A multiscale FE-FFT framework for electro-active materials at finite strains

  • Felix Selim Göküzüm
  • Lu Trong Khiem Nguyen
  • Marc-André KeipEmail author
Original Paper


The present work addresses the fast-Fourier-transform-based computational homogenization of electro-mechanically coupled materials at finite strains. While the macroscopic boundary value problem is solved with finite elements, the solution at microscale is carried out using fast Fourier transforms (FFT). In the context of the FFT-based solution, we propose a general formulation that employs a fully coupled reference medium resulting in fully coupled preconditioning. In order to arrive at an efficient multiscale setting, we provide an algorithmically consistent macroscopic tangent operator for nonlinear electro-mechanical problems derived from the Lippmann–Schwinger equation. We demonstrate the applicability and accuracy of the formulation with some numerical examples. Here, we also investigate both the fully coupled and uncoupled preconditioning as well as the respective impact on the algorithmic solution. It turns out that while the fully coupled scheme leads to quadratic convergence rates, the uncoupled scheme may allow for shorter computation times under certain boundary conditions.


Computational homogenization Fast Fourier transforms Finite elements Electro-mechanics Effective properties 



The financial support of the German Research Foundation (DFG) in the framework of the Cluster of Excellence in “Simulation Technology” (EXC 310/2) at the University of Stuttgart is gratefully acknowledged.


  1. 1.
    Bignonnet F, Hassen G, Dormieux L (2016) Fourier-based strength homogenization of porous media. Comput Mech 58(5):833–859MathSciNetCrossRefGoogle Scholar
  2. 2.
    Brisard S, Dormieux L (2010) FFT-based methods for the mechanics of composites: a general variational framework. Comput Mater Sci 49:663–671CrossRefGoogle Scholar
  3. 3.
    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 217:197–212MathSciNetCrossRefGoogle Scholar
  4. 4.
    Budiansky B (1965) On the elastic moduli of some heterogeneous materials. J Mech Phys Solids 13:223–227CrossRefGoogle Scholar
  5. 5.
    Carpi F, De Rossi D, Kornbluh R, Pelrine RE, Sommer-Larsen P (2011) Dielectric elastomers as electromechanical transducers: fundamentals, materials, devices, models and applications of an emerging electroactive polymer technology. Elsevier, New YorkGoogle Scholar
  6. 6.
    Cooley J, Tukey J (1965) An algorithm for the machine calculation of complex Fourier series. Math Comput 19(90):297–301MathSciNetCrossRefGoogle Scholar
  7. 7.
    Danas K (2017) Effective response of classical, auxetic and chiral magnetoelastic materials by use of a new variational principle. J Mech Phys Solids 105:25–53MathSciNetCrossRefGoogle Scholar
  8. 8.
    De Geus T, Vondřejc J, Zeman J, Peerlings R, Geers M (2017) Finite strain FFT-based non-linear solvers made simple. Comput Methods Appl Mech Eng 318:412–430MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dorfmann A, Ogden RW (2005) Nonlinear electroelasticity. Acta Mater 174:167–183zbMATHGoogle Scholar
  10. 10.
    Estep D (2002) Practical analysis in one variable. Springer Science & Business Media, New YorkzbMATHGoogle Scholar
  11. 11.
    Eyre D, Milton G (1999) A fast numerical scheme for computing the response of composites using grid refinement. Eur Phys J 6:41–47Google Scholar
  12. 12.
    Frigo M (1999) A fast fourier transform compiler. SIGPLAN Not. 34(5):169–180CrossRefGoogle Scholar
  13. 13.
    Gelebart 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
  14. 14.
    Göküzüm FS, Keip M-A (2018) An algorithmically consistent macroscopic tangent operator for FFT-based computational homogenization. Int J Numer Methods Eng 113(4):581–600MathSciNetCrossRefGoogle Scholar
  15. 15.
    Goldberg N, Ospald F, Schneider M (2017) A fiber orientation-adapted integration scheme for computing the hyperelastic tucker average for short fiber reinforced composites. Comput Mech 60(4):595–611MathSciNetCrossRefGoogle Scholar
  16. 16.
    Hashin Z, Shtrikman S (1963) A variational approach to the theory of the elastic behaviour of multiphase materials. J Mech Phys Solids 11(2):127–140MathSciNetCrossRefGoogle Scholar
  17. 17.
    Heideman MT, Johnson D, Burrus C (1984) Gauss and the history of the fast Fourier transform. IEEE ASSP Mag 1:14–21CrossRefGoogle Scholar
  18. 18.
    Hill R (1952) The elastic behaviour of a crystalline aggregate. Proc Phys Soc Sect A 65(5):349CrossRefGoogle Scholar
  19. 19.
    Hill R (1963) Elastic properties of reinforced solids: some theoretical principles. J Mech Phys Solids 11(5):357–372CrossRefGoogle Scholar
  20. 20.
    Hill R (1965) A self-consistent mechanics of composite materials. J Mech Phys Solids 13:213–222MathSciNetCrossRefGoogle Scholar
  21. 21.
    Javili A, Chatzigeorgiou G, Steinmann P (2013) Computational homogenization in magneto-mechanics. Int J Solids Struct 50:4197–4216CrossRefGoogle Scholar
  22. 22.
    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 54:1497–1514MathSciNetCrossRefGoogle Scholar
  23. 23.
    Keip M-A, Rambausek M (2016) A multiscale approach to the computational characterization of magnetorheological elastomers. Int J Numer Methods Eng 107:338–360MathSciNetCrossRefGoogle Scholar
  24. 24.
    Keip M-A, Rambausek M (2017) Computational and analytical investigations of shape effects in the experimental characterization of magnetorheological elastomers. Int J Solids Struct 121:1–20CrossRefGoogle Scholar
  25. 25.
    Keip M-A, Steinmann P, Schröder J (2014) Two-scale computational homogenization of electro-elasticity at finite strains. Comput Methods Appl Mech Eng 278:62–79MathSciNetCrossRefGoogle Scholar
  26. 26.
    Kochmann J, Wulfinghoff S, Ehle L, Mayer J, Svendsen B, Reese S (2017). Efficient and accurate two-scale FE-FFT-based prediction of the effective material behavior of elasto-viscoplastic polycrystals. Comput Mech 1–14Google Scholar
  27. 27.
    Kochmann J, Wulfinghoff S, Reese S, Mianroodi JR, Svendsen B (2016) Two-scale FE-FFT- and phase-field-based computational modeling of bulk microstructural evolution and macroscopic material behavior. Comput Methods Appl Mech Eng 305:89–110MathSciNetCrossRefGoogle Scholar
  28. 28.
    Kröner E (1977) Bounds for effective elastic moduli of disordered materials. J Mech Phys Solids 2(25):137–155CrossRefGoogle Scholar
  29. 29.
    Lefèvre V, Lopez-Pamies O (2017) Nonlinear electroelastic deformations of dielectric elastomer composites: II—non-Gaussian elastic dielectrics. J Mech Phys Solids 99:438–470MathSciNetCrossRefGoogle Scholar
  30. 30.
    Leuschner M, Fritzen F, (2017) Fourier-accelerated nodal solvers (FANS) for homogenization problems. Comput Mech.
  31. 31.
    Lippmann BA, Schwinger J (1950) Variational principles for scattering processes. I. Phys Rev 79:469–480MathSciNetCrossRefGoogle Scholar
  32. 32.
    McMeeking RM, Landis CM (2005) Electrostatic forces and stored energy for deformable dielectric materials. J Appl Mech 72(4):581–590CrossRefGoogle Scholar
  33. 33.
    Miehe C, Schröder J, Schotte J (1999) Computational homogenization analysis in finite plasticity. Simulation of texture development in polycrystalline materials. Comput Meth Appl Mech Eng 171:387–418MathSciNetCrossRefGoogle Scholar
  34. 34.
    Miehe C, Vallicotti D, Teichtmeister S (2016) Homogenization and multiscale stability analysis in finite magneto-electro-elasticity. application to soft matter EE, ME and MEE composites. Comput Methods Appl Mech Eng 300:294–346MathSciNetCrossRefGoogle Scholar
  35. 35.
    Milton GW (2002) The theory of composites. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  36. 36.
    Mori T, Tanaka K (1973) Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Mech 21:571–574Google Scholar
  37. 37.
    Moulinec H, Suquet P (1994) A fast numerical method for computing the linear and nonlinear mechanical properties of composites. Acad Sci 2:1417–1423zbMATHGoogle Scholar
  38. 38.
    Moulinec H, Suquet P (1998) A numerical method for computing the overall response of nonlinear composites with complex microstructure. Comput Methods Appl Mech Eng 157:69–94MathSciNetCrossRefGoogle Scholar
  39. 39.
    Özdemir I, Brekelmans WAM, Geers MGD (2008a) Computational homogenization for heat conduction in heterogeneous solids. Int J Numer Methods Eng 73(2):185–204MathSciNetCrossRefGoogle Scholar
  40. 40.
    Özdemir I, Brekelmans WAM, Geers MGD (2008b) \(\text{ FE }^{2}\) computational homogenization for the thermo-mechanical analysis of heterogeneous solids. Comput Methods Appl Mech Eng 198(3–4):602–613CrossRefGoogle Scholar
  41. 41.
    Pelteret J-P, Davydov D, McBride A, Vu DK, Steinmann P (2016) Computational electro-elasticity and magneto-elasticity for quasi-incompressible media immersed in free space. Int J Numer Methods Eng 108:1307–1342MathSciNetCrossRefGoogle Scholar
  42. 42.
    Polukhov E, Vallicotti D, Keip M-A (2018) Computational stability analysis of periodic electroactive polymer composites across scales. Comput Methods Appl Mech Eng 337:165–197MathSciNetCrossRefGoogle Scholar
  43. 43.
    Ponte Castañeda P, Siboni MH (2012) A finite-strain constitutive theory for electro-active polymer composites via homogenization. Int J Non Linear Mech 47(2):293–306CrossRefGoogle Scholar
  44. 44.
    Reuss A (1929) Berechnung der fließgrenze von mischkristallen auf grund der plastizitätsbedingung für einkristalle. ZAMM J Appl Math Mech (Z Angew Math Mech) 9(1):49–58CrossRefGoogle Scholar
  45. 45.
    Rudykh S, Lewinstein A, Uner G, deBotton G (2013) Analysis of microstructural induced enhancement of electromechanical coupling in soft dielectrics. Appl Phys Lett 102(15):151905CrossRefGoogle Scholar
  46. 46.
    Schneider M, Hofmann T, Andrä H, Lechner P, Ettemeyer F, Volk W, Steeb H (2018) Modelling the microstructure and computing effective elastic properties of sand core materials. Int J Solids Struct 143:1–17CrossRefGoogle Scholar
  47. 47.
    Schneider M, Merkert D, Kabel M (2017) FFT-based homogenization for microstructures discretized by linear hexahedral elements. Int J Numer Methods Eng 109(10):1461–1489MathSciNetCrossRefGoogle 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.
    Schröder J (2009) Derivation of the localization and homogenization conditions for electro-mechanically coupled problems. Comput Mater Sci 46:595–599CrossRefGoogle Scholar
  50. 50.
    Schröder J (2014) A numerical two-scale homogenization scheme: the FE\(^2\)-method. In: Schröder J, Hackl K (eds) Plasticity and Beyond. Vol. 550 of CISM Courses and Lectures. Springer, New York, pp 1–64Google Scholar
  51. 51.
    Schröder J, Keip M-A (2012) Two-scale homogenization of electromechanically coupled boundary value problems. Comput Mech 50:229–244MathSciNetCrossRefGoogle Scholar
  52. 52.
    Spahn J, Andrä H, Kabel M, Müller R (2014) A multiscale approach for modeling progressive damage of composite materials using fast Fourier transforms. Comput Methods Appl Mech Eng 268:871–883MathSciNetCrossRefGoogle Scholar
  53. 53.
    Taylor RL (2017) FEAP—finite element analysis program. University of California, Berkeley. (http://www.ce.berkeley/feap)
  54. 54.
    Toupin RA (1956) The elastic dielectric. J Ration Mech Anal 5(6):849–915MathSciNetzbMATHGoogle Scholar
  55. 55.
    Vidyasagar A, Tan WL, Kochmann DM (2017) Predicting the effective response of bulk polycrystalline ferroelectric ceramics via improved spectral phase field methods. J Mech Phys Solids 106:133–151MathSciNetCrossRefGoogle Scholar
  56. 56.
    Voigt W (1887) Theoretische studien über die Elastizitastsverhältnisse der Kristalle. Königliche Gesellschaft der Wissenschaften zu Göttingen 34Google Scholar
  57. 57.
    Vondřejc J (2016) Improved guaranteed computable bounds on homogenized properties of periodic media by the Fourier-Galerkin method with exact integration. Int J Numer Methods Eng 107(13):1106–1135 nme.5199MathSciNetCrossRefGoogle Scholar
  58. 58.
    Vondřejc J, Zeman J, Marek I (2014) An FFT-based galerkin method for homogenization of periodic media. Comput Math App 68:156–173MathSciNetzbMATHGoogle Scholar
  59. 59.
    Vu DK, Steinmann P, Possart G (2007) Numerical modelling of non-linear electroelasticity. Int J Numer Methods Eng 70(6):685–704MathSciNetCrossRefGoogle Scholar
  60. 60.
    Willis J (1977) Bounds and self-consistent estimates for the overall properties of anisotropic composites. J Mech Phys Solids 25:185–202CrossRefGoogle Scholar
  61. 61.
    Willot F, Abdallah B, Pellegrini Y (2014) Fourier-based schemes with modified Green operator for computing the electrical response of heterogeneous media with accurate local fields. Int J Numer Methods Eng 98:518–533MathSciNetCrossRefGoogle Scholar
  62. 62.
    Zeman J, de Geus TW, Vondřejc J, Peerlings RH, Geers MG (2017) A finite element perspective on nonlinear FFT-based micromechanical simulations. Int J Numer Methods Eng 111(10):903–926MathSciNetCrossRefGoogle Scholar
  63. 63.
    Zeman J, de Geus TWJ, Vondřejc J, Peerlings RHJ, Geers MGD (2016) A finite element perspective on non-linear FFT-based micromechanical simulations. Int J Numer Methods Eng 111:903–926CrossRefGoogle Scholar
  64. 64.
    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 229:8065–8071MathSciNetCrossRefGoogle Scholar
  65. 65.
    Zhang QM, Li H, Poh M, Xia F, Cheng Z-Y, Xu H, Huang C (2002) An all-organic composite actuator material with a high dielectric constant. Nature 419(6904):284–287CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  • Felix Selim Göküzüm
    • 1
  • Lu Trong Khiem Nguyen
    • 1
  • Marc-André Keip
    • 1
  1. 1.Institute of Applied MechanicsUniversity of StuttgartStuttgartGermany

Personalised recommendations