Indicators for the Adaptive Choice of Multi-Scale Solvers Based on Configurational Mechanics

  • Ralf MüllerEmail author
  • Charlotte Kuhn
  • Markus Klassen
  • Heiko Andrä
  • Sarah Staub
Part of the Mathematical Engineering book series (MATHENGIN)


For heterogeneous multi-scale methods,different analytical and numerical homogenisation methods can be applied on the micro-level, where the computational domain is a representative volume element (RVE). Several numerical homogenisation algorithms which are based on boundary element approaches, pseudo spectral discretizations, or finite element schemes are available for RVEs. However, each of these methods is only appropriate in subdomains of the macro-scale domain (component), e.g. in low-stress or highly stressed component regions. Therefore, indicators for the adaptive choice of solvers on the micro-scale are helpful. The proposed indicators make use of ideas from configurational mechanics.First of all, configurational forces are introduced as indicators. Then the multi-scale approach for configurational forces is explained and illustrated with an example. Afterwards the application of the configurational forces as an indicator for a refined homogenisation method is demonstrated. The last section is devoted to the scalability of heterogeneous multi-scale computations on parallel computers. A parallel finite element code is used for the macro-scale, and a PYTHON interface for the coupling with the different micro-scale solvers is described.



The authors thank J. Hebel and Md. Khalaquzzaman for the fruitful discussions and productive cooperation within the project MUlti-scale SImulation of COmposites (MUSIKO).


  1. 1.
    Braun, M.: Configurational forces induced by finite-element discretization. Proc. Estonian Acad. Sci. Phys. Math. 46(1/2), 24–31 (1997)Google Scholar
  2. 2.
    Eshelby, J.D.: Energy relations and the energy-momentum tensor in continuum mechanics, pp. 77–115. In: Kanninen [9] (1970)Google Scholar
  3. 3.
    GeoDict.: Accessed 16 Jan 2019
  4. 4.
    Gross, D., Seelig, T.: Fracture Mechanics: With an Introduction to Micromechanics. Springer (2017)Google Scholar
  5. 5.
    Gürses, E., Miehe, C.: A computational framework of three-dimensional configurational-force-driven brittle crack propagation. Comput. Method Appl. Mech. Eng. 198(15–16), 1413–1428 (2009)CrossRefGoogle Scholar
  6. 6.
    Gurtin, M., Podio-Guidugli, P.: Configurational forces and the basic laws for crack propagation. JMPS 44(6), 905–927 (1996)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gurtin, M.E.: Configurational Forces as Basic Concept of Continuum Physics. Springer, Berlin, New York, Heidelberg (2000)Google Scholar
  8. 8.
    Heintz, P., Larsson, F., Hansbo, P., Runnesson, K.: Adaptive strategies and error control for computing materialforces in fracture mechanics. Int. J. Numer. Method Eng. 60, 1287–1299 (2004)CrossRefGoogle Scholar
  9. 9.
    Hill, R.: On constitutive macro-variables for heterogeneous solids at finite strain. Proc. R. Soc. Lond. A 326(1565), 131–147 (1972)CrossRefGoogle Scholar
  10. 10.
    Kanninen, M.F. (ed.): Inelastic Behaviour of Solids. McGraw Hill, New York (1970)Google Scholar
  11. 11.
    Khalaquzzaman, M., Xu, B.X., Ricker, S., Müller, R.: Computational homogenization of piezoelectric materials using FE\({}^{2}\) to determine configurational forces. Tech. Mech. 32(1), 21–37 (2012)Google Scholar
  12. 12.
    Kienzler, R., Herrmann, G.: Mechanics in Material Space. Springer, New York, Berlin, Heidelberg (2000)CrossRefGoogle Scholar
  13. 13.
    Kuhn, C., Müller, R., Klassen, M., Gross, D.: Numerical homogenization of the Eshelby tensor at small strains. Math. Mech. Solids (2017).
  14. 14.
    Maugin, G.A.: Material Inhomogeneities in Elasticity. Chapman & Hall, London, Glasgow, New York, Tokyo, Melbourne, Madras (1993)CrossRefGoogle Scholar
  15. 15.
    Maugin, G.A.: Configurational Forces—Thermomechanics, Physics, Mathematics, and Numerics. CRC Press, Boca Raton, London, New York (2011)Google Scholar
  16. 16.
    Miehe, C., Gürses, E.: A robust algorithm for configurational force driven brittle crack propagation with r-adaptive mesh alignment. IJNME 72, 127–155 (2007)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Miehe, C., Gürses, E., Birkle, M.: A computational framework of configurational-force-driven brittle fracture based on incremental energy minimization. Int. J. Fract. 145(4), 245–259 (2007)CrossRefGoogle Scholar
  18. 18.
    Molser, J., Ortiz, M.: On the numerical implementation of variational arbitrary Lagrangian–Eulerian (VALE) formulations. Int. J. Numer. Method Eng. 67(9), 1272–1289 (2006)Google Scholar
  19. 19.
    Mueller, R., Maugin, G.: On material forces and finite element discretizations. Comput. Mech. 29(1), 52–60 (2002)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Mueller, R., Gross, D., Maugin, G.: Use of material forces in adaptive finite element methods. Comput. Mech. 33, 421–434 (2004)CrossRefGoogle Scholar
  21. 21.
    Mueller, R., Kolling, S., Gross, D.: On configurational forces in the context of the Finite Element Method. Int. J. Numer. Method Eng. 61(1), 1–21 (2004)Google Scholar
  22. 22.
    Mura, T.: Micromechanics of Defects in Solids. Martinus Nijhoff Publishers (1987)Google Scholar
  23. 23.
    Nemat-Nasser, S., Hori, M.: Micromechanics: Overall Properties of Heterogeneous Materials. North Holland, Amsterdam, London, New York, Tokyo (1993)CrossRefGoogle Scholar
  24. 24.
    Qu, J., Cherkaoui, M.: Fundamentals of Micromechanics of Solids. Wiley (2007)Google Scholar
  25. 25.
    Ricker, S., Mergheim, J., Steinmann, P.: On the multiscale computation of defect driving forces. Int. J. Multiscale Comput. Eng. 7(5) (2009). Scholar
  26. 26.
    Ricker, S., Mergheim, J., Steinmann, P., Müller, R.: A comparison of different approaches in the multi-scale computation of configurational forces. Int. J. Fract. 166, 203–214 (2010)CrossRefGoogle Scholar
  27. 27.
    Steinmann, P.: Application of material forces to hyperelastic fracture mechanics. I. Continuum mechanical setting. Int. J. Solids Struct. 37(48–50), 7371–7391 (2000)Google Scholar
  28. 28.
    Steinmann, P., Ackermann, D., Barth, F.J.: Application of material forces to hyperelastic fracture mechanics. II. Computational setting. Int. J. Solids Struct. 38(32–33), 5509–5526 (2001)Google Scholar
  29. 29.
    Thoutireddy, P., Ortiz, M.: A variational r-adaption and shape-optimization method for finite deformation elasticity. Int. J. Numer. Method Eng. 53, 1557–1574 (2002)Google Scholar

Copyright information

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

Authors and Affiliations

  • Ralf Müller
    • 1
    Email author
  • Charlotte Kuhn
    • 1
  • Markus Klassen
    • 2
  • Heiko Andrä
    • 3
  • Sarah Staub
    • 3
  1. 1.University of KaiserslauternKaiserslauternGermany
  2. 2.RWTH Aachen UniversityAachenGermany
  3. 3.Fraunhofer Institute for Industrial MathematicsKaiserslauternGermany

Personalised recommendations