration with wind by a pulsed infrared tracking system. This measurement system is composed with 6 CCD cameras looking at the structure from the ceiling and giving the displacement of each target, at a rate of 400 data per second, in 3D and with a precision better than 1mm. The time history of the displacement at each target location was analysed in the frequency domain and the PSD's show vibrations at 15 Hz and 18 Hz for an internal pressure of 650Pa, to be compared with calculation results.

Figure 1 :

 1 Figure 1: shape of the existing Jules Verne wind tunnel

Figure 3 :

 3 Figure 3: numerical and experimental approach of the convergent for a narrower section

Figure 4 :

 4 Figure 4: solar Impulse mobile hangar

Figure 6 :

 6 Figure 6: structure of a coated fabric

Figure 7 :

 7 Figure 7: Stresses in a cross-section

Figure 8 :

 8 Figure 8: mesh and boundary conditions

Figure 9 :

 9 Figure 9: results of the inflation

Figure 10 :

 10 Figure 10: Model of inflatable convergent in the high speed section of the wind tunnel, in horizontal and vertical location

Figure 11 :

 11 Figure 11: Shape of the inflatable convergent model at rest and location of pressure taps

Figure 13 :

 13 Figure 13: Pressure distribution along the convergent for various internal pressures and various wind speeds

Figure 14 :

 14 Figure 14: Location of displacement measurement targets on the fabrics.

Figure 15 :

 15 Figure 15: time history of the vertical displacement measured in three locations on the convergent model.

Figure 16 :

 16 Figure 16: PSD of vertical displacement at locations P5, P8 and P11

Paper presented at EACWE 2017, 7th European-African Conference on Wind Engineering, July 4-7, 2017, Liège, BEL


Sylvain Aguinaga
email: sylvain.aguinaga@cstb.fr

Marc Dufresne De Virel

Julien Guilhot

Guillaume Caniot
email: guillaume.caniot@meteodyn.fr

Dominique Dias
email: dominique.dias@urbalterre.fr

Cécile Nguyen

DESIGN OF THE CITADEL OF BONIFACIO URBAN AREA THROUGH EXPERIMENTAL AND NUMERICAL ASSESSMENT OF PEDESTRIAN COMFORT

INTRODUCTION

The city of Bonifacio, located at the extreme south end of Corsica Island, is said to be the windiest city of Europe, and has earned its nickname as "The daughter of The Wind". The Montlaur Citadel is a district of Bonifacio located on a small peninsula. It is an ancient military facility composed of various massive barracks and churches. The peninsula is characterized by its high cliffs (60 m above the sea level) which make the Citadel particularly exposed to high wind. The city of Bonifacio is eager to modernize the location with various services and recreational activities. Urbalterre, an urban architect agency, was in charge of designing the urban area of the future Montlaur Citadel. Aware of the pedestrian comfort issues which may arise from the high wind blowing over the peninsula, Urbalterre wished to design the buildings sizes and locations in the district in order to optimize the pedestrian comfort on the citadel.

To do so, Urbalterre used a mix of numerical and experimental approaches to assess pedestrian comfort using: --Wind tunnel measurements based on Particle Image Velocimetry for a precise assessment of the wind behavior in the district --CFD simulations to optimize the design of the district, most particularly using vegetation

The study was conducted following 4 consecutives steps: 1) Assessment of the statistics of the wind blowing on the Citadel using CFD simulations to extrapolate weather data form a neighbor weather station to the location of the Citadel 2) A first precise assessment of the pedestrian comfort on an early design of the district using PIV measurement in a wind tunnel experiment 3) Successive iterations, using CFD, of the design of the district to enhance the pedestrian wind comfort 4) A final assessment of the pedestrian comfort in wind tunnel of the optimal version of the district.

Many previous studies presented methodologies for the assessment of pedestrian comfort [START_REF] Delpech | Pedestrian wind comfort assessment criteria: A comparative case study[END_REF][2] [START_REF] Blocken | Pedestrian-level wind conditions around buildings: Review of wind-tunnel and CFD techniques and their accuracy for wind comfort assessment[END_REF]. The specificity of the present study is to combine PIV measurements in wind tunnel (very few experiments were presented to our knowledge) and CFD simulations carried out by the urban architect themselves thanks to the Urbawind software to optimize the design of the Citadel.

CALCULATION OF THE WIND STATISTICS AT THE CITADEL OF MONTLAUR

Weather data is available in the weather Station of Cap-Pertusato, located only 3km at the South East of the Bonifacio peninsula. It could be close enough to consider the weather data relevant to the one at Bonifacio. However, considering the high cliffs and the shape of the coast between Cap-Pertusato and Bonifacio, as well as the particular shape of the Peninsula, wind could be channeled by the topography. It was then preferred to use CFD to take into account the orography and roughness effect of the wind blowing between Bonifacio and Cap Pertusato.

To do so CFD simulations were carried out on a computational domain of 20km per 10 km as presented in Figure 1. The orography was taken into account in the CFD mesh with various resolutions, from 10 m on the peninsula till 200 m at the farthest locations of the Peninsula and of Cape Pertusato. Roughness was model as an input for the wall boundary conditions for the CFD code. Special attention was brought to the mesh with a boundary layer mesh generated all the other the terrain model with a first cell size of 2 m.

Figure 1 -Computational domain

Simulations were run with the ANSYS Fluent CFD software using a stationary Reynolds Average Navier Stokes approach combined with a k-Epsilon model. The so-called Beljaars formulation of the k-epsilon model was used to represent the behavior of the wind over the terrain.

Simulations were conducted for 18 wind directions (20° step). For each simulation, the wind "measured" 10 m above the ground at the Montlaur citadel was compared to the one "measured" at the weather station. Those simulations are not here detailed as it is not the main topic of the present paper. Figure 2 presents the flow topology and the wind speed 10 m above the peninsula for one wind directions. Simulations allowed the weather data form Cap Pertusato station to be transferred at the location of the project. Statistics are then computed directly from the "corrected data". Figure 3 presents the wind rose obtained at the citadel locations. One can see that this wind rose is quite uncommon as the majority of wind events are concentrated along two wind directions, 280° and 60°. This is due to the particular location of Bonifacio at the extreme south end of the Corsica Island. With the probability PDi of the wind blowing along direction i and Ai and Ki the "shape" parameters of the Weibull function computed for every direction i. Pi(X) is the probability that the wind exceed value X for wind direction i. Urbalterre transferred wind rose thanks to CFD software for complex terrain called Topowind. Results from Topowind are similar from Fluent.

ASSESSMENT OF PEDESTRIAN CONFIGURATION IN THE INITIAL CONFIGURATION

A first design of the Montlaur district was proposed by Urbalterre and is presented Figure 4. Grey buildings are already built and need to be renovated without changing their shape. Red buildings are the new ones to be constructed. Those buildings are quite massive and may cause wind accelerations between them which might lead to pedestrian discomfort. This first version is then tested in wind tunnel using a 1:100 scale model 7 presents the boundary layer profile generated in the wind tunnel. This profile is relevant to the one occurring on the sea (roughness 0 according to Eurocode regulation). The picture also represents a cross section of the district located ether at sea level (in red), or on top of the peninsula (in blue). One can see that it is necessary to represent the peninsula as it will have a major impact on the way the wind will impact the buildings.

Figure 7 -Boundary layer profile in the wind tunnel

Additional velocity profiles were measured on the top of the peninsula model without the buildings. The velocity measured at 10m full scale (10cm model scale) above the ground will be used as the reference wind speed to extrapolate wind statistics which have been previously computed. Figure 8 plots the wind speed and turbulence intensity profiles measured on the top of the peninsula model. The mean wind speed at pedestrian level (1.5cm model scale) is 6.3 m/s with 28% of turbulence; and 8.8m/s with 21% turbulence at 10m above the ground. PIV is now quite a common technique for velocity measurements in wind tunnels. Previous attempts were performed to use this technique to assess wind speed close to the ground for pedestrian comfort assessment [START_REF] Faure | Pedestrian wind comfort: application of four different measurement techniques[END_REF]. There are two main difficulties for one using PIV for pedestrian comfort assessment. 1) Measurements are carried at 1.5 m full scale above the ground, which means 1.5 cm at 1:100th scale. This means that the laser sheet is very close from the ground and will highlight every roughness of the medium used for the ground. This will alter the contrast of the PIV images and will decrease the efficiency of the correlation algorithm which computes the particles movement.

2) The density of the buildings in the district makes the optical access for the laser sheet difficult to set up. The models were built with Plexiglas faces so that the laser sheet can reach every part of the model where the wind velocity has to be measured. The Plexiglas let the light go through the walls, but will also generate a lot of reflections which will also impair the quality of the PIV images. Special care has to be brought to avoid those reflections adjusting the position of the PIV camera relative to those reflections. The laser sheet is located 1.5 cm above the ground. To enhance the contrast of the images, the ground was painted with several layers of black paint and fluorescent paint. The later one is obtained by mixing Rhodamine dye with a transparent coating. The green wave length of the laser is absorbed by the dye and is re-emitted in an orange wavelength, which is filtered by the PIV camera filter. This technique dramatically increases the quality of the images, and hence of the measurements.

PIV fields size were 22 cm square. Cross correlations windows were 64pixels large, which yields to a 7 mm resolution. For each PIV field, statistics for average and RMS velocities were calculated from 200 instantaneous pair of images. The convergence of those statistics was checked on the most turbulent areas

Results

Two wind directions were tested 60° and 280°, which are the two directions which were identified on the wind rose. 36 PIV fields were measured for wind directions 280° and 41 PIV fields were performed for wind directions 60°, hence respectively 91600 and 106500 measurements points all over the district model. Figure 10 and Figure 11 present the velocity fields measured at pedestrian level, ie 1.5 cm above the ground model scale. Decimation of the vectors was performed for the readability (1 out of 10 vectors is displayed). It is remembered that without the buildings, the mean wind speed is 6 m/s. The PIV measurements show that on some locations, the velocity exceeds this value which means that the buildings will increase the strength of the wind.

Figure 10 -PIV mean wind speed measurements: left for wind direction 280°; right, for wind directions 60°

In order to quantify the discomfort, the Weibull function calculated on the peninsula are combined with the PIV Measurements. The discomfort criterion here used specifies that discomfort is felt when U + σ > 3.6m/s with U the mean wind speed and the RMS value [START_REF] Delpech | Pedestrian wind comfort assessment criteria: A comparative case study[END_REF]. Statistics of the wind speed were computed 10 m above the ground without any building. The function ψi,j is defined for every measurement point j for wind direction i as follows:

With Ui,j and σi,j the values measured at point j for direction i. Uref and Iref are the mean wind speed and turbulence intensity measured above the peninsula 10m (full scale) above the ground without any building.ϒ is the ratio between the velocity measured at pedestrian level and the one measured at 10m (full scale) above the ground. ψi,j can be seen as a transfer function between the reference point were the weather statistics were computed and any point of the model. The probability Pi,j the that Ui,j + σi,j > 3.6m/s for every measurement point j for wind direction i can thus be written as:

Figure 11 present discomfort frequencies calculated from the PIV measurement and the weather statistics from the two wind directions. Results show that for most of the areas, the discomfort frequencies are above 20%. For wind directions 280°, the effect of the buildings increase the discomfort frequencies above 40%. More, one has to keep in mind that the frequencies of both directions should be cumulated. When designing an urban area, CSTB recommends limiting the discomfort frequencies to 2% for the terraces, 10% for strolling areas and less than 20% for fast pace walking. The discomfort frequencies which have been calculated for this first version are dramatically high and need to be reduced.

GEOMETRY OPTIMIZATION USING CFD

Urbalterre's urban architects hence used CFD in order to optimize the design regarding pedestrian wind comfort. They ran the simulations by themselves using the Meteodyn's dedicated CFD software UrbaWind

CFD Computation with Urbawind

The CFD method of UrbaWind consists in solving the Reynolds-Averaged Navier-Stokes equations on an unstructured rectangular grid with automatic refinement of the mesh near obstacles. The CFD tool delivers the tri-dimensional mean velocity field, the turbulence energy field and the mean pressure for each point in the domain. Closure of the RANS equation is achieved using a transport equation on the kinetic energy and an estimation of the eddy length scale based on the wall distance. The equations resolution is based on a finite volume method with a rectangular multi-bloc refined mesh. A very efficient coupled multi-grid solver is used for a fast convergence for every kind of geometry [START_REF] Ferry | New features of the MIGAL solver[END_REF].

The UrbaWind software allows the boundary conditions to be generated automatically. The mean velocity profile at the computational domain inlet is determined by the logarithmic law in the surface layer, and by the Ekman function above [START_REF] Garratt | The atmospheric boundary layer[END_REF]. A 'Blasius'-type ground law is implemented to model frictions (velocity components and turbulent kinetic energy) at the surfaces (ground and buildings). The effect of porous obstacles is modeled by introducing a volume drag force in the cells lying inside the obstacle.

The building definition and topography of Montlaur Citadel were modeled with at least 0.5 m accuracy in order to ensure a correct calculation of speed up and turbulence intensity. The computational domain is a cube of 1 km side with a mesh accuracy of 0.5 m around obstacles and expanding above the urban layer canopy and contains 3.3 million of cells. Mesh and computational domain are presented Figure 12.

The wind profile prescribed at the inlet will depends of the roughness outside of the computational box for a given wind direction. Two main sectors of roughness lengths can be derived from the analysis of the surrounding roughness: water from 135°(South East) to 280° (West by North) and 0.3 m for the other wind directions.

Computational time for a given wind direction vary from 150 min to 400 min on an Intel Xeon E5-2609 at 2.4 GHz. The wind comfort criterion can also be calculated from the CFD results in a similar way as it was computed from the wind tunnel results. Figure 14 presents a comparison between discomfort frequencies obtained with the CFD and with the wind tunnel approach. One can see that the order of magnitude is the same. Discomfort frequencies computed from CFD tends to be lower for some Venturi effects than from wind tunnel results.

Figure 14 -Comparison of Discomfort frequencies

Optimization of the district design

The simulations which were run on the first version of the district showed that the results obtained with UrbaWind yields to the same conclusions regarding pedestrian wind comfort. This tool has hence been used by the urban architect to iterate some solution to enhance the pedestrian wind comfort. Urbalterre's urban architects ran successively 6 designs of the district. For each one, a different windbreak solution has been tested. Table 1 sums up the 6 configurations investigated:

Table 1 -Windbreaks strategy

From the reference case (Figure 15), strategies to reduce wind speed show that:

-The canopy windbreak (case 1 on Figure 16) is efficient for both dominant winds mostly on the North side. Wind is in fact deflected upward by upstream buildings. The canopy prevents wind from diving into the place. -Porous elements located in staggered rows (case 2 on Figure 17) are also very efficient locally for 280° wind direction mostly on the North side. Nevertheless, this windbreak is not very efficient for 60° wind direction. -Building on the West part (case 3 on Figure 18) is only efficient for 280° direction on a limited area of 25% of the total place thanks to the blockage effect -Elongated porous elements (case 4 on Figure 19) is efficient on a very limited area for 280° wind direction only. There is no improvement for wind comfort for 60° wind direction. -Trees in a row (case 5 on Figure 20) are the best strategy here. For 280° wind direction, over 75% of the place mean speed has been divided by 50 % at least. For 60° wind direction, the efficiency is less compared to 280° but the wind speed decreases by at least 25%. -Porous elements located at 4 m above the ground (case 6 on Figure 20) do not provide sufficient protection for the place Except for case n°6, all strategies have positive effect on the mean speed. The case n°5 is the most efficient. Discomfort frequencies which were measured were similar to the ones obtained through CFD. One can observe that the comfort has actually been enhanced compared to the first iteration. Discomfort frequencies remain high, and this is mainly due to strong wind blowing over the peninsula. The main issue for Urbalterre was the corner of the large square between the two ancient barracks highlighted in Figure 25 with as dashed square. Here a Venturi effect can clearly be seen between the two buildings and may cause a severe discomfort for the pedestrians. This effect has been underestimated by the CFD approach. It was the proposed to optimize this issue in the wind tunnel iterating with vegetation and assessing its effect with PIV measurements Figure 26 presents two steps of the iteration process using several lichens. The first step was to create a vegetal porch hanging between the two buildings. The velocity at the pedestrian level was reduced, but it was not considered as sufficient. Then, two additional vegetation patches were set down on both side of the porch in a chicane configuration. Results show that the wind speed is dramatically decreased in this configuration. Again vegetation remains a very interesting leverage for correcting such flow features.

Figure 26 -Optimization of the Venturi effect using vegetation and PIV measurements

CONCLUSION

This paper presented a study on the optimization of the pedestrian comfort in the citadel of Bonifacio, the windiest city of Europe. Urbalterre, the urban architect agency in charge of the project combined Wind Tunnel and CFD simulations to assess and optimize the pedestrian wind confront. The first step was an evaluation of the wind statistics at the district location. Weather data from the nearest weather station was extrapolated to Bonifacio using CFD simulations on a domain of several kilometers to take into account the local orography and roughness. The design of the district was then performed in three steps.

A first assessment was carried out in wind tunnel using Particle Image Velocimetry. To our knowledge, very few studies used this type of measurements to assess wind comfort. This kind of measurements indeed requires special care to obtain sufficient accuracy due to the proximity of the laser sheet from the ground. Velocity measurements were combined with weather data to compute a wind comfort criterion. Results highlighted severe comfort issues which require an optimization of the district. A second step consists of optimization carried out by the urban architects of Urbalterre themselves thanks to the UrbaWind CFD solution. Good correlations between wind tunnel and CFD results were observed. On some locations, CFD results presented lower wind mean speed values for some Venturi effects compared to the wind tunnel results. CFD tool allowed the urban architects to iterate several versions of the district and to convergence towards an optimal solution. The use of vegetation leads to an optimized version of the district which was finally tested again in the wind tunnel. Final adjustments were performed directly in the wind tunnel using PIV measurements to optimize a Venturi effect between two buildings.

The main interest of this study is the combination of PIV measurements, which proved to be a very efficient tool to assess wind comfort in wind tunnel, and CFD simulations which were carried out by the urban architects themselves to iterate various solutions. The configuration of the Montlaur Citadel, due its location above 60m cliffs and the impressive statistics of the wind blowing on the South of Corsica Island, was a tough challenge regarding the optimization of pedestrian wind comfort. Again, the use of the vegetation has proven to be one of the most efficient way to optimize pedestrian wind comfort when designing a urban area.

Figure 2 -

 2 Figure 2 -Left: wind speed at 10 m above the peninsula; right: stream lines of the flow

Figure 3 -

 3 Figure 3 -Wind rose computed at the Montlaur District, wind speeds in m/s and frequency in %

Figure 4 -

 4 Figure 4 -First version of the district

Figure 5 -

 5 Figure 5 -NSA Wind tunnel at CSTB

Figure 6 -

 6 Figure 6 -Model of the district for wind tunnel experiments

Figure

  Figure7presents the boundary layer profile generated in the wind tunnel. This profile is relevant to the one occurring on the sea (roughness 0 according to Eurocode regulation). The picture also represents a cross section of the district located ether at sea level (in red), or on top of the peninsula (in blue). One can see that it is necessary to represent the peninsula as it will have a major impact on the way the wind will impact the buildings.

Figure 8 -

 8 Figure 8 -Velocity profile on the peninsula model, left: mean wind speed, right: turbulence intensity

Figure

  Figure 9 -PIV Measurements

Figure 11 -

 11 Figure 11 -Discomfort frequencies calculated from PIV measurements: left for wind direction 280°; right, for wind direction 60°

Figure 12 -

 12 Figure 12 -Computational domain

Figure 15 -

 15 Figure 15 -Mean speed for case 0: left for wind direction 280°; right, for wind directions 60°

Figure 22 -

 22 Figure 22 -Final version of the district

Figure 23 -

 23 Figure 23 -Discomfort frequencies calculated with Urbawind for the final iteration

Figure 24 -

 24 Figure 24 -Wind Tunnel model of the final version of the district

Figure 25 -

 25 Figure 25 -Discomfort frequencies calculated from PIV measurements: left for wind direction 280°; right, for wind directions 60°

Paper presented at EACWE 2017, 7th European-African Conference on Wind Engineering, July 4-7, 2017, Liège, BEL