task since the transport of multiple LS functions has to be
handled and the normal stress over the grain boundary has
to be computed in a more accurate way. Coupling between
those three main diffusion mechanisms (surface, volume and
grain boundary) could lead to comparisons with experimen-
tal data and calibrated powder compact sintering. Moreover,
the microstructural evolution of the powder compact could
be embedded into macroscopic models.

Additionally, this numerical tool also allows to deal with
the sintering of multi-materials or the study of the sinter-
ing of doped powders. In fact, all the diffusion mecha-
nisms are numerically controlled by the value of the mater-
ial properties—i.e. diffusion coefficients, mechanical prop-
erties, surface tension coefficients—therefore it would be
possible to evaluate different material properties from some
physical considerations to represent the multi-materials sin-
tering or the sintering of doped powders. That corresponds
to a work in progress.

References

1. Adalsteinsson D, Sethian JA (1995) A fast level set method for propagating interfaces. J Comput Phys 118(2):269-277

2. Ashby MF (1974) First report on sintering diagrams. Acta Metalurgica 22(3):275-289

3. Aulisa E, Manservisi S, Scardovelli R, Zaleski S (2007) Interface reconstruction with least-squares fit and split advection in three-dimensional cartesian geometry. J Comput Phys 225(2):2301-2319

4. Ausas RF, Sousa FS, Buscaglia GC (2010) An improved finite element space for discontinuous pressures. Comput Methods Appl Mech Eng 199(17-20):1019-1031

5. Badia S, Codina R (2009) Unified stabilized finite element formulations for the stokes and the darcy problems. SIAM J Numer Anal 47(3):1971-2000

6. Badia S, Codina R (2010) Stabilized continuous and discontinuous galerkin techniques for darcy flow. Comput Methods Appl Mech Eng 199(25-28):1654-1667

7. Bordère S, Vincent S, Caltagirone JP (2010) Stochastic energetic approach devoted to the modeling of static two-phase problems dominated by surface tension. Comput Fluids 39(3):392-402

8. Brackbill JU, Kothe DB, Zemach C (1992) A continuum method for modeling surface tension. J Comput Phys 100(2):335-354

9. Braginsky M, Tikare V, Olevsky E (2005) Numerical simulation of solid state sintering. Int J Solids Struct 42(2):621-636

10. Brezzi F, Fortin M (1991) Mixed and hybrid finite elements methods. Springer-Verlag, New York

11. Brooks AN, Hughes TJR (1982) Streamline upwind/petrov-galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier-stokes equations. Comput Methods Appl Mech Eng 32(1):199-259

12. Bross P, Exner HE (1979) Computer simulation of sintering processes. Acta Metallurgica 27(6):1013-1020

13. Bruchon J, Drapier S, Valdivieso F (2011) 3d finite element simulation of the matter flow by surface diffusion using a level set method. Int J Numer Methods Eng 86(7):845-861

14. Bruchon J, Pino-Muñoz D, Valdivieso F, Drapier S (2012) Finite element simulation of mass transport during sintering of a granular packing. Part i: surface and lattice diffusion. J Am Ceram Soc 95(8):2398-2405

15. Bruchon J, Pino-Muñoz D, Valdivieso F, Drapier S, Pacquaut G et al (2010) 3d simulation of the matter transport by surface diffusion within a level-set context. Eur J Comput Mech 19:281-292

16. Burger Martin, Hauéer Frank, Stöcker Christina, Voigt Axel (2007) A level set approach to anisotropic flows with curvature regularization. J Comput Phys 225(1):183-205

17. Cervera M, Chiumenti M, Codina R (2010) Mixed stabilized finite element methods in nonlinear solid mechanics: Part i: formulation. Comput Methods Appl Mech Eng 199(37-40):2559-2570

18. Chen IW, Hassold GN, Srolovitz DJ (1990) Computer simulation of final-stage sintering. II. Influence of initial pore size. J Am Ceram Soc 73(10):2865-2872

19. Chen LQ (2002) Phase-field models for microstructure evolution. Ann Rev Mater Res 32(1):113-140

20. Ch'ng HN, Pan J (2004) Cubic spline elements for modelling microstructural evolution of materials controlled by solid-state diffusion and grain-boundary migration. J Comput Phys 196(2):724-750

21. Ch'ng HN, Pan J (2005) Modelling microstructural evolution of porous polycrystalline materials and a numerical study of anisotropic sintering. J Comput Phys 204(2):430-461

22. T. Coupez (2006) Réinitialisation convective et locale des fonctions level set pour le mouvement de surfaces et d'interfaces. Journées Activités Universitaires de Mécanique-La Rochelle, 31 août et 1er septembre 2006

23. Coyajee ERA (2007) A front-capturing method for the numerical simulation of dispersed two-phase flow. Ph.D. Thesis, Delft University of Technology

24. Daly BJ, Pracht WE (1968) Numerical study of density-current surges. Phys Fluids 11:15

25. Digonnet H, Silva L, Coupez T (2007) Cimlib: a fully parallel application for numerical simulations based on components assembly. In: Materials processing and design: modeling, simulation and applications, vol. 908, pp 269-274. American Institute of Physics, 2 Huntington Quadrangle, Suite 1 NO 1, Melville, NY, 11747-14502

26. Djohari H, Derby JJ (2009) Transport mechanisms and densification during sintering. II. Grain boundaries. Chem Eng Sci 64(17):3810-3816

27. Djohari H, Martínez-Herrera JI, Derby JJ (2009) Transport mechanisms and densification during sintering. I. Viscous flow versus vacancy diffusion. Chem Eng Sci 64(17):3799-3809

28. Exner HE, Arzt E, Cahn RW, Haasen P (1996) Sintering processes. Physical metallurgy, 4th edn. North-Holland, Oxford, pp 2627-2662

29. Fortin A, Béliveau A, Demay Y (1998) A two-dimensional numerical method for the deformation of drops with surface tension. Int J Numer Methods Fluids 10(3):225-240

30. Garikipati K, Bassman L, Deal M (2001) A lattice-based micromechanical continuum formulation for stress-driven mass transport in polycrystalline solids. J Mech Phys Solids 49(6):1209-1237

31. Gerlach D, Tomar G, Biswas G, Durst F (2006) Comparison of volume-of-fluid methods for surface tension-dominant two-phase flows. Int J Heat Mass Transf 49(3-4):740-754

32. German RM, Lathrop JF (1978) Simulation of spherical powder sintering by surface diffusion. J Mater Sci 13(5):921-929

33. German RM (1996) Sintering theory and practice. Wiley, New York

34. Geuzaine C, Remacle J-F (2009) Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. Int J Numer Methods Eng 79(11):1309-1331

35. Gibbs JW (1928) The collected works of J. Willard Gibbs, vol 1. Longmans, Green, New York

36. Groß S, Reusken A (2007) An extended pressure finite element space for two-phase incompressible flows with surface tension. J Comput Phys 224(1):40-58
37. Harlow FH, Welch JE et al (1965) Numerical calculation of time-
dependent viscous incompressible flow of fluid with free surface.
Phys Fluids 8(12):2182

38. Hassold GN, Chen IW, Srolovitz DJ (1990) Computer simulation of final-stage sintering: I, model kinetics, and microstructure. J Am Ceram Soc 73(10):2857-2864

39. Herring C (1951) Surface tension as a motivation for sintering.
Phys Powder Metall 27(2):143-179

40. Hirt CW, Nichols BD (1981) Volume of fluid (VOF) method for the dynamics of free boundaries. J Comput Phys 39(1):201-225

41. Hitti K (2011) Direct numerical simulation of complex representa-
tive volume elements (RVEs): generation, resolution and homog-
enization. Ph.D. thesis, Ecole Nationale Supérieure des Mines de
Paris

42. Hoge CE, Pask JA (1977) Thermodynamic and geometric consid-
erations of solid state sintering. Ceramurgia Int 3(3):95-99

43. Holm EA, Glazier JA, Srolovitz DJ, Grest GS (1991) Effects of lattice anisotropy and temperature on domain growth in the two-dimensional potts model. Phys Rev A 43(6):2662

44. Howard RE, Lidiard AB (1964) Matter transport in solids. Rep Prog Phys 27:161

45. Hughes TJR (1987) Recent progress in the development and under-
standing of supg methods with special reference to the compress-
ible euler and navier-stokes equations. Int J Numer Methods Fluids
7(11):1261-1275

46. Hughes TJR, Sangalli G (2007) Variational multiscale analysis: the fine-scale Green's function, projection, optimization, localization, and stabilized methods. SIAM J Numer Anal 45(2):539-557

47. Thomas JR (1995) Hughes. Multiscale phenomena: Green's func-
tions, the dirichlet-to-neumann formulation, subgrid scale models,
bubbles and the origins of stabilized methods. Comput Methods
Appl Mech Eng 127(1-4):387-401

48. Hughes TJR, Feijóo GR, Mazzei L, Quincy J-B (1998) The varia-
tional multiscale method-a paradigm for computational mechanics.
Comput Methods Appl Mech Eng 166(1-2):3-24

49. Hysing S (2006) A new implicit surface tension implementation for interfacial flows. Int J Numer Methods Fluids 51(6):659-672

50. Hysing S (2012) Mixed element FEM level set method for numer-
ical simulation of immiscible fluids. J Comput Phys 231(6):2449-
2465. doi:10.1016/j.jcp.2011.11.035

51. Job G, Herrmann F (2006) Chemical potential: a quantity in search of recognition. Eur J Phys 27:353

52. Kucherenko S, Pan J, Yeomans JA (2000) A combined finite ele-
ment and finite difference scheme for computer simulation of
microstructure evolution and its application to pore-boundary sep-
aration during sintering. Comput Mater Sci 18(1):76-92

53. Kuczynski GC (1949) Self-diffusion in sintering of metallic parti-
cles. Trans AIME 185:169-178

54. Lafaurie B, Nardone C, Scardovelli R, Zaleski S, Zanetti G (1994) Modelling merging and fragmentation in multiphase flows with SURFER. J Comput Phys 113(1):134-147

55. Martin CL, Bouvard D, Shima S (2003) Study of particle rearrange-
ment during powder compaction by discrete element. J Mech Phys
Solids 51(4):667-693

56. Merriman B, Bence JK, Osher S (1994) Motion of multiple junc-
tions: a level set approach. J Comput Phys 112(2):334-363

57. Mesri Y, Zerguine W, Digonnet H, Silva L, Coupez T (2008) Dynamic parallel adaption for three dimensional unstructured meshes: Application to interface tracking. In: Proceedings of the 17th International Meshing Roundtable, Springer, pp 195-212

58. Nichols FA, Mullins WW (1965) Morphological changes of a sur-
face of revolution due to capillarity-induced surface diffusion. J
Appl Phys 36:1826-1835

59. Nichols FA, Mullins WW (1965) Morphological changes of a sur-
face of revolution due to capillarity induced surface diffusion. J
Appl Phys 36(6):1826-1835

60. Noh W, Woodward P (1976) Slic (simple line interface calculation). In: Proceedings of the Fifth International Conference on Numerical Methods in Fluid Dynamics June 28-July 2, 1976 Twente University, Enschede, Springer, pp 330-340

61. Osher S, Fedkiw RP (2001) Level set methods: an overview and some recent results. J Comput Phys 169(2):463-502

62. Osher S, Sethian JA (1988) Fronts propagating with curvature-
dependent speed: algorithms based on Hamilton-Jacobi formula-
tions. J Comput Phys 79(1):12-49

63. Pan J, Ch'ng HN, Cocks ACF (2005) Sintering kinetics of large pores. Mech Mater 37(6):705-721

64. Pan J, Cocks ACF (1995) A numerical technique for the analysis of coupled surface and grain-boundary diffusion. Acta Metallurgica et Materialia 43(4):1395-1406

65. Pan J, Cocks ACF, Kucherenko S (1997) Finite element formula-
tion of coupled grain-boundary and surface diffusion with grain-
boundary migration. In: Proceedings of the Royal Society of Lon-
don. Series A: Mathematical, Physical and Engineering Sciences,
453(1965):2161-2184

66. Pan J, Le H, Kucherenko S, Yeomans JA (1998) A model for the sintering of spherical particles of different sizes by solid state diffusion. Acta Materialia 46(13):4671-4690

67. Papadakis G (2008) A novel pressure-velocity formulation and solution method for fluid-structure interaction problems. J Comput Phys 227(6):3383-3404

68. Peng D, Merriman B, Osher S, Zhao H, Kang M (1999) A pde-
based fast local level set method. J Comput Phys 155(2):410-438

69. Pino-Muñoz D (2012) High-performance computing of sintering process at particle scale. Ph.D. Thesis, École Nationale Supérieure des Mines de Saint-Étienne

70. Pino Muñoz D, Bruchon J, Drapier S, Valdivieso F (2013) A finite element-based level set method for fluid-elastic solid interaction with surface tension. Int J Numer Methods Eng 93(9):919-941

71. Popinet S, Zaleski S (1999) A front-tracking algorithm for accu-
rate representation of surface tension. Int J Numer Methods Fluids
30(6):775-793

72. Qiu F, Egerton TA, Cooper IL (2008) Monte carlo simulation of nano-particle sintering. Powder Technol 182(1):42-50

73. Rahama MN (1995) Ceramic processing and sintering. Marcel Dekker, New York

74. Renardy Y, Renardy M (2002) PROST: a parabolic reconstruction of surface tension for the volume-of-fluid method. J Comput Phys 183(2):400-421

75. Rider WJ, Kothe DB (1998) Reconstructing volume tracking. J Comput Phys 141(2):112-152

76. Riedel H, Zipse H, Svoboda J (1994) Equilibrium pore surfaces, sintering stresses and constitutive equations for the intermediate and late stages of sintering-ii. diffusional densification and creep. Acta Metallurgica et Materialia 42(2):445-452

77. Rudman M (1997) Volume-tracking methods for interfacial flow calculations. Int J Numer Methods Fluids 24(7):671-691

78. Shin S, Juric D (2002) Modeling three-dimensional multiphase flow using a level contour reconstruction method for front tracking without connectivity. J Comput Phys 180(2):427-470

79. Silva L, Puaux G, Vincent M, Laure P (2010) A monolithic finite element approach to compute permeabilityatc microscopic scales in LCM processes. Int J Mater Form 3(1):619-622

80. Smith K, Solis F, Chopp D (2002) A projection method for motion of triple junctions by level sets. Interfaces Free Boundaries 4(3):263-276

81. Smolianski A (2005) Finite-element/level-set/operator-splitting (FELSOS) approach for computing two-fluid unsteady flows with free moving interfaces. Int J Numer Methods Fluids 48(3):231-269

82. Sussman M, Smereka P, Osher S (1994) A level set approach for computing solutions to incompressible two-phase flow. J Comput Phys 114(1):146-159
83. Tikare V, Braginsky M, Bouvard D, Vagnon A (2010) Numerical simulation of microstructural evolution during sintering at the mesoscale in a 3D powder compact. Comput Mater Sci 48(2):317-325

84. Tikare V, Braginsky M, Olevsky EA (2003) Numerical simulation of solid-state sintering: I, sintering of three particles. J Am Ceram Soc 86(1):49-53

85. Uskoković DP, Exner HE (1977) Kinetics of contact formation during sintering by diffusion mechanisms. Sci Sinter 9(3):265-303

86. Ville L, Silva L, Coupez T (2011) Convected level set method for the numerical simulation of fluid buckling. Int J Numer Methods Fluids 66(3):324-344

87. Vincent S, Caltagirone JP, Lubin P, Randrianarivelo TN (2004) An adaptative augmented lagrangian method for three-dimensional multimaterial flows. Comput Fluids 33(10):1273-1289

88. Wakai F, Brakke KA (2011) Mechanics of sintering for coupled grain boundary and surface diffusion. Acta Materialia 59(14):5379-5387

89. Wang YU (2006) Computer modeling and simulation of solid-state sintering: a phase field approach. Acta Materialia 54(4):953-961

90. Weaire D, Kermode JP, Wejchert J (1986) On the distribution of cell areas in a Voronoi network. Philos Mag B 53(5):101-105

91. Williams MW, Kothe DB, Puckett EG (1998) Accuracy and convergence of continuum surface tension models: fluid dynamics at interfaces. Cambridge University Press, Cambridge

92. Wu FY (1983) Erratum: the potts model. Rev Mod Phys 55(1):315

93. Youngs DL (1982) Time-dependent multi-material flow with large fluid distortion. Numer Methods Fluid Dyn 24:273-285