FFT-based methods for the mechanics of composites: A general variational framework

For more than a decade, numerical methods for periodic elasticity, based on the fast Fourier transform, have been used successfully as alternatives to more conventional (fem, bem) numerical techniques for composites. These methods are based on the direct, point-wise, discretization of the Lippmann–Schwinger equation, and a subsequent truncation of underlying Fourier series required for the use of the fast Fourier transform. The basic FFT scheme is very attractive, because of its simplicity of implementation and use. However, it cannot handle pores or rigid inclusions, for which a specific (and significantly more involved) treatment is required. In the present paper, we propose a new FFT-based scheme which is as simple as the basic scheme, while remaining valid for infinite contrasts. Since we adopted an energy principle as an alternative to the Lippmann–Schwinger equation, our scheme is derived within a variational framework. As a by-product, it provides an energetically consistent rule for the homogenization of boundary voxels, a question which has been pending since the introduction of Fourier-based methods.