Jean-Baptiste Leblond

Sylvain Patinet

Joël Frelat

Véronique Lazarus

Second-order coplanar perturbation of a semi-infinite crack in an infinite body

Keywords: Semi-infinite crack, coplanar perturbation, second order, stress intensity factor, fundamental kernel

 for the same problem, using a more complex method of solution. Finite element computations performed for sinusoidal perturbations support the new formulae, rather than the older ones. As an application, it is finally shown that the mean value of the energy-release-rate along the front of a perturbed semi-infinite tensile crack is exactly the same, up to second order in the perturbation, as if the front were straight.

Introduction

In the past 25 years, formulae providing the distribution of the stress intensity factors (SIF) for planar cracks with slightly perturbed fronts have received considerable attention. Such formulae have been established for a semi-infinite crack loaded in mode I [START_REF] Rice | First-order variation in elastic fields due to variation in location of a planar crack front[END_REF] or arbitrarily [START_REF] Gao | Shear stress intensity factors for planar crack with slightly curved front[END_REF]; an internal circular crack loaded in mode I (Gao and Rice, 1987a) or arbitrarily [START_REF] Gao | Nearly circular shear mode cracks[END_REF]; an external circular tensile crack (Gao and Rice, 1987b); a tunnel-crack loaded in mode I [START_REF] Leblond | The tensile tunnel-crack with a slightly wavy front[END_REF] or arbitrarily (Lazarus and Leblond, 2002a,b); a semi-infinite interface crack loaded arbitrarily (Lazarus and Leblond, 1998a,b;[START_REF] Piccolroaz | Evaluation of the Lazarus-Leblond constants in the asymptotic model of the interfacial wavy crack[END_REF]; a pair of coplanar, parallel, identical tensile tunnel-cracks [START_REF] Pindra | In-plane perturbation of a system of two coplanar slit-cracks -I: Case of arbitrarily spaced crack fronts[END_REF]; a pair of coplanar semi-infinite tensile cracks (Legrand and Leblond, 2010); and an emerging tensile crack lying on the mid-plane of a semi-infinite plate [START_REF] Legrand | Coplanar perturbation of a crack lying on the mid-plane of a plate[END_REF].

The existence of these formulae has opened the way to theoretical studies of the evolution in time of the deformation of the front of cracks propagating in materials with heterogeneous fracture properties, in various situations: a tunnel-crack loaded in mode I [START_REF] Favier | Statistics of the deformation of the front of a tunnel-crack propagating in some inhomogeneous medium[END_REF] or arbitrarily [START_REF] Pindra | Geometrical disorder of the fronts of a tunnel-crack propagating in shear in some heterogeneous medium[END_REF]; a semi-infinite interface crack loaded arbitrarily [START_REF] Pindra | The deformation of the front of a 3D interface crack propagating quasistatically in a medium with random fracture properties[END_REF]; and a pair of coplanar semi-infinite tensile cracks (Legrand and Leblond, 2010). From a more practical point of view, [START_REF] Rice | First-order variation in elastic fields due to variation in location of a planar crack front[END_REF]'s formula for perturbation of a semi-infinite tensile crack has also been extensively used to assist the interpretation of experiments of separation of bonded plates; see the works of [START_REF] Schmittbuhl | Interfacial crack pinning: effect of nonlocal interactions[END_REF], [START_REF] Tanguy | From individual to collective pinning: effect of long-range elastic interactions[END_REF], [START_REF] Schmittbuhl | Interfacial crack front wandering: influence of quenched noise correlations[END_REF], [START_REF] Krishnamurthy | Dynamic exponent in extremal models of pinning[END_REF], [START_REF] Roux | Effective toughness of heterogeneous brittle materials[END_REF], [START_REF] Schmittbuhl | Slow crack propagation and slip correlations[END_REF], [START_REF] Charles | Material-independent crack arrest statistics[END_REF], [START_REF] Katzav | Roughness of tensile crack fronts in heterogenous materials[END_REF], [START_REF] Bonamy | Cracking dynamics in material failure as the signature of a self-organized dynamic phase transition[END_REF], [START_REF] Laurson | Avalanches and clusters in planar crack front propagation[END_REF], [START_REF] Ponson | Crack propagation in brittle heterogeneous solids: material disorder and crack dynamics[END_REF], and the reviews of [START_REF] Alava | Statistical models of fracture[END_REF] and [START_REF]Perturbation approaches of a planar crack in linear fracture mechanics: a review[END_REF]. Very recently, [START_REF] Patinet | Propagation des fronts de fissure plane dans les matériaux fragiles hétérogènes de dimensions finies[END_REF] have extended these works by accounting for the effect of the finite dimensions of the specimens in such experiments, using [START_REF] Legrand | Coplanar perturbation of a crack lying on the mid-plane of a plate[END_REF]'s extension of [START_REF] Rice | First-order variation in elastic fields due to variation in location of a planar crack front[END_REF]'s formula to perturbation of a tensile crack lying on the mid-plane of a plate.

One drawback of the application of the theoretical formulae mentioned above to the interpretation of various experiments was that all of these formulae were accurate only to first order in the perturbation of the front, whereas there was ample evidence that geometric nonlinearities played an important role in most of the experiments. Clearly, such interpretations of experiments would greatly benefit from use of more accurate, higher-order formulae.

In spite of this, a single work may be found in the literature about the second-order coplanar perturbation of a crack, namely that of [START_REF] Adda-Bedia | Second-order variation in elastic fields of a tensile planar crack with a curved front[END_REF], later summarized by [START_REF] Katzav | Roughness of moving elastic lines: crack and wetting fronts[END_REF], devoted to the case of a semi-infinite tensile crack. Unfortunately these papers contained several incompatible second-order formulae. It therefore seems necessary to re-examine the problem. This paper is devoted to such a re-examination. The method of solution is based on direct and basically straightforward use of earlier results of [START_REF] Rice | Weight function theory for three-dimensional elastic crack analysis[END_REF], and is thus simpler in principle than that of [START_REF] Adda-Bedia | Second-order variation in elastic fields of a tensile planar crack with a curved front[END_REF], which involved calculation of the second derivatives of the Neuber-Papkovich potentials of the problem with respect to the amplitude of the perturbation. It is therefore believed to be more reliable, and this will be confirmed by finite element computations supporting the new formula that will be found, rather than the old ones.

The paper is organized as follows:

• Section 2 presents the results of [START_REF] Rice | Weight function theory for three-dimensional elastic crack analysis[END_REF] which underlie the treatment. These results consist of two formulae providing, for an arbitrary planar tensile crack, the infinitesimal variations of the mode I SIF and fundamental kernel (FK, to be defined below) arising from some infinitesimal coplanar perturbation of the front. • Section 3 then applies [START_REF] Rice | Weight function theory for three-dimensional elastic crack analysis[END_REF]'s second formula to the calculation of the first-order expression of the FK for a semi-infinite crack with a slightly perturbed front. • Section 4 next presents the calculation of the second-order expression of the SIF for the same cracked geometry. The principle consists in deducing from Rice (1989)'s first formula, used with the first-order expression of the SIF and FK, the first-order expression of the derivative of the SIF with respect to the amplitude of the perturbation, and then integrating. • A Fourier transform of the formula found in Section 4 in the direction of the crack front, suggested by the natural translatory invariance of the problem in this direction, is then performed in Section 5. • The result of Section 5 is next specialized in Section 6 to the case of a sinusoidal perturbation. The expression found is compared, together with the earlier ones of [START_REF] Adda-Bedia | Second-order variation in elastic fields of a tensile planar crack with a curved front[END_REF] and [START_REF] Katzav | Roughness of moving elastic lines: crack and wetting fronts[END_REF], to the results of some finite element computations.

• As an application, we finally consider in Section 7 a semi-infinite tensile crack with a periodically but otherwise arbitrarily perturbed front. It is shown that the mean value of the energy-release-rate along this front is exactly the same, up to second order in the perturbation, as if it were straight.

Rice's formulae

Consider a planar crack with arbitrary contour embedded in some isotropic elastic body Ω symmetric about the crack plane (Figure 1). Load the crack through some symmetric system of forces and/or displacements prescribed on Ω and/or ∂Ω. The crack is then in a situation of pure mode I at each point of its front; let K 0 (s), s being some curvilinear abscissa along this front, denote the local SIF.

Ω δa(s)

Fig. 1. An arbitrary planar crack with a slightly perturbed front in an arbitrary body

Now let the front undergo some infinitesimal coplanar perturbation, the loading remaining fixed; let δa(s) denote the local orthogonal distance from the original front to the perturbed one (Figure 1). [START_REF] Rice | Weight function theory for three-dimensional elastic crack analysis[END_REF] has shown that under such conditions, the infinitesimal variation δK of the local SIF is given by the following equation, referred to as Rice's first formula in the sequel:

δK(s 1 ) = [δK(s 1 )] δa(s)≡δa(s 1 ),∀s + P V CF Z(s 1 , s)K 0 (s) [δa(s)δa(s 1 )] ds.

(1)

In this expression,

• the integral is taken over the crack front CF and the symbol P V CF represents the Cauchy principal value of this integral; • [δK(s 1 )] δa(s)≡δa(s 1 ),∀s denotes the value of δK(s 1 ) for a uniform advance of the front equal to δa(s 1 ) (δa(s) ≡ δa(s 1 ), ∀s); • Z(s 1 , s) denotes the fundamental kernel (FK), tied to Bueckner's mode I crack-face weight function.

The FK depends on the cracked geometry considered, but has no dependence upon the loading other than on which portions of Ω and ∂Ω have forces versus displacements imposed. It satisfies the following general properties:

Z(s 1 , s 2 ) = Z(s 2 , s 1 ) ; Z(s 1 , s 2 ) ∼ 1 2π(s 1 -s 2 ) 2 for s 1 -s 2 → 0.
(2)

The second of these properties shows that the integral in equation (1) does make sense as a Cauchy principal value.

In addition, [START_REF] Rice | Weight function theory for three-dimensional elastic crack analysis[END_REF] has shown that provided that the crack advance δa(s) vanishes at points s 1 and s 2 , the infinitesimal variation of the FK is given by the following equation, referred to as Rice's second formula in the sequel:

δZ(s 1 , s 2 ) = P V CF Z(s 1 , s)Z(s, s 2 )δa(s)ds. ( 3
)
Note that there are in fact two principal values here, at s = s 1 and s = s 2 .

Rice's formulae have been used by [START_REF] Bower | Solution of three-dimensional crack problems by a finite perturbation method[END_REF][START_REF] Bower | Solution of three-dimensional crack problems by a finite perturbation method[END_REF][START_REF]Brittle fracture and fatigue propagation paths of 3D plane cracks under uniform remote tensile loading[END_REF] to numerically calculate the SIF along the front of planar tensile cracks with arbitrary contours. The method consisted in starting from some "reference" configuration of this front for which the SIF and FK were known, and then gradually deforming it, updating the SIF and FK at each step, until the configuration of interest was reached. The use made here of Rice's formulae is a kind of analytical equivalent of this numerical method, involving two steps of deformation only.

3 First-order variation of the fundamental kernel

Generalities

We now consider the specific case of a semi-infinite tensile crack embedded in some infinite body (Figure 2). The loading consists only of forces imposed at various locations in the body and/or on the crack faces (no prescribed displacements). The initially straight crack front is displaced within the crack plane Oxz by a distance ∆a(z) of the form

∆a(z) ≡ Aφ(z), (4)
A being a small, but not necessarily infinitesimal "amplitude" and φ(z) a given smooth function.

O y z x a(z) ∆ Fig. 2. A semi-infinite crack with a slightly perturbed front in an infinite body

The FK for this crack configuration is denoted Z(A; z 1 , z 2 ), and our interest here lies in the first-order expression of Z(A; z 1 , z 2 ),

Z(A; z 1 , z 2 ) ≡ Z 0 (z 1 , z 2 ) + AZ 1 (z 1 , z 2 ) + O(A 2 ).
(5)

More precisely, the expression of the FK Z 0 (z 1 , z 2 ) for the unperturbed configuration of the crack is known to be [START_REF] Rice | First-order variation in elastic fields due to variation in location of a planar crack front[END_REF][START_REF] Rice | Weight function theory for three-dimensional elastic crack analysis[END_REF])

Z 0 (z 1 , z 2 ) ≡ 1 2π(z 1 -z 2 ) 2 , (6)
and we are looking for that of Z 1 (z 1 , z 2 ).

Application of Rice's second formula

To be applicable, Rice's second formula (3) requires that the perturbation of the front be zero at those points z 1 , z 2 where the variation of the FK is to be evaluated; thus it does not directly apply to the arbitrary perturbation (4). To make up for this difficulty, it suffices, following Rice (1989)'s suggestion, to additively decompose the perturbation φ(z) into two sub-perturbations:

• one, φ * (z), consisting of a suitable combination of a translatory motion and a rotation bringing points z 1 and z 2 to their correct final positions, given by the two equivalent formulae

φ * (z) ≡          φ(z 1 ) + φ(z 2 ) -φ(z 1 ) z 2 -z 1 (z -z 1 ) φ(z 2 ) + φ(z 1 ) -φ(z 2 ) z 1 -z 2 (z -z 2 ) ; (7)
• the other, φ(z)φ * (z), bringing all points to their correct final positions while leaving points z 1 and z 2 fixed.

Since our interest lies only in the first-order expression of the FK, the variation of this FK may be obtained by simply adding the variations arising from the two sub-perturbations. Now in the first sub-perturbation, the shape of the front is unchanged, so that the FK remains identical, by equation ( 6), to the inverse of 2π times the squared distance between the points considered; but the variation of this distance is zero to first order in A, so that the variation of the FK is zero. On the other hand the second sub-perturbation induces a variation of the FK given by Rice's second formula applied to the perturbation φ(z)-φ * (z) instead of φ(z). It follows that

Z 1 (z 1 , z 2 ) = 1 4π 2 P V +∞ -∞ φ(z) -φ * (z) (z -z 1 ) 2 (z -z 2 ) 2 dz (8)
where use has been made of equation ( 6).

Transformation of the expression of Z 1

Although equation ( 8) may directly be used to evaluate Z 1 (z 1 , z 2 ) numerically, the presence of the rational function 1/[(zz 1 ) 2 (zz 2 ) 2 ] in the integrand makes it unfit for any analytical calculation, whatever the form chosen for the function φ(z). To get a more convenient form, one must decompose this rational function into partial fractions:

1 (z -z 1 ) 2 (z -z 2 ) 2 = 1 (z 1 -z 2 ) 2 1 (z -z 1 ) 2 + 1 (z -z 2 ) 2 + 2 (z 1 -z 2 ) 3 1 z -z 2 - 1 z -z 1 , (9)
and then calculate the integrals of φ(z)φ * (z) times the various fractions, using equation ( 7) for φ * (z). When doing so, one must beware that each of these integrals individually diverges at infinity, so that it is momentarily necessary to integrate between large but finite bounds -L 1 , L 2 before taking the limit L 1 , L 2 → +∞ in the final combination of integrals.1 One thus gets for the fractions 1/(zz 1 ) and 1/(zz 1 ) 2 , using expression (7) 1 of φ * (z):

                 L 2 -L 1 φ(z) -φ * (z) z -z 1 dz = P V L 2 -L 1 φ(z) z -z 1 dz - φ(z 2 ) -φ(z 1 ) z 2 -z 1 (L 1 + L 2 ) -φ(z 1 ) ln L 2 -z 1 L 1 + z 1 P V L 2 -L 1 φ(z) -φ * (z) (z -z 1 ) 2 dz = P V L 2 -L 1 φ(z) -φ(z 1 ) (z -z 1 ) 2 dz - φ(z 2 ) -φ(z 1 ) z 2 -z 1 ln L 2 -z 1 L 1 + z 1
plus, using equation ( 7) 2 , similar expressions for the fractions 1/(zz 2 ) and 1/(zz 2 ) 2 . Using then equation ( 9) and noting that the terms proportional to

L 1 + L 2 in the integrals of [φ(z) -φ * (z)]/(z -z 1 ) and [φ(z) -φ * (z)]/(z -z 2 ) cancel out, one gets P V L 2 -L 1 φ(z) -φ * (z) (z -z 1 ) 2 (z -z 2 ) 2 dz = 1 (z 1 -z 2 ) 2 P V L 2 -L 1 φ(z) -φ(z 1 ) (z -z 1 ) 2 dz + P V L 2 -L 1 φ(z) -φ(z 2 ) (z -z 2 ) 2 dz + 2 (z 1 -z 2 ) 3 P V L 2 -L 1 φ(z) z -z 2 dz -P V L 2 -L 1 φ(z) z -z 1 dz - 1 (z 1 -z 2 ) 2 φ(z 2 ) -φ(z 1 ) z 2 -z 1 ln L 2 -z 1 L 1 + z 1 + φ(z 1 ) -φ(z 2 ) z 1 -z 2 ln L 2 -z 2 L 1 + z 2 + 2 (z 1 -z 2 ) 3 φ(z 1 ) ln L 2 -z 1 L 1 + z 1 -φ(z 2 ) ln L 2 -z 2 L 1 + z 2 .
It is easy to check that the sum of the logarithmic terms here goes to zero in the limit L 1 , L 2 → +∞. One thus gets in this limit, by equation ( 8):

Z 1 (z 1 , z 2 ) = 1 4π 2 (z 1 -z 2 ) 2 P V +∞ -∞ φ(z) -φ(z 1 ) (z -z 1 ) 2 + φ(z) -φ(z 2 ) (z -z 2 ) 2 + 2 z 1 -z 2 φ(z) z -z 2 - φ(z) z -z 1 dz = 1 4π 2 (z 1 -z 2 ) 2 P V +∞ -∞ 1 z -z 1 + 1 z -z 2 φ ′ (z) + 2 z 1 -z 2 1 z -z 2 - 1 z -z 1 φ(z) dz, (10)
where the last expression follows from integration by parts of the first two terms in the preceding integral.

4 Second-order variation of the stress intensity factor

We now study the SIF K(A; z 1 ) for a semi-infinite tensile crack with a front perturbed in the same way as before, equation (4). Unlike the FK, this SIF depends on the loading imposed. For simplicity, this loading will be assumed to be such that the distribution of the SIF along the front be invariant when it is displaced by a uniform distance in the direction x, be it straight or not; the first, non-integral term in Rice's first formula (1) is then zero, whatever the initial configuration of the front considered. Also, the unperturbed SIF will be assumed to be uniform along the front.

The topic of interest here is the second-order expression of K(A; z 1 ),

K(A; z 1 ) ≡ K 0 + AK 1 (z 1 ) + A 2 K 2 (z 1 ) + O(A 3 ). ( 11
)
More specifically, K 0 is assumed to be known, the expression of K 1 (z 1 ) directly results from Rice's first formula:

K 1 (z 1 ) = K 0 2π P V +∞ -∞ φ(z) -φ(z 1 ) (z -z 1 ) 2 dz = K 0 2π P V +∞ -∞ φ ′ (z) z -z 1 dz, (12)
and we are looking for that of K2 (z 1 ).

To determine the expression of K 2 (z 1 ), consider a primary perturbation of the front of the form (4), upon which is added a secondary perturbation

δa(z) ≡ δA φ(z) (13)
having the same function φ(z) as the primary one, but some now infinitesimal amplitude δA. Then Rice's first formula, applied to the variation of the SIF resulting from the additional perturbation, provides, upon division by δA, the value of the derivative (∂K/∂A)(A; z 1 ). The formula involves the SIF and FK for the primarily perturbed configuration of the front. If expressions of these quantities accurate to first order in A are used, the result is the first-order expression of (∂K/∂A)(A; z 1 ). 2

One thus gets

∂K ∂A (A; z 1 ) = P V +∞ -∞ Z 0 (z 1 , z) + AZ 1 (z 1 , z) K 0 + AK 1 (z) [φ(z) -φ(z 1 )] dz + O(A 2 ) = K 0 P V +∞ -∞ Z 0 (z 1 , z) [φ(z) -φ(z 1 )] dz +A P V +∞ -∞ Z 0 (z 1 , z)K 1 (z) + Z 1 (z 1 , z)K 0 [φ(z) -φ(z 1 )] dz + O(A 2 ).
Integration with respect to A then yields

K(A; z 1 ) = K 0 + AK 0 P V +∞ -∞ Z 0 (z 1 , z) [φ(z) -φ(z 1 )] dz + A 2 2 P V +∞ -∞ Z 0 (z 1 , z)K 1 (z) + Z 1 (z 1 , z)K 0 [φ(z) -φ(z 1 )] dz + O(A 3 ), so that K 2 (z 1 ) = 1 2 P V +∞ -∞ Z 0 (z 1 , z)K 1 (z) + Z 1 (z 1 , z)K 0 [φ(z) -φ(z 1 )] dz.
The final expression of K 2 (z 1 ) is then obtained by using the expressions ( 6), ( 10) and ( 12) of Z 0 , Z 1 and K 1 :

K 2 (z 1 ) = K 0 8π 2 P V +∞ -∞ +∞ -∞ 1 z ′ -z 1 + 2 z ′ -z φ ′ (z ′ ) + 2 z -z 1 1 z ′ -z 1 - 1 z ′ -z φ(z ′ ) φ(z) -φ(z 1 ) (z -z 1 ) 2 dzdz ′ . (14)
This expression differs from, and is more complex than, formulae (34) and ( 35) of [START_REF] Adda-Bedia | Second-order variation in elastic fields of a tensile planar crack with a curved front[END_REF] (which are themselves distinct).

5 Formulae in Fourier's space

The potential usefulness of a formula for the Fourier transform of the second-order variation K 2 (z 1 ) in the direction of the crack front is strongly suggested by the natural invariance of the problem in this direction. Such a formula for the first-order variation K 1 (z 1 ) has already been extensively used in the literature and proven very useful.

The definition of the Fourier transform ψ(k) of an arbitrary function ψ(z) adopted here is

ψ(z) ≡ +∞ -∞ ψ(k)e ikz dk ⇔ ψ(k) ≡ 1 2π +∞ -∞ ψ(z)e -ikz dz. (15)
For completeness, we first recall the classical expression of the Fourier transform of the first-order variation (see e.g. Lazarus (2011)):

K 1 (k) = -K 0 |k| 2 φ(k). ( 16
)
This expression is easily established by expressing the function φ in equation ( 12) 2 in terms of its Fourier transform φ, and then calculating the integral

P V +∞ -∞ e ikz z -z 1 dz = e ikz 1 P V +∞ -∞ e ik(z-z 1 ) z -z 1 dz = e ikz 1 P V +∞ -∞ cos(kz ′ ) + i sin(kz ′ ) z ′ dz ′ = i sgn(k)e ikz 1 +∞ -∞ sin u u du = iπ sgn(k)e ikz 1
(17) where sgn(x) denotes the sign of x.

To get the expression of the Fourier transform of the second-order variation, the simplest method again consists in expressing φ(z) and φ(z ′ ) in equation ( 14) in terms of the Fourier transform φ:

K 2 (z 1 ) = K 0 8π 2 +∞ -∞ +∞ -∞ P V +∞ -∞ +∞ -∞ 1 z ′ -z 1 + 2 z ′ -z ik ′ φ(k ′ )e ik ′ z ′ + 2 z -z 1 1 z ′ -z 1 - 1 z ′ -z φ(k ′ )e ik ′ z ′ φ(k) e ikz -e ikz 1 (z -z 1 ) 2 dzdz ′ dkdk ′ .
Performing the integration on z ′ , we obtain upon use of equation ( 17):

K 2 (z 1 ) = K 0 8π +∞ -∞ +∞ -∞ sgn(k ′ ) P V +∞ -∞ -k ′ e ik ′ z 1 + 2e ik ′ z + 2i z -z 1 e ik ′ z 1 -e ik ′ z e ikz -e ikz 1 (z -z 1 ) 2 dz φ(k) φ(k ′ ) dkdk ′ .
To now integrate on z, one must calculate the three integrals

P V +∞ -∞ e ikz -e ikz 1 (z -z 1 ) 2 dz , P V +∞ -∞ e ik ′ z e ikz -e ikz 1 (z -z 1 ) 2 dz , P V +∞ -∞ e ik ′ z 1 -e ik ′ z e ikz -e ikz 1 (z -z 1 ) 3 dz;
this is easily done through integrations by parts and repeated use of formula ( 17). The result for K 2 (z 1 ) reads

K 2 (z 1 ) = K 0 8 +∞ -∞ +∞ -∞ [sgn(k)sgn(k ′ )k(k + k ′ ) +sgn(k ′ )sgn(k + k ′ )(k ′2 -k 2 ) -k ′2 φ(k) φ(k ′ ) e i(k+k ′ )z 1 dkdk ′ .
A nicer formula may be obtained by grouping the terms (k, k ′ ) and (k ′ , k) together in the double integral, which is equivalent to "symmetrizing" the term [...] with respect to k and k ′ :

K 2 (z 1 ) = K 0 8 +∞ -∞ +∞ -∞ F (k, k ′ ) φ(k) φ(k ′ ) e i(k+k ′ )z 1 dkdk ′ (18)
where

F (k, k ′ ) ≡ 1 2 sgn(k)sgn(k ′ )(k + k ′ ) 2 + [sgn(k) -sgn(k ′ )] sgn(k + k ′ )(k 2 -k ′2 ) -k 2 -k ′2 ; ( 19
)
the values of this function in the various regions of the plane (k, k ′ ) are illustrated in Figure 3. Using the change of variable k 1 ≡ k + k ′ , the preceding expression of K 2 (z 1 ) may be rewritten in the form

K 2 (z 1 ) = K 0 8 +∞ -∞ +∞ -∞ F (k, k 1 -k) φ(k) φ(k 1 -k) dk e ik 1 z 1 dk 1 ,
which implies, upon comparison with the definition (15) 1 of the Fourier transform, that

K 2 (k 1 ) = K 0 8 +∞ -∞ F (k, k 1 -k) φ(k) φ(k 1 -k) dk. (20)
Again, this expression differs from formulae (36) of Adda-Bedia et al. ( 2006) and ( 6) of [START_REF] Katzav | Roughness of moving elastic lines: crack and wetting fronts[END_REF] (which are themselves distinct).

Case of a sinusoidal perturbation -Numerical validation

The formulae found for the second-order variation of the SIF (( 14) or equivalently (20)) will now be validated through finite element calculations performed for a sinusoidal perturbation of the front. The first task is to apply these formulae to such a perturbation. We therefore consider the case where the functions φ(z) and φ(k) are of the form

φ(z) ≡ cos(k 0 z) ⇔ φ(k) ≡ 1 2 [δ(k -k 0 ) + δ(k + k 0 )] (21)
where k 0 is a positive number and δ denotes Dirac's function. The first-order variation of the SIF then follows from combination of equations (15) 1 , ( 16) and (21) 2 :

K 1 (z 1 ) K 0 = - +∞ -∞ |k| 4 [δ(k -k 0 ) + δ(k + k 0 )] e ikz 1 dk = - k 0 2 cos(k 0 z 1 ),
and the second-order variation from combination of equations ( 18) (equivalent to (20)), ( 19) and ( 21) 2 :

K 2 (z 1 ) K 0 = 1 32 +∞ -∞ +∞ -∞ F (k, k ′ ) [δ(k -k 0 ) + δ(k + k 0 )] × [δ(k ′ -k 0 ) + δ(k ′ + k 0 )] e i(k+k ′ )z 1 dkdk ′ = 1 32 F (k 0 , k 0 )e 2ik 0 z 1 + F (-k 0 , -k 0 )e -2ik 0 z 1 + F (k 0 , -k 0 ) + F (-k 0 , k 0 ) = k 2 0 32 e 2ik 0 z 1 + e -2ik 0 z 1 -2 = - k 2 0 8 sin 2 (k 0 z 1 ).
The second-order expansion of the SIF follows from these expressions:

K(A; z 1 ) K 0 = 1 -A k 0 2 cos(k 0 z 1 ) -A 2 k 2 0 8 sin 2 (k 0 z 1 ) + O(A 3 ). ( 22
)
The numerical computation will provide the local value of the energy-release-rate G(A; z 1 ) rather than that of the SIF. The second-order expansion of G(A; z 1 ) is readily deduced from that of the square of the SIF, itself resulting from equation ( 22):

K(A; z 1 ) K 0 2 = 1 -Ak 0 cos(k 0 z 1 ) -A 2 k 2 0 4 sin 2 (k 0 z 1 ) + A 2 k 2 0 4 cos 2 (k 0 z 1 ) + O(A 3 ) = 1 + A 2 k 2 0 4 -Ak 0 cos(k 0 z 1 ) -A 2 k 2 0 2 sin 2 (k 0 z 1 ) + O(A 3 ) ⇒ G(A; z 1 ) G 0 = A 0 -A 1 cos(k 0 z 1 ) -A 2 sin 2 (k 0 z 1 ) + O(A 3 ) ,                A 0 ≡ 1 + A 2 k 2 0 4 A 1 ≡ Ak 0 A 2 ≡ A 2 k 2 0 2 (23)
where G 0 denotes the unperturbed energy-release-rate. The comparison with the numerical computations will be performed on the two ratios A 1 /A 0 and A 2 /A 1 , the theoretical values of which are

A 1 A 0 = Ak 0 1 + A 2 k 2 0 /4 ; A 2 A 1 = Ak 0 2 . ( 24
)
Using the finite element code CAST3M developed by the French Commissariat à l'Energie Atomique (CEA), finite element computations are performed for elastic blocks of the type shown in Figure 4. The blocks contain an emerging crack the front of which is perturbed according to equation ( 21) 1 Because of the periodicity of the problem in the z direction, only one slice of material lying between the planes z = 0 and z = λ, where λ ≡ 2π/k 0 denotes the period, is meshed; periodic boundary conditions are imposed on these planes. The other dimensions, that is the depth a of the crack, the length b of the unbroken ligament ahead of the crack front, and the thickness 2h of the block in the vertical direction, are all taken much larger than the period λ in order to simulate an infinite body.

Two types of computations are performed with different boundary conditions on the two halves of the left boundary x = -a: opposite bending moments versus opposite vertical displacements. The two types of conditions must yield identical results for large values of the ratios a/λ, b/λ and h/λ, and this serves as a test that the values chosen are acceptable.

The two types of computations also use different dimensions, meshes and types of elements. In these involving prescribed moments, the values of the ratios a/λ, b/λ and h/λ are 50, 50 and 10 respectively, and the mesh consists of 96,000 bilinear 6-node prismatic elements and 53,346 nodes. In those involving prescribed displacements, the values a/λ = 3, 200,3 b/λ = 3, 200, h/λ 32 are used, and the mesh consists of 530,100 bilinear 8-node parallelepipedic elements and 598,353 nodes. In both cases the elements are strictly prismatic or parallelepipedic only for the unperturbed straight configuration of the front; for the perturbed configuration the same mesh is used with a slight distortion of elements. A value of 0.3 is used for Poisson's ratio; the developments above make it clear that the expansion of the SIF is in fact independent of this value. In view of the linearity of the problem, arbitrary values are chosen for the bending moments or vertical displacements imposed.

Local values of the energy-release-rate along the crack front are evaluated using [START_REF] Destuynder | Quelques remarques sur la mécanique de la rupture élastique[END_REF]'s Gθ method, the accuracy of which is well established.

Figure 5 illustrates the results obtained by prescribing bending moments, with a value of the "normalized amplitude of perturbation" A/λ of 0.1. The quantity on the horizontal axis is the normalized distance along the crack front, z 1 /λ, and that on the vertical axis is the energy-release-rate G(A; z 1 ), expressed in arbitrary units because of the arbitrariness of the magnitude of the loading. The dark blue curve represents numerical results and the red one the sinusoidal approximation of these results having the same maxima and minima. The notable gap between the two is a clear indication that the sinusoidal variation predicted by the first-order expression of G(A; z 1 )/G 0 represents only a mediocre approximation, and that second-order effects are important.

The numerical values of the ratios A 1 /A 0 and A 2 /A 1 may easily be deduced from Figure 5. Indeed the sinusoidal approximation of G(A; z 1 )/G 0 having the same maxima and minima as the true expression (23) of this ratio is simply that obtained by discarding the term -A 2 sin 2 (k 0 z 1 ) in this expression, since this term is precisely zero at the maxima and minima. Hence the red curve simply corresponds to this simplified expression. This means that A 0 and A 1 may be identified to the mean value and amplitude of the sinusoidal function represented by the red curve, that is half of the sum and difference of its extremal values. Also, A 2 may be identified to the maximum vertical gap between the red and dark blue curves.

The numerical values of the ratios A 1 /A 0 and A 2 /A 1 determined in this way, together with the theoretical values predicted by equations ( 24), are as follows:

     (A 1 /A 0 ) num = 0.571 (A 1 /A 0 ) theor = 0.572 ;      (A 2 /A 1 ) num = 0.306 (A 2 /A 1 ) theor = 0.314. (25)
The very good agreement appearing here strongly suggests that all theoretical formulae for the perturbed SIF, ( 12) and ( 16) at order 1, ( 14) and (20) at order 2, are correct. In contrast, formulae (34), ( 35), (36) of [START_REF] Adda-Bedia | Second-order variation in elastic fields of a tensile planar crack with a curved front[END_REF] and ( 6) of [START_REF] Katzav | Roughness of moving elastic lines: crack and wetting fronts[END_REF] yield the following theoretical values of the ratio A 2 /A 1 : Ak 0 , Ak 0 , 3Ak 0 /2, Ak 0 , respectively; these values differ from the numerical result (25) 3 by a factor of 2 or 3.

As a complement, Figures 6 and7 show some results obtained by prescribing vertical displacements. The quantities represented are the numerical values of the ratios A 1 /A 0 and A 2 /A 1 , plotted versus the normalized amplitude of perturbation A/λ and compared to the theoretical values provided by equations ( 24). These equations can be observed to provide very good values of A 1 /A 0 and A 2 /A 1 up to A/λ = 0.20; this means that the second-order formula (23) for G(A; z 1 )/G 0 an accurate representation of this quantity even for normalized amplitudes which are no longer truly small. This is remarkable in view of the fact that for such amplitudes, the variation of G(A; z 1 )/G 0 along the crack front is quite large: see Figure 5 where, even for a lower normalized amplitude A/λ = 0.1, the ratio of the maximum to minimum values of G(A; z 1 )/G 0 is already of the order of 4. 7 An application

As an example of an application, we wish to compare the average values of the energyrelease-rate G(A; z 1 ) along the front of a semi-infinite tensile crack in its unperturbed, straight configuration and some arbitrarily perturbed one. More precisely, we shall show that the two mean values are exactly the same up to second order in the perturbation, provided that they are taken with the same infinitesimal weight dz. (The other possible choice, for the curved front, is the infinitesimal length element ds; the difference between the two quantities is of order O(A 2 ) and therefore significant).

The definition of the average value of a function defined on the entire real line and varying arbitrarily over it raises difficulties. In order to circumvent them, we shall assume the crack front perturbation to be periodic, of period 2π/k 0 where k 0 is a positive number; the functions φ(z) and φ(k) are then of the form

φ(z) ≡ +∞ n=-∞ a n e ink 0 z ⇔ φ(k) = +∞ n=-∞ a n δ(k -nk 0 ). ( 26
)
The energy-release-rate is then also periodic, and its average value over the entire front may be unambiguously defined as its average value over a period.

To derive the expression of this average value, one must use the second-order expansion of the square of the SIF, which reads by equation (11):

K(A; z 1 ) K 0 2 = 1 + 2A K 1 (z 1 ) K 0 + 2A 2 K 2 (z 1 ) K 0 + A 2 K 1 (z 1 ) K 0 2 + O(A 3 ). ( 27
)
It is thus necessary to evaluate the average values of K 1 (z 1 )/K 0 , [K 1 (z 1 )/K 0 ] 2 and K 2 (z 1 )/K 0 .

To calculate the first one, note that by equations (15) 1 , ( 16) and ( 26) 2 ,

K 1 (z 1 ) K 0 = - +∞ -∞ |k| 2 φ(k)e ikz 1 dk = - 1 2 +∞ n=-∞ |n|a n k 0 e ink 0 z 1 . ( 28
)
Now the average value of e ink 0 z 1 is zero or unity depending on whether n differs from, or is equal to zero. Thus in the sum above, only the term n = 0 could yield a nonzero contribution to the average value, but it does not because of the factor |n| = 0; therefore

K 1 (z 1 ) K 0 = 0. ( 29
)
Also, equation (28) implies that

K 1 (z 1 ) K 0 2 = 1 4 +∞ m=-∞ +∞ n=-∞
|mn| a m a n k 2 0 e i(m+n)k 0 z 1 .

In this double sum, the sole nonzero contributions to the average value arise from the pairs (m, n) = (-n, n). It follows that

K 1 (z 1 ) K 0 2 = 1 4 +∞ n=-∞ n 2 |a n | 2 k 2 0 ( 30
)
where use has been made of the fact that a -n = a n since φ is a real function.

Finally, combination of equations ( 18) and ( 26)

2 yields K 2 (z 1 ) K 0 = 1 8 +∞ -∞ +∞ -∞ F (k, k ′ ) × +∞ m=-∞ +∞ n=-∞ a m δ (k -mk 0 ) a n δ (k ′ -nk 0 ) e i(k+k ′ )z 1 dkdk ′ = 1 8 +∞ m=-∞ +∞ n=-∞ F (mk 0 , nk 0 ) a m a n e i(m+n)k 0 z 1 .
Again, the sole nonzero contributions to the average value arise from the pairs (m, n) = (-n, n) so that

K 2 (z 1 ) K 0 = 1 8 +∞ n=-∞ F (-nk 0 , nk 0 ) a -n a n = - 1 8 +∞ n=-∞ n 2 |a n | 2 k 2 0 ( 31
)
where equation (19) (or Figure 3) has been used.

Combining equations ( 27), ( 29), ( 30) and ( 31), one sees that

K(A; z 1 ) K 0 2 ≡ G(A; z 1 ) G 0 = 1 + O(A 3 ) (32)
where G 0 denotes the unperturbed energy-release-rate; this establishes the property announced.

The meaning of this property is that when the crack front propagates without any deformation over some pre-specified distance a under constant loading, the total (elas-tic+potential) energy released is independent of whether this front is straight or (periodically) curved.

This result could be anticipated using a simple argument sketched in a paper of [START_REF] Gao | A first-order perturbation analysis of crack trapping by arrays of obstacles[END_REF] and explained in more detail below. Figure 8 shows the trace of the crack plane within a slice of material of thickness equal to the period λ of the crack front perturbation, and four possible configurations of the front, two straight ones S 1 , S 2 separated by a distance of a, and two identically curved ones C 1 , C 2 separated by the same distance.

The loading being fixed, let W 1 S and W 2 S the total (elastic+potential) energies of

W 1 S -W 2 S = G 0 aλ ; W 1 C -2 C = G(A; z 1 ) aλ
where the average value is taken with the infinitesimal weight dz (not ds). Taking the difference between these equations, one gets

W 1 C -W 1 S -(W 2 C -W 2 S ) = G(A; z 1 ) -G 0 aλ.
But the difference W 1 C -W 1 S represents the variation of energy of the slice caused by the deformation of the front from the straight configuration S 1 to the curved one C 1 ; and W 2 C -W 2 S admits a similar interpretation. Now the invariance of the problem in the direction x implies that such a difference is unaffected by translatory motions of the front in this direction, so that W

1 C -W 1 S = W 2 C -W 2 S .
The preceding equation then implies that

G(A; z 1 ) = G 0 . ( 33
)
This reasoning makes it clear that the property discussed is related to the invariance of the problem considered in the direction of crack propagation, and would not subsist for other geometries and/or loadings. On the other hand it may be observed to be independent of the magnitude of the perturbation, and thus hold at all orders in A.

A final observation is that unlike formulae ( 14) and ( 20), none of those of Adda-Bedia et al. (2006) and Katzav et al. (2007) satisfies property (32). This brings additional support to the former formulae since the correctness of the property has been established by independent means.

Conclusion

This paper was devoted to some re-derivation of the expression of the second-order variation of the local stress intensity factor (SIF) resulting from coplanar perturbation of the front of a semi-infinite tensile crack in some infinite body. The task seemed necessary in view of the fact that the earlier work of [START_REF] Adda-Bedia | Second-order variation in elastic fields of a tensile planar crack with a curved front[END_REF] on the topic, as well as [START_REF] Katzav | Roughness of moving elastic lines: crack and wetting fronts[END_REF]'s summary of it, contained inconsistencies.

The method of derivation, which consisted in some basically straightforward application of some results of [START_REF] Rice | Weight function theory for three-dimensional elastic crack analysis[END_REF], was simpler in principle than that of [START_REF] Adda-Bedia | Second-order variation in elastic fields of a tensile planar crack with a curved front[END_REF], which implied calculation of the second derivatives of the Neuber-Papkovich potentials of the problem with respect to the amplitude of the perturbation.

The results of [START_REF] Rice | Weight function theory for three-dimensional elastic crack analysis[END_REF] used were twofold. The first one was a formula for the infinitesimal variation of the local mode I SIF resulting from infinitesimal coplanar perturbation of an arbitrary planar crack, involving the fundamental kernel (FK) of the cracked geometry considered. The second one was a similar formula for the infinitesimal variation of the FK.

In a first step, Rice's second formula was used to derive an expression of the FK for a semi-infinite crack with a slightly curved front, accurate to first order in the deviation from straightness.

In a second step, Rice's first formula was applied to the same perturbed configuration of the front with the first-order expressions of the SIF and FK, to derive the first-order of the derivative of the SIF with respect to the amplitude of the perturbation. Integration with respect to this amplitude then yielded the second-order expression of the perturbed SIF.

A Fourier transform of this second-order expression in the direction of the crack front, suggested by the natural invariance of the problem in this direction, was then performed. next step consisted in applying the formula for the Fourier transform of the secondorder variation of the local SIF to some sinusoidal perturbation of the front. Finite element computations were then performed for solid blocks of large dimensions, simulating an infinite body, and containing such a sinusoidally perturbed crack. The numerical results were found to bring support to the new formula for the second-order variation of the local SIF, rather than the earlier ones of [START_REF] Adda-Bedia | Second-order variation in elastic fields of a tensile planar crack with a curved front[END_REF] and [START_REF] Katzav | Roughness of moving elastic lines: crack and wetting fronts[END_REF].

As a final application, we considered a semi-infinite crack with a periodically perturbed front, and calculated the average value of the energy-release rate along this front up to second order in the deviation from straightness. This average value was found to exactly coincide with that for a straight front, and a simple rationale for the coincidence was provided. This observation and explanation brought additional support to the new formula the second-order variation of the local SIF.

Acknowledgement

Fig. 3 .

 3 Fig. 3. Values of the function F (k, k ′ ) in the plane (k, k ′ )

Fig. 4 .

 4 Fig. 4. Geometry considered for the numerical validation

Fig. 5 .

 5 Fig. 5. Distribution of the energy-release-rate along the front -Numerical results and sinusoidal approximation

Fig. 6 .

 6 Fig. 6. Ratio A 1 /A 0 as a function of the normalized amplitude -Numerical results versus theoretical expression

Fig. 7 .

 7 Fig. 7. Ratio A 2 /A 1 as a function of the normalized amplitude -Numerical results versus theoretical expression

Fig. 8 .

 8 Fig. 8. Translational motions of a straight front and a periodically perturbed one

The limit will be perfectly defined since the integral in equation (8) is convergent at infinity, its integrand behaving like |z| -3 by equation (7).

Note that Rice's formula may safely be applied using the infinitesimal distance between the fronts and the infinitesimal length element measured perpendicularly to, and along the Oz axis, respectively, instead of the true ones measured perpendicularly to, and along the primarily perturbed front; indeed the errors made are of second order in A.

Imposing vertical displacements rather than bending moments makes it necessary to use a much larger value of a/λ, in order to ensure invariance of the SIF in translatory motions of the front in the x direction.

The authors wish to express their sincere thanks to Drs. Mokhtar Adda-Bedia, Laurent Ponson and Damien Vandembroucq for several stimulating discussions.


Using Constraint Programming for the Urban Transit Crew Rescheduling Problem

Xavier Lorca, Charles Prud'homme, Aurélien Questel, and Benoît Rottembourg TASC -École des Mines de Nantes, Université de Nantes, INRIA, LINA UMR 6241 {firstname.lastname}@mines-nantes.fr EURODECISION Versailles {firstname.lastname}@eurodecision.com Abstract. Scheduling urban and trans-urban transportation is an important issue for industrial societies. The Urban Transit Crew Scheduling Problem is one of the most important optimization problem related to this issue. It mainly relies on scheduling bus drivers' workday respecting both collective agreements and the bus schedule needs. If this problem has been intensively studied from a tactical point of view, its operational aspect has been neglected while the problem becomes more and more complex and more and more prone to disruptions. In this way, this paper presents how the constraint programming technologies are able to recover the tactical plans at the operational level in order to efficiently help in answering regulation needs after disruptions.

Context and opportunities

Scheduling urban and trans-urban transportation is an important issue for industrial societies. Several aspects are considered by territorial collectivities and transportation operators: human and material resource for, environmental and social constraint enforcement, user need requirements. Basically, there exists two central problems: transit scheduling (vehicles -buses, trains, tramways -planning on routes) and driver duty planning (assigning crew to those routes). These problems become more and more complex (regulation, network expansion) and more and more prone to disruptions (city events, accidents, resource failures) in the operational phase. This is the context in which the Urban Transit Crew Scheduling Problem (UTCSP) has been introduced [START_REF] Desrochers | A column generation approach to the urban transit crew scheduling problem[END_REF]. In our proposal, we have to schedule bus drivers' workdays according to several constraints mainly related to: (1) collective agreements, e.g. breaking rules (basically, how long a bus driver can work before a break) ; (2) the bus schedule itself, e.g. chaining rules (geographic position, schedule compatibility between two tasks, etc.). From a tactical point of view, the UTCSP has been thoroughly studied both at the academic and industrial levels. Primarily, the technologies derived from mathematical programming (from integer linear programming to column generation and dynamic programming) dominate the literature and offer satisfying results. The problem is mainly solved using a set covering approach that represents bus schedule as a set of tasks, linked together by chaining rules and respecting the breaking rules. From an operational point of view, these technologies become useless due to their time consumption. Such a pitfall leads the operators to manually repair the tactical solutions after a disruption (at the operational level), and to consequently derive the applied schedule quality.

Focusing on the operational point of view of the UTCSP, a.k.a. Urban Transit Crew Rescheduling Problem (UTCRP), this paper presents how the Constraint Programming (CP) technologies are able to recover the tactical plans at the operational level in order to efficiently help answer regulation needs after disruptions. Precisely, it is shown how a constraint model of the UTCSP can be simply modified to address the UTCRP, its operational reformulation.

The paper is composed of six parts. The present one has introduced the context of the UTCSP and has motivated the opportunities for the CP technologies to tackle the rescheduling problem, namely the UTCRP. Next, Section 2 is dedicated to the related works and the operational context of the UTCSP. Section 3 is the main section of the paper. A CP model for the UTCSP is first presented. Next, it is shown how simple modifications of the UTCSP lead to a model for the UTCRP. Then, Section 4 introduces a common search strategy for UTCSP and its operational version. Section 5 reports the empirical evaluations of both UTCSP and UTCRP on industrial instances. Finally, Section 6 concludes.

Related Works and Operational context

The most widely used approach for this problem is related to a set partitioning problem [START_REF] Caprara | Algorithms for the set covering problem[END_REF]. It is mainly solved by a Branch-and-Bound algorithm where the lower bound is computed through a column generation procedure [START_REF] Desaulniers | Column generation[END_REF]. In this case, the subproblem corresponds to an Elementary Shortest Path Problem with Resource Constraints (ESP-PRC). Most of the time, it is solved using dynamic programming [START_REF] Feillet | An exact algorithm for the elementary shortest path problem with resource constraints: Application to some vehicle routing problems[END_REF][START_REF] Irnich | Shortest path problems with resource constraints[END_REF]. However, according to the number and the nature of the resource constraints, generating a pool of column during a preprocessing may sometimes be more efficient [START_REF] Jacquet-Lagrèzel | Horaires de chauffeurs de bus[END_REF]. The resulting Branch-and-Price algorithm is designed to be an exact model which, nevertheless, may fail to optimally solve very large problems (with more than thousands tasks) [START_REF] Chen | An improved column generation algorithm for crew scheduling problems[END_REF][START_REF] Desrochers | A column generation approach to the urban transit crew scheduling problem[END_REF][START_REF] Peixoto | A study of different metaheuristics to solve the urban transit crew scheduling problem[END_REF]. To bypass this issue, a common approach consists in truncating the search tree [START_REF] Franck | Truncated branch-and-bound, schedule-construction, and schedule-improvement procedures for resource-constrained project scheduling[END_REF], the lower bound quality is ensured then to be the nearest possible from optimal solutions. Even if many meta-heuristic algorithms have been proposed for the UTCSP [START_REF] Peixoto | A study of different metaheuristics to solve the urban transit crew scheduling problem[END_REF][START_REF] Forsyth | An ant system for bus driver scheduling1[END_REF][START_REF] Helena R Lourenc ¸o | Multiobjective metaheuristics for the bus driver scheduling problem[END_REF][START_REF] Shen | Tabu search for driver scheduling[END_REF], the most efficient industrial softwares such as Austrics, 1 Hastus,2 GoalDriver3 or LP-EasyDriver4 are based on a truncated version of the exact Branch-and-Price algorithm.

Constraint programming attempt. The UTCSP has not been many studied by the CP community. The most related work is [START_REF] Tallys | A hybrid approach for solving large scale crew scheduling problems[END_REF]. It introduced a pure CP model for a variant of the UTCSP, as well as a hybrid approach composing a column generation algorithm with a CP model dedicated to generate the columns. Nevertheless, the proposed pure CP approach is quite poor because it does not embed global constraints, like REGULAR, to express the regulation needs. Moreover, the search strategy is limited to the first-fail principle which is clearly inappropriate. Consequently, not more than 30 tasks can be scheduled to optimality and feasible solutions can be provided with only at most 125 tasks. However, the main contribution is related to the hybridization. Thus, it is shown that the hybrid approach (combining column generation and CP) can be efficient up to 150 tasks.

Operational context. To the best of our knowledge, there is no literature dealing with the rescheduling of the UTCSP. So, we present an industrial point of view which motivates our proposal. Disruptions can occur a few days before the buses run. Typically, some special event is programmed in the city, like a football match or local roadworks and the bus schedules must be adapted "accordingly". Without loss of generality, tasks to be performed by the buses can either be added, deleted or their duration modified. In some cases buses have connections with trains at the coach station, and a slight change of the train timetable (imposed by French Railways for instance) can have an impact on the bus task at this place in the network. From an operational point of view, a compromise must be found by the planning team: on the one hand, building a new cost-efficient schedule covering the modified tasks and, on the other hand, building a schedule not "too different" from the regular daily schedules. For bus drivers, new workdays might have an impact on either security or comfort, for instance when the new workday ends later than usual.

Today, tools are mainly focussed on finding optimal solutions and tends to produce -when tasks are altered -bran-new schedules that "destroy" the daily ones. In practice, human operators adopt a workaround strategy and manually fix large parts of the schedule, asking the solver tool to optimize only subparts of the schedule. This approach clearly avoids too much perturbation but consequently offer suboptimal solutions. It is also a time consuming effort. The UTCRP consists then in managing a good balance between cost optimality and schedule updates, with dedicated metric and constraints on top on the classical ones. On the converse to pure tactical crew scheduling tools, the computational time of rescheduling must be short, as various trials and errors can have to be experimented by the planning team which is evaluating the impact of what-if scenarios.

Constraint programming model

In the UTCSP, tasks have to be assigned to bus drivers. Formally, given n tasks that are to be assigned to at most m bus drivers, the objective is to find a full assignment that minimizes the cost and satisfies breaking rules and chaining rules. A bus driver is mainly characterized by a unique identifier, a skill level, among novice and expert, and a workday duration and a hourly cost induced by its skill. A novice can only execute low level skill tasks while an expert can execute any type of tasks and can work longer but at a higher hourly cost. The maximal number of novices, nbNovices and experts, nbExperts, needed to trivially satisfy the problem is determined from the input. A same bus driver can perform many tasks, but one task is only processed by a single bus driver. Each task is defined by its fixed beginning time, B i , its fixed duration, D i , its fixed end time, E i such that, E i = B i + D i , ∀i ∈ 1..n. The initially unknown bus driver performing the task is denoted A i . In addition, a task requires exactly one skill, K i : either high level skill, only experts can do it, or low level skill, anyone can do it.

A first constraint is that tasks assigned to the same bus driver should not overlap in time. A second constraint deals with the allocation of breaks to each bus driver. This is done by stating a SHIFT constraint, which we now describe.

Definition 1 (Shift).

A shift is a maximum sequence of tasks assigned to a same bus driver such that the time gap between any two consecutive tasks is shorter than a given threshold minBreak .

Two consecutive shifts of a bus driver are separated by a break of minimum duration minBreak and define its workday. Two consecutive tasks of a shift are separated by a gap shorter than minBreak . The span of a shift is the difference between the end time of its last task and the beginning time of its first task. It is bounded by a given threshold maxSpan.

Modeling the SHIFT constraint

Concisely expressing constraints like SHIFT is hard and has been recently studied.

In [START_REF] Beldiceanu | A modelling pearl with sortedness constraints[END_REF], a possible model is presented to manage the SHIFT problem based on the REGU-LAR [START_REF] Pesant | A regular language membership constraint for finite sequences of variables[END_REF] and GLOBAL CARDINALITY [START_REF] Régin | Generalized arc consistency for global cardinality constraint[END_REF] constraints. We do not report such a model in this paper because of its time generation and memory consumption. Another interesting model, also introduced in [START_REF] Beldiceanu | A modelling pearl with sortedness constraints[END_REF], addresses most of the pitfalls of the previous one: a STABLEKEYSORT-based model. In such a model, a decomposition of the SHIFT constraint is expressed as a conjunction of a STABLEKEYSORT constraint and simple arithmetical and logical constraints. The STABLEKEYSORT(L, P, S, k) constraint is declared with:

-L = A i , B i , D i , E i | i ∈ 1.
.n]: a list of task attribute tuples, -P : an optional permutation list (not required here),

-S = A i , B i , D i , E i | i ∈ 1.
.n]: a stable and non decreasing rearrangement of L, k = 2: number of first positions to consider in the tuples. Doing so, the STABLEKEYSORT(L, P, S, k) provides a view of tasks of L in which tasks are sorted by workdays. Hence, it eases the expression of the required constraints which can be directly expressed on sorted variables, while their expression could be more tedious otherwise.

Decomposition 1 depicts how the SHIFT constraint is expressed using a STABLE-KEYSORT constraint in the state-of-the-art. Constraint (1) ensures tasks integrity. Constraint (2) maintains the rearrangement of task attribute tuples, here only A i and B i are considered to sort tuples. Constraint (3) ensures that two consecutive tasks either belong to the different workdays or are chronologically ordered in the same workday and thus do not overlap in time. Constraints (4) and constraint (5) introduce auxiliary 0..1 variables: Y i indicates whether two consecutive tasks i -1 and i are in the same workday, X i indicates if two tasks i -1 and i are in the same shift. Finally, a last set of auxiliary 1..maxSpan variables R i are needed in constraint [START_REF] Feillet | An exact algorithm for the elementary shortest path problem with resource constraints: Application to some vehicle routing problems[END_REF], they compute the shift length up to the end of task i.

SHIFT([ Ai, Bi, Di, Ei | i ∈ 1..n], [ A i , B i , D i , E i , Yi | i ∈ 1..n], minBreak , maxSpan) ⇔ (1) ∧ (2) ∧ (3) ∧ (4) ∧ (5) ∧ (6) Ei = Bi + Di, ∀i ∈ 1..n (1) STABLEKEYSORT([ Ai, Bi, Di, Ei | i ∈ 1..n], [ A i , B i , D i , E i | i ∈ 1..n], 2) (2)
A i-1 < A i ∨ E i-1 ≤ B i , ∀i ∈ 2..n (3)
Yi = 1 ⇔ false if i = 1 A i = A i-1 if i ∈ 2..n (4) Xi = 1 ⇔ false if i = 1 Yi ∧ B i -E i-1 < minBreak if i ∈ 2..n (5) Ri = D i if i = 1 D i + Xi • (Ri-1 + B i -E i-1 ) if i ∈ 2..n (6)
where Ri ∈ 1..maxSpan, ∀i ∈ 1..n Decomposition 1: Decomposition of the SHIFT constraint [START_REF] Beldiceanu | A modelling pearl with sortedness constraints[END_REF].

A constraint-based model for the UTCSP

The targeted problem is based on a central SHIFT constraint but also comes with the following additional variables and constraints, presented in Model 1. For modeling purpose, bus drivers whom unique identifier is in 1..nbNovices are novices, those with greater identifier, up to nbNovices + nbExperts, are experts. Hence, constraint (8) introduces the auxiliary 0..1 variables K i which indicate whether the bus driver performing the task i is an expert. P i , W i , C i denote respectively the period between two consecutive tasks i -1 and i, stated by constraint [START_REF] Gent | Data structures for generalised arc consistency for extensional constraints[END_REF], the duration of a shift including task i, stated by constraint [START_REF] Jacquet-Lagrèzel | Horaires de chauffeurs de bus[END_REF], and the cost of a bus driver's workday, stated by constraint [START_REF] Helena R Lourenc ¸o | Multiobjective metaheuristics for the bus driver scheduling problem[END_REF]. Note that the workday duration is bounded thanks to that last constraint. It is expressed with a TABLE [START_REF] Gent | Data structures for generalised arc consistency for extensional constraints[END_REF] constraint wherein possible combinations of skill and workday and costs are listed. Constraint [START_REF] Irnich | Shortest path problems with resource constraints[END_REF] ensures that, in a shift, the period between two tasks i -1 and i is less than or equal to the maxSpan. Based on an a priori analysis of the tasks network, ALLDIFFERENT [START_REF] Régin | A filtering algorithm for constraints of difference in csps[END_REF] constraints (13) make sure that tasks belonging to the same clique are performed by different bus drivers. Finally, the model is improved by considering symmetries and their related symmetry breaking constraints, depicted by Model 2. First, we introduce M R i and M E i which maintain, for each skill, the list of identifier of used bus drivers (constraints ( 14) and ( 15)), they also help to count the number of bus drivers per skill. Then, constraints ( 16), ( 17), ( 18) and ( 19) break symmetries.

A constraint-based model for the UTCRP

In order to stay close to the operational context of the bus networks, we have built instances for the UTCRP based on disruptions of UTCSP instances. Indeed, a daily schedule is considered as known, and the tasks' disruptions occur locally. So, the disruptions can be easily simulated from the UTCSP instances. We have to keep in mind that these disruptions can either be:

Minimize n i=1 Ci (7) subject to SHIFT([ Ai, Bi, Di, Ei | i ∈ 1..n], [ A i , B i , D i , E i , Yi | i ∈ 1..n], minBreak , maxSpan) Ki = 1 ⇔ A i > nbNovices, ∀i ∈ 1..n (8)
Pi = 0 if i = 1 B i -E i-1 if i ∈ 2..n (9)
Yi ⇒ Pi ≤ maxSpan, ∀i ∈ 2..n (10)

Wi = D i if i = 1 D i + Xi • (Wi-1 + Pi) if i ∈ 2..n (11)
Ci = (1 -Yi+1) • cost K i ,W i if i ∈ 1..n -1 cost K i ,W i if i = n (12) Redundant constraint ALLDIFFERENT([A k | k ∈ clique(A)]) (13)
where Consequently, UTCRP aims at producing new schedules which: a) cover all the tasks (previous and new); b) do not differ too much in terms of cost from the initial daily schedule; and c) ensure that the workday modifications are minimal for the bus drivers. These schedules have to be produced within a few seconds of computational time so that the planning team can test different scenarios and choose the most convenient one.

From a constraint programming point of view, Model 3 presents how to turn the UTCSP model, depicted in Section 3.2, into UTCRP. Given I s , an assignment of n tasks to m bus drivers, let us denote C its cost. Consider that some of the tasks have been disrupted and that I s has to be repaired. First, the previous objective function, equation [START_REF] Forsyth | An ant system for bus driver scheduling1[END_REF] in Model 1, is turned into the hard constraint [START_REF] Tallys | A hybrid approach for solving large scale crew scheduling problems[END_REF]: new solutions have to be at most % above C . Then, for each set of tasks performed by the same bus driver in I s , the number of bus drivers needed to perform the same tasks anew is maintained by constraint (22). Finally, the objective function [START_REF] Peixoto | A study of different metaheuristics to solve the urban transit crew scheduling problem[END_REF] aims at computing solutions similar to I s in term of unmodified workdays.

M R i =      A i if Ki = 0 0 if Ki = 1 ∧ i = 1 M R i-1 if Ki = 1 ∧ i ∈ 2..n (14)
M E i =      A i if Ki = 1 nbNovices if Ki = 0 ∧ i = 1 M E i-1 if Ki = 0 ∧ i ∈ 2..n (15) M R i-1 ≤ M R i , ∀i ∈ 2..n (16
)

M E i-1 ≤ M E i , ∀i ∈ 2..n (17)
INTVALUEPRECEDECHAIN(A, [j | j ∈ 1..nbNovices]) (18)
INTVALUEPRECEDECHAIN(A, [j | j ∈ nbNovices..nbNovices + nbExperts]) (19)
where ∀i ∈ 1..n,

M R i ∈ 1..nbNovices, M E i ∈ nbNovices..nbNovices + nbExperts

Search strategy

A constructive search strategy is defined to dive to a first solution quickly without failure. The A i variables are selected in lexicographic order. The bus driver that performs a task A i is computed as follow. Bus drivers already performing at least one task are considered first. Those whose workday is not directly compatible with the task to assign (w.r.t. either break rules or chaining rules cannot be satisfied) are ruled out. Remaining ones are then tried sequentially. Some more tries are finally considered, allowing the addition of at most one bus driver per type of skill required. Next solutions are obtained by plugging Large Neighborhood Search (LNS) [START_REF] Shaw | Principles and Practice of Constraint Programming -CP[END_REF] in, a straightforward two-phase local search-like approach. It partially relaxes a given solution and tries to repair it. Given an input solution, the relaxation phase builds a partial solution: some variables are selected to be relaxed to their initial domain, while the other ones are assigned to their value in the solution. The reparation phase tries to extend the partial solution to a complete one that improves the objective function.

The efficiency of LNS lies in the way variables are selected to be relaxed. In our case, it tries to find a better solution by locally rearranging bus drivers' workday. So, the workdays are first extracted from the A variables in a solution. Then, up to θ ∈ [ [START_REF] Caprara | Algorithms for the set covering problem[END_REF][START_REF] Desaulniers | Column generation[END_REF]] workdays are randomly selected to be rearranged in such way that they overlap in time at least one of the other selected one. The corresponding A variables are relaxed, the other ones are assigned to the same bus driver as declared in the solution. To consolidate even more the partial solution, bus drivers' identifier corresponding of fixed A variables are removed from relaxed A variables' domain. The process is completed with a fast restart strategy [START_REF] Perron | Fast restart policies and large neighborhood search[END_REF], which limits the reparation phase to 2n failures and avoids spending too much time in hard-to-repair partial solutions.

The same search strategy is applied to both the UTCSP and the UTCRP. The strategy was initially designed to produce few dense workdays, with a local reasoning. When the objective changes to repair assignments, the model is constrained enough to guide the process towards workdays similar to the initial ones.

Practical Experiments

For an empirical evaluation, we instantiate the UTCSP model of Section 3.2. The SHIFT constraint holds for a 15-minute minBreak and 2-hour maxSpan. A novice cannot work more than 8 hours and its hourly cost is fixed to 10. An expert cannot work more than 9 hours, and its hourly cost is fixed to 17. We consider here the following additional rule: any working bus driver has a minimal 6-hour pay, even if its workday lasts for less than 6 hours. This constraint is directly encoded into constraint [START_REF] Helena R Lourenc ¸o | Multiobjective metaheuristics for the bus driver scheduling problem[END_REF]. Finally, the UTCRP model depicted in Section 3.3 also instantiates , the distance to C , to 10.

There are two sets of instances. A first set aims at comparing the constraint-based approach with the state-of-the-art one on the UTCSP. It is composed of real-world problem instances involving up to 3,200 tasks. A second set aims at evaluating how repairing a disrupted assignment is made easy with a constraint-based approach using the UTCRP (Model 3). First of all, the best solution found by EURODECISION for the instance with 800 tasks is selected, it is composed of 160 workdays. Then, this instance is disrupted applying the following process: a task j is randomly selected to be removed. Starting from j, within a range of β ∈ {30, 60, 90, 120} minutes before B j and after E j , tasks on a path to j, that is w.r.t. breaking rules and chaining rules, are removed. Tasks that overlap the range in time are reduced (either B i or E i is modified). The selection-and-removing phase is repeated α ∈ [ [1..5]] times. This results in a set of 20 instances to be repaired with up to 25% disrupted tasks, which corresponds to our real life context.

Protocol. The experiments were run on a Mac Pro with 8-core Intel Xeon E5 at 3 GHz under MacOS 10.11.3 and Java 1.8.0 25. Each instance of the first set was run with a 2-hour limit on its own core, and each instance of the second set was run with a 1minute limit on its own core for CP approach. LP-EasyDriver was evaluated in a 2-hour limit for both sets. All of them were run with up to 4 GB of memory. The tools used are: Choco [START_REF] Prud'homme | Choco Documentation[END_REF] for the constraint programming part, while EURODECISION provides LP-Taskplanner, a generic framework for ESPPRC based column generation models, relying on CPLEX 12.6.2. LP-Taskplanner is embedded in LP-EasyDriver: the latter handles every business aspects while the former deals with the optimization parts. In the following, this tool will be referred to as LP-EasyDriver.

Solving the UTCSP

In this section, we compare the constraint-based approach for the UTCSP with the one being used by EURODECISION and introduced in Section 2. The results are given in Table 1 which is divided into four parts. The first part indicates the instance size n. The second part reports information related to the constraint-based approach: the time to get the first solution (time, in seconds), its cost (first) and the best cost obtained in 120 minutes (best@120). The third part is about LP-EasyDriver: the time (time, in seconds) to compute the lower bound (LB) and a first upper bound (UB) then the best upper bound in 120 minutes (UB@120). Finally, the last part reports, when possible, the ratio

g(a, b) = a b × 100
where, here, a = (best@120 -UB@120) and b = UB@120. In addition, "N/A" denotes that the information is not available or applicable, "OOT" denotes that a particular approach runs out of time. 1. Empirical results on the UTCSP: time (time, in seconds) for finding the first solution (first) and the best solution found in 120 minutes (best@120), and for finding the root lower bound (LB), the first upper bound (UB), and the best upper bound in 120 minutes (UB@120).

The ratio of best@120 to UB@120 is also reported (Gap, in %).

We observe that:

-The truncated Branch-and-Price approach fails at solving large problems. This observation confirms the state-of-the-art (Section 2). The bounds produced are very sharp considering a 2-hour time limit. Indeed, such a time limit is below from what is commonly allocated at the tactical level. -The constraint-based approach is able to find a first solution for any problem size.

Moreover, the ability of the model to scale up combined with a constructive search strategy makes possible to provide a first solution in a very short time. Nonetheless, the quality of the best solution found in the given time limit to decrease when the instance size n becomes larger. In the end, without any evaluation -neither propagation -of the lower bound, the approach fails at proving optimality.

At a tactical level, where the run time matters less than the quality of the bounds, and up to mid-size problems, the state-of-the-art approach is suitable. On any-size problems, the constraint-based approach is responsive, yet less accurate. At an operational level, when rescheduling tasks is needed, responsiveness becomes indispensable. This is the point we want to evaluate in Section 5.2. e

Solving the UTCRP

When a disrupt occurs, its extent is measurable thanks to three indicators: the number of removed or modified tasks (n ), the number of bus drivers whose workday is disrupted (m ) and the total number of affected tasks, gathering all tasks of disrupted workdays (n ). In practice, when dealing with rescheduling, not all the tasks are set as input of the model depicted in Section 3.3. Indeed, only affected tasks are considered. The remaining workdays are immutable. Recall that the solution selected to be disrupted was made of n = 800 tasks and m = 160 bus drivers (or workdays).

We report the results of constraint-based approach for the UTCRP. The results are given in Table 2 which is composed of four parts. The first part indicates the disruption parameters β and α and the indicators of its extent n , m and n . The second part reports information related to the constraint-based approach: the time to get the first solution (time, in seconds) and gap@first that is the ratio g(m-m +m 1 , m) where m 1 is the number of workdays covering the n tasks (recall that the unaffected workdays are kept immutable). This qualifies the distance from the first solution found to the initial one in terms of number of modified workdays needed to cover the affected tasks. The ratios based on the best solution found in ten seconds (gap@10) and 60 seconds (gap@60) relative to the initial solution are also reported. Then, the cost of the best solution found in 60 seconds is indicated (cost@60). For indication purpose, the results found with LP-EasyDriver when solving the problems from scratch, without any time limit, are reported in the third part (best). Finally, the ratio g(cost@60best, best) can be observed in the last part (Gap). In addition, "-" denotes that no better solution was found in the elapsed time.

Our observations are threefold:

-The constraint-based approach is responsive as expected. Its ability to quickly reschedule disrupted workdays does not depend on the extent of the disruption.

Inst. parameters

Constraint-based approach LP-EasyDriver Gap β α n m n time gap@first gap@10 gap@60 cost@60 best Table 2. Empirical results on the UTCRP: range of the disruption (β) along a path involving a randomly selected task, number of randomly selected tasks (α), number of directly affected tasks (n ) and bus drivers (m ) and total number of affected tasks (n ), time (time, in seconds) for finding the first solution, the ratio of the number of workdays needed in, respectively, the first solution (gap@first, in %), the solution found in 10 seconds to (gap@10, in %) and the solution found in 60 seconds (gap@60, in %) to the initial number of workdays required, the cost of the solution found in 60 seconds (cost@60), and the ratio of cost@60 to the best solution (best) computed from scratch with the LP-EasyDriver (Gap, in %).

30
Nevertheless, the quality of the reparation is related to the total number of the affected tasks n . -The search strategy is able to compute good quality solutions in term of number of workdays. When less than 83 tasks are affected by the disruption, ten seconds are enough to provide solutions at most 5% above the number of workdays needed before the disruption. When more tasks are affected, sixty seconds are always enough to find solution at most 5% above the initially required number of workdays. -The search strategy is able to compute good quality solution in terms of cost. The best solutions found with CP are very close to the ones computed from scratch by LP-EasyDriver. Nevertheless, the EURODECISION approach fails at maintaining the workdays which are not impacted by disruptions. Consequently, such approach does not fit the needs of the planning teams.

Conclusion and Further Works

In the context of scheduling bus drivers' workdays, this paper first described a constraint-based model for the UTCSP. We empirically confirmed that this CP model scaled up to 3,200 tasks and was able to quickly build good quality solutions. Nevertheless, due to a lack of any lower bound integration, it struggled to improve them enough to reach and prove optimality. Second, we highlighted CP flexibility by turning the UTCSP into its operational counterpart, the UTCRP. The ability of the constraint-based approach to quickly reschedule workdays has been assessed on randomly disrupted instances. Indeed, the closeness of the new scheduling to the initial one in terms of both workdays and cost bears out that CP is a serious runner for that problem.

The responsiveness and the good quality of the solutions found by the CP approach for UTCRP offer new perspectives and advocates for field experiments. EURODECI-SION will launch pilots with groups of customers in order to measure the ease of use of repaired solutions together with the real life metrics used by the planning teams. Our goal is to reduce the hassle imposed to the planners by the disruptions without loosing too much cost compared to optimality.

Model 1 :----

 1 ∀i ∈ 1..n, Pi ∈ -dayDuration..dayDuration, Wi ∈ 0..dayDuration, Ci ∈ 0..dayDuration × costExpert, Formulation of the Urban Transit Crew Scheduling Problem. Creation of new tasks: the bus line has to serve the stadium station at 1 o'clock in the morning, due to a rock concert; Task deletion: a roadwork prevents the bus to use the road and stop at the station; Duration change: roadworks slow down the traffic; Start time change: a train timetable has been modified, and the connection forces a change of the bus departure time.

Model 2 :Model 3 :

 23 Improving Model 1 with symmetry breaking constraints. Modifications to bring to the CP model of UTCSP to turn it into UTCRP.

http://www.trapezegroup.com

http://www.giro.ca

http://www.goalsystems.com

http://www.eurodecision.fr

Acknowledgements

The authors are supported by the French common-laboratory grant "TransOp" involving the TASC team and EURODECISION.


Jade Georis-Creuseveau
email: jade.georis-creuseveau@univ-brest.fr

Roger Longhorn
email: rlonghorn@gsdi.org

Joep Crompvoets
email: joep.crompvoets@soc.kuleuven.be

Survey of National Coastal and Marine geoportals: European developments

HAL is

Introduction

The intensity and magnitude of current changes that characterize marine and coastal areas demand the implementation of collective information processes and tools to enhance knowledge and management of these territories [START_REF] Cicin-Sain | Integrated coastal and ocean management: concepts and practices[END_REF].

Since the 1990's, the role of spatial data and information in decision making in many sectors (including the marine and coastal environment) has led to the development of Spatial Data Infrastructures (SDIs) to better manage and share spatial data [START_REF] Crompvoets | Assessing the worldwide developments of national spatial data clearinghouses[END_REF]. The development of SDIs responds to international needs to facilitate access, exchange, and sharing of spatial data held by many stakeholders to maximize its use and re-use, while reducing the cost of management and production. Coastal and Marine SDIs have been gradually implemented in order to improve the accessibility and the availability of spatial data related to marine and coastal areas at various levels [START_REF] Longhorn | Coastal spatial data infrastructure[END_REF][START_REF] Canessa | Spatial Information Infrastructure for Integrated Coastal and Ocean Management in Canada[END_REF]. A favourable context for increasing efficiency in spatial data production and improving availability and accessibility is being put in place to support the international concepts of Integrated Coastal Zone Management (ICZM) [START_REF] Cicin-Sain | Integrated coastal and ocean management: concepts and practices[END_REF] and Maritime Spatial Planning (MSP) [START_REF] Douvere | The importance of marine spatial planning in advancing ecosystem-based sea use management[END_REF]. These concepts aim to deliver a sustainable approach to the management of coastal zones, oceans and seas, across sectors, between different levels of government, and across jurisdictional boundaries. These concepts and related regulations (e.g. European Commission (2008), European Commission (2013), European Commission ( 2014)) act as the catalyst for increasing production, access, sharing, use and integration of coastal and maritime spatial data in order to inform ICZM/MSP decision making [START_REF] Iho | Spatial Data Infrastructures[END_REF]. The needs for coastal and marine SDIs are becoming a priori evident.

Within this context, a European Web survey aims to assess worldwide developments of existing national marine and coastal geoportals for SDIs or similar Web services, as such inventory does not exist today. The geoportal is the central web gateway targeting the facilitation of spatial data discovery, access, and related added-value services [START_REF] Crompvoets | Assessing the worldwide developments of national spatial data clearinghouses[END_REF][START_REF] Maguire | The emergence of geoportals and their role in spatial data infrastructures[END_REF]. The existence and status of such portals represents one of the ways to assess the processes implemented by a country to enable the access and use of coastal and marine geographic data. This paper presents the first preliminary results of the European survey undertaken.

Scope and methodology

The survey focuses on geoportals implemented by national public bodies in Europe enabling the access and the use of geographic data specifically related to marine and/or coastal zones. Within the framework of the survey, the term "data" encompasses a broad range of items such as real-time observations, time series data, GIS data layers, digital maps, etc.

Geoportal inventory

The inventory was conducted in November 2014. The list of geoportals was gathered by browsing of the Internet with the help of specific monitoring tools: Google alert, Mention and Netvibes. Based on a literature review, several keywords were used in English, Spanish and French to parameterize the tools in order to encompass the diversity of vocabulary used (e.g. coastal, marine, geoportal, SDI, data centre, etc...). The approach also included a scan of various types of information sources related to geomatics and coastal/marine domains (e.g. fromGlobal Spatial Data Infrastructure Association (GSDI), International Coastal Atlas Network(ICAN), International Hydrographic Organization (IHO), International Oceanographic Data and Information Exchange (IODE), INSPIRE conference, COASTGIS conference, etc.)

Geoportal characteristics measurements

In order to provide a general overview of the existing geoportals, the following eleven characteristics were sourced from the geoportal Web pages in November 2014 and again in March 2015 in order to monitor current developments. The choice of the characteristics is based on similar geoportal assessments [START_REF] Crompvoets | Assessing the worldwide developments of national spatial data clearinghouses[END_REF]EUROGI/eSDI-Net+, 2011) and on specificities recognized useful for ICZM and MSP processes [START_REF] Dwyer | Potentials and limitations of Coastal Web Atlases[END_REF][START_REF] Wright | Coastal informatics: web atlas design and implementation[END_REF].

Results

Thirty-fivesurveyed geoportals are implemented by national public organizations providing access to coastal and/or marine spatial data. Mostof the surveyed geoportals were implemented around mid 2000's1 . It appears that most (94 %) of the geoportals of the non-English countries provide information in two (88 %) or more languages (12 %).

Typology

Given the diversity of the geoportals inventoried, the first step of the survey was to establish a typology to guide the ongoing analysis of the different characteristics of the geoportals.

The comparison of characteristics 4, 6, 8 and 9 (see above) suggests a typology resulting into the following four types of geoportal: Atlas-like geoportals, (2) Hydrographic Office geoportals, (3) Oceanographic/Marine Data Centre geoportals, and (4) Hybrid geoportals.

Atlas-like geoportals

Within the framework of the IOC IODE International Coastal Atlas Network (ICAN) Project,a coastal web atlas (CWA) is defined as "a collection of digital maps and datasets with supplementary tables, illustrations and information that systematically illustrate the coast, often with cartographic and decision support tools, all of which are accessible via the Internet" [START_REF] Cummins | Report on Coastal Mapping and Informatics Trans-Atlantic Workshop 1: Potentials and Limitations of Coastal Web Atlases[END_REF]. This type gathers the 3 geoportals of national atlases of the ICAN network together with 6 other Atlas-like geoportals (26 % of the total number of surveyed geoportals).

Hydrographic Office geoportals

The second type includes 9 geoportals (26 % of the total number of surveyed geoportals) that are mainly implemented by national Hydrographic Offices, organizations which are historically devoted to surveying and charting seas, oceans and navigable waters for purposes of maintaining safety of life at sea. International cooperation between these Hydrographic Offices led to the creation of the International Hydrographic Organization (IHO) in 1921.

3.1.3.Oceanographic/Marine Data Centre geoportals

In the third category,13 geoportals were identified (37 % of the total number of surveyed geoportals). They correspond mainly to National Oceanographic Data Centres(NODC)and other Marine Data Centres. The NODCs have been progressively implemented by the IOC's International Oceanographic Data and Information Exchange (IODE) since the 1960's. At the European level, the NODC and other Marine Data Centreshave been gathered in theSeaDataNet network, a Pan-European network providing on-line integrated databases since the end of the 2000's.

Hybrid geoportals

In addition to the 31 geoportals classified into these three first types, 4 geoportals (11 % of the total number of surveyed geoportals) were identified as Hybrid geoportals sharing the characteristics of the types 1 and 3.

Characteristics

In the sections below, the results of the measurements of the 11 characteristics are providedaccordingly to the five components proposed by [START_REF] Rajabifard | Future directions for SDI development[END_REF].In addition for the general portrayal of the 35analyzed geoportals,thesecharacteristics allowa detailed description of the four types of geoportal. The characteristicswere measured two times (November 2014 andMarch 2015). Except for the number of datasets which are only slightly increasing, the other characteristics are very stable over time.

Data

The component Data is described by the data theme(4) and the number ofgeographicaldatasets (5).

The analyzed geoportals provide variousdigital data within diverse domains including administration (63 %), physical (77 %), biological (54 %) and human (46 %) aspects (Figure 1).They combine reference data as well as business data. Atlas-like geoportals provide a large diversity of data themes describing administrative (100 %) and biophysical (100 % and 67 %) aspects (e.g. marine biology, biodiversity, etc.) of the coastal and marine zones along with human uses (89 %) (e.g. pollution related topics, tourism, etc.).The main data accessible on the Hydrographic Office geoportals are related to administrative (100 %) (e.g. Electronic Nautical Charts, lists of lights, sailing directions) and a few to physical aspects (11 %)(e.g. tidal stream data, etc).This data is also often made available via distribution agents. Oceanographic/Marine Data Centre geoportals enable access to data mainly related to physical, chemical, geology, oceanography (100 %) or marine biology (69 %) collected from marine cruises, buoys, tide gauge stations, and satellites. The data provided by Hybrid geoportals are variable with a majority of data related to the physical (100 %) and biological (100%) themes (Figure 1).

Notes: For each data theme, 100 % means that all the geoportals of this type (e.g. 9 for the Atlas-like geoportals) provide data of this theme

Figure 1. Data Themes

The variety in the number of datasets between different geoportals is high: 67 %of Atlas-like geoportals give access to 100-500 datasets while 69 % of the Oceanographic/Marine Data Centre geoportalsallow the user access to more than 1000 datasets and 67 % of the Hydrographic Office geoportals to less than 500 datasets.

These differences can be explained by the specificity of the data in each type of geoportal.The Oceanographic/Marine Data Centre geoportals gather datasets corresponding mainly to vertical profiles of physical, chemical and biological data from numerous cruises and collected by different equipment. In the Atlas-like geoportals, the datasets correspond to GIS layers while in Hydrographic Office geoportals, dataset are mainly nautical chart cells with defined limitations.The total number of datasets is not available for 9 % (November 2014)and 6 % (March 2015)of the analyzed geoportals. The numbers of datasets are slightly increasing between November 2014 and March 2015 research.(Notes: for each class of number of datasets, 100 % means that all the geoportals of this type (e.g. 4 for the Hybrid geoportals) provide datasetin this class Figure 2).

Policy

The component Policy is described by the level of openness (6) and licensing (7).

Referring to figure 3, 43 % of the analyzed SDIs provide free access for all. The remaining geoportals provide free access only for registered users (9 %), free access by using request forms (23 %), or provide access subject to fees (26 %). Although personal registration may be required for the portal (22 %), access to the data of the Atlas-like geoportals is mostly free for all (78 %). Registration gives the user access to various additional functionalities (e.g. online map saving, search request saving). Access to the products of the Hydrographic Office geoportals is fee-paying (100 %). For the Oceanographic/Marine Data Centre geoportals, the data are freely accessible for all (38 %) or through a request form (62 %). For the Hybrid geoportals, the data are freely accessible for all (75 %) or through a request form (25 %). Regarding licensing, the data policy depends of the type of geoportal.The Atlas-like geoportals provide the data under Open Licenses (22 %) or Specific Data Use Agreements (78 %). The data of the Hydrographic geoportals are under the General Conditions ofSale of the distribution agents (see 3.2.1).The data policy of the Oceanographic/Marine Data Centre geoportals is compliant with IOC Oceanographic Data Exchange Policy (IOC, 2015)or the SeaDataNet Data Policy and User license [START_REF] Seadatanet | SeaDataNet Data Policy and User License[END_REF].

Technology

The component Technology is described by the mechanisms for searching (8) and accessing (9).

Searching for spatial data a geoportal can be done through different mechanisms: catalog interfaces (60 %) that allow searching by means of keywords, production time, data theme, providers, etc.; map interfaces (70 %) for locating an area of interest or by clicking on an area with predefined boundaries and a list of products in hypertext (46 %) (Figure 4).

Access of the data over the geoportals can be performed in diverseways:viewing interfaces (71 %), downloadingservices (51%),web services which allow the user to have direct access to the geographic data (20%), data transmission via e-mail (37 %) or data purchasefrom certified distribution agents (e.g. retail stores) (26 %) (Figure 5).

To consult the data, the user of the Atlas-like geoportals has mainly access to map search mechanisms (100 %) and catalogue interfaces (67 %). Data access services are mainly based on viewing (100 %), downloading (89 %) and other web services (44%). Oceano. DC ( 13)

Hybrid ( 4)

The data of the Hydrographic geoportals are mainly accessible via a list of products in hypertext (89 %). Access to the products is generally from certified distribution agents (100 %).

For the Oceanographic/Marine Data Centre geoportals, the search page is in the form of a map (77 %) or a catalogue interface (69 %) based on criteria such as parameters, time period, cruise,etc. The data can be transferred by the data centre (100%), viewed (69 %) or downloaded (54%).

The Hybrid geoportalshave mainly map search mechanisms (100 %) and view access interface (75 %) (Figure 4 and5). Oceano. DC ( 13) Hybrid (4)

People

The People componentis characterized by the number of monthly visitors (10)and by the number of data suppliers (11).

The monthly number of users was not available online, expect for one Atlas-like geoportal.

Number of data suppliers is not for 54 % of the analyzed geoportals. Based on the available data, 43 % of the surveyed geoportals have less than 50 data suppliers.

Figure 6. Number of data suppliers

Standards

The Standard component is represented by the application of a metadata standard (12) in the geoportal.

All the Hydrographic Office geoportals provide metadata according to theIHO transfer standard for digital hydrographic data (S-57) [START_REF] Iho | [END_REF]. The remaining geoportals provide metadata in the INSPIRE Metadata standard (European Commission, 2009).

Discussion and preliminary conclusion

The preliminary results suggest that European developments are underway for geoportals enabling users to have access to various data concerning coastal and marine zones. At the present stage of the study, the 35 geoportals assessed can be classified in four types: Atlas-like geoportals, Hydrographic Office geoportals, Thesekinds of data and mechanisms are very stable between November 2014 and March 2015or are slightly increasing (e.g. number of datasets).These characteristics are very similar to the 'terrestrial-oriented' geoportals [START_REF] Crompvoets | Assessing the worldwide developments of national spatial data clearinghouses[END_REF]except the data themes and standards used.

Despite the integrated approach promoted by ICZM and MSP concepts and related regulations, the results indicate that platforms allowing access to a wide range of data related to marine, coastal and land territories are not commonly found. True data harmonisation and services interoperability, which are the underpinning principles for SDIs, need to be improved.

The main limitation of the survey concerns the fact that some information needed for assessment is not available online.In the near future, a questionnaire will be sent to the geoportal coordinators in order to assess real usage of the geoportals.

The research also provides some insights for further developments. The proposedgeoportals coordinators'survey can be extended to geoportal users in order to analyze what they do with the data in their day-to-day responsibilities and what are their needs.The combination of these approaches (geoportals, coordinators'and users' survey) should contribute to a Multi-View Framework [START_REF] Crompvoets | A Multi-View Framework to Assess Spatial Data Infrastructures[END_REF] in order to assess SDIs and their ability to matchthe sustainable approach to the management of the coastal zones, oceans and seas.

Figure 2 .

 2 Figure 2. Number of datasets

Figure 3 .

 3 Figure 3. Level of openness

Figure

  Figure 4. Search Mechanisms

  Data Centre geoportals, and Hybrid geoportals. Each type enables access to different kinds of data through various mechanisms.

This information is not available from the geoportal Web pages for 77 % of the surveyed geoportals.

Jade Georis-Creuseveau, Research fellow, French National Centre for Scientific Research (CNRS)

The implementation of SDIs in Natural Resource Management has been the focus of the career of Jade Georis-Creuseveau. She spent her first ten years in the professional sector working in this area. This experience led her to complete a PhD at the European Institute of Marine Studies (France). Her thesis focused on SDI usage among coastal and marine organizations. Currently, she is a Research fellow at the French National Centre for Scientific Research (CNRS). Her research focuses on the Information Governance for Sustainability/Environment by the analysis of spatial data infrastructures and volunteered geographic information.

Several key questions drive her research: Who are the SDI users? What do they do with available data? What are their needs? The framework of her experimental domain is coastal and marine management in France and at the international level. Jade is a member of several working groups focusing on (marine and coastal) geographical information usage: AFIGEO, GDR MAGIS Contact: jade.georis-creuseveau@univ-brest.fr and jade.georis-creuseveau@cnrs.fr