Cybernetics and Systems Analysis

, Volume 54, Issue 2, pp 284–294 | Cite as

A System Approach to Mathematical and Computer Modeling of Geomigration Processes Using Freefem++ and Parallelization of Computations

  • V. A. Herus
  • N. V. Ivanchuk
  • P. M. Martyniuk


A method is described for constructing mathematical models of interrelated processes in porous media that are complex multicomponent systems. The performance capabilities of the package FreeFem++ are shown as applied to solving corresponding free boundary-value problems for systems of quasilinear parabolic equations by the finite element method using the parallelization of computations.


mathematical and computer modeling multicomponent porous medium chemical and mechanical suffosion filtration consolidation heat-and-mass transfer free movable boundary system of quasilinear parabolic equations finite element method FreeFem++ parallel computing 


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  1. 1.
    V. M. Bulavatsky and V. V. Skopetskii, “Generalized mathematical model of the dynamics of consolidation processes with relaxation,” Cybernetics and Systems Analysis, Vol. 44, No. 5, 646–654 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    V. M. Bulavatsky and V. V. Skopetsky, “On an unconventional mathematical model of geoinformatics,” Journal of Automation and Information Sciences, Vol. 42, No. 10, 16–25 (2010).CrossRefzbMATHGoogle Scholar
  3. 3.
    A. Ya. Bomba, V. M. Bulavatsky, and V. V. Skopetsky, Nonlinear Mathematical Models of Geohydrodynamics Processes [in Ukrainian], Naukova Dumka, Kyiv (2007).Google Scholar
  4. 4.
    A. P. Vlasyuk and P. M. Martinyuk, Mathematical Modeling of Soil Consolidation during Filtration of Saline Solutions under Nonisothermal Conditions [in Ukrainian], NUWEE, Rivne (2008).Google Scholar
  5. 5.
    O. Kolditz, U.-J. Gorke, H. Shao, W. Wang, and S. Bauer, Thermo-Hydro-Mechanical-Chemical Processes in Fractured Porous Media: Modelling and Benchmarking: Benchmarking, Springer, Cham (2016).Google Scholar
  6. 6.
    E. Ballarini, B. Graupner, and S. Bauer, “Thermal-hydraulic-mechanical behavior of bentonite and sand-bentonite materials as seal for a nuclear waste repository: Numerical simulation of column experiments,” Applied Clay Science, Vol. 135, 289–299 (2017).Google Scholar
  7. 7.
    C. Beyer, S. Popp, and S. Bauer, “Simulation of temperature effects on groundwater flow, contaminant dissolution, transport and biodegradation due to shallow geothermal use,” Environmental Earth Sciences, Vol. 75, 12–44 (2016).CrossRefGoogle Scholar
  8. 8.
    H. Bayesteh and A. A. Mirghasemi, “Numerical simulation of pore fluid characteristic effect on the volume change behavior of montmorillonite clays,” Computers and Geotechnics, Vol. 48, 146–155 (2013).CrossRefGoogle Scholar
  9. 9.
    V. A. Jambhekar, R. Helmig, N. Schröder, and N. Shokri, “Free-flow-porous-media coupling for evaporation-driven transport and precipitation of salt in soil,” Transp. Porous Med., Vol. 110, No. 2, 251–280 (2015).CrossRefGoogle Scholar
  10. 10.
    F. Golay and S. Bonelli, “Numerical modeling of suffusion as an interfacial erosion process,” European Journal of Environmental and Civil Engineering, Vol. 15, No. 8, 1225–1241 (2011).CrossRefGoogle Scholar
  11. 11.
    A. Seghir, A. Benamar, and H. Wang, “Effects of fine particles on the suffusion of cohesionless soils. Experiments and modeling,” Transp. Porous Med., Vol. 103, No. 2, 233–247 (2014).CrossRefGoogle Scholar
  12. 12.
    J. K. Mitchell and K. Soga, Fundamentals of Soil Behavior, Wiley, Hoboken (2005).Google Scholar
  13. 13.
    V. A. Herus and P. M. Martyniuk, “Generalization of soil consolidation equation taking into account the influence of physico-chemical factors,” Bulletin of V. Karazin Kharkiv National University, Ser. “Mathematical modeling. Information technology. Automated control systems,” Iss. 27, 41–52 (2015).Google Scholar
  14. 14.
    V. A. Herus, P. M. Martyniuk, and O. R. Michuta, “General kinematic boundary condition in the theory of soil filtration consolidation,” Physico-Mathematical Modeling and Information Technologies, Iss. 22, 23–30 (2015).Google Scholar
  15. 15.
    N. G. Bourago and V. N. Kukudzhanov, “A review of contact algorithms,” Bulletin of RAS, Mechanics of Solids, No. 1, 44–85 (2005).Google Scholar
  16. 16.
    I. V. Sergienko, V. V. Skopetsky, and V. S. Deineka, Mathematical Modeling and Analysis of Processes in Inhomogeneous Media [in Russian], Naukova Dumka, Kyiv (1991).Google Scholar
  17. 17.
    V. M. Trushevsky, G. A. Shynkarenko, and N. M. Shcherbyna, Finite Element Method and Artificial Neural Networks: Theoretical Aspects and Application [in Ukrainian], Ivan Franko National University of Lviv, Lviv (2014).Google Scholar
  18. 18.
    F. Hecht, S. Auliac, O. Pironneau, J. Morice, A. Le Hyaric, and K. Ohtsuka, FreeFem++, Univ. Pierre et Marie Curiex, Paris (2013).Google Scholar
  19. 19.
    I. V. Sergienko, I. N. Molchanov, and A. N. Khimich, “Intelligent technologies of high-performance computing,” Cybernetics and Systems Analysis, Vol. 46, No. 5, 833-844 (2010).CrossRefGoogle Scholar
  20. 20.
    A. Khimich, E. Nikolaevskaya, and T. Chistyakova, Programming with Multiple Precision, Springer, Berlin (2012).Google Scholar
  21. 21.
    V. M. Trushevsky and G. A. Shynkarenko, “Concurrent approximation of elliptic boundary value problems by radial basis function neural network,” Visnyk of the Lviv University, Series Applied Mathematics and Computer Science, Iss. 22, 108–117 (2014).Google Scholar
  22. 22.
    A. Yu. Baranov, M. V. Bilous, I. V. Sergienko, and A. N. Khimich, “Hybrid algorithms to solve linear systems for finite-element modeling of filtration processes,” Cybernetics and Systems Analysis, Vol. 51, No. 4, 594–602 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    I. M. Smith, D. V. Griffiths, and L. Margetts, Programming the Finite Element Method, Wiley, Chichester (2014).zbMATHGoogle Scholar
  24. 24.
    I. S. Duff, A. M. Erisman, and J. K. Reid, Direct Methods for Sparse Matrices, Oxford Univ. Press, Oxford (2017).Google Scholar
  25. 25.
    O. R. Michuta, P. M. Martyniuk, and V. A. Herus, Mathematical Modeling of Processes of Chemical and Contact Suffusions in Soils [in Ukrainian], NUWEE, Rivne (2016).Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • V. A. Herus
    • 1
  • N. V. Ivanchuk
    • 1
  • P. M. Martyniuk
    • 1
  1. 1.National University of Water and Environmental Engineering (NUWEE)RivneUkraine

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