Fourier-based schemes for computing the mechanical response of composites with accurate local fields

Abstract 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.

[1]  Hervé Moulinec,et al.  A Computational Method Based on Augmented Lagrangians and Fast Fourier Transforms for Composites with High Contrast , 2000 .

[2]  Luc Dormieux,et al.  FFT-based methods for the mechanics of composites: A general variational framework , 2010 .

[3]  V. Levin,et al.  Self-Consistent Methods for Composites , 2008 .

[4]  Hervé Moulinec,et al.  Comparison of three accelerated FFT‐based schemes for computing the mechanical response of composite materials , 2014 .

[5]  L. Dormieux,et al.  Combining Galerkin approximation techniques with the principle of Hashin and Shtrikman to derive a new FFT-based numerical method for the homogenization of composites , 2012 .

[6]  F. Willot,et al.  Fourier‐based schemes with modified Green operator for computing the electrical response of heterogeneous media with accurate local fields , 2013, 1307.1015.

[7]  F. Willot,et al.  Fast Fourier Transform computations and build-up of plastic deformation in 2D, elastic-perfectly plastic, pixelwise disordered porous media , 2008, 0802.2488.

[8]  Hervé Moulinec,et al.  Comparison of FFT-based methods for computing the response of composites with highly contrasted mechanical properties , 2003 .

[9]  Julien Yvonnet,et al.  A critical comparison of several numerical methods for computing effective properties of highly heterogeneous materials , 2013, Adv. Eng. Softw..

[10]  Jaroslav Vondrejc,et al.  An FFT-based Galerkin method for homogenization of periodic media , 2013, Comput. Math. Appl..

[11]  I. Cohen.  Simple algebraic approximations for the effective elastic moduli of cubic arrays of spheres , 2004 .

[12]  Wolfgang H. Müller,et al.  Mathematical vs. Experimental Stress Analysis of Inhomogeneities in Solids , 1996 .

[13]  A. Rollett,et al.  Modeling the viscoplastic micromechanical response of two-phase materials using Fast Fourier Transforms , 2011 .

[14]  Hervé Moulinec,et al.  A computational scheme for linear and non‐linear composites with arbitrary phase contrast , 2001 .

[15]  H. Moulinec,et al.  A fast numerical method for computing the linear and nonlinear mechanical properties of composites , 1994 .

[16]  D. Jeulin,et al.  Elastic behavior of composites containing Boolean random sets of inhomogeneities , 2009 .

[17]  Graeme W. Milton,et al.  A fast numerical scheme for computing the response of composites using grid refinement , 1999 .

[18]  W. Müller,et al.  Discrete Fourier transforms and their application to stress—strain problems in composite mechanics: a convergence study† , 2002, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[19]  D. Jeulin,et al.  Microstructure-induced hotspots in the thermal and elastic responses of granular media , 2013 .

[20]  D. Jeulin,et al.  Elastic and electrical behavior of some random multiscale highly-contrasted composites , 2011 .

[21]  G. Milton.  The Theory of Composites , 2002 .

[22]  J. Michel,et al.  Analysis of Inhomogeneous Materials at Large Strains using Fast Fourier Transforms , 2003 .

[23]  Dominique Jeulin,et al.  Estimation of local stresses and elastic properties of a mortar sample by FFT computation of fields on a 3D image , 2011 .

[24]  Frederic Legoll,et al.  Periodic homogenization using the Lippmann--Schwinger formalism , 2014, 1411.0330.

[25]  M Faessel,et al.  Segmentation of 3D microtomographic images of granular materials with the stochastic watershed , 2010, Journal of microscopy.

[26]  M. Thorpe,et al.  Elastic moduli of two‐dimensional composite continua with elliptical inclusions , 1985 .

[27]  Songhe Meng,et al.  A non-local fracture model for composite laminates and numerical simulations by using the FFT method , 2012 .

[28]  Jan Novák,et al.  Accelerating a FFT-based solver for numerical homogenization of periodic media by conjugate gradients , 2010, J. Comput. Phys..