Skip to main content
SpringerLink
Log in

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

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

Abstract

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.

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

Similar content being viewed by others

Notes

  1. The fast Fourier transform is a highly efficient implementation of the discrete Fourier transform which traces back to the seminal work of Gauss, see Heideman et al. [17]. The algorithm by Cooley and Tukey [6] is the most common one used. A very powerful code for computing the FFT exists in the form of the FFTW-library [12], which is also used in the present contribution.

  2. Note that in the context of the FFT-based solution at microscale, the primary fields \(\widetilde{{\varvec{F}}}\) and \({\widetilde{{{\varvec{D}}}}}\) appear as unknows and the properties (18)\(_{2,4}\) are ensured by the construction of suitable Green operators [28], see “Appendix B”.

  3. It is instructive to note that in FFT-based methods the periodic trigonometric polynomials employed for the interpolation of \({\widetilde{{{\varvec{F}}}}}\) and \({\widetilde{{{\varvec{D}}}}}\) automatically fulfill conditions (21) and (22). Furthermore, it holds that

    $$\begin{aligned} \int _{{\mathcal {B}}^0} {\widetilde{{{\varvec{F}}}}} \, {\mathrm{d}}V = {{{\varvec{0}}}}\quad \text {and}\quad \int _{{\mathcal {B}}^0} {\widetilde{{{\varvec{D}}}}} \, {\mathrm{d}}V = {{{\varvec{0}}}}. \end{aligned}$$
    (21)

  4. We refer to Bignonnet et al. [1], Goldberg et al. [15], Schneider et al. [46] for recent FFT-based approaches to compute effective properties of heterogeneous aggregates.

  5. Note that the ordering of the wave numbers might change depending on the used implementation.

References

  1. Bignonnet F, Hassen G, Dormieux L (2016) Fourier-based strength homogenization of porous media. Comput Mech 58(5):833–859

    Article  MathSciNet  MATH  Google Scholar 

  2. Brisard S, Dormieux L (2010) FFT-based methods for the mechanics of composites: a general variational framework. Comput Mater Sci 49:663–671

    Article  Google Scholar 

  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 217:197–212

    Article  MathSciNet  MATH  Google Scholar 

  4. Budiansky B (1965) On the elastic moduli of some heterogeneous materials. J Mech Phys Solids 13:223–227

    Article  Google Scholar 

  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 York

    Google Scholar 

  6. Cooley J, Tukey J (1965) An algorithm for the machine calculation of complex Fourier series. Math Comput 19(90):297–301

    Article  MathSciNet  MATH  Google Scholar 

  7. Danas K (2017) Effective response of classical, auxetic and chiral magnetoelastic materials by use of a new variational principle. J Mech Phys Solids 105:25–53

    Article  MathSciNet  Google Scholar 

  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 318:412–430

    Article  MathSciNet  Google Scholar 

  9. Dorfmann A, Ogden RW (2005) Nonlinear electroelasticity. Acta Mater 174:167–183

    MATH  Google Scholar 

  10. Estep D (2002) Practical analysis in one variable. Springer Science & Business Media, New York

    MATH  Google Scholar 

  11. Eyre D, Milton G (1999) A fast numerical scheme for computing the response of composites using grid refinement. Eur Phys J 6:41–47

    Google Scholar 

  12. Frigo M (1999) A fast fourier transform compiler. SIGPLAN Not. 34(5):169–180

    Article  Google Scholar 

  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 77:430–439

    Article  Google Scholar 

  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):581–600

    Article  MathSciNet  Google Scholar 

  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):595–611

    Article  MathSciNet  MATH  Google Scholar 

  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):127–140

    Article  MathSciNet  MATH  Google Scholar 

  17. Heideman MT, Johnson D, Burrus C (1984) Gauss and the history of the fast Fourier transform. IEEE ASSP Mag 1:14–21

    Article  MATH  Google Scholar 

  18. Hill R (1952) The elastic behaviour of a crystalline aggregate. Proc Phys Soc Sect A 65(5):349

    Article  Google Scholar 

  19. Hill R (1963) Elastic properties of reinforced solids: some theoretical principles. J Mech Phys Solids 11(5):357–372

    Article  MathSciNet  MATH  Google Scholar 

  20. Hill R (1965) A self-consistent mechanics of composite materials. J Mech Phys Solids 13:213–222

    Article  Google Scholar 

  21. Javili A, Chatzigeorgiou G, Steinmann P (2013) Computational homogenization in magneto-mechanics. Int J Solids Struct 50:4197–4216

    Article  Google Scholar 

  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 54:1497–1514

    Article  MathSciNet  MATH  Google Scholar 

  23. Keip M-A, Rambausek M (2016) A multiscale approach to the computational characterization of magnetorheological elastomers. Int J Numer Methods Eng 107:338–360

    Article  MathSciNet  MATH  Google Scholar 

  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 121:1–20

    Article  Google Scholar 

  25. Keip M-A, Steinmann P, Schröder J (2014) Two-scale computational homogenization of electro-elasticity at finite strains. Comput Methods Appl Mech Eng 278:62–79

    Article  MathSciNet  MATH  Google Scholar 

  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 1–14

  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 305:89–110

    Article  MathSciNet  MATH  Google Scholar 

  28. Kröner E (1977) Bounds for effective elastic moduli of disordered materials. J Mech Phys Solids 2(25):137–155

    Article  MATH  Google Scholar 

  29. Lefèvre V, Lopez-Pamies O (2017) Nonlinear electroelastic deformations of dielectric elastomer composites: II—non-Gaussian elastic dielectrics. J Mech Phys Solids 99:438–470

    Article  MathSciNet  Google Scholar 

  30. Leuschner M, Fritzen F, (2017) Fourier-accelerated nodal solvers (FANS) for homogenization problems. Comput Mech. https://doi.org/10.1007/s00466-017-1501-5

  31. Lippmann BA, Schwinger J (1950) Variational principles for scattering processes. I. Phys Rev 79:469–480

    Article  MathSciNet  MATH  Google Scholar 

  32. McMeeking RM, Landis CM (2005) Electrostatic forces and stored energy for deformable dielectric materials. J Appl Mech 72(4):581–590

    Article  MATH  Google Scholar 

  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 171:387–418

    Article  MathSciNet  MATH  Google Scholar 

  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 300:294–346

    Article  MathSciNet  MATH  Google Scholar 

  35. Milton GW (2002) The theory of composites. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

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

    Google Scholar 

  37. Moulinec H, Suquet P (1994) A fast numerical method for computing the linear and nonlinear mechanical properties of composites. Acad Sci 2:1417–1423

    MATH  Google Scholar 

  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 157:69–94

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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):602–613

    Article  MATH  Google Scholar 

  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 108:1307–1342

    Article  MathSciNet  Google Scholar 

  42. Polukhov E, Vallicotti D, Keip M-A (2018) Computational stability analysis of periodic electroactive polymer composites across scales. Comput Methods Appl Mech Eng 337:165–197

    Article  MathSciNet  Google Scholar 

  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):293–306

    Article  Google Scholar 

  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):49–58

    Article  MATH  Google Scholar 

  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):151905

    Article  Google Scholar 

  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 143:1–17

    Article  Google Scholar 

  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):1461–1489

    Article  MathSciNet  MATH  Google Scholar 

  48. Schneider M, Ospald F, Kabel M (2016) Computational homogenization of elasticity on a staggered grid. Int J Numer Methods Eng 105(9):693–720

    Article  MathSciNet  Google Scholar 

  49. Schröder J (2009) Derivation of the localization and homogenization conditions for electro-mechanically coupled problems. Comput Mater Sci 46:595–599

    Article  Google Scholar 

  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 1–64

    Google Scholar 

  51. Schröder J, Keip M-A (2012) Two-scale homogenization of electromechanically coupled boundary value problems. Comput Mech 50:229–244

    Article  MathSciNet  MATH  Google Scholar 

  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 268:871–883

    Article  MathSciNet  MATH  Google Scholar 

  53. Taylor RL (2017) FEAP—finite element analysis program. University of California, Berkeley. (http://www.ce.berkeley/feap)

  54. Toupin RA (1956) The elastic dielectric. J Ration Mech Anal 5(6):849–915

    MathSciNet  MATH  Google Scholar 

  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 106:133–151

    Article  MathSciNet  Google Scholar 

  56. Voigt W (1887) Theoretische studien über die Elastizitastsverhältnisse der Kristalle. Königliche Gesellschaft der Wissenschaften zu Göttingen 34

  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.5199

    Article  MathSciNet  MATH  Google Scholar 

  58. Vondřejc J, Zeman J, Marek I (2014) An FFT-based galerkin method for homogenization of periodic media. Comput Math App 68:156–173

    MathSciNet  MATH  Google Scholar 

  59. Vu DK, Steinmann P, Possart G (2007) Numerical modelling of non-linear electroelasticity. Int J Numer Methods Eng 70(6):685–704

    Article  MathSciNet  MATH  Google Scholar 

  60. Willis J (1977) Bounds and self-consistent estimates for the overall properties of anisotropic composites. J Mech Phys Solids 25:185–202

    Article  MATH  Google Scholar 

  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 98:518–533

    Article  MathSciNet  MATH  Google Scholar 

  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):903–926

    Article  MathSciNet  Google Scholar 

  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 111:903–926

    Article  Google Scholar 

  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 229:8065–8071

    Article  MathSciNet  MATH  Google Scholar 

  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):284–287

    Article  Google Scholar 

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

  1. Institute of Applied Mechanics, University of Stuttgart, Pfaffenwaldring 7, 70569, Stuttgart, Germany

    Felix Selim Göküzüm, Lu Trong Khiem Nguyen & Marc-André Keip

Authors
  1. Felix Selim Göküzüm
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lu Trong Khiem Nguyen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Marc-André Keip
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marc-André Keip.

Additional information

Publisher's Note

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

Appendices

Appendix

Macroscopic tangent computation

In order to set up the macroscopic tangent \(\overline{{\mathbb {C}}}{}^{algo}\), let us reconsider the increment of the dual fields (32)

$$\begin{aligned} \left[ \begin{array}{c} \Delta {\overline{{{\varvec{T}}}}} \\ \Delta {\overline{{{\varvec{E}}}}} \end{array}\right] = \overline{{\mathbb {C}}}{}^{algo} \diamond \left[ \begin{array}{c} \Delta {\overline{{{\varvec{F}}}}} \\ \Delta {\overline{{{\varvec{D}}}}} \end{array}\right] = \left[ \begin{array}{cc} \frac{\partial \overline{{{\varvec{T}}}}}{\partial {\overline{{{\varvec{F}}}}}} &{}\quad \frac{\partial \overline{{{\varvec{T}}}}}{\partial \overline{{{\varvec{D}}}}}\\ \frac{\partial \overline{{{\varvec{E}}}}}{\partial {\overline{{{\varvec{F}}}}}} &{}\quad \frac{\partial \overline{{{\varvec{E}}}}}{\partial \overline{{{\varvec{D}}}}} \end{array}\right] \diamond \left[ \begin{array}{c} \Delta {\overline{{{\varvec{F}}}}} \\ \Delta {\overline{{{\varvec{D}}}}} \end{array}\right] . \end{aligned}$$
(A.1)

Having the relations (16) in mind, we immediately see that the macroscopic dual fields \(\overline{{\mathbb {S}}}\) are dependent on the microscopic primary fields \({\mathbb {G}}\). Along with the decomposition (18), we can apply the chain rule and rewrite (A.1) as

$$\begin{aligned} \left[ \begin{array}{c} \Delta {\overline{{{\varvec{T}}}}} \\ \Delta {\overline{{{\varvec{E}}}}} \end{array}\right] = \left[ \begin{array}{cc} \dfrac{1}{V}\int _{{\mathcal {B}}^0}\bigg (\frac{\partial {{\varvec{T}}}}{\partial {{\varvec{F}}}}:\frac{\partial {{\varvec{F}}}}{\partial {\overline{{{\varvec{F}}}}}} + \frac{\partial {{\varvec{T}}}}{\partial {{\varvec{D}}}}\cdot \frac{\partial {{\varvec{D}}}}{\partial {\overline{{{\varvec{F}}}}}}\bigg ) {\mathrm{d}}V &{}\quad \dfrac{1}{V}\int _{{\mathcal {B}}^0}\bigg (\frac{\partial {{\varvec{T}}}}{\partial {{\varvec{D}}}}\cdot \frac{\partial {{\varvec{D}}}}{\partial \overline{{{\varvec{D}}}}} + \frac{\partial {{\varvec{T}}}}{\partial {{\varvec{F}}}}:\frac{\partial {{\varvec{F}}}}{\partial \overline{{{\varvec{D}}}}}\bigg ) {\mathrm{d}}V \\ \dfrac{1}{V}\int _{{\mathcal {B}}^0}\bigg (\frac{\partial {{\varvec{E}}}}{\partial {{\varvec{F}}}}:\frac{\partial {{\varvec{F}}}}{\partial {\overline{{{\varvec{F}}}}}} + \frac{\partial {{\varvec{E}}}}{\partial {{\varvec{D}}}}\cdot \frac{\partial {{\varvec{D}}}}{\partial {\overline{{{\varvec{F}}}}}} \bigg ) {\mathrm{d}}V &{}\quad \dfrac{1}{V}\int _{{\mathcal {B}}^0}\bigg (\frac{\partial {{\varvec{E}}}}{\partial {{\varvec{D}}}}\cdot \frac{\partial {{\varvec{D}}}}{\partial \overline{{{\varvec{D}}}}} + \frac{\partial {{\varvec{E}}}}{\partial {{\varvec{F}}}}:\frac{\partial {{\varvec{F}}}}{\partial \overline{{{\varvec{D}}}}} \bigg ) {\mathrm{d}}V \end{array}\right] \diamond \left[ \begin{array}{c} \Delta {\overline{{{\varvec{F}}}}} \\ \Delta {\overline{{{\varvec{D}}}}} \end{array}. \right] \end{aligned}$$
(A.2)

Having the decomposition (18) in mind, we further simplify

$$\begin{aligned} \left[ \begin{array}{c} \Delta {\overline{{{\varvec{T}}}}} \\ \Delta {\overline{{{\varvec{E}}}}} \end{array}\right] = \left[ \begin{array}{cc} \dfrac{1}{V}\int _{{\mathcal {B}}^0} {\mathbb {A}}+ {\mathbb {A}}:\frac{\partial {\widetilde{{{\varvec{F}}}}}}{\partial {\overline{{{\varvec{F}}}}}} + {{\varvec{q}}}\cdot \frac{\partial {\widetilde{{{\varvec{D}}}}}}{\partial {\overline{{{\varvec{F}}}}}} \, {\mathrm{d}}V &{}\quad \dfrac{1}{V}\int _{{\mathcal {B}}^0} {{\varvec{q}}}+ {{\varvec{q}}}\cdot \frac{\partial {\widetilde{{{\varvec{D}}}}}}{\partial \overline{{{\varvec{D}}}}} + {\mathbb {A}}: \frac{\partial {\widetilde{{{\varvec{F}}}}}}{\partial \overline{{{\varvec{D}}}}} \, {\mathrm{d}}V\\ \dfrac{1}{V}\int _{\mathcal {B}^0} {{\varvec{q}}}^T + {{\varvec{q}}}^T:\frac{\partial {\widetilde{{{\varvec{F}}}}}}{\partial {\overline{{{\varvec{F}}}}}} + {\mathbb {K}}\cdot \frac{\partial {\widetilde{{{\varvec{D}}}}}}{\partial {\overline{{{\varvec{F}}}}}} \, {\mathrm{d}}V &{}\quad \dfrac{1}{V}\int _{{\mathcal {B}}^0} {\mathbb {K}}+ {\mathbb {K}}\cdot \frac{\partial {\widetilde{{{\varvec{D}}}}}}{\partial \overline{{{\varvec{D}}}}} + {{\varvec{q}}}^T : \frac{\partial {\widetilde{{{\varvec{F}}}}}}{\partial \overline{{{\varvec{D}}}}} \, {\mathrm{d}}V \end{array}\right] \cdot \left[ \begin{array}{c} \Delta {\overline{{{\varvec{F}}}}} \\ \Delta {\overline{{{\varvec{D}}}}} \end{array}\right] , \end{aligned}$$
(A.3)

where we used the abbreviations introduced in (37). By comparing (A.3) with the increment of the macroscopic dual fields (38), we see that the fluctuation variable takes the form

$$\begin{aligned} {\mathbb {C}}\diamond \dfrac{\partial {\widetilde{{\mathbb {G}}}}}{\partial {\overline{{\mathbb {G}}}}} = \left[ \begin{array}{cc} {\mathbb {A}}:\frac{\partial {\widetilde{{{\varvec{F}}}}}}{\partial {\overline{{{\varvec{F}}}}}} + {{\varvec{q}}}\cdot \frac{\partial {\widetilde{{{\varvec{D}}}}}}{\partial {\overline{{{\varvec{F}}}}}} &{}\quad {{\varvec{q}}}\cdot \frac{\partial {\widetilde{{{\varvec{D}}}}}}{\partial \overline{{{\varvec{D}}}}} + {\mathbb {A}}: \frac{\partial {\widetilde{{{\varvec{F}}}}}}{\partial \overline{{{\varvec{D}}}}}\\ {{\varvec{q}}}^T:\frac{\partial {\widetilde{{{\varvec{F}}}}}}{\partial {\overline{{{\varvec{F}}}}}} + {\mathbb {K}}\cdot \frac{\partial {\widetilde{{{\varvec{D}}}}}}{\partial {\overline{{{\varvec{F}}}}}} &{}\quad {\mathbb {K}}\cdot \frac{\partial {\widetilde{{{\varvec{D}}}}}}{\partial \overline{{{\varvec{D}}}}} + {{\varvec{q}}}^T : \frac{\partial {\widetilde{{{\varvec{F}}}}}}{\partial \overline{{{\varvec{D}}}}} \end{array}\right] . \end{aligned}$$
(A.4)

Green operator for electro-mechanical coupling

We briefly discuss the explicit form of the Green operators used in (29) and (43). For the sake of clarity and reproducability, we will use index notation for all derivations in this part of the appendix. The differential equations (28) then take the form

$$\begin{aligned} \begin{aligned}&({\mathbb {A}}^{0}_{ijkl}\, F_{kl} + q^{0}_{ijk}\, D_{k})_{,j} = -\tau ^{m}_{ij,j}, \\&\epsilon _{ijk}(g^{0}_{klm}\, F_{lm} + {\mathbb {K}}^{0}_{kl}\, D_{l})_{,j}= -\epsilon _{ijk}\tau ^{e}_{k,j}, \end{aligned} \end{aligned}$$
(B.1)

where \(\epsilon _{ijk}\) is the Levi-Civita symbol. The basic idea of obtaining the explicit Green operators is to evaluate the differential equations (B.1) in Fourier space. This is done by using the property of the discrete Fourier transform that computes the derivatives according to

$$\begin{aligned} (\bullet )_{,j} = \mathrm {fft}^{-1}\{ \mathrm {fft}(\bullet )\, \mathrm {i} k_{j} \} \quad \text {with} \quad \mathrm {i}^{2} = - 1, \end{aligned}$$
(B.2)

having \(k_j\) as the wave vector. Algebraically reformulating (B.1) in Fourier space into the form of (29) gives

$$\begin{aligned} \begin{aligned} \widehat{\Gamma }^{m}_{ijkl}&= -k_{j} k_{l} (\kappa _{ik}^{-1} - \kappa ^{-1}_{ir}\varrho _{rn} \omega _{np} \varsigma _{ps} \kappa ^{-1}_{sk}), \\ \widehat{\Gamma }^{m,e}_{ijk}&= -k_{j} k_{r} \kappa _{is}^{-1} \varrho _{sn} \omega _{np} \epsilon _{prk}, \\ \widehat{\Gamma }^{e,m}_{ijk}&= -k_{k} k_{l} \epsilon _{iln}\omega ^{-1}_{nm}\varsigma _{mr}\kappa _{rj}^{-1}, \\ \widehat{\Gamma }^{e\phantom {m,}}_{ij}&= k_{l} k_{r} \epsilon _{iln}\epsilon _{mrj}\omega ^{-1}_{nm}, \end{aligned} \end{aligned}$$
(B.3)

where we used the following abbreviations

$$\begin{aligned} \begin{aligned} \kappa _{ik}&= {\mathbb {A}}^{0}_{ijkl} k_{l} k_{j}, \quad \quad \quad&\varrho _{in}&= q^{0}_{ijk}\epsilon _{kmn}k_{m} k_{j}, \\ \varsigma _{ik}&= \epsilon _{ijl}g^{0}_{lkm} k_{m} k_{j},&\eta _{in}&= \epsilon _{ijk} \epsilon _{lmn} {\mathbb {K}}^{0}_{kl} k_{m} k_{j}, \\ \omega _{in}&= \varsigma _{ik}\kappa ^{-1}_{kr}\varrho _{rn} - \eta _{in}. \end{aligned} \end{aligned}$$
(B.4)

Note that for the solution of a three-dimensional problem, the uniqueness of the vector potential is not assured. This is due to the identity \({{\varvec{D}}}= \mathrm{Curl\,}( {{\varvec{A}}}+ \nabla \theta ) = \mathrm{Curl\,}{{\varvec{A}}}\), which allows us to add arbitrary gradient fields to the solution of \({{\varvec{A}}}\). We therefore augment the differential equations (B.1) by a gauge restriction

$$\begin{aligned} \begin{aligned}&({\mathbb {A}}^{0}_{ijkl}\, F_{kl} + q^{0}_{ijk}\, D_{k})_{,j} = -\tau ^{m}_{ij,j}, \\&\epsilon _{ijk}(g^{0}_{klm}\, F_{lm} + {\mathbb {K}}^{0}_{kl}\, D_{l})_{,j} = -\epsilon _{ijk}\tau ^{e}_{k,j} - \Lambda A_{k,ki}, \end{aligned} \end{aligned}$$
(B.5)

where \(\Lambda \) is a penalty parameter that enforces the divergence of the vector potential to be a constant. Unsurprisingly, the derivation of \(\widehat{\varvec{\Gamma }}\) remains almost the same with only one adaption in (B.4) we need to make according to

$$\begin{aligned} \eta ^{gauge}_{in} =\epsilon _{ijk} \epsilon _{lmn} {\mathbb {K}}^{0}_{kl} k_{m} k_{j} - \Lambda k_{i} k_{n}. \end{aligned}$$
(B.6)

Fluctuation sensitivities

In this section we recapitulate on how to obtain (43). We therefore start with the explicit expression for the fluctuations (40)

$$\begin{aligned} \begin{aligned} {\widetilde{{{\varvec{F}}}}}&={\varvec{\Gamma }}^{m} *{\varvec{\tau }}^{m} + {\varvec{\Gamma }}^{m,e}*{\varvec{\tau }}^{e}, \\ {\widetilde{{{\varvec{D}}}}}&={\varvec{\Gamma }}^{e,m} *{\varvec{\tau }}^{m} + {\varvec{\Gamma }}^{e}*{\varvec{\tau }}^{e}, \end{aligned} \end{aligned}$$
(C.1)

which we want to differentiate with respect to the macroscopic deformation gradient \({\overline{{{\varvec{F}}}}}\) and electric displacement \(\overline{{{\varvec{D}}}}\). As the perturbation tensors \({\varvec{\tau }}^m\) and \({\varvec{\tau }}^e\) are linked to the stresses through (27), they are dependent on the microscopic deformation gradient \({{\varvec{F}}}\) and electric displacement \({{\varvec{D}}}\). We obtain

$$\begin{aligned} \begin{aligned} \frac{\partial {\widetilde{{{\varvec{F}}}}}}{\partial {\overline{{{\varvec{F}}}}}}&={\varvec{\Gamma }}^{m\phantom {,e}} *\left[ \frac{\partial {\varvec{\tau }}^{m}}{\partial {{\varvec{F}}}}: \frac{\partial {{\varvec{F}}}}{\partial {\overline{{{\varvec{F}}}}}} + \frac{\partial {\varvec{\tau }}^{m}}{\partial {{\varvec{D}}}}\cdot \frac{\partial {{\varvec{D}}}}{\partial {\overline{{{\varvec{F}}}}}}\right]&+\quad&{\varvec{\Gamma }}^{m,e}*\left[ \frac{\partial {\varvec{\tau }}^{e}}{\partial {{\varvec{F}}}}: \frac{\partial {{\varvec{F}}}}{\partial {\overline{{{\varvec{F}}}}}} + \frac{\partial {\varvec{\tau }}^{e}}{\partial {{\varvec{D}}}}\cdot \frac{\partial {{\varvec{D}}}}{\partial {\overline{{{\varvec{F}}}}}}\right] , \\ \frac{\partial {\widetilde{{{\varvec{F}}}}}}{\partial \overline{{{\varvec{D}}}}}&={\varvec{\Gamma }}^{m\phantom {,e}} *\left[ \frac{\partial {\varvec{\tau }}^{m}}{\partial {{\varvec{F}}}}: \frac{\partial {{\varvec{F}}}}{\partial \overline{{{\varvec{D}}}}} + \frac{\partial {\varvec{\tau }}^{m}}{\partial {{\varvec{D}}}}\cdot \frac{\partial {{\varvec{D}}}}{\partial \overline{{{\varvec{D}}}}}\right]&+\quad&{\varvec{\Gamma }}^{m,e}*\left[ \frac{\partial {\varvec{\tau }}^{e}}{\partial {{\varvec{F}}}}: \frac{\partial {{\varvec{F}}}}{\partial \overline{{{\varvec{D}}}}} + \frac{\partial {\varvec{\tau }}^{e}}{\partial {{\varvec{D}}}}\cdot \frac{\partial {{\varvec{D}}}}{\partial \overline{{{\varvec{D}}}}}\right] , \\ \frac{\partial {\widetilde{{{\varvec{D}}}}}}{\partial \overline{{{\varvec{D}}}}}&={\varvec{\Gamma }}^{e,m} *\left[ \frac{\partial {\varvec{\tau }}^{m}}{\partial {{\varvec{F}}}}: \frac{\partial {{\varvec{F}}}}{\partial \overline{{{\varvec{D}}}}} + \frac{\partial {\varvec{\tau }}^{m}}{\partial {{\varvec{D}}}}\cdot \frac{\partial {{\varvec{D}}}}{\partial \overline{{{\varvec{D}}}}}\right]&+\quad&{\varvec{\Gamma }}^{e\phantom {,m}}*\left[ \frac{\partial {\varvec{\tau }}^{e}}{\partial {{\varvec{F}}}}: \frac{\partial {{\varvec{F}}}}{\partial \overline{{{\varvec{D}}}}} + \frac{\partial {\varvec{\tau }}^{e}}{\partial {{\varvec{D}}}}\cdot \frac{\partial {{\varvec{D}}}}{\partial \overline{{{\varvec{D}}}}}\right] , \\ \frac{\partial {\widetilde{{{\varvec{D}}}}}}{\partial {\overline{{{\varvec{F}}}}}}&={\varvec{\Gamma }}^{e,m} *\left[ \frac{\partial {\varvec{\tau }}^{m}}{\partial {{\varvec{F}}}}: \frac{\partial {{\varvec{F}}}}{\partial {\overline{{{\varvec{F}}}}}} + \frac{\partial {\varvec{\tau }}^{m}}{\partial {{\varvec{D}}}}\cdot \frac{\partial {{\varvec{D}}}}{\partial {\overline{{{\varvec{F}}}}}}\right]&+\quad&{\varvec{\Gamma }}^{e\phantom {,m}}*\left[ \frac{\partial {\varvec{\tau }}^{e}}{\partial {{\varvec{F}}}}: \frac{\partial {{\varvec{F}}}}{\partial {\overline{{{\varvec{F}}}}}} + \frac{\partial {\varvec{\tau }}^{e}}{\partial {{\varvec{D}}}}\cdot \frac{\partial {{\varvec{D}}}}{\partial {\overline{{{\varvec{F}}}}}}\right] . \end{aligned} \end{aligned}$$
(C.2)

Lastly, inserting the definition (27) and further using the decompositions \({{\varvec{F}}}= {\overline{{{\varvec{F}}}}} + {\widetilde{{{\varvec{F}}}}}\) and \({{\varvec{D}}}= \overline{{{\varvec{D}}}} + {\widetilde{{{\varvec{D}}}}}\), we arrive at the final form

$$\begin{aligned} \frac{\partial {\widetilde{{{\varvec{F}}}}}}{\partial {\overline{{{\varvec{F}}}}}}= & {} {\varvec{\Gamma }}^{m\phantom {,e}} *\left[ {\mathbb {A}}^\Delta + {\mathbb {A}}^\Delta :\frac{\partial {\widetilde{{{\varvec{F}}}}}}{\partial {\overline{{{\varvec{F}}}}}} + {{\varvec{q}}}^\Delta \cdot \frac{\partial {\widetilde{{{\varvec{D}}}}}}{\partial {\overline{{{\varvec{F}}}}}} \right] + {\varvec{\Gamma }}^{m,e} *\left[ ({{\varvec{q}}}^\Delta )^T + ({{\varvec{q}}}^\Delta )^T:\frac{\partial {\widetilde{{{\varvec{F}}}}}}{\partial {\overline{{{\varvec{F}}}}}} + {\mathbb {K}}^{\Delta } \cdot \frac{\partial {\widetilde{{{\varvec{D}}}}}}{\partial {\overline{{{\varvec{F}}}}}} \right] ,\nonumber \\ \frac{\partial {\widetilde{{{\varvec{F}}}}}}{\partial \overline{{{\varvec{D}}}}}= & {} {\varvec{\Gamma }}^{m\phantom {,e}} *\left[ {{\varvec{q}}}^\Delta + {{\varvec{q}}}^\Delta \cdot \frac{\partial {\widetilde{{{\varvec{D}}}}}}{\partial \overline{{{\varvec{D}}}}} + {\mathbb {A}}^\Delta : \frac{\partial {\widetilde{{{\varvec{F}}}}}}{\partial \overline{{{\varvec{D}}}}} \right] + {\varvec{\Gamma }}^{m,e} *\left[ {\mathbb {K}}^\Delta + {\mathbb {K}}^\Delta \cdot \frac{\partial {\widetilde{{{\varvec{D}}}}}}{\partial \overline{{{\varvec{D}}}}} + ({{\varvec{q}}}^\Delta )^T :\frac{\partial {\widetilde{{{\varvec{F}}}}}}{\partial \overline{{{\varvec{D}}}}} \right] ,\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad ~~~~~\\ \frac{\partial {\widetilde{{{\varvec{D}}}}}}{\partial \overline{{{\varvec{D}}}}}= & {} {\varvec{\Gamma }}^{e,m} *\left[ {{\varvec{q}}}^\Delta + {{\varvec{q}}}^\Delta \cdot \frac{\partial {\widetilde{{{\varvec{D}}}}}}{\partial \overline{{{\varvec{D}}}}} + {\mathbb {A}}^\Delta :\frac{\partial {\widetilde{{{\varvec{F}}}}}}{\partial \overline{{{\varvec{D}}}}} \right] + {\varvec{\Gamma }}^{e\phantom {,m}} *\left[ {\mathbb {K}}^\Delta + {\mathbb {K}}^\Delta \cdot \frac{\partial {\widetilde{{{\varvec{D}}}}}}{\partial \overline{{{\varvec{D}}}}} + ({{\varvec{q}}}^\Delta )^T :\frac{\partial {\widetilde{{{\varvec{F}}}}}}{\partial \overline{{{\varvec{D}}}}} \right] ,\nonumber \\ \frac{\partial {\widetilde{{{\varvec{D}}}}}}{\partial {\overline{{{\varvec{F}}}}}}= & {} {\varvec{\Gamma }}^{e,m} *\left[ {\mathbb {A}}^\Delta + {\mathbb {A}}^\Delta :\frac{\partial {\widetilde{{{\varvec{F}}}}}}{\partial {\overline{{{\varvec{F}}}}}} + {{\varvec{q}}}^\Delta \cdot \frac{\partial {\widetilde{{{\varvec{D}}}}}}{\partial {\overline{{{\varvec{F}}}}}} \right] + {\varvec{\Gamma }}^{e\phantom {,m}} *\left[ ({{\varvec{q}}}^\Delta )^T + ({{\varvec{q}}}^\Delta )^T:\frac{\partial {\widetilde{{{\varvec{F}}}}}}{\partial {\overline{{{\varvec{F}}}}}} + {\mathbb {K}}^{\Delta } \cdot \frac{\partial {\widetilde{{{\varvec{D}}}}}}{\partial {\overline{{{\varvec{F}}}}}} \right] ,\nonumber \end{aligned}$$
(C.3)

where \((\bullet )^{\Delta }\) according to (42) denotes the difference of the moduli w.r.t. the linear reference medium. Equation (C.4) corresponds to the generic operator introduced in (43) as

$$\begin{aligned} -{\varvec{\Gamma }}*{\mathbb {C}}^\Delta = {\varvec{\Gamma }}*\left( {\mathbb {C}}^\Delta \diamond \dfrac{\partial {\widetilde{{\mathbb {G}}}}}{\partial {\overline{{\mathbb {G}}}}} \right) - \dfrac{\partial {\widetilde{{\mathbb {G}}}}}{\partial {\overline{{\mathbb {G}}}}}. \end{aligned}$$
(C.4)

Note that neglecting the macroscopic coupled sensitivities by setting \({\partial {\widetilde{{{\varvec{F}}}}}}/{\partial \overline{{{\varvec{D}}}}} = {\partial {\widetilde{{{\varvec{D}}}}}}/{\partial {\overline{{{\varvec{F}}}}}} = {{{\varvec{0}}}}\) leads to a reduced system.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Göküzüm, F.S., Nguyen, L.T.K. & Keip, MA. A multiscale FE-FFT framework for electro-active materials at finite strains. Comput Mech 64, 63–84 (2019). https://doi.org/10.1007/s00466-018-1657-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-018-1657-7

Keywords

Search

Navigation