S. Chen, G. Guillemot, and C. Gandin, 3D Coupled Cellular Automaton (CA)???Finite Element (FE) Modeling for Solidification Grain Structures in Gas Tungsten Arc Welding (GTAW), ISIJ International, vol.54, issue.2, pp.401-407, 2014.
DOI : 10.2355/isijinternational.54.401

URL : https://hal.archives-ouvertes.fr/hal-00974737

S. K. Choudhary and S. Ganguly, Morphology and Segregation in Continuously Cast High Carbon Steel Billets URL: https, ISIJ International, vol.474712, issue.12, pp.1759-1766, 2007.
DOI : 10.2355/isijinternational.47.1759

T. W. Clyne and W. Kurz, Solute redistribution during solidification with rapid solid state diffusion, Metallurgical Transactions A, vol.100, issue.6, 1981.
DOI : 10.1007/BF02644078

. Combeau, Modeling of microsegregation in macrosegregation computations, Metallurgical and Materials Transactions A, pp.2314-2327, 1996.
DOI : 10.1016/0921-5093(93)90236-8

H. Combeau, M. Zalo?nik, S. Hans, R. , and P. E. , Prediction of Macrosegregation in Steel Ingots: Influence of the Motion and the Morphology of Equiaxed Grains, Metallurgical and Materials Transactions B, vol.47, issue.4, pp.289-304, 2009.
DOI : 10.1007/s11663-008-9178-y

. Copley, The origin of freckles in unidirectionally solidified castings, Metallurgical Transactions, vol.245, issue.8, pp.2193-2204, 1970.
DOI : 10.1007/BF02643435

Q. Du, D. Li, Y. Li, R. Li, and P. Zhang, Simulating a double casting technique using level set method, Computational Materials Science, vol.22, issue.3-4, pp.3-4, 2001.
DOI : 10.1016/S0927-0256(01)00190-2

. Easton, Grain Morphology of As-Cast Wrought Aluminium Alloys, MATERIALS TRANSACTIONS, vol.52, issue.5, pp.842-847, 2011.
DOI : 10.2320/matertrans.L-MZ201118

J. Ettrich, A. Choudhury, O. Tschukin, E. Schoof, A. August et al., Modelling of transient heat conduction with diffuse interface methods, Modelling and Simulation in Materials Science and Engineering, pp.850060965-0393085006, 2014.
DOI : 10.1088/0965-0393/22/8/085006

S. D. Felicelli, J. C. Heinrich, and D. R. Poirier, Simulation of freckles during vertical solidification of binary alloys, Metallurgical Transactions B, vol.10, issue.6, pp.847-859, 1991.
DOI : 10.1007/BF02647732

. Felicelli, Modeling freckle formation in three dimensions during solidification of multicomponent alloys, Metallurgical and Materials Transactions B, vol.39, issue.3, pp.847-855, 1998.
DOI : 10.1007/BF02914651

I. L. Ferreira, A. Garcia, and B. Nestler, On macrosegregation in ternary Al???Cu???Si alloys: numerical and experimental analysis, Scripta Materialia, vol.50, issue.4, pp.407-411, 2004.
DOI : 10.1016/j.scriptamat.2003.11.012

. Ferreira, Modeling and experimental analysis of macrosegregation during transient solidification of a ternary Al???6wt%Cu???1wt%Si alloy, Philosophical Magazine Letters, vol.24, issue.12, pp.769-777, 2009.
DOI : 10.1016/j.ijheatmasstransfer.2008.07.003

M. C. Nereo and G. E. , Macrosegregation: Part I " . Transactions of the Metallurgical Society of AIME, pp.1449-1461, 1967.

M. C. Nereo and G. E. , Macrosegregation: Part III " . Transactions of the Metallurgical Society of AIME, pp.50-55, 1968.

. Flemings, Macrosegregation: Part II " . Transactions of the Metallurgical Society of AIME, pp.41-49, 1968.

V. H. Gada and A. Sharma, On Derivation and Physical Interpretation of Level Set Method???Based Equations for Two-Phase Flow Simulations, Numerical Heat Transfer, Part B: Fundamentals, vol.56, issue.4, pp.307-322, 2009.
DOI : 10.1016/0021-9991(92)90240-Y

C. A. Gandin, G. Guillemot, B. Appolaire, and N. T. Niane, Boundary layer correlation for dendrite tip growth with fluid flow, Materials Science and Engineering: A, vol.342, issue.1-2, pp.44-50, 2003.
DOI : 10.1016/S0921-5093(02)00261-7

URL : https://hal.archives-ouvertes.fr/hal-01435720

B. Genereux, P. D. Genereux, C. A. Borg, P. K. Warrendale, M. Green et al., Characterization of Freckles in a High Strength Wrought Nickel Superalloy, Superalloys 2000 (Ninth International Symposium), pp.19-27, 2000.
DOI : 10.7449/2000/Superalloys_2000_19_27

A. F. Giamei and B. H. Kear, On the nature of freckles in nickel base superalloys, Metallurgical Transactions, vol.1, issue.11, pp.2185-2192, 1970.
DOI : 10.1007/BF02643434

F. Gibou, R. Fedkiw, R. Caflisch, and S. Osher, A Level Set Approach for the Numerical Simulation of Dendritic Growth, Journal of Scientific Computing, vol.19, issue.1/3, pp.183-199, 2003.
DOI : 10.1023/A:1025399807998

S. Gouttebroze, Modélisation 3d par éléments finis de la macroségrégation lors de la solidification d'alliages binaires, p.11, 2005.

C. Gruau and T. Coupez, 3D tetrahedral, unstructured and anisotropic mesh generation with adaptation to natural and multidomain metric, Computer Methods in Applied Mechanics and Engineering, vol.194, issue.48-49, pp.48-49, 2005.
DOI : 10.1016/j.cma.2004.11.020

URL : https://hal.archives-ouvertes.fr/hal-00517639

G. Guillemot, G. Guillemot, and C. Gandin, Analytical model for equiaxed globular solidification in multicomponent alloys, Acta Materialia, vol.97, pp.419-434, 2015.
DOI : 10.1016/j.actamat.2015.04.030

URL : https://hal.archives-ouvertes.fr/hal-01181256

J. Guo and C. Beckermann, Three-Dimensional Simulation of Freckle Formation During Binary Alloy Solidification: Effect of Mesh Spacing " . Numerical Heat Transfer, Part A: Applications, pp.559-576, 2003.

L. Hachani, B. Saadi, X. D. Wang, A. Nouri, K. Zaidat et al., Experimental analysis of the solidification of Sn???3 wt.%Pb alloy under natural convection, International Journal of Heat and Mass Transfer, vol.55, issue.7-8, pp.0017931011007009-134, 1986.
DOI : 10.1016/j.ijheatmasstransfer.2011.11.054

URL : https://hal.archives-ouvertes.fr/hal-00728106

E. Hachem, B. Rivaux, T. Kloczko, H. Digonnet, C. et al., Stabilized finite element method for incompressible flows with high Reynolds number, Journal of Computational Physics, vol.229, issue.23, pp.229-8643, 2010.
DOI : 10.1016/j.jcp.2010.07.030

URL : https://hal.archives-ouvertes.fr/hal-00521881

E. Hachem, Stabilized finite element method for heat transfer and turbulent flows inside industrial furnaces URL: https://tel.archives-ouvertes, p.443532, 2009.
DOI : 10.1016/j.simpat.2012.07.013

W. Harlow, F. H. Harlow, and J. E. Welch, Numerical Calculation of Time-Dependent Viscous Incompressible Flow of Fluid with Free Surface, Physics of Fluids, vol.149, issue.12, pp.2182-2189, 1958.
DOI : 10.1098/rsta.1952.0006

D. J. Hebditch and J. D. Hunt, Observations of ingot macrosegregation on model systems, Metallurgical Transactions, vol.4, issue.7, pp.1557-1564, 1974.
DOI : 10.1007/BF02646326

C. W. Hirt, An arbitrary Lagrangian-Eulerian computing technique, Proceedings of the Second International Conference on Numerical Methods in Fluid Dynamics, pp.350-3553, 1971.
DOI : 10.1007/3-540-05407-3_50

C. W. Hirt and B. Nichols, Volume of fluid (VOF) method for the dynamics of free boundaries, Journal of Computational Physics, vol.39, issue.1, pp.201-2250021999181901455, 1981.
DOI : 10.1016/0021-9991(81)90145-5

K. Hitti, Direct numerical simulation of complex Representative Volume Elements (RVEs): Generation, Resolution and Homogenization, p.35, 2011.
URL : https://hal.archives-ouvertes.fr/pastel-00667428

. Hysing, Quantitative benchmark computations of two-dimensional bubble dynamics, International Journal for Numerical Methods in Fluids, vol.3, issue.5, pp.1259-1288, 2009.
DOI : 10.1002/nme.1620150502

G. E. Jannoun, Space-Time accurate anisotropic adaptation and stabilized finite element methods for the resolution of unsteady CFD problems, 2014.
URL : https://hal.archives-ouvertes.fr/tel-01146245

S. Karagadde, L. Yuan, N. Shevchenko, S. Eckert, L. et al., 3-D microstructural model of freckle formation validated using in situ experiments, Acta Materialia, vol.79, pp.168-180, 2014.
DOI : 10.1016/j.actamat.2014.07.002

A. Karma and W. Rappel, Phase-field method for computationally efficient modeling of solidification with arbitrary interface kinetics, Physical Review E, vol.33, issue.4, pp.3017-3020, 1996.
DOI : 10.1103/PhysRevA.33.2704

S. Kobayashi, Solute redistribution during solidification with diffusion in solid phase: A theoretical analysis, Journal of Crystal Growth, vol.88, issue.1, pp.87-96, 1988.
DOI : 10.1016/S0022-0248(98)90010-0

F. Kohler, Peritectic solidification of Cu???Sn alloys: Microstructural competition at low speed, Acta Materialia, vol.57, issue.1, 2008.
DOI : 10.1016/j.actamat.2008.08.058

. Koshikawa, Computation of Phase Transformation Paths in Steels by a Combination of the Partial- and Para-equilibrium Thermodynamic Approximations, ISIJ International, vol.54, issue.6, pp.1274-1282, 2014.
DOI : 10.2355/isijinternational.54.1274

URL : https://hal.archives-ouvertes.fr/hal-01025814

M. A. Beckermann, C. Gandin, and C. , A solutal interaction mechanism for the columnarto-equiaxed transition in alloy solidification, Metallurgical and Materials Transactions A, issue.8, pp.34-1657, 2003.

Y. Mesri, H. Digonnet, C. , and T. , Advanced parallel computing in material forming with CIMLib, Revue europ??enne de m??canique num??rique, vol.18, issue.7-8, 2009.
DOI : 10.3166/ejcm.18.669-694

URL : https://hal.archives-ouvertes.fr/hal-00836212

S. Mosbah, Multiple scales modeling of solidification grain structures and segregation in metallic alloys, 2008.
URL : https://hal.archives-ouvertes.fr/tel-00349885

J. Ni and C. Beckermann, A volume-averaged two-phase model for transport phenomena during solidification, Metallurgical Transactions B, vol.21, issue.3, pp.349-361, 1991.
DOI : 10.1007/BF02671947

N. T. Niane, Étude des macroségrégations induites par le retrait et la déformation thermomécanique lors de la solidification de la peau des produits sidérurgiques, pp.102-127, 2004.

S. Onodera and Y. Arakida, Effect of Gravity on the Macro-Segregation of Larger Steel Ingots URL: http://eprints.nmlindia.org, pp.358-368358, 1959.

S. Osher and R. Fedkiw, Signed Distance Functions, Level Set Methods and Dynamic Implicit Surfaces, 2003.
DOI : 10.1007/0-387-22746-6_2

S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, Journal of Computational Physics, vol.79, issue.1, pp.12-490021999188900022, 1988.
DOI : 10.1016/0021-9991(88)90002-2

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.413.5254

. Peng, A PDE-Based Fast Local Level Set Method, Journal of Computational Physics, vol.155, issue.2, 1999.
DOI : 10.1006/jcph.1999.6345

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.23.928

E. J. Pickering, Macrosegregation in Steel Ingots: The Applicability of Modelling and Characterisation Techniques, ISIJ International, vol.53, issue.6, pp.935-949, 2013.
DOI : 10.2355/isijinternational.53.935

D. R. Poirier, Permeability for flow of interdendritic liquid in columnar-dendritic alloys, Metallurgical Transactions B, vol.32, issue.1, pp.245-255, 1987.
DOI : 10.1007/BF02670426

M. Pollock, T. M. Pollock, and W. H. Murphy, The breakdown of single-crystal solidification in high refractory nickel-base alloys, Metallurgical and Materials Transactions A, pp.1081-1094, 1996.
DOI : 10.1098/rspa.1959.0141

. Prescott, Convective Transport Phenomena and Macrosegregation During Solidification of a Binary Metal Alloy: II???Experiments and Comparisons With Numerical Predictions, Journal of Heat Transfer, vol.116, issue.3, pp.742-749, 1994.
DOI : 10.1115/1.2910930

A. Prosperetti, Navier-Stokes Numerical Algorithms for Free-Surface Flow Computations: An Overview, 2002.
DOI : 10.1007/978-3-7091-2594-6_8

J. C. Ramirez and C. Beckermann, Evaluation of a rayleigh-number-based freckle criterion for Pb-Sn alloys and Ni-base superalloys, Metallurgical and Materials Transactions A, vol.5, issue.7, pp.1525-1536, 2003.
DOI : 10.1007/BF02645803

M. Bellet, M. Deville, and M. , Numerical Modeling in Materials Science and Engineering, pp.6-41, 2003.
URL : https://hal.archives-ouvertes.fr/hal-00510518

H. Resk, L. Delannay, M. Bernacki, T. Coupez, and R. Logé, Adaptive mesh refinement and automatic remeshing in crystal plasticity finite element simulations, Modelling and Simulation in Materials Science and Engineering, pp.750120965-0393, 2009.
DOI : 10.1088/0965-0393/17/7/075012

URL : https://hal.archives-ouvertes.fr/hal-00509476

B. Rivaux, Simulation 3D éléments finis des macroségrégations en peau induites par déformations thermomécaniques lors de la solidification d'alliages métalliques, pp.637168-637179, 2011.

. Bibliography and . Saad, Temperature-based energy solver coupled with tabulated thermodynamic properties ? Application to the prediction of macrosegregation in multicomponent alloys, 2015.

J. R. Sarazin, A. M. Hellawell, and . Worster, Studies of Channel-Plume Convection during Solidification " . Interactive Dynamics of Convection and Solidification, 1992.

M. C. Gu, J. P. Beckermann, C. Boettinger, W. J. Kattner, and U. R. , Modeling of microand macrosegregation and freckle formation in single-crystal nickel-base superalloy directional solidification, 1997.

J. A. Sethian, A fast marching level set method for monotonically advancing fronts., Proceedings of the National Academy of Sciences, vol.93, issue.4, pp.1591-15951591, 1996.
DOI : 10.1073/pnas.93.4.1591

URL : https://www.ncbi.nlm.nih.gov/pmc/articles/PMC39986/pdf

J. A. Sethian, Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science, 1999.

M. Shah, N. A. Shah, and J. J. Moore, Effect of thermal conditions and alloying constituents (Ni, Cr) on macrosegregation in continuously cast high-carbon (0.8 Pct), low-alloy steel, Metallurgical Transactions B, vol.1, issue.11, pp.893-910, 1989.
DOI : 10.1139/p64-205

. Shakoor, An efficient and parallel level set reinitialization method ??? Application to micromechanics and microstructural evolutions, Applied Mathematical Modelling, vol.39, issue.23-24, 2015.
DOI : 10.1016/j.apm.2015.03.014

URL : https://hal.archives-ouvertes.fr/hal-01139858

. Shevchenko, Chimney Formation in Solidifying Ga-25wt pct In Alloys Under the Influence of Thermosolutal Melt Convection, Metallurgical and Materials Transactions A, vol.50, issue.8, pp.3797-3808, 2013.
DOI : 10.1201/9781439808030

. Strotos, Numerical investigation of the cooling effectiveness of a droplet impinging on a heated surface, International Journal of Heat and Mass Transfer, vol.51, issue.19-20, pp.4728-4742, 2008.
DOI : 10.1016/j.ijheatmasstransfer.2008.02.036

E. Süli, Lecture Notes on Finite Element Methods for Partial Differential Equations, 2000.

C. Beckermann, Diffuse interface modeling of two-phase flows based on averaging: mass and momentum equations, Physica D: Nonlinear Phenomena, vol.198, issue.3, pp.281-308, 2004.

. Sussman, A Level Set Approach for Computing Solutions to Incompressible Two-Phase Flow, Journal of Computational Physics, vol.114, issue.1, pp.146-159, 1994.
DOI : 10.1006/jcph.1994.1155

K. Suzuki and T. Miyamoto, The Mechanism of Formation of the V Segregation in Steel Ingot, Tetsu-to-Hagane, vol.59, issue.3, pp.431-445, 1973.
DOI : 10.2355/tetsutohagane1955.59.3_431

C. R. Swaminathan and V. R. Voller, ON THE ENTHALPY METHOD, International Journal of Numerical Methods for Heat & Fluid Flow, vol.8, issue.3, pp.233-244, 1993.
DOI : 10.1007/BF02667334

L. Tan and N. Zabaras, A level set simulation of dendritic solidification of multi-component alloys, Journal of Computational Physics, vol.221, issue.1, pp.9-40, 2007.
DOI : 10.1016/j.jcp.2006.06.003

. Tezduyar, Incompressible flow computations with stabilized bilinear and linear equal-order-interpolation velocity-pressure elements, Computer Methods in Applied Mechanics and Engineering, vol.95, issue.2, pp.221-242, 1992.
DOI : 10.1016/0045-7825(92)90141-6

T. E. Tezduyar and Y. Osawa, Finite element stabilization parameters computed from element matrices and vectors, Computer Methods in Applied Mechanics and Engineering, vol.190, issue.3-4, pp.411-430, 2000.
DOI : 10.1016/S0045-7825(00)00211-5

L. Thuinet and H. Combeau, Prediction of macrosegregation during the solidification involving a peritectic transformation for multicomponent steels, Journal of Materials Science, vol.39, issue.24, pp.7213-7219, 2004.
DOI : 10.1023/B:JMSC.0000048734.34597.1e

. Tiller, The redistribution of solute atoms during the solidification of metals, Acta Metallurgica, vol.1, issue.4, pp.428-4370001616053901266, 1953.
DOI : 10.1016/0001-6160(53)90126-6

D. Tourret and C. A. Gandin, A generalized segregation model for concurrent dendritic, peritectic and eutectic solidification, Acta Materialia, vol.57, issue.7, pp.2066-2079, 2009.
DOI : 10.1016/j.actamat.2009.01.002

URL : https://hal.archives-ouvertes.fr/hal-00509262

. Tourret, Multiple non-equilibrium phase transformations: Modeling versus electro-magnetic levitation experiment, Acta Materialia, vol.59, issue.11, pp.59-4665, 2011.
DOI : 10.1016/j.actamat.2011.04.013

URL : https://hal.archives-ouvertes.fr/hal-00595250

P. Vigneaux, Méthodes Level Set pour des problèmes d'interface en microfluidique, 2007.

L. Ville, L. Silva, C. , and T. , Convected level set method for the numerical simulation of fluid buckling, International Journal for Numerical Methods in Fluids, vol.4, issue.3, pp.324-344, 2011.
DOI : 10.1142/S0218202594000327

URL : https://hal.archives-ouvertes.fr/hal-00595325

. Voller, The modelling of heat, mass and solute transport in solidification systems, International Journal of Heat and Mass Transfer, vol.32, issue.9, pp.1719-17310017931089900549, 1989.
DOI : 10.1016/0017-9310(89)90054-9

C. Y. Wang and C. Beckermann, A multiphase solute diffusion model for dendritic alloy solidification, Metallurgical Transactions A, vol.21, issue.12, 1993.
DOI : 10.1007/BF02656584

J. I. Wanqi and Z. Yaohe, Formation of hot-top segregation in steel ingot and effect of steel compositions, Metallurgical Transactions B, vol.32, issue.9, pp.723-730, 1989.
DOI : 10.1007/BF02655930

L. Xu, D. Xu, and Q. Li, Gravity-and Solidification-Shrinkage-Induced Liquid Flow in a Horizontally Solidified Alloy Ingot " . Numerical Heat Transfer, Part A: Applications, pp.203-221, 1991.
DOI : 10.1080/10407789108944817

L. Yuan and P. D. Lee, A new mechanism for freckle initiation based on microstructural level simulation, Acta Materialia, vol.60, issue.12, pp.4917-4926, 2012.
DOI : 10.1016/j.actamat.2012.04.043

N. Zabaras and D. Samanta, A stabilized volume-averaging finite element method for flow in porous media and binary alloy solidification processes, International Journal for Numerical Methods in Engineering, vol.60, issue.6, pp.1103-1138, 2004.
DOI : 10.1002/nme.998

H. Zhang, C. Gandin, B. Hamouda, H. Tourret, D. Keiji et al., Prediction of Solidification Paths for Fe???C???Cr Alloys by a Multiphase Segregation Model Coupled to Thermodynamic Equilibrium Calculations, ISIJ International, vol.50, issue.12, pp.1859-1866, 2010.
DOI : 10.2355/isijinternational.50.1859

URL : https://hal.archives-ouvertes.fr/hal-00570414