Skip to main content
SpringerLink
Log in

Spectral Solvers for Crystal Plasticity and Multi-physics Simulations

  • Reference work entry
  • First Online:
Handbook of Mechanics of Materials

Abstract

The local and global behavior of materials with internal microstructure is often investigated on a (representative) volume element. Typically, periodic boundary conditions are applied on such “virtual specimens” to reflect the situation in the bulk of the material. Spectral methods based on Fast Fourier Transforms (FFT) have been established as a powerful numerical tool especially suited for this task. Starting from the pioneering work of Moulinec and Suquet, FFT-based solvers have been significantly improved with respect to performance and stability. Recent advancements of using the spectral approach to solve coupled field equations enable also the modeling of multiphysical phenomena such as fracture propagation, temperature evolution, chemical diffusion, and phase transformation in conjunction with the mechanical boundary value problem. The fundamentals of such a multi-physics framework, which is implemented in the Düsseldorf Advanced Materials Simulation Kit (DAMASK), are presented here together with implementation aspects. The capabilities of this approach are demonstrated on illustrative examples.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 919.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 1,299.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    To simplify the notation, in the following, the argument x is dropped whenever it is possible, i.e., F(x) is denoted as F only

  2. 2.

    Quantities in real space and Fourier space are distinguished by notation Q(x) and Q(k), respectively, with x the position in real space, k the frequency vector in Fourier space, and i2 =  − 1.  ℱ−1[⋅] denotes the inverse Fourier transform.

  3. 3.

    The solution for the deformation gradient field, i.e., the actual spectral method procedure, is performed in parallel to these iterations.

References

  1. Lebensohn RA, Brenner R, Castelnau O, Rollett AD. Orientation image-based micromechanical modelling of subgrain texture evolution in polycrystalline copper. Acta Mater. 2008;56(15):3914–26. https://doi.org/10.1016/j.actamat.2008.04.016.

    Article  Google Scholar 

  2. Suquet P, Moulinec H, Castelnau O, Montagnat M, Lahellec N, Grennerat F, Duval P, Brenner R. Multi-scale modeling of the mechanical behavior of polycrystalline ice under transient creep. In: Procedia IUTAM: IUTAM symposium on linking scales in computation: from microstructure to macroscale properties, vol. 3, p. 64–78. 2012. https://doi.org/10.1016/j.piutam.2012.03.006.

    Article  Google Scholar 

  3. Ma D, Eisenlohr P, Epler E, Volkert CA, Shanthraj P, Diehl M, Roters F, Raabe D. Crystal plasticity study of monocrystalline stochastic honeycombs under in-plane compression. Acta Mater. 2016;103:796–808. https://doi.org/10.1016/j.actamat.2015.11.016.

    Article  Google Scholar 

  4. De Geus TWJ, Maresca F, Peerlings RHJ, Geers MGD. Microscopic plasticity and damage in two-phase steels: on the competing role of crystallography and phase contrast. Mech Mater. 2016;101:147–59. https://doi.org/10.1016/j.mechmat.2016.07.014.

    Article  Google Scholar 

  5. Zhang H, Diehl M, Roters F, Raabe D. A virtual laboratory for initial yield surface determination using high resolution crystal plasticity simulations. Int J Plast. 2016;80:111–38. https://doi.org/10.1016/j.ijplas.2016.01.002.

    Article  Google Scholar 

  6. Arul Kumar M, Beyerlein IJ, McCabe RJ, Tomé CN. Grain neighbour effects on twin transmission in hexagonal close-packed materials. Nat Commun. 2016;7:13826. https://doi.org/10.1038/ncomms13826.

    Article  Google Scholar 

  7. Diehl M, Shanthraj P, Eisenlohr P, Roters F. Neighborhood influences on stress and strain partitioning in dual-phase microstructures. An investigation on synthetic polycrystals with a robust spectral-based numerical method. Meccanica. 2016;51(2):429–41. https://doi.org/10.1007/s11012-015-0281-2.

    Article  MathSciNet  Google Scholar 

  8. Moulinec H, Suquet P. A fast numerical method for computing the linear and nonlinear properties of composites. C R Acad Sci Sér II Mécanique Phys Chim Astronomie. 1994;318:1417–23.

    MATH  Google Scholar 

  9. Monchiet V, Bonnet G. A polarization-based FFT iterative scheme for computing the effective properties of elastic composites with arbitrary contrast. Int J Numer Methods Eng. 2012;89(11):1419–36. https://doi.org/10.1002/nme.3295.

    Article  MathSciNet  MATH  Google Scholar 

  10. Moulinec H, Silva F. Comparison of three accelerated FFT-based schemes for computing the mechanical response of composite materials. Int J Numer Methods Eng. 2014;97(13):960–85. https://doi.org/10.1002/nme.4614.

    Article  MathSciNet  MATH  Google Scholar 

  11. Schneider M. Convergence of FFT-based homogenization for strongly heterogeneous media. Math Methods Appl Sci. 2015;38(13):2761–2778. https://doi.org/10.1002/mma.3259.

    Article  MathSciNet  Google Scholar 

  12. Zeman J, Vondřejc J, Novák J, Marek I. Accelerating a FFT-based solver for numerical homogenization of periodic media by conjugate gradients. J Comput Phys. 2010;229(21):8065–71. https://doi.org/10.1016/j.jcp.2010.07.010.

    Article  MathSciNet  MATH  Google Scholar 

  13. Zeman J, De Geus TWJ, Vondřejc J, Peerlings RHJ, Geers MGD. A finite element perspective on non-linear FFT-based micromechanical simulations. Int J Numer Methods Eng. 2017;111(10):903–926. https://doi.org/10.1002/nme.5481.

    Article  MathSciNet  Google Scholar 

  14. Roters F, Eisenlohr P, Kords C, Tjahjanto DD, Diehl M, Raabe D. DAMASK: the Düsseldorf advanced Material simulation kit for studying crystal plasticity using an FE based or a spectral numerical solver. In: Cazacu O, editor. Procedia IUTAM: IUTAM symposium on linking scales in computation: from microstructure to macroscale properties, vol. 3. Amsterdam: Elsevier; 2012. p. 3–10. https://doi.org/10.1016/j.piutam.2012.03.001.

    Chapter  Google Scholar 

  15. Roters F, Diehl M, Shanthraj P, Eisenlohr P, Reuber C, Wong SL, Maiti T, Ebrahimi A, Hochrainer T, Fabritius H-O, Nikolov S, Friák M, Fujita N, Grilli N, Janssens KGF, Jia N, Kok PJJ, Ma D, Meier F, Werner E, Stricker M, Weygand D, Raabe D. DAMASK – the Düsseldorf advanced material simulation Kit for modelling multi-physics crystal plasticity, damage, and thermal phenomena from the single crystal up to the component scale. Comput. Mater. Sci. 2018. (article in press). https://doi.org/10.1016/j.commatsci.2018.04.030.

    Article  Google Scholar 

  16. Trefethen LN. Finite differences and spectral methods for ordinary and partial differtential equations. Ithaca, 1996. https://www.math.hmc.edu/~dyong/math165/trefethenbook.pdf

  17. Boyd JP. Chebyshev and Fourier spectral methods. 2nd ed. New York: Dover Publications; 2000.

    Google Scholar 

  18. Trefethen LN. Spectral methods in MATLAB. Soc Ind Appl Math. 2012. https://doi.org/10.1137/1.9780898719598.

  19. Shanthraj P, Svendsen B, Sharma L, Roters F, Raabe D. Elasto-viscoplastic phase field modelling of anisotropic cleavage fracture. J Mech Phys Solids. 2017;99:19–34. https://doi.org/10.1016/j.jmps.2016.10.012.

    Article  MathSciNet  Google Scholar 

  20. Hutchinson JW. Bounds and self-consistent estimates for creep of polycrystalline materials. Proc R Soc A. 1976;348:101–27. https://doi.org/10.1098/rspa.1976.0027.

    Article  MATH  Google Scholar 

  21. Peirce D, Asaro RJ, Needleman A. An analysis of nonuniform and localized deformation in ductile single crystals. Acta Metall. 1982;30(6):1087–119. https://doi.org/10.1016/0001-6160(82)90005-0.

    Article  Google Scholar 

  22. Lebensohn RA. N-site modeling of a 3D viscoplastic polycrystal using fast Fourier transform. Acta Mater. 2001;49(14):2723–37. https://doi.org/10.1016/S1359-6454(01)00172-0.

    Article  Google Scholar 

  23. Lebensohn RA, Castelnau O, Brenner R, Gilormini P. Study of the antiplane deformation of linear 2-D polycrystals with different microstructures. Int J Solids Struct. 2005;42(20):5441–59. https://doi.org/10.1016/j.ijsolstr.2005.02.051.

    Article  MATH  Google Scholar 

  24. Lefebvre G, Sinclair CW, Lebensohn RA, Mithieux J-D. Accounting for local interactions in the prediction of roping of ferritic stainless steel sheets. Model Simul Mater Sci Eng. 2012;20(2):024008. https://doi.org/10.1088/0965-0393/20/2/024008.

    Article  Google Scholar 

  25. Lebensohn RA, Kanjarla AK, Eisenlohr P. An elasto-viscoplastic formulation based on fast Fourier transforms for the prediction of micromechanical fields in polycrystalline materials. Int J Plast. 2012;32–33:59–69. https://doi.org/10.1016/j.ijplas.2011.12.005.

    Article  Google Scholar 

  26. Grennerat F, Montagnat M, Castelnau O, Vacher P, Moulinec H, Suquet P, Duval P. Experimental characterization of the intragranular strain field in columnar ice during transient creep. Acta Mater. 2012;60(8):3655–66. https://doi.org/10.1016/j.actamat.2012.03.025.

    Article  Google Scholar 

  27. Michel JC, Moulinec H, Suquet P. A computational scheme for linear and non-linear composites with arbitrary phase contrast. Int J Numer Methods Eng. 2001;52(12):139–60. https://doi.org/10.1002/nme.275.

    Article  Google Scholar 

  28. Eyre DJ, Milton GW. A fast numerical scheme for computing the response of composites using grid refinement. Eur Phys J Appl Phys. 1999;6(1):41–7. https://doi.org/10.1051/epjap:1999150.

    Article  Google Scholar 

  29. Brisard S, Dormieux L. FFT-based methods for the mechanics of composites: a general variational framework. Comput Mater Sci. 2010;49(3):663–71. https://doi.org/10.1016/j.commatsci.2010.06.009.

    Article  Google Scholar 

  30. Schneider M, Merkert D, Kabel M. FFT-based homogenization for microstructures discretized by linear hexahedral elements. Int J Numer Methods Eng. 2016;109(10):1461–89. https://doi.org/10.1002/nme.5336.

    Article  MathSciNet  MATH  Google Scholar 

  31. De Geus TWJ, Vondřejc J, Zeman J, Peerlings RHJ, Geers MGD. Finite strain FFT-based non-linear solvers made simple. Comput Methods Appl Mech Eng. 2017;12:032. https://doi.org/10.1016/j.cma.2016.

    Article  MathSciNet  Google Scholar 

  32. Vinogradov V, Milton GW. An accelerated FFT algorithm for thermoelastic and non-linear composites. Int J Numer Methods Eng. 2008;76(11):1678–95. https://doi.org/10.1002/nme.2375.

    Article  MathSciNet  MATH  Google Scholar 

  33. Shanthraj P, Sharma L, Svendsen B, Roters F, Raabe D. A phase field model for damage in elasto-viscoplastic materials. Comput Methods Appl Mech Eng. 2016;312:167–85. https://doi.org/10.1016/j.cma.2016.05.006.

    Article  MathSciNet  Google Scholar 

  34. Eisenlohr P, Diehl M, Lebensohn RA, Roters F. A spectral method solution to crystal elasto-viscoplasticity at finite strains. Int J Plast. 2013;46:37–53. https://doi.org/10.1016/j.ijplas.2012.09.012.

    Article  Google Scholar 

  35. Shanthraj P, Eisenlohr P, Diehl M, Roters F. Numerically robust spectral methods for crystal plasticity simulations of heterogeneous materials. Int J Plast. 2015;66:31–45. https://doi.org/10.1016/j.ijplas.2014.02.006.

    Article  Google Scholar 

  36. Lahellec N, Michel JC, Moulinec H, Suquet P. Analysis of inhomogeneous materials at large strains using fast Fourier transforms. In: Miehe C, editor. IUTAM symposium on computational mechanics of solid materials at large strains, Solid mechanics and its applications, vol. 108. Dordrecht: Kluwer; 2001. p. 247–58. https://doi.org/10.1007/978-94-017-0297-3_22.

    Chapter  MATH  Google Scholar 

  37. Mura T. Micromechanics of defects in solids. 2nd ed. Dordrecht: Martinus Nijhoff Publishers; 1987.

    Book  Google Scholar 

  38. Michel JC, Moulinec H, Suquet P. A computational method based on augmented Lagrangians and fast Fourier transforms for composites with high contrast. Comput Model Eng Sci. 2000;1(2):79–88. https://doi.org/10.3970/cmes.2000.001.239.

    Article  MathSciNet  Google Scholar 

  39. Kachanov L. Introduction to continuum damage mechanics. Dodrecht: Springer Science+Business Media; 1986.

    Book  Google Scholar 

  40. Chaboche JL. Continuum damage mechanics 1. General concepts. J Appl Mech Trans ASME. 1988;55(1):59–64.

    Article  Google Scholar 

  41. Bourdin B, Francfort GA, Marigo J-J. Numerical experiments in revisited brittle fracture. J Mech Phys Solids. 2000;48(4):797–826. https://doi.org/10.1016/S0022-5096(99)00028-9.

    Article  MathSciNet  MATH  Google Scholar 

  42. Miehe C, Welschinger F, Hofacker M. Thermodynamically consistent phase-field models of fracture: Variational principles and multi-field FE implementations. Int J Numer Methods Eng. 2010;83(10):1273–311. https://doi.org/10.1002/nme.2861.

    Article  MathSciNet  MATH  Google Scholar 

  43. Frigo M, Johnson SG. The design and implementation of FFTW3. Proc IEEE. 2005;93(2):216–31. https://doi.org/10.1109/JPR0C.2004.840301.

    Article  Google Scholar 

  44. Gottlieb D, Shu C-W. On the Gibbs phenomenon and its resolution. SIAM Rev. 1997;39(4):644–68. http://www.jstor.org/stable/2132695.

    Article  MathSciNet  Google Scholar 

  45. Gelb A, Gottlieb S. Chapter: The resolution of the Gibbs phenomenon for fourier spectral methods. In: Jerri, A. (Ed.), Advances in the Gibbs phenomenon. Potsdam, New York: Sampling Publishing; 2007.

    Google Scholar 

  46. Willot F. Fourier-based schemes for computing the mechanical response of composites with accurate local fields. C R Mécanique. 2015;343(3):232–45.

    Article  MathSciNet  Google Scholar 

  47. Kaßbohm S, Müller WH, Feßler R. Improved approximations of Fourier coefficients for computing periodic structures with arbitrary stiffness distribution. Comput Mater Sci. 2006;37(1–2):90–3. https://doi.org/10.1016/j.commatsci.2005.12.010.

    Article  Google Scholar 

  48. Schneider M, Ospald F, Kabel M. Computational homogenization of elasticity on a staggered grid. Int J Numer Methods Eng. 2015;105(9):693–720. https://doi.org/10.1002/nme.5008.

    Article  MathSciNet  Google Scholar 

  49. Kabel M, Fliegener S, Schneider M. Mixed boundary conditions for FFT-based homogenization at finite strains. Comput Mater Sci. 2016;57(2):193–210. https://doi.org/10.1007/s00466-015-1227-1.

    Article  MathSciNet  MATH  Google Scholar 

  50. Moulinec H, Suquet P. A numerical method for computing the overall response of nonlinear composites with complex microstructure. Comput Methods Appl Mech Eng. 1998;157(1–2):69–94. https://doi.org/10.1016/S0045-7825(97)00218-1.

    Article  MathSciNet  MATH  Google Scholar 

  51. 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. 2013;77:430–9. https://doi.org/10.1016/j.commatsci.2013.04.046.

    Article  Google Scholar 

  52. Balay S, Brown J, Buschelman K, Eijkhout V, Gropp WD, Kaushik D, Knepley MG, McInnes LC, Smith BF, Zhang H. PETSc users manual. Technical report, 2013.

    Google Scholar 

  53. Kelley CT. Iterative methods for linear and nonlinear equations, Frontiers in applied mathematics, vol. 16. Philadelphia: Society for Industrial and Applied Mathematics; 1995. www.siam.org/books/textbooks/fr16_book.pdf.

    Book  Google Scholar 

  54. Oosterlee CW, Washio T. Krylov subspace acceleration of nonlinear multigrid with application to recirculating flows. SIAM J Sci Comput. 2000;21(5):1670–90. https://doi.org/10.1137/S1064827598338093.

    Article  MathSciNet  MATH  Google Scholar 

  55. Chen Y, Shen C. A Jacobian-free Newton-GMRES(m) method with adaptive preconditioner and its application for power flow calculations. IEEE Trans Power Syst. 2006;21(3):1096–103. https://doi.org/10.1109/TPWRS.2006.876696.

    Article  Google Scholar 

  56. Eisenstat SC, Walker HF. Choosing the forcing terms in an inexact Newton method. SIAM J Sci Comput. 1996;17(1):16–32. https://doi.org/10.1137/0917003.

    Article  MathSciNet  MATH  Google Scholar 

  57. Andreev VI, Cybin NY. The inhomogeneous plate with a hole: Kirsch’s problem. Procedia Eng. 2014;91:26–31. https://doi.org/10.1016/j.proeng.2014.12.006.

    Article  Google Scholar 

Download references

Acknowledgments

PS and FR acknowledge funding through SFB 761 Steel – ab initio by the Deutsche Forschungsgemeinschaft (DFG). MD acknowledges the funding of the TCMPrecipSteel project in the framework of the SPP 1713 Strong coupling of thermo-chemical and thermo-mechanical states in applied materials by the DFG.

Author information

Authors and Affiliations

  1. Max-Planck-Institut für Eisenforschung GmbH, Düsseldorf, Germany

    Pratheek Shanthraj, Martin Diehl, Franz Roters & Dierk Raabe

  2. Chemical Engineering and Materials Science, Michigan State University, East Lansing, MI, USA

    Philip Eisenlohr

  3. The School of Materials, The University of Manchester, Manchester, UK

    Pratheek Shanthraj

Authors
  1. Pratheek Shanthraj
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martin Diehl
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Philip Eisenlohr
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Franz Roters
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Dierk Raabe
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pratheek Shanthraj .

Editor information

Editors and Affiliations

  1. Institute for Materials Testing, Materials Science and Strength of Materials, University of Stuttgart, Stuttgart, Germany

    Siegfried Schmauder

  2. Department of Civil Engineering, National Taiwan University, Taipei, Taiwan

    Chuin-Shan Chen

  3. UAB Mail BEC 254, Birmingham, AL, USA

    Krishan K. Chawla

  4. Fulton School of Engineering, Arizona State University , Tempe, AZ, USA

    Nikhilesh Chawla

  5. Engineering Mechanics, Zhejiang University , Hangzhou, China

    Weiqiu Chen

  6. Katayanagi Advanced Research Laboratories, Tokyo University of Technology, Tokyo, Japan

    Yutaka Kagawa

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Singapore Pte Ltd.

About this entry

Check for updates. Verify currency and authenticity via CrossMark

Cite this entry

Shanthraj, P., Diehl, M., Eisenlohr, P., Roters, F., Raabe, D. (2019). Spectral Solvers for Crystal Plasticity and Multi-physics Simulations. In: Schmauder, S., Chen, CS., Chawla, K., Chawla, N., Chen, W., Kagawa, Y. (eds) Handbook of Mechanics of Materials. Springer, Singapore. https://doi.org/10.1007/978-981-10-6884-3_80

Download citation

Publish with us

Policies and ethics

Search

Navigation