A computational method based on augmented Lagrangians and fast Fourier Transforms for composites with high contrast

An iterative numerical method based on Fast Fourier Transforms has been pro posed by Moulinec and Suquet (1998) to investigate the effective properties of periodic composites. This iterative method is based on the exact expres sion of the Green function for a linear elastic, homogeneous reference mate rial. When dealing with linear phases, the number of iterations required to reach convergence is proportional to the contrast between the phases prope rties, and convergence is therefore not ensured in the case of composites w ith infinite contrast (those containing voids or rigid inclusions or highly nonlinear materials). It is proposed in this study to overcome this diffic ulty by using an augmented Lagrangian method. The resulting saddle-point pr oblem involves three steps. The first step consists of solving a linear ela stic problem, using the Fourier Transform method. In the second step, a non linear problem is solved at each individual point in the volume element. Th e third step consists of updating the Lagrange multiplier. This method was applied successfully to composites with high or infinite contrast. The firs t case presented here is that of a linear elastic material containing voids . The second example is that of a two-phase composite with power-law consti tuents. The third example involves voided rigid-plastic materials.