Efficient algorithm for solving the system of Allen-Cahn and Cahn-Hilliard equations: modeling the sintering process





sintering, phase-field, Cahn-Hilliard equation, Allen-Cahn equation


In this work, we present an algorithm for solving the system of Allen–Cahn and Cahn–Hilliard equations, which describes the process of sintering. The algorithm does not require significant computational resources and makes possible the sintering simulating of a large number of grains using a computation node with an Intel Xeon E5 2697 v3 CPU and an NVIDIA K40 GPU in a reasonable time. Experiments were carried out to simulate the sintering of sorbent-like structures (packings of spherical particles), for which the efficiency of the algorithm was shown.

Author Biographies

Dmitry I. Prokhorov

Yaroslav V. Bazaikin

Vadim V. Lisitsa


  1. H. Tanaka, “Sintering of Silicon Carbide and Theory of Sintering,” J. Ceramic Soc. JAPAN 110 (1286), 877-883 (2002).
    doi 10.2109/jcersj.110.877
  2. J. Poetschke, V. Richter, T. Gestrich, and A. Michaelis, “Grain Growth during Sintering of Tungsten Carbide Ceramics,” Int. J. Refract. Met. Hard Mater. 43, 309-316 (2014).
    doi 10.1016/j.ijrmhm.2014.01.001
  3. N. Florin and P. Fennell, “Synthetic CaO-based Sorbent for CO_2 Capture,” Energy Procedia 4, 830-838 (2011).
    doi 10.1016/j.egypro.2011.01.126
  4. Ya. V. Bazaikin, E. G. Malkovich, V. S. Derevschikov, et al., “Evolution of Sorptive and Textural Properties of CaO-based Sorbents during Repetitive Sorption/Regeneration Cycles,” Chem. Eng. Sci. 152, 709-716 (2016).
    doi 10.1016/j.ces.2016.06.064
  5. R. K. Bordia, S.-J. L. Kang, and E. A. Olevsky, “Current Understanding and Future Research Directions at the Onset of the Next Century of Sintering Science and Technology,” J. Am. Ceram. Soc. 100 (6), 2314-2352 (2017).
    doi 10.1111/jace.14919
  6. R. E. White, “An Enthalpy Formulation of the Stefan Problem,” SIAM J. Numer. Anal. 19 (6), 1129-1157 (1982).
    https://www.jstor.org/stable/2157200 . Cited April 10, 2022.
  7. A. W. Date, “A Strong Enthalpy Formulation for the Stefan Problem,” Int. J. Heat Mass Transf. 34 (9), 2231-2235 (1991).
    doi 10.1016/0017-9310(91)90049-K
  8. D. Tarwidi and S. R. Pudjaprasetya, “Godunov Method for Stefan Problems with Enthalpy Formulations,” East Asian J. Appl. Math. 3 (2), 107-119 (2013).
    doi 10.4208/eajam.030513.200513a
  9. S. Molins, D. Trebotich, C. Steefel, and C. Shen, “An Investigation of the Effect of Pore Scale Flow on Average Geochemical Reaction Rates Using Direct Numerical Simulation,” Water Resour. Res. 48 (3) (2012).
    doi 10.1029/2011WR011404
  10. S. Molins, D. Trebotich, L. Yang, et al., “Pore-Scale Controls on Calcite Dissolution Rates from Flow-through Laboratory and Numerical Experiments,” Environ. Sci. Technol. 48 (13), 7453-7460 (2014).
    doi 10.1021/es5013438
  11. C. I. Steefel and A. C. Lasaga, “A Coupled Model for Transport of Multiple Chemical Species and Kinetic Precipitation/Dissolution Reactions with Application to Reactive Flow in Single Phase Hydrothermal Systems,” Am. J. Sci. 294 (5), 529-592 (1994).
    doi 10.2475/ajs.294.5.529
  12. D. Trebotich, M. F. Adams, S. Molins, et al., “High-Resolution Simulation of Pore-Scale Reactive Transport Processes Associated with Carbon Sequestration,” Comput. Sci. Eng. 16 (6), 22-31 (2014).
    doi 10.1109/MCSE.2014.77
  13. X. Li, H. Huang, and P. Meakin, “Level Set Simulation of Coupled Advection-Diffusion and Pore Structure Evolution due to Mineral Precipitation in Porous Media,” Water Resour. Res. 44 (12) (2008).
    doi 10.1029/2007WR006742
  14. X. Li, H. Huang, and P. Meakin, “A Three-Dimensional Level Set Simulation of Coupled Reactive Transport and Precipitation/Dissolution,” Int. J. Heat Mass Transf. 53 (13), 2908-2923 (2010).
    doi 10.1016/j.ijheatmasstransfer.2010.01.044
  15. S. Osher and R. P. Fedkiw, “Level Set Methods: An Overview and Some Recent Results,” J. Comput. Phys. 169 (2), 463-502 (2001).
    doi 10.1006/jcph.2000.6636
  16. S. Marella, S. Krishnan, H. Liu, and H. S. Udaykumar, “Sharp Interface Cartesian Grid Method I: An Easily Implemented Technique for 3D Moving Boundary Computations,” J. Comput. Phys. 210 (1), 1-31 (2005).
    doi 10.1016/j.jcp.2005.03.031
  17. R. Mittal and G. Iaccarino, “Immersed Boundary Methods,” Annu. Rev. Fluid Mech. 37 (1), 239-261 (2005).
    doi 10.1146/annurev.fluid.37.061903.175743
  18. C. S. Peskin, “Flow Patterns around Heart Valves: A Numerical Method,” J. Comput. Phys. 10 (2), 252-271 (1972).
    doi 10.1016/0021-9991(72)90065-4
  19. F. Sotiropoulos and X. Yang, “Immersed Boundary Methods for Simulating Fluid-Structure Interaction,” Prog. Aerosp. Sci. 65, 1-21 (2014).
    doi 10.1016/j.paerosci.2013.09.003
  20. Y.-H. Tseng and J. H. Ferziger, “A Ghost-Cell Immersed Boundary Method for Flow in Complex Geometry,” J. Comput. Phys. 192 (2), 593-623 (2003).
    doi 10.1016/j.jcp.2003.07.024
  21. K. A. Gadylshina, T. S. Khachkova, and V. V. Lisitsa, “Numerical Modeling of Chemical Interaction between a Fluid and Rocks,” Vychisl. Metody Program. 20 (4), 457-470 (2019).
    doi 10.26089/NumMet.v20r440
  22. D. Prokhorov, V. Lisitsa, T. Khachkova, et al., “Topology-Based Characterization of Chemically-Induced Pore Space Changes Using Reduction of 3D Digital Images,” J. Comput. Sci. 58 (2022).
    doi 10.1016/j.jocs.2021.101550
  23. D. P. Munoz, J. Bruchon, F. Valdivieso, and S. Drapier, “Solid-State Sintering Simulation: Surface, Volume and Grain-Boundary Diffusions,” in Proc. ECCOMAS 2012: European Congress on Computational Methods in Applied Sciences and Engineering, Vienna, Austria, September 10-14, 2012 ,
    https://www.researchgate.net/publication/235673306_Solid-state_sintering_simulation_Surface_volume_and_grain-boundary_diffusions . Cited April 10, 2022.
  24. P. Smereka, “Semi-Implicit Level Set Methods for Curvature and Surface Diffusion Motion,” J. Sci. Comput. 19 (1-3), 439-456 (2003).
    doi 10.1023/A: 1025324613450
  25. N. Moelans, B. Blanpain, and P. Wollants, “An Introduction to Phase-Field Modeling of Microstructure Evolution,” Calphad 32 (2), 268-294 (2008).
    doi 10.1016/j.calphad.2007.11.003
  26. J. W. Cahn and J. E. Hilliard, “Free Energy of a Nonuniform System. I. Interfacial Free Energy,” J. Chem. Phys. 28 (2), 258-267 (1958).
    doi 10.1063/1.1744102
  27. S. M. Allen and J. W. Cahn, “Ground State Structures in Ordered Binary Alloys with Second Neighbor Interactions,” Acta Metall. 20 (3), 423-433 (1972).
    doi 10.1016/0001-6160(72)90037-5
  28. Yu. U. Wang, “Computer Modeling and Simulation of Solid-State Sintering: A Phase Field Approach,” Acta Mater. 54 (4), 953-961 (2006).
    doi 10.1016/j.actamat.2005.10.032
  29. S. G. Kim, D. I. Kim, W. T. Kim, and Y. B. Park, “Computer Simulations of Two-Dimensional and Three-Dimensional Ideal Grain Growth,” Phys. Rev. E 74 (2006).
    doi 10.1103/PhysRevE.74.061605
  30. J. Hötzer, M. Jainta, P. Steinmetz, et al., “Large Scale Phase-Field Simulations of Directional Ternary Eutectic Solidification,” Acta Mater. 93, 194-204 (2015).
    doi 10.1016/j.actamat.2015.03.051
  31. R. Zhang, Z. Chen, W. Fang, and X. Qu, “Thermodynamic Consistent Phase Field Model for Sintering Process with Multiphase Powders,” Trans. Nonferrous Met. Soc. China 24 (3), 783-789 (2014).
    doi 10.1016/S1003-6326(14)63126-5
  32. J. Hötzer, M. Seiz, M. Kellner, et al., “Phase-Field Simulation of Solid State Sintering,” Acta Mater. 164, 184-195 (2019).
    doi 10.1016/j.actamat.2018.10.021
  33. N. Moelans, F. Wendler, and B. Nestler, “Comparative Study of Two Phase-Field Models for Grain Growth,” Comput. Mater. Sci. 46 (2), 479-490 (2009).
    doi 10.1016/j.commatsci.2009.03.037
  34. J. W. Cahn, “On Spinodal Decomposition,” Acta Metall. 9 (9), 795-801 (1961).
    doi 10.1016/0001-6160(61)90182-1
  35. A. Fick, “Ueber Diffusion,” Ann. Phys. 170 (1), 59-86 (1855).
    doi 10.1002/andp.18551700105
  36. Ya. E. Geguzin, Physics of Sintering (Nauka, Moscow, 1984) [in Russian].
  37. K. Ahmed, J. Pakarinen, T. Allen, and A. El-Azab, “Phase Field Simulation of Grain Growth in Porous Uranium Dioxide,” J. Nucl. Mater. 446 (1-3), 90-99 (2014).
    doi 10.1016/j.jnucmat.2013.11.036
  38. M. A. Spears and A. G. Evans, “Microstructure Development during Final/Intermediate Stage Sintering—II. Grain and Pore Coarsening,” Acta Metall. 30 (7), 1281-1289 (1982).
    doi 10.1016/0001-6160(82)90146-8
  39. C. Shen, Q. Chen, Y. H. Wen, et al., “Increasing Length Scale of Quantitative Phase Field Modeling of Growth-Dominant or Coarsening-Dominant Process,” Scr. Mater. 50 (7), 1023-1028 (2004).
    doi 10.1016/j.scriptamat.2003.12.029
  40. R. M. German, Sintering Theory and Practice (Wiley, New York, 1996).
  41. Ya. V. Bazaikin, E. G. Malkovich, D. I. Prokhorov, and V. S. Derevschikov, “Detailed Modeling of Sorptive and Textural Properties of CaO-Based Sorbents with Various Porous Structures,” Sep. Purif. Technol. 255 (2021).
    doi 10.1016/j.seppur.2020.117746
  42. B. D. Lubachevsky and F. H. Stillinger, “Geometric Properties of Random Disk Packings,” J. Stat. Phys. 60, 561-583 (1990).
    doi 10.1007/BF01025983
  43. Y.-S. Liu, J. Yi, H. Zhang, et al., “Surface Area Estimation of Digitized 3D Objects Using Quasi-Monte Carlo Methods,” Pattern Recognit. 43 (11), 3900-3909 (2010).
    doi 10.1016/j.patcog.2010.06.002
  44. V. S. Derevschikov, J. V. Veselovskaya, T. Yu. Kardash, et al., “Direct CO_2 Capture from Ambient Air Using K_2CO_3/Y_2O_3 Composite Sorbent,” Fuel 127, 212-218 (2014)
    doi 10.1016/j.fuel.2013.09.060



How to Cite

Прохоров Д. И., Базайкин Я. В., Лисица В. В. Efficient Algorithm for Solving the System of Allen-Cahn and Cahn-Hilliard Equations: Modeling the Sintering Process // Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie). 2022. 23. 75-94. doi 10.26089/NumMet.v23r206



Methods and algorithms of computational mathematics and their applications