We propose a new discrete FFT-based method for computational homogenization of micromechanics on a regular grid that is simple, fast and robust. The discretization scheme is based on a tetrahedral stencil that displays three crucial properties. First, and most importantly, the Fourier representation of the associated Green operator is defined for any finite q-vector generated by the periodic boundary conditions and that does not belong to the Reciprocal Lattice of the discrete grids. As shown in the paper, this property guaranties that, for any elastic contrats, even infinite, mechanical equilibrium is always mathematically stable, i.e. free of any unphysical patterns, such as oscillations, ringing or checkerboarding, a property which is not shared by the original Moulinec-Suquet method \cite{moulinec1994fast,moulinec1998numerical} nor by the rotated scheme proposed by Willot \cite{willot2015fourier}. Second, the components of tensorial quantities are all defined on the same location, which permits the use of any elastic anisotropy and any spatial variation of the material fields. Third, convergence to equilibrium using the simplest iterative scheme, the "basic scheme", is fast and the number of iterates stabilizes at high contrasts, so that infinite contrast is obtained without additional computational cost.

翻译：我们提出一种基于离散快速傅里叶变换的新方法，用于在规则网格上进行微观力学的计算均匀化。该方法具有简单、快速和鲁棒的特点。离散化方案采用四面体模板，该模板展现出三个关键特性。首先，也是最重要的，相关格林算子的傅里叶表示可针对由周期性边界条件生成且不属于离散网格倒易晶格的任意有限q-矢量进行定义。如本文所示，该性质保证了无论弹性对比度多大（甚至无穷大），力学平衡始终在数学上保持稳定，即不会出现任何非物理模式（如振荡、振铃或棋盘格现象）。这一特性是原始Moulinec-Suquet方法 \cite{moulinec1994fast,moulinec1998numerical} 及Willot提出的旋转格式 \cite{willot2015fourier} 所不具备的。其次，所有张量分量的定义位置相同，这允许使用任意弹性各向异性及材料场的任意空间变化。第三，采用最简单的迭代方案（"基本格式"）可快速收敛至平衡态，且在高对比度情况下迭代次数趋于稳定，因此获得无穷对比度无需额外计算成本。