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Sintering at Particle Scale: An Eulerian Computing Framework to Deal with Strong Topological and Material Discontinuities

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Abstract

This work presents a numerical modelling approach of particle packing consolidation, at the particle scale, based on specific numerical methods implemented in a high-performance computing framework. Typically, the sintering process triggers several mass transport paths, thermally activated, that are driven by geometrical as well as physical gradients and laplacians. Computing precisely such major characteristics is of paramount importance but represents a real scientific challenge, which have not been fully solved yet but which must however be tackled to gain precious insights into sintering mechanisms which are seldom accessible at this scale. An Eulerian-based formulation is then proposed here to deal with the strong topological changes related to particle sintering. Also, a specific attention is paid to the precise and robust computation of high-order derivatives which are known to control the physics of surface solid diffusion, namely the surface laplacian of the curvature. Besides, the hydrostatic pressure gradient is known to control the volume diffusion path, it results from the coupled fluid-solid mechanical equilibrium, including surface tension, which must be solved precisely. Furthermore, a mesh adaptation technique allows the particles surface description to be improved, while the number of mesh elements is kept reasonable. Once verified on test-cases, this numerical approach is applied to several 3D granular packings undergoing micro-structural changes under combined surface and volume diffusion.

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Notes

  1. That depend on the amount of material of the system.

  2. Sometimes also known as surface tension coefficient or for the sake of simplicity just surface tension.

  3. The Boltzmann constant (\(k_B\)) present in this equation can be different for every physical phenomenon, therefore it is a parameter that has to be set.

  4. This kind of flux path is considered in order to ensure the mass conservation; if only the diffusion paths were considered, the mass could, in some cases, not be conserved [88, 89].

  5. Also known as characteristic functions.

  6. The specific surface is the ratio between the total free surface of the compact powder and its mass.

  7. An analytical integration can also be carried out.

References

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

  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

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Book  Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  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

    Article  MATH  Google Scholar 

  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

    Article  Google Scholar 

  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

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

  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

    Article  Google Scholar 

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

    Article  Google Scholar 

  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

    Article  MATH  Google Scholar 

  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

    Article  MATH  Google Scholar 

  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

    Article  MATH  Google Scholar 

  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

    Article  Google Scholar 

  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

    Article  Google Scholar 

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

    Chapter  Google Scholar 

  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

    MATH  Google Scholar 

  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

    Article  MATH  Google Scholar 

  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

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

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

    Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

  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

    Article  MATH  Google Scholar 

  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

    Article  Google Scholar 

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

    Google Scholar 

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

    Article  MATH  Google Scholar 

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

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

    Article  Google Scholar 

  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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

  47. Thomas JR (1995) Hughes. Multiscale phenomena: Green’s functions, 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

    MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  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 surface of revolution due to capillarity-induced surface diffusion. J Appl Phys 36:1826–1835

    Article  Google Scholar 

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

    Article  Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  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

    Article  Google Scholar 

  65. Pan J, Cocks ACF, Kucherenko S (1997) Finite element formulation of coupled grain-boundary and surface diffusion with grain-boundary migration. In: Proceedings of the Royal Society of London. 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

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

  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

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

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

    Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  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

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  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

    Article  MATH  Google Scholar 

  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

    Article  Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

  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

    Article  MATH  Google Scholar 

  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

    Article  Google Scholar 

  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

    Article  Google Scholar 

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

    Google Scholar 

  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

    Article  MATH  Google Scholar 

  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

    Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  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

    Article  Google Scholar 

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

    Google Scholar 

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

    Article  Google Scholar 

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

    Google Scholar 

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  1. D. Pino-Muñoz

    Present address: Computational Solid Mechanics Laboratory (LSMS), EPFL, Lausanne, Switzerland

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  1. Lab. G. Friedel, UMR CNRS 5307, École des Mines de Saint-Étienne, Saint-Étienne, France

    D. Pino-Muñoz, J. Bruchon, S. Drapier & F. Valdivieso

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Pino-Muñoz, D., Bruchon, J., Drapier, S. et al. Sintering at Particle Scale: An Eulerian Computing Framework to Deal with Strong Topological and Material Discontinuities. Arch Computat Methods Eng 21, 141–187 (2014). https://doi.org/10.1007/s11831-014-9101-4

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