International audienceThe local and overall responses of nonlinear composites are classically investigated by the Finite Element Method. We propose an alternate method based on Fourier series which avoids meshing and which makes direct use of microstructure images. It is based on the exact expression of the Green function of a linear elastic and homogeneous comparison material. First, the case of elastic nonhomogeneous constituents is considered and an iterative procedure is proposed to solve the Lippman-Schwinger equation which naturally arises in the problem. Then, the method is extended to non-linear constituents by a step-by-step integration in time. The accuracy of the method is assessed by varying the spatial resolution of the microstructures. The flexibility of the method allows it to serve for a large variety of microstructures. (C) 1998 Elsevier Science S.A

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SUMMARYA numerical method making use of fast Fourier transforms has been proposed in Suquet (1994, 1998) to investigate the e ective properties of linear and non-linear composites. This method is based on an iterative scheme the rate of convergence of which is proportional to the contrast between the phases. Composites with high contrast (typically above 10 4 ) or inÿnite contrast (those containing voids or rigid inclusions or highly non-linear materials) cannot be handled by the method. This paper presents two modiÿed schemes. The ÿrst one is an accelerated scheme for composites with high contrast which extends to elasticity a scheme initially proposed in Eyre and Milton (1999). Its rate of convergence varies as the square root of the contrast. The second scheme, adequate for composites with inÿnite contrast, is based on an augmented Lagrangian method. The resulting saddle-point problem involves three steps. The ÿrst step consists of solving a linear elastic problem, using the fast Fourier transform method. In the second step, a non-linear problem is solved at each individual point in the volume element. The third step consists of updating the Lagrange multiplier. Applications of this scheme to rigidly reinforced and to voided composites are shown.

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To cite this version:Hervé Moulinec, Fabrice Silva. Comparison of three accelerated FFT-based schemes for computing the mechanical response of composite materials. International Journal for Numerical Methods in Engineering, Wiley, 2014Wiley, , 97 (13), pp.960-985. 10.1002 Comparison of three accelerated FFT-based schemes for computing the mechanical response of composite materials This study shows that the scheme which minimizes this upper bound is the scheme of Eyre-Milton. The paper discusses the choice of the convergence test used in the schemes.