An FFT framework which preserves a good numerical performance in the case of domains with large regions of empty space is proposed and analyzed for its application to lattice based materials. Two spectral solvers specially suited to resolve problems containing phases with zero stiffness are considered (1) a Galerkin approach combined with the MINRES linear solver and a discrete differentiation rule and (2) a modification of a displacement FFT solver which penalizes the indetermination of strains in the empty regions, leading to a fully determined equation. The solvers are combined with several approaches to smooth out the lattice surface, based on modifying the actual stiffness of the voxels not fully embedded in the lattice or empty space. The accuracy of the resulting approaches is assessed for an octet-lattice by comparison with FEM solutions for different relative densities and discretization levels. It is shown that the adapted Galerkin approach combined with a Voigt surface smoothening was the best FFT framework considering accuracy, numerical efficiency and $h$-convergence. With respect to numerical efficiency it was observed that FFT becomes competitive compared to FEM for cells with relative densities above $\approx$7%. Finally, to show the real potential of the approaches presented, the FFT frameworks are used to simulate the behavior of a printed lattice by using direct 3D tomographic data as input. The approaches proposed include explicitly in the simulation the actual surface roughness and internal porosity resulting from the fabrication process. The simulations allowed to quantify the reduction of the lattice stiffness as well as to resolve the stress localization of $\approx$ 50% near large pores.

提出并分析了在具有大空间区域的域的情况下保持良好数值性能的 FFT 框架在基于晶格的材料中的应用。考虑了两个特别适用于解决包含零刚度相的问题的谱求解器 (1) Galerkin 方法与 MINRES 线性求解器和离散微分规则相结合和 (2) 位移 FFT 求解器的修改，该求解器惩罚了应变的不确定性空白区域，导致完全确定的方程。基于修改未完全嵌入晶格或空白空间的体素的实际刚度，求解器与多种方法相结合来平滑晶格表面。通过与不同相对密度和离散化级别的 FEM 解决方案进行比较，对八位位组格子的结果方法的准确性进行了评估。结果表明，考虑到准确性、数值效率和$H$-收敛。关于数值效率，观察到 FFT 与 FEM 相比，对于相对密度高于$\approx$7%。最后，为了展示所提出方法的真正潜力，FFT 框架用于通过使用直接 3D 断层扫描数据作为输入来模拟打印晶格的行为。所提出的方法在模拟中明确包括由制造过程产生的实际表面粗糙度和内部孔隙率。模拟允许量化晶格刚度的降低以及解决应力局部化$\approx$ 50% 接近大毛孔。