Emmanuel Clouet

Ab Initio Models of Dislocations *

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Introduction

Dislocations are line defects which control the development of the plastic deformation in crystals. These defects induce a long range stress field, which is well described by elasticity, and dislocation elasticity theory offers a powerful framework to model dislocations and their interaction with their surrounding environment [START_REF] Hirth | Theory of Dislocations[END_REF][START_REF] Bacon | Anisotropic continuum theory of lattice defects[END_REF]. But some of their fundamental properties, like Emmanuel Clouet DEN-Service de Recherches de Métallurgie Physique, CEA, Université Paris-Saclay, F-91191 Gifsur-Yvette, France, e-mail: emmanuel.clouet@cea.fr their glide plane and their mobility, highly depend on their core, i.e. the region in the immediate vicinity of the defect where the perturbation of the crystal is too important to be described by elasticity. The modeling of this core region necessitates an atomic description and atomistic simulations have thus become a valuable tool to study dislocation properties. Among such simulations, ab initio calculations based on the density functional theory (DFT), as they rely on an electronic description of the atomic bonding, appear as the most accurate and predictive. But as these calculations are still limited in the size of the system they can handle, typically at most a few hundred atoms, the ab initio modeling of dislocations need special attention. Specific methodologies have been therefore developed to study dislocation core properties with ab initio calculations. The purpose of this chapter is to review the different modeling approaches for the ab initio study of dislocations, starting from a quick overview of DFT formalism, before describing more thoroughly boundary conditions specific to dislocation models, then the analysis of the atomic structure in the dislocation core and finally the extraction of meaningful energy properties.

Beyond the examples illustrated in this chapter, results which have been obtained from such ab initio studies for the dislocation core properties in different metals and semi-conductors can be found in the recent review of [START_REF] Rodney | Ab initio modeling of dislocation core properties in metals and semiconductors[END_REF].

Ab initio calculations

Ab initio calculations describe the bonding between atoms thanks to the resolutions of the Schrödinger equation for the electrons of the system. These are firstprinciples approaches as they do not use any experimental data and allows the modeling of atomic interaction only from the atomic number and other fundamental quantities. Compared to empirical interatomic potentials, such approaches are completely transferable, without any parameterization depending on the environment under study, but at the expense of a much higher CPU time. Although ab initio in nature and usually very accurate, these approaches nevertheless rely on different approximations, the validity of which needs generally to be assessed.

The most fundamental approximation is the Born-Oppenheimer approximation. As atom nuclei have a much higher mass than electrons, one can assume that the electrons are always equilibrated with respect to the positions of the nuclei which are considered as immobile. The resolution of the Schrödinger equation for the electrons therefore leads to the energy of the system as a function of the atomic positions. Knowing this function and also its first derivatives, i.e. the atomic forces, standard algorithms of atomic simulations can then be used. For the ab initio modeling of dislocations, this is usually restricted to molecular statics, including energy barrier calculations, because of the high CPU burden of the energy and force calculation.

Most ab initio calculations of dislocations are relying on the density functional theory (DFT). This makes use of the [START_REF] Hohenberg | Inhomogeneous electron gas[END_REF] theorem showing that the ground-state energy is the minimum of a functional depending only on the electronic density. This dramatically simplifies the problem as the electronic density depends only on the position, whereas the many-electron wave function entering Schrödinger equation is a function depending on the 3N electron coordinates, with N the number of electrons in the system. The [START_REF] Kohn | Self-consistent equations including exchange and correlations effects[END_REF] approach allows then a practical implementation, where the Schrödinger equation is solved for an equivalent system of non-interacting electrons. This necessitates the definition of an unknown contribution to the Hamiltonian, the exchange and correlation potential. Most of dislocation calculations are performed with the local density (LDA) or the generalized gradient (GGA) approximations, assuming that this contribution depends only locally on the electronic density or also its gradient.

For dislocation calculations, it is enough to consider that only the electrons of the outer shells participate to the atomic bonding. Electrons of the inner shells are not sensitive to the atom environment and can be assumed to have the same ground state as for the isolated atom. Kohn-Sham equations are then solved only for valence electrons. One can further reduce the CPU overhead by replacing with a pseudopotential the interaction potential of the valence electrons with the ionic core. This pseudopotential aims to reduce the strong oscillations of the electronic wave functions close to the dislocation core, because the description of these oscillations necessitates a large basis set, while still leading to the correct wave functions outside this core region. Different pseudoization schemes, norm-conserving or ultrasoft pseudopotentials as well as the projected augmented wave (PAW) method, are available.

Ab initio codes used for dislocations are relying on Born-von Karman periodic boundary conditions to model the solid, whatever the boundary conditions used to incorporate a dislocation in the simulation cell (Section 3). Electronic wave functions are thus a superposition of Bloch waves with wavevectors spanning the first Brillouin zone. Integration in the reciprocal space is performed on a regular grid sampling the first Brillouin zone, using smearing functions to broaden the electronic density of states. Different basis sets can be used to describe the Bloch waves, with plane waves being the most popular choice for dislocations.

Ab initio approaches devoted to the study of dislocations are thus not specific: they are making use of standard DFT implementations which are now current modeling tools in solid state physics. Feature specific to dislocation modeling, as described in the next section, is the necessity to use a supercell large enough to let the dislocation core adopt its fully relaxed configuration, with boundary conditions compatible with the long range distortion induced by the defect. A high accuracy is also generally needed for such calculations as the energy variations involved by the dislocation core are usually small. For instance, the Peierls energy barrier opposing the glide of 1/2 1 1 1 screw dislocations in BCC transition metals does not exceed 100 meV/b, where b, the norm of the Burgers vector, corresponds to the height of the simulation cell necessary to model such a dislocation.

Boundary conditions

The ab initio modeling of dislocations needs special care in the way the boundary conditions are handled. First, a dislocation creates a long-range elastic field which needs to be taken into account. Second, it is not possible to include a single dislocation in a simulation box with full periodic boundary conditions which usually constitute the paradigm in the modeling of bulk materials: the dislocation opens a displacement discontinuity and another defect is needed to close the discontinuity and allow for periodicity. As a result, different boundary conditions compatible with ab initio calculations have been developed to model dislocations.

All approaches enforce periodicity in the direction of the dislocation line. In pure metals, one usually uses the shortest periodicity vector to define the dimension of the simulation cell in this direction, thus modeling an infinite straight dislocation. But this size needs to be increased if one wants to introduce a solute atom on the dislocation line, so as to minimize the interaction of the solute atom with its periodic images and truly study the interaction of the dislocation with a single foreign atom. A larger size is also needed to model a kinked dislocation. This is usually possible only in covalent crystals where the atomic bonds are highly directional, leading to abrupt kinks experiencing a non negligible energy barrier when migrating along the dislocation line. In metallic systems with less directional atomic bonding, kinks are usually spread over a larger distance and are highly mobile, making it hard to stabilize them in a simulation cell whose size is compatible with ab initio calculations.

Cluster approach

The easiest way to model a dislocation is to use an infinite cylinder whose axis coincides with the dislocation line. Periodicity is enforced only along the dislocation line. The dislocation is created by displacing all atoms according to the Volterra solution given by anisotropic elasticity theory for the dislocation displacement field [START_REF] Stroh | Dislocations and cracks in anisotropic elasticity[END_REF][START_REF] Stroh | Steady state problems in anisotropic elasticity[END_REF]. Atoms at the cylinder surface (region 2 in Fig. 1a) are kept fixed in their initial positions and only atoms inside the cylinder are relaxed. One thus models an isolated dislocation in an infinite continuum.

But this modeling approach has severe drawbacks. The elastic solution used to fix the atoms at the boundary is only approximate as it relies on linear elasticity, thus neglecting crystal anharmonicity which can be strong close to the dislocation line. Moreover, the Volterra elastic solution, used to fix the atoms at the boundary, only corresponds to the long-range elastic field of the dislocation. Close to the dislocation line some additional contributions, the dislocation core field, need to be accounted for [START_REF] Eshelby | Anisotropic elasticity with applications to dislocation theory[END_REF]. A spreading of the dislocation core can be the reason for the existence of such a core field, but even dislocations with a compact core, like 111 screw dislocations in BCC metals, possess a non negligible core field. Although this core field decays more rapidly than the Volterra elastic field, the size of the simulation boxes that can be handled by ab initio calculations are never large enough to neglect it. The rigid boundary conditions do not allow the correct development of this core field and thus perturb the relaxation of the dislocation core.

The fixed atomic positions imposed at the boundary also prevent use of this method to determine the lattice friction opposing dislocation motion. If the dislocation moves during the simulation, this boundary condition will not be compatible anymore with the new dislocation position. This induces indeed a back-stress opposing the dislocation motion. As a result, any simulation relying on this boundary condition will overestimate the dislocation Peierls stress, which is the minimum stress necessary to move the dislocation at 0 K.

Flexible boundary conditions

To remove the artifacts induced by the rigid boundary conditions, dislocation modeling with flexible boundary conditions has been developed. The proposed method relies either on the use of the lattice Green's function [START_REF] Sinclair | Flexible boundary conditions and nonlinear geometric effects in atomic dislocation modeling[END_REF][START_REF] Woodward | First-principles simulations of dislocation cores[END_REF] or on the coupling with an empirical potential [START_REF] Liu | An improved QM/MM approach for metals[END_REF][START_REF] Chen | Solute and dislocation junction interactions[END_REF].

The lattice Green's function G i j (r) expresses, in the crystal harmonic approximation, the displacement u induced on an atom in position r by a force F acting on an atom at origin:

2 u i (r) = G i j (r)F j . (1)
This lattice Green's function can be obtained by inversion of the force-constant matrices of the perfect crystal [START_REF] Yasi | Prediction of thermal cross-slip stress in magnesium alloys from a geometric interaction model[END_REF][START_REF] Tan | Computation of the lattice Green function for a dislocation[END_REF] or can be tabulated from direct calculations in a perfect lattice [START_REF] Sinclair | Flexible boundary conditions and nonlinear geometric effects in atomic dislocation modeling[END_REF][START_REF] Rao | Green's function boundary conditions in two-dimensional and three-dimensional atomistic simulations of dislocations[END_REF]. In the long range limit, G i j (r) converges to the elastic Green's function given by anisotropic elasticity theory.

(a)

(2)

(1)

d (b) (1) (2) (3)
Fig. 1 Boundary conditions used to model an isolated straight dislocation in the cluster approach. The outer boundary is either (a) rigid or (b) flexible and controlled by lattice Green's functions or by coupling with an empirical potential.

Flexible boundary conditions based on lattice Green's functions still make use of a cylinder geometry to model a single dislocation, but three zones are now defined (Fig. 1b). Atoms in the inner zone (1) are relaxed with the ab initio code until the forces acting on them are smaller than a fixed threshold, while atoms in zones (2) and (3) are kept fixed. At the end of this step, atomic forces have appeared in zone (2), because the dislocation elastic field deviates from the Volterra solution used as an initial guess. The lattice Green's function is then used to displace atoms in all three zones according to Eq. 1 using all atomic forces in zone (2). This leads to the cancellation of forces in zone (2) but makes new forces appear in zone (1). The procedure is thus iterated until all forces in zones ( 1) and ( 2) are null. This self-consistent cycle is necessary because the lattice Green's function of the perfect crystal only approximates the linear response of the dislocated crystal. Atoms in zone (3) serve as a buffer to prevent any perturbation by the external boundary of forces building in zone (2). As shown by [START_REF] Segall | Ab initio and finitetemperature molecular dynamics studies of lattice resistance in tantalum[END_REF], this buffer region may need to be quite large in metals to obtain negligible perturbations in the inner regions. This can be minimized by removing the surfaces delineating zone (3) and using periodic boundary conditions in all directions. Interface defects are then present at the boundary between two periodic simulations cells. But these defects lead to a smaller perturbation of the electronic density than the vacuum layer of the surfaces [START_REF] Woodward | First-principles simulations of dislocation cores[END_REF]. One thus perfectly models an isolated dislocation in an infinite crystal taking full account of the dislocation core field. It is possible to study dislocation cores with a reduced number of atoms in the simulation cell, a size usually compatible with ab initio calculations.

A similar approach relies on the coupling of the ab initio calculations with an empirical potential [START_REF] Liu | An improved QM/MM approach for metals[END_REF][START_REF] Chen | Solute and dislocation junction interactions[END_REF]. The simulation cell is still divided in three regions (Fig. 1b). Ab initio calculations are performed only for a smaller simulation cell corresponding to regions (1) and (2). Atoms in regions ( 2) and (3) are relaxed according to the forces calculated with the empirical potential, whereas atoms in region (1) are relaxed according to ab initio forces plus a correction to withdraw the perturbation caused by the external boundary of the ab initio box. The buffer region (2) has been added to the original approach [START_REF] Choly | Multiscale simulations in simple metals: A densityfunctional-based methodology[END_REF] to minimize this correction by protecting atoms from the external boundary. To operate, this method needs therefore an empirical potential which perfectly reproduces the lattice parameters given by ab initio calculations, which can generally be done by rescaling the distances. Besides, the potential has also to match as best as possible the ab initio linear response, i.e. at least the elastic constants and, ideally, the whole phonon spectrum.

As it will be discussed in the last section, the main drawback of this ab initio dislocation model using flexible boundaries arises from the difficulty of extracting dislocation energy. The problem may be actually less sensitive with the second approach relying on a coupling with an empirical potential where an energy formulation exists. In this case, one can obtain a reasonable estimation of the dislocation energy provided the potential gives an accurate description of the boundary energy compared to the ab initio calculations. While these flexible boundaries truly allow the modeling of an isolated dislocation, thus predicting its core structure and its evo-lution under an applied stress without any a priori artifact induced by the small size of the simulation cell inherent to ab initio calculations, the approach is still under active development to also provide information on the dislocation energy.

Periodic boundary conditions

To get rid of the external boundary and to use periodic boundary conditions in all three directions without the introduction of a defective interface, one needs to introduce in the simulation cell a dislocation dipole, i.e. two dislocations with opposite Burgers vectors. One thus models a 2D periodic array of dislocations (Fig. 2).

Several arrangements of dislocation arrays can be though off, but they are not all equivalent. Among all of them, the ones which are quadrupolar display strong advantages. A periodic array is quadrupolar, if the vector d linking the two dislocations of opposite signs is equal to 1/2 (u 1 + u 2 ), where u 1 and u 2 are the periodicity vectors of the simulation cell (Fig. 2). This ensures that every dislocation is a symmetry center of the array: fixing, as a convention, the origin at a dislocation center, if a dislocation b is located at the position r, there will also be a dislocation b in -r. The stress created by these two dislocations will cancel to first order at the origin thanks to the symmetry property of the Volterra elastic field. 3 As a consequence, this quadrupolar periodic array minimizes the elastic interaction between the dislocations, and hence the Peach-Koehler force acting on each dislocation because of the image dislocations associated with periodic boundaries. It is the best-suited periodic array to extract dislocation core properties from ab initio calculations. Linear elasticity is still used to build the initial configuration, displacing all atoms according to the superposition of the displacement fields created by each dislocation composing the periodic array. The summation on periodic images can be either performed in reciprocal space [START_REF] Daw | Elasticity effects in electronic structure calculations with periodic boundary conditions[END_REF] or in direct space after regularization of the conditionally convergent sums [START_REF] Cai | Periodic image effects in dislocation modelling[END_REF]. The crystal orientation used in such elastic calculations should be chosen so as to fix the displacement discontinuity exactly in-between the two dislocations composing the dipole, thus preventing the propagation of this discontinuity to infinity. The cut vector A defining this discontinuity (Fig. 2) is therefore given by A = l × d, where l is the line vector of the dislocations and d the vector joining the centers of the +b dislocation to the -b one. If the scalar product A.b is non-null, i.e. if the dislocation dipole has an edge component and the displacement discontinuity does not coincide with the dislocation glide plane, it is also necessary to insert atoms into or delete them from the original lattice at the discontinuity location, thus following the Volterra operation.

+b -b A U 2 U 1 d U 1 U 1 U 2 U 2 +b +b -b -b A A d d
A homogeneous strain needs also to be applied to accommodate the plastic strain created by the dipole [START_REF] Daw | Elasticity effects in electronic structure calculations with periodic boundary conditions[END_REF][START_REF] Cai | Periodic image effects in dislocation modelling[END_REF] and ensure that the average stress in the simulation cell is null. This can be easily demonstrated by considering the variation of the elastic energy when a homogeneous strain ε i j is applied to a simulation cell containing a dislocation dipole defined by its Burgers vector b and its cut vector

A ∆ E(ε) = 1 2 SC i jkl ε i j ε kl +C i jkl b i A j ε kl ,
where energies are defined per dislocation unit length and have been thus normalized by the height of the simulation cell in the direction of the dislocation line. S is the area of the simulation cell perpendicular to this direction and C i jkl are the elastic constants. The average stress existing in the simulation cell is then given by

σ i j = 1 S ∂ ∆ E ∂ ε i j = C i jkl ε kl -ε 0 kl , (2)
with the plastic strain defined by

ε 0 kl = - b i A j + b j A i 2S . (3)
One thus sees that the stress given by Eq. 2 is null when the applied strain ε i j is equal to the plastic strain ε 0 i j . When this applied strain is different, a Peach-Koehler force acting on the dislocations may exist. This allows studying properties of the dislocation core under an applied stress, to determine its Peierls stress, for instance.

Finally, when a stress variation is observed in ab initio calculations, Eq. 2 allows to deduce the plastic strain from this stress, and thus the dislocations' relative positions via the cut vector A (Eq. 3). For instance, the trajectories of the screw dislocations gliding between two neighbouring Peierls valleys have been determined thanks to this method in HCP Zr [START_REF] Chaari | First-principles study of secondary slip in zirconium[END_REF] and in BCC transition metals [START_REF] Dezerald | Plastic anisotropy and dislocation trajectory in BCC metals[END_REF].

With these periodic boundary conditions, all the excess energy contained in the simulation cell is due to the dislocations. As it will be shown in the last section, elasticity theory can be used to isolate the contribution of a single dislocation. These periodic boundary conditions offer thus a convenient way to extract dislocation energy from ab initio calculations. But the dislocation core structure, and hence the associated excess energy, can be perturbed by the presence of the periodic images. In practice, one will therefore need to check how sensitive are the obtained dislocation properties with the size of the simulation cell.

Dislocation core structures

Different representations can be used to image and analyse the relaxed dislocation core structure obtained by atomic simulations. This allows, for instance, highlighting a spreading of or a dissociation of the dislocation.

Differential displacement maps

Differential displacement maps were introduced by [START_REF] Vitek | The core structure of 1/2 111 screw dislocations in b.c.c. crystals[END_REF]. Two examples are shown in Fig. 3 for a screw dislocation in a body centered cubic (bcc) crystal and a hexagonal close packed (hcp) crystal. In these maps, the crystal is projected in the plane perpendicular to the dislocation line, using for atomic columns the positions in the perfect crystal. The differential displacement caused by the dislocation is calculated by considering the difference between the vector connecting two neighbour atoms in the relaxed dislocated crystal and the same connecting vector in the perfect crystal. One then plots the projection of this differential displacement along the direction of the Burgers vector with an arrow pointing from one atomic column to the other, centered in the middle of the two columns and with an amplitude proportional to the differential displacement. As the arrows are proportional to the displacement difference, they are a representation of the discrete derivative of the displacement field, i.e. of the strain created by the dislocation.

The differential displacement map of the 1/2 1 1 1 screw dislocation in bcc Fe shown in Fig. 3a highlights the compactness and the 3-fold symmetry of the core. Arrows have been normalized so that an arrow linking the centers of two atomic columns corresponds to a differential displacement of b/3. One can thus draw Burgers circuits on this map and obtain the norm of the Burgers vector of the enclosed dislocation by summing arrows. The only non-null Burgers vector is obtained for circuits containing the dislocation center indicated by a cross, in particular for the triangle connecting the three central [1 1 1] atomic rows, with a norm equal to b. The dislocation is thus well localized.

The picture is quite different for the 1/3 1 2 1 0 screw dislocation in hcp Zr shown in Fig. 3b. The differential displacement map shows a non isotropic distribution with a spreading of the dislocation core in the (1 0 1 0) prismatic plane. The normalization here ensures that the maximal arrows correspond to a b/2 differential displacement. The presence of a ribbon with arrows having almost the same length therefore corresponds to a b/2 prismatic stacking fault which is known to be stable in this transition metal. The differential displacement map thus clearly evidences the dissociation of the screw dislocation in two 1/6 1 2 1 0 partial dislocations separated by a prismatic stacking fault. We only give here the salient points of the method to extract the Nye tensor from atomic simulations and the reader is referred to the original publication for the practical implementation.

Dislocation density

The first step is to define the elastic distortion, i.e. the gradient of the elastic displacement, at each atomic position. Note that this differs from the gradient of the total displacement. One cannot simply compare the atomic positions after and before the introduction of the dislocation to obtain this elastic distortion, but one needs to find for each position the closest undistorted environment corresponding to a zero stress state. This is performed by comparing, for each atom, the positions of its nearest neighbours in the dislocated relaxed crystal with the ones in a perfect crystal. Knowing the two sets of neighbour positions, each bond in the dislocated crystal, defined by its vector P (γ) , is identified with the corresponding Q (β ) bond in the perfect crystal, which is the perfect bond leading to the smallest angle Φ (γβ ) between the vectors P (γ) and Q (β ) . Only the bonds which are not too much distorted and for which the angle Φ (γβ ) is smaller than a chosen threshold are kept. The elastic distortion F e is then locally defined through the relation P

(γ) i = F e i j Q (β )
j . This cannot be satisfied for each set of associated bond (γ β ) as the system of equations is overdetermined and the matrix F e is obtained by the pseudo-inverse method, i.e. a least-square fitting. The Nye tensor α is then defined through the rotational of the inverse transpose of the distortion, α = -∇ × ( + F e ) -1 , using finite differences between neighbour atoms for derivation. This defines the Nye tensor on a set of discrete points, generally atomic positions, which can be then interpolated with cubic-splines or Fourier series, or smeared with Gaussian-like spreading functions.

The dislocation density obtained for the 1/2 1 1 1 screw dislocation in bcc Fe (Fig. 3a) illustrates the compactness of the core: the distribution has only one peak. On the other hand, the dislocation distribution for the 1/3 1 2 1 0 screw dislocation in hcp Zr (Fig. 3b) shows two well-separated peaks which correspond to the two partial dislocations. To obtain the Nye tensor in this latter case, the neighbourhood of each atom in the dislocated crystal is compared not only to the two different neighbourhoods existing in the perfect hcp crystal, but also to the ones of the unrelaxed prismatic stacking fault, to identify the closer reference from which the elastic distortion is calculated.

Disregistry

The extraction of the disregistry offers another way to characterize the dislocation core structure, particularly convenient when the core is planar. The disregistry is the difference of displacement induced by the dislocation between the plane just above and the one just below the dislocation glide plane. It is thus obtained from the relaxed configuration through

D(x) = u + (x) -u -(x),
where u + (x) and u -(x) are the displacements of the atoms belonging respectively to the upper and lower planes and located at the position x in the direction perpendicular to the dislocation line. This disregistry varies4 from 0 for x → -∞ to b for x → ∞, thus corresponding to the dislocation glide plane being locally sheared by Peierls and Nabarro built a model that leads to a simple analytical expression of the disregistry [START_REF] Lu | The Peierls-Nabarro model of dislocations: a venerable theory and its current development[END_REF]. According to this model, the disregistry 5 is given by

D(x) = b π arctan x -x D ζ + π 2 ,
where x D is the dislocation position and ζ its spreading in the glide plane. Fitting of these two parameters to the data extracted from the atomistic simulations allows thus defining the dislocation position and characterizing the spreading of its core.

For dissociated dislocations, the disregistry is the sum of the contributions of the two partial dislocations, i.e., assuming that each partial dislocation has the same Burgers vector b/2 and the same spreading ζ ,

D(x) = b 2π arctan x -x D -d/2 ζ + arctan x -x D + d/2 ζ + π ,
where d is the dissociation distance. As shown in Fig. 4 for the 1/3 1 2 1 0 screw dislocation in hcp Zr, such an analytical expression generally perfectly describes the disregistry extracted from the atomic simulations. One can also notice that the positions in the glide plane of the partial dislocations deduced from the disregistry agree which what can be inferred from the differential displacement and the Nye tensor maps (Fig. 3b). Some consequences of the periodic boundary conditions used to model this dislocation are visible on these disregistry plots. The dislocation density slightly depends, through the dissociation distance d and the partial spreading ζ , on the simulation cell, not only its size but also its shape. One also sees that the density of the periodic dislocation arrays (solid line in Fig. 4), obtained by summation of the contributions of the periodic images in the glide plane, slightly differs from the one of the isolated dislocation (dashed line in Fig. 4), especially in the distribution tail. Flexible boundary conditions, as discussed in Section 3.2, have been developed to solve such limitations of periodic boundary conditions.

Dislocation energy

Ab initio calculations give access to the dislocation core energy and its variations. This core energy is the part of the dislocation excess energy which arises from the strong perturbation of the atomic interactions in the immediate vicinity of the disdone as the Burgers vector b of a perfect dislocation is a periodicity vector of the lattice and adding a displacement nb (n ∈ Z) to an atom does not change the configuration. 5 For simplicity, we consider scalar quantities by projecting the displacement in the direction of the Burgers vector.

location line and which cannot be described by linear elasticity. Contrary to the dislocation elastic energy, this is an intrinsic property which only depends on the dislocation and not on the surrounding environment. When several configurations exist for the same dislocation, this core energy controls their relative stability. Its variations with the position of the dislocation in the crystal lattice is at the origin of the lattice friction opposing dislocation glide.

Core energy

Among the different boundary conditions introduced in section 3 to model a dislocation at an atomic scale, only periodic boundary conditions allow for an unambiguous determination of the dislocation core energy with ab initio calculations. This is a consequence of the energy formulation inherent to ab initio calculations. Because of the non-locality of the electronic energy, which contains a contribution which needs to be evaluated in reciprocal space, one cannot easily partition the excess energy of the simulation cell between the dislocation and the external boundary contributions when a defective boundary has been introduced like in cluster approaches using either fixed ( § 3.1) or flexible boundaries ( § 3.2). Ab initio methods to project the energy on atoms have been proposed: they theoretically allow for such a partition but the application to the calculation of a dislocation core energy still remains to be done. Even if the absolute value of the core energy appears difficult to determine with cluster approaches, methods to estimates its variation are nevertheless possible. One can, for instance, calculate the difference of core energy between two configurations of the same dislocations by simply considering the difference of ab initio total energies. But such an approach assumes that the contribution of the external boundary will cancel in the difference, an assumption which may be hard to validate. Variation of the dislocation energy with its position in the crystal lattice can also be estimated by considering the work of the atomic forces during the motion [START_REF] Swinburne | Computing energy barriers for rare events from hybrid quantum/classical simulations through the virtual work principle[END_REF]).

On the other hand, with periodic boundary conditions, all the excess energy arises from the dislocations. This excess energy ∆ E is defined as the energy difference per unit of height between the supercell with and without the dislocation dipole. 6 It is given by the sum of the core energy E core of the two dislocations, of the elastic energy E elas dipole of the dipole contained in the supercell and of its elastic interaction with its periodic images:

∆ E = 2 E core + E elas dipole + 1 2 ∑ n,m E elas inter (n u 1 + m u 2 ). (4)
The factor 1/2 appears in front of this last contribution as only one half of the interaction is attributed to each interacting dipole. When partitioning the excess energy into a core and an elastic contribution, it is necessary to introduce a cutoff distance to isolate the dislocation cores. Close to the dislocation lines, strains are much too high to be described by linear elasticity. As a consequence, elastic fields diverge at the origin and one needs to exclude the core region from the elastic description. The elastic contribution to the excess energy is thus obtained by integrating the elastic energy density on the whole supercell except two cylinders of radius r c which isolate this elastic divergence. The core energy corresponds to the excess energy contained in these cylinders. The total excess energy ∆ E does not depend on the choice for this core radius, but the partition between a core and an elastic contribution depends on r c . The elastic energy of the dipole and its interaction with its periodic images can be computed by considering the Volterra elastic field created by the dislocations. This calculation can be performed either in reciprocal space [START_REF] Daw | Elasticity effects in electronic structure calculations with periodic boundary conditions[END_REF] or in direct space using classical results of dislocation elastic theory [START_REF] Bacon | Anisotropic continuum theory of lattice defects[END_REF]. In this last case, one uses the decomposition of Eq. ( 4), with the contribution of the dipole contained in the supercell and its interaction with the periodic images calculated separately. The dipole elastic energy is obtained by the volume integral

E elas dipole = 1 2 V σ (1) i j + σ (2) i j ε (1) i j + ε (2) i j dV ,
where σ (n) and ε (n) are the stress and strain created by the dislocation n. This is transformed into a surface integral thanks to Gauss' theorem

E elas dipole = 1 2 S σ (1)
i j + σ (2) i j u (1) i + u
(2) i dS j , with u (n) the displacement field associated with dislocation n. The integration surface is composed of the two cylinders S

(1) c and S

(2) c of radii r c removing the elastic divergence at the dislocation cores, and of the two surfaces S 0 -and S 0 + removing the displacement discontinuity along the dislocation cut (Fig. 5). The integration on both core cylinders leads to the same contribution 5 Definition of the contour surface used to calculate the elastic energy of a dislocation dipole

E elas c (φ ) = 1 2 S (1) c σ (1)
i j u

(1)

i dS j = 1 2 S (2) c σ (2) i j u
(2) i dS j .

(5)

This contributions of the core tractions to the elastic energy [START_REF] Clouet | Elastic energy of a straight dislocation and contribution from core tractions[END_REF] should not be forgotten as it ensures that the elastic energy is a state variable compatible with the work of the Peach-Koehler forces. Besides, in ab initio calculations where the distance d between the two dipole dislocations is small, this can lead to a non negligible contribution compared to the one associated with the integral along the cut surface, even for a screw orientation. The elastic energy of the dislocation dipole is then

E elas dipole = 2E elas c (φ ) + b i K 0 i j b j ln d r c , (6)
where the tensor K 0 defined by [START_REF] Stroh | Dislocations and cracks in anisotropic elasticity[END_REF][START_REF] Stroh | Steady state problems in anisotropic elasticity[END_REF] only depends on the elastic constants. The total elastic energy is finally obtained by adding the interaction of the dipole with its periodic images. But, one should realize that the summation on periodic images appearing in Eq. ( 4) is only conditionally convergent: it can be regularized with the method of [START_REF] Cai | Periodic image effects in dislocation modelling[END_REF].

As shown in Fig. 6 for the 1/2 1 1 1 screw dislocation in bcc iron, once this elastic energy is subtracted from the dislocation excess energy given by ab initio calculations, one obtains a constant core energy which does not depend on the size of the supercell. Some slight variations of the core energy are nevertheless still observed with the type of periodic arrangement used for the atomic simulations. These variations arise because only the Volerra elastic field has been considered in the calculation of the elastic energy. Dislocations also cause a core elastic field, which decays more rapidly than the Volterra elastic field. Because of the small size of the supercell used in ab initio calculations, this core field may also lead to an elastic interaction between the different dislocations composing the periodic array. This contribution to the elastic energy can be computed to improve the convergence of core energies [START_REF] Clouet | Dislocation core energies and core fields from first principles[END_REF]. A quadrupolar periodic arrangement minimizes this contribution of the core field. This is why such an arrangement is preferred when periodic boundary conditions are used. The neglect of anharmonic effects in the calculation of the elastic energy can also be a reason for the variation of the core energy with the supercell. Knowing higher order elastic constants, one can use non-linear elasticity theory in principle to calculate more precisely this elastic contribution [START_REF] Teodosiu | Elastic Models of Crystal Defects[END_REF]. But this leads to much cumbersome calculations. In practice, as anharmonicity is important only close to the dislocation core, the consideration of the dislocation core field offers a way to incorporate anhamonic effects while still relying on linear elasticity.

Peierls energy barrier

The Peierls energy is the energy barrier opposing dislocation glide. It corresponds to a variation of the dislocation core energy as the elastic energy is not dependent upon the dislocation position in the crystal lattice. It can be calculated by finding the minimum energy path linking two neighbouring stable positions of the dislocation using either constrained minimization or nudged elastic band (NEB) calculations [START_REF] Henkelman | Methods for finding saddle points and minimum energy paths: theoretical methods in condensed phase chemistry[END_REF].

With periodic boundary conditions, the Peierls energy is directly obtained by considering a path where both dislocations composing the dipole are displaced by one Peierls valley in the same direction. If the two dislocations are moved simultaneously along the path, their separation distance does not vary and the elastic energy is constant. This ensures that the energy variation given by the constrained minimization or the NEB calculations directly corresponds to the Peierls energy. However, this is possible only if crystal symmetry ensures that the path is symmetrical as the two dislocations are traversing their Peierls barriers in the opposite direction. This is the case, for instance, for the 1/2 1 1 1 screw dislocation in a bcc lattice gliding in a {1 1 0} plane (Fig. 7a). If the path is not symmetrical, either because of the lack of crystal symmetries or because of an applied stress, it is not possible anymore to move both dislocations simultaneously in the same direction. One needs either to move them in opposite directions or to keep one dislocation fixed when the second one is moving. As a consequence, the separation distance, and thus the elastic interaction energy, is varying along the path. One can calculate this variation of the elastic energy and subtract it from the excess energy in order to obtain the Peierls energy. To be able to perform this elastic calculation, one needs first to determine the exact dislocation position x D for each reaction coordinate ζ along the path. This can be done using the dislocation disregistry (cf. §4.3) if the motion is planar or by fitting the atomic displacements with the Volterra elastic solution. As the stress is directly linked to the applied strain and the dislocation positions (Eqs. 2 and 3), one can also use the stress variation observed along the dislocation path to determine the dislocations position. The example of Fig. 7b shows that, with this correction for the variation of the elastic energy, the same Peierls energy is obtained under zero applied stress when one dislocation is fixed or when both dislocations are moved (Fig. 7a).

Peierls stress

The Peierls stress is the applied resolved shear stress necessary to cancel the Peierls barrier so that the dislocations can glide freely without the need of thermal activation, i.e. the stress necessary to move the dislocation at 0 K. For an applied stress τ, the Peierls barrier is given by the enthalpy variation

∆ H P (x D , τ) = ∆ E P (x D ) -τ b x D ,
which corresponds to the Peierls energy barrier plus the work of the applied stress when the dislocation has glided a distance x D . The Peierls stress is thus the maximum applied stress τ for which the function ∆ H P (x D , τ) goes through a maximum in the range 0 ≤ x D ≤ λ P . If one assumes that the energy barrier ∆ E P (x D ) does not depend on the applied stress τ, it is given by

τ P = 1 b Max ∂ ∆ E P ∂ x D . (7)
The Peierls stress can thus be theoretically obtained from the calculation of the Peierls energy barrier under zero applied stress. But, the evaluation of the derivative in Eq. 7 requires to know the variation of the energy as a function of the dislocation position and not only of the reaction coordinate. In practice, the obtained value for τ P will sensitively vary with the method chosen to estimate the dislocation position along the path.

One can also directly calculate with ab initio calculations the Peierls barrier under an applied stress so as to estimate at which stress the barrier cancels (Fig. 7).

In such calculations, one does not really apply a stress but a strain corresponding to the target stress (Eq. 2). With periodic boundary conditions, as the distance between the two dislocation is varying, the applied stress is also varying along the path. Eqs.

(2) and (3) show that the stress variation is directly proportional to the dislocation displacement and to the inverse of the surface S of the simulation cell perpendicular to the dislocation line. If only one dislocation is moving along the path, this stress variation therefore does not exceed

δ τ = µ bλ P S ,
where µ is the shear modulus in the dislocation glide plane.

If one is only interested in the calculation of the Peierls stress and not in the variation of the Peierls barrier with the applied stress, one can simply perform static relaxation of a dislocation under an applied stress to see at which applied stress the dislocation glides by at least one Peierls valley. With periodic boundary conditions one still needs to take into account the variation of the elastic interaction and of the applied stress when the dislocation is moving to interpret the results. On the other hand, no such artifact exists with a cluster approach using flexible boundary conditions which truly models a single isolated dislocation under an applied stress. Straining homogeneously the simulation cell to obtain the targeted applied stress, the Peierls stress is defined as the stress for which the dislocation cannot be stabilized anymore and escapes from the cluster. If one is only interested in the evolution of the dislocation core structure under an applied stress and on the determination of the Peierls stress, this cluster approach therefore appears as the method of choice. Nevertheless, whatever the boundary conditions, determination of the Peierls stress by such an instability condition of the dislocation core under an applied stress necessitates a strict threshold criterion on the atomic forces to obtain a meaningful value.

Conclusions

Dislocation core properties can now be routinely studied with ab initio calculations thanks to the different methodological developments summarized in this chapter. This usually necessitates a coupling between atomistic model and elasticity theory, for which different already available tools can be used: see, for instance, D. R.

Trinkle website7 for an implementation of the lattice Greens functions or the Babel package8 for handling dislocations in atomistic simulation cells and elastic energy calculations. Useful information on the dislocation core structure are thus obtained. Such calculations can, for instance, characterize possible dissociation or spreading of the core, or evidence the existence of several stable configurations for the same dislocation. One gets access to the different energy barriers opposing the dislocation motion and to their variation with the applied stress. It is also possible to study how these core properties are altered by the interaction with solute atoms.

Because of the limited size that can be handled by ab initio calculations, such studies are usually limited to the study of straight dislocation, and only few ab initio calculations have considered until now the presence of kinks on the dislocation lines. Upscaling modeling approaches, relying, for instance, to the line tension approximation to describe kink nucleation, are therefore needed to go from these fundamental core properties determined at 0 K with ab initio calculations to dislocation mobility laws at finite temperature. Larger atomistic simulations are also possible using empirical potentials to describe atomic interactions. These simulations allow studying more complex situations and simulating different dislocation mechanisms, like glide, cross-slip and interaction with other elements of the microstructure, without assuming a priori the elementary mechanism. In such a context, ab initio calculations are useful to validate and also help the development of empirical potentials which correctly reproduce dislocation fundamental properties.

Fig. 2

 2 Fig.2Simulation of a dislocation dipole with periodic boundary conditions, using a quadrupolar arrangement. The dipole is defined by its Burgers vector b, the dipole vector d joining the two dislocation centers, and the cut vector A, with the corresponding discontinuity surface indicated by a double black line. u 1 and u 2 are the periodicity vectors of the simulation cell perpendicular to the dislocation line. The example on the right corresponds to the simulation cell used for the modeling of the 1/2 1 1 1 screw dislocation in bcc iron. The dislocation core structures are shown through their differential displacement maps and their density (cf. Fig.3afor a details).

Fig. 3

 3 Fig. 3 Core structure of a b screw dislocation (a) in bcc iron (Dezerald et al 2016) and (b) in hcp zirconium (Clouet et al 2015). In these projections perpendicular to the dislocation line, atoms are sketched by symbols with a colour depending on their (a) (1 1 1) and (b) (1 2 1 0) plane in the original perfect crystal. In (b), different symbols are used for atoms depending on their neighbourhood in the dislocated crystal, i.e. close to the perfect hcp crystal (circles) or to the unrelaxed prismatic stacking fault (squares). The arrows between atomic columns are proportional to the differential displacement created by the dislocation in the direction of the Burgers vector. The colour map show the dislocation density ρ b normalized by the lattice parameter (Nye tensor). The center of the dislocation is indicated by a + cross. The × crosses in (b) correspond to the positions of the partial dislocations deduced from the disregistry in Fig. 4.

Fig. 4

 4 Fig. 4 Disregistry D(x) created by a 1/3 1 2 1 0 screw dislocation in its (1 0 1 0) prismatic glide plane in hcp Zr, and corresponding dislocation density ρ(x) = ∂ D(x)/∂ x. Symbols correspond to ab initio calculations and lines to the fit of the Peierls-Nabarro model, considering periodicity or not (straight and dashed lines respectively). Results are shown for different n × m periodic arrangements corresponding to the periodicity vectors u 1 = n/2 [1 0 1 0] and u 2 = m [0 0 0 1] (see Clouet (2012) for details). For clarity, disregistries D(x) have been shifted by 0.2 between different data sets. The obtained dissociation distance d and spreading ζ of the partial dislocations are indicated for the 6 × 8 periodic array whose core structure is shown in Fig. 3b.

Fig. 6

 6 Fig.6Decomposition of the excess energy ∆ E of a 1/2 1 1 1 screw dislocation dipole in bcc Fe in an elastic contribution E elas and a core energy E core , using a core radius r c = b/2. Different symbols correspond to different periodic arrangements (see[START_REF] Clouet | Dislocation core energies and core fields from first principles[END_REF] for details).

Fig. 7

 7 Fig. 7 Peierls barrier of a 1/2 1 1 1 screw dislocation in bcc Mo. (a) The energy variation ∆ E is shown as a function of the reaction coordinate ζ , and (b) the enthalpy variation ∆ H as a function of the dislocation position x D normalized by the distance λ P between two Peierls valleys (see[START_REF] Dezerald | Ab initio modeling of the two-dimensional energy landscape of screw dislocations in bcc transition metals[END_REF][START_REF] Dezerald | Plastic anisotropy and dislocation trajectory in BCC metals[END_REF] for details). For the Peierls barriers under stress (b), only one half of the barrier has been computed, with one dislocation of the dipole being displaced while the second one remains fixed. The open symbol is the enthalpy variation in the middle of the pathway (x D /λ P = 1/2) before correcting for the variation of the elastic interaction energy for the τ = 0 calculation.

We use the Einstein implicit summation convention on repeated indexes appearing in all expressions.

σ V (-r) = -σ V (r) with σ V the Volterra stress field of a single dislocation.

If the cut plane used to introduce the dislocation in the simulation cell does not correspond to its glide plane, it is necessary to define the atomic displacement in the 0 to b interval. This can be

If atoms have been removed or inserted during the creation of the dipole, the energy of the perfect supercell needs to be normalized by the correct number of atoms.

http://dtrinkle.matse.illinois.edu

http://emmanuel.clouet.free.fr/Programs/Babel

Acknowledgements Drs. Nermine Chaari, Lucile Dezerald, and Lisa Ventelon are acknowledged for their contributions to the works presented here. Dr. Antoine Kraych is thanked for fruitful discussions. Parts of this work have been performed using HPC resources from GENCI-CINES and -TGCC (Grant 2017-096847).


Antoine Marot

Balthazar Donon

Isabelle Guyon

Benjamin Donnot

Learning To Run a Power Network Competition

We present the design of a new competition "Learning To Run a Power Network" to test the potential of Reinforcement Learning to solve a real world problem of great practical importance: controlling electricity transportation in smart grids while keeping people and equipment safe.

Context

Power grids transporting electricity across states, countries, or continents, are vital components of modern societies, playing a central economical and societal role, by supplying reliably power to industry, services, and consumers. Electricity blackouts may lead to significant losses and delay in public services and strategical industries, de facto freezing society. Grid operators are in charge of ensuring that a reliable supply of electricity is provided everywhere, at all times. However, their task is increasingly difficult in the digital era because they have to examine in real time massive amounts of data despite Artificial Intelligence's nascent developments. Power systems are in some sense archaic complex artificial (intelligent) systems currently in operation, in great need for innovation to cope with the increasingly complex task of satisfying electricity demand while using renewable energies and open market exchanges. Indeed, wind, solar, and the like, are not very dependable sources of energy (because they vary with the day/night cycle and weather conditions). Also, while providing opportunities, electricity markets may be plagued by speculations. This places new flexibility and reactivity requirements on the smart grids of the future.

Rather than using more hardware (e.g. more transportation lines), new software that could operate the grid with latest AI could be a game changer to optimize usage of existing assets [START_REF] Donnot | Antoine Marot, Introducing machine learning for power system operation support[END_REF]. In the community of power systems, optimal control methods [START_REF]Multistage adaptive robust optimization for the unit commitment problem[END_REF] have been explored but have failed short until now, and humans are still in charge of operating the grid. With the latest breakthroughs from AlphaGo at go [START_REF] Silver | Mastering the game of Go with deep neural networks and tree search[END_REF] and Libratus at poker [START_REF] Brown | Safe and Nested Subgame Solving for Imperfect-Information Games[END_REF], Reinforcement Learning (RL) seems a promising new avenue to develop an artificial agent able to operate a complex power system in real-time, assisted by high performance physical grid simulators. In line with recent work [START_REF] Dalal | Hierarchical Decision Making In Electricity Grid Management[END_REF] using RL within the power system community, we propose a competition "Learning to run a Power Network" using a challenging game-like environment we have designed, to enable the RL community to join forces with the power system community in order to tackle this burning smart grid issue at larger scale.

Competition Protocol

This competition is inspired by the "Learning to run" competition at NIPS 20171 , as could be guessed from its name "Learning to run a power network". By analogy, its goal is somewhat similar in nature: competitors will have to "run" (manage) a power grid as long as possible, before "falling" into a blackout (cascading failure). It will be also similar in terms of platform, since we are making available a virtual environment of a power grid that relies both on a "generator of scenarios", including inputs (so-called "injections", provided by power generators and consumers) and obstacles (broken transmission lines), and on a physical grid simulator computing power flows in every transmission line (the state of the system). This platform uses the opengym RL framework, developed by OpenAI2 and the open source PyPower simulator3 . Competitors will then have to create a proper controller in this environment, most likely (but not necessarily) using RL techniques. Aside from addressing a problem in a totally different application domain with a important economic impact, compared to the Learning to Run challenge, which inspired us, our setting has many novel and challenging features from the RL point of view :

• A state space described by both continuous variables (flows in lines) and discrete variables (line interconnection patterns).

• A discrete action space (change in line interconnection patterns).

• A stochastic component in the environments: "obstacles" present themselves to the agents as scheduled operations on the grid (maintenance), random variations in injections due e.g. to weather conditions (changing loads and productions) and unpredictable incidents (lines broken by a thunderstorm).

• Various reward models created to adjust the game difficulty level.

Hence, this competition offers novel aspects, which should attract RL researchers and ML people at large. The competition will take place in 2019.

https://www.crowdai.org/challenges/nips-2017-learning-to-run

https://gym.openai.com/

https://pypi.org/project/PYPOWER/

Acknowledgments

This work would not have been possible without the support of RTE (French Power Grid Operator), INRIA as well as Chalearn, and without the contribution of many advisors and students, including Patrick Panciatici, Marc Schoenauer, Louis Wehenkel, Olivier Pietquin, Joao Araujo, Marvin Lerousseau, and Kimang Khun.


Natalia Grabar

Olga Kanishcheva

Thierry Hamon

Multilingual aligned corpus with Ukrainian as the target language

Introduction

Creation of linguistic resources is a very important issue for linguistics and Natural Language Processing. Their availability provides the possibility to design, develop and evaluate methods and tools specific to several contexts and applications (information retrieval, acquisition of lexica, machine translation, categorization of documents...).

The purpose of this work is to describe multilingual parallel and aligned corpus, in which the target language is Ukrainian, while the current source languages are Polish, French and English. This means that the corpus is built with texts from these three languages (Polish, French and English) and their translations in Ukrainian.

Ukrainian language is currently provided with little freely available resources. We are mostly interested here by the available corpora, including parallel corpora. Among the existing work, let's notice the national corpus of the Ukrainian language (Дарчук, 2010) available online (http://www.mova.info/corpus.aspx?l1=209) and literary corpus with the work by Ivan Franko (Бук, 2010) built for the research and educational purposes, and a corpus with dialectal texts (Сірук, 2012). Besides, several parallel corpora involving Ukrainian have been proposed, such as Polish-Ukrainian [START_REF] Kotsyba | Polukr (a Polish-Ukrainian parallel corpus) as a testbed for a parallel corpora toolbox[END_REF] and Bulgarian-Ukrainian [START_REF] Siruk | Linguistic corpora as international cultural heritage: The corpus of Bulgarian and Ukrainian parallel texts[END_REF] corpora.

Collection of texts

For the building of our corpus, we use two kinds of texts. The first source is composed of literary corpus in Ukrainian collected from the УкрЛіт and UkrLib websites. The purpose of these websites is to promote literature in Ukrainian. According to the policy of these websites, these works are publicly available and can be used as far as they are cited. For the translated works, we collected publicly available originals from websites like Project Gutenberg (https://www.gutenberg.org/). Three source languages are covered: Polish, French and English.

The second source is composed of medical documents from the MedlinePlus website (https://medlineplus.gov/). These documents contain patient-oriented brochures on several medical topics. These brochures have been created in English and translated in several languages, among which Ukrainian. Here again, Ukrainian is the target language.

Corpus

Occurrence This dataset contains parallel texts. These source languages have been chosen for their representativity and relation with the Ukrainian language. Polish is also a Slavic language, and is close to Ukrainian. Polish is now quite well researched within the NLP field. We assume that the methods and tools developed for the Polish language can be adapted to Ukrainian provided that there are suitable corpora and resources. English and French languages are well researched from the NLP point of view. We assume, it is possible to take advantage of this research using the transfer methodologies [START_REF] Yarowsky | Inducing multilingual text analysis tools via robust projection across aligned corpora[END_REF][START_REF] Lopez | Word-level alignment for multilingual resource acquisition[END_REF], provided that there are suitable parallel and aligned corpora, and resources.

Building of corpus

The original documents may be in different formats (pdf, word, text, html). They are all converted in the text format. Besides, the documents are also converted in the UTF-8 encoding. Then, the text files are automatically segmented in sentences in each language, for which we use strong punctuation and upper-cased characters. Ideally, such segmentation should provide corpus aligned at the sentence level. Yet, it is necessary to verify the correctness of the segmentation in sentences and the parallelism between the source and target versions of a given document. Indeed, during the translation process, the organization of the sentences and their segmentation can be modified by the translator in order to better convey the meaning. Besides, some sentences can also be omitted.

Hence, the manual control and correction during the alignment at the sentence level is necessary. This is a very long and thorough yet necessary process, as it guarantees the quality of the aligned corpora. Only part of the whole set of texts available is aligned. Table above indicates the size of the currently aligned texts, each of which has undergone manual verification. On the whole, the aligned corpus provides 1,151,593 word occurrences in the target Ukrainian language. As we can see, all medical texts and all literary texts in the Polish/Ukrainian pair has been aligned and verified, while only part of French and English source texts is operational up to now. The current version of this parallel and aligned corpus is intended to grow with new texts: other texts are being checked for the correct alignment.

Corpus

Current usage of the corpus

Up to now, the medical part of the corpus has been used for the acquisition of medical terminology in three languages (English, French and Ukrainian) using the transfer methodology [START_REF] Hamon | Adaptation of cross-lingual transfer methods for the building of medical terminology in Ukrainian[END_REF]. This work relied on terminology acquisition methods and tools in English and French. It permitted to build a set with 4,588 terms in Ukrainian and the corresponding terms in French and English, for 34,267 relations in the acquired network.

Conclusion and Future Work

In this work, we proposed a description of parallel corpus in which Ukrainian is the target language, while the source languages are Polish, French and English. The corpus mainly contains fiction work but also some texts from the medical field. This corpus is partly aligned at the level of sentences. There are some current exploitations of the corpus for the acquisition of medical terminology.

Future exploitations of this corpus may be related to the machine translation, to the acquisition of cross-lingual paraphrases and disambiguation, to various contrastive studies (including stylistics and discourse). An important issue is the creation of tools for the linguistic processing of texts in Ukrainian, like POS-tagging and syntactic parsing.

A subset of the texts is being aligned by two annotators, so that the inter-annotator agreement can be computed. Besides, tools for the automatic alignment of sentences are being investigated, which may allow to enrich the set with the aligned sentences.

  In Tableabove, we indicate the size of the collected corpora (number of texts and number of word occurrences) for each language: Ukrainian (UK), French (FR), Polish (PL), and English (EN). Overall, we have over 3M word occurrences in Ukrainian.

	MedlinePlus/EN	46,544	129
		of words	Number of texts
	Literature/UK	3,111,656	110
	Literature/FR	1,310,732	29
	Literature/EN	2,203,350	51
	Literature/PL	260,536	30
	MedlinePlus/UK	43,184	129


Natalia Grabar
email: natalia.grabar@univ-lille3.fr

Thierry Hamon
email: hamon@limsi.fr

Automatic Detection of Temporal Information in Ukrainian General-language Texts

Keywords:

emporl informtion provides importnt nd preise indiE tions on fts nd eventsF etD when utomtilly proessing unstruE tured doumentsD it my e omplited to extrt temporlityErelted informtionF fesidesD suh tools re not ville for mny lngugesF e propose to dpt n existing tool for the utomti detetion nd nnottion of temporl informtion in krininF he tool is ruleEsedF st permits to detets temporl expressionsD e they solute or reltiveD nd to normlize themF e test the dpted tool on two orpor from di'erent genres nd evlute the resultsF

Introduction

Unstructured documents are the most common source of information, and they may represent the majority of information available on a particular question and domain. For instance, in the biomedical area, several documents are unstructured, such as clinical discharge summaries, scientific literature, patient brochures, informed consents, or clinical trial protocols. The situation is similar in other areas (law, energy, economics, politics, history, etc.). Hence, when working with unstructured narrative texts, the process is very demanding on automatic methods for detecting, extracting, formalizing and organizing information contained in these documents. Information extraction (IE), which is part of Natural Language Processing (NLP), proposes such methods and aims at detecting and extracting relevant pieces of information from textual data. Different types of information can be searched, such as (1) entities and events, which are traditionally detected thanks to the exploitation of terminological resources and thesauri, when available. The process is dedicated to the recognition of terms and is a very challenging issue related to the computing of variants of these terms in documents [14,12,2,16,6]; (2) or detection and extraction of contextual information, such as temporality related to the concepts. If the detection of entities and events provides factual information, extraction of contextual data permits to describe these facts with more detail. For instance, examples (1) to (7) below contain precise contextual temporal information on various events. Temporal information is important for several tasks and areas, as it allows to structure the entities and events according to their chronological occurrence. This is important in several situations. For instance, in historical studies, the events are usually ordered and then taught and studied in this order; in medical area, events related to a given patient may be ordered and thus provide a clearer view of his disease and its evolution. As a matter of fact, temporality has become an important research field in the NLP domain and several challenges addressed this task up to now, such as: ACE [1], SemEval [23, 24, 22], I2B2 2012 [21].

In our work, we propose to contribute to this research and to concentrate on the description of temporal information and on its automatic detection and annotation in Ukrainian. This implies that we have to design suitable methods, resources and tools for this language.

In what follows, we first present some related work (Sec. 2). We then precise our objectives (Sec. 3), introduce the material used (Sec. 4) and the proposed method (Sec. 5). Our results and their discussion are presented in Section 6. Finally, we conclude with some directions for future work (Sec. 7).

Related Work

Work on temporal information relies on three important steps when processing unstructured narrative documents: identification of linguistic expressions that are indicative of the temporality and their normalization [23,5,19,10], and modeling and chaining of temporal information [3,13,15,21,7]. Identification of temporal expressions, which corresponds to the first step, provides basic knowledge for further tasks aiming at the processing of the temporality. The existing available automatic systems such as HeidelTime [19] or SUTIME [5] exploit rule-based approaches, which makes them adaptable to new data, areas, and languages. Such tools usually encode temporal information with the TimeML standard.

TimeML1 [15] is an annotation standard for temporal expressions proposed in 2010. Since then, it has became the reference for encoding temporal information in different languages. For instance, it has been used in several contexts: for encoding temporal data in challenge corpora such as TempEval [24,22,4] In addition to the annotation of temporal expressions, TimeML also allows to describe events as well as relations between temporal expressions and/or events. In this paper, we only focus on the annotation of temporal expressions (TIMEX3) related to dates and durations. Description and detection of other temporal information will addressed in later work.

Objectives

The purpose of our work is to automatically detect and annotate temporal expressions in corpora in Ukrainian language. We aim particularly at the description of dates and durations, such as in Examples ( 1)-( 3) and ( 5)-( 7). During a preliminary study, we tested several existing systems for identification of temporal expressions and found out that HeidelTime [19] has the best combination of performance and adaptability. We propose to exploit this automatic system, to adapt it and to test it on general-language texts in Ukrainian.

Material

We use two types of texts from two different genres: newspaper and encyclopedic articles. Both of them have the potential to contain temporal information.

Newspaper articles

Newspaper articles are obtained from the online news journal from Ukraine Українська правда3 (Ukrainian truth). This journal covers any news related to the events which happen in Ukraine and also to the events which happen in other places but which may be important to Ukraine. The journal has been founded in 2010 and shows good popularity and objectivity. The main interest in using this type of articles is that they typically contain dates associated to events. We use 40 articles (over 31,000 word occurrences) for the development of the system and 40 articles (over 35,000 word occurrences) for its tests and evaluation.

Encyclopedic articles

Encyclopedic articles are obtained from the Wikipedia resource4 , which is a free and collaborative resource. This encyclopedia contains information on a great variety of topics. We have chosen to work with the articles related to wars, as part of the WikiWars corpus 5 [11]. This corpus is a collection of texts issued from Wikipedia articles. These texts describe the course of the most famous wars in history, including the biggest wars that happened in the 20th century. The corpus contains 22 articles (such as WW1, WW2, Vietnamese war, Russo-Japanese war, or Punic wars). The main interest in working with these articles is that they contain several dates, as they are typically associated with battles, meetings, armistices, etc. The initial project contains articles in English. It has been extended to three other languages (German, Vietnamese and Croatian) [18,9]. For our work, we compiled the corpus with the corresponding articles in Ukrainian (66,474 word occurrences). The articles have been collected similarly to the building of the original WikiWars corpus. Hence, we use these 22 articles (66,479 word occurrences) for the evaluation of the automatic system.

Methods

The methods are composed of several steps: pre-processing of texts, adaptation of HeidelTime to Ukrainian, and evaluation of the automatic annotations.

Pre-processing

All source documents are in the html format because they are obtained from online resources. The documents are converted in the text format. The characters are encoded with the UTF-8 characterset.

Adaptation of HeidelTime

HeidelTime is a cross-domain temporal tagger that extracts temporal expressions from documents and normalizes them according to the TIMEX3 annotation standard, which is part of the markup language TimeML [15]. This is a rule-based system. Because the source code and the resources (patterns, normalization information, and rules) are strictly separated, it is possible to develop and implement resources for additional languages and areas using HeidelTime rule syntax. HeidelTime is provided with modules for processing documents in several languages (English, French, Italian, Spanish...). Recently, an attempt has been made to extend it to over 200 other languages using existing multilingual resources [20], and more particularly Wiktionary 6 , which provides data for 170 languages. Ukrainian is part of the languages which has been added to the system. Exploitation of this automatically built system produced no results when applied to the Ukrainian data: fully automatic collection of suitable resources is a complicated task. Even if it permits to go faster, it still requires human processing for the validation, disambiguation and enrichment of the resources, as well as the setting of the normalization process.

Hence, adaptation of the HeidelTime resources to Ukrainian is the main step of the current work. The detection and normalization of temporal information by HeidelTime relies on three kinds of resources:

-linguistic patterns, which describe linguistic elements of the temporality (days of the week, months, numbers, etc.). This type of resources is used for the detection of temporality in texts; -normalization resources, which are created to permit the normalization of the detected elements. In this way, all the detected units are normalized. Thanks to these resources, normalization can be performed for absolute (Example (8)) and relative (Example ( 9)) dates, durations and sets. Thus, the normalized values of Examples ( 8) and ( 9) are 2015-05-07 and 2017-05-09, respectively if we consider that these two dates are related; -rules for composing more sophisticated detection of temporality, such as periods, intervals and specific expressions.

Через два днi. (Two days later.)

6 https://www.wiktionary.org/

Evaluation of automatic annotation

A subset of the corpus (22 newspaper articles) is used for the development and tuning of HeidelTime. The rest of the corpus is used for the evaluation. For all the processed document, we used the default parameters, namely no postagger and the type of document are considered as narrative which leads to solve relative dates regarding the previous absolute date. The results generated are evaluated manually with two classical evaluation measures [17]: true positives T P : number of correctly extracted or normalized temporal expressions; precision P: percentage of the relevant temporal expressions extracted and normalized divided by the total number of the temporal expressions extracted and normalized.

6 Results and Discussion 1, we present the evaluation results obtained on the two processed corpora. The results are indicated in terms of true positives T P and precision P. We also indicate the total number of temporal expressions occurring in each corpus (total).

The system was adapted to Ukrainian on a subset of newspaper articles. We can see that the results obtained on the two subsets of newspaper corpus are comparable and close to 90% precision. This is a very good performance for the first version of the system. Besides, when working with newspaper articles, both detection and normalization show good results. Transposition of the system on another genre, encyclopedia articles, permits to test the same system on different data. As we can see, the results are slightly lower, especially for the normalization. For the detection, we keep the precision values high (86%), while the normalization process is more complicated: it shows 78% precision. By comparison with similar work in other languages [8], we obtain higher results in French (0.90-0.95 precision) and lower in English (0.80-0.85 precision). We will illustrate below typical cases of success and failure of the system.

Examples below illustrate successful annotation of temporal values and of their normalization. In these examples, the TIMEX3 values are annotated in the XML format, which is the native format of HeidelTime. We present here examples for dates and durations and explain them. Our main interest is to show the results obtained when normalizing relative, ambiguous or imprecise expressions: -<TIMEX3 type="DATE" value="2017-03-01">1 березня 2017</TIMEX3> ... У <TIMEX3 type="DATE" value="2016-09">вереснi</TIMEX3> керiвник Спецiалiзованої антикорупцiйної прокуратури Назар Холодницький заявив, що Чаус перебуває в анексованому Росiєю Криму... <TIMEX3 type="DATE" value="2016-11-11">11 листопада</TIMEX3> Iнтерпол оголосив Чауса в мiжнародний розшук. In this example, the starting date 2017-03-01 is first recorded by the system. Then, the next two dates (2016-09 and 2016-11-11) are positioned during the previous year, which is correct. This example illustrates the possibility of the system to disambiguate the chronology of events even if the years of the events are not indicated precisely. -S&P прогнозує зростання ВВП України на 1,9% <TIMEX3 type="DATE" value="2017">цього року</TIMEX3>. Similarly to previous example, the system records that the current year is 2017 and can disambiguate expression цього року (this year) through its correct normalization.

-<TIMEX3 type="DATE" value="XXXX-XX-XX">Середа </TIMEX3>, <TIMEX3 type="DATE" value="2017-03-08">8 березня 2017</TIMEX3>. ...Про це повiдомив генпрокурор Юрiй Луценко у <TIMEX3 type="DATE" value="2017-03-08">середу</TIMEX3> в Facebook. In this example, the system can normalize the date by referring it to the day of week Середа (Wednesday).

-S&P: за <TIMEX3 type="DURATION" value="P3Y">три роки </TIMEX3> Україна повинна вiддати 20 мiльярдiв доларiв боргiв. This sentence provides an example of the detection and normalization of durations. -запобiжний захiд у виглядi арешту на <TIMEX3 type="DURATION" value="P60D">60 дiб</TIMEX3>. This is another example of the detection and normalization of durations.

We have also found several cases in which the system is not successful, such as those presented below:

-радник президента США з нацiональної безпеки Майкл Флiнн подав у вiдставку <TIMEX3 type="TIME" value="2017-02-24TEV">ввечерi </TIMEX3> <TIMEX3 type="DATE" value="2017-02-13">13 лютого </TIMEX3>. This is a typical example of temporal expressions in which the normalization process may fail. Here, the right normalization value is 2017-02-13. Yet, for the normalization of the first part of the expression ввечерi (in the evening), the systems exploits the last date recorded, which is 2017-02-24. Such temporal expressions are very frequent in the encyclopedia corpus since several military actions are happening in the evening. For mending this kind of errors, more sophisticated patterns and rules will be created. -У <TIMEX3 type="TIME" value="1776-12-07TNI">нiч</TIMEX3> з 25 на <TIMEX3 type="DATE" value="1776-12-26">26 грудня</TIMEX3> крадькома перетнув Делавер i розбив британський загiн у битвi бiля Трентона, захопивши майже 1000 здивованих i неукрiплених гессенських найманцiв. This example is similar to the previous example, but the situation is more complicated because we have in addition the interval of dates з 25 на 26 грудня (from December 25th to 26th). As we can see, currently we did not fit the system to the detection of intervals and usually only the last date is detected. This is the main source of current errors as the intervals are very frequent in presentation of historical and political information. They frequently occur in both corpora processed. Description and encoding of intervals will be added to the system in future. -Уряд планує додати до пенсiй вiд 200 до <TIMEX3 type="DATE" value= "1000">1000</TIMEX3> гривень. Here, the system detects wrong information because of the preposition до (up to), which can also have temporal meaning, followed by numbers 1000, yet not related to the temporality. This kind of errors can be reduced with the creation of specific exception patterns and rules. -У спробах полегшити тиск з <TIMEX3 type="TIME" value="1967-06-12T24:00">пiвночi</TIMEX3>, <TIMEX3 type="DATE" value="1967-08-09">9 серпня</TIMEX3> мобiльна бригада армiї Бiафри у складi 3000 осiб за пiдтримки артилерiї та бронемашин переправилася на захiдний берег Нiгера. In this example, we can see that the word пiвночi is ambiguous as it can mean midnight and north. This temporality marker causes several errors in the processed corpora. Its disambiguation will need additional analysis of texts, as pre-processing or post-processing step.

These are certainly the main detection and normalization errors which we can find in the corpora processed. Another difficulty which we currently face is related to very specific temporal expressions. The system does not take them into account, which causes several silences (false negatives) in the output. They will be described and encoded in our future work. Here are some example:

-Relative temporal expressions like в той же день, цього ж дня (the same day, that day); -Specific temporal expressions like За минулий з тих пiр мiсяць (during the month that passed since);

-Specific forms for expressing the temporality (combination of numbers and characters) like 21-го столiття, на початку 1990-х рокiв (XXI century, at the beginning of 1990s);

-Specific calendars like the one introduced during the French revolution. На вимогу Робесп'єра 14 фрiмера другого року (<TIMEX3 type="DATE" value="1793-12-04">4 грудня 1793</TIMEX3>) був органiзований уряд iз винятковими повноваженнями. In this example, the system correctly detects and normalizes 4 грудня 1793, but is not sensible to 14 фрiмера другого року (14 of Frimaire of the second year). For such cases, additional resources must be created, so that their detection, conversion and normalization become possible.

We presented our work on creation of automatic system for the detection and normalization of temporal information in Ukrainian. In information extraction applications, temporal information is indeed important. For performing this task, we proposed to use an existing tool HeidelTime. This system has undergone automatic adaptation of its resources to Ukrainian but this showed to be not efficient: the system produced no results with these resources. Hence, the main purpose of our work was to create the suitable resources.

We used two corpora from two genres (newspaper and encyclopedia). 22 newspaper articles were used to develop the system. The rest of our data was used for the tests. Encyclopedia articles correspond to the WikiWars project. The Ukrainian WikiWars corpus has been compiled for our study. The evaluation of the system shows that up to 90% of temporal units in newspaper articles are detected and normalized correctly. In encyclopedia articles, the detection shows 86% precision and the normalization 78% precision. We also present and discuss some examples of the current failures of the system.

In future, the system will be further developed and fitted to Ukrainian, so that it detects and normalizes other temporal expressions. It will be made freely available to the research community. Besides, the two used corpora will be fully annotated with temporal expressions and also made freely available to the research community.

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Table 1 .

 1 reision @PA for the detetion nd normliztion of temporl expressions otined on development nd test sets in the two proessed orpor @newspper nd enylopediAF T P is the numer of dte orretly identi(ed or normlizedF

	Development Detection Normalization total TP P TP P	Test Detection Normalization total TP P TP P
	Newspaper TSS SWV WIFQH SUI VUFIV Encyclopedia E E E E E	UHQ TQS WHFQQ TPP VVFRV PDPPT IDWIV VTFIT IDURS UVFQW
	In Table

http://www.timeml.org

http://timexportal.wikidot.com/

https://www.pravda.com.ua/news/

https://uk.wikipedia.org

http://timexportal.wikidot.com/wikiwars