Skip to main content
SpringerLink
Log in

An efficient finite element based multigrid method for simulations of the mechanical behavior of heterogeneous materials using CT images

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

X-ray tomography techniques give researchers the full access to material inner structures. With such ample information, employing numerical simulation on real material images becomes more and more common. Material behavior, especially, of heterogeneous materials, e.g. polycrystalline, composite, can be observed at the microscopic scale with the computed tomography (CT) techniques. In this work, an efficient strategy is proposed to carry out simulations on large 3D CT images. A proposed matrix free type finite element based MultiGrid method is applied to improve convergence speed and to reduce memory space requirements. Homogenization techniques are used to obtain specific operators to enhance the convergence of the MultiGrid method when large material property variations are present. Hybrid parallel computing is implemented for memory space and computing time reasons. Free edge effects in a laminate composite with more than 16 billion degrees of freedom and crack opening in a cast iron are studied using the proposed strategy.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21

Similar content being viewed by others

References

  1. Alcouffe R, Brandt A, Dendy J Jr, Painter J (1981) The multi-grid method for the diffusion equation with strongly discontinuous coefficients. SIAM J Sci Stat Comput 2(4):430–454

    Article  Google Scholar 

  2. Arbenz P, van Lenthe GH, Mennel U, Müller R, Sala M (2008) A scalable multi-level preconditioner for matrix-free \(\mu \)-finite element analysis of human bone structures. Int J Numer Methods Eng 73(7):927–947

    Article  MathSciNet  Google Scholar 

  3. Augarde C, Ramage A, Staudacher J (2006) An element-based displacement preconditioner for linear elasticity problems. Comput Struct 84(31–32):2306–2315

    Article  Google Scholar 

  4. Benveniste Y (1987) A new approach to the application of Mori-Tanaka’s theory in composite materials. Mech Mater 6(2):147–157

    Article  Google Scholar 

  5. Bessho M, Ohnishi I, Matsuyama J, Matsumoto T, Imai K, Nakamura K (2007) Prediction of strength and strain of the proximal femur by a CT-based finite element method. J Biomech 40(8):1745–1753

    Article  Google Scholar 

  6. Boffy H, Baietto M-C, Sainsot P, Lubrecht AA (2012) An efficient 3d model of heterogeneous materials for elastic contact applications using multigrid methods. J Tribol 134(2):021401

    Article  Google Scholar 

  7. Boffy H, Venner CH (2014) Multigrid solution of the 3d stress field in strongly heterogeneous materials. Tribol Int 74:121–129

    Article  Google Scholar 

  8. Brandt A (1973) Multi-level adaptive technique (MLAT) for fast numerical solution to boundary value problems. In: Proceedings of the third international conference on numerical methods in fluid mechanics. Springer, pp 82–89

  9. Brandt A (1977) Multi-level adaptive solutions to boundary-value problems. Math Comput 31(138):333–390

    Article  MathSciNet  Google Scholar 

  10. Carey GF, Jiang B-N (1986) Element-by-element linear and nonlinear solution schemes. Commun Appl Numer Methods 2(2):145–153

    Article  Google Scholar 

  11. Ferrant M, Warfield SK, Guttmann CR, Mulkern RV, Jolesz FA, Kikinis R (1999) 3d image matching using a finite element based elastic deformation model. In: International conference on medical image computing and computer-assisted intervention. Springer, pp 202–209

  12. Gu H, Réthoré J, Baietto M-C, Sainsot P, Lecomte-Grosbras P, Venner CH, Lubrecht AA (2016) An efficient multigrid solver for the 3d simulation of composite materials. Comput Mater Sci 112:230–237

    Article  Google Scholar 

  13. Hashin Z (1979) Analysis of properties of fiber composites with anisotropic constituents. J Appl Mech 46(3):543–550

    Article  Google Scholar 

  14. Hoekema R, Venner K, Struijk JJ, Holsheimer J (1998) Multigrid solution of the potential field in modeling electrical nerve stimulation. Comput Biomed Res 31(5):348–362

    Article  Google Scholar 

  15. Hughes TJ, Ferencz RM, Hallquist JO (1987) Large-scale vectorized implicit calculations in solid mechanics on a cray X-MP/48 utilizing ebe preconditioned conjugate gradients. Comput Methods Appl Mech Eng 61(2):215–248

    Article  MathSciNet  Google Scholar 

  16. Kronbichler M, Ljungkvist K (2019) Multigrid for matrix-free high-order finite element computations on graphics processors. ACM Trans Parallel Comput (TOPC) 6(1):1–32

    Article  Google Scholar 

  17. Lecomte-Grosbras P, Paluch B, Brieu M, De Saxcé G, Sabatier L (2009) Interlaminar shear strain measurement on angle-ply laminate free edge using digital image correlation. Compos A Appl Sci Manuf 40(12):1911–1920

    Article  Google Scholar 

  18. Lecomte-Grosbras P, Réthoré J, Limodin N, Witz J-F, Brieu M (2015) Three-dimensional investigation of free-edge effects in laminate composites using X-ray tomography and digital volume correlation. Exp Mech 55(1):301–311

    Article  Google Scholar 

  19. Lengsfeld M, Schmitt J, Alter P, Kaminsky J, Leppek R (1998) Comparison of geometry-based and CT voxel-based finite element modelling and experimental validation. Med Eng Phys 20(7):515–522

    Article  Google Scholar 

  20. Liu X, Réthoré J, Baietto M-C, Sainsot P, Lubrecht AA (2019) An efficient strategy for large scale 3d simulation of heterogeneous materials to predict effective thermal conductivity. Comput Mater Sci 166:265–275

    Article  Google Scholar 

  21. Michailidis N, Stergioudi F, Omar H, Tsipas D (2010) An image-based reconstruction of the 3d geometry of an al open-cell foam and fem modeling of the material response. Mech Mater 42(2):142–147

    Article  Google Scholar 

  22. Mori T, Tanaka K (1973) Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall 21(5):571–574

    Article  Google Scholar 

  23. Moulinec H, Suquet P (1995) A FFT-based numerical method for computing the mechanical properties of composites from images of their microstructures. In: IUTAM symposium on microstructure-property interactions in composite materials. Springer, pp 235–246

  24. Nguyen T-T, Rethore J, Yvonnet J, Baietto M-C (2017) Multi-phase-field modeling of anisotropic crack propagation for polycrystalline materials. Comput Mech 60(2):289–314

    Article  MathSciNet  Google Scholar 

  25. Özdemir I, Brekelmans W, Geers M (2008) Computational homogenization for heat conduction in heterogeneous solids. Int J Numer Methods Eng 73(2):185–204

    Article  MathSciNet  Google Scholar 

  26. Pipes RB, Pagano N (1994) Interlaminar stresses in composite laminates under uniform axial extension. In: Reddy JN (ed) Mechanics of composite materials. Springer, Dordrecht, pp 234–245

    Chapter  Google Scholar 

  27. Proudhon H, Li J, Wang F, Roos A, Chiaruttini V, Forest S (2016) 3d simulation of short fatigue crack propagation by finite element crystal plasticity and remeshing. Int J Fatigue 82:238–246

    Article  Google Scholar 

  28. Rannou J, Limodin N, Réthoré J, Gravouil A, Ludwig W, Baïetto-Dubourg M-C, Buffiere J-Y, Combescure A, Hild F, Roux S (2010) Three dimensional experimental and numerical multiscale analysis of a fatigue crack. Comput Methods Appl Mech Eng 199(21–22):1307–1325

    Article  Google Scholar 

  29. Tezduyar T, Aliabadi S, Behr M, Johnson A, Mittal S (1993) Parallel finite-element computation of 3d flows. Computer 26(10):27–36

    Article  Google Scholar 

  30. van Rietbergen B, Weinans H, Huiskes R, Polman B (1996) Computational strategies for iterative solutions of large fem applications employing voxel data. Int J Numer Methods Eng 39(16):2743–2767

    Article  Google Scholar 

  31. Venner CH, Lubrecht AA (2000) Multi-level methods in lubrication. Elsevier, Amsterdam

    Google Scholar 

  32. Watremetz B, Baietto-Dubourg M, Lubrecht A (2007) 2d thermo-mechanical contact simulations in a functionally graded material: a multigrid-based approach. Tribol Int 40(5):754–762

    Article  Google Scholar 

Download references

Acknowledgements

We gratefully thank for the support of Région Pays de la Loire and Nantes Métropole through a Grant Connect Talent IDS.

Author information

Authors and Affiliations

  1. GeM, CNRS UMR 6183, Centrale Nantes, 44321, Nantes, France

    Xiaodong Liu & Julien Réthoré

  2. INSA-Lyon, CNRS UMR5259, LaMCoS, Univ Lyon, 69621, Lyon, France

    Marie-Christine Baietto, Philippe Sainsot & Antonius Adrianus Lubrecht

Authors
  1. Xiaodong Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Julien Réthoré
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Marie-Christine Baietto
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Philippe Sainsot
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Antonius Adrianus Lubrecht
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Julien Réthoré.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

The following Eshelby’s tensor is for the spherical inclusion of a linear elastic problem in macroscopically isotropic materials.

$$\begin{aligned} \varvec{S_{s}}=\frac{1}{15(1-\nu )}\left( \begin{matrix} 7-5\nu &{} 5\nu -1 &{} 5\nu -1 &{} 0 &{} 0 &{} 0 \\ 5\nu -1 &{} 7-5\nu &{} 5\nu -1 &{} 0 &{} 0 &{} 0 \\ 5\nu -1 &{} 5\nu -1 &{} 7-5\nu &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 4-5\nu &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 4-5\nu &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 4-5\nu \end{matrix} \right) \end{aligned}$$

where \(\nu \) is the Poisson’s ratio of the inclusion.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, X., Réthoré, J., Baietto, MC. et al. An efficient finite element based multigrid method for simulations of the mechanical behavior of heterogeneous materials using CT images. Comput Mech 66, 1427–1441 (2020). https://doi.org/10.1007/s00466-020-01909-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-020-01909-y

Keywords

Search

Navigation