Parallel Solvers for Numerical Upscaling

  • R. Blaheta
  • O. Jakl
  • J. Starý
  • Erhan Turan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7782)


This contribution deals with numerical upscaling of the elastic material behaviour, namely of geocomposites, from microscale to macroscale through finite element analysis. This computationally demanding task raises many algorithmic and implementation issues related to efficient parallel processing. On the solution of the arising boundary value problem, considered with either Dirichlet or Neumann boundary conditions, we discuss various parallelization strategies, and compare their implementations in the specialized in-house finite element package GEM and through the general numerical solution framework Trilinos.


upscaling in elasticity large scale linear systems singular systems iterative solvers aggregation based preconditioners parallel computation Trilinos 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arbenz, P., van Lenthe, G.H., Mennel, U., Müller, R., Sala, M.: A scalable multi-level preconditioner for matrix-free μ-finite element analysis of human bone structures. Int. J. Numer. Methods Eng. 73(7), 927–947 (2008)MATHCrossRefGoogle Scholar
  2. 2.
    Blaheta, R.: A multilevel method with overcorrection by aggregation for solving discrete elliptic problems. J. Comput. Appl. Math. 24, 227–239 (1988)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Blaheta, R.: Displacement Decomposition - Incomplete Factorization Preconditioning Techniques for Linear Elasticity Problems. Numerical Linear Algebra with Applications 1, 107–128 (1994)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Blaheta, R.: GPCG - generalized preconditioned CG method and its use with non-linear and non-symmetric displacement decomposition preconditioners. Numerical Linear Algebra with Applications 9, 527–550 (2002)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Blaheta, R.: Space Decomposition Preconditioners and Parallel Solvers. In: Feistauer, M., Dolejíší, V., Knobloch, P., Najzar, K. (eds.) ENUMATH 2003, pp. 20–38. Springer, Berlin (2004)Google Scholar
  6. 6.
    Blaheta, R., Jakl, O., Kohut, R., Starý, J.: GEM – A Platform for Advanced Mathematical Geosimulations. In: Wyrzykowski, R., Dongarra, J., Karczewski, K., Wasniewski, J. (eds.) PPAM 2009, Part I. LNCS, vol. 6067, pp. 266–275. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  7. 7.
    Blaheta, R., Sokol, V.: Multilevel Solvers with Aggregations for Voxel Based Analysis of Geomaterials. In: Lirkov, I., Margenov, S., Wasniewski, J. (eds.) LSSC 2011. LNCS, vol. 7116, pp. 489–497. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  8. 8.
    Blaheta, R., Hrtus, R., Kohut, R., Axelsson, O., Jakl, O.: Material Parameter Identification with Parallel Processing and Geo-applications. In: Wyrzykowski, R., Dongarra, J., Karczewski, K., Wasniewski, J. (eds.) PPAM 2011, Part I. LNCS, vol. 7203, pp. 366–375. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  9. 9.
    Blaheta, R., Kohut, R., Kolcun, A., Souček, K., Staš, L., Vavro, L.: Digital image based numerical micromechanics of geocomposites with application to chemical grouting (submitted)Google Scholar
  10. 10.
    Bochev, P., Lehoucq, R.B.: On the Finite Element Solution of the Pure Neumann Problem. SIAM Review 47, 50–66 (2005)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Gee, M.W., Seifert, C.M., Hu, J.J., Tuminaro, R.S., Sala, M.G.: ML 5.0 Smoothed Aggregation User’s Guide. Sandia Report SAND2006-2649 (2006)Google Scholar
  12. 12.
    Nečas, J., Hlaváček, I.: Mathematical theory of elastic and elasto-plastic bodies: an introduction. Elsevier, Amsterdam (1981)MATHGoogle Scholar
  13. 13.
    Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia (2003)Google Scholar
  14. 14.
    Vaněk, P., Mandel, J., Brezina, M.: Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems. Computing 56, 179–196 (1996)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Zohdi, T.J., Wriggers, P.: An Introduction to Computational Micromechanics. Springer, Berlin (2005, 2008)MATHGoogle Scholar
  16. 16.
    The Trilinos Project Home Page. Sandia National Laboratories (2012),
  17. 17.
    IFPACK Home. IFPACK Object-Oriented Algebraic Preconditioner Package, Sandia National Laboratories (2012),
  18. 18.
    ML Home. ML: Multi Level Preconditioning Package, Sandia National Laboratories (2012),

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • R. Blaheta
    • 1
  • O. Jakl
    • 1
  • J. Starý
    • 1
  • Erhan Turan
    • 2
  1. 1.IT4Innovations DepartmentInstitute of Geonics AS CROstravaCzech Republic
  2. 2.Department of Computer ScienceETH ZürichZürichSwitzerland

Personalised recommendations