Advertisement

Parallel Inelastic Heterogeneous Multi-Scale Simulations

  • Ramin Shirazi Nejad
  • Christian WienersEmail author
Chapter
Part of the Mathematical Engineering book series (MATHENGIN)

Abstract

We recall the heterogeneous multi-scale method for elasticity and its extension to inelasticity within a two-scale energetic approach, where the fine-scale material properties are evaluated in Representative Volume Elements. These RVEs are located at Gauß points of a coarse finite element mesh. Within this \(\text {FE}^2\) method the displacement is approximated on a coarse-scale, and depending on the strain at the Gauß points in every RVE a periodic micro-fluctuation and the internal variables describing the material history in this RVE are computed. Together, this defines the global energy and the dissipation functional, both depending on coarse-scale displacements as well as on fluctuations and internal variables on the micro-scale. Here we introduce a parallel realisation of this method which allows the computation of 3D micro-structures with fine resolution. It is based on the parallel representation of the RVE with distributed internal variables associated to each Gauß points, and a parallel multigrid solution method in the nonlinear computation of the micro-fluctuations and for the up-scaling of the algorithmic tangent within the incremental loading steps of the macro-problem. The efficiency of the method is demonstrated for a simple damage model combined with elasto-plasticity describing a PBT matrix material with glass fibre inclusions. For this investigation we use the material models in J. Spahn (Ph.D. thesis Kaiserslautern 2015) and the software developed by R. Shirazi Nejad (Ph.D. thesis Karlsruhe 2017).

References

  1. 1.
    Allaire, G.: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23(6), 1482–1518 (1992)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bakhvalov, N., Panasenko, G.: Homogenisation: Averaging Processes in Periodic Media : Mathematical Problems in the Mechanics of Composite Materials. Springer, Dordrecht (1989)zbMATHGoogle Scholar
  3. 3.
    Balzani, D., Gandhi, A., Klawonn, A., Lanser, M., Rheinbach, O., Schröder, J.: One-way and fully-coupled FE2 methods for heterogeneous elasticity and plasticity problems: Parallel scalability and an application to thermo-elastoplasticity of dual-phase steels. In: Software for Exascale Computing-SPPEXA 2013–2015, pp. 91–112. Springer (2016)Google Scholar
  4. 4.
    Diebels, S., Jung, A., Chen, Z., Seibert, H., Scheffer, T.: Experimentelle Mechanik: Von der Messung zum Materialmodell. Rundbrief GAMM (2015)Google Scholar
  5. 5.
    Feyel, F., Chaboche, J.L.: FE2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials. Comput. Methods Appl. Mech. Eng. 183(3), 309–330 (2000)CrossRefGoogle Scholar
  6. 6.
    Fritzen, F., Hodapp, M.: The finite element square reduced (FE2R) method with gpu acceleration: towards three-dimensional two-scale simulations. Int. J. Numer. Methods Eng. 107(10), 853–881 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Fritzen, F., Hodapp, M., Leuschner, M.: GPU accelerated computational homogenization based on a variational approach in a reduced basis framework. Comput. Methods Appl. Mech. Eng. 278, 186–217 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    GeoDict: The digital material laboratory. http://www.geodict.de/ (2014)
  9. 9.
    Ju, J.: On energy-based coupled elastoplastic damage theories: constitutive modeling and computational aspects. Int. J. Solids Struct. 25(7), 803–833 (1989)CrossRefGoogle Scholar
  10. 10.
    Kachanov, L.: Introduction to Continuum Damage Mechanics. Springer, Mechanics of Elastic Stability (1986)CrossRefGoogle Scholar
  11. 11.
    Lippmann, H., Lemaitre, J.: A Course on Damage Mechanics. Springer, Berlin Heidelberg (1996)zbMATHGoogle Scholar
  12. 12.
    Maurer, D., Wieners, C.: A parallel block LU decomposition method for distributed finite element matrices. Parallel Comput. 37(12), 742–758 (2011)CrossRefGoogle Scholar
  13. 13.
    Miehe, C., Schotte, J., Schröder, J.: Computational micro-macro transitions and overall moduli in the analysis of polycrystals at large strains. Comput. Mater. Sci. 16(1), 372–382 (1999)CrossRefGoogle Scholar
  14. 14.
    Miehe, C., Schröder, J., Schotte, J.: Computational homogenization analysis in finite plasticity. Comput. Methods Appl. Mech. Eng. 171, 3–4 (1999)CrossRefGoogle Scholar
  15. 15.
    Mielke, A.: Evolution of rate-independent systems. In: Handbook of Differential Equations: Evolutionary Equations, vol. 2, chap. 6, pp. 461–559. North-Holland (2005)Google Scholar
  16. 16.
    Mielke, A., Roubíček, T.: Rate-Independent Systems: Theory and Application. Applied Mathematical Sciences. Springer, New York (2015)CrossRefGoogle Scholar
  17. 17.
    Mielke, A., Timofte, A.M.: Two-scale homogenization for evolutionary variational inequalities via the energetic formulation. SIAM J. Math. Anal. 39(2), 642–668 (2007)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Papanicolau, G., Bensoussan, A., Lions, J.: Asymptotic Analysis for Periodic Structures. Studies in Mathematics and its applications. Elsevier Science (1978)Google Scholar
  19. 19.
    Rabotnov, Y.: Creep Problems in Structural Members. Elsevier, Applied Mathematics and Mechanics Series (1969)zbMATHGoogle Scholar
  20. 20.
    Röhrig, C.: Personal communication (2016)Google Scholar
  21. 21.
    Röhrig, C., Scheffer, T., Diebels, S.: Mechanical characterization of a short fiber-reinforced polymer at room temperature: experimental setups evaluated by an optical measurement system. In: Continuum Mechanics and Thermodynamics, pp. 1–19 (2017)Google Scholar
  22. 22.
    Sanchez-Palencia, E., Zaoui, A.: Homogenization techniques for composite media: lectures delivered at the CISM International Center for Mechanical Sciences, Udine, Italy, July 1–5, 1985. In: Lecture Notes in Physics. Springer (1987)Google Scholar
  23. 23.
    Schröder, J.: A numerical two-scale homogenization scheme: the FE2-method. In: Plasticity and Beyond: Microstructures. Crystal-Plasticity and Phase Transitions, pp. 1–64. Springer, Vienna (2014)CrossRefGoogle Scholar
  24. 24.
    Shirazi Nejad, R.: A parallel elastic and inelastic heterogeneous multiscale method for rate-independent materials. Ph.D. thesis, Karlsruhe Institute of Technology (2017)Google Scholar
  25. 25.
    Simo, J., Hughes, T.: Computational Inelasticity. Interdisciplinary Applied Mathematics. Springer, New York (2000)Google Scholar
  26. 26.
    Smit, R., Brekelmans, W., Meijer, H.: Prediction of the mechanical behavior of nonlinear heterogeneous systems by multi-level finite element modeling. Comput. Methods Appl. Mech. Eng. 155(1–2), 181–192 (1998)CrossRefGoogle Scholar
  27. 27.
    Spahn, J.: An efficient multiscale method for modeling progressive damage in composite materials. Ph.D. thesis, Technische Universität Kaiserslautern (2015)Google Scholar
  28. 28.
    Spahn, J., Andrä, H., Kabel, M., Müller, R.: A multiscale approach for modeling progressive damage of composite materials using fast Fourier transforms. Comput. Methods Appl. Mech. Eng. 268, 871–883 (2014)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Thomas, M., Mielke, A.: Damage of nonlinearly elastic materials at small strain—existence and regularity results. Zeitschrift Angewandte Mathematik und Mechanik 90, 88–112 (2010)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Weinan, E., Engquist, B., et al.: The heterogeneous multiscale methods. Commun. Math. Sci. 1(1), 87–132 (2003)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Wieners, C.: A geometric data structure for parallel finite elements and the application to multigrid methods with block smoothing. Comput. Vis. Sci. 13(4), 161–175 (2010)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Karlsruhe Institute of TechnologyKarlsruheGermany

Personalised recommendations