Numerical Homogenization of Heterogeneous Anisotropic Linear Elastic Materials

  • S. Margenov
  • S. Stoykov
  • Y. Vutov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8353)


The numerical homogenization of anisotropic linear elastic materials with strongly heterogeneous microstructure is studied. The developed algorithm is applied to the case of trabecular bone tissue. In our previous work [1], the orthotropic case was considered. The homogenized anisotropic tensor is transformed according to the principle directions of anisotropy (PDA). This provides opportunities for better interpretation of the results as well as for classification of the material properties.

The upscaling procedure is described in terms of six auxiliary elastic problems for the reference volume element (RVE). Rotated trilinear Rannacher-Turek finite elements are used for discretization of the involved subproblems. A parallel PCG method is implemented for efficient solution of the arising large-scale systems with sparse, symmetric, and positive semidefinite matrices. Then, the bulk modulus tensor is computed from the upscaled stiffness tensor and its eigenvectors are used to define the transformation matrix. The stiffness tensor of the material is transformed with respect to the PDA which gives a canonical (unique) representation of the material properties.

Numerical experiments for two different RVEs from the trabecular part of human bones are presented.


Preconditioned Conjugate Gradient Stiffness Tensor Fictitious Domain Nonconforming Finite Element Numerical Homogenization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This work is supported in part by Grants DFNI I01/5 and DCVP-02/1 from the Bulgarian NSF and the Bulgarian National Center for Supercomputing Applications (NCSA), giving access to the IBM Blue Gene/P computer.

The research is also partly supported by the project AComIn “Advanced Computing for Innovation”, grant 316087, funded by the FP7 Capacity Programme (Research Potential of Convergence Regions)


  1. 1.
    Margenov, S., Vutov, Y.: Parallel MIC(0) preconditioning for numerical upscaling of anisotropic linear elastic materials. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds.) LSSC 2009. LNCS, vol. 5910, pp. 805–812. Springer, Heidelberg (2010) CrossRefGoogle Scholar
  2. 2.
    Wirth, A.J., Mueller, T.L., Vereecken, W., Flaig, C., Arbenz, P., Mller, R., van Lenthe, G.H.: Mechanical competence of bone-implant systems can accurately be determined by image-based micro-finite element analyses. Arch. Appl. Mech. 80(5), 513–525 (2010)CrossRefzbMATHGoogle Scholar
  3. 3.
    Fung, Y.C.: Foundations of Solid Mechanics. Prentice-Hall, Englewood Cliffs (1965)Google Scholar
  4. 4.
    Nayfeh, A., Pai, P.: Linear and Nonlinear Structural Mechanics. Wiley, New York (2004)CrossRefzbMATHGoogle Scholar
  5. 5.
    Sokolonikoff, I.: Mathematical Theory of Elasticity. Mc-Graw-Hill, New York (1956)Google Scholar
  6. 6.
    Hoppe, R.H.W., Petrova, S.I.: Optimal shape design in biomimetics based on homogenization and adaptivity. Math. Comput. Simul. 65(3), 257–272 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Bensoussan, A., Lions, J.L., Papanicolaou, G.: Asymptotic Analysis for Periodic Structures. Elsevier, Amsterdam (1978)zbMATHGoogle Scholar
  8. 8.
    Rannacher, R., Turek, S.: Simple nonconforming quadrilateral Stokes element. Numer. Meth. Partial Differ. Equ. 8(2), 97–112 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Arnold, D.N., Brezzi, F.: Mixed and nonconforming finite element methods: Implementation, postprocessing and error estimates. RAIRO. Model. Math. Anal. Numer. 19, 7–32 (1985)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Malkus, D., Hughes, T.: Mixed finite element methods – reduced and selective integration techniques: an uniform concepts. CMAME 15, 63–81 (1978)zbMATHGoogle Scholar
  11. 11.
    Blaheta, R.: Displacement decomposition-incomplete factorization preconditioning techniques for linear elasticity problems. NLAA 1(2), 107–128 (1994)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Arbenz, P., Margenov, S., Vutov, Y.: Parallel MIC(0) preconditioning of 3D elliptic problems discretized by Rannacher-Turek finite elements. Comput. Math. Appl. 55(10), 2197–2211 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Rand, O., Rovenski, V.: Analytical Methods in Anisotropic Elasticity: With Symbolic Computational Tools. Birkhauser, Boston (2004)Google Scholar
  14. 14.
    Walker, D., Dongarra, J.: MPI: a standard Message Passing Interface. Supercomputer 63, 56–68 (1996)Google Scholar
  15. 15.
    Beller, G., Burkhart, M., Felsenberg, D., Gowin, W., Hege, H.-C., Koller, B., Prohaska, S., Saparin, P.I., Thomsen, J.S.: Vertebral body data set esa29-99-l3.
  16. 16.
    Cowin, S.: Bone poroelasticity. J Biomech. 32, 217–238 (1999)CrossRefGoogle Scholar
  17. 17.
    Wolff, J.: The Law of Bone Remodeling. Springer, Heidelberg (1986)CrossRefGoogle Scholar
  18. 18.
    Kosturski, N., Margenov, S.: Numerical homogenization of bone microstructure. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds.) LSSC 2009. LNCS, vol. 5910, pp. 140–147. Springer, Heidelberg (2010) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Institute of Information and Communication TechnologiesBulgarian Academy of SciencesSofiaBulgaria

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