A Della Corte

A Battista

F Dell'isola

P Seppecher

Large deformations of Timoshenko and Euler beams under distributed load

Keywords: Mathematics Subject Classification. 74B20, 34B15, 49J45. Keywords. Nonlinear elasticity, Timoshenko beam, Euler beam, Stability of solutions of nonlinear ODEs

In this paper, the general equilibrium equations for a geometrically nonlinear version of the Timoshenko beam are derived from the energy functional. The particular case in which the shear and extensional stiffnesses are infinite, which correspond to the inextensible Euler beam model, is studied under a uniformly distributed load. All the global and local minimizers of the variational problem are characterized, and the relative monotonicity and regularity properties are established.

Introduction

Since the first mathematically consistent theory of Elastica, by Leonhard Euler and Bernoulli brothers (see [START_REF] Euler | Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes Sive Solutio Problematis Isoperimetrici Latissimo Sensu Accepti[END_REF][START_REF] Bernoulli | The 26th letter to Euler[END_REF][START_REF] Bernoulli | Quadratura curvae, e cujus evolutione describitur inflexae laminae curvatura[END_REF]), a great deal of effort has been devoted to the study of beam models, due to their theoretical relevance as a 1D elastic model as well as their importance in structural engineering both as a static and a dynamic element. A rich literature exists considering both linearized and geometrically nonlinear models [START_REF] Antman | Nonlinear problems of elasticity[END_REF][START_REF] Steigmann | Finite Elasticity Theory[END_REF][START_REF] Bisshopp | Large deflection of cantilever beams[END_REF][START_REF] Fertis | Nonlinear Structural Engineering[END_REF][START_REF] Ladevèze | Nonlinear Computational Structural Mechanics: New Approaches and Non-incremental Methods of Calculation[END_REF][START_REF] Steigmann | Invariants of the stretch tensors and their application to finite elasticity theory[END_REF] together with generalized 1D elastic models (see [START_REF] Nizette | Towards a classification of Euler-Kirchhoff filaments[END_REF][START_REF] Goriely | On the dynamics of elastic strips[END_REF]), among which beam models capable to take into account other deformations than deflection, extension and shear (see for instance [START_REF] Hamdouni | An asymptotic non-linear model for thin-walled rods with strongly curved open cross-section[END_REF][START_REF] Luongo | Mathematical Models of Beams and Cables[END_REF][START_REF] Piccardo | A perturbation approach to the nonlinear generalized beam theory[END_REF][START_REF] Taig | An unconstrained dynamic approach for the generalised beam theory[END_REF][START_REF] Piccardo | A complete dynamic approach to the generalized beam theory cross-section analysis including extension and shear modes[END_REF]). However, two aspects in geometrically nonlinear beam theory especially require further investigation: the behavior of the system under a distributed load and the related multiplicity of arising solutions. An analysis in these directions has been started in the work [START_REF] Della Corte | Equilibria of a clamped euler beam (elastica) with distributed load: large deformations[END_REF], where an inextensible Euler beam has been considered. In the present work, we extend the results presented in [START_REF] Dell'isola | Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium[END_REF], introducing a geometrically nonlinear generalization of an extensible Timoshenko beam under a distributed load and characterizing all the stable equilibrium configurations of the inextensible Elastica. The necessity of considering distributed loads in large deformation arises, for instance, in the field of fluid-structure interaction [START_REF] Bungartz | Fluid-Structure Interaction: Modelling, Simulation, Optimisation[END_REF][START_REF] Bazilevs | Computational Fluid-Structure Interaction: Methods and Applications[END_REF][START_REF] Solaria | A numerical algorithm for the aerodynamic identification of structures[END_REF][START_REF] Pagnini | A numerical approach for the evaluation of wind-induced effects on inclined, slender structural elements[END_REF][START_REF] Liberge | Reduced-order modelling by pod-multiphase approach for fluid-structure interaction[END_REF] or in the framework of microstructured continua [START_REF] Pideri | A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium[END_REF][START_REF] Forest | Nonlinear microstrain theories[END_REF], in particular when the microstructure can be modeled as an array of fibers that can be individually modeled as beams undergoing large deformations, while the interaction with the remaining part of the array can be modeled as a distributed load acting on the beam. Possible examples are the structures described in [START_REF] Dell'isola | Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium[END_REF][START_REF] Atai | On the nonlinear mechanics of discrete networks[END_REF][START_REF] Boutin | Linear pantographic sheets: asymptotic micro-macro models identification[END_REF][START_REF] Placidi | Identification of two-dimensional pantographic structure via a linear d4 orthotropic second gradient elastic model[END_REF][START_REF] Scerrato | Three-dimensional instabilities of pantographic sheets with parabolic lattices: numerical investigations[END_REF][START_REF] Turco | Large deformations induced in planar pantographic sheets by loads applied on fibers: experimental validation of a discrete Lagrangian model[END_REF][START_REF] Turco | Pantographic lattices with non-orthogonal fibres: experiments and their numerical simulations[END_REF][START_REF] Turco | Can a Hencky-type model predict the mechanical behaviour of pantographic lattices?[END_REF][START_REF] Placidi | A review on 2d models for the description of pantographic fabrics[END_REF], which contributed to motivate the present study. The importance of fibrous microstructured systems is increasing in current literature, especially since these objects can be manufactured with great precision and relatively limited costs by means of computer-aided manufacturing (see for instance [START_REF] Kalpakjian | Manufacturing Engineering and Technology[END_REF][START_REF] Misra | A review of presentations and discussions of the workshop computational mechanics of generalized continua and applications to materials with microstructure that was held in Catania 29-31 October 2015[END_REF] for useful reference works). Therefore, investigating the behavior of their elementary constituents in geometrically nonlinear deformation is more relevant nowadays. Moreover, equations similar to those describing the equilibrium of beams can also be found in different contexts: That is the case for instance when describing the separation line of self-adhesive polymeric films [START_REF] Nase | Identifying tractionseparation behavior of self-adhesive polymeric films from in situ digital images under t-peeling[END_REF].

Surprisingly, enough there are not many results in the literature, in particular, rigorous ones, concerning nonlinear deformation of beams under distributed load. The classical reference work [START_REF] Antman | Nonlinear problems of elasticity[END_REF] only covers the case of concentrated load. Afterward, some numerical results for the inextensible Euler beam under distributed load were published in [START_REF] Faulkner | On the use of a segmental shooting technique for multiple solutions of planar elastica problems[END_REF][START_REF] Raboud | Multiple three-dimensional equilibrium solutions for cantilever beams loaded by dead tip and uniform distributed loads[END_REF]. The paper is organized as follows: in Sect. 2 a nonlinear version of the extensible Timoshenko beam model is introduced and the problem of a clamped-free beam is formulated. Euler-Lagrange equations are formally derived. In Sect. 3, some numerical results concerning curled equilibrium configurations are shown. These results motivate the analytical study of the properties of the equilibrium solutions done in Sect. 4, in which the Euler-Lagrange equation is studied in the particular case of an infinite shear stiffness, which leads to the nonlinear Euler beam.

The model

In this section, we introduce a general Timoshenko model and formally derive the Euler-Lagrange equations associated with the minimization of the total energy.

Kinematics and deformation energy

We consider a beam lying in a two-dimensional space E in which we fix once and for all an orthonormal reference system {e 1 , e 2 }. The beam can be understood as the section of a plate which is in planar strain or stress state. The placement of the beam is described by a pair (χ, ϕ) of functions defined on the interval [0, L] and taking values, respectively, in E and R : χ(s) represents the position of a point s of the beam while ϕ is an extra kinematic variable attached to point s. In the classical presentation of Timoshenko linear model in the early 1920s (see [START_REF] Timoshenko | On the correction for shear of the differential equation for transverse vibrations of prismatic bars[END_REF][START_REF] Timoshenko | On the transverse vibrations of bars of uniform cross-section[END_REF]), ϕ is thought to represent the angle of rotation of the sections of the beam (supposed rigid) with respect to the normal to the neutral axis. It is well known that this standard Timoshenko beam model is a particular case of a linear Cosserat 1D continuum in which ϕ is thought to represent an internal rotational degree of freedom (the original work by Cosserat brothers is [START_REF] Cosserat | Théorie des corps déformables[END_REF]; see for instance [START_REF] Altenbach | Cosserat-type rods[END_REF][START_REF] Altenbach | Cosserat-type rods[END_REF][START_REF] Altenbach | On generalized cosserat-type theories of plates and shells: a short review and bibliography[END_REF] for interesting applications). The chosen mechanical interpretation of ϕ is in fact irrelevant. Anyway, we classically call "couple" the dual variable to ϕ while we call "force" the dual variable to χ.

We assume that, at rest, the beam lies along e 1 so that its placement (χ 0 , ϕ 0 ) is given ∀s ∈ [0, L] by χ 0 (s) = s e 1 and ϕ 0 (s) = 0 so that L stands for the length of the beam at rest. The beam is clamped at the extremity point s = 0 (in the sense that χ(0) = 0 and ϕ(0) = 0) and this point coincides with the origin of the reference axes.

We introduce the tangent vector to the actual configuration and write it under the form

χ (s) = α(s)e(θ(s)) (1)
where α := χ ≥ 0, and e(θ) := cos(θ)e 1 + sin(θ)e 2

The quantity α accounts for the elongation of the beam, while θ(s) is the angle that χ (s) forms with e 1 .

As at rest α = 1 and ϕ = θ = 0, we assume that the deformation of the beam can be measured by the quantities α -1, ϕ and ϕ -θ and that the associated energy reads

L 0 k e f (α(s)) + k b 2 (ϕ (s)) 2 + k t 2 (ϕ(s) -θ(s)) 2 ds ( 2
)
where f is a C 1 function on ]0, + ∞[ which is: i) positive; ii) convex; iii) tending to + ∞ when x tends to 0 or + ∞. We extend it on R by setting f (x) = + ∞ when x ≤ 0 (in the numerical section we will use the function f (α) = -0.01 log(α) + (α -1) 2 for x > 0); as it is easily seen, this function has the desired properties and is very close to (α -1) 2 except in the vicinity of the origin, where it diverges rapidly). We also introduce the derivative g of f on ]0, + ∞[. The three addends in the previous integral are, respectively, called "extensional energy," "flexural energy" and "shear energy" and the positive material parameters k e , k b and k t are the associated stiffnesses.

It is well known that, in the classical Timoshenko formulations in terms of deflection and rotation, shear energy due to shear strain (i.e., the difference between rotation and gradient of deflection) leads to the socalled numerical shear locking phenomena in standard finite element methods. Traditionally, shear locking has been avoided by introducing reduced numerical integration for the shear term, whereas recently the problem has been overcome by reformulations based on change of variables, for both Timoshenko beams and Reissner-Mindlin plates [START_REF] Balobanov | Locking-free variational formulations and isogeometric analysis for the Timoshenko beam models of strain gradient and classical elasticity[END_REF][START_REF] Beirao Da Veiga | A locking-free model for Reissner-Mindlin plates: analysis and isogeometric implementation via nurbs and triangular nurps[END_REF]. The present three-variable formulation (2), with shear strain of the form ϕ -θ, should not be prone to numerical locking either.

The natural functional space for energy (2) is the set of functions (α, θ, ϕ) ∈ V where

V := L 2 (0, L) × L 2 (0, L) × H 1 0 (0, L) (3)
where H 1 0 stands for set of functions in the Sobolev space H 1 which vanish at s = 0. Note that the energy takes values in [0, + ∞] on this functional space as f • α may not be integrable. Of course, this will never happen for equilibrium solutions which must have a finite energy.

We suppose that the beam is submitted to a distributed dead load represented by b ∈ L 2 ((0, L), E) and to concentrated load and couple R and M at the free extremity s = L, so that the total energy reads

L 0 k e f (α(s)) + k b 2 (ϕ (s)) 2 + k t 2 (ϕ(s) -θ(s)) 2 -b(s) • χ(s) ds -R • χ(L) -Mϕ(L) (4)
Let us introduce the vector B(s), which may be interpreted as the force exerted by the right side of the beam on the left side at point s, by setting Using [START_REF] Euler | Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes Sive Solutio Problematis Isoperimetrici Latissimo Sensu Accepti[END_REF], this identity reads

B(s) := R + L s b(σ)dσ (5)
L 0 b(s) • χ(s) ds + R • χ(L) = L 0 α(s)B(s) • e(θ(s)) ds.
Therefore, the total energy can be written as a function of α, θ and ϕ only.

Remark 1. In the following of the paper, we will assume that the distributed load is transverse and uniform (b(s) := b e 2 for some positive constant b) and that no concentrated load or couple are applied :

R = 0, M = 0 and B(s) = B(s) e 2 with B(s) := b (1 -s). ( 6
)
Remark 2. The number of parameters in the equilibrium problem can be reduced by adimensionalizing the problem, i.e., by choosing L to be the physical length unit and by choosing the unit of energy such that k b = 1. Hence, in the sequel, we fix L = 1 and k b =1.

Euler-Lagrange equations

The equilibrium states correspond to stationary points in V of energy [START_REF] Antman | Nonlinear problems of elasticity[END_REF]. The first variation with respect to ϕ, α and θ leads to the following boundary value problem

1 : ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ -ϕ + k t (ϕ -θ) = 0 k e g(α) -B • e(θ) = 0 k t (ϕ -θ) + α B • e ⊥ (θ) = 0 ϕ (1) = 0 ϕ(0) = 0 (7)
Remark 3. In the great majority of the textbooks of structural mechanics, the search of stationary points of ( 4) is performed by directly looking for the solutions of the differential problem [START_REF] Fertis | Nonlinear Structural Engineering[END_REF], assumed as fundamental, without introducing the variational principle (4). The description of contact actions for a Timoshenko beam becomes necessary in this formulation, while starting from a variational principle this delicate issue can be skipped.

Finally, we can add that the variational formulation has several computational advantages (see for instance [START_REF] Capobianco | Time finite element based Moreau-type integrators[END_REF] for a general discussion and [START_REF] Eugster | Director-based beam finite elements relying on the geometrically exact beam theory formulated in skew coordinates[END_REF] for applications to beam problems).

Particular case of inextensible Euler beam

In the case k e = k t = + ∞, we have α = 1 and ϕ(s) = θ(s) for all s. The equilibrium problem, in a nondimensional form, then reduces to the minimization of

E(θ) = 1 0 1 2 (θ (s)) 2 -b (1 -s) sin(θ(s)) ds (8)
which corresponds to the case of a clamped inextensible Euler beam in large deformations when submitted to a uniformly distributed transverse force field (see, e.g., [START_REF] Eugster | Geometric Continuum Mechanics and Induced Beam Theories[END_REF]). The first variation of ( 8) with respect to θ gives the boundary value problem:

θ (s) + b(1 -s) cos(θ(s)) = 0, θ(0) = 0, θ (1) = 0. (9)
This boundary value problem will be studied first numerically in Sect. 3 and then analytically in Sect. 4.

Equilibrium configurations of Euler and Timoshenko beams: some numerical results

We focus here on problem [START_REF] Fertis | Nonlinear Structural Engineering[END_REF] and we numerically solve it. Instead of considering problem (7) directly, we remark that, as the function g is the derivative of the convex potential f , it is a bijective function from ]0, + ∞[ onto R. Hence, from the second equation, we can write α as a function of θ. From the third equation, we can also easily write ϕ in terms of θ. The first equation becomes a second-order differential equation for θ. As there generally exist more than one value of θ(0) which ensures ϕ(0) = 0 we select the smallest one in absolute value a. It remains to satisfy the condition ϕ (1) = 0. We do this by means of a shooting technique : we replace this last condition by θ (0) = k, numerically solve the resulting Cauchy problem

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ k e g(α) -B • e(θ) = 0 k t (ϕ -θ) + α B • e ⊥ (θ) = 0 -ϕ + k t (ϕ -θ) = 0 θ(0) = a θ (0) = k (10)
by a standard explicit Euler method for different values of the parameter k and finally select the solutions which satisfy (with prescribed accuracy) ϕ (1) = 0. Note that, in the particular case of an inextensible Euler beam model with uniformly distributed load, this parametric Cauchy problem reduces to:

P k = ⎧ ⎪ ⎨ ⎪ ⎩ θ = -b (1 -s) cos θ θ(0) = 0 θ (0) = k. ( 11
)
In the sequel, we indicate with θ k (s) the solution of ( 10) or [START_REF] Goriely | On the dynamics of elastic strips[END_REF]. In 9) are, in the case of the inextensible Euler beam, the zeros of the function c(k). We found them numerically with a standard application of the bisection method. These graphs suggest that the number of solutions grows as b increases. Moreover, they suggest that these solutions belong to a neighborhood of the origin depending on b. Indeed, we have: Let us now consider the deformed shapes of the beam corresponding to equilibrium configurations. We start by showing a solution of the boundary value problem (9) (with b = 600) displaying a complex behavior, with three monotonicity intervals for the variable θ. As it is possible to see from Fig. 3, the angle θ never reaches the values π 2 or -3π 2 (in the last part it comes quite close to the first value). This is consistent with Proposition 1 of the following section. In Fig. 4, we show the deformed shapes corresponding to the three solutions relative to the right panel of Fig. 1 for an inextensible Euler beam, while in Fig. 5 we show analogous equilibrium solutions for an inextensible Timoshenko beam. A comparison between the two, under the same adimensional load b = 60, shows2 that the second model presents an overall decrease in stiffness as expected due to the presence of shear deformation; in the presented simulations the shear stiffness is set at k t = 7 × 10 3 .

c(k) = (θ k ) | s=1 = (θ k )| s=0 + 1 0 θ k (t) dt = k - 1 0 b(1 -t) cos(θ k (t)) dt. ( 12
k t ∞ ∞ 1.8 × 10 3 1.8 × 10 3 k e ∞ 5 × 10 4 ∞ 2 × 10 3
In Fig. 6, we show a set of deformed shapes of an inextensible Euler beam. The deformed shapes all belong to a branch of solutions turning around the clamped edge (see the following section for a precise definition of branches). We start from b = 50 and gradually increase the load up to b = 450. In Fig. 7, we show a set of deformed shapes of an extensible Euler beam starting from b = 50, and gradually increasing the load up to b = 450. In this case, another parameter enters the description of the problem, i.e., the extensional stiffness k e . We recall that the function f describing how the energy density depends on α assumed for all the simulations involving extensibility is f = -0.01 log(α) + (α -1) 2 . In Fig. 8, we show the plot of the functions θ and α relative to the last simulation presented in Fig. 7.

In particular, it is emphasized in the figure that α attains its minimum (corresponding to the maximal local compression) in the point s that verify θ(s) = -π/2, and α = 1 in the point s such that θ(s) = -π. In Fig. 9, we show a set of deformed shapes of an inextensible Timoshenko beam starting from b = 50, and gradually increasing the load up to b = 450. Here again, another parameter is needed for the description of the problem, i.e., k t , the shear stiffness. In these simulations, we have set k t = 1.8 × 10 3 . In Fig. 10, we show a set of deformed shapes of an extensible Timoshenko beam starting from b = 50, and gradually increasing the load up to b = 450. In this case, the two parameters introduced above, k e and k t are necessary. In these simulations, we have set k t = 1.8 × 10 3 and k e = 2 × 10 3 . In Fig. 11, we show the plot of the functions θ, ϕ and α relative to the last simulation presented in Fig. 10. As for the extensible Euler case, it is emphasized in the figure that α is minimum at the point s that verify θ(s) = -π/2, and α = 1 at the point s such that θ(s) = -π. Summarizing our results, we have numerical evidence that there exists a family of "curled" equilibrium configurations, that is configurations in which |θ(s)| becomes larger than π for some s. It has to be pointed out that the configurations are shown in Figs. 6, 7, 9 and 10 can be also found by a numerical minimization procedure applied to a discretized beam of the type presented in the numerical section of [START_REF] Alibert | Extensional elastica in large deformation as Γ-limit of a discrete 1D mechanical system[END_REF]. This suggests the conjecture that the minimizers of the energy (i.e., stable equilibria) are characterized by the fact that θ ≤ 0 everywhere (and never reaches θ = -3π/2). In the following section, we formalize and prove these results in the case of an inextensible Euler beam. We extend the results of [START_REF] Della Corte | Equilibria of a clamped euler beam (elastica) with distributed load: large deformations[END_REF] showing that there can exist only two branches of stable equilibrium configurations.

Inextensible Euler beam under distributed load: analytical results

In this section, we prove what we have seen in the numerical simulations shown in Sect. 3, i.e., that there exist a family of curled stable equilibrium configurations for the Elastica under distributed load. Our starting point will be the boundary value problem [START_REF] Steigmann | Invariants of the stretch tensors and their application to finite elasticity theory[END_REF]. We classify the different types of solutions by the number of zeros of the function sin(θ) on the interval [0, 1]. We say that a solution belongs to branch n of the set of equilibrium solutions if there are exactly n distinct values 0 = s 1 < s 2 < • • • < s n satisfying sin(θ(s i )) = 0.

A priori bound for the range of stationary points

Here, we prove that, for any stationary point of the functional (8), have - 3 2 π < θ < π 2 . This means, informally speaking, that the beam cannot make a complete turn around the clamped point at the equilibrium (see Fig. 12). More precisely, we establish the following:

Proposition 1. If θ is a solution of (9), then sin(θ(s)) = 1 for every s ∈ [0, 1].
Proof. Let us define the function:

V (s) := 1 2 (θ (s)) 2 + b(1 -s) sin θ(s) -b(1 -s)
A direct computation gives

V (s) = θ (s)[θ (s) + b(1 -s) cos θ(s)] + b(1 -sin θ(s))
and, using (9),

V (s) = b(1 -sin θ(s)) ≥ 0.
Thus V is nondecreasing and, as V (1) = 0, V is nonpositive in [0, 1]. Assume by contradiction that there exists s ∈ (0, 1) : sin(θ(s)) = 1. Then V (s) = 1 2 (θ (s)) 2 ≤ 0 and thus θ (s) = 0. The constant function θ = θ 0 is clearly solution of the Cauchy problem made by Eq. ( 9) with data θ(s) = θ 0 with sin(θ 0 ) = 1 and θ (s) = 0. Uniqueness stated by Cauchy-Lipschitz theorem proves that no such a solution can satisfy θ(0) = 0.

A graphical representation of a deformed shape that is prohibited as an equilibrium configuration for the clamped Elastica under uniform distributed load is shown in Fig. 12.

Remark 4. The function V (s) allows us to give an estimate of the curvature at s = 0. Indeed, from

V (0) = 1 2 (θ (0)) 2 -b ≤ 0, we get |θ (0)| ≤ √ 2b.
Remark 5. The previous reasoning holds true as well if we replace b(1 -s) with any C 1 , positive and decreasing function, which corresponds to a density of force which has always the same verse (i.e., it is pointing always "upwards" or always "downwards").

Study of equilibrium configurations for an inextensible Euler beam

In the following, we will set B(s) := b(1 -s). As sin(θ) is a continuous function which does not reach 1 when θ takes values in (-3π 2 , π 2 ), θ(s i ) is either 0 or -π. On an interval [s i , s i+1 ] four situations can arise: • S1: θ(s i ) = 0 and θ (s i ) > 0 : then θ is positive on the considered interval and θ(s i+1 ) = 0. Owing to (9), we know that θ is strictly concave, it reaches a local maximum at some unique t i ∈ (s i , s i+1 ) and θ (s i+1 ) < 0.

Fig. 12. Example of an "impossible" equilibrium shape for an Euler beam under distributed load: Proposition 1 proves that such a deformed shape, in which the beams turns two times around the clamp, cannot be an equilibrium configuration

• S2: θ(s i ) = 0 and θ (s i ) ≤ 0 : by integrating (9) we get

(θ (s)) 2 2 = (θ (s i )) 2 2 -B(s) sin(θ(s)) - s si B (t) sin(θ(t)) dt and thus (θ (s)) 2 > (θ (s i )) 2 s ∈ (s i , s i+1 ].
Therefore, θ cannot vanish, θ is strictly decreasing, θ(s i+1 ) = -π and θ (s i+1 ) < 0. • S3: θ(s i ) = -π and θ (s i ) ≤ 0 : then θ belongs to (-3π 2 , -π) on the considered interval and θ(s i+1 ) = -π. Owing to (9), we know that θ is strictly convex, it reaches a local minimum at some unique t i ∈ (s i , s i+1 ) and θ (s i+1 ) > 0.

• S4: θ(s i ) = -π and θ (s i ) > 0 : For the same reason as in situation (S2), θ cannot vanish, θ is strictly increasing, θ(s i+1 ) = 0 and θ (s i+1 ) > 0.

On the interval [s n , 1], only two situations can arise which allow for θ to vanish.

• S1 : θ(s n ) = 0 and θ (s n ) > 0 : then θ is positive and strictly concave on the considered interval. It reaches its maximum at s = 1 where θ (1) = 0.

• S3 : θ(s n ) = -π and θ (s n ) ≤ 0 : then θ belongs to (-3π
2 , -π), is strictly concave on the considered interval and reaches its minimum at s = 1 where θ (1) = 0.

Therefore, an equilibrium solution in branch n is made by a sequence of intervals in the order S1 -S2 -S3 -S4 -S1 . . . ending with S1 or S3 and starting, if it is made by more than one interval, with S1 or S2. For instance, branches 1, 2 and 3 have, respectively, the structure S1 , S2 -S3 and S1 -S2 -S3 . In general, critical points of a functional do not correspond to local minima: In that case, they correspond to unstable equilibrium solutions. To check if a solution is unstable one usually checks, if it is linearly unstable by computing the second Gateaux differential of the functional (8):

1 0 (h (s)) 2 + 2b(1 -s) sin(θ(s))(h(s)) 2 ds ( 14
)
The fact that this quadratic form is nonnegative is a necessary condition for stability. Actually, in the particular case of our functional, the fact that this bilinear form is positive is a sufficient condition for stability. Indeed, we have:

Lemma 1.
Let θ be a critical point of [START_REF] Ladevèze | Nonlinear Computational Structural Mechanics: New Approaches and Non-incremental Methods of Calculation[END_REF]. If there exists a constant C > 0 such that, for every h ∈ H 1 verifying h(0) = 0,

1 (h (s)) 2 2 + b(1 -s) sin(θ(s)) (h(s)) 2 2 ds ≥ C h 2 H 1 (15)
then θ is a local minimizer of E (defined by [START_REF] Ladevèze | Nonlinear Computational Structural Mechanics: New Approaches and Non-incremental Methods of Calculation[END_REF]) for the H 1 topology and therefore for the uniform norm too.

Proof. A direct computation gives

E(θ + h) = E(θ) + 1 0 [θ h -B cos(θ)h] ds + 1 0 (h ) 2 2 + B sin(θ) h 2 2 -B sin(θ) cos(h) -1 + h 2 2 + cos(θ) sin(h) -h ds
Integrating by parts, the second addend can be rewritten

1 0 [θ h -B cos(θ)h] ds = θ (1)h(1) -θ (0)h(0) - 1 0 [θ (s) -B cos(θ(s))] h(s) ds
Let θ be a critical point for E. It satisfies [START_REF] Steigmann | Invariants of the stretch tensors and their application to finite elasticity theory[END_REF] and, as h(0) = 0, all terms in the above sum vanish. Using moreover inequality [START_REF] Taig | An unconstrained dynamic approach for the generalised beam theory[END_REF], we get the estimation

E(θ + h) -E(θ) -C h 2 H 1 ≥ - 1 0 B sin(θ) cos(h) -1 + h 2 2 + cos(θ) sin(h) -h ds ≥ -b 1 0 cos(h) -1 + h 2 2 + sin(h) -h ds
Let η be such that 0 < η < C 2b . There exists > 0 such that 0 < x < implies | cos(x) -1 + x 2 2 | < ηx 2 and | sin(x) -x| < ηx 2 . Hence, for every h such that h H 1 < , which implies h L ∞ < , we have:

1 0 cos(h) -1 + h 2 2 + sin(h) -h ds < 2η 1 0 h 2 < 2η h 2 H 1 .
Therefore, we obtain that, for any h with h H 1 < ,

E(θ + h) -E(θ) ≥ C -2b η h 2 H 1
In the following propositions, we will completely characterize the stable equilibrium configurations of the Elastica under distributed load. Indeed, we show that branch 1 corresponds to global minimizers of the functionals and thus to stable equilibrium solutions, that branch 2 corresponds, for b large enough, to other stable equilibrium solutions while all other branches are unstable.

Proposition 2. For any b, there exists a unique corresponding solution in branch 1. This solution is the global minimizer of the functional [START_REF] Ladevèze | Nonlinear Computational Structural Mechanics: New Approaches and Non-incremental Methods of Calculation[END_REF].

Proof. The existence of a global minimizer θ of the functional ( 8) is assured by the coercivity and lower semi-continuity of the functional. The function θ := -π 2 + π 2 + θ has the same energy as θ and takes values in (-π 2 , π 2 ). The truncated function θ := max(0, θ) (which clearly is in H 1 0 ) has a lower energy (strictly lower if θ does not belong to branch 1) and takes values in [0, π 2 ). Hence, any global minimizer takes values in (0, π 2 ): It must belong to branch 1. Let us now prove that, for a given b, there exists at most one solution in branch 1. By contradiction, let us consider two distinct solutions θ 1 and θ 2 . We have θ 1 (0) = θ 2 (0) owing to Cauchy-Lipschitz uniqueness property. We can therefore suppose that θ 1 (0) < θ 2 (0). We then introduce s 1 > 0 as the maximum value such that θ 1 < θ 2 on the interval (0, s 1 ). On (0, s 1 ) we have cos(θ 1 ) > cos(θ 2 ), thus -B cos(θ 1 ) < -B cos(θ 2 ), θ 1 < θ 2 and so θ 1 < θ 2 . As a consequence θ 1 (s 1 ) < θ 2 (s 1 ) which is in contradiction with the definition of s 1 and the continuity of functions θ 1 and θ 2 unless s 1 = 1. But in that case θ 1 < θ 2 implies also θ 1 (1) < θ 2 (1) which is in contradiction with the fact that θ 1 (1) = θ 2 (1) = 0.

To prove that any solution θ in branch n for n > 2 is unstable, let us remark that it must contain an interval of type S1 or S3. Hence, there are at least two distinct points t i , t j such that θ (t i ) = θ (t j ) = 0. This prevents linear stability. Indeed, we have: Proposition 3. Any solution containing a part defined over [s 1 , s 2 ] ⊂ [0, 1] such that θ (s 1 ) = θ (s 2 ) = 0 is unstable. Thus only branches 1 and 2 can be stable.

Proof. Let us check the second variation

V := 1 0 (h (s)) 2 2 + b(1 -s) sin(θ(s)) (h(s)) 2 2 ds with h = θ 1 [s1,s2
] (here by 1 X we denote the characteristic function of the set X). We have:

V = s2 s1 (θ (s)) 2 2 + 1 2 (b(1 -s)θ (s))(sin(θ(s))θ (s)) ds
and after integrating by parts the second term:

V = s2 s1 (θ (s)) 2 2 + 1 2 (b(1 -s)θ (s) -bθ (s)) cos(θ(s)) ds
Recalling Eq. ( 7), we have:

V = - s2 s1 1 2 bθ (s) cos(θ(s)) ds = - s2 s1 1 2 b(1 -s) 1 -s θ (s) cos(θ(s)) ds
Recalling again Eq. ( 7) and integrating by parts once more:

V = + s2 s1 1 2 θ (s)θ (s) 1 -s ds = - s2 s1 1 4 (θ (s)) 2 (1 -s) 2 ds < 0
It remains to study branch 2. We prove that, for b large enough, the energy possesses some local minimizers which belong to branch 2 and therefore are stable solutions.

F (u) := 1 0 (u (s)) 2 + 2b(1 -s) cos 2 (u(s)) ds
and the set S of admissible u is now the set of functions in H 1 (0, 1) taking values in (-π 2 , π 2 ) and satisfying u(0) = π 4 . Let us assume b 1/4 > 100 (which is not an optimal value) and denote δ := b -1/4 . Consider the set O of functions u in S satisfying u(δ) < 0 and its closure O, i.e., the set of functions u in S satisfying u(δ) ≤ 0. We will prove that there exists a local minimizer of the energy in O. Such a local minimizer cannot correspond to a critical solution in branch n > 2 as we already know that such a solution is unstable. Neither it can belong to branch 1 as u(δ) ≤ 0. Therefore, it must belong to branch 2. To prove that there exists a local minimizer of the energy in O, we consider a minimizer ū of the energy in O (its existence is clearly ensured as O is closed with respect to the H 1 topology) and check that it actually belongs to O.

We first establish an upperbound for F (ū). Setting s 0 := ln( Thus, there exists some t 1 ∈ (0, δ) such that cos which completes the proof.

Conclusion

We have classified the equilibria of nonlinear Euler and Timoshenko beams subjected to uniformly distributed load. We have identified sequences of equilibria among which two at most are stable. The methods and techniques used here are rather simple and could be efficiently generalized to attack more general theories of beams formulated in order to take into account different kinds of effects. We think of beams with deformation of the section or differentiated deformations of different layers in composite beams [START_REF] Bîrsan | Deformation analysis of functionally graded beams by the direct approach[END_REF], strain concentration in thin-walled beams, piezoelectric activated deformations [START_REF] Chróścielewski | Nonlinear finite element modeling of vibration control of plane rodtype structural members with integrated piezoelectric patches[END_REF].

Another possible extension of the presented result could be the study of the problem introduced in [START_REF] Javili | Thermomechanics of solids with lower-dimensional energetics: on the importance of surface, interface, and curve structures at the nanoscale. a unifying review[END_REF] of equilibria of beams constrained to remain on a given smooth surface.

0 B

 0 so that B (s) = -b(s) and B(L) = R. As we assumed χ(0) = 0 and as, by definition B(L) = R, integrating by parts gives L 0 b(s) • χ(s) ds + R • χ(L) = L (s) • χ (s) ds.

Fig. 1 ,

 1 we represent c(k) := (θ k ) | s=1 as a function of the parameter k. The cases b = 30 and b = 60 are represented. In Fig. 2, we show the same for b = 200 and b = 1000. It is clear that the values of k such that θ k is a solution of (

)

  Hence |c(k) -k| ≤ b, which implies that the zeroes of the function c must belong to [-b, b] and that lim k→∞ c(k) k = 1 (13) This explains why the graph in Figs. 1 and 2 tends to a straight line with slope 1 as k diverges. Remarkably, this happens independently of the value of b.

Fig. 1 .

 1 Fig. 1. Parametric shooting: c(k) is plotted for two different values of the adimensional distributed load in the case of an inextensible Euler beam. The plots provide numerical evidence that the boundary value problem (9) admits one solution for b = 30 and three solutions for b = 60

Fig. 2 .

 2 Fig. 2. Parametric shooting: c(k) is plotted for two different values of the adimensional distributed load in the case of an inextensible Euler beam. The plots provide numerical evidence that the boundary value problem (9) admits five solutions for b = 200 and nine solutions for b = 1000

Fig. 3 .Fig. 4 .Fig. 5 .

 345 Fig. 3. A solution of the boundary value problem (9) with b = 600. The solution is consistent with Proposition 1. Indeed, it is possible to see that the angle θ never reaches the values π 2 or -3π 2 (in the last part it comes quite close to the first value)

Fig. 6 .

 6 Fig. 6. Parametric study along a branch of solutions for an inextensible Euler beam model varying the transversal load b. The values of the load are indicated in the figure

Fig. 7 . 4 Fig. 8 .

 748 Fig. 7. Parametric study along a branch of solutions for an extensible Euler beam, varying the transversal load b. The values of the load are indicated in the figure. In these simulations, we have set ke = 5 × 10 4

Fig. 9 .Fig. 10 .

 910 Fig. 9. Parametric study along a branch of solutions for an inextensible Timoshenko beam model varying the transversal load b. The values of the load are indicated in the figure

Proposition 4 . 1 0

 41 For b large enough, branch 2 contains stable solutions. Proof. We first remark that the energy minimization problem min θ (θ (s)) 2 2 + b(1 -s)(1sin(θ(s)) ds can be rewritten, by setting u = π 4 + θ 2 and rescaling, as min u∈S F (u) where

, 2 -F (v) ≤ 1 0 2 1 0 √

 2120 the function v defined by v(s) := π 2 arctan(e √ b(s-s0) ) belongs to O. Indeed, v(s) < 0 as soon as s > s 0 and this is the case for s = δ.Moreover, noticing that v = -√ b cos(v), we can upperbound F (v) by (v (s)) 2 + b cos 2 (v(s)) ds = -b cos(v(s))v (s) ds ≤ 4 √ b. Therefore F (ū) ≤ F (v) ≤ 4 √ b.Let us now assume, by contradiction, that ū ∈ O. This means ū(δ) = 0. Using previous estimation, we get δ 0 b(1 -δ) cos 2 (ū(s)) ds ≤ δ 0 b(1 -s) cos 2 (ū(s)) ds ≤ 4 √ b.

2 y x b( 1 -

 21 s)) 2 + b(1 -s) cos 2 (ū(s)) ds ≥ 2 y x (ū (s)) 2 + b(1 -y) cos 2 (ū(s)) ds ≥ y) |cos(ū(s))ū (s)| ds ≥ 2 b(1 -y)| sin(ū(y))sin(ū(x))|.We obtain the contradiction :F (ū) ≥ 2 b(1 -2δ) | sin(ū(t 1 ))sin( π 4 )| + | sin(ū(t 1 ))| + | sin(ū(t 2 ))|

  2 (ū(t 1 )) ≤ | sin(ū(t 1 ))| ≥ 95 99 ≥ 95 99 . For a similar reason, there exists some t 2 ∈ (δ, 2δ) such that | sin(ū(t 2 ))| ≥ 94 98 . We can estimate F (ū) on the different intervals [0, t 1 ], [t 1 , δ], [δ, t 2 ]. On each one, we use the estimate:

	4 √ b(1 -δ)δ b	≤	4 99	.

and

We introduce here the transverse unit vector e ⊥ (θ) :=sin(θ)e 1 + cos(θ)e

.

It has to be remarked that another parameter enters the problem for the Timoshenko model, i.e., shear stiffness.

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In life science over the years, studies of all domains gathered lots of data in different places and as fields of studies grows the number of databases also increases. Lots of research teams have their own specific database and update it at different speed. Some initiative to integrate databases have emerge through the year such as Intermine [START_REF] Kalderimis | InterMine: extensive web services for modern biology[END_REF] and BioMart [START_REF] Smedley | The BioMart community portal: an innovative alternative to large, centralized data repositories[END_REF].

Semantic Web technologies

Semantic Web technologies are a broader initiative (i.e. not limited to life science) of the W3C for addressing data integration 1 . They provide a unified technical framework called RDF, allowing to describe entities and relations between them across several datasets (called endpoints). SPARQL is the associated query language. This will be further explained in the Background section.

Over the last two decades, they have been increasingly adopted by the Lifes science community [START_REF] Anguita | NCBI2RDF: Enabling Full RDF-Based Access to NCBI Databases[END_REF]. Currently, most major databases are available in RDF or via SPARQL endpoints.

Moreover, the databases now have explicit references to entities from other databases, which reflects the need for integration that we had identified previously. They now constitute the largest and most connected part of the LOD cloud 2 .

Query federation

To accomodate the fact that the description of an entity in a dataset can reference other entities in another dataset, SPARQL supports query federation. This allows to write a single query in which a part concerns the first dataset, and another part concerns the second dataset as if they were one single virtual dataset.

The SPARQL engine processes such federated queries automatically. This is an important difference with relational databases, where the user would have to send the two subqueries to the respective databases, retrieve the results and perform their integration.

A correct but naive approach for processing SPARQL queries consists in sending each subquery to all endpoints. Of course, this leads to a combinatorial explosion and is not 1 https://www.w3.org/standards/semanticweb/ 2 https://lod-cloud.net/ 1 practical in terms of performance. State of the art query engine all follow variations of a more elaborate approach composed of three main steps:

• First, the Query parsing is the part where the query is divided in small parts, called statement patterns.

• Then, the Source selection where the federation engine chooses where to send each statement patterns.

• Finally, the Query planning regroups statements patterns that have the same sources and reorganise the query to optimise the join operations.

However, their performances are still not good enough to handle typical life science queries, that are complex and involve large datasets [START_REF] Saleem | How Representative is a SPARQL Benchmark? An Analysis of RDF Triplestore Benchmarks[END_REF].

General objective

The main objective is to improve the steps of federated query processing. Section 2 provides background information on RDF and SPARQL, analyses the difficulty of processing federated queries and surveys the state of the art query engines and benchmarks. Section 3 defines the internships objectives. Sections 4 and 5 describre the materials and methods and results:

we performed an in depth analysis of the query LS6 from LargeRDFbench, we proposed an improved scoring for query planning, implemented it in the Corese query engine and measured the gain of performance. Section 6 is a discussion and synthesis. Therefore URLs are a subset of URIs.

Triple

An RDF triple is a list of a subject, a predicate and an object (Table 1b). The subject is the URI of the resource being described. The predicate is the URI of a relation. The object is one of the values of the predicate for the subject. It can be either an URI if the relation designates another entity, or a string (Table 1a). Table 1: Data and a part of the corresponding triples.

Graph

A set of RDF triples constitutes a directed graph where the nodes are the resources and the arcs are the triples (Figure 1).

SPARQL

SPARQL is the language to query RDF data, defined by the W3C consortium3 .

One DB at a time

Users can send a query to an endpoint and retrieve the results. A query is a set of triples where the subject, the predicate or the object can also be variables (with their name starting by a question mark). The query engine computes all the combinations of the variables' values that satisfy the query. For example, if you want the molecular weight and name of all drugs, your query will be as Figure 2 and the results will display in no particular order (Figure 3).

Combining graphs

Because some databases have complementary information of the same resources, they can posses the same URI. This common URI creates a connection (or an "overlap") between the two databases, this is called entity matching (Figure 4 & 5). Most of the time, the queries needed by the scientist are complexes and needs to be sent to multiple databases because of that linking.

Federation

This shared information means that there is a chance the results will not be complete. As the Figure 6 shows, the results will be endpoint dependent and the user will not have the possible combinations between them. The first solution that comes in mind is to merge the endpoints together but the new dataset size will be too large to be useful. It will also make no sense since the concept of linked open data will not be preserved.

Naive method

The naive way is to send each triple to each endpoint and join every results (Figure 7).

Classical method Decompose the query in subqueries and only send these subqueries to the endpoints that could potentially return some results.

Figure 6: Federated queries: sending the whole query to each endpoint and computing the union of their results misses all the solutions obtained by combining some triples from an endpoint with other triples of another endpoint Figure 7: A correct but naive approach for processing a federated query consists in for each triple of the query (1) send it to each endpoint, (2) compute the union of the results, (3) iterate ( 1) and ( 2) over all the triples of the query, (4) compute the join of the results of (3).

Splitting the query increases the number of queries sent, the number of join operations and the evaluation time will increase but there will be no more incorrect results. The objectives of the federation engine are to gather the query fragments to reduces joins, select the relevant endpoints to send them and determine the order of processing these fragments.

Tools and benchmarks 2.3.1 Endpoints

To manage the data at each endpoint and store the triples, the most used software are Virtuoso4 , Fuseki5 , Corese [2]6 , Sesame/RDF4J7 . Their role is to receive the queries, evaluate it and return the results. Most of them have an interface where users can directly send queries. Here we will use Virtuoso.

FedX

Developed in 2011, FedX [START_REF] Schwarte | FedX: Optimization Techniques for Federated Query Processing on Linked Data[END_REF] is implemented in Java and extends the Sesame framework.

This modification adds a new source selection method that works with ASK queries sent at each endpoints. This query's result only tells if the endpoint can answer the main one, which inquires fewer searching process. Because every ASK query can be send individually, FedX introduces parallelization, a way that allows multiple tasks to be executed at the same time.

This is also used to improve the joining operation.

HiBISCus

HiBISCus [START_REF] Saleem | HiBISCuS: Hypergraph-Based Source Selection for SPARQL Endpoint Federation[END_REF] is a modification of FedX made by Fluidops. It creates indexes which retain information from ASK queries. The difference with FedX is that the engine searches inside this indexes instead of sending queries, it suppresses the response time induced by ASK queries. The company does not exist anymore and their website is down resulting in the precise documentation being unreachable and little to no support.

PPinSS

Designed and implemented by Guillaume Alviset in 2018, PPinSS is a modification of HiBIS-Cus. This tools generates different indexes with more information. It manages to select fewer sources while giving the same results and being faster as shown in Figure 8. His benchmarks were made with the Life Science (LS) queries.

Grafeque

Vijay Ingalalli worked on Grafeque in 2018 for the Federated Query Scaler project of Dyliss' team 8 . It is a modification of HiBISCus' query planning. The indexes changes again and a second view on the hyper-graph that represents the triples changes the source selection.

Benchmarks

A lots of benchmarks exist but they do not explore the federation with linked data the same ways [START_REF] Saleem | How Representative is a SPARQL Benchmark? An Analysis of RDF Triplestore Benchmarks[END_REF]. Here we will focus on the queries from FedBench [START_REF] Schmidt | FedBench: A Benchmark Suite for Federated Semantic Data Query Processing[END_REF] and its newer version, LargeRDFBench [START_REF] Saleem | LargeRDFBench: A billion triples benchmark for SPARQL endpoint federation[END_REF]. These queries have been created in a way that should cover every needs 8 http://www.irisa.fr/dyliss/public/federatedQueryScaler/

Internship objectives

Thanks to PPinSS the queries evaluation is faster. Its new indexes allows the source selection method to be more strict and grants faster queries evaluation with fewer intermediate results to join. But because of the error on LS6 for HiBISCus, the comparison is not complete. I will continue the benchmarks with HiBISCus and PPinSS, as it needs to be fully functional with each tools and each query and I will add Grafeque. Finally, because FedX is outdated, a migration to the Corese federation engine is considered.

4 Material and methods

Benchmarks

To compare the performances between HiBISCus, PPinSS and Grafeque, we used queries from FedBench [START_REF] Schmidt | FedBench: A Benchmark Suite for Federated Semantic Data Query Processing[END_REF] and LargeRDFBench [START_REF] Saleem | LargeRDFBench: A billion triples benchmark for SPARQL endpoint federation[END_REF]. The focus of the analysis are the queries for the life science domain from FedBench. A bash script has to launch every query with each version of the tools we wanted and specified within the arguments how many times the test needs to be run. The endpoints are created on virtual machine on the GenOuest cluster This setup allows us to access these endpoints without any conflict in case of external connections.

Scoring change

The whole analysis of LS6 brought to us the idea that the query planning of HiBISCus could be improved. No new scripts were written but the debug mode of Eclipse and lots of printing in the terminal were useful.

To change the query planning on HiBISCus, I proposed a new scoring function for the patterns and a new query planning algorithm based on this score.

Corese

Here Corese will replace FedX in PPinSS. Thanks to Guillaume, PPinSS also comes in a separate package that can be added to any federated engine. It will be used to implement PPinSS to Corese. After the complete migration, a benchmark has been planned, mainly focusing on the Life Science domain queries.

Results

Benchmarks

Once the job on the GenOuest cluster is launched, a log file appears in the user's home directory containing any texts output. For our benchmarks, this log file's format is as follow:

• Current query's name.

• Name of the tool.

• Number of sources selected, Sources selection time (in milliseconds), Number of results, Query evaluation time (in milliseconds).

Having every result in one line separated by a coma allows us to extract the results and to facilitate analysis. We can see in Figure 9 that Grafeque does not get results for LS4 and LS5 even though it got the right sources. The reason has not yet been found.

For PPinSS the lack of results for LS7 may be an error in the joining process because of the keyword FILTER inside the query. Without it, the query sends the correct results.

LS6

This variation can be explained by the joining order of the query. HiBISCus joins every statements pattern one after the other, but one joining operation asks for more than 25,000 results to be passed on the next part of the query. A simple swap of two triple patterns can reduce it to 164 results because the first patterns return fewer results than the second. That trick is not written on the scientific paper of HiBISCus and it is not how the query was first designed. PPinSS does not have this particular problem thanks to its source selection. It scans the variables used in the three triple patterns and gathers them in an exclusive group statement assigned to a single endpoint. The tests with ASP were promising at first, but the limitation over arrays with the ASP solver used and the complexity of the rules applied for our operations leads us to simply use Python.

To represent a statement, I have created a class named Bloc which contains an array with 0 and 1 that adds up when two statements are joined, a name and the number of result that statement should return. The name is needed to keep tracks of the join order and the number of results will shows us the worst number of operations which is linked to the query evaluation time.

Figure 10 shows that there is one order for which the join operation will take a minimum time. Even if the times displayed is not the real ones, it gives us an idea of the worst cases and demonstrate the importance of the query planning.

To confirm the engine capability, tests were realised with the other queries and the other available domains: Crossed domain and Linked data.

B1 = Bloc([1,0,0,0],"B1",48) B2 = Bloc([1,1,0,0],"B2",2240) B3 = Bloc([0,1,1,0],"B3",102343) B4 = Bloc([0,0,1,0],"B4",8117) B5 = Bloc([0,0,1,1],"B5",121158)
Listing 2: Initialisation of the statements for the python's scripts.

Scoring changes

To stop HiBISCus from hitting 120 seconds for LS6, I have searched for a way where its statements pattern have a better joining order. Initially, HiBISCus increase the score of the statement when there are no common variables with any statements previously analysed. The statement with the smaller score is then placed in the ordered list for joining. If a common variable were found, no action was taken. Therefore I have added a clause to decrease the score when has a common variable is encountered. This simple change managed to reorganise the query joining order and improve the time on LS6.

Corese

The implementation of PPinSS may be delayed, researches about how to do it continues. As it uses differents indexes, the source selection method will changes. We will use of the .jar files to launch the life science queries for the benchmarks without PPinSS.

Conclusion

Most of the internship was trial and error, trying to find the option that can make a difference when launching benchmarks. The lack of documentation made the development much slower but with the migration to Corese, it will be easier for future works. The benchmarks shows that PPinSS is slightly better that HiBISCus at evaluating queries for the Life Science domain. It also allows the discovery of an improvement to HiBISCus with its score modification. Unfortunately, in its actual state, Grafeque is not an improvement but its time for sources selection is a good sign. Finally, all the debugging, all the time spent on searching in the code and the documentations being not always relevant indicates that a migration to Corese, an engine with a live working team, will be helpful. Cela nous a permis de trouver une modification du calcul du score pour l'étape de planification de la requête. En outre, le développement de FedX n'étant plus maintenu en raison du rachat de l'entreprise qui le développait, une migration vers le moteur Corese est donc préparée.

Mots clés: Web sémantique, SPARQL, RDF, requêtes fédérés.

Internship abstract

For several years we have observed the increase of data in the life science domain and the multiplication of databases. Overlapping data between these databases is forcing scientists to develop new ways to evaluate queries to save time and cover all databases. Being able to send a query on several databases is the basis of query federation but many methods exists. Many federation engines have been developed, we will work on a few of them: FedX, HiBISCus (a modification of FedX), PPinSS (a modification of HiBISCus), Grafeque (a modification of HiBISCus). The objective of the internship was to continue the tests with HiBISCus and PPinSS and add Grafeque to the tools used. The benchmarks already carried out showed that HiBISCus did not evaluate LS6 correctly, so we studied this query and its evaluation by the different tools. Reading through the entire source code was quite tedious, python scripts were used to simulate the joints by changing the order of triples. This allowed us to find a modification of the scoring method for the query planning. In addition, FedX is no longer maintained in development due to the company that developed it being bought, a migration to the Corese engine is prepared.

Key words: Semantic Web, SPARQL, RDF, federated queries.

Figure 1 :

 1 Figure 1: Example of a graph.

Figure 2 :

 2 Figure 2: Example of a graph for a simple query.

Figure 3 :

 3 Figure 3: Part of the results for a simple query.

Figure 4 :Figure 5 :

 45 Figure 4: Illustration of linked data between two databases

Figure 8 :

 8 Figure 8: Comparisons of query execution time after source selection for each approach. LS6 has timed out for both HiBISCuS original and character index and also for FedX original.Figure taken from Guillaume's internship report.

Figure 9 :

 9 Figure 9: Queries results. #S is the number of sources selected. TS is the time for source selection. #R is the number of results. TQ is the time of the query evaluation and source selection for HiBISCus and PPinSS and only query evaluation for Grafeque.

Figure 10 :

 10 Figure 10: Results of the Join searching. From left to right: The 4 steps of the joining, the full joining time, the number of operations for each steps of the joining and the total number of operations.

1

  Introduction 1.1 Life science databases and the need to integrate them . . . . . . . . . . . . . . . . . 1.2 Semantic Web technologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Query federation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 General objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . RDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 URI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Triple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 SPARQL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 One DB at a time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Combining graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Federation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Tools and benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	1 Introduction
	2 Background
	2.1 6 Conclusion
	I

2.3.1 Endpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 FedX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 HiBISCus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 PPinSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.5 Grafeque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.6 Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Internship objectives 4 Material and methods 4.1 Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 LS6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Scoring change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Corese . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Results 5.1 Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 LS6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Scoring changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Corese . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Life science databases and the need to integrate them

Table 2 :

 2 Endpoints specifications. TCGA-A was added for HiBISCus and PPinSS but because the summaries for Grafeque did not included any TCGA, they were not used for further benchmarks.

	9 . They

1

  Example of a graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Example of a graph for a simple query. . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Part of the results for a simple query. . . . . . . . . . . . . . . . . . . . . . . . . . . ) over all the triples of the query, (4) compute the join of the results of (3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data and a part of the corresponding triples. . . . . . . . . . . . . . . . . . . . . . . 2 Endpoints specifications. TCGA-A was added for HiBISCus and PPinSS but because the summaries for Grafeque did not included any TCGA, they were not used for further benchmarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Depuis plusieurs années nous observons l'augmentation des données dans domaine de la vie et la multiplication des bases de données. Le chevauchement des données entre ces bases pousse les scientifiques à développer de nouvelles manières d'évaluer les requêtes pour gagner du temps et recouvrir toutes les bases de données. Pouvoir envoyer une requête sur plusieurs bases de données est la base de la fédération des requêtes. De nombreux moteurs de fédération ont été développé, nous allons travailler avec: FedX, HiBISCus (une modification de FedX), PPinSS (une modification de HiBISCus), Grafeque (une modification de HiBISCus). L'objectif du stage était de reprendre les tests avec HiBISCus et PPinSS et ajouter Grafeque aux outils utilisés. Les benchmarks déjà effectués ont montré qu'HiBISCus n'évaluait pas correctement LS6, nous avons donc étudié cette requêtes et son traitement par les différents outils. L'exploration du code source étant assez fastidieux, des scripts en python ont servis à simuler les jointures en changeant l'ordre des triplets.

	Master de Bioinformatique de l'Université de Rennes 1, années 2018-2019.
		ABEL Antoine
	Résumé du stage
	4	Illustration of linked data between two databases . . . . . . . . . . . . . . . . . . . .
	5	Graph of linked data between two databases . . . . . . . . . . . . . . . . . . . . . . .
	6	Federated queries: sending the whole query to each endpoint and computing the
		union of their results misses all the solutions obtained by combining some triples
		from an endpoint with other triples of another endpoint . . . . . . . . . . . . . . . .
	7	A correct but naive approach for processing a federated query consists in for each
		triple of the query (1) send it to each endpoint, (2) compute the union of the results,
	(3) iterate (1) and (2List of Tables
	1
		II

8

Comparisons of query execution time after source selection for each approach. LS6 has timed out for both HiBISCuS original and character index and also for FedX original. Figure taken from Guillaume's internship report. . . . . . . . . . . . . . . . 9 Queries results. #S is the number of sources selected. TS is the time for source selection. #R is the number of results. TQ is the time of the query evaluation and source selection for HiBISCus and PPinSS and only query evaluation for Grafeque. . 10 Results of the Join searching. From left to right: The 4 steps of the joining, the full joining time, the number of operations for each steps of the joining and the total number of operations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

https://www.w3.org/TR/rdf-sparql-query/

https://virtuoso.openlinksw.com/

https://jena.apache.org/documentation/fuseki2/

https://project.inria.fr/corese/

https://rdf4j.eclipse.org/

https://www.genouest.org/

LS6

The conclusion of Guillaume's work opens to the analysis of LS6, as we can see in the Figure 8. LS6's evaluation with FedX and HiBISCus returned a time-out error with no apparent reason even though their performances with the other queries were acceptable.

First, we have searched for an optimisation of the joining operations by finding every achievable joining order. Only the joining between statements that has common variables are made, otherwise the number of results is increased and the federation engine already optimises it. Given that the ASP is useful for solving search problems, an attempt was made.

Then most of the scripts used were written in Python 3, easier and faster to implement.

SELECT ?drug ?title WHERE { ?drug DB:drugCategory DB:micronutrient .

?drug DB:casRegistryNumber ?id .

?keggDrug w3:22-rdf-syntax-ns\#type bio2rdf:kegg\#Drug .

?keggDrug bio2rdf:bio2rdf\#xRef ?id .

?keggDrug <http://purl.org/dc/elements/1.1/title> ?title . } Listing 1: LS6. Statements from line 2 to 6 were named B1-B5 for better identification when researching with the joining operations. URI were shortened.

The particularity on this query is that HiBISCus selects 7 sources while PPinSS selects only 5 sources, one per statement. In fact, the statement on lines 5 and 6 can be retrieved from two different endpoints. We can see in the query (Listing 1) that the lines 3 and 5 have in common the variable "?id", the fist objective was to bring them closer.

Then, the idea of switching every statement of the query leads to new scripts. To be able to do benchmarks with it, I have made a script that finds and rewrites the query in every possible order then launches the evaluation of every new query. This leads to 120 new queries to launch.