Comptes Rendus
no. 3
Fourier-based schemes for computing the mechanical response of composites with accurate local fields
François Willot1 
1 Mines ParisTech, PSL Research University, Centre for mathematical morphology, 35, rue Saint-Honoré, 77300 Fontainebleau, France
Comptes Rendus. Mécanique, Volume 343 (2015) no. 3, pp. 232-245.

We modify the Green operator involved in Fourier-based computational schemes in elasticity, in 2D and 3D. The new operator is derived by expressing continuum mechanics in terms of centered differences on a rotated grid. Using the modified Green operator leads, in all systems investigated, to more accurate strain and stress fields than using the discretizations proposed by Moulinec and Suquet (1994) [1] or Willot and Pellegrini (2008) [2]. Moreover, we compared the convergence rates of the “direct” and “accelerated” FFT schemes with the different discretizations. The discretization method proposed in this work allows for much faster FFT schemes with respect to two criteria: stress equilibrium and effective elastic moduli.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crme.2014.12.005
Mots clés : FFT methods, Homogenization, Heterogeneous media, Linear elasticity, Computational mechanics, Spectral methods
François Willot 1

1 Mines ParisTech, PSL Research University, Centre for mathematical morphology, 35, rue Saint-Honoré, 77300 Fontainebleau, France 
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François Willot. Fourier-based schemes for computing the mechanical response of composites with accurate local fields. Comptes Rendus. Mécanique, Volume 343 (2015) no. 3, pp. 232-245. doi : 10.1016/j.crme.2014.12.005. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2014.12.005/

[1] H. Moulinec; P. Suquet A fast numerical method for computing the linear and non linear mechanical properties of the composites, C. R. Acad. Sci. Paris, Ser. 2, Méc. Phys. Chim. Astron., Volume 318 (1994) no. 11, pp. 1417-1423

[2] F. Willot; Y.-P. Pellegrini Fast Fourier transform computations and build-up of plastic deformation in 2D, elastic–perfectly plastic, pixelwise-disordered porous media, CMDS 11 (D. Jeulin; S. Forest, eds.), École des Mines, Paris (2008), pp. 443-449

[3] S.-B. Lee; R.A. Lebensohn; A.D. Rollett Modeling the viscoplastic micromechanical response of two-phase materials using fast Fourier transforms, Int. J. Plast., Volume 27 (2011) no. 5, pp. 707-727

[4] J. Li; S. Meng; X. Tian; F. Song; C. Jiang A non-local fracture model for composite laminates and numerical simulations by using the FFT method, Composites, Part B, Eng., Volume 43 (2011) no. 3, pp. 961-971

[5] M. Faessel; D. Jeulin Segmentation of 3D microtomographic images of granular materials with the stochastic watershed, J. Microsc., Volume 239 (2010) no. 1, pp. 17-31

[6] F. Willot; L. Gillibert; D. Jeulin Microstructure-induced hotspots in the thermal and elastic responses of granular media, Int. J. Solids Struct., Volume 50 (2013) no. 10, pp. 1699-1709

[7] C.F. Dunant; B. Bary; A.B. Giorla; C. Péniguel; J. Sanahuja; C. Toulemonde; A.B. Tran; F. Willot; J. Yvonnet A critical comparison of several numerical methods for computing effective properties of highly heterogeneous materials, Adv. Eng. Softw., Volume 58 (2013), pp. 1-12

[8] F. Willot; D. Jeulin Elastic and electrical behavior of some random multiscale highly-contrasted composites, Int. J. Multiscale Comput. Eng., Volume 9 (2011) no. 3, pp. 305-326

[9] J. Escoda; F. Willot; D. Jeulin; J. Sanahuja; C. Toulemonde Estimation of local stresses and elastic properties of a mortar sample by FFT computation of fields on a 3D image, Cem. Concr. Res., Volume 41 (2011) no. 5, pp. 542-556

[10] Morphhom software, October 4, 2014 http://cmm.ensmp.fr/morphhom (accessed)

[11] CraFT software, October 4, 2014 http://craft.lma.cnrs-mrs.fr (accessed)

[12] GeoDict software, October 4, 2014 http://www.geodict.de (accessed)

[13] J. Vondřejc; J. Zeman; I. Marek An FFT-based Galerkin method for homogenization of periodic media, Comput. Math. Appl., Volume 68 (2014) no. 3, pp. 156-173

[14] S. Brisard; L. Dormieux 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., Volume 217 (2012) no. 220, pp. 197-212

[15] J. Zeman; J. Vodrejc; J. Novak; I. Marek Accelerating a FFT-based solver for numerical homogenization of a periodic media by conjugate gradients, J. Comput. Phys., Volume 229 (2010) no. 21, pp. 8065-8071

[16] S. Brisard; L. Dormieux FFT-based methods for the mechanics of composites: a general variational framework, Comput. Mater. Sci., Volume 49 (2010) no. 3, pp. 663-671

[17] S. Brisard; F. Legoll Periodic homogenization using the Lippmann–Schwinger formalism, 2014 | arXiv

[18] F. Willot; B. Abdallah; Y.-P. Pellegrini Fourier-based schemes with modified green operator for computing the electrical response of heterogeneous media with accurate local fields, Int. J. Numer. Methods Eng., Volume 98 (2014) no. 7, pp. 518-533

[19] W.H. Müller Mathematical vs. experimental stress analysis of inhomogeneities in solids, J. Phys., Volume 6 (1996) no. C1, pp. 139-148

[20] C.M. Brown; W. Dreyer; W.H. Müller Discrete Fourier transforms and their application to stress–strain problems in composite mechanics: a convergence study, Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci., Volume 458 (2002), pp. 1967-1987

[21] M.F. Thorpe; P.N. Sen Elastic moduli of two-dimensional composite continua with elliptic inclusions, J. Acoust. Soc. Am., Volume 77 (1985) no. 5, pp. 1674-1680

[22] G.W. Milton The Theory of Composites, Cambridge University Press, Cambridge, UK, 2002

[23] S. Kanaun; V. Levin Self-Consistent Methods for Composites, Springer, Dordrecht, The Netherlands, 2008

[24] H. Moulinec; P. Suquet A numerical method for computing the overall response of nonlinear composites with complex microstructure, Comput. Methods Appl. Mech. Eng., Volume 157 (1998) no. 1, pp. 69-94

[25] D.J. Eyre; G.W. Milton A fast numerical scheme for computing the response of composites using grid refinement, Eur. Phys. J. Appl. Phys., Volume 6 (1999) no. 1, pp. 41-47

[26] J.-C. Michel; H. Moulinec; P. Suquet A computational scheme for linear and non-linear composites with arbitrary phase contrast, Int. J. Numer. Methods Eng., Volume 52 (2001) no. 1–2, pp. 139-160

[27] H. Moulinec; F. Silva Comparison of three accelerated FFT-based schemes for computing the mechanical response of composite materials, Int. J. Numer. Methods Eng., Volume 97 (2014) no. 13, pp. 960-985

[28] I. Cohen Simple algebraic approximations for the effective elastic moduli of cubic arrays of spheres, J. Mech. Phys. Solids, Volume 52 (2004) no. 9, pp. 2167-2183

[29] F. Willot; D. Jeulin Elastic behavior of composites containing boolean random sets of inhomogeneities, Int. J. Eng. Sci., Volume 47 (2009) no. 2, pp. 313-324

[30] H. Moulinec; P. Suquet Comparison of FFT-based methods for computing the response of composites with highly contrasted mechanical properties, Physica B, Condens. Matter, Volume 338 (2003) no. 1–4, pp. 58-60

[31] N. Lahellec; J.-C. Michel; H. Moulinec; P. Suquet Analysis of inhomogeneous materials at large strains using fast Fourier transforms, Proc. IUTAM Symposium on Computational Mechanics of Solid Materials at Large Strains, Kluwer Academic Publishers, 2001, pp. 247-258

[32] J.-C. Michel; H. Moulinec; P. Suquet A computational method based on augmented Lagrangians and fast Fourier transforms for composites with high contrast, Comput. Model. Eng. Sci., Volume 1 (2000) no. 2, pp. 79-88

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