Olivier Desmaison
email: olivier.desmaison@mines-paristech.fr

Gildas Guillemot

Michel Bellet

Modelling of the bead formation during multi pass hybrid laser/gas metal arc welding

A three dimensional finite element model has been developed to simulate weld bead formation in multi pass hybrid laser/gas metal arc welding. The model considers both a gas metal arc welding (GMAW) electrode and a laser beam moving along a workpiece. A Eulerian approach is used in which the interface between the metal and the surrounding gas or plasma is defined by a level set function. Therefore heat transfer boundary conditions are applied through a "Continuum Surface Force" model. An original method has been settled to model material supply. A volume expansion source term is added to the right hand side of the mass conservation equation for certain liquid finite elements in the fusion zone. A compressible Navier Stokes equation is then solved. The new obtained velocity field is used to solve the transport of the level set function for the updating of the gas/metal interface. A new topological reinitialization method has been developed to hold the Eikonal property in the compressible framework of the level set transport. The efficiency of this global model enables to simulate industrial multi pass hybrid welding processes.

Introduction

The large use of welding processes in heavy industries has lead scientists to take an interest in the welding modelling. The understanding of weld pool dynamics and solidification during gas metal arc welding (GMAW) is of great importance to master the shape of fusion and heat affected zones, the shape of weld beads, and the formation of deformations and stresses in the neighbourhood of the fusion zone. Moreover, these stresses may lead to hot tearing for some critical alloys and highlights the need for accurate numerical modelling. Recently the efficiency of GMAW processes has been enhanced through the coupling with a laser beam. Located ahead of the torch, this new heat source provides additional energy to the workpiece. The resulting weld pool spreading enables to supply more material, to mix it more homogeneously and consequently to weld better and faster.

Since the development of 2D-axi stationary models for gas tungsten arc welding (GTAW) processes at the beginning of the 90's [START_REF] Choo | On the calculation of the free surface temperature of gas-tungsten-arc weld pools from first principles: Part i. modelling the welding arc[END_REF][START_REF] Choo | On the calculation of the free surface temperature of gas-tungsten-arc weld pools from first principles: Part ii. modelling the weld pool and comparison with experiments[END_REF][START_REF] Fan | Heat transfer and fluid flow in a partially or fully penetrated weld pool in gas tungsten arc welding[END_REF][START_REF] Fenggui | Modelling and finite element analysis on gtaw arc and weld pool[END_REF] significant progress have been made. Recent models are now able to realistically simulate GMAW processes taking into account the main phenomena inside both the arc plasma and the weld pool [START_REF] Hu | Heat and mass transfer in gas metal arc welding. part i: The arc[END_REF][START_REF] Hu | Heat and mass transfer in gas metal arc welding. part ii: The metal[END_REF][START_REF] Xu | Three-dimensional modelling of arc plasma and metal transfer in gas metal arc welding[END_REF]. The heat and mass transfer models are continuously improved. The simplified Goldak approach and the laminar flows analysis in the weld pool tend to be replaced by more physical coupled models including a Gaussian heat flux and turbulent flows without any arbitrary enhanced viscosity [START_REF] Hong | Modelling turbulent thermofluid flow in stationary gas tungsten arc weld pools[END_REF][START_REF] Jaidi | Three-dimensional turbulent weld pool convection in gas metal arc welding process[END_REF][START_REF] Chakraborty | Modelling of turbulent molten pool convection in laser welding of a copper-nickel dissimilar couple[END_REF][START_REF] Ribic | Numerical simulation of heat transfer and fluid flow in gta/laser hybrid welding[END_REF]. Regarding free surface deformation, surface tension and arc pressure are also taken into account in the minimization of the surface energy [START_REF] Kumar | Heat transfer and fluid flow during gas-metal-arc fillet welding for various joint configurations and welding positions[END_REF][START_REF] Xu | Numerical analysis of weld pool geometry in globular-transfer gas metal arc welding[END_REF][START_REF] Kim | Modelling of temperature field and solidified surface profile during gas-metal arc fillet welding[END_REF]. Because of the progress in hybrid welding modelling and the research focus on keyhole phenomenon these numerical methods have been carried over laser/GMAW modelling [START_REF] Ki | Modelling of laser keyhole welding: Part i. mathematical modelling, numerical methodology, role of recoil pressure, multiple reflections, and free surface evolution[END_REF][START_REF] Ki | Modelling of laser keyhole welding: Part ii. simulation of keyhole evolution, velocity, temperature profile, and experimental verification[END_REF][START_REF] Zhao | Macroporosity free aluminum alloy weldments through numerical simulation of keyhole mode laser welding[END_REF][START_REF] Rai | A convective heat-transfer model for partial and full penetration keyhole mode laser welding of a structural steel[END_REF][START_REF] Zhang | Thermomechanical analysis for laser + gmaw-p hybrid welding process[END_REF]. As a result, the numerically obtained bead shapes better fit the experimental ones [START_REF] Traidia | Numerical and experimental study of arc and weld pool behaviour for pulsed current gta welding[END_REF]. Nevertheless, only a very few models consider the entirety of the previous phenomena. Kumar and DebRoy [START_REF] Kumar | Heat transfer and fluid flow during gas-metal-arc fillet welding for various joint configurations and welding positions[END_REF], for instance, propose an advanced GMAW model in 3D, but limited to stationary and laminar conditions. Recently Wang et al. [START_REF] Wang | Numerical simulation of transient temperature field during laser keyhole welding of 304 stainless steel sheet[END_REF] have developed a 3D transient model for laser keyhole welding through the volume of fluid (VOF) approach to study the temperature field evolution. Results are promising but no material supply is taken into account.

The present paper reports on a transient three dimensional finite element model that has been developed to simulate the weld bead forming during multi pass hybrid laser/GMAW. The model considers both a consumable electrode and a laser beam moving along a workpiece (Figure 1-a). However the electrode, the arc plasma, the droplets and the laser beam are not explicitly represented. The model includes the workpiece and a surrounding gaseous domain with which different boundary conditions are expressed. The plasma over the fusion zone, and the mix of air and protection gas surrounding the formed joint are parts of this domain. In the present study, the laser is supposed defocused and therefore does not give birth to the keyhole phenomenon.

The paper will present first the mathematical formulation of the finite element problem. The three main governing conservation equations -mass, energy and momentum equations -will be detailed. Secondly a multi pass hybrid laser/GMAW operation will be described. Lastly, the numerical results will be analyzed and the performance and efficiency of the model will be discussed.

Mathematical formulation 2.1. Level Set approach

A Eulerian approach is used in which the interface between the metal and the surrounding air or plasma is defined by a level set function (Figure 1-b/c). Contrary to a Lagrangian approach, the chosen one enables to overcome the numerical difficulties induced by the onset of material/material contacts in multi pass welding. A nodal signed distance function φ is defined to represent the interface between gas and metal. To ensure the finite element resolution of the governing equations the material properties are averaged near the interface, using a mixed law defined as a sinusoidal Heaviside function H(φ).

( ) ( ) [ ] ε πϕ π ε ϕ ϕ sin 1 1 2 1 + + = H for [ ] ε ε ϕ + - ∈ , (1)
The mixed law is restricted to a [-ε, +ε] domain perpendicular to the surface defined by ϕ = 0. Being α gas and α metal the values of a physical variable in both domains, the averaged or mixed value α is equal, in the whole model, to:

( ) ( ) gas metal H H α ϕ ϕ α α + - = ] 1 [ (2) All model's physical parameters such as η λ ρ , , , p C
are computed according to this equation, using the same Heaviside function.

Heat transfer and energy equation

The non steady-state energy equation is:

( ) Q T T v t T C p & r r r r = ∇ ⋅ ∇ -       ∇ ⋅ + ∂ ∂ λ ρ (3)
ρ, C p and λ are respectively the density, the equivalent specific heat and the thermal conductivity. The energy equation is solved for the temperature field. Therefore, in order to take into account the latent heat associated with fusion and solidification, the equivalent specific heat is computed following Morgan's model:

( ) ( ) t t t t t t p T T H H C ∆ - ∆ - - - = (4)
where the enthalpy curve H(T) is an input of the model. In the present model, the flows in the weld pool are not taken into account and the velocity field v r is supposed null. To offset the flows and natural convection lack, the material mixing is modelled by an enhanced conductivity in the weld pool. Q corresponds to a heat source term. The time discretization of the energy equation is fully implicit. Once the new temperature obtained, the thermo-mechanical properties are updated.

The arc plasma supplies a surface heat flux on the interface which can be modelled by a Gaussian distribution Q P . However, the level set framework makes impossible to impose this boundary condition on the interface as a Neumann condition because the interface is not explicitly discretized. So the Continuum Surface Force method, developed by Brackbill [START_REF] Brackbill | A continuum method for modelling surface tension[END_REF], has to be used to transform the surface boundary condition into a volumetric source term, as in [START_REF] Ki | Modelling of laser keyhole welding: Part i. mathematical modelling, numerical methodology, role of recoil pressure, multiple reflections, and free surface evolution[END_REF][START_REF] Ki | Modelling of laser keyhole welding: Part ii. simulation of keyhole evolution, velocity, temperature profile, and experimental verification[END_REF]. This method is based on the following equality:

( ) ∫ ∫ Ω = Γ V P S P d Q d Q ϕ δ & & (5)
with S and V respectively the surface of the interface and the volume of the model. The Dirac function δ(ϕ) is the derivative of the Heaviside function. In order to respect the equality of Eq. ( 5) Dirac function's integral in the [-ε, +ε] domain and in the normal direction of the interface has to be equal to one.

The same method is used for the heat source induced by the defocused laser beam. A second Gaussian distribution Q L is applied on the interface and multiplied by the Dirac function. Heat convection and radiation effects are also taken into account through a third volume distribution Q T .

Finally, an additional volumetric source term is defined to model the impingement of molten droplets of filler metal in the pool. According to the initial model of Lancaster [START_REF] Lancaster | The physics of welding[END_REF], modified in [START_REF] Kumar | Three-dimensional finite element modelling of gas metal-arc welding[END_REF], all the nodes included in a cylindrical cavity defined just below the torch beam get an extra heat source term Q D as in [START_REF] Xu | Three-dimensional modelling of arc plasma and metal transfer in gas metal arc welding[END_REF][START_REF] Jaidi | Three-dimensional turbulent weld pool convection in gas metal arc welding process[END_REF][START_REF] Kumar | Heat transfer and fluid flow during gas-metal-arc fillet welding for various joint configurations and welding positions[END_REF][START_REF] Kim | Modelling of temperature field and solidified surface profile during gas-metal arc fillet welding[END_REF]. The value of Q D mainly depends on the temperature of the droplets and the current intensity. Thus, the total volumetric heat source term Q of Eq. ( 3) is equal to:

( ) ( ) D T L P Q Q Q Q Q + - + = ϕ δ (6)

Material supply and deformation surface forces

In the mechanical problem, an original method has been settled to model material supply [START_REF] Bellet | Direct modelling of material deposit and identification of energy transfer in gas metal arc welding[END_REF]. A volume expansion source term θ & is added to the right hand side of the mass conservation equation for elements contained in the cylindrical cavity described in §2.2. It is computed as a function of the geometrical dimensions of the electrode and the welding parameters. This approach to model the bead shaping is not sufficient to be realistic and the volumetric forces which deform the free surface have to be taken into account. In the present model, only two phenomena are considered: gravity F g and surface tension F γ . The arc pressure is neglected. Gravity is applied on the whole model as a volumetric force but to consider the surface tension force F γ , once again, the Continuum Surface Force method must be used. The corresponding surface tension volumetric force, F γ-vol , is given as a function of the mean curvature κ and the surface tension term γ, using the Dirac function:

( ) ϕ δ γ κ γ n F vol r r = - (7)
An asset of the level framework is to easily compute the normal vector and the curvature, as follows:

ϕ ϕ ∇ ∇ = r r n and n r ⋅ -∇ = κ (8)

Fluid problem resolution and level set transport

Once the energy equation solved and the thermo physical properties of the liquid/solid phases updated, the fluid problem resolution is carried over. In addition to both bead shaping forces which appear in the right hand side of the Navier Stokes equation, the expansion term is added to the right hand side of the mass conservation equation to model the bead forming. Thus, the full fluid problem to be solved is the following: [START_REF] Jaidi | Three-dimensional turbulent weld pool convection in gas metal arc welding process[END_REF] In the context of the present study, and as a first approach, a Newtonian behaviour is assumed for both the gas and the metal. The compressible Navier Stokes equations are solved using P1+/P1 stabilized elements, with a coupled SUPG and a variational multiscale method [START_REF] Hachem | Stabilized finite element solution to handle complex heat and fluid flows in industrial furnaces using the immersed volume method[END_REF]. Then in the level set approach, the signed distance function must be convected to take into account the interface deformation and, by this way, to model the bead shaping. The velocity field v r , obtained from the Navier Stokes equations resolution, is used to solve the convection equation of the level set [START_REF] Ville | Convected level set method for the numerical simulation of fluid buckling[END_REF]: [START_REF] Chakraborty | Modelling of turbulent molten pool convection in laser welding of a copper-nickel dissimilar couple[END_REF] Thus, the new zero value level set represents the new metal/gas interface and takes into account the material supply. In total, this method is comparable to the resolution of the deformation surface energy minimization problem as presented in [START_REF] Kumar | Heat transfer and fluid flow during gas-metal-arc fillet welding for various joint configurations and welding positions[END_REF][START_REF] Xu | Numerical analysis of weld pool geometry in globular-transfer gas metal arc welding[END_REF][START_REF] Kim | Modelling of temperature field and solidified surface profile during gas-metal arc fillet welding[END_REF][START_REF] Ki | Modelling of laser keyhole welding: Part i. mathematical modelling, numerical methodology, role of recoil pressure, multiple reflections, and free surface evolution[END_REF][START_REF] Ki | Modelling of laser keyhole welding: Part ii. simulation of keyhole evolution, velocity, temperature profile, and experimental verification[END_REF][START_REF] Traidia | Numerical and experimental study of arc and weld pool behaviour for pulsed current gta welding[END_REF].

( ) θ η σ σ ρ γ & r r r r r r r r r r r r r = ⋅ ∇ ∇ + ∇ = + = ⋅ ∇ -       ∇ ⋅ + ∂ ∂ - v v v with F F v v t v T g vol
0 = ∇ ⋅ + ∂ ∂ ϕ ϕ r r v t

Reinitialization of the level set

The level set transport is always coupled with a reinitialization method which holds the respect of the Eikonal property: after the level set advection, the norm of the distance function gradient has to be maintained equal to one,∇φ=1. For that, a modified convection equation is solved. Usually it is coupled to the transport equation for more efficiency and less computational time resolution [START_REF] Ville | Convected level set method for the numerical simulation of fluid buckling[END_REF]. Nevertheless, in the compressible approach of the material supply method described above, the common level set reinitialization does not provide acceptable results. A new topological method has been settled. Based on the works of Elias et al. [START_REF] Elias | Simple finite element-based computation of distance functions in unstructured grids[END_REF], it consists in propagating the level set in the whole mesh from the zero isovalue level set. The method takes the advantage of being only geometrical and relies on the mesh topology. An enhancement concerning the sorting of the "to solve elements list" of the initial method has been proposed to allow the level set propagation through the heterogeneous mesh of welding simulations. The finite element model consists of a parallelepiped 90 mm high, 120 mm long and 90 mm wide. A chamfer of 50 mm depth and 12 mm wide passes, lengthwise, through the workpiece, Figure 2. The interface between the chamfer inside and the gas is defined by the level set function. A hybrid heat source moves along the workpiece to form the weld bead. It is composed of a GMAW electrode and a 3 mm radius laser beam located 3 mm ahead. To fill one layer two passes are needed. Three passes are modelled. The power of the laser beam is 6000 W and the torch power is 10000 W, assumed to be divided in two parts: 70% for the arc plasma source and 30% for melting the electrode metal into droplets. The velocity of the torch is 13.3 mm/s, the velocity and the diameter of the wire are respectively v w = 200 mm/s and φ w = 1.2 mm. The material is a low C mid-alloyed steel 16MnNiMo or ASME SA533. Material properties, found in the literature, [START_REF] Coret | Experimental study of the phase transformation plasticity of 16mnd5 low carbon steel induced by proportional and nonproportional biaxial loading paths[END_REF][START_REF] Coret | Experimental study of the phase transformation plasticity of 16mnd5 low carbon steel under multiaxial loading[END_REF][START_REF] Vincent | Thermo-mechanical consequences of phase transformations in the heat-affected zone using a cyclic uniaxial test[END_REF], are given in Table 1. According to literature models where the flows are not taken into account, a thermal conductivity factor L f λ is used to model the material mixing into the weld pool.

Model description

Solid and liquid thermal conductivity

L S λ λ L f K m W λ × - - 5 . 32 . . 37 1 1

Latent heat of fusion

Results and discussion

The weld bead forming during the first pass is presented in Figure 3. The view is shot from a longitudinal cut plane inside the chamfer and only the metal/gas interface is visible (such as shown in Figure 2). For three different steps, the temperature field is displayed and the weld pool is drawn by a white line. Step by step the bead is forming. The temperature field follows correctly the hybrid torch and the weld pool stays ahead the bead. The computed temperatures are in the range of experimental welding ones. Adaptive meshing tools are used to catch the interface close to the bead: refined mesh areas are computed around the bead zero isovalue level set and in the weld pool. That is why the interface which models the chamfer bottom is not smooth far from the torch. As the torch goes along, the mesh is refined ahead. The progressive filling of the chamfer is shown in Figure 4. The use of the level set function allows easily managing the overlapping of the beads resulting from each pass. The three orthogonal bead profiles are superimposed in Figure 5. The shape of the beads qualitatively corresponds to the ones experimentally observed. The computed material supply height after two passes is around 4 mm matching the measured height. A full campaign of instrumented welding tests is planned in a near future and will permit a more quantitative evaluation of the accuracy of the model predictions.

The first pass has been simulated without the laser beam in order to analyze the assets of a hybrid welding. The comparison between the two situations is presented in Figure 6. The white contour lines represent the two heat sources. As expected, the additional heat flux source from the laser beam causes the spreading of the weld pool in both longitudinal and orthogonal directions. As a consequence, the hybrid welding bead is larger but also forms more ahead of the torch in the hybrid case. This can be explained by the fact that although the cavity that is supposed to be the impingement zone for droplets is located in the same position in both cases, the supplied material spreads throughout the weld pool. It will be interesting, in a near future, to compare such prediction with weld pool surface profiles recorded through high speed cameras. It might be expected that such a comparison would reveal the need to take into account the arc pressure effect in the numerical model.

Conclusion

A new transient three dimensional finite element model has been developed to simulate hybrid laser/GMAW industrial processes. The Eulerian level set approach chosen to overcome the numerical difficulties induced by bimaterial contacts from multi pass welding is coupled to a new material supply modelling method. The addition of a volume expansion term in the mass conservation equation provides, after resolving Navier-Stokes equations, a velocity field which models the bead forming. The use of the Continuum Surface Force Method enables to transform surface boundary conditions into volumetric ones around the interface. To complete the model, a new level set reinitialization method has been implemented: the transport of the distance function is now possible in a compressible fluid area. The simulation of an industrial process from this model has highlighted realistic results regarding the weld bead shapes and the temperature field distribution. Moreover the efficiency of the model is reinforced by the adaptative meshing tools: the weld pool and the interface of the new beads are well represented through refined mesh areas. Moreover, this model turns out helpful to understand the assets of a hybrid welding.

Acknowledgment

This work has been supported by the French Agence Nationale de la Recherche (ANR) in the framework of the project "SISHYFE" (ANR-09-MAPR-0019).

Figure 1 .

 1 Figure 1. Hybrid laser -GMAW model description according to the governing equations: schematic industrial process (a), energy equation model (b), material supply model (c).

Figure 2 .

 2 Hybrid welding model.

Figure 3 .

 3 Figure 3. Weld bead forming and temperature field at: t = 1.5 s (a), t = 3.0 s (b), t = 4.5 s (c).

Figure 4 .

 4 Figure 4. Filling of the chamfer from 3 passes at the same time t = 1.5 s after each pass has started. Temperature field distribution on the metal/gas interface for the first pass (a), the second one (b) and the third one (c).

Figure 5 .

 5 Figure 5. Over lapping of the three weld beads in an orthogonal section plane.

Figure 6 .

 6 Figure 6. Comparison of two beads obtained from (a) a GMAW simulation and (b) a hybrid Laser/GMAW simulation.

Table 1 .

 1 Material properties for 16MnNiMo (ASME AS533)

	Nomenclature Symbol	Value	Nomenclature Symbol	Value
	Solidus and liquidus temperature	T T	L S	1690 1740	K	Solid and liquid specific heat

& & & & &


Olivier Desmaison
email: olivier.desmaison@mines-paristech.fr

Gildas Guillemot
email: gildas.guillemot@mines-paristech.fr

Michel Bellet
email: michel.bellet@mines-paristech.fr

Numerical modelling of hybrid arc/laser welding: a Level Set approach for weld bead formation and residual stresses

The joining of high thickness steel sheets by means of hybrid Laser/GMAW welding processes is studied in this paper. A three dimensional finite element model has been developed to simulate this process. Through an ALE framework, a level set approach is used to model the interface between the metal and the surrounding gas. Even though the physics of the plasma is not modelled, both thermal and material supply phenomena are taken into account: (i) The laser and GMAW heat sources are simulated and applied on the interface through the Continuum Surface Force method, (ii) An original method of volume element expansion has been chosen to simulate the material supply and the bead formation. A thermo mechanical problem resolution has been settled in this model. Depending on the thermal evolution of the mechanical parameters and on the velocity field, the material behaviour will be elastic, elasto-visco-plastic or visco-plastic. The ALE approach enables to compute the stresses inside the workpiece and to obtain the displacements of the workpiece borders. Two finite elements models are presented to illustrate: (i) A hybrid arc/laser welding simulation through the thermal and material supply resolution, (ii) A TIG welding simulation through the stress and strain mechanical resolution.

INTRODUCTION

The range of welding process's use in joining operations has recently expanded thanks to the emergence of hybrid arc / laser processes. Indeed, the coupling of a laser beam with a gas metal arc torch enhances the efficiency of gas metal arc welding (GMAW) processes. Located ahead of the torch, this new heat source provides additional energy to the workpiece. The resulting weld pool spreading enables to supply more material, to mix it more homogeneously and consequently to weld better and faster.

Nevertheless, the complexity of coupled thermo-mechanical phenomena induced by this new process increases. Therefore, a deeper understanding is required regarding the heat transfer from the heat sources and the molten droplets to the weld pool and the heat affected zone. Actually, the neighbourhood of the fusion zone is where deformations and stresses form first. These stresses may lead to hot tearing for some critical alloys and highlights the need for an accurate numerical modelling.

Recent models have shown their ability to realistically simulate GMAW processes taking into account the main thermal and fluid flow phenomena inside both the arc plasma and the weld pool [START_REF] Hu | Heat and transfer in gas metal arc welding. part i: The arc[END_REF][START_REF] Hu | Heat and mass transfer in gas metal arc welding. part ii: The metal[END_REF][START_REF] Xu | Three-dimensional modelling of arc plasma and metal transfer in gas metal arc welding[END_REF]. Such heat and fluid flow approaches have been recently applied to hybrid arc/laser melting, especially with the objective of simulating the effect of laser, through direct or indirect modelling of keyhole phenomena [START_REF] Ki | Modelling of laser keyhole welding: Part i. mathematical modelling, numerical methodology, role of recoil pressure, multiple reflections, and free surface evolution[END_REF][START_REF] Ki | Modelling of laser keyhole welding: Part ii. simulation of keyhole evolution, velocity, temperature profile, and experimental verification[END_REF][START_REF] Zhao | Macroporosity free aluminum alloy weldments through numerical simulation of keyhole mode laser welding[END_REF][START_REF] Rai | A convective heat-transfer model for partial and full penetration keyhole mode laser welding of a structural steel[END_REF][START_REF] Zhang | Thermomechanical analysis for laser + gmaw-p hybrid welding process[END_REF]. As a result, the calculated shapes of weld beads better fit the experimental ones.

Research in hybrid welding process simulations has not been yet extended to stress/strain computation. The analysis of stress/strain formation during and after welding is generally achieved through uncoupled thermalmechanical calculations to, further, study hot cracking phenomena induced by the process [START_REF] Danis | Numerical modeling of inconel 738lc deposition welding: Prediction of residual stress induced cracking[END_REF][START_REF] Wei | Three-dimensional numerical simulation of weld solidification cracking[END_REF] or residual stresses in the final workpiece [START_REF] Preston | Physically-based constitutive modelling of residual stress development in welding of aluminium alloy 2024[END_REF][START_REF] Taljat | Numerical analysis of gta welding process with emphasis on post-solidification phase transformation effects on residual stresses[END_REF][START_REF] Kong | Modeling of heat transfer and fluid flow in the laser multilayered cladding process[END_REF]. In fact, while the thermal field strongly impacts the residual stress field, the stress field affects with a low influence the thermal field.

The present paper reports on a transient three dimensional finite element model that has been developed to simulate the weld bead forming during hybrid laser/GMAW process. The model considers both a consumable electrode and a laser beam moving along a workpiece. However, the electrode, the arc plasma, the droplets and the laser beam are not explicitly represented. The model includes the workpiece and a surrounding gaseous domain for which different boundary conditions are expressed. During the formation of the weld bead, firstly, the energy equation is solved. The material supply is taken into account thanks to the level set approach, the interface between the surrounding gas/air and the workpiece being defined by a signed distance function. The Proc. JOM17, Int. Conf. on Joining Materials, May 5-8, 2013, Helsingor, Denmark, International Institute of Welding (2013) 11 pages resolution of Navier-Stokes equations, including first a mass source term to account for material supply, and second surface tension forces, gives access to the velocity field, which is used to transport and update the level set interface. The uncoupled resolution of the thermal-mechanical problem begins then. From the temperature distribution, the mechanical properties of the 316L steel are derived. According to the material state, elastovisco-plastic, visco-plastic or Newtonian, the stresses and strains in the heat affected zone and in the base metal are computed.

The paper presents first the mathematical formulation for the heat & material supply. A simulation of a hybrid arc/laser welding process is carried out to analyse the model efficiency. A comparison between the weld bead cut planes from simulation and experimentation is proposed. Next, the mechanical solver is detailed. A simulation of a TIG welding process is presented. To the best of our knowledge, it is the first time that Level Set approach and stress & strain computation are coupled. To validate the presented model a comparison between both approaches with and without LS is carried out.

MATHEMATICAL FORMULATION FOR A GMAW SIMULATION

Level Set approach

A Eulerian approach is used in which the interface between the metal and the surrounding air or plasma is defined by a level set function. This choice is justified by the modelling of multi-pass hybrid arc/laser welding processes [START_REF] Desmaison | Modelling of the bead formation during multi pass hybrid laser/gas metal arc welding[END_REF] and the objective to model arc plasma in a future extended version of the model. A nodal signed distance function φ is defined to represent the interface between gas and metal. Because the governing mass, energy and momentum equations are solved in the entire finite element domain, as illustrated in FIGURE 1, a smooth interface transition of the thermal properties is needed. Near the interface a mixed property law is defined using a sinusoidal Heaviside function H(φ) :

( ) ( ) [ ] ε πϕ π ε ϕ ϕ sin 1 1 2 1 + + = H for [ ] ε ε ϕ + - ∈ , (1)
The mixed law is restricted to a [-ε,+ε] domain perpendicular to the surface defined by φ(t)=0. Denoting α gas and α metal the values of a physical variable in both domains, the averaged or mixed value α is defined by:

( ) ( ) gas metal H H α ϕ ϕ α α + - = ] 1 [ (2)
All model's physical parameters such as the density ρ, the specific heat C p , the thermal conductivity λ or the viscosity η, are computed according to this mixing rule, using the same Heaviside function.

Heat transfer: energy equation

The transient energy equation solved to obtain the temperature T in the whole domain is the following:

( ) Q T T v t T C p & r r r r = ∇ ⋅ ∇ -       ∇ ⋅ + ∂ ∂ λ ρ (3)
where ρ, C p and λ are the thermal properties computed from Eq. ( 2). All of them are temperature dependant and their value varies with the liquid fraction. v r is the velocity vector. In this model the temperature homogeneisation resulting from the fluid flow in the weld pool is only taken into account through an enhanced liquid thermal conductivity. Nevertheless the velocity field v r is not equal to zero (see §2.4). The study of flows in the weld pool from the model described in this paper is detailed in [START_REF] Desmaison | A multi-physics level set approach for the simulation of the hybrid laser / gmaw process[END_REF]. Two heat sources have to be considered in a hybrid laser/arc welding simulation. The first one consists in the heat supplied by the arc plasma for which a surface Gaussian distribution of the heat input is supposed. The model relies on thermal radiation physics: the input heat flux q P through the surface depends on the opening angle of the arc plasma, the distance to the electrode and the orientation of the surface normal vector with respect to the direction of the torch. The second heat source is associated with the laser beam. In the present model, a defocused laser beam of moderate power is supposed for which no keyhole phenomenon is observed.

The heat input is then modelled through a surface heat flux q L with a Gaussian distribution and an area limited to the laser radius. Both sources -plasma and laser -are expressed through surface boundary conditions. In the present context of level set formulation, the Continuum Surface Force developed by Brackbill et al. [START_REF] Brackbill | A continuum method for modelling surface tension[END_REF] has to be used to transform them into volumetric heat input conditions, appearing as right hand side terms Q & in Eq. ( 3). Therefore the Dirac function δ(φ) -that is the derivative of the Heaviside function H(φ) (Eq. ( 1), the integral of which over [-ε,+ε] is 1) -is used to define the two volumetric heat sources:

) (ϕ δ P P q Q = & and ) (ϕ δ L L q Q = & .
The heat provided to the fusion zone by the impingement of molten droplets represents the last heat source

D Q & .
Based on the volumetric model of Lancaster [START_REF] Lancaster | The physics of welding[END_REF] and improved later by Kumar and Bhaduri [START_REF] Kumar | Three-dimensional finite element modelling of gas metal-arc welding[END_REF], it consists in defining a cylinder below the arc/plasma in the weld pool into where an additional source term is imposed. The appropriate dimensions of the cylinder are discussed in [START_REF] Bellet | Direct modelling of material deposit and identification of energy transfer in gas metal arc welding[END_REF]. Finally three heat source terms are added to the right hand side of the energy equation ( 3):

( ) ( ) D L P Q q q Q & & + + = ϕ δ (4)
Convective heat transfer and emissivity are taken into account too. They are imposed as a Fourier condition along the borders of the finite element model. On the interface, they are computed as:

( ) ) ( ) ( ) ( 4 4 ϕ δ εσ ext ext T T T T T h Q - + - = & ( 5
)
where h is the heat transfer coefficient, ε the emissivity, σ the Boltzmann constant and T ext the ambient temperature. The heat losses T Q & are subtracted to the final heat source term of Eq. ( 4). FIGURE 1 represents the different heat sources applied on the model.

Material supply: mass & momentum equation

An original method has been settled to model material supply [START_REF] Bellet | Direct modelling of material deposit and identification of energy transfer in gas metal arc welding[END_REF]. A volume expansion source term θ & is added to the right hand side of the mass conservation equation for finite elements contained in the cylindrical zone introduced in the previous paragraph. θ & is computed as a function of the geometrical dimensions of the electrode and the welding parameters. The coupled mass and momentum equations are the compressible Navier-Stokes ones:

     = ⋅ ∇ = ∇ + ⋅ ∇ -         ∇ ⋅ + ∂ ∂ θ ρ θ θ θ θ θ θ & r r r r r r r r r v F p s v v t v (6)
The computed velocity field θ & r v will be used then to transport the interface as explained next. A Newtonian behaviour is considered for the metal such as:

θ θ η & & & e s 2 = with ( ) ( ) I v v tr v v e t t       ∇ + ∇ - ∇ r r r r r r r r & 2 1 2 1 ( 7
)
where η is the viscosity and θ & & e the strain rate tensor computed from the velocity θ & r v . As previously, the viscosity is computed from Eq. ( 2) and it is temperature dependant. An enhanced value of the liquid viscosity is used for a stronger stabilization of the mechanical resolution. The right hand side term of the momentum equation, F r , includes two forces which apply onto the free surface. The first one is the gravity, g F r . The second one is the surface tension. First a surface force vector is defined as a function of the distance function:

n T r r γ κ γ = (8)
where the mean curvature κ and the normal vector n r are computed as:

ϕ ϕ ∇ ∇ = r r r n and n r ⋅ -∇ = κ (9)
As this force is a surface force, once again, the Continuum Surface Force has to be applied. The final term F r is equal to:

( ) ϕ δ γ T F F g r r r + = (10)
The arc pressure is neglected in this model.

Weld bead shaping: transport & reinitialization of the Level Set

Once the velocity field associated with material supply is computed, the interface has to be transported. In this way, the following convection equation is solved:

0 = ∇ ⋅ + ∂ ∂ ϕ ϕ θ r r & v t ( 11
)
The new Level Set φ=0 is the updated interface. A reinitialization step is then realized to respect the Eikonal property on the field φ:

1 = ∇ϕ r
. The modelling of the material supply step ( §2.3 & §2.4 ) is illustrated in FIGURE 2. Moreover, because of the Eulerian approach used in this resolution, the temperature field must to be transported too. Therefore, the velocity field v r of the energy equation ( 3) is equal to θ & r v .

FIGURE 2:

Illustration of the different steps of the supply material modelling.

HYBRID LASER/GMAW PROCESS SIMULATION

Model description

The simulation of a hybrid laser/GMAW welding is proposed to illustrate the previous mathematical simulation. The simulated process consists in the filling of a narrow chamfer (FIGURE 3). The metal is the low alloyed 18MND5 steel. Its properties, thermal dependant, are taken from the commercial software ThermoCalc. The interface between the machinated chamfer and the gas is defined by the level set function. A hybrid heat source moves along the workpiece to form the weld bead. It is composed of a GMAW electrode and a 3 mm radius laser beam located 3 mm ahead. The power of the laser beam is 6000 W and the torch power is around 10000 W, assumed to be divided in two parts: 70% for the arc plasma source and 30% for melting the electrode metal into droplets [START_REF] Bellet | Direct modelling of material deposit and identification of energy transfer in gas metal arc welding[END_REF]. The velocity of the torch is 13.3 mm/s, the velocity and the diameter of the wire are respectively v w = 200 mm/s and Ф w = 1.2 mm. A value of 20 [START_REF] Hamide | Modélisation numérique du soudage à l'arc des aciers[END_REF] is used for the enhanced conductivity factor to model the weld pool mixture. Only one pass is simulated and the results are presented in the next section. model, the flows inside the weld pool are not taken into account. So, the mixture of the material does not occur which explains the curved shape of the weld pool.

Nevertheless the similarity between the simulation and the experimentation keeps very good. It proves the efficiency of the present model. A fluid flows model could be improve the results.

RESIDUAL STRESSES: BEHAVIOUR LAW & MECHANICAL EQUATION

Like all models found in the literature, this model is uncoupled. The thermal distribution field and the weld bead shaping computations are separated from the mechanical resolution. Indeed the residual stress computation is carried out from the final model state after energy, mass and momentum equations resolution at each time step.

The mechanical problem to be solved is the following:

( ) ( )

     + = ⋅ ∇ = ∇ + ⋅ ∇ -       ∇ ⋅ + ∂ ∂ th el S S S S S S tr tr v F p s v v t v ε ε ρ & & r r r r r r r r (12)
The solutions are the velocity of the material S v r and the pressure p. The right hand side term of the equation, F r , corresponds to the body forces. Here, only the gravity force is considered. The term el ε& represents the elastic part of the strain rate tensor and the term th ε& the thermal one. The deviatoric stress tensor S s in Eq. ( 12) directly derives from the velocity field S v r through the material behaviour law. As this is done at the centre of tetrahedral elements, no mixing law is required here. Indeed, according to the state of the element (gas or metal) and its centre temperature, the kind of resolution will not be the same. Three different behaviours are defined in this model: -The gas or surrounding air is considered as an incompressible fluid. Its behaviour is Newtonian (N). The deviatoric stress tensor S s is defined as:

e s gas S & η 2 = ( 13
)
where η gas is the dynamic viscosity of the gas. Its value is very low and, as expected, stresses in the gas part of the model are negligible. The strain rate tensor e & is computed as in Eq. ( 7) from the velocity

S v r .
-If the metal has a temperature lower than a critical temperature T C (usually chosen as the solidus temperature) its behaviour is elastic-visco-plastic (EVP). A multiplicative behaviour law is used in the present model. The equivalent stress is computed as:

Y n m EVP EVP K σ ε ε σ + = & (14)
where K EVP , m EVP and n are respectively the EVP consistency, the EVP strain-rate sensitivity and the hardening coefficient. σ y is the yield stress. These four material parameters are temperature dependant. The equivalent strain ε is an input from the previous time step whereas the equivalent strain rate ε & depends on the velocity

S v r .
-If the metal has a temperature higher than the critical temperature T C , its behaviour is visco-plastic (VP). The Norton-Hoff law is applied to express the deviatoric stress tensor:

( ) ( ) e K s VP m P VP S & & 1 5 . 0 3 2 - = ε (15)
where K VP , m VP are respectively the VP consistency and the VP strain-rate sensitivity which are temperature dependant.

Moreover, the temperature influence on the residual stress computation is also taken into account through the mass conservation equation of Eq. ( 12). Indeed, the thermal part of the strain rate tensor th ε& depends on the thermal evolution of the density:

( ) ( ) ( ) ( ) ( ) t T T t T T T T T tr th ∂ ∂ - = ∂ ∂         ∂ ∂ - = = ρ ρ ρ ρ α ε 1 3 1 3 3 & & ( 16
)
This mechanical problem is non linear due to the velocity dependence of the behaviour law. So a Newton-Raphson method has to be used. After convergence, the iterative resolution provides the nodal velocity and pressure fields

M S t t t t v x x ∆ + = ∆ + r (17)
Whereas in the material supply resolution a Eulerian approach is used (the velocity field θ & r v is used to transport the interface and the temperature), in this part, a Lagrangian approach is considered. In fact, the mesh is convected by the velocity field S v r . So the impact of the mechanical resolution on the level set function and the interface is also taken into account.

RESIDUAL STRESSES COMPUTATION IN TIG WELDING

Model description

The 3D finite elements model consists in a thick plate of austenitic steel 316L. The dimensions are given on the FIGURE 7. Two domains coexist: the gas in the top part and the metal below. Around the interface plane, φ = 0, the mesh is refined for a correct definition of the mixing law (Eq. ( 1)) and to be able to efficiently use the Continuum Surface Force method (Eq. ( 4)).

The torch is moving along the x-axis on the centre of the plate with a velocity of 1mm/s. The plate sides, which are parallel to the welding direction, are fixed: their displacement is null,

0 = S v r
. This welding configuration has been proposed by Hamide, [START_REF] Hamide | Modélisation numérique du soudage à l'arc des aciers[END_REF]. In order to illustrate the presented mechanical resolution of this model, a comparison of numerical results with a LS approach and without is proposed here. Indeed, to the best of our knowledge it is the first time that a stress and strain computation is carried out from a welding simulation including a level set approach. The thermal and mechanical material properties of the 316L are given in [START_REF] Hamide | Modélisation numérique du soudage à l'arc des aciers[END_REF]. The TIG heat source is circular with a power P P of 975W and a radius r P of 5mm:

2 p P P r P Q π = & (18)

Numerical validation of the mechanical computation

A sensor is located at a distance of 95mm ahead the plate front side, at a depth of φ = -5mm and on the weld bead axis. The numerical longitudinal (x direction) and transversal (y direction) stresses from both models with and without Level Set are compared on the FIGURE 8. The curves tendency is respected. Only the maximum values of tensile and compressive stresses are bigger with the Level Set approach. This is due to the meshes which are different. In fact, as the mechanical computation is realized at the element centre, according to position of the sensor in the element, the result could be a bit different. The thermal resolution is solved at each integration point of the element. So, the mesh has a lowest impact on the thermal measurement: both curves are identical. The computation of stresses from the present model with a level set approach is so considered validate. This result is significant because it is the first time that mechanical computation, level set approach and welding simulation are gathered in a same FE model. The residual stresses distribution on the plate top face after a welding of 95s is presented on the FIGURE 9. The lines represent the isothermal distribution. The maximal temperature is around 1660K which corresponds to a liquid fraction of 55%. The main stresses are located in the isothermal line 720K, the second one. To highlight the different behaviour of the steel plate, a last graph is proposed (FIGURE 10). The behaviour is visco-plastic where the temperature is upper than the solidus temperature, 1598K. A thick band at each part of the weld pool has an elasto-visco-plastic behaviour where the compressive stresses are the highest. Everywhere else the behaviour is elastic.

CONCLUSION

In this paper, a 3D finite element model has been described. Its main faculties are to allow the simulation of hybrid arc/laser welding, from the heat and mass supply to the residual stresses computation. Two cases have been proposed to illustrate both the parts of the paper.

An industrial hybrid welding process simulation has been carried out to highlight the model efficiency to model the weld bead growth. Weld pool dimensions and shape have been compared to the experiment. The Level Set approach used in this model will next allow realizing multi-pass welding simulation. Only one pass has been shown in this paper, but several ones could be presented during the speaking.

The mechanical computation is also possible with this model. Indeed, from a non coupled resolution and the thermal distribution as input, the model is able to compute the stresses induced by the welding heat source supply. No industrial case has been proposed to validate this aspect of the model. A comparison between the model with a Level Set approach and without has sooner be shown. In fact, it quite unusual to compute mechanical stresses from a Level Set approach and it seem important to show that the model meets the criteria. A case with material supply could be presented during the speaking.

FIGURE 1 :

 1 FIGURE 1: Schematics of the heat sources considered in the finite element model, in level set formulation.

FIGURE 5 :

 5 FIGURE 5: Comparison of the weld pool size between experimentation (a) and simulation (b).

FIGURE 6 :

 6 FIGURE 6: Superimposition of the weld bead from the experimentation (Exp) and the numerical simulation (Num).

r

  and p. From the velocity field S v r and the mechanical time step ∆t M the nodal displacements are computed:

FIGURE 7 :

 7 FIGURE 7: Model description.

FIGURE 8 :

 8 FIGURE 8: Comparison of the numerical stresses from both with and without Level Set models.

FIGURE 9 :

 9 FIGURE 9: Longitudinal (a) and transversal (b) stress distribution on the interface.

FIGURE 10 :

 10 FIGURE 10: Evolution of the interface stresses according to the torch distance.

& & & & &

& & & & &


Vincent Archer
email: vincent.archer@imag.fr

Using Conceptual Vectors to get Magn Collocations (and using contrastive properties to get their translations)

This paper presents a semi-automatic approach for extraction of collocations from corpora which uses the results of Conceptual Vectors as a semantic filter. First, this method estimates the ability of each co-occurrence to be a collocation, using a statistical measure based on the fact that it occurs more often than by chance. Then the results are automatically filtered (with conceptual vectors) to retain only one given semantic kind of collocations. Finally we perform a new filtering based on manually entered data. Our evaluation on monolingual and bilingual experiments shows the interest to combine automatic extraction and manual intervention to extract collocations (to fill multilingual lexical databases). It proves especially that the use of conceptual vectors to filter the candidates allows us to increase the precision noticeably.

Introduction

Natural language processing needs linguistic knowledge, especially in machine translation: current systems have bad results because of parsing errors and collocations. In fact, some expressions can not be translated word-for-word because the meaning of a whole expression is not necessarily the combination of the meanings of its components. This problem could be easily solved for recognized locutions by considering them as an unique lexical object, but it is really more difficult for collocations (expressions where one term is chosen in function of the other one, like driving rain for the intensification of rain). It is more difficult to know and recognize collocations than idioms, because they are more numerous and their meaning is not fully independent from the components. In the Meaning-Text Theory, the lexical functions [START_REF] Mel' Uk | Introduction à la lexicologie explicative et combinatoire, Louvain-la-Neuve: Duculot. Using Conceptual Vectors to get Magn Collocations[END_REF] provide a good representation of the collocations, for example č Magn(rain)={heavy, driving} for the intensification. If a translation system knows that heavy rain is an intensification of rain, and that, in French, the intensification of pluie (translation of rain) is pluie battante, it will be able to translate this expression correctly. learning, interaction with non-specialists, etc. In this paper, we will present the use of linguistic knowledge (conceptual vectors) to filter the results of an extraction of collocations.

Modeling -the Lexical Functions

Before starting extraction, we have to define what we exactly consider as a collocation: different researchers who worked on collocations covered different notions. [START_REF] Sinclair | English Lexical Studies: Report to OSTI on Project C/LP/08[END_REF] defined it as "the occurrence of two items in a context within a specified environment. Significant collocation is a regular collocation between two items, such as they co-occur more often than their respective frequencies", with no remarks on the dependency between the items. Here we use the definition given in [START_REF] Kahane | Formal foundation of lexical functions[END_REF]): a collocation is "a linguistic expression made up of at least two components: 1. the base of the collocation: a full lexical unit which is "freely" chosen by the speaker; 2. the collocate: a lexical unit or a multilexical expression which is chosen in a (partially) arbitrary way to express a given meaning and/or a grammatical structure contingent upon the choice of the base".

There are many models of the lexicon, and some have been implemented to create lexical database. Some contain informations about co-occurrence, like the co-occurrence dictionary in EDR, the qualia structure in the Generative Lexicon [START_REF] Pustejovsky | The Generative Lexicon[END_REF] or troponyms in Wordnet, which may sometimes be collocations, but there is no manner to distinguish a collocation from another co-occurrence. The only representation of the lexicon that really aims to model collocations is the Lexical Functions (LF), a part of Mel' uk's Meaning-Text č Theory: a given lexical function links a lexical unit with a set of lexical units which have a particular relation with it. These relations can be paradigmatic (like synonymy, derivation, etc.) or syntagmatic (combinatory links, collocations). Moreover, this theory (Mel' uk et al., č 1995) has been implemented by the realization of explanatory and combinatory dictionaries for French (DEC, on paper), and in automatic database (DiCo 1 , a simplification of the DEC). Our research is related to the Papillon project. Its aim is to build a multilingual lexical database [START_REF] Mangeot | Construction collaborative de données lexicales multilingues, le projet Papillon[END_REF] which may be consulted at http://www.papillondictionary.org, and be edited in a collaborative way. This database can be viewed as a dictionary made of several volumes: one for each language, and one for the interlingual pivot structure. The macro-structure links the entries from the volumes using an abstract pivot made up of interlingual acceptions, or axies [START_REF] Sérasset | Interlingual Lexical Organisation for Multilingual Lexical Databases in NADIA[END_REF], modeling differences of semantic refinement 2 . The structure of the lexical units is similar to the one used in DiCo, using Lexical Functions.

Conceptual Vectors

In the theory of Conceptual Vectors, there is a finite set of concepts that could be used to generate the terms of the whole language: each meaning of the language could be considered as a linear combination of those concepts (Schwab et al., 2002). Conceptual vectors represents the language terms, and the dimensions of these vectors are the basic concepts of the language. Using a mathematical representation of meaning like vectors, allows to consider the distance of meaning between terms as the angular distance of the correspondent vectors. An 1 available online with the interface DicOuèbe at http://olst.ling.umontreal.ca/dicouebe 2 For instance, river can be translated in French by 2 non-synonyms words: rivière (flows into a river) and fleuve (flows into the sea) ; so the axie for river is refined into 2 axies, one for rivière, the other one for fleuve experimental implementation of this theory on French is made at the LIRMM (Montpellier, France), where 873 concepts are identified, like vie, mort, recherche, fin (respectively life, death, research, end), etc., using a French thesaurus [START_REF] Larousse ; Larousse Léon | Acquisition semi-automatique de relations lexicales bilingues (français-anglais) à partir du web[END_REF]. It uses existing data to refine vectors: new vectors are computed from definitions from different sources (dictionaries, synonym lists, manual indexations, etc.) which are parsed, and from existing vectors. It needs a bootstrap: you must have a kernel made of pre-computed vectors (generally manually indexed) to begin the process.

Using Conceptual Vectors to filter our results

A collocation can be viewed as made of three components: base, collocate, and meaning. After the extraction of collocations from a corpus, there are lots of candidates with supposed base and collocate but no meaning. The use of existing lexical resources, like [START_REF] Pearce | Synonymy in collocation extraction[END_REF] made with Wordnet, allows to improve the quality of extraction: those resources contain semantic informations that can be really interesting in such tasks. Here we need a semantic filter to consider only collocates that express intensification: we want to get a class of such collocates. We can use Conceptual Vectors to get it: we believe that the set of nearest conceptual vectors (according to angular distance) from the concept intensité (intensity) could be this wanted class of intensifiers. We assume the fact that this filter would find the more transparent collocations (where the collocate has always a meaning close to intensification) but will miss the less decodable collocations: we will increase precision but decrease recall. But even decodable collocations are of great interest because of their unpredictability, a great problem in machine translation: the meaning is generally insufficient to generate the whole collocation from the base ; for instance, gravement and grièvement are French synonyms, you can generate gravement malade (seriously ill) and grièvement blessé (seriously hurt), but not *grièvement malade. So, even if we get decodable collocations, it will be useful to know that we should use one particular collocate and not another one to express intensification.

Collocation acquisition

In our research, our aim is to get intensification collocations. [START_REF] Claveau | Discovering and Organizing Noun-Verb Collocations in a Specialized Corpora Using Inductive Logic Programming[END_REF] showed the interest of inferring rules from the contexts of known collocations to find other collocations. (Wanner et al., 2006) also used machine learning techniques to label collocations with semantic tags. We do not have a learning base to perform such a task. Furthermore, we want to propose a method that could be easily implemented. That's why we propose an acquisition of collocations by extraction. The X-tract system [START_REF] Smadja | Retrieving collocations from text: Xtract[END_REF], even if it did not aim to extract the same things as we do (Smadja considers that ``a collocation is an arbitrary and recurrent word combination'', there is nothing about the fact that the use of a term depends on the other term), showed the interest of using an hybrid method that combines a linguistic (syntactic) analysis and a statistical filter. That's why our approach is also hybrid.

Syntactic and semantic aspects: Contexts, Conceptual Vectors

The co-occurrence of two terms in the same phrase is not sufficient to recognize a collocation: base and collocate must have a particular relation (like modification). So we use syntactic analysis, considering different kinds of context: the first one is the fact that a term is immediately followed by an other term (linear context) ; the second one is the fact that an analyzer says that there is a relation of modification between the two terms (dependency Vincent Archer context): [START_REF] Lin | Extracting Collocations from Text Corpora[END_REF] obtained quite good results using that kind of context. We also use a stoplist to eliminate stative verbs which make noise because they can never be intensified.

As we explained before, we decided to use conceptual vectors to filter our candidates, in order to get Magn collocations. We download at http://www.lirmm.fr/~lafourcade the 500 nearest words from c4.intensité (according to the angular distance), this set will be used to filter our results: we will only keep the co-occurrences of which the supposed collocate is part of the set. [START_REF] Larousse ; Larousse Léon | Acquisition semi-automatique de relations lexicales bilingues (français-anglais) à partir du web[END_REF] acquire bilingual lexical relations using a simple manual validation of English lexical relations to increase the precision of their final results from 7,5% to 83,3%: it shows that it is really interesting to have a human intervention to complete automatic extraction. Our method is based on the same idea: we want to apply a simple manual filter (and so use human knowledge about language) to our automatic acquisition (based on the cooccurrence of terms in corpora) and to our automatic filter (conceptual vectors).

Collocability

An essential property of collocations is that these co-occurrences are more frequent than by chance. The mutual information MI  x , y =log  P  x , y /[ P  x . P  y ] 3 seems to be convenient to model that. [START_REF] Lin | Extracting Collocations from Text Corpora[END_REF] proposed an adaptation of this measure to triples (2 terms and 1 relation): MI  w , r , w =log[ P w , r , w /P  r . P w |r  . P w |r ]4 . However, mutual information has a drawback for its use in NLP tasks: it tends to overestimate the association between two words with low frequencies. That's why [START_REF] Fung | A technical word-and term-translation aid using noisy parallel corpora across language groups[END_REF]) introduced the weighted mutual information, with ponderation: wMI w , w =P  w , w . log[ P w , w / P w . P w ] . [START_REF] Wu | Synonymous collocation extraction using translation information[END_REF] proposed the adaptation of this last measure to triples (with the relation) which we use in our approach: WMI  w , r , w =P  w , r , w . log P w , r , w  P w |r . P w |r . P r 

Bi-collability

As multilingual information on collocations is very useful for translation systems, we are also interested to extract bi-collocations (two collocations which are translations of each other): we shall use bilingual corpora to get such an information. As the choice of the collocate depends on the base, it is frequent that the collocate chosen for the translation of the base is not the translation of the collocate chosen for the base. As a bi-collocation is not really useful when collocates are translations (the translation by MT systems would be correct), we want to extract contrastive bi-collocations, where the bases are translations but the collocates are not necessarily translations. We want to express the fact that a bi-collocation is a couple of collocations which often appears in similar (comparable, aligned) documents ; we adapt the cos measure cos x , y =|X ∩Y|/|X|.|Y| (where X and Y are the documents where x and y occur) to similar distinct sets: cos bilingual c fr , c en =|BI -DOCS  c fr , c en |/  |C FR |.|C EN | 5 . It is insufficient to compute the association between the two parts of the bi-collocation candidates: 3 where P(w) is the probability of the occurrence of w, P(x, y) the co-occurrence of the terms in a given context the final measure must also model that these two parts are collocations. So we use a ponderation which should be maximized when the collocability of each monolingual property is high and minimized when it is low. Our final measure to rank bi-collocations candidates is: Bicollocability c fr , c en =WMI c fr WMI c en ×cos bilingual c fr , c en  .

Evaluation

Experiments

We choose to extract candidates for Magn collocations with a verbal base and a adverbial collocate. We conduct three different experiments in order to evaluate the effectiveness of filtering using conceptual vectors. The first one is monolingual (acquisition of Magn collocations for French) ; the other ones are bilingual (acquisition of French-English Magn bicollocations -couples of Magn collocations which are translations): one task is made using comparable corpora, the other one is made using parallel corpora. The monolingual task uses the corpus LeMonde95 ; the first bilingual task uses LeMonde95 and GH95 (two newspapers corpora) as comparable corpora ; finally we use French and English parts of EuroParl (proceedings of the debates at the European Parliament) aligned by sentences. We supposed that GH95 and LeMonde95 were comparable, but we did not have any correspondance at the level of documents, so we computed a comparability measure between French and English documents. The criteria we used to determine if documents speak about the same topic were: the proximity in time (less than 2 days between the publications of the articles), the same named entities in the two documents, and the fact that nominal syntagms (which express the thema of a document) are translated. That's why we compute a very simple "comparability measure" between every potential couple of documents CFR and CEN are the sets of documents where cfr and cen, appear ; BI-DOCS(cfr,cen) is the set of bi-documents (comparable or aligned documents) where cfr appears in the French document and cen in the English document comparability value if their topics are similar). Using a minimal threshold of 0.2, we obtained 63 621 associations (1.34 per French document, 1.13 per English document).

Experiments

Evaluation method

We compute precision for each experiment. We can not compute recall because we do not have a reference base at our disposal: there is no standard evaluation measure for collocation extraction. Moreover, we do not aim to extract the same things as other researchers, because we consider a different definition of collocation: [START_REF] Smadja | Retrieving collocations from text: Xtract[END_REF] tried to extract all recurrent co-occurrences, [START_REF] Lin | Extracting Collocations from Text Corpora[END_REF] aimed to obtain all "habitual word combinations", etc., whereas we consider collocations like co-occurrences with particular linguistic properties. In addition there is one more difficulty to present an objective comparison with existing works: we tried to extract collocations that express one particular meaning. For each monolingual experiment, we evaluate the 1000 first produced couple candidates, ranked by their WMI value. For the "comparable" bilingual experiment we evaluate the 200 first candidates ; for the "parallel" bilingual one we evaluate the 43 candidates we get using Conceptual Vectors (plus the 200 first candidates without using Conceptual Vectors). The first experiment was a statistical extraction with no filtering, so we got low precision (17%). Using an automatically produced list of adverbs to filter the results, the precision is multiplied by 2,5 (41% or 44%, depending the context). Moreover, we can already retrieve more collocations in top candidates, like régner sans partage or réduire considérablement (intensifications of rule and reduce). But we still have candidates in which the adverb never expresses intensification. We can observe that the kind of context seems not to be determining: informations on dependency do not allow to increase precision, we even obtain slightly better results with linear context, because dependency analysis retrieves more adverbs far from the verbs (we increase recall) but is more sensible to noise (we decrease precision). At last, a simple operation, the introduction of a new filter on adverbs (manually defined from the results of the precedent experiment: we remove adverbs like trop (too much), très (very), tant (so much), etc.: the last ones can express intensification but are so frequent that they are not interesting) allows us to eliminate 47% of candidates: then the precision increases from 44% to 83%. This shows the effectiveness of manual intervention on collocation extraction. Even with manual filtering, we get 17% of noise in the results because some adverbs may be intensifiers with a given verb and express another meaning with another one. The filtering allows a gain in precision but implies a loss of recall ; we loose in this case the less decodable collocations like défendre bec et ongles or reprendre de plus belle (intensifications of defend and resume): as the locutions do not express clearly the notion of intensification by

Monolingual

Filtering

Using Conceptual Vectors to get Magn Collocations themselves, they are unfortunately not enough near to the concept of intensification (in the conceptual vectors) to be kept here.

Bilingual

Experiment The precision of produced candidates is higher in the experiment using parallel corpora, and this is not surprising: the chance to get translations of co-occurences is higher using aligned documents than using comparable ones. In the comparable experiment (and in a lesser extent in the parallel one), we can have non-collocations candidates where their two components are really collocations ; it could be a problem of polysemy (the intensified verb is used in different acceptions) or the fact that the intensification is not made on the same argument of the predicate.

Conclusion and Future Work

In this paper, we have described a semi-automatic method of collocation extraction that uses Conceptual Vectors to produce a semantic filter (which is then refined manually). We proved that a human intervention on such a process is necessary to obtain high-quality results, and that the results of conceptual vectors are a good semantic filter for the extraction of one particular kind of collocation, especially when they are completed by a manual intervention.

We also showed that this method is more efficient in the monolingual case because it is much harder to find bi-collocations than collocations in corpora (so the recall is much lower). We will make experiments to find the best corpus size to extract collocations. Our current objective is to realize programs that could be easily used by people who are not computer scientists (especially by linguists) to produce candidates for collocations from the corpora they have at their disposal, allowing them to guide the process manually. We also want to explore other ways to get collocations, like machine learning (learn the characteristics of collocations, and retrieve co-occurrences with these characteristics), expansion of results with thesaurus (by instance, retrieve driving snow from driving rain). Another track is to interact with non-specialists (every native speaker of English can say that a driving rain is an intense rain) using games: questions with known answers allow to evaluate the player, and we keep the answers from good players for the other ones ; we can also have a two-player party (it is a good proof of collocability when two different players give the same answer).

  comp  D fr , D en =overlap [NS  D fr  , NS  D en ]overlap[trans en  NS  D fr  , NS  D en ]/ 6 (we use the overlap measure to allow a short document and a long one to have a great 5

Table 1 :

 1 Characteristics of corpora

	Documents Sentences	Words

Table 2 :

 2 Evaluation of the monolingual experiments (top 1000 candidates)

		No	Conceptual Vectors Conceptual Vectors Conc. Vectors + Manual
	Context Dependency	Dependency	Linear	Linear
	Precision	17%	41%	44%	83%

Table 3 :

 3 Top 5 candidates for bicollocations, in the two bilingual experiments

	on comparable corpora	Experiment on parallel corpora
	mettre beaucoup / take seriously	soutenir pleinement / support wholy
	jouer pleinement / work hard	jouer pleinement / work together
	vouloir particulièrement / want really	changer radicalement / change radically
	accomplir particulièrement / perform strongly	modifier radicalement / modify radically
	regretter énormément / regret deeply	jouer pleinement / play right
	Is the English couple a Magn collocation ?	Yes		No
	Are the 2 couples in translation ?	Yes	No	Yes
	1-100 / 101-200	17% / 5%	23% / 18%	60% / 77%

Table 4 :

 4 Evaluation of the bilingual comparable experiment (top 200 candidates)

  At this point, we should comment the number of candidates produced: in the first experiment (comparable corpora), we get 80 298 candidates before any filter, 3 973 after applying Conceptual Vectors, and 201 after manual intervention. In the second one (parallel corpora), we get 15 583 candidates before any filter, 1995 after applying Conceptual Vectors, and 43 after manual interventions. This last number is very low and insufficient. It seems not the best way to get bi-collocations: it is interesting to filter monolingually to reduce the number of candidates to several thousands, but the bilinguality combines English and French filters and we finally obtain to few collocations. If we want to get more collocations, we have to know that even big bilingual corpora contain few bi-collocations.

	Which collocations are correct ? French+English	French	English	No
	Translation	19 / 36,5%	0 / 1%	1 / 1,5%	0 / 13,5%
	No translation	2 / 2,5%	5 / 21%	2 / 12%	10 / 12%
	Tableau 5: Evaluation of the bilingual parallel experiment

where P(w1|r) and P(w2|r) are the respective probabilities of the occurrence of w1 as the first element of a relation r, and the occurrence of w2 as the second element of a relation r, and P(w1, r, w2) the probability of co-occurrence of w1 and w2 to be in relation r.

Where Dfr and Den are French and English documents, NS(D) is the set of nominal syntagms in document D, and trans(NS(D)) is the set of the translations of the nominal syntagms in document D


Takao Koshikawa
email: takao.koshikawa@mines-paristech.fr

Michel Bellet

Charles-André Gandin

Study of hot tearing through ingot bending test: thermomechanical and solute transport analysis

. It allows simulating solute transport phenomena leading to macrosegregation induced by deformation shrinkage and advection. The effect of micro-and macro-segregation on hot tearing sensitivity is then discussed.

Introduction

Hot tearing is one of the critical defects in industrial casting processes such as continuous casting and ingot casting. It comes from a complicated combination of thermomechanical and solidification phenomena. This defect basically corresponds to the incapacity of liquid feeding in the presence of tensile stresses perpendicular to dendrite growth direction in a semi-solid zone. It develops when the solid fraction is typically between 0.9 and 1.0, called brittle temperature range (BTR). In this range, the ductility and strength of the alloy are decreased due to residual liquid film in between dendrites. A schematic summary of these phenomena is explained in reference [3].

In this study, the macroscopic hot tearing criterion of Won et al. [5] is evaluated, which is based on cumulated strain perpendicular to dendrite growth direction. The corresponding strain rate is denoted as ε & , as explained in reference [3].

The strain based criterion is expressed as follows:

          - = ∫ C BTR WYSO dt HCC ε ε & with ( ) * * ˆn m C BTR ε ϕ ε & = (1)
where C εˆ is the strain limit and the three parameters are deduced by a non-linear data fitting with several hot tearing experiments: φ = 0.02821, m* = 0.3131, n* = 0.8638. It is obvious that, since BTR appears in the criterion, micro-and macro-segregation have an impact on the hot tearing sensitivity.

In order to identify hot tearing, a hot tearing experiment, so-called ingot bending test has been developed at Nippon Steel & Sumitomo Metal. For thermal mechanical analyses, two numerical codes are used, THERCAST and R2SOL developed at CEMEF. The micro-and macro-segregation effect on the hot tearing sensitivity is then discussed.

Ingot bending test

The schematic of the ingot bending test, is shown in Fig. 1. It has been designed to mimic real process thermomechanical solicitation; for instance, cooling rate and deformation. The procedure of the experiment is as follows: Molten metal is prepared in a ladle and the temperature is controlled as 1640 °C just before pouring. The molten metal is filled into the mould from the top through a tundish. The filling duration is about 70 sec. The mass of SteelSim2013, 5 th Int. Conf. on Modelling and Simulation of Metallurgical Processes in Steelmaking, Ostrava, September 10-12, 2013, DVD published by the Czech metallurgical Society (2013) 9 pages metal is approximately 450 kg. The size of the ingot is 0.16 m thickness, 0.5 m width and 0.75 m height. After about 5 minutes from filling, the part of the mould (labeled as Right_mould_sup in Fig. 1) is removed. The ingot is then deformed during solidification by the external bending tool, located at the height of 0.45 m from the ingot bottom. The tool velocity and displacement are controlled and measured during the test. The test condition is summarized in Tab. 1, together with measured nominal compositions in the ladle before filling. To know temperature evolution during the test, thermocouples are positioned in the ingot and the left side mould (labeled as Left_mould in Fig. 1). Furthermore, the ingot surface temperature has been measured in case of trial N-4, using a pyrometer before and after deformation. After experiment, the ingot is cut at the ingot width center and the micrograph is investigated after appropriate etching to verify whether the specimen has hot tears or not. Tab. Fig. 2 Comparison of micrographs close to the bending region; trial N-1 (left) and N-4 (right). In both cases, there are hot tears indicated by pink square marks. In addition, a white band can be seen in N-1 (left), corresponding to less solute containing zone; negative macro-segregation Fig. 2 shows two typical micrographs in case of N-1 and N-4 close to the bending region. In case of N-1, the surface deformation due to the bending tool is quite marked, while the micrograph of N-4 does not show much because of the difference of displacement. One can find dendritic microstructure which grows from the ingot surface (right hand side) toward the centre of the ingot (left hand side). Around the centre of the ingot, the morphology is changed to equiaxed type, which is due to the decrease of cooling rate close to the core. In both cases, there are hot tears, which are marked as black collar in the micrographs. It is seen that the hot tears exist in interdendritic region, which is expected and explained in previous section. In addition, in case of N-1, white band is observed in the micrograph which corresponds to less solute containing zone; negative macro-segregation.

Thermo-mechanical modeling

Standard thermomechanical modeling

In the standard thermomechanical modelling approach [1,2], the mushy zone (composed of liquid and solid phases) is considered as a homogenized continuum so that a unique velocity field is solved for. The numerical simulation is based on the thermomechanical stress/strain analysis. It is conducted using the 3D finite element code THERCAST. The thermal problem and the mechanical problem are solved sequentially at each time increment. First, the energy conservation is solved on all domains taking into account heat exchange between them. In this resolution, the solidification path is defined a priori. Second, the mechanical problem is solved. The alloy is modelled with a hybrid constitutive equation. Over the solidus temperature, the alloy is seen as a non-Newtonian fluid obeying a temperaturedependent viscoplastic multiplicative law, as follows:

( ) m m K ε σ & 1 3 + = (2)
where σ is the equivalent von Mises stress, K the viscoplastic consistency, ε & the generalized strain rate and m the strain rate sensitivity. The value of m over the liquidus temperature is equal to 1 (Newtonian fluid). Note that the liquidus viscosity is defined as 10 Pas, for numerical stability reasons. Below the solidus temperature, the alloy obeys an elastic viscoplastic constitutive equation, expressed by: ( )

m n m y K ε ε σ σ & 1 3 + + = (3)
in which σ y is the plastic yield stress, ε the cumulated plastic strain and n the strain hardening coefficient. Physical properties and constitutive parameters are temperature dependent and determined with experimental data.

Concerning predefined solidification path, to evaluate the effect on hot tearing sensitivity, three paths are used for this type of simulation; one is based on lever rule approximation (LR), meaning that thermodynamic equilibrium is achieved in all existing phases at given temperature. Second is the partial-equilibrium approximation together with the paraequilibrium approximation (PE+PA) in which an interstitial element; i.e. carbon has finite diffusivity in all phases, whereas substitutional elements are frozen in solid phases. In addition, the model can account for peritectic reaction, taking place in the steel grade in Tab. 1, under para-equilibrium condition. Last one is the measurement result by means of unidirectional solidification experiment. The details are explained in reference [6]. The simulation of LR and PE+PA has been performed with Thermo-Calc and the TQ-interface [7] using TCFE6 database [START_REF] Shi | TCS steels/Fe-alloys database V6.0 Thermo-Calc Software AB[END_REF]. The comparison of solidification paths are shown in Fig. 3. As illustrated in Fig. 1, the 3D model is prepared with a consideration of symmetry; half of the setup is modelled. The initial mesh size of the ingot is 8 mm constant and the number of elementis is approximately 600,000. For better accuracy in the bending region, remeshing is performed to obtain finer mesh (4 mm) just before bending, leading to 1,000,000 elements. Filling stage is not taken into account for so that the thermo mechanical simulation starts at 70 s from the start of filling. Concerning the initial temperature of the ingot, uniform temperature as 1550 °C, which is estimated using measurement data, is defined on the ingot domain. For other domains, 20 °C is the initial temperature.

The thermal boundary condition in between ingot and mould is defined as air gap dependent heat resistance. When the ingot contacts the mould, the heat transfer coefficient is used as 850 W/m 2 /K which is calibrated with measurement results. When air gap generates, the air gap width is considered and the heat exchange depends on radiation between ingot and mould surface and heat conduction in the air, which depends on the gap width. Regarding the ingot top, powder is put after the end of filling. Thus, adiabatic condition is used for the top. After mould removal, the free surface heat exchange is defined as radiation and convection, where the emissivity is taken as 0.8 and the heat transfer coefficient for convection is assumed as 15 W/m 2 /K.

"Two phase" modeling and macrosegregation analysis

The second approach is an effective "two phase" modeling in which the liquid and solid velocity fields are computed separately in the mushy zone. It allows simulating solute transport phenomena leading to macro-segregation induced by deformation, shrinkage and advection. It is performed with the 2D finite element code R2SOL. Details of the formulation can be found on reference [4]. The assumptions are listed as follows:

The liquid phase is considered as an incompressible Newtonian fluid.

The solid phase is seen as a compressible non-Newtonian generalized fluid.. The momentum interaction between solid and liquid is taken into consideration with an isotropic Darcy law. Shrinkage term is considered by a different specific mass for liquid and solid phases (ρ S and ρ L ). However, these are constant in solidification interval.

Local thermal equilibrium holds within representative element volume (REV); T=T l =T s . Simply, a binary alloy system is extended to a multi component system under lever rule approximation, meaning that partition coefficient k i and liquidus slope m i are constant for each solute element i. The above mentioned assumptions lead to the set of conservation equations, expressed as follows:

( )

0 1 2 = + - + ∇ - ⋅ ∇ - g v v Σ s s s l l l l s s g g p g ρ κ µ (4) ( ) ( ) ( ) (
) [START_REF] Shi | TCS steels/Fe-alloys database V6.0 Thermo-Calc Software AB[END_REF] in which v s and v l are the solid and liquid velocity vector, p l is the liquid phase pressure, <ρ> and <λ> are the average density and the average heat conductivity of the solid and liquid mixture, <h> is the average enthalpy of the mixture. f s and f l are respectively the solid and liquid mass fraction, defined as g k ρ k /<ρ> with each phase k. <w i > and w l,i are respectively the average solute concentration and the liquid phase concentration for solute component i. In the momentum equations, the Darcy term is seen with the permeability κ, deriving from liquid phase volume fraction and predefined secondary dendrite arm spacing (Carman-Kozeny model). Constitutive models for the liquid and the solid phases can be seen in reference [4] Concerning solidification path, it is not defined a priori since solute concentration varies due to solute transport. To model the microsegregation phenomena, we simply extend binary alloy system under lever rule approximation to multi component system. Therefore, constant partition coefficient k i and liquidus slope m i for each solute component are assumed. Considering the average enthalpy, one obtains the following relations:

l l l L l l L L l s l l l l l l g t g g g p g v v v g v v s × ⋅ ∇ + ∂ ∂ = + - - ∇ - ⋅ ∇ - ρ ρ ρ κ µ 1 2 (5) ( ) ( ) t g g g s L S L s s L S l l ∂ ∂         - = ⋅ ∇ + ⋅ ∇ ρ ρ ρ ρ ρ v v (6) ( ) ( ) ( ) 0 = ∇ ⋅ ∇ - - ⋅ ∇ + ⋅ ∇ + ∂ ∂ T f f L h t h s l s l λ ρ ρ ρ v v v (7) ( ) ( ) ( ) ( ) 0 , , = ∇ ⋅ ∇ - - ⋅ ∇ + ⋅ ∇ + ∂ ∂ i l l l s l i l l s i i w D g w g w t w v v v
( ) ∑ + = i i l i m w m T T , (9) ( ) ( ) [ ] i l i l l i s l l i l i w k g g w g g w w , , , 1 1 -
+ = - + = (10) L f T c h l p + = (11)
where T m is melting temperature of pure iron, c p is a constant heat capacity and L is the latent heat. Defining the number of solutes as N S , the number of equations is N S +2. <h> and <w i > are known since these variables are obtained thanks to the global energy and solute conservation resolutions. So, the unknowns are T, g l and w l,i ; the number of unknowns is N S +2. Thus, the set of equations is completed and solved iteratively. This effective two phase modeling is applied to the case of N-1, in which the ingot centre is fully mushy state during bending. The calculation is carried out in two steps. At first, a standard thermo mechanical calculation is conducted up to just before bending stage. The simulation is then restarted using the "two phase" model. It is assumed that there is no macro segregation at that time. Fig. 4 shows the comparison of temperature evolution in the ingot. There are two thermo couples in the ingot; one is close to the ingot center as blue collar in the schematic in Fig. 4, while another is located in the close to the narrow surface of the ingot. TC-2 shows less solidification time than TC-1 because it is close to the narrow surface but also the surface of the ingot. So, solidification is preceded, compared to TC-1. The calculated result is in good agreement with measurement data, especially during bending. Fig. 5 (left) shows the comparison of surface temperature evolution at the bending tool height. Looking at the calculated result, one can see the change of the evolution at 300 s. At that moment, the mould located in front of the bending tool is removed, prior bending. Consequently, the cooling rate is decreased and the surface temperature is then increased. The calculated temperature evolution is in good agreement with the measured data. So it is concluded that the thermal modeling regarding the boundary condition and thermo physical parameters are well calibrated.

Results and discussion

Hot tearing sensitivity

Fig. 6 demonstrates hot tearing sensitivity field in horizontal section at bending tool height after bending in case of N-4. A positive band can be seen in front of the bending tool. It is in good agreement with the micrograph shown in Fig. 6 where hot tears are observed in the similar position. Fig. 7 shows the comparison of hot tearing sensitivity profile and cumulated strain profile as a function of the distance from the ingot surface together with the hot tear location from the micrograph. First remark is that the calculated profiles are in good agreement with the measured crack position. The solidification path effect on the location of the high risk is slightly different from each other; in case of LR, the peak is close to the core, while in case of PE+PA, it is located near the surface. The second is the difference of the magnitude of the sensitivity; especially LR case shows lower value than others. It is associated with the difference of BTR. BTR are 17.2 °C, 57.3 °C and 44.5 °C respectively for LR, PE+PA and EX. When BTR is small, the strain limit C εˆ is increased and the hot tearing sensitivity is then decreased. The magnitude of the cumulated strain is almost the same in three cases (Fig. 7 (right)) so that the difference of hot tearing sensitivity is directly linked to the difference of BTR. 8 shows the relative velocity vector field with liquid fraction field during bending. Liquid flow is from the centre to the top, the bottom but also the periphery region because of the compression of the mushy solid phase. It is explained in Fig. 9 that the divergence of solid velocity is negative at the centre. Regarding the periphery region, tensile deformation along the vertical direction takes place so that liquid flow turns to the periphery in order to conserve mass.

Solute concentration field

Calculated relative solute concentration field is presented in Fig. 10. Because of the relative velocity field discussed at the previous section, negative segregation is created at the centre of the ingot and the positive segregation generates at the top, the bottom and the periphery. Concerning bending region, the calculated negative segregation is in good agreement with the white band position in the micrograph. Note that the white band indicates less solute elements. Since the partition coefficient is less than 1.0 in both cases (k Mn = 0.70, k P = 0.28), macro-segregation pattern is similar but the magnitude is different; P favors segregation as expected. On the periphery, positive concentration can be seen but the magnitude is quite small. Therefore, macro-segregation may not have a big impact on hot tearing sensitivity. For better discussion, quantitative comparison is useful and it is in progress now. Now, the volume average mass conservation equation (Eq. ( 6)) is introduced to the volume average solute conservation equation (Eq. ( 8)). In addition, the time derivative of solute and solid fraction decomposed into a time derivative with respect to the mesh, whose velocity is v s , and an advection term are employed, leading to:

w i i i i v v v ⋅ ∇         - + ∂ ∂         - + ∇ ⋅ ∇ + - ⋅ -∇ = ∂ ∂ ρ ρ ρ ρ ρ , , , , (14)
The first term of the second member is linked to convection, the next is diffusion, the third is shrinkage and the last is solid phase compression. In order to understand each contribution to macrosegregation formation, four calculations have been conducted in which only one term is accounted for respectively; for instance, the convection term is tested where other terms are deactivated. The results are summarized in Tab. 2. As expected, solid phase compression is dominant for the formation of macrosegregation.

Tab. 2 Summary of each term contribution to macrosegregation with respect to P relative concentration ((<wi> -wi,0)/wi,0 [%])

Solid phase compression

Ingot centre ˜ 0 % ˜ 0 % +0.3 % +6 % Periphery -0.3 % ˜ 0 % ˜ 0 % +1 % Convection Diffusion Shrinkage

Conclusions

An ingot bending test has been modeled by two numerical codes. Using a standard thermomechanical model, it is then found that strain based criteria have a capability to predict crack occurrence. In addition, micro-segregation model has an impact on hot tearing sensitivity, as expected. Through effective "two phase" simulation, it is demonstrated that negative segregation generates at the centre of the ingot mainly due to solid phase deformation, which is observed in the cast ingot. Quantitative comparison with respect to solute concentration is in progress and the first results should be available soon. Also, all experiments are being modeled and the results will be discussed soon.

Fig. 1

 1 Fig. 1 Schematic of the hot tearing experiment developed at Nippon Steel & Sumitomo Metal: (a) initial state, (b) bending stage after mould removal. The shown configuration is a half of the setup, considering symmetry with respect to the width direction.

Fig. 3

 3 Fig.3Comparison of solidification paths; LR: calculated solidification path based on lever rule approximation, PE+PA: calculated solidification path based on partial-and para-equilibrium approximation, EX: measurement result by mean of unidirectional solidification experiment.

4. 1

 1 Standard thermomechanical simulation result 4.1.1 Temperature evolution in case of N-4

Fig. 4

 4 Fig.4Comparison temperature evolution in the ingot (left) and a schematic of thermocouple positi ons in the ingot (right). There are two thermocouples, denoted TC-1 and TC-2. TC-1 is close to the center of the ingot, while TC-2 is close to the narrow surface.

Fig. 5

 5 Fig. 5 Comparison of ingot surface temperature evolution (left) and a schematic of measurement position in the ingot surface (right). The measurement has been performed after mould removal.

Fig. 6 Fig. 7

 67 Fig. 6 Hot tearing sensitivity field in horizontal section at the bending tool height with the microsegregation model PE+PA. One can find a positive band along lateral direction in front of the bending tool.

4. 2 "

 2 Fig.8shows the relative velocity vector field with liquid fraction field during bending. Liquid flow is from the centre to the top, the bottom but also the periphery region because of the compression of the mushy solid phase. It is explained in Fig.9that the divergence of solid velocity is negative at the centre. Regarding the periphery region, tensile deformation along the vertical direction takes place so that liquid flow turns to the periphery in order to conserve mass.

Fig. 8 Fig. 9

 89 Fig. 8 Liquid fraction field before 2 phase simulation restart (a), liquid fraction field with relative velocity vector (vl -vs) during bending (b) and (c). (c) is close to the bending region with liquid fraction isotherms as eight vertical black lines.

Fig. 10 ( 4 )

 104 Fig.10Relative concentration field after bending; (a) is Mn, (b) is P. (c) is a comparison of calculated relative concentration profile at the bending tool height along horizontal direction with the micrograph. The position of the calculated negative concentration peak is in good agreement with the white band in the micrograph which indicates less solute elements. Note that relative solute concentration is defined as (<wi> -w0)/w0.

Acknowledgement This work has been supported by Nippon Steel & Sumitomo Metal Corporation in a collaboration project with ArcelorMittal. The authors are deeply grateful to Dr. O. Jaouen and Dr. F. Costes, from

Transvalor SA, for their kind help and discussion about thermomechanical modeling using THERCAST software.