A computational scheme for linear and non-linear composites with arbitrary phase contrast

A numerical method making use of fast Fourier transforms has been proposed in Moulinec and 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 104) or in?nitecontrast (thosecontaining voids or rigid inclusions or highly non-linear materials) cannot be handled by the method. This paper presents two modi?ed schemes. The ?rst 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 in?nite contrast, is based on an augmented Lagrangian method. The resulting saddle-point problem involves three steps. The ?rst 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