Transport in Porous Media

, Volume 125, Issue 1, pp 59–80 | Cite as

On the Importance of Simulated Annealing Algorithms for Stochastic Reconstruction Constrained by Low-Order Microstructural Descriptors

  • Pavel ČapekEmail author


In the past two decades, simulated annealing has played an important role in stochastic reconstruction of porous media. In this study, we compare simulated annealing algorithms constrained by low-order microstructural descriptors and controlled by four annealing schedules that use different ways of temperature reduction. Besides the plain exponential decay of temperature, three adaptive schedules deducing the optimum cooling speed from statistical measures, such as mean system energy and standard deviation of energy, are investigated. Unlike the first three schedules, which modify temperature by a stepwise manner, the fourth one takes into account the effect of move generation strategies and decreases temperature continuously maintaining quasi-equilibrium. The performance of the algorithms is exemplified by reconstructing pore structures of five porous samples. The fourth annealing schedule can significantly reduce the total time required for the reconstruction process itself and for parameter tuning because the same values of annealing parameters are used for both 2D and 3D reconstruction. None of the schedules affects the quality of the reconstructed pore structures.


Stochastic reconstruction Simulated annealing Annealing schedule Move generation method Microstructural descriptor 



This work was carried out within the Operational Programme Prague—Competitiveness (CZ.2.16/3.1.00/24501) and “National Program of Sustainability” (NPU I LO1613) MSMT-43760/2015. The author is very grateful to Prof. Muhammad Sahimi and Prof. Pejman Tahmasebi for their kind invitation to this special issue.


  1. Aarts, E., Korst, J.: Simulated Annealing and Boltzmann Machines: A Stochastic Approach to Combinatorial Optimization and Neural Computing. Wiley, Chichester (1989)Google Scholar
  2. Biswal, B., Manwart, C., Hilfer, R., Bakke, S., Øren, P.E.: Quantitative analysis of experimental and synthetic microstructures for sedimentary rock. Phys. A 273(3–4), 452–475 (1999)CrossRefGoogle Scholar
  3. Čapek, P., Hejtmánek, V., Brabec, L., Zikánová, A., Kočiřík, M.: Stochastic reconstruction of particulate media using simulated annealing: improving pore connectivity. Transp. Porous Media 76(2), 179–198 (2009)CrossRefGoogle Scholar
  4. Čapek, P., Hejtmánek, V., Kolafa, J., Brabec, L.: Transport properties of stochastically reconstructed porous media with improved pore connectivity. Transp. Porous Media 88(1), 87–106 (2011)CrossRefGoogle Scholar
  5. Čapek, P., Veselý, M., Bernauer, B., Sysel, P., Hejtmánek, V., Kočiřík, M., Brabec, L., Prokopová, O.: Stochastic reconstruction of mixed-matrix membranes and evaluation of effective permeability. Comput. Mater. Sci. 89, 142–156 (2014)CrossRefGoogle Scholar
  6. Davis, M.A., Walsh, S.D.C., Saar, M.O.: Statistically reconstructing continuous isotropic and anisotropic two-phase media while preserving macroscopic material properties. Phys. Rev. E 83(2), 026706 (2011)CrossRefGoogle Scholar
  7. Frost, R., Heineman, P.: Simulated annealing: a heuristic for parallel stochastic optimization. In: Arabnia, H.R. (ed.) Proceedings of the International Conference on Parallel and Distributed Processing Techniques and Applications, PDPTA 1997, June 30–July 3, 1997, pp. 1595–1604. CSREA Press, Las Vegas (1997)Google Scholar
  8. Gao, M., He, X., Teng, Q., Zuo, C., Chen, D.: Reconstruction of three-dimensional porous media from a single two-dimensional image using three-step sampling. Phys. Rev. E 91(1), 013308 (2015)CrossRefGoogle Scholar
  9. Gerke, K.M., Karsanina, M.V.: Improving stochastic reconstructions by weighting correlation functions in an objective function. EPL (Europhys. Lett.) 111(5), 56002 (2015)CrossRefGoogle Scholar
  10. Gerke, K.M., Karsanina, M.V., Vasilyev, R.V., Mallants, D.: Improving pattern reconstruction using directional correlation functions. EPL (Europhys. Lett.) 106(6), 66002 (2014)CrossRefGoogle Scholar
  11. Gommes, C.J., Jiao, Y., Torquato, S.: Microstructural degeneracy associated with a two-point correlation function and its information content. Phys. Rev. E 85(5), 051140 (2012)CrossRefGoogle Scholar
  12. Havelka, J., Kučerová, A., Sýkora, J.: Compression and reconstruction of random microstructures using accelerated lineal path function. Comput. Mater. Sci. 122, 102–117 (2016)CrossRefGoogle Scholar
  13. Jiao, Y., Stillinger, F.H., Torquato, S.: Modeling heterogeneous materials via two-point correlation functions. II. Algorithmic details and applications. Phys. Rev. E 77(3), 031135 (2008)CrossRefGoogle Scholar
  14. Jiao, Y., Stillinger, F.H., Torquato, S.: A superior descriptor of random textures and its predictive capacity. Proc. Natl. Acad. Sci. USA 106(42), 17634–17639 (2009)CrossRefGoogle Scholar
  15. Jiao, Y., Stillinger, F.H., Torquato, S.: Geometrical ambiguity of pair statistics. II. Heterogeneous media. Phys. Rev. E 82(1), 011106 (2010)CrossRefGoogle Scholar
  16. Kikkinides, E.S., Politis, M.G.: Linking pore diffusivity with macropore structure of zeolite adsorbents. Part I: three dimensional structural representation combining scanning electron microscopy with stochastic reconstruction methods. Adsorption 20(1), 5–20 (2014a)CrossRefGoogle Scholar
  17. Kikkinides, E.S., Politis, M.G.: Linking pore diffusivity with macropore structure of zeolite adsorbents. Part II: simulation of pore diffusion and mercury intrusion in stochastically reconstructed zeolite adsorbents. Adsorption 20(1), 21–35 (2014b)CrossRefGoogle Scholar
  18. Lam, J., Delosme, J.M.: An efficient simulated annealing schedule: derivation. Tech. Rep. 8816, Yale University, Yale, New Haven (1988a)Google Scholar
  19. Lam, J., Delosme, J.M.: An efficient simulated annealing schedule: implementation and evaluation. Tech. Rep. 8817, Yale University, Yale, New Haven (1988b)Google Scholar
  20. L’Ecuyer, P.: Good parameters and implementations for combined multiple recursive random number generators. Oper. Res. 47(1), 159–164 (1999)CrossRefGoogle Scholar
  21. Lee, S.B., Torquato, S.: Pair connectedness and mean cluster size for continuum-percolation models: computer-simulation results. J. Chem. Phys. 89(10), 6427–6433 (1988)CrossRefGoogle Scholar
  22. Lin, W., Li, X., Yang, Z., Wang, J., Xiong, S., Luo, Y., Wu, G.: Construction of dual pore 3-D digital cores with a hybrid method combined with physical experiment method and numerical reconstruction method. Transp. Porous Media 120(1), 227–238 (2017)CrossRefGoogle Scholar
  23. Manwart, C., Torquato, S., Hilfer, R.: Stochastic reconstruction of sandstones. Phys. Rev. E 62(1), 893–899 (2000)CrossRefGoogle Scholar
  24. Øren, P.E., Bakke, S.: Reconstruction of Berea sandstone and pore-scale modelling of wettability effects. J. Pet. Sci. Eng. 39(3–4), 177–199 (2003)CrossRefGoogle Scholar
  25. Panneton, F., L’Ecuyer, P., Matsumoto, M.: Improved long-period generators based on linear recurrences modulo 2. ACM Trans. Math. Softw. 32(1), 1–16 (2006)CrossRefGoogle Scholar
  26. Politis, M.G., Kikkinides, E.S., Kainourgiakis, M.E., Stubos, A.K.: A hybrid process-based and stochastic reconstruction method of porous media. Microporous Mesoporous Mater. 110(1), 92–99 (2008)CrossRefGoogle Scholar
  27. Rintoul, M.D., Torquato, S.: Reconstruction of the structure of dispersions. J. Colloid Interface Sci. 186(2), 467–476 (1997)CrossRefGoogle Scholar
  28. Rozman, M.G., Utz, M.: Efficient reconstruction of multiphase morphologies from correlation functions. Phys. Rev. E 63(6), 066701 (2001)CrossRefGoogle Scholar
  29. Sahimi, M.: Flow and Transport in Porous Media and Fractured Rock: From Classical Methods to Modern Approaches, 2nd edn. Wiley-VCH, Weinheim (2011)CrossRefGoogle Scholar
  30. Sheehan, N., Torquato, S.: Generating microstructures with specified correlation functions. J. Appl. Phys. 89(1), 53–60 (2001)CrossRefGoogle Scholar
  31. Tahmasebi, P., Sahimi, M.: Reconstruction of three-dimensional porous media using a single thin section. Phys. Rev. E 85(6), 066709 (2012)CrossRefGoogle Scholar
  32. Tahmasebi, P., Sahimi, M.: Cross-correlation function for accurate reconstruction of heterogeneous media. Phys. Rev. Lett. 110(7), 078002 (2013)CrossRefGoogle Scholar
  33. Tahmasebi, P., Sahimi, M.: Geostatistical simulation and reconstruction of porous media by a cross-correlation function and integration of hard and soft data. Transp. Porous Media 107(3), 871–905 (2015a)CrossRefGoogle Scholar
  34. Tahmasebi, P., Sahimi, M.: Reconstruction of nonstationary disordered materials and media: watershed transform and cross-correlation function. Phys. Rev. E 91, 032401 (2015b)CrossRefGoogle Scholar
  35. Talukdar, M.S., Torsaeter, O., Ioannidis, M.A.: Stochastic reconstruction of particulate media from two-dimensional images. J. Colloid Interface Sci. 248(2), 419–428 (2002a)CrossRefGoogle Scholar
  36. Talukdar, M.S., Torsaeter, O., Ioannidis, M.A., Howard, J.J.: Stochastic reconstruction of chalk from 2D images. Transp. Porous Media 48(1), 101–123 (2002b)CrossRefGoogle Scholar
  37. Torquato, S.: Random Heterogeneous Materials: Microstructure and Macroscopic Properties. Springer, New York (2002)CrossRefGoogle Scholar
  38. Veselý, M., Bultreys, T., Peksa, M., Lang, J., Cnudde, V., Van Hoorebeke, L., Kočiřík, M., Hejtmánek, V., Šolcová, O., Soukup, K., Gerke, K., Stallmach, F., Čapek, P.: Prediction and evaluation of time-dependent effective self-diffusivity of water and other effective transport properties associated with reconstructed porous solids. Transp. Porous Media 110(1), 81–111 (2015)CrossRefGoogle Scholar
  39. Yeong, C.L.Y., Torquato, S.: Reconstructing random media. Phys. Rev. E 58(1), 495–506 (1998a)CrossRefGoogle Scholar
  40. Yeong, C.L.Y., Torquato, S.: Reconstructing random media. II. Three-dimensional media from two-dimensional cuts. Phys. Rev. E 58(1), 224–233 (1998b)CrossRefGoogle Scholar
  41. Zhao, X., Yao, J., Yi, Y.: A new stochastic method of reconstructing porous media. Transp. Porous Media 69(1), 1–11 (2007)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Chemical TechnologyUniversity of Chemistry and Technology, PraguePrague 6Czech Republic

Personalised recommendations