Computational Mechanics

, Volume 64, Issue 4, pp 1073–1095 | Cite as

On polarization-based schemes for the FFT-based computational homogenization of inelastic materials

  • Matti Schneider
  • Daniel WichtEmail author
  • Thomas Böhlke
Original Paper


We revisit the polarization-based schemes introduced to FFT-based computational homogenization by Eyre–Milton, Michel–Moulinec–Suquet and Monchiet–Bonnet. When applied to nonlinear problems, these polarization-based methods suffer from two handicaps. Firstly, the optimal choice of algorithmic parameters is only known for the linear elastic case. Secondly, in its original version each iteration of the polarization scheme requires solving a nonlinear system of equations for each voxel. We overcome both difficulties for small-strain elastic–viscoplastic materials. In particular, we show how to avoid solving the nonlinear system. As a byproduct, we identify a computationally efficient convergence criterion enabling a fair comparison to gradient-based solvers (like the basic scheme). The convergence behavior of the polarization schemes is compared to the basic scheme of Moulinec–Suquet and fast gradient methods, based on numerical demonstrations.


Computational homogenization FFT Douglas–Rachford splitting Elasto-viscoplasticity 



The authors M. Schneider and T. Böhlke acknowledge partial financial support by the German Research Foundation (DFG) within the International Research Training Group “Integrated engineering of continuous–discontinuous long fiber reinforced polymer structures” (GRK 2078). D. Wicht gratefully acknowledges the financial support by the Helmholtz Association of German Research Centers under the framework of the Helmholtz Research School on “Integrated Materials Development for Novel High Temperature Alloys (IMD)”, Grant No. VH-KO-610. We thank the anonymous reviewers for their constructive criticism and R. Lebensohn for providing the value of \({\dot{\gamma }}_0\) in Table 10.


  1. 1.
    Andrä H, Combaret N, Dvorkin J, Glatt E, Han J, Kabel M, Keehm Y, Krzikalla F, Lee M, Madonna C, Marsh M, Mukerji T, Saenger EH, Sain R, Saxena N, Ricker S, Wiegmann A, Zhan X (2013) Digital rock physics benchmarks-part II: computing effective properties. Comput Geosci 50:33–43CrossRefGoogle Scholar
  2. 2.
    Ayachit U (2015) The paraview guide: a parallel visualization application. Kitware, Clifton ParkGoogle Scholar
  3. 3.
    Behnel S, Bradshaw R, Citro C, Dalcin L, Seljebotn DS, Smith K (2011) Cython: the best of both worlds. Comput Sci Eng 13(2):31–39CrossRefGoogle Scholar
  4. 4.
    Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  5. 5.
    Brisard S, Dormieux L (2010) FFT-based methods for the mechanics of composites: a general variational framework. Comput Mater Sci 49(3):663–671CrossRefGoogle Scholar
  6. 6.
    Brisard S, Dormieux L (2012) Combining Galerkin approximation techniques with the principle ofHashin and Shtrikman to derive a new FFT-based numerical method for the homogenization of composites. Comput Methods Appl Mech Eng 217–220:197–212CrossRefzbMATHGoogle Scholar
  7. 7.
    Cooley JW, Tukey JW (1965) An algorithm for the machine calculation of complex Fourier series. Math Comput 19:297–301MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Davis D (2015) Convergence rate analysis of primal–dual splitting schemes. SIAM J Optim 225(3):1912–1943MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Doghri I, Brassart L, Adam L, Gérard J-S (2011) A second-moment incremental formulation for the mean-field homogenization of elasto–plastic composites. Int J Plast 27:352–371CrossRefzbMATHGoogle Scholar
  10. 10.
    Douglas J, Rachford HH (1956) On the numerical solution of heat conduction problems in two and three space variables. Trans Am Math Soc 82:421–439MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Eisenlohr P, Diehl M, Lebensohn RA, Roters F (2013) A spectral method solution to crystal elasto-viscoplasticity at finite strains. Int J Plast 46:37–53CrossRefGoogle Scholar
  12. 12.
    Eyre DJ, Milton GW (1999) A fast numerical scheme for computing the response of composites using grid refinement. Eur Phys J Appl Phys 6(1):41–47CrossRefGoogle Scholar
  13. 13.
    Frigo M, Johnson SG (2005) The design and implementation of FFTW3. Proc IEEE 93:216–231CrossRefGoogle Scholar
  14. 14.
    Gélébart L, Mondon-Cancel R (2013) Non-linear extension of FFT-based methods accelerated by conjugate gradients to evaluate the mechanical behavior of composite materials. Comput Mater Sci 77:430–439CrossRefGoogle Scholar
  15. 15.
    Giselsson P, Boyd S (2017) Linear convergence and metric selection for Douglas–Rachford splitting and ADMM. IEEE Trans Autom Control 62:532–544MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gomersall H pyFFTW. Accessed 30 July 2018
  17. 17.
    Haupt P (2002) Continuum mechanics and theory of materials. Springer, BerlinCrossRefzbMATHGoogle Scholar
  18. 18.
    Herrera P pyEVTK. Accessed 30 July 2018
  19. 19.
    Hutchinson JW (1976) Bounds and self-consistent estimates for creep of polycrystalline materials. Proc R Soc Lond A 348:101–127CrossRefzbMATHGoogle Scholar
  20. 20.
    Kabel M, Böhlke T, Schneider M (2014) Efficient fixed point and Newton-Krylov solvers for FFT-based homogenization of elasticity at large deformations. Comput Mech 54(6):1497–1514MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kabel M, Fliegener S, Schneider M (2016) Mixed boundary conditions for FFT-based homogenization at finite strains. Comput Mech 57(2):193–210MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lahellec N, Suquet P (2007) On the effective behavior of nonlinear inelastic composites: I. Incremental variatonal principles. J Mech Phys Solids 55:1932–1963MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lebensohn RA (2001) N-site modeling of a 3D viscoplastic polycrystal using Fast Fourier Transform. Acta Mater 49(14):2723–2737CrossRefGoogle Scholar
  24. 24.
    Lebensohn RA, Kanjarla AK, Eisenlohr P (2012) An elasto-viscoplastic formulation based on fast Fourier transforms for the prediction of micromechanical fields in polycrystalline materials. Int J Plast 32–33:59–69CrossRefGoogle Scholar
  25. 25.
    Lessard L, Recht B, Packard A (2016) Analysis and design of optimization algorithms via integral quadratic constraints. SIAM J Optim 26(1):57–95MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Li G, Pong TK (2015) Global convergence of splitting methods for nonconvex composite minimization. SIAM J Optim 25(4):2434–2460MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Li G, Pong TK (2016) Douglas–Rachford splitting for nonconvex optimization with application to nonconvex feasibility problems. Math Program 159(1):371–401MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Lions P-L, Mercier B (1979) Splitting algorithms for the sum of two nonlinear operators. SIAM J Numer Anal 16(6):964–979MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Lowekamp BC, Chen DT, Ibáñez L, Blezek D (2013) The design of SimpleITK. Front Neuroinform 7:45CrossRefGoogle Scholar
  30. 30.
    Matouš K, Geers MGD, Kouznetsova VG, Gillman A (2017) A review of predictive nonlinear theories for multiscale modeling of heterogeneous materials. J Comput Phys 330:192–220MathSciNetCrossRefGoogle Scholar
  31. 31.
    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 52:139–160CrossRefGoogle Scholar
  32. 32.
    Milton GW (1990) On characterizing the set of possible effective tensors of composites: the variational method and the translation method. Commun Pure Appl Math 43:63–125MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Milton GW (2002) The theory of composites. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  34. 34.
    Minty GJ (1962) Monotone (nonlinear) operators in Hilbert space. Duke Math J 29(3):341–346MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Mishra N, Vondřejc J, Zeman J (2016) A comparative study on low-memory iterative solvers for FFT-basedhomogenization of periodic media. J Comput Phys 321(321):151–168MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    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 89:1419–1436MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Moulinec H, Silva F (2014) Comparison of three accelerated FFT-based schemes for computing the mechanical response of composite materials. Int J Numer Methods Eng 97:960–985MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    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 318(11):1417–1423zbMATHGoogle Scholar
  39. 39.
    Moulinec H, Suquet P (1998) A numerical method for computing the overall response of nonlinear composites with complex microstructure. Comput Methods Appl Mech Eng 157:69–94MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Müller V, Kabel M, Andrä H, Böhlke T (2015) Homogenization of linear elastic properties of short-fiber reinforced composites—a comparison of mean field and voxel-based methods. Int J Solids Struct 67–68:56–70CrossRefGoogle Scholar
  41. 41.
    Nesterov Y (1983) A method for solving the convex programming pproblem with convergence rate \(O(1/k^2)\). Doklady Akademii Nauk SSSR 269(3):543–547MathSciNetGoogle Scholar
  42. 42.
    Nesterov Y (2004) Introductory lectures on convex optimization: a basic course, mathematics and its applications. Kluwer Academic Publishers, DordrechtCrossRefzbMATHGoogle Scholar
  43. 43.
    Peaceman DW, Rachford HH (1955) The numerical solution of parabolic and elliptic differential equations. J Soc Ind Appl Math 3(1):28–41MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Polyak BT (1964) Some methods of speeding up the convergence of iteration methods. USSR Comput Math Math Phys 4(5):1–17CrossRefGoogle Scholar
  45. 45.
    Quey R, Dawson PR, Barbe F (2011) Large-scale 3D random polycrystals for the finite element method: generation, meshing and remeshing. Comput Methods Appl Mech Eng 200(17–20):1729–1745CrossRefzbMATHGoogle Scholar
  46. 46.
    Schneider M (2015) Convergence of FFT-based homogenization for strongly heterogeneous media. Math Methods Appl Sci 38(13):2761–2778MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Schneider M (2017) An FFT-based fast gradient method for elastic and inelastic unit cell homogenization problems. Comput Methods Appl Mech Eng 315:846–866MathSciNetCrossRefGoogle Scholar
  48. 48.
    Schneider M, Ospald F, Kabel M (2016) Computational homogenization of elasticity on a staggered grid. Int J Numer Methods Eng 105(9):693–720MathSciNetCrossRefGoogle Scholar
  49. 49.
    Segurado J, Llorca J, González C (2002) On the accuracy of mean-field approaches to simulate the plastic deformation of composites. Scripta Mater 46(7):525–529CrossRefGoogle Scholar
  50. 50.
    Shantraj P, Eisenlohr P, Diehl M, Roters F (2015) Numerically robust spectral methods for crystal plasticity simulations of heterogeneous materials. Int J Plast 66:31–45CrossRefGoogle Scholar
  51. 51.
    Simmons G, Wang H (1971) Single crystal elastic constants and calculated aggregate properties: a handbook. MIT Press, BostonGoogle Scholar
  52. 52.
    Simo JC, Hughes TJR (1998) Computational inelasticity. Springer, BerlinzbMATHGoogle Scholar
  53. 53.
    Vondřejc J, Zeman J, Marek I (2015) Guaranteed upper–lower bounds on homogenized properties by FFT-based Galerkin method. Comput Methods Appl Mech Eng 297:258–291MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Williams S, Philipse A (2003) Random packings of spheres and spherocylinders simulated by mechanical contraction. Phys Rev E 67:1–9CrossRefGoogle Scholar
  55. 55.
    Willot F (2015) Fourier-based schemes for computing the mechanical response of composites with accurate local fields. C R Méc 343(3):232–245MathSciNetCrossRefGoogle Scholar
  56. 56.
    Zeman J, Vondřejc J, Novak J, Marek I (2010) Accelerating a FFT-based solver for numerical homogenization of periodic media by conjugate gradients. J Comput Phys 229(21):8065–8071MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Institute of Engineering MechanicsKarlsruhe Institute of Technology (KIT)KarlsruheGermany

Personalised recommendations