Computational Mechanics

, Volume 54, Issue 6, pp 1497–1514 | Cite as

Efficient fixed point and Newton–Krylov solvers for FFT-based homogenization of elasticity at large deformations

Original Paper

Abstract

In recent years the FFT-based homogenization method of Moulinec and Suquet has been established as a fast, accurate and robust tool for obtaining effective properties in linear elasticity and conductivity problems. In this work we discuss FFT-based homogenization for elastic problems at large deformations, with a focus on the following improvements. Firstly, we exhibit the fixed point method introduced by Moulinec and Suquet for small deformations as a gradient descent method. Secondly, we propose a Newton–Krylov method for large deformations. We give an example for which this methods needs approximately 20 times less iterations than Newton’s method using linear fixed point solvers and roughly \(100\) times less iterations than the nonlinear fixed point method. However, the Newton–Krylov method requires 4 times more storage than the nonlinear fixed point scheme. Exploiting the special structure we introduce a memory-efficient version with 40 % memory saving. Thirdly, we give an analytical solution for the micromechanical solution field of a two-phase isotropic St.Venant–Kirchhoff laminate. We use this solution for comparison and validation, but it is of independent interest. As an example for a microstructure relevant in engineering we discuss finally the application of the FFT-based method to glass fiber reinforced polymer structures.

Keywords

Composite materials Finite deformations Lippmann–Schwinger equation FFT 

Mathematics Subject Classification

74B20 45G10 65T50 

Notes

Acknowledgments

The authors benefited from many fruitful discussions with Heiko Andrä and Andreas Günnel.

References

  1. 1.
    Advani SG, Tucker III, CL (1987) The use of tensors to describe and predict fiber orientation in short fiber composites. J Rheol 31(8), 751–784, doi:10.1122/1.549945. URL http://link.aip.org/link/?JOR/31/751/1
  2. 2.
    Agoras M, Castañeda PP (2012) Multi-scale homogenization of semi-crystalline polymers. Phil Mag 92(8):925–958. doi:10.1080/14786435.2011.637982 CrossRefGoogle Scholar
  3. 3.
    Axelsson O, Kaporin IE (2000) On the sublinear and superlinear rate of convergence of conjugate gradient methods. Numer Algorithm 25(1–4):1–22. doi:10.1023/A:1016694031362 CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Bertram A, Böhlke T, Šilhavý M (2007) On the rank 1 convexity of stored energy functions of physically linear stress-strain relations. J Elast 86(3):235–243. doi:10.1007/s10659-006-9091-z CrossRefMATHGoogle Scholar
  5. 5.
    Boyd JP (1989) Chebyshev and Fourier spectral methods. Springer-Verlag, BerlinGoogle Scholar
  6. 6.
    Braess D (2007) Finite elements: theory. Fast solvers and applications in solid mechanics. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  7. 7.
    Brighi B, Bousselsal M (1995) On the rank-one-convexity domain of the Saint Venant-Kirchhoff stored energy function. Rendiconti del Seminario Matematico della Università di Padova 94, 25–45. URL http://eudml.org/doc/108375
  8. 8.
    Brisard S, Dormieux L (2010) FFT-based methods for the mechanics of composites: A general variational framework. Comput Mater Sci 49(3), 663–671. doi:10.1016/j.commatsci.2010.06.009. URL: http://www.sciencedirect.com/science/article/pii/S0927025610003563
  9. 9.
    Brisard S, Dormieux L (2012) Combining Galerkin approximation techniques with the principle of Hashin and Shtrikman to derive a new FFT-based numerical method for the homogenization of composites. Comput Methods Appl Mech Eng 217–220(0):197–212. doi:10.1016/j.cma.2012.01.003. URL http://www.sciencedirect.com/science/article/pii/S0045782512000059
  10. 10.
    Castañeda PP (1996) Exact second-order estimates for the effective mechanical properties of nonlinear composite materials. J Mech Phys Solids 44(6):827–862CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Castañeda PP, Suquet P (1998) Nonlinear composites. Adv Appl Mech 34(998):171–302Google Scholar
  12. 12.
    Ciarlet PG (1988) Mathematical elasticity: three-dimensional elasticity, vol I. Elsevier, AmsterdamMATHGoogle Scholar
  13. 13.
    Eisenlohr P., Diehl, M., Lebensohn, R., Roters, F.: A spectral method solution to crystal elasto-viscoplasticity at finite strains. Int J Plast 46(0), 37–53 (2013). doi:10.1016/j.ijplas.2012.09.012. URL http://www.sciencedirect.com/science/article/pii/S0749641912001428
  14. 14.
    Eyre DJ, Milton GW (1999) A fast numerical scheme for computing the response of composites using grid refinement. Eur Phys J 6(01):41–47Google Scholar
  15. 15.
    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(3–4), 309–330 (2000). doi:10.1016/S0045-7825(99)00224-8. URL http://www.sciencedirect.com/science/article/pii/S0045782599002248
  16. 16.
    Francfort G (1983) Homogenization and linear thermoelasticity. SIAM J Math Anal 14(4):696–708. doi:10.1137/0514053 CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Gélébart, L., Mondon-Cancel, R.: Non-linear extension of FFT-based methods accelerated by conjugate gradients to evaluate the mechanical behavior of composite materials. Comput Mater Sci 77(0), 430–439 (2013). doi:10.1016/j.commatsci.2013.04.046. URL http://www.sciencedirect.com/science/article/pii/S0927025613002188
  18. 18.
    Hestenes MR, Stiefel E (1952) Methods of conjugate gradients for solving linear systems. J Res Nat Bureau Stand 49:409–436CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Hill R (1972) On constitutive macro-variables for heterogeneous solids at finite strain. Proc R Soc Lond A 326:131–147CrossRefMATHGoogle Scholar
  20. 20.
    Johnson SG, Frigo M (2007) A modified split-radix FFT with fewer arithmetic operations. Signal Process IEEE Trans on 55(1):111–119Google Scholar
  21. 21.
    Kocks UF, Tome CN, Wenk HR (1998) Texture and anisotropy: preferred orientations in polycrystals and their effect on materials properties. Cambridge University Press, CambridgeMATHGoogle Scholar
  22. 22.
    Krawietz A (1986) Materialtheorie. Springer-Verlag, BerlinCrossRefMATHGoogle Scholar
  23. 23.
    Kröner E (1971) Statistical continuum mechanics. Springer, WienCrossRefGoogle Scholar
  24. 24.
    Kröner, E.: Bounds for effective elastic moduli of disordered materials. J Mech Phys Solids 25(2), 137–155 (1977). doi:10.1016/0022-5096(77)90009-6. URL http://www.sciencedirect.com/science/article/pii/0022509677900096
  25. 25.
    Lahellec, N., Michel, J.C., Moulinec, H., Suquet, P.: Analysis of inhomogeneous materials at large strains using fast Fourier transforms. In: C. Miehe (ed.) IUTAM Symposium on computational mechanics of solid materials at large strains, Solid mechanics and its applications, vol. 108, pp. 247–258. Springer, Netherlands (2003). doi:10.1007/978-94-017-0297-3_22. URL http://dx.doi.org/10.1007/978-94-017-0297-3_22
  26. 26.
    Michel JC, Moulinec H, Suquet P (2001) A computational scheme for linear and non-linear composites with arbitrary phase contrast. Int J Numer Methods Eng 52(12):139–160. doi:10.1002/nme.275 CrossRefGoogle Scholar
  27. 27.
    Milton GW (2002) The theory of composites. Cambridge University Press, CambridgeCrossRefMATHGoogle Scholar
  28. 28.
    Monchiet V, Bonnet G (2012) A polarization-based FFT iterative scheme for computing the effective properties of elastic composites with arbitrary contrast. Int J Numer Methods Eng 89(11):1419–1436. doi:10.1002/nme.3295 CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Monchiet, V., Bonnet, G.: Numerical homogenization of nonlinear composites with a polarization-based FFT iterative scheme. Comput Mater Sci 79(0), 276–283 (2013). doi:10.1016/j.commatsci.2013.04.035. URL http://www.sciencedirect.com/science/article/pii/S0927025613002073
  30. 30.
    Moulinec H, Suquet P (1994) A fast numerical method for computing the linear and nonlinear mechanical properties of composites. Comptes Rendus de l’Académie des Sciences. Série II, Mécanique, Physique, Chimie, Astronomie 318(11):1417–1423MATHGoogle Scholar
  31. 31.
    Moulinec H, Suquet P (1998) A numerical method for computing the overall response of nonlinear composites with complex microstructure. Comput Methods Appl Mech Eng 157(1–2):69–94. doi:10.1016/s0045-7825(97)00218-1 CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Moulinec, H., Suquet, P.: Comparison of FFT-based methods for computing the response of composites with highly contrasted mechanical properties. Physica B: Condens Matter 338(1–4), 58–60 (2003). doi:10.1016/S0921-4526(03)00459-9. URL http://www.sciencedirect.com/science/article/pii/S0921452603004599. Proceedings of the Sixth International Conference on Electrical Transport and Optical Properties of Inhomogeneous Media
  33. 33.
    Mura T (1987) Micromechanics of defects in solids. Mechanics of elastic and inelastic solids, 2nd edn. Martinus Nijhoff Publishers, DordrechtGoogle Scholar
  34. 34.
    Nemat-Nasser S (1993) Micromechanics: overall properties of heterogeneous materials, North-Holland series in applied mathematics and mechanics. Elsevier Science Publishers B.V, AmsterdamGoogle Scholar
  35. 35.
    Ortega JM (1968) The Newton-Kantorovich theorem. Am Math Mon 75(6):658–660. doi:10.2307/2313800 CrossRefMATHGoogle Scholar
  36. 36.
    Ortega JM, Rheinboldt W (1970) Iterative solution of nonlinear equations in several variables. Academic Press, New YorkMATHGoogle Scholar
  37. 37.
    Paige CC, Saunders MA (1975) Solution of sparse indefinite systems of linear equations. SIAM J Numer Anal 12(4):617–629. doi:10.1137/0712047 CrossRefMATHMathSciNetGoogle Scholar
  38. 38.
    Schladitz K, Peters S, Reinel-Bitzer D, Wiegmann A, Ohser J (2006) Design of acoustic trim based on geometric modeling and flow simulation for non-woven. Comput Mater Sci 38(1), 56–66. doi:10.1016/j.commatsci.2006.01.018. URL http://www.sciencedirect.com/science/article/pii/S092702560600019X
  39. 39.
    Schneider M (2014) Convergence of FFT-based homogenization for strongly heterogeneous media. Mathematical methods in the applied sciences n/a(n/a), n/a-n/a. doi:10.1002/mma.3259. URL http://dx.doi.org/10.1002/mma.3259
  40. 40.
    Šilhavý M (1997) The mechanics and thermodynamics of continuous media. Springer, New YorkMATHGoogle Scholar
  41. 41.
    Smit RJM, Brekelmans WAM, Meijer HEH (1998) 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. doi:10.1016/S0045-7825(97)00139-4. URL http://www.sciencedirect.com/science/article/pii/S0045782597001394
  42. 42.
    Spahn J, Andrä H, Kabel M, Müller R (2014) A multiscale approach for modeling progressive damage of composite materials using fast Fourier transforms. Compu Methods Appl Mech Eng 268(0):871–883. doi:10.1016/j.cma.2013.10.017. URL http://www.sciencedirect.com/science/article/pii/S0045782513002697
  43. 43.
    Truesdell C, Noll W (1965). The non-linear field theories of mechanics, encyclopedia of physics, vol. III. Springer URL http://books.google.de/books?id=dp84F_odrBQC
  44. 44.
    Vinogradov V, Milton GW (2008) An accelerated FFT algorithm for thermoelastic and non-linear composites. Int J Numer Methods Eng 76(11):1678–1695. doi:10.1002/nme.2375 CrossRefMATHMathSciNetGoogle Scholar
  45. 45.
    Vondrejc J, Zeman J, Marek I (2011) Analysis of a fast Fourier transform based method for modeling of heterogeneous materials. In: Lirkov I, Margenov S, Wasniewski J (eds) LSSC Lecture Notes Computer Science. Springer, Berlin, pp 515–522. doi:10.1007/978-3-642-29843-1_58
  46. 46.
    Vondřejc J (2013) FFT-based method for homogenization of periodic media: theory and applications. Ph.D. thesis, Department of Mechanics, Faculty of Civil Engineering, Czech Technical University, Czech Republic, Prague (2013).Google Scholar
  47. 47.
    Zeller R, Dederichs PH (1973) Elastic constants of polycrystals. Physica Status Solidi (b) 55(2):831–842. doi:10.1002/pssb.2220550241 CrossRefGoogle Scholar
  48. 48.
    Zeman J, Vondřejc J, Novák J, Marek I (2010) Accelerating a FFT-based solver for numerical homogenization of periodic media by conjugate gradients. J Comput Phys 229:8065–8071. doi:10.1016/j.jcp.2010.07.010. URL http://www.sciencedirect.com/science/article/pii/S0021999110003931

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Matthias Kabel
    • 1
  • Thomas Böhlke
    • 2
  • Matti Schneider
    • 3
  1. 1.Department of Flow and Material SimulationFraunhofer ITWMKaiserslauternGermany
  2. 2.Intitute of Engineering MechanicsKarlruhe Institute of TechnologyKarlsruheGermany
  3. 3.Faculty of Mechanical EngineeringChemnitz University of TechnologyChemnitzGermany

Personalised recommendations