An Algorithm for Tensor Product Approximation of Three-Dimensional Material Data for Implicit Dynamics Simulations

  • Krzysztof Podsiadło
  • Marcin Łoś
  • Leszek SiwikEmail author
  • Maciej Woźniak
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10861)


In the paper, a heuristic algorithm for tensor product approximation with B-spline basis functions of three-dimensional material data is presented. The algorithm has an application as a preconditioner for implicit dynamics simulations of a non-linear flow in heterogeneous media using alternating directions method. As the simulation use-case, a non-stationary problem of liquid fossil fuels exploration with hydraulic fracturing is considered. Presented algorithm allows to approximate the permeability coefficient function as a tensor product what in turn allows for implicit simulations of the Laplacian term in the partial differential equation. In the consequence the number of time steps of the non-stationary problem can be reduced, while the numerical accuracy is preserved.



This work was supported by National Science Centre, Poland, grant no. 2014/15/N/ST6/04662. The authors would like to acknowledge prof. Maciej Paszyński for his help in this research topic and preparation of this paper.


  1. 1.
    Łoś, M., Woźniak, M., Paszyński, M., Dalcin, L., Calo, V.M.: Dynamics with matrices possessing kronecker product structure. Proc. Comput. Sci. 51, 286–295 (2015). Scholar
  2. 2.
    Łoś, M., Woźniak, M., Paszyński, M., Lenharth, A., Amber-Hassan, M., Pingali, K.: IGA-ADS: isogeometric analysis FEM using ADS solver. Comput. Phys. Commun. 217, 99–116 (2017). Scholar
  3. 3.
    Woźniak, M., Łoś, M., Paszyński, M., Dalcin, L., Calo, V.M.: Parallel fast isogeometric solvers for explicit dynamics. Comput. Inf. 36(2), 423–448 (2017). Scholar
  4. 4.
    Siwik, L., Łoś, M., Kisiel-Dorohinicki, M., Byrski, A.: Hybridization of isogeometric finite element method and evolutionary mulit-agent system as a tool-set for multi-objective optimization of liquid fossil fuel exploitation with minimizing groundwater contamination. Proc. Comput. Sci. 80, 792–803 (2016). Scholar
  5. 5.
    Łoś, M.: Fast isogeometric L2 projection solver for non-linear flow in non-homogenous media, Master Thesis, AGH University, Krakow, Poland (2015)Google Scholar
  6. 6.
    Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng. 194(39), 4135–4195 (2005). Scholar
  7. 7.
    Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y.: Isogeometric Analysis: Toward Unfication of CAD and FEA. Wiley, New York (2009). The Attrium, Southern Gate, Chichester, West SussexCrossRefGoogle Scholar
  8. 8.
    Courant, R., Friedrichs, K., Lewy, H.: On the partial difference equations of mathematical physics. In: AEC Research and Development Report, NYO-7689. AEC Computing and Applied Mathematics Centre-Courant Institute of Mathematical Sciences, New York (1956)Google Scholar
  9. 9.
    Paszyński M, Łoś, M., Calo, V.M.: Fast isogeometric solvers for implicit dynamics. Comput. Math. Appl. (2017, submitted to)Google Scholar
  10. 10.
    Alotaibi, M., Calo, V.M., Efendiev, Y., Galvis, J., Ghommem, M.: Global-local nonlinear model reduction for flows in heterogeneous porous media. Comput. Methods Appl. Mech. Eng. 292, 122–137 (2015). Scholar
  11. 11.
    Efendiev, Y., Ginting, V., Hou, T.: Multiscale finite element methods for nonlinear problems and their applications. Commun. Math. Sci. 2(4), 553–589 (2004). Scholar
  12. 12.
    Warrick, A.W.: Time-dependent linearized in filtration: III. Strip and disc sources. Soil Sci. Soc. Am. J. 40, 639–643 (1976)CrossRefGoogle Scholar
  13. 13.
    Nebro, A.J., Durillo, J.J., Vergne, M.: Redesigning the jMetal Multi-objective optimization framework. In: Proceedings of the Companion Publication of the 2015 Annual Conference on Genetic and Evolutionary Computation, GECCO Companion 2015 (2015)Google Scholar
  14. 14.
    Siwik, L., Los, M., Kisiel-Dorohinicki, M., Byrski, A.: Evolutionary multiobjective optimization of liquid fossil fuel reserves exploitation with minimizing natural environment contamination. In: Rutkowski, L., Korytkowski, M., Scherer, R., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds.) ICAISC 2016. LNCS (LNAI), vol. 9693, pp. 384–394. Springer, Cham (2016). Scholar
  15. 15.
    Goreinov, S.A., Tyrtyshnikov, E.E., Zamarashkin, N.L.: A theory of pseudoskeleton approximations. Linear Algebra Appl. 261(1–3), 1–21 (1997). Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Krzysztof Podsiadło
    • 1
  • Marcin Łoś
    • 1
  • Leszek Siwik
    • 1
    Email author
  • Maciej Woźniak
    • 1
  1. 1.AGH University of Science and TechnologyKrakowPoland

Personalised recommendations