A Computational Method Based on Augmented Lagrangians and Fast Fourier Transforms for Composites with High Contrast
J.C. Michel, H. Moulinec, P. Suquet

Source CMES: Computer Modeling in Engineering & Sciences, Vol. 1, No. 2, pp. 79-88, 2000
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Keywords computational method, augmented lagrangians, fast fourier transforms, nonlinear composites
Abstract An iterative numerical method based on Fast Fourier Transforms has been proposed by \cite{MOU98} to investigate the effective properties of periodic composites. This iterative method is based on the exact expression of the Green function for a linear elastic, homogeneous reference material. When dealing with linear phases, the number of iterations required to reach convergence is proportional to the contrast between the phases properties, and convergence is therefore not ensured in the case of composites with infinite contrast (those containing voids or rigid inclusions or highly nonlinear materials). It is proposed in this study to overcome this difficulty by using an augmented Lagrangian method. The resulting saddle--point problem involves three steps. The first step consists of solving a linear elastic problem, using the Fourier Transform method. In the second step, a nonlinear problem is solved at each individual point in the volume element. The third step consists of updating the Lagrange multiplier. This method was applied successfully to composites with high or infinite contrast. The first 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 constituents. The third example involves voided rigid-plastic materials.
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